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1,116,691,497,715 | arxiv | \section{Introduction}
\label{sec:intro}
Object category recognition has long been a central topic in computer vision research.
Traditionally, object recognition has been addressed by supervised learning using a large dataset of image-label pairs~\cite{Deng2009ImageNetAL}.
However, with supervised approaches, the model can only recognize a frozen set of object classes and is not suitable for real-world object recognition, where numerous object classes exist.
Recently, image recognition methods based on contrastive learning using image-text pair datasets have emerged~\cite{clip, align}.
By training on hundreds of millions of image-text pairs, these models have acquired remarkable zero-shot recognition capabilities for a wide variety of objects.
However, these models can recognize objects that commonly appear in the pre-training dataset but are not as effective for rare objects~\cite{Shen2022KLITELT}.
Collecting new data and retraining the entire model to make these models recognize novel objects is impractical considering the cost of data collection and computation.
Therefore, it is essential to develop a method that enables the model to recognize novel objects while maintaining low data collection costs and avoiding model retraining as much as possible.
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\linewidth]{images/intro.pdf}
\caption{
Conceptual illustration of our proposed pipeline.
If the model is confident about the predicted knowledge, question generation is performed in confirmation mode.
If the model is not confident, question generation is performed in exploration mode.
}
\label{fig:intro}
\end{figure*}
When humans acquire knowledge about the world, asking questions and explicitly acquiring knowledge are important skills involved~\cite{chouinard2007children,Ronfard2018QuestionaskingIC}.
Inspired by this, we explored methods to dynamically increase knowledge in image recognition by asking questions.
This approach has several advantages over the traditional supervised learning method: (1) it requires only a small amount of data to acquire knowledge because the system acquires only the knowledge it needs, and (2) it has a low data collection cost because the system itself seeks the required data.
We propose a pipeline consisting of a knowledge-based object classifier (OC) and a question generator (QG) for knowledge acquisition.
Following previous research on structured knowledge~\cite{Ji2022ASO}, we represent knowledge as a knowledge triplet, that is, a list of three words or phrases: \textit{head}, \textit{relation}, and \textit{tail}, such as $\langle$dog, IsA, mammal$\rangle$.
We train the OC to retrieve knowledge from knowledge sources, which outputs the corresponding head in the knowledge source as the predicted object class (e.g., \textlangle IsA, mammal\textrangle$\,\to\,$dog).
The QG model then generates questions to add new knowledge to the knowledge source for novel object recognition.
In the QG model, we use two modes in question generation: \textbf{confirmation} and \textbf{exploration}, as illustrated in Figure~\ref{fig:intro}.
First, ``confirmation'' is used when the unknown object is relatively close to a known object category.
For example, if the model knows about ``dog,'' then a novel category ``chihuahua'' is considered to be a close concept to ``dog.''
In this case, the model can infer reasonable knowledge (e.g., both ``chihuahua'' and ``dog'' are a type of mammal) and ask questions to confirm it, such as ``What is the mammal on the left side of the image?''
In contrast, the ``exploration'' mode is used when the unknown object is far from the existing object category (e.g., ``teddy-bear'' may not resemble any known object class).
In this case, the model is unable to estimate the proper knowledge and attempts to obtain all the necessary knowledge by asking questions (``What is the object sitting next to the dog made of?'').
Our contributions and findings can be summarized as follows:
\begin{itemize}
\setlength{\parskip}{-2pt}
\setlength{\itemsep}{3pt}
\item We propose a novel pipeline to acquire knowledge about novel objects by asking questions. We designed the OC model based on CLIP~\cite{clip} and the QG model as a Transformer~\cite{transformer} based text generation model.
\item We built a novel dataset to train the QG model, namely, \textbf{Professional K-VQG}. This dataset contains a variety of annotations such as object labels, bounding boxes, knowledge, and knowledge-aware questions.
\item We compare our proposed pipeline with several baselines and show that the knowledge acquired through question generation is effective for novel object recognition.
\end{itemize}
\section{Related Work}\label{sec:related-works}
\paragraph{Novel object recognition}
Increasing the number of recognizable object classes is a widely studied problem in object recognition.
A typical approach in novel object recognition is to train a model that computes the similarity between the visual and semantic features of objects.
To compute semantic features of a novel object, external knowledge about the object (e.g., attributes~\cite{Lampert2009LearningTD,Farhadi2009DescribingOB, Jayaraman2014ZeroshotRW, Akata_2016_CVPR,Li2021ZeroshotRW}, class hierarchy~\cite{Rohrbach2011EvaluatingKT, Wang2018ZeroShotRV}, or textual description~\cite{Ba2015PredictingDZ,Qiao2016LessIM,Reed2016LearningDR,Zareian2021OpenVocabularyOD}) is often used.
Recently proposed vision-and-language contrastive learning methods, such as CLIP~\cite{clip} or ALIGN~\cite{align}, use extremely large-scale image caption data to learn the relationship between images and their textual descriptions.
With the help of the prefix-tuning technique, these models exhibited a strong zero-shot recognition ability.
However, the abovementioned studies have a problem in that they require either a well-prepared knowledge database on novel objects or a large number of image-text pair datasets and appropriately designed prompts, both of which are labor-intensive tasks for humans.
In our method, once the question generation model is trained, the model dynamically acquires the necessary knowledge, thereby reducing human effort.
\paragraph{Learning by asking (LBA)}
LBA generates questions to collect additional data to train a model.
With the development of natural language generation methods, several studies using question generation to acquire the information necessary to solve a task (e.g., reading comprehension~\cite{Du2017LearningTA,Yuan2017MachineCB} or question answering~\cite{scialom-staiano-2020-ask}) have been conducted.
In addition, in vision-and-language fields, LBA is applied to VQA tasks~\cite{lba} or image captioning tasks~\cite{vqg_caption}.
However, our research is the first attempt to apply LBA to a novel object recognition task, which is a more real-world-oriented visual recognition task.
\paragraph{Visual question generation (VQG)}
The first studies to address VQG used a simple method of inputting image features into a text decoder and generating questions~\cite{vqg}.
Subsequent research has focused on how to better control the content of the questions to be generated.
In general, the control of question generation is achieved by providing additional information to the text decoder in addition to the image features.
The main methods include providing the answers~\cite{iqan,ivqa} and providing categories of the answers~\cite{info-vqg,c3vqg,vqg_unknown}.
In addition, similar to this study, another study targeted knowledge that can be acquired through questioning~\cite{Uehara2022KVQGKV}.
They created a knowledge-aware VQG dataset (\textbf{K-VQG}) using AMT and used UNITER~\cite{uniter}, which is a state-of-the-art vision-and-language transformer, as an encoder for images and knowledge to successfully generate questions for the knowledge.
We designed a question generation model based on their work and created a new dataset as an extension of their dataset.
\section{Method}\label{sec:method}
Our system consists of an \textbf{O}bject \textbf{C}lassifier (OC) and a \textbf{Q}uestion \textbf{G}enerator (QG).
First, we describe the overall pipeline of our system (Figure~\ref{fig:overall}).
Then, we describe the detail of each module in the following sections.
\begin{figure*}[t]
\centering
\includegraphics[width=0.7\linewidth]{images/overall_v2}
\caption{
Overall pipeline of our method.
The OC model performs knowledge-based object recognition using knowledge sources.
The QG model generates questions target the knowledge needed for novel object recognition.
Answers to the questions are provided by the Oracle Answerer and added to the knowledge source.
With the newly added knowledge, the OC model is able to recognize novel objects.
}
\label{fig:overall}
\end{figure*}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{images/object_classifier_v2.pdf}
\caption{
Architecture of the OC model.
Based on the knowledge encoded by BERT and the similarity calculation of the object image encoded by CLIP, the prediction of the knowledge required for object recognition is performed.
}
\label{fig:classifier}
\end{figure}
The OC model explicitly utilizes knowledge when object recognition is performed.
In other words, for a given image of the object, the OC retrieves the appropriate knowledge $k = [h,\;r,\;t] \in \mathcal{K}$, where $h$, $r$, and $t$ are the head, relation, and tail, respectively, and $\mathcal{K}$ is a knowledge source.
Here, $h$ corresponds to the object label.
Thus, once the model predicts the knowledge of the object image, it can search $\mathcal{K}$ using the predicted knowledge and obtain the object label.
This design allows the model to instantly recognize novel objects when it obtains knowledge about them, simply by updating the knowledge source, without retraining the model.
The QG model is responsible for generating questions about the objects in the image and acquiring knowledge that is useful for novel object recognition.
To this end, we conditioned the QG with partial knowledge, which masks part of the knowledge.
Once the answers to the generated questions are obtained, the acquired knowledge $\mathcal{K}'$ is added to the model's original knowledge source $\mathcal{K}$.
Consequently, the OC's knowledge source is updated to $\mathcal{K}^+ = \mathcal{K}\;\cup\;\mathcal{K}'$.
Then, in the next inference phase, the OC makes predictions about the knowledge and labels by referring to the updated knowledge source $\mathcal{K}^+$.
\subsection{Object Classifier}\label{subsec:object-classifier}
The OC model (Figure~\ref{fig:classifier}) predicts object-related knowledge by computing the similarity between the object feature $\bm{f}_o \in \mathbb{R}^{d}$ and the knowledge feature $\bm{f}_{k} \in \mathbb{R}^{d}$ of the associated knowledge, that is, $p(k) = \mathrm{sim} (\bm{f}_o,\;\bm{f}_k)$, where $d$ is the dimension of the object features and knowledge features.
We adopted a model based on CLIP~\cite{clip}, which is one of the latest visual recognition models, as the object knowledge predictor.
CLIP is a model consisting of an image encoder and text encoder, each of which can be used to compute the similarity between images and text.
The image encoder of CLIP $f_{\theta}$ receives a cropped image $I_{\textrm{crop}}$ as input and outputs the visual feature $\bm{f}_o$.
Knowledge features of knowledge $\bm{f}_k$ is also computed using pre-trained CLIP text encoder $f_{\phi}$.
Here, we converted the knowledge from a triplet representation (e.g., <cat, IsA, mammal>) to the representation of a single sentence with a masked head (e.g., ``[MASK] is a mammal'') before feeding it into the text encoder.
We used cosine similarity to compute the similarity between the object and knowledge features as follows:
\begin{align}
\bm{f}_o = f_{\theta}(I_{crop}),\quad
\bm{f}_k = f_{\phi}(k) \\
\mathrm{sim}(\bm{f}_o,\;\bm{f}_k) = \frac{\bm{f}_o^{\top} \bm{f}_k}{\|\bm{f}_o\| \|\bm{f}_k\|}
\end{align}
The OC model is trained to minimize the binary cross-entropy loss as follows:
\begin{multline}
L_{\mathrm{OC}} = -\sum^{|\mathcal{K}|}_i \bigl(
y_i \cdot \log \sigma (\mathrm{sim}(\bm{f}_o,\;\bm{f}_{k_i})) + \\ (1-y_i) \cdot \log (1-\sigma (\mathrm{sim}(\bm{f}_o,\;\bm{f}_{k_i}))) \bigr)
\end{multline}
where $y_i \in \{0, 1\}$ indicates the ground-truth label for the $i$-th knowledge.
If the model successfully predicts the knowledge, then it has the relation and tail of the knowledge about the object.
Then, when inferring labels from the predicted knowledge $\hat{k}$, we search the knowledge source $\mathcal{K}$ for knowledge that satisfies the predicted relation and tail conditions and treat the matching head as the predicted label.
\subsection{Question Generator}\label{subsec:question-generator}
The QG model takes an image $I$ and target knowledge $k$ as input and generates question $q$.
First, we explain how the target knowledge is determined.
In knowledge acquisition, it is important to acquire knowledge that is ``appropriate'' and ``useful'' for recognition, that is, to acquire correct knowledge at the lowest possible cost.
Here, ``low cost'' implies that retraining the OC model should be avoided as much as possible.
Therefore, we propose using two different modes of question generation: ``confirmation'' and ``exploration.''
As described in Sec.~\ref{sec:intro}, the ``confirmation'' mode is used when the unknown object is relatively close to a known object category, whereas the ``exploration'' mode is used when the unknown object is far from the existing object category.
The target knowledge $k$ in each case is defined as follows:
\begin{equation}
k = \left\{
\begin{array}{ll}
\lbrack\,\texttt{MASK},\;\hat{r},\;\hat{t}\,\rbrack & \mathrm{(confirmation)}\\
\lbrack\,\texttt{MASK},\;r^*,\;\texttt{MASK}\,\rbrack & \mathrm{(exploration)}
\end{array}
\right.\label{eq:knowledge}
\end{equation}
Here, $\hat{r}$ and $\hat{t}$ are the predicted relation and tail, respectively, and $r^*$ is an arbitrarily chosen relation based on the frequency of the relation in the data.
We control the policy of mode selection based on the expected value of the utility that the model can obtain from the answer.
Here, we define the policy selection function $\pi$, which takes the value 1 for the confirmation mode and 0 for the exploration mode.
Then, we adopt a policy that maximizes the expected utility of the model with utility function $u_{\theta}$ for the training data $\mathcal{X}$:
\begin{equation}
\frac{1}{|\mathcal{X}|}\sum (\pi \, u_{\theta} (\bm{f}_o) + (1 - \pi) \, u_{\theta} (\bm{f}_o))\label{eq:expected_utility}
\end{equation}
We define the utility function as the sum of the ``correctness'' and ``informativeness'' of the expected answer.
The ``correctness'' represents the estimated correctness of the knowledge expected to be acquired by the answer.
For simplicity, we assume that the oracle answer should be correct and suppose that the expected correctness is 1.0 when the mode is ``exploration.''
On the other hand, when the mode is ``confirmation,'' the expected correctness depends on the confidence of the model $\mathrm{conf}(\hat{k})$; thus, we set the expected correctness as the predicted score output by the OC model.
The ``informativeness'' is the value representing how useful the acquired knowledge is to the model.
For the ``exploration'' mode, we estimate the informativeness using the similarity between the input image feature and target knowledge feature $\mathrm{sim} (\bm{f}_o,\;\bm{f}_{\hat{k}})$.
For the ``confirmation'' mode, we use the expected value of the similarity based on the mean similarity of the training data, i.e., $\mathbf{E}[I] \simeq \frac{1}{|\mathcal{X}|}\sum \mathrm{sim} (\bm{f}_{o},\;\bm{f}_{\hat{k})}$.
The utility function is expressed as follows:
\begin{equation}
u_{\theta}(\bm{f}_o) = \left\{
\begin{array}{ll}
\mathrm{conf}(\hat{k}) + \mathrm{sim} (\bm{f}_o,\;\bm{f}_{\hat{k}}) & \mathrm{(conf.)}\\
1 + \frac{1}{|\mathcal{X}|} \sum \mathrm{sim} (\bm{f}_{o},\;\bm{f}_{\hat{k}}) & \mathrm{(exp.)}
\end{array}
\right.\label{eq:utility}
\end{equation}
Once the input knowledge $k$ is determined, question generation is performed using it as input.
In the question generation model, we used the ViLT~\cite{vilt} encoder, which is one of the state-of-the-art vision-and-language model, and GPT-2~\cite{gpt2} based decoder.
The ViLT encoder $\mathrm{Enc}(\cdot)$ takes two inputs: (1) the input image $I$ and the masked region image $I_M$ and (2) knowledge triplets in sentence form, i.e., ``[MASK] is a mammal.''
Here, a masked region image is one in which all pixel values outside the target region are replaced by zero.
A learnable type embedding vector is added to the masked image and the original image, respectively.
The model is trained using the following loss function:
\begin{equation}
L = - \sum_{i} \log P(y_i \; | \; y_{<i},\; \mathrm{Enc}([I, I_M],\;k))\label{eq:vqg_loss}
\end{equation}
Here, $y_i$ is the $i$-th word of the question and $\mathrm{Enc}(\cdot)$ is the ViLT encoder, which produces fused feature of the visual and textual knowledge features.
\subsection{Oracle Answerer}\label{subsec:oracle-answerer}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{images/oracle}
\caption{The architecture of the oracle answerer.}
\label{fig:oracle_answerer}
\end{figure}
Given an image and a generated question, the Oracle Answerer predicts the answer knowledge to the question.
We implement this module as a composition of three sub-modules; (1) Head classifier, (2) Relation classifier, and (3) Region classifier.
Each module checks whether the generated question is ``valid'', and if all modules agree that the question is ``valid,'' Oracle Answerer searches the oracle knowledge source and outputs the knowledge that matches the targeted head and relation.
Oracle knowledge source is a knowledge source that merges ConceptNet~\cite{conceptnet} and \textsc{Atomic$^{20}_{20}$}~\cite{comet2020}, which were used in the creation of the datset (see Sec.~\ref{subsec:dataset}).
\paragraph{Head classifier}
The head classifier $\mathcal{H}$ predicts the head of the target knowledge from the generated question, i.e., $h = \mathcal{H}(I, Q)$.
We implement this module following the standard VQA methodology, that is, as a multi-class classification problem that outputs the proper entity given an image and a question.
For this module, we fine-tuned pre-trained ViLT-VQA~\cite{vilt} model.
This module returns ``valid'' if the predicted head is equal to the object in the target region.
\paragraph{Relation classifier}
The relation classifier $\mathcal{R}$ predicts the relation of the target knowledge from the generated question, i.e., $r = \mathcal{R}(Q)$.
Since this problem can be formulated as a sentence classification problem, we use a fine-tuned Distil-BERT~\cite{distilbert} as the relation classifier.
This module returns ``valid'' if the predicted relation matches the target relation ($r$ in the eq. \ref{eq:knowledge}).
\paragraph{Region classifier}
The region classifier $\mathcal{G}$ predicts the target region, i.e., $g = \mathcal{G}(I, Q)$.
We design this module as a model that, given a question and a set of candidate regions, outputs the region that is most relevant to the question.
This problem setup is similar to that of the Referring Expression Comprehension (RE Comprehension)~\cite{refcoco}.
Therefore, we used a fine-tuned version of UNITER grounding model~\cite{uniter}, which has achieved high performance in RE Comprehension task.
This module returns ``valid'' if the predicted region is sufficiently close to the target region.
We calculated the IoBB (Intersection over Bounding Box) between the predicted region and the target region and considered two regions sufficiently close if the value was greater than 0.4.
\subsection{Knowledge Expansion}\label{subsec:knowledge-expansion}
When an answer knowledge $k`$ is obtained for a generated question $q$ by the model, it is added to the model's knowledge source $\mathcal{K}$, that is, $\mathcal{K}^+ = \mathcal{K}\;\cup\;\{k'\}_{i=1}^M$, where $M$ is the number of newly acquired knowledge.
\section{Experiments}\label{sec:experiments}
\subsection{Dataset}\label{subsec:dataset}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{images/wordcloud.pdf}
\caption{Word clouds for the questions and answers in the Professional K-VQG dataset.
}
\label{fig:wordcloud}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.6\linewidth]{images/relation_count.pdf}
\caption{The distribution of relations in the Professional K-VQG dataset.
}
\label{fig:relation-count}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{images/pkvqg_pickup}
\caption{Some examples of the Professional K-VQG dataset.
}
\label{fig:pkvqg-pickup}
\end{figure}
\begin{table}[t]
\centering
\scalebox{0.8}{
\begin{tabular}{@{}lccc@{}}
\toprule
& K-VQG & \begin{tabular}[c]{@{}c@{}}Processional\\ K-VQG\end{tabular} & Merged \\ \midrule
\# questions & 16,098 & 10,431 & 22,212 \\
-- head answers & 11,588 & 5,047 & 13,916 \\
-- tail answers & 4,510 & 5,384 & 8,296 \\
\# images & 13,648 & 9,494 & 9,210 \\ \midrule[0.3pt]
\# unique answers & 2,819 & 3,687 & 4,953 \\
\# unique knowledge & 6,084 & 5,242 & 7,808 \\ \midrule[0.3pt]
\# unique head & 527 & 371 & 533 \\
\# unique tail & 4,922 & 4,257 & 6,199 \\ \bottomrule
\end{tabular}
}
\caption{Detailed statistics of the Professional K-VQG dataset.}
\label{tab:dataset-stats}
\end{table}
To train the QG model, we used the K-VQG dataset~\cite{Uehara2022KVQGKV}, which contains knowledge-aware questions about objects in images, and the newly created \textbf{Professional K-VQG} dataset.
The Professional K-VQG dataset contains knowledge-aware visual questions related to objects annotated by professional annotators.
The source of the images is the Visual Genome~\cite{visualgenome}, and the sources of knowledge are ConceptNet~\cite{conceptnet} and \textsc{Atomic$^{20}_{20}$}~\cite{comet2020}.
We selected 371 object classes from the Visual Genome dataset that were commonly included as heads in the knowledge sources.
In the annotation process, for each candidate object in the image, we obtained the corresponding knowledge triplets from the knowledge sources and asked the annotator to write a knowledge-aware question, with the head or tail of the knowledge triplet as the answer.
As a result, we obtained 10,431 questions for 9,210 images and 5,242 unique knowledge.
We show the word clouds and the relation distribution in the Professional K-VQG dataset in Figure~\ref{fig:wordcloud} and~\ref{fig:relation-count} to visualize several features of the dataset.
We can see that our dataset contains questions on a wide variety of topics about food, clothing, furniture, and so on.
The most and second most frequent relations, ``UsedFor'' and ``IsA'', account for nearly 50\% and 25\% of the total, respectively, which seems a little biased, but this is due to the fact that these relations are also frequent in our knowledge sources.
We show some examples of the Professional K-VQG dataset in Figure~\ref{fig:pkvqg-pickup}.
Because K-VQG and Professional K-VQG share the same annotation format, we merged them to train our QG model.
To use region-based image features obtained from the Faster R-CNN~\cite{butd,Ren2015FasterRT} in the QG module, we excluded samples with target regions that were not detected as objects by the Faster R-CNN (i.e., the IoU between the detected bounding box and the target bounding box was less than 0.5).
We split the merged K-VQG into train, query, and test sets.
The train set was used to train the OC and QG models and the Oracle Answerer.
The query set was used by the OC model to acquire new knowledge, that is, to evaluate the OC model, generate questions to acquire new knowledge using the QG model, and provide answers using the Oracle Answerer.
The test set was used to evaluate the final performance of the OC model using knowledge acquired from the query set.
The train / query / test splits contained 13,334 / 3,693 / 1,414 images and 16,110 / 4,561 / 1,541 questions, respectively.
The number of known knowledge (i.e., knowledge contained in the train set) was 460, and the number of novel knowledge (i.e., knowledge not contained in the train set) was 49.
We show some detailed statistics of the Professional K-VQG dataset in Table~\ref{tab:dataset-stats}.
To train the OC model, we used a dataset consisting of object annotations in the Visual Genome as well as the knowledge triplets obtained from the aforementioned knowledge sources.
The train and validation splits contained 52,321 / 13,080 images and 111,027 / 35,430 objects, respectively, with an average of 52.18 knowledge for each object.
We took care not to contaminate the images and object classes in the train split with other splits in the K-VQG dataset.
\subsection{Training}\label{subsec:training}
We used the same text encoder as CLIP~\cite{clip} and ViT-B/32~\cite{dosovitskiy2020vit} as the visual encoder in the OC model.
All the models were trained using the AdamW optimizer~\cite{adam}, where the learning rate was set to 8e-5 for the OC model and 5e-5 for the QG model.
We trained the OC model for 200 epochs and the QG model for 50 epochs with 8$\times$Tesla A100 GPUs.
\subsection{Baselines}\label{subsec:baselines}
We compared our approach with four baselines: \textbf{CLIP-Ret.}
In this setting, no knowledge acquisition is performed by the QG model, and the performance of the OC model trained with only the training set is evaluated.
\textbf{All Exp. / All Conf.}
In these settings, the question generation policy is fixed to ``exploration'' and ``confirmation,'' respectively.
\textbf{Random Policy.}
The question generation policy is chosen randomly.
We tested this method three times with different random seeds.
\subsection{Results}\label{subsec:results}
\setlength{\tabcolsep}{1.7mm}
\begin{table*}[]
\centering
\scalebox{0.8}{
\begin{tabular}{@{}lcccccc|cc@{}}
\toprule
& \multicolumn{2}{c}{Overall acc.} & \multicolumn{2}{c}{Known acc.} & \multicolumn{2}{c|}{Novel acc.} & & \\
& zero-shot & fine-tune & zero-shot & fine-tune & zero-shot & fine-tune & \#valid Q. & \#knowledge \\ \midrule
Baseline (CLIP-Ret.) & 38.9 & - & 59.9 & - & 0.0 & - & - & - \\
All Conf. & \textbf{56.2} & 59.7 & \textbf{65.8} & 67.5 & \textbf{38.5} & 45.3 & 9756 & 8272 \\
All Exp. & 41.5 & 45.5 & 57.1 & 54.7 & 12.5 & 28.4 & 1057 & 8431 \\
Random Policy & 48.9 $\pm$ 3.8 & 53.2 $\pm$ 0.39 & 60.6 $\pm$ 2.4 & 60.8 $\pm$ 0.71 & 27.4 $\pm$ 7.1 & 39.2 $\pm$ 0.56 & 2788 $\pm$ 116 & 8521 $\pm$ 169 \\
Ours & 56.1 & \textbf{62.0} & \textbf{65.8} & \textbf{67.5} & 38.3 & \textbf{51.9} & \textbf{9864} & 8354 \\ \bottomrule
\end{tabular}
}
\caption{
Results of the OC model.
The top row shows the baseline results without knowledge acquisition through question generation, and the subsequent rows show the results using different question generation policies.
}
\label{tab:main-results}
\end{table*}
The main results are summarized in Table~\ref{tab:main-results}.
We report results separately for the ``zero-shot'' and ``fine-tune'' settings, the former in which the obtained knowledge is simply added to the OC and tested without additional training, and the latter in which the OC is additionally trained with the obtained knowledge.
Known acc. is the accuracy for data where the correct answers are the objects already contained in the training data (i.e., known objects), and Novel acc. is the accuracy for data where the correct answers are the objects not contained in the training data (i.e., novel objects).
Overall, the proposed methods outperformed the baseline CLIP-Ret on all metrics.
In the zero-shot setting, All Conf. performed best, while Ours performed best in the fine-tune setting.
This may be due to the nature of the confirmation mode.
In the confirmation mode, the question target of the QG model is the partial knowledge (i.e., head masked knowledge) that scored highest in the prediction of OC.
Therefore, when no additional training is performed, the confirmation mode, which allows the acquisition of knowledge that is known to be "useful" in advance, tends to perform better.
However, in the confirmation mode, it is not possible to obtain knowledge with a combination of relation and tail that does not exist in the training data, and the variety of knowledge to be obtained is limited.
Thus, the number of knowledge (\#knowledge) obtained in the confirmation mode was less than in the other modes, which may have resulted in inferior performance after the additional training.
On the other hand, in settings All Exp. and Random Policy, the number of knowledge acquired was larger than in Ours, but the number of valid questions (\#valid Q.) was smaller.
Unlike the confirmation mode, the exploration mode is a mode that is more likely to yield completely new knowledge.
However, in the exploration mode, less information is input to the question generation model and the question generation performance is degraded.
For these reasons, the number of valid novel data available for training has decreased, and the performance of the OC model cannot be adequately improved.
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\linewidth]{images/result_vilt.pdf}
\caption{
Examples of generated questions (left) and the novel object recognition with newly acquired knowledge (right).
The left column of the question generation results shows examples of successfully acquiring new knowledge.
In the object recognition results, both ``coat'' and ``airplane'' are novel objects, but the model succeeds to recognize them by using the newly acquired knowledge.
}
\label{fig:results}
\end{figure*}
\subsection{Qualitative Results}\label{subsec:qualitative-results}
In Figure~\ref{fig:results}, we show several examples of the generated questions and novel object recognition using the newly acquired knowledge.
As shown in the left column of the question generation results, the QG model generated questions that could successfully acquire new knowledge.
However, in some cases, the QG model failed to generate valid questions.
This is mainly because VQG targeting knowledge is still a challenging task, and the model generates questions that are not relevant to the content of the image or the targeted knowledge.
We leave further improvement of knowledge-targeted question generation as a future issue.
\section{Conclusion}\label{sec:conclusion}
In this study, we presented a novel method for knowledge acquisition through VQG in object classification tasks.
By using a method that appropriately determines the target of the question, we achieved novel object recognition based on knowledge, while reducing the cost of retraining.
A future challenge would be to study better question-targeting policies.
Our work is related to active learning, in which the main topics are to be labeled by a human.
However, when active learning is applied to complicated tasks such as those with multimodal inputs, it is known that the performance gains are smaller than those for random baselines~\cite{Karamcheti2021MindYO}.
We believe that an interesting issue to be addressed in the future is how to apply active learning methods to our proposed task of knowledge acquisition through question generation.
\section*{Acknowledgments}
This work was partially supported by JST AIP Acceleration Research JPMJCR20U3, Moonshot R\&D Grant Number JPMJPS2011, CREST Grant Number JPMJCR2015, JSPS KAKENHI Grant Number JP19H01115, JP20H05556 and Basic Research Grant (Super AI) of Institute for AI and Beyond of the University of Tokyo.
We would like to thank Yusuke Mori for their helpful discussions.
|
1,116,691,497,716 | arxiv | \section{Introduction}
Immunotherapy seeks to arm the patient with an increased number of effective immune cells that are specifically trained to seek out and destroy harmful cells. One target for immunotherapy that has earned a place in the clinical spotlight recently is melanoma, since specific peptides have been isolated from melanoma cells that can be used in cancer vaccines. A subset of immune cells, called dendritic cells, are able to stimulate the production of immune effector cells that are capable of recognizing and killing specific tumor cells. Researchers have been able to extract these cells from patients and culture them {\it ex vivo} to create a vaccine that can boost a patient's response against their own cancerous cells, \cite{PilonThomas2004}. The success of clinical trials of dendritic cell (DC) vaccines has resulted in the recent FDA approval of the first cancer vaccine for prostate cancer, \cite{ABCNews}
Despite promising clinical responses in vaccine trials, it remains difficult to predict which patients will actually respond to these vaccines and why,
\cite{Boon2006, {Trefzer2005}}. This may be due to the complicated kinetics of the immune response to the presentation of antigen by the dendritic cells. In this paper, we analyze the effect of the delay between the time the DCs come into contact with the T-cell population and the initiation of the expansion, or proliferating phase. We focus on the dynamics describing the active effector T-cell population in the spleen, using a phenomenological description that captures the effect of the delay. Our goal here is to isolate the role of this delay term in order to analyze its effect on the sustainability of T-cell production.
The model we present is very simple, and is not intended to realistically capture the entire cascade of immune events in the response to antigen presentation by dendritic cells. Nor does this model describe the trafficking of immune cells between compartments of the body, in particular between lymphoid organs, such as the spleen, and the tumor site. These effects are crucial to the understanding of the immune response and must be considered in the design of vaccine therapies. One group of experiments that looks at the trafficking of DCs between compartments is described in \cite{Preynat2007}, and a mathematical model of DC trafficking and T-cell activation is presented in \cite{Ludewig2004}. Elsewhere we combine these ideas, along with previous work on cell-lysis rates, \cite{dePillis2005}, in a more realistic model that includes several types of immune cells, and trafficking between compartments.
In section \ref{ModelDescription} we describe the model itself, and then we show stability switching as a function of the delay in Section \ref{Stability}. We conclude with some discussion of the model simulations and results in Section \ref{Conclusion}.
\section{Description of the Mathematical Model} \label{ModelDescription}
After encountering antigen, such as tumor cells or tumor peptides, dendritic cells migrate to lymph organs such as the spleen. As they migrate, they mature and, upon arrival in the spleen, they are able to activate the proliferation of T-cells that have the ability to seek out and destroy (\emph{lyse}) the specific target cells that provided the antigen. This activation requires some contact time between the dendritic cells and the naive T-cells; this connection time is called the ``synaptic connection time." In our model, we represent this time as a delay, $\tau$. Once proliferation is initiated, new antigen-specific T-cells are created. A somewhat simplified description of the fate of these new cells is as follows: The cells must go through a check-point during which the new cells are tested for efficacy against the antigen. If they pass this test, they either move into the blood stream to migrate to the target cells, they become memory cells in order to protect against future challenges by the same antigen, or they return to the proliferating compartment to produce more specific T-cells. Proliferation continues only while certain cytokines are present. These cytokines are produced by previously activated T-cells that have been in contact with dendritic cells for the synaptic connection time, $\tau$. When the dendritic cells have done their job ``educating" the T-cells, it is believed that they die, \cite{PilonThomas2004}, and are then ``cleared" from the spleen compartment. When the DCs are no longer present, cell proliferation ceases, and the activated T-cells either move into the blood stream or become memory cells. This process is represented graphically in Figure \ref{Fig:ModelProcess}.
In this model, the concentration of the dendritic cells at a given time, $t$, is given by $a(t)$, while the number of activated T-cells is given by $x(t)$. The activation/proliferation rate is then given by $$a(t - \tau) f(x(t - \tau) ) x(t),$$ where $f$ is represents the ``feedback" function. The ``feedback" function depends on the number of T-cells present in the proliferating compartment time $\tau$ earlier in the presence of the DCs; therefore they are producing the necessary cytokines. The current number of proliferating T-cells is represented by $x(t)$. We assume that the ``feedback" function, $f$, is a positive function that increases to some maximum level, $r$, and then decreases. The justification for this functional form is that activation by DCs is maximized at a certain concentration of DCs in the spleen. The addition of more DCs has the effect of keeping the activated cells in the area. If they are present for too long, the effectiveness of the T-cells begins to diminish. This is sometimes called the ``decreasing potential hypothesis" in the literature, \cite{Joshi2008, Kalia2006}. One simple function that captures these dynamics is given by:
\begin{center}
\vskip .3in \hbox{\hskip .5in$ f(x_{\tau} ) = \displaystyle{\frac{ r x_{\tau}}{1 + x_{\tau}^4} } $ } \vskip -.5in \hskip .2in \hbox{\includegraphics[scale = .15]{FeedbackFunQuartic.pdf} }
\end{center}
where $x_{\tau} = x(t - \tau)$ represents the population of active T-cells from time $\tau$ ago.
\begin{figure}
\includegraphics[scale = .33]{ReducedModelFig.pdf}
\caption{A graphical representation of the process underlying the mathematical model.} \label{Fig:ModelProcess}
\end{figure}
We assume that T-cells leave the proliferating compartment through death, moving into the blood stream or the memory compartment at an average per-cell rate, $\mu > 0 $. Putting these pieces together results in the following single delay equation:
\begin{equation} \label{Eqn:DDE}
\frac{dx}{dt} = a(t - \tau ) \, f(x_{\tau}) x - \mu x
\end{equation}
where for simplicity:
$$ a(t) = \begin{cases} 1 & \hbox{ when DCs are present} \\ 0 & \hbox{otherwise} \end{cases} .$$
By rescaling the time variable, we can assume that $r = 1$, (the decay parameter, $\mu$, must also be appropriately scaled). Simulated solutions are shown in Figures \ref{Fig:SampleTrajectories_tauzero} and \ref{Fig:SwitchingTimes}. These solutions show that changing the value of $\tau$, as well as the initial history (the values of $x(t)$ for $- \tau < t < 0$), can have a strong effect on the long-term behavior of the system. We verify this in the next section by performing a stability analysis.
\section{Stability Anlaysis} \label{Stability}
When dendritic cells are present in the spleen, $a(t) = 1$, so that the differential equation becomes:
\begin{equation} \label{Eqn:ScaledDDE}
\frac{dx}{dt} = \frac{x_{\tau} }{1 + x_{\tau}^4 } x - \mu x.
\end{equation}
\subsection{Equilibria and stability when $\tau = 0$.}
Equation \ref{Eqn:ScaledDDE} potentially has three equilibria. One is the zero fixed point: $x_E = 0$, and the other two are solutions to
$$ \frac{x}{1 + x^4} = \mu \Leftrightarrow x ( 1 - \mu x^3) = \mu .$$
These two solutions are depicted graphically in Figure \ref{Fig:TwoEquil}. Note that the function $x - \mu x^4$ has a maximum which occurs at $x_{Max} = (4\mu)^{-1/3}$. When the maximum of this function is greater than $\mu$ the two additional equilibria will exist. This occurs for $\mu \in \left({ 0, \displaystyle{\frac{3^{3/4}}{4}} }\right)$. We denote the greater of these two equilibria by $x^*$, and the smaller by $x^-$, if they exist.
\begin{figure}
\includegraphics[scale = .5]{TwoNonZeroEquilibria.pdf}
\caption{The non-zero equilibria of the DDE given in \ref{Eqn:ScaledDDE} are the solutions of
$x (1 - \mu x^3) = \mu$. These are positive for $\mu$-values in $(0, \frac{3^{3/4}}{4})$.}
\label{Fig:TwoEquil}
\end{figure}
If $x^*$ is stable, then proliferation is sustained, and active effector cells continue to be produced and sent into the blood stream and into the memory compartment. However, if $x^*$ is unstable, then production may die off and no new effector cells will be produced, even when DCs are still present in the spleen.
Our goal in this section is to show that changing the value of the delay, $\tau$, can change the stability of $x^*$.
Linearization around an equilibrium, $x_E$ gives:
$$ \frac{dz}{dt} = \alpha z_\tau + \beta z, $$
where $z = x - x_E $, \quad $\alpha = x \frac{\partial f}{\partial x_{\tau}} $, evaluated at the equilibrium $x_E$, and $ \beta = f(x_E) - \mu$. The characteristic equation is:
$$ \lambda = \alpha e^{- \lambda \tau} + \beta.$$
Evaluating $\alpha$ and $\beta$ at $x = x_E$ gives:
$$ \alpha = \frac{ \mu^2}{x_E} \left({ 1 - 3 x_E^4}\right), \quad \hbox{ when } x_E = x^*, x^-, \quad \hbox{and} \quad \alpha = 0 \hbox{ when } \, x_E = 0 $$
$$ \beta = 0, \quad x_E = x^*, x^-, \quad \hbox{and} \quad \beta = - \mu, \, x_E = 0 .$$
When we consider the characteristic equation in the case of the equilibrium $x_E = 0$, we always obtain $\lambda = -\mu < 0$, \emph{independent of $\tau$.} Thus we see that the equilibrium at $x_E =0$ is {\bf always stable}.
On the other hand,linearizing around the upper equilibrium at $x^*$ in the case when $\tau = 0$ gives:
$$ \lambda = \alpha = \frac{ \mu^2}{x} \left({ 1 - 3 x^4}\right)< 0 \Rightarrow x > 4 \mu /3 .$$
Using the fact that $(x^*)^4 = \frac{x^*}{\mu} - 1$, we see that stability of the upper equilibrium is ensured if
\begin{equation} \label{Eqn:xstarcondition}
x^* > \frac{1}{3^{1/4}} .
\end{equation}
We saw earlier that the maximum of $x - \mu x^4$ occurs at $x_{Max} = \left({ 4 \mu}\right) ^{-1/3}$, and we know that $x^*
> x_{Max}$. Because we assume $\mu < \displaystyle{ \frac{3^{3/4}}{4} } $, we see that Equation \ref{Eqn:xstarcondition} is satisfied. Therefore, when $\tau = 0$, $x^*$ is {\bf stable}.
Since $\tau = 0$ corresponds to the ODE case, and $x^-$ lies between the two stable equilibria $x_E = 0$ and $x_E = x^*$, we know that $x^-$ is {\bf unstable}. Sample trajectories are shown in Figure \ref{Fig:SampleTrajectories_tauzero}.
\begin{figure}
\includegraphics[scale = .4]{ThreeEquilibria.pdf}
\caption{Sample trajectories in the case $\tau = 0$ showing two stable equilibria at $x^*$ and $0$, and an unstable equilibrium at $x^-$. In these simulations $\mu = .5$. } \label{Fig:SampleTrajectories_tauzero}
\end{figure}
\subsection{Stability switching as $\tau$ increases.}
Changes in stability can occur when
the real part of $\lambda$ switches sign, i.e. when $\lambda$ is purely imaginary. In this case, the characteristic equation becomes
\begin{equation} \label{Eqn:om}
\lambda = i \omega = \alpha e^{- i \omega \tau} + \beta .
\end{equation}
At the upper equilibrium, $x^*$, $\beta = 0$. Equating the real and imaginary parts in Equation \ref{Eqn:om} in this case gives:
\begin{eqnarray}
\alpha \cos ( \omega \tau) &=& 0 , \\
- \alpha \sin ( \omega \tau ) &=& \omega .
\end{eqnarray}
The first equation implies that $\omega \tau = (2n + 1) \frac{\pi}{2}$. Substituting this into the second equation gives:
$$ \omega = \pm \alpha \Rightarrow \tau = \pm \frac{(2n+1) \pi }{2 \alpha} = \pm \frac{(2n+1) \pi x^*}{2 \mu^2 (1 - 3 {x^*}^4 )} . $$
For a fixed value of $\mu$, this gives a sequence of possible switching times, at which the equilibrium at $x^*$ can change stability. For example, taking $\mu = \frac{1}{2}$, we see that $x^* = 1$, and possible switching times occur when
$$ \tau = (2n + 1) \pi, \quad n \in {\mathbb Z} . $$
Figure \ref{Fig:SwitchingTimes} shows numerical solutions for increasing values of $\tau$. We see that the equilibrium point at
$x^* = 1$ loses stability when $\tau = \pi$, when a limit cycle appears. Increasing $\tau$ destroys this limit cycle and, for larger values of $\tau$, solutions that once tended to $x^*$ converge to the zero equilibrium.
\begin{figure}[h!]
\hskip -.6in \includegraphics[scale = .4]{CompareTausFig5.pdf}
\caption{As $\tau$ is increased, the equilibrium at $x^* = 1$ loses stability. First, periodic solutions appear for $\tau$ greater than $\pi$. Further increases in $\tau$ result in a complete loss of stability, and solutions tend towards the zero equilibrium.}
\label{Fig:SwitchingTimes}
\end{figure}
\section{Discussion and Conclusions} \label{Conclusion}
We have presented a simplified model of the interaction between dendritic cells and T-cells at the initiation of a specific immune response. The model contains one delay that represents the synaptic connection time between dendritic cells and T-cells which is necessary for the development of cytokines that stimulate proliferation. There is experimental evidence that prolonged contact between dendritic cells and T-cells results in a decrease in viable T-cell production. The model presented here shows a change in stability of the high T-cell equilibrium as the delay is increased. This equilibrium goes from being stable when the delay, $\tau$, is small to being unstable for larger values of $\tau$. These results support the experimental evidence for the ``decreasing potential hypothesis". The model also suggest a role for dendritic cell therapies in boosting the immune defense against certain diseases such as cancer. However, the model indicates that levels of dendritic cells in the spleen should not be maintained at high levels for too long in order to optimize the expansion of active effector T-cells.
\section*{Acknowledgments}
We would like to thank the organizers of the 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications that took place in Dresden, Germany, May 25-28, 2010, at which this work was presented.
|
1,116,691,497,717 | arxiv | \section{Introduction}
\label{sec:intro}
Machine learning generally amounts to solving an optimization problem where a loss function has to be minimized. As the problems tackled are getting more and more complex (nonlinear, nonconvex, etc.), fewer efficient algorithms exist, and the best recourse seems to rely on iterative schemes that exploit first-order or second-order derivatives of the loss function to get successive improvements and converge towards a local minimum. This explains why variants of gradient descent are becoming increasingly ubiquitous in machine learning and have been made widely available in the main deep learning libraries, being the tool of choice to optimize deep neural networks. Other types of local algorithms exist when no derivatives are known \citep{sigaud2018policy}, but in this paper we assume that some derivatives are available and only consider first order gradient-based or second order Hessian-based methods.
Among these methods, vanilla gradient descent strongly benefits from its computational efficiency as it simply computes partial derivatives at each step of an iterative process. Though it is widely used, it is limited for two main reasons: it depends on arbitrary parameterizations and may diverge or converge very slowly if the step size is not properly tuned.
To address these issues, several lines of improvement exist. Here, we focus on two of them. On the one hand,
first-order methods such as the natural gradient introduce particular metrics to restrict gradient steps and make them independent from parametrization choices \citep{amari98natural}. On the other hand, second-order methods use the Hessian matrix of the loss or its approximations to take into account its local curvature.
Both types of approaches enhance the vanilla gradient descent update, multiplying it by the inverse of a large matrix (of size $d^2$, where $d$ is the dimensionality of the parameter space).
We propose a simple framework that unifies these first-order or second-order improvements of the gradient descent,
and use it to study precisely the similarities and differences between the 6 aforementioned methods.
This general framework uses a first-order approximation of the loss and constrains the step with a quadratic norm. Therefore,
each modification $\mymath{\delta \vth}$ of the vector of parameters $\mymath{\vec \theta}$ is computed via an optimization problem of the following form:
\begin{equation}
\label{eq:framework}
\left\{
\begin{array}{l}
\min_{\mymath{\delta \vth}} \nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta})^T \mymath{\delta \vth}\\
\mymath{\delta \vth}^T M(\mymath{\vec \theta}) \mymath{\delta \vth} \leq \epsilon^2,
\end{array}
\right.
\end{equation}
where $\nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta})$ is the gradient of the loss $L(\mymath{\vec \theta})$, and $M(\mymath{\vec \theta})$ a symmetric positive-definite matrix.
The 6 methods differ by the matrix $M(\mymath{\vec \theta})$, which has an effect not only on the size of the steps, but also on the direction of the steps,
as illustrated in Figure~\ref{fig:metric_effect}.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.67\textwidth]{metric_effect.pdf}
\caption{Different metrics affect both the gradient step size and direction.
Here, a different $\mymath{\delta \vth}$ is obtained with $M=I$ or $M$ an arbitrary symmetric positive-definite matrix.}
\label{fig:metric_effect}
\end{figure}
The solution of the minimization problem~\eqref{eq:framework} has the following form (see Appendix~\ref{ap:solution}):
$$\mymath{\delta \vth} = -\alpha M(\mymath{\vec \theta})^{-1} \nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta}).$$
In Section~\ref{sec:first}, we show how the vanilla gradient descent, the classical Gauss-Newton method and the
natural gradient descent method fit into the proposed framework. It can be noted that these 3 approaches constrain the steps in a way that
is independent from the loss function.
In Section~\ref{sec:second}, we consider approaches that depend on the loss, namely the gradient covariance matrix approach, Newton's method
and the generalized Gauss-Newton method, and show that they also fit into the framework.
Table~\ref{summary} summarizes the different values of $M(\mymath{\vec \theta})$ for all 6 approaches.
\begin{table}
\begin{center}
\begin{tabular}{c|l}
$M(\mymath{\vec \theta})$ & Corresponding approach\\
\hline
& \\
$I$ & vanilla gradient descent \\[2pt]
${{\rm I\!E}}{}_{\mymath{\vec s}}{\left[J_{\mymath{\vec x}}(\mymath{\vec \theta})^TJ_{\mymath{\vec x}}(\mymath{\vec \theta})\right]} + \lambda I$ & classical Gauss-Newton \\[4pt]
${{\rm I\!E}}{}_\mymath{\vec s}\left[ \mymath{\nabla_{\vth}} \log( p_\mymath{\vec \theta}(\mymath{\vec y}|\mymath{\vec x}) ) \mymath{\nabla_{\vth}}\log( p_\mymath{\vec \theta}(\mymath{\vec y}|\mymath{\vec x}) )^T \right] + \lambda I$ & natural gradient (with empirical Fisher matrix) \\[4pt]
${{\rm I\!E}}{}_{\mymath{\vec s}}\left[ \nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s}) \nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s})^T \right] + \lambda I$ & gradient covariance matrix \\[4pt]
$H(\mymath{\vec \theta}) + \lambda I$ & Newton's method\\[2pt]
${{\rm I\!E}}{}_{\mymath{\vec s}}\left[ J_{\mymath{\vec x}}(\mymath{\vec \theta})^T \mathcal{H}_{\mymath{\vec y}}(\mymath{\vec h_{\vth}}(\mymath{\vec x})) J_{\mymath{\vec x}}(\mymath{\vec \theta}) \right] + \lambda I$ & generalized Gauss-Newton\\[2pt]
\hline
\end{tabular}
\end{center}
\caption{\label{summary} The matrices $M(\mymath{\vec \theta})$ associated to 6 popular variants of the gradient descent, when interpreted
as instances of the optimization problem~\eqref{eq:framework}.
See Section~\ref{sec:not} for the definitions of the notations.}
\end{table}
Providing a unifying view of several first-order and second-order variants of the gradient descent,
the framework presented in this paper
makes the connections between different approaches more obvious,
and can hopefully give new insights on these connections and help clarifying
some of the literature. Finally, we believe that it can facilitate the selection between these
methods given a specific problem.
\section{Problem statement and notations}
\label{sec:not}
\begin{tabular}{c|l}
Notation & Description\\
\hline
& \\
$\mymath{\vec s} = (\mymath{\vec x}, \mymath{\vec y})$ & a sample, with an input variable $\mymath{\vec x}$, and an output variable $\mymath{\vec y}$\\[2pt]
$L(\mymath{\vec \theta})$ & the scalar loss to minimize, $\mymath{\vec \theta}$ being the vector of parameters\\[2pt]
$p_\mymath{\vec \theta}(\cdot|\mymath{\vec x})$ & p.d.f. estimating the conditional distribution of the output variable\\[2pt]
$\mymath{\vec h_{\vth}}(\mymath{\vec x})$ & a finite-dimensional representation of the distribution $p_\mymath{\vec \theta}(\cdot|\mymath{\vec x})$\\[2pt]
$J_{\mymath{\vec x}}(\mymath{\vec \theta})$ & Jacobian of the function $\mymath{\vec \theta} \mapsto \mymath{\vec h_{\vth}}(\mymath{\vec x})$\\[2pt]
${{\rm I\!E}}{}_{\mymath{\vec s}}[\cdot]$ & expectation over the samples\\[2pt]
$l_\mymath{\vec \theta}(\mymath{\vec s}) = l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x}))$ & the atomic loss of which $L$ is the average over the samples: $L(\mymath{\vec \theta}) = {{\rm I\!E}}{}_{\mymath{\vec s}}{\left[ l_\mymath{\vec \theta}(\mymath{\vec s}) \right]}$\\[2pt]
$\mymath{\delta \vth}$ & small update of $\mymath{\vec \theta}$ computed at every iteration \\[2pt]
$(\cdot)^T$ & transpose operator \\[2pt]
$\| \vec v \|$ & Euclidean norm of the vector $\vec v$: $\| \vec v \| = \sqrt{\vec v^T \vec v}$ \\[2pt]
${{\rm I\!E}}{}_{a \sim p_\mymath{\vec \theta}(\cdot|\mymath{\vec x})}[f(a)]$ & expected value of $f(a)$, when $a$ follows the distribution $p_\mymath{\vec \theta}(\cdot|\mymath{\vec x})$\\[2pt]
$\nabla_\mymath{\vec \theta} f(\mymath{\vec \theta})$ & gradient w.r.t. $\mymath{\vec \theta}$: $\nabla_\mymath{\vec \theta} f(\mymath{\vec \theta}) = \frac{\partial f(\mymath{\vec \theta})}{\partial \mymath{\vec \theta}}$ \\[2pt]
$M_{CGN}(\mymath{\vec \theta})$ & Classical Gauss-Newton matrix: $
M_{CGN}(\mymath{\vec \theta}) = {{\rm I\!E}}{}_{\mymath{\vec s}}{\left[J_{\mymath{\vec x}}(\mymath{\vec \theta})^TJ_{\mymath{\vec x}}(\mymath{\vec \theta})\right]}
$\\[2pt]
$\mathcal{I}_\mymath{\vec x}(\mymath{\vec \theta})$ & Fisher information matrix of $p_\mymath{\vec \theta}(\cdot|\mymath{\vec x})$\\[2pt]
$F(\mymath{\vec \theta})$ & empirical Fisher matrix: $ F(\mymath{\vec \theta}) = {{\rm I\!E}}{}_\mymath{\vec s}\left[ \mymath{\nabla_{\vth}} \log{\left( p_\mymath{\vec \theta}(\mymath{\vec y}|\mymath{\vec x}) \right)} \mymath{\nabla_{\vth}}\log{\left( p_\mymath{\vec \theta}(\mymath{\vec y}|\mymath{\vec x}) \right)}^T \right]$\\[2pt]
$KL\big( p_1, p_2 \big)$ & Kullback-Leibler divergence between the probability distributions $p_1$ and $p_2$\\[2pt]
$H(\mymath{\vec \theta})$ & Hessian of the loss $L$, defined by $\left[H(\mymath{\vec \theta})\right]_{i,j} = \frac{\partial^2 L}{\partial \mymath{\vec \theta}_i \partial \mymath{\vec \theta}_j}(\mymath{\vec \theta})$\\[2pt]
$\mathcal{H}_{\mymath{\vec y}}(\mymath{\vec h_{\vth}}(\mymath{\vec x}))$ & Hessian of the function $\vec h \mapsto l(\mymath{\vec y}, \vec h)$ at $\vec h = \mymath{\vec h_{\vth}}(\mymath{\vec x})$\\[2pt]
\hline
\end{tabular}
We consider a context of regression analysis in which, based on samples $\vec{s} = (\mymath{\vec x}, \mymath{\vec y})$, the objective is to estimate the conditional distribution of $\mymath{\vec y}$ given $\mymath{\vec x}$.
More formally, this conditional distribution is estimated by a parametrized
probability density function (p.d.f.) $p_\mymath{\vec \theta}(\mymath{\vec y} | \mymath{\vec x})$, and the goal of the learning is to progressively optimize the vector $\mymath{\vec \theta}$ to improve the accuracy of this probability estimation. We furthermore assume that the
p.d.f. $p_\mymath{\vec \theta}( \cdot | \mymath{\vec x})$ can be represented by a finite-dimensional vector $\mymath{\vec h_{\vth}}(\mymath{\vec x})$. For instance, in many applications, $p_\mymath{\vec \theta}(\cdot | \mymath{\vec x})$ is
a multivariate Gaussian distribution, and in this case the vector $\mymath{\vec h_{\vth}}(\mymath{\vec x})$ would typically contain the mean and covariance matrix components.
The accuracy of $p_\mymath{\vec \theta}( \cdot | \mymath{\vec x})$ is measured via a loss function
$L$ estimated over a dataset of samples $\mymath{\vec s}$. $L$ depends only on $\theta$ and we assume that it is
expressed as the expected value (over the sample distribution) of an atomic loss, a loss \emph{per sample} $l_\mymath{\vec \theta}(\mymath{\vec s})$:
$$
L(\mymath{\vec \theta}) = {{\rm I\!E}}{}_{\mymath{\vec s}}{\left[ l_\mymath{\vec \theta}(s) \right]}.
$$
In practice, the expected value over the samples is estimated with an empirical mean over a batch
$\mathcal{B} = (\mymath{\vec s}_1, \mymath{\vec s}_2, \dots, \mymath{\vec s}_N)$, so the loss actually used can be written
$\hat{L}_\mathcal{B}(\mymath{\vec \theta}) = \frac{1}{N}\sum_i{ l_\mymath{\vec \theta}(\mymath{\vec s}_i) }$, the gradient of
which being directly expressible from the gradient of the atomic loss (w.r.t. $\mymath{\vec \theta}$). In the remainder of the paper,
we keep expressions based on the expected value $ {{\rm I\!E}}{}_{\mymath{\vec s}}{\left[ \cdot \right]}$, knowing that
at every iteration it is replaced by an empirical mean over a (new) batch of samples.
The dependency of the atomic loss to $\mymath{\vec \theta}$ is via $p_\mymath{\vec \theta}( \cdot | \mymath{\vec x})$, so we can also express it as a function of the finite-dimensional representation $\mymath{\vec h_{\vth}}(\mymath{\vec x})$:
$$
l_\mymath{\vec \theta}(\mymath{\vec s}) = l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x})).
$$
Remark: choosing a concrete context of regression analysis helps
us to simplify the notations, and give examples, but the results obtained are not specific to this setting, as the 6 gradient descent variants considered can also be useful for other types of learning tasks, leading to similar expressions that can be brought back to our general framework.
\section{Vanilla, classical Gauss-Newton and natural gradient descent}
\label{sec:first}
All variants of the gradient descent are iterative numerical optimization methods: they start
with a random initial \mymath{\vec \theta} and attempt to decrease the value of $L(\mymath{\vec \theta})$ over iterations by adding a
small increment vector \mymath{\delta \vth} to \mymath{\vec \theta} at each step. The core of all these algorithms
is to determine the direction and magnitude of \mymath{\delta \vth}.
\subsection{Vanilla gradient descent}
\label{sec:vanilla}
The so-called ``vanilla'' gradient descent is a first-order method that relies on a first-order Taylor approximation of the loss function $L$:
\begin{equation}
\label{eq:loss_taylor_first_order}
L(\mymath{\vec \theta} + \mymath{\delta \vth}) \simeq L(\mymath{\vec \theta}) + \nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta})^T \mymath{\delta \vth}.
\end{equation}
At each iteration, the objective is the minimization of $\nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta})^T \mymath{\delta \vth}$ with the variable $\mymath{\delta \vth}$.
If the gradient is non-zero, the value of this term is unbounded below: it suffices for instance
to set $\mymath{\delta \vth} = -\alpha \nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta})$ with $\alpha$ arbitrarily large.
As a result, constraints are needed to
avoid making excessively large steps.
In vanilla approaches, the Euclidean metric ($\|\mymath{\delta \vth}\| = \sqrt{\mymath{\delta \vth}^T \mymath{\delta \vth}}$) is used to bound the increments $\mymath{\delta \vth}$.
The optimization problem solved at every step of the scheme is:
\begin{equation}
\label{eq:vanilla}
\left\{
\begin{array}{l}
\min_{\mymath{\delta \vth}} \nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta})^T \mymath{\delta \vth} \\
\mymath{\delta \vth}^T \mymath{\delta \vth} \leq \epsilon^2,
\end{array}
\right.
\end{equation}
\noindent where $\epsilon$ is a user-defined upper bound. It is
the most trivial instance of the general framework~(\ref{eq:framework}). As
shown in Appendix~\ref{ap:solution}, the solution
of this problem is $\mymath{\delta \vth} = -\alpha \mymath{\nabla_{\vth}} L(\mymath{\vec \theta})$, with $\alpha = \frac{\epsilon}{\|\nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta})\|}$.
To set the size of the step, instead of tuning $\epsilon$, the most common approach is
to use the expression $-\alpha \mymath{\nabla_{\vth}} L(\mymath{\vec \theta})$ and directly tune $\alpha$, which is called the \emph{learning rate}.
An interesting property with this approach is that, as $\mymath{\vec \theta}$ gets closer to an optimum, the norm of the gradient $\|\nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta})\|$ decreases, so
the $\epsilon$ corresponding to the fixed $\alpha$ decreases as well.
This means that the steps tend to become smaller and smaller, which is a
necessary property to make asymptotic convergence possible.
\subsection{Classical Gauss-Newton}
\label{sec:cgn}
As mentioned in Section~\ref{sec:not}, the atomic loss function $l_\mymath{\vec \theta}(\mymath{\vec s}) = l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x}))$ depends indirectly on the parameters $\mymath{\vec \theta}$ via the vector $\mymath{\vec h_{\vth}}(\mymath{\vec x})$, which is a finite-dimensional representation of the p.d.f. $p_\mymath{\vec \theta}( \cdot | \mymath{\vec x})$. So it may be more meaningful to measure the modifications arising from an update $\mymath{\delta \vth}$ by looking at the change on $\mymath{\vec h_{\vth}}(\mymath{\vec x})$, not simply on $\mymath{\delta \vth}$ as with the vanilla gradient descent approach.
The constraint $\mymath{\delta \vth}^T \mymath{\delta \vth} \leq \epsilon^2$ acts as if all components of $\mymath{\delta \vth}$
had the same importance, which is not necessarily the case. Some components of $\mymath{\vec \theta}$ might have much smaller effects on $\mymath{\vec h_{\vth}}(\mymath{\vec x})$
than others, and this will not be taken into account with the vanilla gradient descent method, which typically performs badly with unbalanced parametrizations.
Measuring and bounding the change on the vector $\mymath{\vec h_{\vth}}(\mymath{\vec x})$ makes the updates independent from the way $\mymath{\vec h_{\vth}}$ is parametrized. To do this, a natural choice is to bound the expected squared Euclidean distance between $\mymath{\vec h_{\vth}}(\mymath{\vec x})$ and $\mymath{\vec h_{\vth + \delth}}(\mymath{\vec x})$:
$$
{{\rm I\!E}}{}_{\mymath{\vec s}}{\left[\|\mymath{\vec h_{\vth + \delth}}(\mymath{\vec x}) - \mymath{\vec h_{\vth}}(\mymath{\vec x})\|^2\right]}\leq \epsilon^2.
$$
Using again a first-order approximation, we have
$\mymath{\vec h_{\vth + \delth}}(\mymath{\vec x}) - \mymath{\vec h_{\vth}}(\mymath{\vec x}) \simeq J_{\mymath{\vec x}}(\mymath{\vec \theta})\mymath{\delta \vth}$,
where $J_{\mymath{\vec x}}(\mymath{\vec \theta})$ is the Jacobian of the function $\mymath{\vec \theta} \mapsto \mymath{\vec h_{\vth}}(\mymath{\vec x})$.
The constraint can be rewritten:
$$
{{\rm I\!E}}{}_{\mymath{\vec s}}{\left[\|J_{\mymath{\vec x}}(\mymath{\vec \theta})\mymath{\delta \vth}\|^2\right]} = \mymath{\delta \vth}^T{{\rm I\!E}}{}_{\mymath{\vec s}}{\left[J_{\mymath{\vec x}}(\mymath{\vec \theta})^TJ_{\mymath{\vec x}}(\mymath{\vec \theta})\right]}\mymath{\delta \vth} \leq \epsilon^2,
$$
resulting in the optimization problem:
\begin{equation}
\label{eq:jacobian}
\left\{
\begin{array}{l}
\min_{\mymath{\delta \vth}} \nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta})^T \mymath{\delta \vth}\\
\mymath{\delta \vth}^T {{\rm I\!E}}{}_{\mymath{\vec s}}{\left[J_{\mymath{\vec x}}(\mymath{\vec \theta})^TJ_{\mymath{\vec x}}(\mymath{\vec \theta})\right]} \mymath{\delta \vth} \leq \epsilon^2,
\end{array}
\right.
\end{equation}
which fits into the general framework~\eqref{eq:framework} if the matrix $M_{CGN}(\mymath{\vec \theta}) = {{\rm I\!E}}{}_{\mymath{\vec s}}{\left[J_{\mymath{\vec x}}(\mymath{\vec \theta})^TJ_{\mymath{\vec x}}(\mymath{\vec \theta})\right]}$ is symmetric positive-definite.
\paragraph{Damping.}
The structure of the matrix $M_{CGN}(\mymath{\vec \theta})$ makes it symmetric and
positive semi-definite, but not necessarily definite-positive. To ensure the definite-positiveness,
a regularization or damping term $\lambda I$ can be added, resulting
in the constraint $\mymath{\delta \vth}^T \big(M_{CGN}(\mymath{\vec \theta}) + \lambda I \big) \mymath{\delta \vth} \leq \epsilon^2$, which can be
rewritten:
$$
\mymath{\delta \vth}^T M_{CGN}(\mymath{\vec \theta}) \mymath{\delta \vth} + \lambda \mymath{\delta \vth}^T \mymath{\delta \vth} \leq \epsilon^2.
$$
We see that this kind of damping, often called Tikhonov damping
\citep{martens2012training}, regularizes the constraint with
a term proportional to the squared Euclidean norm of $\mymath{\delta \vth}$.
It must be noted that with a regularization term, the constraint
is not independent to the parametrization in $\mymath{\vec \theta}$ anymore. And if a large value of $\lambda$ is chosen (which also usually requires
increasing $\epsilon$), the method becomes similar
to the vanilla gradient descent.
\paragraph{A more common definition of the classical Gauss-Newton method concerns atomic losses expressed as squared errors.}
We assume that the atomic loss $l_\mymath{\vec \theta}(\mymath{\vec s})$ is defined as follows:
$$l_\mymath{\vec \theta}(\mymath{\vec s}) = \frac{1}{2}\|\mymath{\Delta_{\vth}}(\mymath{\vec s})\|^2,$$
where $\Delta_\mymath{\vec \theta}(\mymath{\vec s})$ is a vector-valued function.
Functions of the form $\Delta_\mymath{\vec \theta}(\mymath{\vec s}) = \mymath{\vec y} - f_\mymath{\vec \theta}(\mymath{\vec x})$ are typical examples in the context of regression.
Denoting by $\mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta})$ the Jacobian of $\mymath{\vec \theta} \mapsto \mymath{\Delta_{\vth}}(\mymath{\vec s})$:
$\mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta}) = \left(\frac{\partial \Delta_{\mymath{\vec \theta}_i}(\mymath{\vec s})}{\partial \mymath{\vec \theta}_j}\right)_{i,j}$, we have:
\begin{equation*}
\begin{split}
l_{\mymath{\vec \theta} + \delta\mymath{\vec \theta}}(\mymath{\vec s}) & = \frac{1}{2}
\left( \Delta_\mymath{\vec \theta}(\mymath{\vec s}) + \mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta}) \mymath{\delta \vth} + O(\mymath{\delta \vth}^2) \right)^T \left( \Delta_\mymath{\vec \theta}(\mymath{\vec s}) + \mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta})\mymath{\delta \vth} + O(\mymath{\delta \vth}^2) \right)\\
& = l_\mymath{\vec \theta}(\mymath{\vec s}) + \Delta_\mymath{\vec \theta}(\mymath{\vec s})^T \mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta})\mymath{\delta \vth} + \frac{1}{2}\mymath{\delta \vth}^T \mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta})^T \mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta})\mymath{\delta \vth} + O(\mymath{\delta \vth}^3).
\end{split}
\end{equation*}
$\mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta})^T\Delta_\mymath{\vec \theta}(\mymath{\vec s})$ is the gradient of the loss $l$ in $\mymath{\vec \theta}$, so the equality can be rewritten:
$$
l_{\mymath{\vec \theta} + \delta\mymath{\vec \theta}}(\mymath{\vec s}) = l_\mymath{\vec \theta}(\mymath{\vec s}) + \mymath{\nabla_{\vth}} l_\mymath{\vec \theta}(\mymath{\vec s})^T \mymath{\delta \vth} + \frac{1}{2}\mymath{\delta \vth}^T \mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta})^T \mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta})\mymath{\delta \vth} + O(\mymath{\delta \vth}^3).
$$
By averaging over the samples, we get:
$$
L(\mymath{\vec \theta} + \mymath{\delta \vth}) = L(\mymath{\vec \theta}) + \mymath{\nabla_{\vth}} L(\mymath{\vec \theta})^T\mymath{\delta \vth} + \frac{1}{2}\mymath{\delta \vth}^T {{\rm I\!E}}{}_{\mymath{\vec s}}{\left[ \mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta})^T \mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta}) \right]} \mymath{\delta \vth} + O(\mymath{\delta \vth}^3).
$$
The classical Gauss-Newton method is often motivated by the minimization of this second-order Taylor approximation (see \cite{bottou2018optimization}).
Assuming that ${{\rm I\!E}}{}_{\mymath{\vec s}}{\left[ \mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta})^T \mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta}) \right]}$ is positive-definite,
as shown in Appendix~\ref{ap:quad} the minimum is reached with
$$
\mymath{\delta \vth} = -{{\rm I\!E}}{}_{\mymath{\vec s}}{\left[ \mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta})^T \mathcal{J}^\Delta_{\mymath{\vec s}}(\mymath{\vec \theta}) \right]}^{-1} \mymath{\nabla_{\vth}} L(\mymath{\vec \theta}).
$$
As in the update obtained with the optimization problem~(\ref{eq:jacobian}), the matrix with a structure of type Jacobian transpose-times-Jacobian is characteristic of the classical Gauss-Newton approach. To ensure positive-definiteness, damping can be added in the exact same way.
The derivation that lead to~(\ref{eq:jacobian}) shows that this kind of update does not only make sense with a squared error-type of loss, so in some sense it is
a generalization of the context in which a classical Gauss-Newton approach may be useful.
If the dependency of the loss to $\mymath{\vec \theta}$ is naturally expressed via a finite-dimensional vector $\vec v(\mymath{\vec \theta})$ (e.g. $\mymath{\vec h_{\vth}}(\mymath{\vec x})$ or $\mymath{\Delta_{\vth}}(\mymath{\vec s})$ in the above cases), then
measuring the quantity $\|\vec v(\mymath{\vec \theta} + \mymath{\delta \vth}) - \vec v(\mymath{\vec \theta})\|$ to evaluate the magnitude of the modifications induced by $\mymath{\delta \vth}$ is likely to be more meaningful than using the vanilla approach (i.e. simply measuring $\|(\mymath{\vec \theta} + \mymath{\delta \vth}) - \mymath{\vec \theta}\| = \|\mymath{\delta \vth} \| $).
\paragraph{Learning rate.}
As shown in Appendix~\ref{ap:solution}, the general framework~\eqref{eq:framework}
has a unique solution $\mymath{\delta \vth}=-\alpha M(\mymath{\vec \theta})^{-1} \mymath{\nabla_{\vth}} L(\mymath{\vec \theta})$,
with $\alpha = \frac{\epsilon}{\sqrt{\mymath{\nabla_{\vth}} L(\mymath{\vec \theta})^T M(\mymath{\vec \theta})^{-1}\mymath{\nabla_{\vth}} L(\mymath{\vec \theta})}}$. The classical Gauss-Newton approach corresponds to $M(\mymath{\vec \theta}) = M_{CGN}(\mymath{\vec \theta}) + \lambda I$, or $M(\mymath{\vec \theta}) = M_{CGN}(\mymath{\vec \theta})$ if we ignore the damping.
With the approach based on the second-order approximation of the loss expressed as a squared error, the resulting update has a form $\mymath{\delta \vth}=- M(\mymath{\vec \theta})^{-1} \mymath{\nabla_{\vth}} L(\mymath{\vec \theta})$,
which is similar to the above expression except that $\alpha = 1$. However, this theoretical difference in $\alpha$ (referred to as the learning rate in Section~\ref{sec:vanilla}) is not very significant in practice since its value is usually redefined separately. It is very common to use heuristics to set $\alpha$ to smaller values so as to increase the stability of the iterative method. Indeed,
in the proposed framework, if $\epsilon$ is constant, the updates are not getting smaller and smaller, which means that no convergence is possible.
Another example of motivation for the redefinition of $\alpha$ is that when $M(\mymath{\vec \theta})$ is a very large matrix, $M(\mymath{\vec \theta})^{-1}$ is often estimated via drastic approximations. In that case, it can be preferable to only keep the update direction of the solution of~\eqref{eq:framework}, and then perform a line search to find a value of $\alpha$ for which it is reverified that the corresponding step size is reasonable.
This line search is an important component of the popular reinforcement learning
algorithm TRPO \citep{schulman2015trust}.
\subsection{Natural gradient}
\label{sec:nat}
To go one step further in terms of independence to the parametrization, it is possible to measure directly the
change from $p_\mymath{\vec \theta}( \cdot | \mymath{\vec x})$ to $p_{\mymath{\vec \theta} + \mymath{\delta \vth}}( \cdot | \mymath{\vec x})$ with a metric on probability density functions.
This way, the updates do not even depend on the choice of
finite-dimensional representation via $\mymath{\vec h_{\vth}}$.
Amari \citep{amari1997neural, amari98natural} proposed and
popularized the notion of natural gradient, which is based on a matrix called the
Fisher information matrix, defined for the p.d.f $p_\mymath{\vec \theta}( \cdot | \mymath{\vec x})$ by:
$$
\mathcal{I}_\mymath{\vec x}(\mymath{\vec \theta}) = {{\rm I\!E}}{}_{a\sim p_\mymath{\vec \theta}(\cdot|\mymath{\vec x})}{\left[ \mymath{\nabla_{\vth}} \log{\left( p_\mymath{\vec \theta}(a|\mymath{\vec x}) \right)} \mymath{\nabla_{\vth}}\log{\left( p_\mymath{\vec \theta}(a|\mymath{\vec x}) \right)}^T \right]}.
$$
It can be used to measure a ``distance'' $d\ell$ between two infinitesimally close
probability distributions $ p_\mymath{\vec \theta}(\cdot|\mymath{\vec x})$ and $p_{\mymath{\vec \theta}+\mymath{\delta \vth}}(\cdot|\mymath{\vec x})$ as follows:
$$
d\ell^2(p_\mymath{\vec \theta}(\cdot|\mymath{\vec x}), p_{\mymath{\vec \theta}+\mymath{\delta \vth}}(\cdot|\mymath{\vec x})) = \mymath{\delta \vth}^T \mathcal{I}_\mymath{\vec x}(\mymath{\vec \theta}) \mymath{\delta \vth}.
$$
Averaging over the samples, we extrapolate a measure of distance between $\mymath{\vec \theta}$ and $\mymath{\vec \theta} + \mymath{\delta \vth}$:
$$
DL^2(\mymath{\vec \theta}, \mymath{\vec \theta} + \mymath{\delta \vth}) = \mymath{\delta \vth}^T {{\rm I\!E}}{}_\mymath{\vec s}\left[ \mathcal{I}_\mymath{\vec x}(\mymath{\vec \theta}) \right] \mymath{\delta \vth},
$$
where ${{\rm I\!E}}{}_\mymath{\vec s}\left[ \mathcal{I}_\mymath{\vec x}(\mymath{\vec \theta}) \right] = {{\rm I\!E}}{}_\mymath{\vec s}\left[ {{\rm I\!E}}{}_{a\sim p_\mymath{\vec \theta}(\cdot|\mymath{\vec x})}{\left[ \mymath{\nabla_{\vth}} \log{\left( p_\mymath{\vec \theta}(a|\mymath{\vec x}) \right)} \mymath{\nabla_{\vth}}\log{\left( p_\mymath{\vec \theta}(a|\mymath{\vec x}) \right)}^T \right]} \right]$ is
the averaged Fisher information matrix.
It is common to approximate ${{\rm I\!E}}{}_\mymath{\vec s}\left[ {{\rm I\!E}}{}_{a \sim p_\mymath{\vec \theta}(\cdot|\mymath{\vec x})}[\cdot] \right]$ with the
empirical mean over the samples,
which reduces the above expression to
$$
DL^2(\mymath{\vec \theta}, \mymath{\vec \theta} + \mymath{\delta \vth}) \approx \mymath{\delta \vth}^T {{\rm I\!E}}{}_\mymath{\vec s}\left[ \mymath{\nabla_{\vth}} \log{\left( p_\mymath{\vec \theta}(\mymath{\vec y}|\mymath{\vec x}) \right)} \mymath{\nabla_{\vth}}\log{\left( p_\mymath{\vec \theta}(\mymath{\vec y}|\mymath{\vec x}) \right)}^T \right] \mymath{\delta \vth}.
$$
$ {{\rm I\!E}}{}_\mymath{\vec s}\left[ \mymath{\nabla_{\vth}} \log{\left( p_\mymath{\vec \theta}(\mymath{\vec y}|\mymath{\vec x}) \right)} \mymath{\nabla_{\vth}}\log{\left( p_\mymath{\vec \theta}(\mymath{\vec y}|\mymath{\vec x}) \right)}^T \right]$ is called the \emph{empirical}
Fisher matrix \citep{martens2014new}. We denote it by $F(\mymath{\vec \theta})$.
Putting an upper bound on $\mymath{\delta \vth}^T F(\mymath{\vec \theta}) \mymath{\delta \vth}$ results in the following optimization problem:
\begin{equation}
\label{eq:natural}
\left\{
\begin{array}{l}
\min_{\mymath{\delta \vth}} \nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta})^T \mymath{\delta \vth}\\
\mymath{\delta \vth}^T F(\mymath{\vec \theta}) \mymath{\delta \vth} \leq \epsilon^2,
\end{array}
\right.
\end{equation}
which yields \emph{natural} gradient steps of the form
$$
\mymath{\delta \vth} = -\alpha F(\mymath{\vec \theta})^{-1} \mymath{\nabla_{\vth}} L(\mymath{\vec \theta}),
$$
provided that $F(\mymath{\vec \theta})$ is invertible. $F(\mymath{\vec \theta})$ is always positive semi-definite.
Therefore, as in Section~\ref{sec:cgn} with the classical Gauss-Newton approach, a damping term
can be added to ensure invertibility (but again, by doing so the independence to the parametrization is lost).
The Fisher information matrix is in some sense uniquely defined by the property of
invariance to reparametrization of the metric it induces \citep{cencov1982}, and it can be obtained
from many different derivations. But a particularly interesting fact is that
$d\ell^2(p_\mymath{\vec \theta}(\cdot|\mymath{\vec x}), p_{\mymath{\vec \theta}+\mymath{\delta \vth}}(\cdot|\mymath{\vec x}))$ corresponds
to the second-order approximation of the Kullback-Leibler divergence $KL(p_\mymath{\vec \theta}(\cdot|\mymath{\vec x}),p_{\mymath{\vec \theta}+\mymath{\delta \vth}}(\cdot|\mymath{\vec x}))$
\citep{kullback97, akimoto2013objective}.
Hence, the terms $\mymath{\delta \vth}^T \mathcal{I}_\mymath{\vec x}(\mymath{\vec \theta}) \mymath{\delta \vth}$ and $\mymath{\delta \vth}^T F(\mymath{\vec \theta}) \mymath{\delta \vth}$ share some of the properties of the Kullback-Leibler divergence. For instance, when the variance of the probability distribution $p_\mymath{\vec \theta}(\cdot|\mymath{\vec x})$ decreases,
the same parameter modification $\mymath{\delta \vth}$ tends to result in increasingly large
measures $\mymath{\delta \vth}^T \mathcal{I}_\mymath{\vec x}(\mymath{\vec \theta}) \mymath{\delta \vth}$ (see Figure~\ref{fig:kl}).
\begin{figure}[H]
\includegraphics[width=\textwidth]{kl_new.pdf}
\caption{The same parameter change (here, a constant shift of the mean to the right) yields a larger Kullback-Leibler divergence when the variance is small.}
\label{fig:kl}
\end{figure}
Consequently, if the bound $\epsilon^2$ of Equation~(\ref{eq:natural}) is kept constant, the possible modifications of \mymath{\vec \theta} become somehow smaller
when the variance of the parametrized distribution decreases. Thus the natural gradient iterations slow down when the variance becomes small, which is a desirable
property when keeping some amount of variability is important.
Typically, in the context of reinforcement learning, this variability can be related to exploration, and it should not vanish early.
This is one of the reasons why several reinforcement learning algorithms benefit from the use of natural gradient
steps \citep{PetersNAC,schulman2015trust,wu2017scalable}.
\paragraph{Relation between natural gradient and classical Gauss-Newton approaches.}
Let us consider a very simple case where
$p_\mymath{\vec \theta}(\cdot|\mymath{\vec x})$ is a multivariate normal distribution with fixed covariance matrix $\Sigma = \beta^2 I$.
The only variable parameter on which the distribution $p_\mymath{\vec \theta}(\cdot|\mymath{\vec x})$ depends is its mean $\vec \mu_\mymath{\vec \theta}$, so we can use it as
representation of the distribution itself and write
$$
\mymath{\vec h_{\vth}}(\mymath{\vec x}) = \vec \mu_\mymath{\vec \theta}.
$$
It can be shown that the Kullback-Leibler divergence between two normal distributions
of equal variance and different means is proportional
to the squared Euclidean distance between the means.
More precisely, the Kullback-Leibler divergence between
$p_\mymath{\vec \theta}(\cdot|\mymath{\vec x})$ and $p_{\mymath{\vec \theta}+\mymath{\delta \vth}}(\cdot|\mymath{\vec x})$ is equal to
$\frac{1}{2\beta^2}\| \mymath{\vec h_{\vth + \delth}}(\mymath{\vec x}) - \mymath{\vec h_{\vth}}(\mymath{\vec x})\|^2$.
For small values of $\mymath{\delta \vth}$, this expression is
approximately equal to the measure
obtained with the true Fisher information
matrix:
$$
\frac{1}{2\beta^2}\| \mymath{\vec h_{\vth + \delth}}(\mymath{\vec x}) - \mymath{\vec h_{\vth}}(\mymath{\vec x})\|^2 \approx \mymath{\delta \vth}^T \mathcal{I}_\mymath{\vec x}(\mymath{\vec \theta}) \mymath{\delta \vth}.
$$
Bounding the average over the samples of the right term is the motivation of the natural gradient descent method.
Besides, we have seen in Section~\ref{sec:cgn} that the
classical Gauss-Newton method can be considered as a way to bound
${{\rm I\!E}}{}_{\mymath{\vec s}}[\| \mymath{\vec h_{\vth + \delth}}(\mymath{\vec s}) - \mymath{\vec h_{\vth}}(\mymath{\vec x})\|^2]$, which is equal to the average of the left term over the samples,
up to a multiplicative constant.
Hence, even though both methods introduce slightly different approximations,
we can conclude that in this context,
the classical Gauss-Newton and natural gradient descent methods are very similar.
This property is used in \cite{pascanu2013revisiting} to perform a
natural gradient descent on deterministic neural networks, by interpreting
their outputs as the mean of a conditional Gaussian distribution with fixed variance.
\section{Gradient covariance matrix, Newton's method and generalized Gauss-Newton}
\label{sec:second}
The approaches seen in Section~\ref{sec:first} all fit the general framework~\eqref{eq:framework}:
\begin{equation*}
\left\{
\begin{array}{l}
\min_{\mymath{\delta \vth}} \nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta})^T \mymath{\delta \vth}\\
\mymath{\delta \vth}^T M(\mymath{\vec \theta}) \mymath{\delta \vth} \leq \epsilon^2,
\end{array}
\right.
\end{equation*}
with matrices $M(\mymath{\vec \theta})$ that do not depend on the loss function. But since
the loss is typically based on quantities that are relevant for the task to achieve,
it can be a good idea to exploit it to constrain the steps.
We present 3 approaches that fit into the same framework but with matrices $M(\mymath{\vec \theta})$
that do depend on the loss, namely the gradient covariance matrix method,
Newton's method, and the generalized Gauss-Newton method.
\subsection{Gradient covariance matrix}
The simplest way to use the loss to measure the
magnitude of a change due to parameter modifications
is to consider the expected squared difference between
$l_\mymath{\vec \theta}(\mymath{\vec s})$ and $l_{\mymath{\vec \theta}+\mymath{\delta \vth}}(\mymath{\vec s})$:
$$
{{\rm I\!E}}{}_{\mymath{\vec s}}\left[ \big(l_{\mymath{\vec \theta}+\mymath{\delta \vth}}(\mymath{\vec s}) - l_\mymath{\vec \theta}(\mymath{\vec s})\big)^2 \right].
$$
For a single sample \mymath{\vec s}, changing slightly the object $\mymath{\vec \theta}$
does not necessarily modify the atomic loss $l_\mymath{\vec \theta}(\mymath{\vec s})$,
but in many cases it can be assumed that this loss becomes
different for at least some of the samples, yielding a positive
value for
$
{{\rm I\!E}}{}_{\mymath{\vec s}}\left[ \big(l_{\mymath{\vec \theta}+\mymath{\delta \vth}}(\mymath{\vec s}) - l_\mymath{\vec \theta}(\mymath{\vec s})\big)^2 \right]
$ which quantifies in some sense the amount of change introduced by $\mymath{\delta \vth}$ with
respect to the objective. It is often a meaningful measure as it usually depends
on the most relevant features for the task to achieve.
Let us replace $l_{\mymath{\vec \theta}+\mymath{\delta \vth}}(\mymath{\vec s})$ by a first-order approximation:
$$ l_{\mymath{\vec \theta}+\mymath{\delta \vth}}(\mymath{\vec s}) \simeq l_\mymath{\vec \theta}(\mymath{\vec s}) + \nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s})^T \mymath{\delta \vth}.$$
The above expectation simplifies to
$$
{{\rm I\!E}}{}_{\mymath{\vec s}}\left[ \big(\nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s})^T \mymath{\delta \vth}\big)^2 \right]
= \mymath{\delta \vth}^T {{\rm I\!E}}{}_{\mymath{\vec s}}\left[ \nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s}) \nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s})^T \right] \mymath{\delta \vth}.
$$
${{\rm I\!E}}{}_{\mymath{\vec s}}\left[ \nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s}) \nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s})^T \right]$ is called the
gradient covariance matrix \citep{bottou2008tradeoffs}.
It can also be called the outer product metric~\citep{ollivier:hal-00857982}.
Putting a bound on $\mymath{\delta \vth}^T {{\rm I\!E}}{}_{\mymath{\vec s}}\left[ \nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s}) \nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s})^T \right] \mymath{\delta \vth}$,
the iterated optimization becomes:
\begin{equation}
\label{eq:covgrad}
\left\{
\begin{array}{l}
\min_{\mymath{\delta \vth}} \nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta})^T \mymath{\delta \vth} \\
\mymath{\delta \vth}^T {{\rm I\!E}}{}_{\mymath{\vec s}}\left[ \nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s}) \nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s})^T \right] \mymath{\delta \vth} \leq \epsilon^2.
\end{array}
\right.
\end{equation}
It results in updates of the form:
$$
\mymath{\delta \vth} = -\alpha {{\rm I\!E}}{}_{\mymath{\vec s}}\left[\nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s}) \nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s})^T \right]^{-1} \mymath{\nabla_{\vth}} L(\mymath{\vec \theta}).
$$
Again, a regularization term may be added to ensure the invertibility of the matrix.
\paragraph{Link with the natural gradient.}
Let us assume that the atomic loss on a sample $\mymath{\vec s} = (\mymath{\vec x}, \mymath{\vec y})$ is
the negative log-likelihood (a very common choice):
$$
l_\mymath{\vec \theta}(\mymath{\vec s}) = -\log( p_\mymath{\vec \theta}(\mymath{\vec y}|\mymath{\vec x}) ).
$$
It follows that the empirical Fisher matrix, as defined in Section~\ref{sec:nat}, is equal
to ${{\rm I\!E}}{}_{\mymath{\vec s}}\left[ \nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s}) \nabla_{\mymath{\vec \theta}} l_\mymath{\vec \theta}(\mymath{\vec s})^T \right]$,
which is exactly the definition of the gradient covariance matrix. Therefore, in this
case, the two approaches are identical. Several algorithms use this identity for the natural gradient
computation, e.g. \cite{george2018fast}.
\subsection{Newton's method}
\label{sec:Newton}
Let us consider now a second-order approximation of the loss:
$$
L(\mymath{\vec \theta} + \mymath{\delta \vth}) \approx L(\mymath{\vec \theta}) + \nabla_\mymath{\vec \theta} L(\mymath{\vec \theta})^T \mymath{\delta \vth} + \frac{1}{2}\mymath{\delta \vth}^T H(\mymath{\vec \theta}) \mymath{\delta \vth} ,
$$
where $H(\mymath{\vec \theta})$ is the Hessian matrix: $\left[H(\mymath{\vec \theta})\right]_{i,j} = \frac{\partial^2 L}{\partial \mymath{\vec \theta}_i \partial \mymath{\vec \theta}_j}(\mymath{\vec \theta})$.
Although there are obvious counterexamples, one can argue that the first-order approximation, i.e. $L(\mymath{\vec \theta} + \mymath{\delta \vth}) \approx L(\mymath{\vec \theta}) + \nabla_\mymath{\vec \theta} L(\mymath{\vec \theta})^T \mymath{\delta \vth}$
(which is used as minimization objective for gradient descents), is most likely good
as long as the second-order term $\frac{1}{2}\mymath{\delta \vth}^T H(\mymath{\vec \theta}) \mymath{\delta \vth}$ remains small. Therefore,
it makes sense to directly put an upper bound on this quantity to restrict $\mymath{\delta \vth}$, as follows:
$$
\mymath{\delta \vth}^T H(\mymath{\vec \theta}) \mymath{\delta \vth} \leq \epsilon^2.
$$
This constraint defines a trust region, i.e. a neighborhood of $\mymath{\vec \theta}$
in which the first-order approximation of $L(\mymath{\vec \theta} + \mymath{\delta \vth})$ is supposed to be reasonably accurate. The trust region is bounded and well defined if the matrix $H(\mymath{\vec \theta})$ is symmetric positive-definite.
However, $H(\mymath{\vec \theta})$ is symmetric but not even necessarily positive semi-definite, unlike the matrices
obtained with the previous
approaches. Therefore the required damping to make it definite-positive may be larger than with other methods.
It leads to the following optimization problem solved at every iteration:
\begin{equation}
\label{eq:hess}
\left\{
\begin{array}{l}
\min_{\mymath{\delta \vth}} \nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta})^T \mymath{\delta \vth} \\
\mymath{\delta \vth}^T (H(\mymath{\vec \theta}) + \lambda I) \mymath{\delta \vth} \leq \epsilon^2,
\end{array}
\right.
\end{equation}
and to updates of the form:
$$
\mymath{\delta \vth} = -\alpha (H(\mymath{\vec \theta}) + \lambda I)^{-1} \mymath{\nabla_{\vth}} L(\mymath{\vec \theta}).
$$
\paragraph{The more usual derivation of Newton's method.} The same update
direction is obtained by directly minimizing the damped second-order approximation:
$$
L(\mymath{\vec \theta}) + \nabla_\mymath{\vec \theta} L(\mymath{\vec \theta})^T \mymath{\delta \vth} + \frac{1}{2}\mymath{\delta \vth}^T (H(\mymath{\vec \theta}) + \lambda I) \mymath{\delta \vth}.
$$
When $(H(\mymath{\vec \theta}) + \lambda I)$ is symmetric positive-definite, as shown in Appendix~\ref{ap:quad} the minimum of this
expression is obtained for:
$$
\mymath{\delta \vth} = - (H(\mymath{\vec \theta}) + \lambda I)^{-1} \mymath{\nabla_{\vth}} L(\mymath{\vec \theta}).
$$
\subsection{Generalized Gauss-Newton}
$L(\mymath{\vec \theta})$ is equal to ${{\rm I\!E}}{}_{\mymath{\vec s}}{\left[ l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x})) \right]}$: it
does not depend directly on $\mymath{\vec \theta}$ but on the outputs of $\mymath{\vec h_{\vth}}$, which are vectors of finite dimension.
Posing $\delta \vec h = \mymath{\vec h_{\vth + \delth}}(\mymath{\vec x}) - \mymath{\vec h_{\vth}}(\mymath{\vec x})$, a second-order Taylor expansion
of $l(\mymath{\vec y}, \mymath{\vec h_{\vth + \delth}}(\mymath{\vec x}))$ can be written:
$$
l(\mymath{\vec y}, \mymath{\vec h_{\vth + \delth}}(\mymath{\vec x})) = l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x})) + \frac{\partial l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x}))}{\partial \vec h}^T \delta \vec h
+ \frac{1}{2}\delta \vec h^T \mathcal{H}_\mymath{\vec y}(\mymath{\vec h_{\vth}}(\mymath{\vec x})) \delta \vec h + O(\delta \vec h^3),
$$
where $\mathcal{H}_\mymath{\vec y}(\mymath{\vec h_{\vth}}(\mymath{\vec x}))$ is the Hessian matrix of the atomic loss $l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x}))$
with respect to variations of $\mymath{\vec h_{\vth}}(\mymath{\vec x})$, and $\frac{\partial l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x}))}{\partial \vec h}$
is the gradient of $l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x}))$ w.r.t. variations of $\mymath{\vec h_{\vth}}(\mymath{\vec x})$.
Using the equality $\delta \vec h = J_{\mymath{\vec x}}(\mymath{\vec \theta}) \mymath{\delta \vth} + O(\mymath{\delta \vth}^2)$ (with $J_{\mymath{\vec x}}(\mymath{\vec \theta})$ the Jacobian of the function $\mymath{\vec \theta} \mapsto \mymath{\vec h_{\vth}}(\mymath{\vec x})$):
\begin{equation*}
\begin{split}
l(\mymath{\vec y}, \mymath{\vec h_{\vth + \delth}}(\mymath{\vec x})) = & \ l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x})) + \frac{\partial l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x}))}{\partial \vec h}^T J_{\mymath{\vec x}}(\mymath{\vec \theta}) \mymath{\delta \vth}
+ \frac{\partial l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x}))}{\partial \vec h}^T O(\mymath{\delta \vth}^2) \\
& + \frac{1}{2}\mymath{\delta \vth}^T J_{\mymath{\vec x}}(\mymath{\vec \theta})^T \mathcal{H}_\mymath{\vec y}( \mymath{\vec h_{\vth}}(\mymath{\vec x})) J_{\mymath{\vec x}}(\mymath{\vec \theta}) \mymath{\delta \vth} + O(\mymath{\delta \vth}^3).
\end{split}
\end{equation*}
The generalized Gauss-Newton approach is an approximation that consists in dropping
the term $\frac{\partial l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x}))}{\partial \vec h}^T O(\mymath{\delta \vth}^2)$.
Averaging over the samples yields:
$$
L(\mymath{\vec \theta} + \mymath{\delta \vth}) \approx L(\mymath{\vec \theta}) + {{\rm I\!E}}{}_{\mymath{\vec s}}\left[\frac{\partial l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x}))}{\partial \vec h}^T J_{\mymath{\vec x}}(\mymath{\vec \theta})\right]^T \mymath{\delta \vth} +
\frac{1}{2}\mymath{\delta \vth}^T {{\rm I\!E}}{}_{\mymath{\vec s}}\left[ J_{\mymath{\vec x}}(\mymath{\vec \theta})^T \mathcal{H}_\mymath{\vec y}(\mymath{\vec h_{\vth}}(\mymath{\vec x})) J_{\mymath{\vec x}}(\mymath{\vec \theta}) \right] \mymath{\delta \vth}.
$$
Noticing that
${{\rm I\!E}}{}_{\mymath{\vec s}}\left[\frac{\partial l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x}))}{\partial \vec h}^T J_{\mymath{\vec x}}(\mymath{\vec \theta}) \right] = \nabla_\mymath{\vec \theta} L(\mymath{\vec \theta})$,
it results in the following approximation:
$$
L(\mymath{\vec \theta} + \mymath{\delta \vth}) \approx L(\mymath{\vec \theta}) + \nabla_\mymath{\vec \theta} L(\mymath{\vec \theta})^T \mymath{\delta \vth} +
\frac{1}{2}\mymath{\delta \vth}^T {{\rm I\!E}}{}_{\mymath{\vec s}}\left[ J_{\mymath{\vec x}}(\mymath{\vec \theta})^T \mathcal{H}_\mymath{\vec y}(\mymath{\vec h_{\vth}}(\mymath{\vec x})) J_{\mymath{\vec x}}(\mymath{\vec \theta}) \right] \mymath{\delta \vth}.
$$
As for Newton's method, the usual way to derive the generalized Gauss-Newton method
is to directly minimize this expression (see \cite{martens2014new}), but we can
also put a bound on the quantity $\mymath{\delta \vth}^T {{\rm I\!E}}{}_{\mymath{\vec s}}\left[ J_{\mymath{\vec x}}(\mymath{\vec \theta})^T \mathcal{H}_\mymath{\vec y}(\mymath{\vec h_{\vth}}(\mymath{\vec x})) J_{\mymath{\vec x}}(\mymath{\vec \theta}) \right] \mymath{\delta \vth}$ to define a trust region for the validity of the first-order approximation (as in Section~\ref{sec:Newton}),
provided that ${{\rm I\!E}}{}_{\mymath{\vec s}}\left[ J_{\mymath{\vec x}}(\mymath{\vec \theta})^T \mathcal{H}_\mymath{\vec y}(\mymath{\vec h_{\vth}}(\mymath{\vec x})) J_{\mymath{\vec x}}(\mymath{\vec \theta}) \right]$ is symmetric positive-definite.
If the loss $l(\mymath{\vec y}, \mymath{\vec h_{\vth}}(\mymath{\vec x}))$ is convex in $\mymath{\vec h_{\vth}}(\mymath{\vec x})$ (which is often true),
the matrix is at least positive semi-definite, so a small damping term suffices to make it positive-definite.
If a non-negligible portion of the matrices $J_{\mymath{\vec x}}(\mymath{\vec \theta})$ are full rank,
the damping term may be added to $\mathcal{H}_\mymath{\vec y}(\mymath{\vec h_{\vth}}(\mymath{\vec x}))$ rather than to the full matrix.
See \cite{martens2012training} for an extensive discussion on different options for damping
and their benefits and drawbacks. With the damping on the full matrix,
the optimization problem to solve at every iteration becomes:
\begin{equation}
\label{eq:GGN}
\left\{
\begin{array}{l}
\min_{\mymath{\delta \vth}} \nabla_\mymath{\vec \theta} L(\mymath{\vec \theta})^T \mymath{\delta \vth}\\
\mymath{\delta \vth}^T \left( {{\rm I\!E}}{}_{\mymath{\vec s}}\left[ J_{\mymath{\vec x}}(\mymath{\vec \theta})^T \mathcal{H}_\mymath{\vec y}(\mymath{\vec h_{\vth}}(\mymath{\vec x})) J_{\mymath{\vec x}}(\mymath{\vec \theta}) \right] + \lambda I\right) \mymath{\delta \vth} \leq \epsilon^2,
\end{array}
\right.
\end{equation}
resulting in updates of the form:
$$
\mymath{\delta \vth} = - \alpha \left( {{\rm I\!E}}{}_{\mymath{\vec s}}\left[ J_{\mymath{\vec x}}(\mymath{\vec \theta})^T \mathcal{H}_\mymath{\vec y}(\mymath{\vec h_{\vth}}(\mymath{\vec x})) J_{\mymath{\vec x}}(\mymath{\vec \theta}) \right] + \lambda I\right)^{-1} \mymath{\nabla_{\vth}} L(\mymath{\vec \theta}).
$$
\section{Summary and conclusion}
\label{sec:summ}
In sections~\ref{sec:first} and \ref{sec:second} we motivated and derived 6 different ways to compute parameter updates, that can all be interpreted as
solving an optimization problem of this type:
\begin{equation*}
\left\{
\begin{array}{l}
\min_{\mymath{\delta \vth}} \nabla_{\mymath{\vec \theta}} L(\mymath{\vec \theta})^T \mymath{\delta \vth}\\
\mymath{\delta \vth}^T M(\mymath{\vec \theta}) \mymath{\delta \vth} \leq \epsilon^2,
\end{array}
\right.
\end{equation*}
resulting in updates of the form:
$$
\mymath{\delta \vth} = - \alpha M(\mymath{\vec \theta})^{-1} \mymath{\nabla_{\vth}} L(\mymath{\vec \theta}),
$$
$M(\mymath{\vec \theta})$ being symmetric positive-definite. The quadratic term
of the inequality corresponds to a specific metric defined by $M(\mymath{\vec \theta})$
used to measure the magnitude of the modification induced by $\mymath{\delta \vth}$.
To evaluate this magnitude, the focus can simply be on the norm of $\mymath{\delta \vth}$,
or on the effect of $\mymath{\delta \vth}$ on the loss, or on the effect
of $\mymath{\delta \vth}$ on $\mymath{\vec h_{\vth}}(\mymath{\vec x})$ or on $p_\mymath{\vec \theta}(\cdot|\mymath{\vec x})$, resulting in various approaches,
with various definitions of $M(\mymath{\vec \theta})$. In a context of probabilistic regression, we gave 6 examples that
correspond to popular variants of the gradient descent, summarized in Table~\ref{summary}. All methods except the natural gradient can be declined to deterministic cases.
Unifying several first-order or second-order variants of the gradient descent method
enabled us to reveal links between these different approaches,
and contexts in which some of them are equivalent.
The proposed framework gives a compact overview
of common variants of the gradient descent, and
hopefully can help
choosing adequately between them depending on the problem to solve.
Perhaps, it can also help designing new variants or combining
existing ones to obtain new desired features.
\section*{Acknowledgements}
This research was partially supported by the French National Research Agency (ANR), Project \mbox{ANR-18-CE33-0005} HUSKI.
|
1,116,691,497,718 | arxiv | \section{Introduction}
\label{sec:intro}
Recent years have seen the proliferation of deceptive information online.
With the increasing necessity to validate the information from the Internet, \emph{automatic fact checking} has emerged as an important research topic. It is at the core of multiple applications, e.g., discovery of fake news, rumor detection in social media, information verification in question answering systems, detection of information manipulation agents, and assistive technologies for investigative journalism. At the same time, it touches many aspects, such as credibility of users and sources, information veracity, information verification, and linguistic aspects of deceptive language.
\noindent In this paper, we present an approach to fact-checking with the following design principles: ({\em i})~ generality, ({\em ii})~ robustness, ({\em iii})~ simplicity, ({\em iv})~ reusability, and ({\em v})~ strong machine learning modeling. Indeed, the system makes very few assumptions about the task, and looks for supportive information directly on the Web.
Our system works fully automatically. It does not use any heavy feature engineering and can be easily used in combination with task-specific approaches as well, as a core subsystem. Finally, it combines the representational strength of recurrent neural networks with kernel-based classification.
\begin{figure*
\centering
\includegraphics[width=1.0\textwidth]{retrieval_pipeline.png}
\vspace{2mm}
\scriptsize
\begin{tabular}{p{15.8cm}}
$^a$\url{http://www.snopes.com/2015/09/16/ahmed-mohamed/} \\
$^b$\url{http://www.businessinsider.com/ahmed-mohamed-arrested-irving-texas-clock-bomb-2015-9} \\
$^c$\url{ http://www.snopes.com/chipotle-closing/} \\
$^d$\url{http://www.chicagotribune.com/business/ct-chipotle-burger-restaurant-20160728-story.html} \\
\end{tabular}
\caption{Example claims
and the information we use to predict whether they are factually true or false.\label{fig:example_claim}}
\end{figure*}
The system starts with a claim to verify. First, we automatically convert the claim into a query, which we execute against a search engine in order to obtain a list of potentially relevant documents. Then, we take both the snippets and the most relevant sentences in the full text of these Web documents, and we compare them to the claim. The features we use are dense representations of the claim, of the snippets and of related sentences from the Web pages, which we automatically train for the task using
Long Short-Term Memory networks (LSTMs).
We also use the final hidden layer of the neural network as a task-specific embedding of the claim, together with the Web evidence. We feed all these representations as features, together with pairwise similarities, into a Support Vector Machine (SVM) classifier using an RBF kernel to classify the claim as True or False.
Figure~\ref{fig:example_claim} presents a real example from one of the datasets we experiment with. The left-hand side of the figure contains a True example, while the right-hand side shows a False one. We show the original claims from \url{snopes.com}, the query generated by our system, and the information retrieved from the Web (most relevant snippet and text selection from the web page).
The veracity of the claim can be inferred from the textual information.
\noindent Our contributions can be summarized as follows:
\begin{itemize}
\item We propose a general-purpose light-weight framework for fully-automatic fact checking using evidence derived from the Web.
\item We propose a deep neural network with LSTM encoding to combine semantic kernels with task-specific embeddings that encode a claim together with pieces of potentially-relevant text fragments from the Web, taking the source reliability into account.
\item We further study factuality in community Question Answering (cQA), and we create a new high-quality dataset, which we release to the research community. To the best of our knowledge, we are the first to study factuality of answers in cQA forums, and our dataset is the first dataset specifically targeting factuality in a cQA setting.
\item We achieve strong results on two different tasks and datasets ---rumor detection and fact checking of the answers to a question in community question answering forums---, thus demonstrating the generality of the approach and its potential applicability to different fact-checking problem formulations.
\end{itemize}
\noindent The remainder of this paper is organized as follows.
Section~\ref{sec:method} introduces our method for fact checking claims using external sources.
Section~\ref{sec:experiments} presents our experiments and discusses the results.
Section~\ref{sec:application2cqa} describes an application of our approach to a different dataset and a slightly different task: fact checking in community question answering forums.
Section~\ref{sec:related} presents related work.
Finally, Section~\ref{sec:conclusions} concludes and suggests some possible directions for future work.
\section{The Fact-Checking System}
\label{sec:method}
Given a claim, our system searches for support information on the Web in order to verify whether the claim is likely to be true. The three steps in this process are
({\em i})~ external support retrieval, ({\em ii})~ text representation, and ({\em iii})~ veracity prediction.
\subsection{External Support Retrieval}
\begin{figure*
\centering
\includegraphics[width=0.65\textwidth]{nn_architecture_2_.jpg}\vspace*{4mm}
\includegraphics[width=0.93\textwidth]{BiLstm-processed.jpg}
\caption{Our general neural network architecture (top) and detailed LSTM representation (bottom).
Each blue box in the top consists of the bi-LSTM structure in the bottom.\label{fig:NN}}
\end{figure*}
This step consists of generating a query out of the claim and querying a search engine (here, we experiment with Google and Bing) in order to retrieve supporting documents.
Rather than querying the search engine with the full claim (as on average, a claim is two sentences long), we generate a shorter query following the lessons highlighted in \cite{potthast2013overview}.
\noindent As we aim to develop a general-purpose fact checking system, we use an approach for query generation that does not incorporate any features that are specific to claim verification (e.g., no temporal indicators).
We rank the words by means of \emph{tf-idf}. We compute the \emph{idf} values on a 2015 Wikipedia dump and the English Gigaword.\footnote{\url{catalog.ldc.upenn.edu/ldc2011t07}} \citet{potthast2013overview} suggested that a good way to perform high-quality search is to only consider the verbs, the nouns and the adjectives in the claim; thus, we exclude all words in the claim that belong to other parts of speech. Moreover, claims often contain named entities (e.g., names of persons, locations, and organizations); hence, we augment the initial query with all the named entities from the claim's text. We use IBM's AlchemyAPI\footnote{\url{www.ibm.com/watson/alchemy-api.html}} to identify named entities. Ultimately, we generate queries of 5--10 tokens, which we execute against a search engine.
We then collect the snippets and the URLs in the results, skipping any result that points to a domain that is considered unreliable.\footnote{We created such a list by manually checking the 100 most frequent domains in the results, which we accummulated across many queries and experiments.}
Finally, if our query has returned no results, we iteratively relax it by dropping the final tokens one at a time.
\subsection{Text Representation
\label{sub:representation}
Next, we build the representation of a claim and the corresponding snippets and Web pages.
First, we calculate three similarities (a)~between the claim and a snippet, or (b)~between the claim and a Web page: ({\em i})~ cosine with \emph{tf-idf}, ({\em ii})~ cosine over embeddings, and ({\em iii})~ containment~\cite{lyon2001detecting}.
We calculate the embedding of a text as the average of the embeddings of its words; for this, we use pre-trained embeddings from GloVe \cite{pennington-socher-manning:2014:EMNLP2014}.
Moreover, as a Web page can be long, we first split it into a set of rolling sentence triplets, then we calculate the similarities between the claim and each triplet, and we take the highest scoring triplet.
Finally, as we have up to ten hits from the search engine, we take the maximum and also the average of the
three similarities over the snippets
and over the Web pages.
We further use as features the embeddings of the claim, of the best-scoring snippet, and of the best-scoring sentence triplet from a Web page. We calculate these embeddings ({\em i})~ as the average of the embeddings of the words in the text, and also ({\em ii})~ using LSTM encodings, which we train for the task as part of a deep neural network (NN).
We also use a task-specific embedding of the claim together with all the above evidence about it, which comes from the last hidden layer of the NN.
\subsection{Veracity Prediction}
Next, we build classifiers: neural network (NN), support vector machines (SVM), and a combination thereof (SVM+NN).
\paragraph{NN.}
The architecture of our NN is shown on top of Figure~\ref{fig:NN}. We have five LSTM
sub-networks, one for each of the text sources from two search engines: \emph{Claim}, \emph{Google Web page}, \emph{Google snippet}, \emph{Bing Web page}, and \emph{Bing snippet}. The claim is fed into the neural network as-is. As we can have multiple snippets, we only use the best-matching one as described above. Similarly, we only use a single best-matching triple of consecutive sentences from a Web page.
We further feed the network with the similarity features described above.
\noindent All these vectors are concatenated and fully connected to a much more compact hidden layer that captures the task-specific embeddings. This layer is connected to a softmax output unit to classify the claim as true or false.
The bottom of Figure~\ref{fig:NN} represents the generic architecture of each of the LSTM components.
The input text is transformed into a sequence of word embeddings, which is then passed to the bidirectional LSTM layer to obtain a representation for the full sequence.
\begin{table*}[t
\centering
\footnotesize
\begin{tabular}{l|ccc|ccc|ccr}
\multicolumn{1}{c}{ } & \multicolumn{3}{c}{\bf False Claims} & \multicolumn{3}{c}{\bf True Claims} & \multicolumn{3}{c}{\bf Overall}\\
\bf Model & \bf P & \bf R & \bf F1 & \bf P & \bf R & \bf F1 & \bf AvgR & \bf AvgF$_1$ & \bf Acc\\
\hline
SVM + NN & 84.1 & 86.3 & 85.2 & 71.1 & 67.5 & 69.2 & \bf 76.9 & \bf 77.2 & \bf 80.0\\
NN & 79.6 & 92.5 & 85.5 & 77.8 & 52.5 & 62.7 & \bf 72.5 & \bf 74.1 & \bf 79.2 \\
SVM & 75.0 & 86.3 & 80.2 & 60.7 & 42.5 & 50.0 & \bf 64.4 & \bf 65.1 & \bf 71.7\\
\hline
all \emph{false} & 66.7 & 100.0 & 80.0 & -- & 0.0 & 0.0 & 50.0 & 40.0 & 66.7\\
all \emph{true} & -- & 0.0 & 0.0 & 33.3 & 100.0 & 50.0 & 50.0 & 25.0 & 33.3\\
\hline
\end{tabular}
\caption{\label{table:resultsRumor} Results on the \emph{rumor detection dataset} using Web pages returned by the search engines.}
\end{table*}
\paragraph{SVM.}
Our second classifier is an SVM with an RBF kernel.
The input is the same as for the NN: word embeddings and similarities.
However, the word embeddings this time are calculated by averaging rather than using a bi-LSTM.
\paragraph{SVM + NN.}
Finally, we combine the SVM with the NN by augmenting the input to the SVM with the values of the units in the hidden layer. This represents a task-specific embedding of the input example, and in our experiments it turned out to be quite helpful. Unlike in the SVM only model, this time we use the bi-LSTM embeddings as an input to the SVM. Ultimately, this yields a combination of deep learning and task-specific embeddings with RBF kernels.
\section{Experiments and Evaluation}
\label{sec:experiments}
\subsection{Dataset}
\label{sub:dataset}
We used part of the rumor detection dataset created by~\citet{ma2016detecting}. While they analyzed a claim based on a set of potentially related tweets, we focus on the claim itself and on the use of supporting information from the Web.
\noindent The dataset consists of 992 sets of tweets, 778 of which are generated starting from a claim on \url{snopes.com}, which~\newcite{ma2016detecting} converted into a query. Another 214 sets of tweets are tweet clusters created by other researchers~\cite{Castillo:2011:ICT:1963405.1963500,Kwon:2013} with no claim behind them.
\newcite{ma2016detecting} ignored the claim and did not release it as part of their dataset.
We managed to find the original claim for 761 out of the 778 \url{snopes.com}-based clusters.
Our final dataset consists of 761 claims from \url{snopes.com}, which span various domains including politics, local news, and fun facts. Each of the claims is labeled as factually \emph{true} (34\%) or as a \emph{false} rumor (66\%).
We further split the data into 509 for training, 132 for development, and 120 for testing.
As the original split for the dataset was not publicly available, and as we only used a subset of their data, we had to make a new training and testing split.
Note that we ignored the tweets, as we wanted to focus on a complementary source of information: the Web.
Moreover, \newcite{ma2016detecting} used manual queries, while we use a fully automatic method.
Finally, we augmented the dataset with Web-retrieved snippets, Web pages, and sentence triplets from Web pages.\footnote{All the data, including the splits, is available at \url{github.com/gkaradzhov/FactcheckingRANLP}}
\subsection{Experimental Setup}
We tuned the architecture (i.e., the number of layers and their size) and the hyper-parameters of the neural network on the development dataset.
The best configuration uses a bidirectional LSTM with 25 units. It further uses a RMSprop optimizer with 0.001 initial learning rate, L2 regularization with $\lambda$=0.1, and 0.5 dropout after the LSTM layers. The size of the hidden layer is 60 with \emph{tanh} activations.
We use a batch of 32 and we train for 400 epochs.
For the SVM model, we merged the development and the training dataset, and we then ran a 5-fold cross-validation with grid-search, looking for the best kernel and its parameters. We ended up selecting an RBF kernel with $c=16$ and $\gamma=$0.01.
\subsection{Evaluation Metrics}
The evaluation metrics we use are P~(precision), R~(recall), and F$_1$, which we calculate with respect to the false and to the true claims. We further report AvgR (macro-average recall), AvgF$_1$ (macro-average F$_1$), and Acc (accuracy).
\subsection{Results}
Table~\ref{table:resultsRumor} shows the results on the test dataset.
We can see that both the NN and the SVM models improve over the majority class baseline (all false rumors) by a sizable margin. Moreover, the NN consistently outperforms the SVM by a margin on all measures. Yet, adding the task-specific embeddings from the NN as features of the SVM yields overall improvements over both the SVM and the NN in terms of avgR, avgF$_1$, and accuracy.
We can see that both the SVM and the NN overpredict the majority class (false claims); however, the combined SVM+NN model is quite balanced between the two classes.
Table~\ref{table:results:rumor:detailed} compares the performance of the SVM with and without task-specific embeddings from the NN, when training on Web pages vs. snippets, returned by Google vs. Bing vs. both. The NN embeddings consistently help the SVM in all cases. Moreover, while the baseline SVM using snippets is slightly better than when using Web pages, there is almost no difference between snippets vs. Web pages when NN embeddings are added to the basic SVM. Finally, gathering external support from either Google or Bing makes practically no difference,
and using the results from both together does not yield much further improvement.
Thus, ({\em i})~ the search engines already do a good job at generating relevant snippets, and one does not need to go and download the full Web pages, and ({\em ii})~ the choice of a given search engine is not an important factor. These are good news for the practicality of our approach.
Unfortunately, direct comparison with respect to~\cite{ma2016detecting} is not possible. First, we only use a subset of their examples: 761 out of 993 (see Section~\ref{sub:dataset}),
and we also have a different class distribution.
More importantly, they have a very different formulation of the task: for them, the claim is not available as input (in fact, there has never been a claim for 21\% of their examples); rather an example consists of a set of tweets retrieved using \emph{manually} written queries.
\noindent In contrast, our system is fully automatic and does not use tweets at all.
Furthermore, their most important information source is the change in tweets volume over time, which we cannot use. Still, our results are competitive to theirs when they do not use temporal features.
\begin{table
\centering
\footnotesize
\resizebox{\linewidth}{!}{%
\begin{tabular}{llcccc}
\bf Model & \bf External support & \bf AvgR & \bf AvgF$_1$ & \bf Acc\\
\hline
\hline
SVM + NN& Bing+Google; pages & 76.9 & 77.2 & \bf 80.0\\
SVM & Bing+Google; pages & 64.4 & 65.1 & \bf 71.7\\
\hline
SVM + NN & Bing+Google; snippets & 75.6 & 75.6 & \bf 78.3\\
SVM & Bing+Google; snippets & 68.1 & 69.0 & \bf 74.2\\
\hline
\hline
SVM + NN & Bing; pages & 77.5 & 77.0 & \bf79.2\\
SVM & Bing; pages & 66.9 & 67.5 & \bf 72.5\\
\hline
SVM + NN & Bing; snippets & 76.3 & 76.4 & \bf 79.2\\
SVM & Bing; snippets & 68.8 & 69.7 & \bf 75.0\\
\hline
\hline
SVM + NN & Google; pages & 73.1 & 74.2 & \bf 78.3\\
SVM & Google; pages & 63.1 & 63.8 & \bf 71.7\\
\hline
SVM + NN & Google; snippets & 73.1 & 74.2 & \bf 78.3\\
SVM & Google; snippets & 65.6 & 66.6 & \bf 73.3\\
\hline
\hline
\multicolumn{2}{l}{baseline (all false claims)} & 50.0 & 40.0 & \bf 66.7\\
\hline
\end{tabular}
}
\caption{\label{table:results:rumor:detailed} Results using an SVM with and without task-specific embeddings from the NN on the \emph{Rumor detection dataset}. Training on Web pages vs. snippets vs. both.}
\end{table}
To put the results in perspective,
we can further try to make an indirect comparison to the very recent paper by \citet{Popat:2017:TLE:3041021.3055133}. They also present a model to classify true vs. false claims extracted from \url{snopes.com}, by using information extracted from the Web. Their formulation of the task is the same as ours, but our corpora and label distributions are not the same, which makes a direct comparison impossible. Still, we can see that regarding overall classification accuracy they improve a baseline from 73.7\% to 84.02\% with their best model, i.e., a 39.2\% relative error reduction. In our case, we go from 66.7\% to 80.0\%, i.e., an almost identical 39.9\% error reduction. These results are very encouraging, especially given the fact that our model is much simpler than theirs
regarding the sources of information used (they model the stance of the text, the reliability of the sources, the language style of the articles, and the temporal footprint).
\section{Application to cQA}
\label{sec:application2cqa}
Next, we tested the generality of our approach by applying it to a different setup:
fact-checking the answers in community question answering (cQA) forums.
As this is a new problem, for which no dataset exists, we created one.
We augmented with factuality annotations the cQA dataset from SemEval-2016 Task 3 (CQA-QA-2016)~\cite{nakov-EtAl:2016:SemEval}. Overall, we annotated 249 question--answer, or $q$-$a$, pairs (from 71 threads):
128~factually true and 121 factually false answers.
\begin{figure
\centering
\includegraphics[width=0.40\textwidth]{cqa_example.png}
\caption{\label{fig:example} Example from the cQA forum dataset.}
\label{fig:example1}
\end{figure}
Each question in CQA-QA-2016 has a subject, a body, and meta information: ID, category (e.g., \emph{Education}, and \emph{Moving to Qatar}), date and time of posting, user name and ID.
We selected
only the factual questions such as ``\emph{What is Ooredoo customer service number?}'', thus filtering out all
(\emph{i})~socializing, e.g., ``\emph{What was your first car?}'', (\emph{ii})~requests for opinion/advice/guidance, e.g., ``\emph{Which is the best bank around??}'', and (\emph{iii})~questions containing multiple sub-questions, e.g., ``\emph{Is there a land route from Doha to Abudhabi. If yes; how is the road and how long is the journey?}''
Next, we annotated for veracity the answers to the retained questions. Note that in CQA-QA-2016, each answer has a subject, a body, meta information (answer ID, user name and ID), and a judgment about how well it addresses the question of its thread: \textsc{Good}\,\ vs.\ \textsc{Potentially Useful}\,\ vs.\ \textsc{Bad}\,.
We only annotated the \textsc{Good}\,\ answers.\footnote{See \cite{nakov-EtAl:2017:SemEval} for an overview of recent approaches to finding \textsc{Good}\,\ answers for cQA.}
We further discarded answers whose factuality was very time-sensitive (e.g., ``\emph{It is Friday tomorrow.}'', ``\emph{It was raining last week.}'')\footnote{Arguably, many answers are somewhat time-sensitive, e.g., ``\emph{There is an IKEA in Doha.}'' is true only after IKEA opened, but not before that. In such cases, we just used the present situation as a point of reference.},
or for which the annotators were unsure.
\begin{table
\footnotesize
\centering
\begin{tabular}{llccc}
& \textbf{Label} & \textbf{Answers} \\
\hline
$+$ & \textsc{Factual - True} & 128 \\
\hline
$-$ & \textsc{Factual - Partially True} & 38 \\
$-$ & \textsc{Factual - Conditionally True} & 16 \\
$-$ & \textsc{Factual - False} & 22 \\
$-$ & \textsc{Factual - Responder Unsure} & 26 \\
$-$ & \textsc{NonFactual} & 19 \\
\hline
& \bf TOTAL & \bf 249 \\
& $+$ \bf \textsc{Positive} & \bf 128\\
& $-$ \bf \textsc{Negative} & \bf 121\\
\hline
\end{tabular}
\caption{Distribution of the answer labels.}
\label{table:comment-labels-distribution}
\end{table}
\begin{table*}[tbh]
\centering
\footnotesize
\begin{tabular}{l|ccc|ccc|ccr}
\multicolumn{1}{c}{ } & \multicolumn{3}{c}{\bf False Claims} & \multicolumn{3}{c}{\bf True Claims} & \multicolumn{3}{c}{\bf Overall}\\
\bf Model & \bf P & \bf R & \bf F1 & \bf P & \bf R & \bf F1 & \bf AvgR & \bf AvgF1 & \bf Acc\\
\hline
SVM + NN & 72.2 & 76.5 & 74.3 & 73.3 & 68.8 & 71.0 & \bf 72.7 & \bf 72.7 & \bf 72.7\\
SVM & 70.6 & 70.6 & 70.6 & 68.8 & 68.8 & 68.8 & \bf 69.7 & \bf 69.7 & \bf 69.7\\
NN & 61.1 & 64.7 & 62.9 & 60.0 & 56.3 & 58.1 & \bf 60.5 & \bf 60.5 & \bf 60.6 \\
\hline
all \emph{false} & 51.5 & 100.0 & 68.0 & -- & 0 & 0 & \bf 50.0 & \bf 34.0 & \bf 51.5\\
all \emph{true} & -- & 0.0 & 0.0 & 48.5 & 100.0 & 65.3 & \bf 50.0 & \bf 32.7 & \bf 48.5\\
\hline
\end{tabular}
\caption{\label{table:results:QL} Results on the cQA answer fact-checking problem.
}
\end{table*}
\noindent We targeted very high quality, and thus we did not use crowdsourcing for the annotation, as pilot annotations showed that the task was very difficult and that it was not possible to guarantee that \textit{Turkers} would do all the necessary verification, e.g.,~gathering evidence from trusted sources. Instead, all examples were first annotated independently by four annotators,
and then \emph{each example} was discussed in detail to come up with a final label.
We ended up with 249 \textsc{Good} answers to 71 different questions, which we annotated for factuality: 128 \textsc{Positive} and 121 \textsc{Negative} examples. See Table~\ref{table:comment-labels-distribution} for details.
We further split our dataset into 185 $q$--$a$ pairs for training, 31 for development, and 32 for testing, preserving the general positive:negative ratio, and
making sure that the questions for the $q$--$a$ pairs did not overlap between the splits.%
Figure~\ref{fig:example1} presents an excerpt of an example from the dataset, with one question and three answers selected from a longer thread. Answer $a_1$ contains false information, while $a_2$ and $a_3$ are true, as can be checked on an official governmental website.\footnote{\url{https://www.moi.gov.qa/site/english/departments/PassportDept/news/2011/01/03/23385.html}}
We had to fit our system for this problem, as here we do not have claims, but a question and an answer. So, we constructed the query from the concatenation of $q$ and $a$.
Moreover, as Google and Bing performed similarly, we only report results using Google. We limited our run to snippets only, as we have found them rich enough above (see Section~\ref{sec:experiments}).
Also, we had a list of reputed and Qatar-related sources for the domain, and we limited our results to these sources only.
This time, we had more options to calculate similarities compared to the rumors dataset: we can compare against $q$, $a$, and $q$--$a$;
we chose to go with the latter.
For the LSTM representations, we use both the question and the answer.
\noindent Table~\ref{table:results:QL} shows the results on the cQA dataset. Once again, our models outperformed all baselines by a margin. This time, the predictions of all models are balanced between the two classes, which is probably due to the dataset being well balanced in general. The SVM model performs better than the NN by itself. This is due to the fact that the cQA dataset is significantly smaller than the \textit{rumor detection} one. Thus, the neural network could not be trained effectively by itself. Nevertheless, the task-specific representations were useful and combining them with the SVM model yielded consistent improvements on all the measures once again.
\section{Related Work}
\label{sec:related}
Journalists, online users, and researchers are well aware of the proliferation of false information on the Web, and topics such as information credibility and fact checking are becoming increasingly important as research directions.
For example, there was a recent 2016 special issue of the ACM Transactions on Information Systems journal on Trust and Veracity of Information in Social Media \cite{Papadopoulos:2016:OSI}, there was a SemEval-2017 shared task on Rumor Detection \cite{derczynski-EtAl:2017:SemEval}, and there is an upcoming lab at CLEF-2018 on Automatic Identification and Verification of Claims in Political Debates \cite{RANLP2017:debates}.
The credibility of contents on the Web has been questioned by researches for a long time.
While in the early days the main research focus was on online news portals~\cite{brill2001online,finberg2002digital,Hardalov2016}, the interest has eventually shifted towards social media \cite{Castillo:2011:ICT:1963405.1963500,PlosONE:2016,Popat:2017:TLE:3041021.3055133,RANLP2017:clickbait},
which are abundant in
sophisticated malicious users such as opinion manipulation \emph{trolls}, paid \cite{Mihaylov2015ExposingPO} or just perceived \cite{Mihaylov2015FindingOM,mihaylov-nakov:2016:P16-2}, \emph{sockpuppets} \cite{Maity:2017:DSS:3022198.3026360}, \emph{Internet water army} \cite{Chen:2013:BIW:2492517.2492637}, and \emph{seminar users} \cite{SeminarUsers2017}.
\noindent For instance, \citet{Canini:2011} studied the credibility of Twitter accounts (as opposed to tweet posts), and found that both the topical content of information sources and social network structure affect source credibility.
Other work, closer to ours, aims at addressing credibility assessment of rumors on Twitter as a problem of finding false information about a newsworthy event~\cite{Castillo:2011:ICT:1963405.1963500}. This model considers
user reputation, writing style, and various time-based features, among others.
Other efforts have focused on news communities. For example, several truth discovery algorithms are
combined in an ensemble method for veracity estimation in the VERA system~\cite{Ba:2016:VERA}. They proposed a platform for end-to-end truth discovery from the Web: extracting unstructured information from multiple sources, combining information about single claims, running an ensemble of algorithms, and visualizing and explaining the results.
They also explore two different real-world application scenarios for their system:
fact checking for crisis situations and evaluation of trustworthiness of a rumor. However, the input to their model is structured data, while here we are interested in unstructured text as input.
Similarly, the task defined by~\citet{mukherjee2015leveraging} combines three objectives: assessing the credibility of a set of posted articles, estimating the trustworthiness of sources, and predicting user's expertise.
They considered a manifold of features characterizing language, topics and Web-specific statistics (e.g., review ratings) on top of a continuous conditional random fields model.
In follow-up work, \citet{popat2016credibility} proposed a model to support or refute claims from \url{snopes.com} and Wikipedia
by considering supporting information gathered from the Web. They used the same task formulation for claims as we do, but different datasets.
In yet another follow-up work, \newcite{Popat:2017:TLE:3041021.3055133} proposed a complex model that considers stance, source reliability, language style, and temporal information.
\noindent Our approach to fact checking is related: we verify facts on the Web. However, we use a much simpler and feature-light system, and a different machine learning model.
Yet, our model performs very similarly to this latter work (even though a direct comparison is not possible as the datasets differ), which is a remarkable achievement given the fact that we consider less knowledge sources, we have a conceptually simpler model, and we have six times less training data than \newcite{Popat:2017:TLE:3041021.3055133}.
Another important research direction is on using tweets and temporal information for checking the factuality of rumors.
For example, \citet{Ma:2015:DRU} used temporal patterns of rumor dynamics to detect false rumors and to predict their frequency.
\citet{Ma:2015:DRU} focused on detecting false rumors in Twitter using time series.
They used the change of social context features over a rumor's life cycle in order to detect rumors at an early stage after they were broadcast.
A more general approach for detecting rumors is explored by~\citet{ma2016detecting}, who used recurrent neural networks to learn hidden representations that capture the variation of contextual information of relevant posts over time. Unlike this work, we do not use microblogs, but we query the Web directly in search for evidence. Again, while direct comparison to the work of \citet{ma2016detecting} is not possible, due to differences in dataset and task formulation, we can say that our framework is competitive when temporal information is not used. More importantly, our approach is orthogonal to theirs in terms of information sources used, and thus, we believe there is potential in combining the two approaches.
In the context of question answering, there has been work on assessing the credibility of an answer, e.g., based on intrinsic information~\cite{banerjee-han:2009:NAACLHLT09-Short}, i.e., without any external resources. In this case, the reliability of an answer is measured by computing the divergence between language models of the question and of the answer.
The spawn of community-based question answering Websites also allowed for the use of other kinds of information. Click counts, link analysis (e.g., PageRank), and user votes have been used to assess the quality of a posted answer~\cite{Agichtein:2008:FHC:1341531.1341557,Jeon:2006:FPQ:1148170.1148212,Jurczyk:2007:DAQ:1321440.1321575}. Nevertheless, these studies address the answers' credibility level just marginally.
\noindent Efforts to determine the credibility of an answer in order to assess its overall quality required the inclusion of content-based information~\cite{Su-EtAl:2010:PACLIC2010}, e.g., verbs and adjectives such as \emph{suppose} and \emph{probably}, which cast doubt on the answer.
Similarly, \citet{lita2005qualitative}
used source credibility (e.g., does the document come from a government Website?),
sentiment analysis, and answer contradiction compared to other related answers.
Overall, \emph{credibility} assessment for question answering has been mostly modeled at the feature level, with the goal of assessing the quality of the answers. A notable exception is the work of \cite{RANLP2017:credibility:trolls}, where credibility is treated as a task of its own right.
Yet, note that \emph{credibility} is different from \emph{factuality} (our focus here) as the former is a subjective perception about whether a statement is credible, rather than verifying it as true or false as a matter of fact; still, these notions are often wrongly mixed in the literature.
To the best of our knowledge, no previous work has targeted fact-checking of answers in the context of community Question Answering by gathering external support.
\section{Conclusions and Future Work}
\label{sec:conclusions}
We have presented and evaluated a general-purpose method for fact checking that relies on retrieving supporting information from the Web and comparing it to the claim using machine learning. Our method is lightweight in terms of features and can be very efficient because it shows good performance by only using the snippets provided by the search engines. The combination of the representational power of neural networks with the classification of kernel-based methods has proven to be crucial for making balanced predictions and obtaining good results.
Overall, the strong performance of our model across two different fact-checking tasks confirms its generality and potential applicability for different domains and for different fact-checking task formulations.
In future work, we plan to test the generality of our approach by
applying it to these and other datasets in combination with complementary methods, e.g., those focusing on microblogs and temporal information in Twitter to make predictions about rumors~\cite{Ma:2015:DRU,ma2016detecting}.
We also want to explore the possibility of providing justifications for our predictions, and we plan to integrate our method into a real-world application.
\section*{Acknowledgments}
This research was performed by the Arabic Language Technologies group at Qatar Computing Research Institute, HBKU\@, within the Interactive sYstems for Answer Search project ({\sc Iyas}).
\bibliographystyle{acl_natbib}
|
1,116,691,497,719 | arxiv | \chapter{Introduction}
\section{Motivation}
Homomorphic Encryption (HE) is a technology which allows operations to be performed on encrypted data without needing to decrypt it. This concept has long been referred to as the "holy grail" of cryptography \cite{micciancio2010first}, but has historically struggled with speed and accessibility issues. Recent improvements in these areas has led to HE being proposed as a practical solution to real-world problems, such as in the healthcare \cite{prakaashini2021comprehensive} or finance \cite{han2019logistic} domains. I initially read about HE while working at encryption services startup Evervault, and decided to search for some existing problems which it could potentially be a solution for. The intention was to become familiar with the state-of-the-art practices in HE, while solving an important problem. Given the sensitive nature of credit card (CC) data, it stood out as a domain which could benefit from private computation.
CC fraud is the use of someone's payment card fraudulently, typically involving the theft of funds via transfer, or the purchasing of goods. The European Central Bank found that in 2019 alone, the value of fraudulent transactions amounted to over €1.8 billion \cite{ecb2021}. CC fraud is an important issue for banks, as they must make large amounts of payouts to customers each year. It is also a great frustration for customers when one's card has had funds stolen from it, or is frozen due to fraudulent activities. This makes CC fraud detection a vital issue for banks and customers, and a difficult problem to be solved in the financial world. Existing fraud detectors have found success by using rule-based methods or Machine Learning (ML) methods on data surrounding the transactions. This data may include information such as the location at which the transaction was made, or the type of goods or services being purchased.
Although the detectors have resulted in a relative reduction in CC fraud, it has led to the need for users to share additional data, in plaintext form. The upward trend of data breaches motivates a reduction in the amount of sensitive data gathered from users. Additionally, some users with high demands for privacy may not want to share this data with their bank, let alone with potential hackers. This calls for a solution in which user data can be kept secure, while also allowing it to be used in fraud detectors. This paper explores using HE to build a CC fraud detector which meets this need for privacy. As the main focus is on the privacy preservation, the aim is not to make reach state-of-the-art performance at the fraud detection task, but to make a model which performs well as a fraud detector in the HE domain.
ML models in various domains process vast amounts of sensitive user data each day. This provides an additional motivation to produce research which converts standard ML models to models which use HE, as parts of this research can then be used in domains unrelated to CC fraud. Hence, the methods implemented in this paper will be ML based.
An additional motivation is to produce an accessible, open source codebase for this project. It is currently quite difficult to find tutorials for ML inference using HE, or information on how to adapt on previous work. Novel technologies such as HE are accelerated by community contributions. It is hoped that the codebase released with this paper will be valuable for further research.
\section{Approach}
The approach of the paper is to create HE implementations of CC fraud detectors and make latency and storage comparisons with their original plaintext versions. The produced HE models will also be compared, to see which would be most practical to use in the real-world. Comparisons made include transaction encryption latency, model size, transaction size, and inference latency.
The choice of which methods to implement in HE is based on how effective their plaintext versions are at detecting fraud. The number of models chosen for comparison, two in this case, is limited by the amount of time allocated to the project.
\section{Objectives}
\label{sec: objectives}
The objective of this research is to produce a model which, on the selected datasets, achieves the following standards:
\begin{enumerate}
\item Transaction data is encrypted at 128 bits of security, and operated on using Homomorphic Encryption.
\item The model achieves high performance detecting the fraudulant transactions in the dataset.
\item The HE implementation has the same detection performance as the plaintext model.
\item The inference time of the model is reasonable for use in a service requiring real-time performance.
\item The time taken to encrypt the model is reasonable for use in a service which may have many users.
\item The size of the encrypted model is reasonable for use in a service which may have many users.
\end{enumerate}
If possible, multiple models which achieve this standard will be produced. Having models with different characteristics, pros, and cons can open up opportunities for different use cases.
\section{Report Structure}
Section \ref{sec: background} outlines the necessary background information on the topics used, and previous relevant work. Section \ref{sec: pp-cc-fd} details the design of the proposed privacy-preserving CC fraud detection system, with section \ref{sec: implementation} describing the technical details of the implementation. In section \ref{sec: evaluation} the results of CC fraud detection performance are shown, along with latency and storage size benchmarks. Section \ref{sec: further work} discusses the performance of the models, limitations of the system, and proposals of further work on the system. The conclusion of the paper is stated in section \ref{sec: conclusion}.
\chapter{Background}
\label{sec: background}
\section{Credit Card Fraud Detection}
Credit card (CC) fraud is an umbrella term used for any time a payment card (debit or credit) is used against the will of the owner. CC fraud consists of card-present fraud, which is the use of a physical card, and card-not-present fraud, which involves the use of stolen card details for online transactions \cite{ecb2021}. CC fraud detection is the problem of predicting whether a given transaction is fraudulent or legitimate. Designing an effective detection system has been a consistent subject of research for the past three decades \cite{leonard1993detecting,maes2002credit,patidar2011credit,yang2019ffd}.
\section{Transaction Data}
Access to previous transaction data allows for detection models to be based on patterns found in existing data, and then tested on that data. Datasets typically contain a list of transactions in chronological order. Each transaction contains a number of transaction features, such as transaction amount, or product type purchased. They are also labelled as fraudulent or legitimate transactions, represented by 1 or 0 respectively.
Datasets which are made publicly available must undergo a process of anonymization to ensure that information about who made the transactions cannot be inferred. A typical method for anonymization is Principal Component Analysis (PCA) \cite{abdi2010principal}. Some prior works use anonymized, publicly available datasets \cite{ge2020credit}, and others use private commercial datasets \cite{fu2016credit}. A simulator for CC fraud data has also recently been implemented \cite{leborgne2022fraud}, which holds potential to enable further testing of models on generated data.
A small percentage of CC transactions are fraudulent, for instance, only 0.024\% of SEPA transactions \cite{ecb2021}. This skew is also typically present in datasets used for fraud detection, in which often less than 5\% of transactions are fraudulent. Managing this unbalanced data is a challenging problem which must be dealt with by practitioners \cite{dal2014learned}.
\section{Machine Learning Techniques}
In this section, Machine Learning (ML) methods and techniques used in previous works on CC fraud detection are discussed, with two families of ML models being the focus: Decision Trees and Neural Networks. Previous work on CC fraud detetors have also used classical ML algorithms such as Support Vector Machine (SVM) \cite{sahin2010detecting} and logistic regression \cite{sahin2011detecting}. Some ML terminology is required, with definitions provided for Loss Function \ref{def: loss-func}, Overfitting \ref{def: overfitting}, and Feature Engineering \ref{def: feature-engineering}.
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\begin{definition}[Loss Function]
\label{def: loss-func}
When training ML models, the goal of the training process is to decrease the value of a function which usually represents the difference between the current outputs of the model and the intended outputs. This function is known as the loss function.
\end{definition}
\begin{definition}[Overfitting]
\label{def: overfitting}
If an ML model is trained on existing data, this data is known as the \emph{training set}. Overfitting is a term used for when a model learns to perform correctly on the training set, but performs significantly worse when introduced to new data.
\end{definition}
\begin{definition}[Feature Engineering]
\label{def: feature-engineering}
Feature engineering is the process of adding extra features to a dataset with the intention of improving model training.
\end{definition}
\subsection{Decision Trees}
A Classification and Regression Tree \cite{loh2008classification} is a tree structure in which a comparison is made at each \emph{node}, and the \emph{leaf node} reached contains a result. These are often referred to as CARTs, regression trees or decision trees. For simplicity, in this paper are referred to as \emph{trees}. In 2007, Shen et al. attempted to use a single tree to detect CC fraud \cite{shen2007application}, but it was outperformed by other models.
An ensemble of trees is when the result of multiple trees is combined to produce the output. Random Forest, an ensemble method, was proposed as a method for CC fraud detection, but was unable to adequately handle the unbalanced data \cite{xuan2018random}. Gradient boosting \cite{friedman2000additive} is a method which has been used to improve the training of the trees. Models using gradient boosting have found success for CC fraud detection, such as XGBoost \cite{priscilla2020influence} and LightGBM \cite{ge2020credit}. An example of how a tree ensemble model could be used to predict a probability for if a transaction is fraudulent is shown in Figure \ref{fig: tree-ensemble}.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{2-background/tree-ensemble.png}
\caption{Decision tree ensemble for predicting probability of CC fraud}
\label{fig: tree-ensemble}
\end{figure}
\subsection{Neural Networks}
An Artificial Neural Network (NN) is an ML model which processes data using a combination of functions called \emph{neurons}, which calculate a learned linear function on its input. The simplest form of NN is called a feedforward NN. In a feedforward NN, numerical inputs are passed through a series of \emph{layers} to produce an output. The \emph{depth} of a feedforward NN corresponds to the number of layers it has. Each layer contains a set of neurons. The output of each layer passes through an \emph{activation function} before being passed to the next layer. Activation functions are used to introduce non-linearity to the model. The most popular activation function is the Rectified Linear Unit (ReLU) \cite{xu2015empirical}.
A Classifier NN is one which produces a single output which is either a binary value or a continuous probability. This architecture was used in the first implementation of an NN for CC fraud detection \cite{ghosh1994credit}. An example of how a feedforward classifier NN could be used to predict a probability for if a transaction is fraudulent is shown in Figure \ref{fig: feedforward}.
\begin{figure}[H]
\centering
\includegraphics[width=0.6\textwidth]{2-background/feedforward.png}
\caption{Feedforward Classifier NN for predicting probability of CC fraud}
\label{fig: feedforward}
\end{figure}
Other types of NNs previously used for CC fraud detection include Autoencoder \cite{aleskerov1997cardwatch,al2019credit}, Convolutional Neural Network (CNN) \cite{fu2016credit} and Long Short Term Memory \cite{jurgovsky2018sequence}.
\section{Homomorphic Encryption}
Encryption is the process of converting data into another form such that the original value can only be revealed by the holder of a \emph{secret key}. The encrypted form is called a \emph{ciphertext} and the plain form is called a \emph{plaintext}. Homomorphic Encryption (HE) is a way of encrypting data such that some or all standard operations (addition, multiplication, division etc.) can be performed using ciphertexts, with the decrypted result showing the correct value. An example of multiplication using HE is shown in Figure \ref{fig: homomorphic-multiplication}, as a demonstration of this concept.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{2-background/homomorphic-multiplication.png}
\caption{Secure multiplication using Homomorphic Encryption}
\label{fig: homomorphic-multiplication}
\end{figure}
HE was originally proposed by Rivest et. al in 1978 \cite{rivest1978data}, with Fully Homomorphic Encryption (FHE) being the end goal. FHE schemes are those which allow for any operation to be evaluated any number of times. The FHE problem was cracked in 2009 by Gentry \cite{gentry2009fully}, but choosing an FHE scheme is not always optimal for performance. \emph{Partially} or \emph{leveled} HE schemes are often chosen, which have limitations on, respectively, the type of operations allowed or the number of operations allowed, but may provide lower latency. A comprehensive survey by Acar et al. contains information on the history and specifics of various schemes \cite{acar2018survey} and also found that the dominant topics of research for HE are in the health and finance domains.
Definitions for Symmetric Encryption \ref{def: symmetric} and Asymmetric Encryption \ref{def: asymmetric} are given to aid in understanding the uses of the HE schemes discussed.
\begin{definition}[Symmetric Encryption]
\label{def: symmetric}
In a symmetric encryption scheme a single secret key is used to encrypt and decrypt.
\end{definition}
\begin{definition}[Asymmetric Encryption]
\label{def: asymmetric}
In an asymmetric encryption scheme a public key and a secret key are generated. The public key can be safely sent to anyone, and anyone can use it to encrypt values. However, only the holder of the secret key can decrypt.
\end{definition}
\section{Decision Tree Inference using Homomorphic Encryption}
The first implementation of a Decision Tree using HE was by Bost et al. in 2014 \cite{bost2014machine}. They used the Quadratic Residuosity scheme \cite{goldwasser2019probabilistic} and a leveled HE scheme implemented by HElib \cite{halevi2014algorithms}, with optional ensembling using the Paillier scheme \cite{paillier1999public}. Inference time of approximately 1 second, depending on security parameters and tree depth.
In 2015, Khedr et al. demonstrated an encrypted tree ensemble model \cite{khedr2015shield} which used a Ring Learning With Errors variant of the GSW scheme \cite{gentry2013homomorphic}. They achieved a much better inference time of 0.027s.
In 2020, X. Meng \& J. Feigenbaum released Privacy-Preserving XGBoost Inference (PPXGB) \cite{meng2020privacy}, which uses the Order Preserving Encryption (OPE) \cite{boldyreva2009order} scheme and Paillier in an encrypted implementation of XGBoost. They demonstrated an inference time of 0.32s.
\section{Neural Network Inference using Homomorphic Encryption}
Significantly more research has been done on HE implementations of NNs than Decision Trees. This is likely due to the trend of deep learning, and the applicability of NNs to problems in various domains. CryptoNets \cite{gilad2016cryptonets} in 2016 was the first implementation, which used the YASHE scheme \cite{bos2013improved} and a square activation function. CryptoNets performed well on its classification task, but still had a high latency of 250 seconds.
Latency improvements have since been made by using a GPU in HCNN \cite{al2018towards}, and by more efficient encoding in LoLa \cite{brutzkus2019low}, which both used the BFV scheme \cite{fan2012somewhat}. nGraph-HE2 \cite{boemer2019ngraph} then achieved state-of-the-art throughput using the CKKS scheme \cite{cheon2017homomorphic}, which is a fast HE scheme for approximate computation.
Clet et al. made a comparison on the BFV, CKKS and TFHE \cite{chillotti2020tfhe} schemes and found that the CKKS scheme was the fastest for inference using a small architecture \cite{clet2021bfv}. Some works have used polynomial approximations of activation functions such as ReLU \cite{chabanne2017privacy} for improved performance in exchange for added latency.
\section{Alternatives to Homomorphic Encryption}
\subsection{Different methods for private computation}
Alternative methods to HE for private computation have also been proposed. A Trusted Execution Environment (TEE) \cite{sabt2015trusted} is a compute environment in which code can be run in a confidential manner and the code running can be remotely attested. Secure Multi-Party Computation (MPC) \cite{goldreich1998secure} is a method for sharing the compute of a function between parties, without any single party knowing the full information.
\subsection{Anonymization}
If data can be anonymized in such a way that detection models using the anonymized data reach the same level of performance of models which use private data, then simply anonymizing the data would result in privacy-preserving detection.
\subsection{Using Non-Sensitive Data}
Similarly to anonymization, if adequate performance can be reached using data which is deemed non-sensitive, then there is no need for privacy-preserving methods.
\section{Related Work}
HE for CC fraud detection was explored in two previous works. Canillas et al. used the original HE encrypted tree implementation \cite{canillas2018exploratory} to implement an encrypted single-tree CC fraud detector, with a latency of approximately 1 second. Vazquez-Saavedra et al. used the CKKS scheme to implement an encrypted SVM model. This achieved very low latency (<1ms in some cases), but the encrypted model did not have the same detection performance as the plaintext \cite{vazquez202154}.
\chapter{Privacy-Preserving Credit Card Fraud Detection}
\label{sec: pp-cc-fd}
This section proposes a system for private CC fraud detection using HE, including use cases, the methods implemented, and the system architecture of each method.
\section{Use Cases}
\label{sec: introduction-use-cases}
It is important to understand the use cases of the proposed system to ensure that it has practical applications. Each use case has (1) Host, which is the party hosting the HE CC fraud detetor, and (2) Client, which is the party sending an encrypted transaction to Host in order to find out if the transaction(s) is fraudulent or not.
\subsection{Customer \& Bank}
\label{sec: use-cases-customer-bank}
In this use case, a bank is Host. Customers of the bank are Clients. A customer attempts to make a CC transaction. Before the transaction goes through, its features are encrypted and sent to Host for private fraud detection. If the transaction is predicted as legitimate, the transaction goes through. If not, the transaction is blocked.
Every customer of the bank could potentially be a Client, or perhaps only a subset of customers who are willing to accept extra transaction latency or payment costs, in exchange for a higher guarantee of privacy. A limitation on this use case is discussed in Section \ref{sec: limitation-encrypted}.
\subsection{Bank \& Third-Party}
In this use case, a bank is Client. Host is any third-party which has created a (plaintext) CC fraud detection model which has better detection performance than the model currently available to the bank. The bank does not trust the third-party with plaintext customer data, so the third-party hosts a HE implementation of their model. The bank can then send encrypted transaction data to the third-party to make use of the superior fraud detector. The third-party could be another bank, or a company set up to provide this service. Note that in this use case, the Client bank could theoretically store their customers transaction data in plaintext, which would not protect from data breaches.
\section{Methods Implemented}
A Decision Tree model (XGBoost) and a NN model (Feedforward Classifier) were both selected for use in the system. Selecting from two different families of ML algorithms can demonstrate the usability of each and can reveal features of each which may be more suited to certain use cases. If both are performant, it also opens up further opportunities for a combination via ensemble.
\subsection{HE XGBoost}
XGBoost was chosen due to the high performance of its plaintext implementation on CC fraud detection tasks. In addition, the open-source implementation of PPXGB\footnote{https://github.com/awslabs/privacy-preserving-xgboost-inference} meant there would be little experimentation required for successfully converting a model to HE.
An example of what the XGBoost model previously shown in Figure \ref{fig: tree-ensemble} would look like after conversion to HE using PPXGB is shown in Figure \ref{fig: he-xgboost}.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{3-system-design/he-xgboost.png}
\caption{XGBoost model after conversion to HE. OPE represents a value encrypted using an Order Preserving Encryption scheme, P represents a value encrypted using the Paillier encryption scheme.}
\label{fig: he-xgboost}
\end{figure}
\subsection{HE Feedforward Classifer}
A Feedforward Classifier NN model was chosen due to its previous success for CC fraud detection, and its simple architecture. The HE implementation used is the CKKS scheme, as it was found to be have the fastest inference time for a similar architecture \cite{clet2021bfv}, and open-source implementations are supported in various programming languages. The CKKS scheme works well for NNs, as it has fast latency by using approximate arithmetic. As Feedforward Classifer NNs produce a fuzzy output, using approximate arithmetic can result in a slightly different output, but show the same prediction after conversion to binary.
An example of a CKKS implementation of the NN previously shown in Figure \ref{fig: feedforward} is shown in Figure \ref{fig: ckks-feedforward}.
\begin{figure}[H]
\centering
\includegraphics[width=0.6\textwidth]{3-system-design/ckks-feedforward.png}
\caption{CKKS implementation of feedforward NN model. C represents a value encrypted using the CKKS scheme. CX,C+ represent multiplications and additions using the CKKS scheme.}
\label{fig: ckks-feedforward}
\end{figure}
\section{System Architecture}
\label{sec: sys-arch}
The sequence of events for a CC fraud detection system using the HE NN is shown Protocol \ref{alg: he-nn}, with a visualisation shown in Figure \ref{fig: nn-sequence-diagram}. Once step 1 is completed, any Client can take part in steps 2-6 an infinite number of times.
\begin{algorithm}[H]
\caption{HE NN for CC fraud detection}
\label{alg: he-nn}
\begin{algorithmic}
\STATE $H$: Host
\STATE $C$: Client
\end{algorithmic}
\algsetup{linenosize=\small,linenodelimiter=.}
\begin{algorithmic}[1]
\STATE $H$ trains NN model on plaintext transaction data.
\STATE $C$ encrypts a transaction.
\STATE $C$ sends encrypted transaction to $H$.
\STATE $H$ processes encrypted transaction using HE NN, producing encrypted result.
\STATE $H$ sends encrypted result to $C$.
\STATE $C$ decrypts to view the result.
\end{algorithmic}
\end{algorithm}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{3-system-design/nn-sequence-diagram.png}
\caption{Sequence of operations in CC Fraud detection system using CKKS NN}
\label{fig: nn-sequence-diagram}
\end{figure}
The sequence of events for a CC fraud detection system using the HE XGBoost model is shown in Protocol \ref{alg: he-xgboost}, with a visualisation shown in Figure \ref{fig: xgboost-sequence-diagram}. As PPXGB uses symmetric encryption, the model must be encrypted in a \emph{Secure Middle Server}. In a production setting, this middle server would be run in a TEE. Steps 2-5 must be done once \emph{per Client}. Once completed, they can take part in steps 6-10 an infinite number of times.
\begin{algorithm}[H]
\caption{HE XGBoost for CC fraud detection}
\label{alg: he-xgboost}
\begin{algorithmic}
\STATE $H$: Host
\STATE $C$: Client
\STATE $SMS$: Secure Middle Server
\end{algorithmic}
\algsetup{linenosize=\small,linenodelimiter=.}
\begin{algorithmic}[1]
\STATE $H$ trains XGBoost model on plaintext transaction data.
\STATE $H$ sends XGBoost model to $SMS$.
\STATE $SMS$ generates $K$ and encrypts XGBoost model.
\STATE $SMS$ sends HE XGBoost model to $H$.
\STATE $SMS$ sends keys to $C$.
\STATE $C$ encrypts a transaction using keys.
\STATE $C$ sends encrypted transaction to $H$.
\STATE $H$ processes encrypted transaction using HE XGBoost, producing encrypted result.
\STATE $H$ sends encrypted result to $C$.
\STATE $C$ decrypts to view the result.
\end{algorithmic}
\end{algorithm}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{3-system-design/xgboost-sequence-diagram.png}
\caption{Sequence of operations in CC Fraud detection system using encrypted XGBoost}
\label{fig: xgboost-sequence-diagram}
\end{figure}
\section{Datasets Chosen}
The datasets chosen for model training are the Vesta and ULB datasets. The reasons for choosing these is that they are the two largest public datasets available, and have both been widely used in previous work since their respective releases.
No extra feature engineering was done on the datasets, as the focus of this paper is not on analysing the datasets. Using the datasets in their existing form with little pre-processing demonstrates how the models can be implemented directly on existing data.
Both of the datasets used have undergone some level of anonymization. It may seem impractical to use them in a privacy-preserving way, as they don't contain any private data. However, they still work as a proof-of-concept for privacy-preserving inference. The reason for this is that anonymizing the data can only result in a level of entropy which is \emph{less than} or \emph{equal to} the amount of entropy in the original data. Hence, if the level of entropy in the anonymized data can be used to accurately predict if transactions are fraudulent, it follows that there would be enough entropy present in the original data to reach at least the same level of performance.
\subsection{"Credit Card Fraud Detection" (ULB)}
The ULB dataset\footnote{https://www.kaggle.com/datasets/mlg-ulb/creditcardfraud} contains 284,807 transactions, 492 of which are labelled as fraudulent (0.172\%). Each transaction has 28 numerical features V1-V28, along with the relative time since the first transaction and a label of whether it was fraudulent or legitimate. PCA has been applied to the dataset before release for anonymization, with V1-V28 being the first 28 principal components.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{3-system-design/ulb-head.png}
\caption{ULB dataset sample}
\label{fig: ulb-sample}
\end{figure}
\subsection{"IEEE CIS Fraud Detection" (Vesta)}
\label{sec: comp}
The Vesta dataset was released for use in the IEEE CIS Fraud Detection competition\footnote{https://www.kaggle.com/competitions/ieee-fraud-detection/}. It contains 590,540 transactions, 20,663 of which are fraudulent (3.5\%). Each transaction has 431 features (400 numerical, 31 categorical), along with the relative timestamp and a label of whether it was fraudulent or legitimate. For anonymization purposes, the names of the identity features have been masked, along with the names of the extra features engineered by Vesta. Some features are \emph{sparse}, meaning the feature is empty empty or \emph{Not a Number} (NaN) in many transactions. Figure \ref{fig: vesta-sample} shows the first 5 rows in the dataset.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{3-system-design/vesta-head.png}
\caption{Vesta dataset sample}
\label{fig: vesta-sample}
\end{figure}
\chapter{Implementation}
\label{sec: implementation}
This section explains the technical details of the implementation. The code written for this implementation is open-sourced in a GitHub repository\footnote{https://github.com/davidnugent2425/he-cc-fraud-detection.git}. All scripts and notebooks mentioned are located in the \verb|src| directory.
\section{Core Language \& Tools}
\subsection{Python}
Python 3 was chosen as the programming language for the implementation, as this is the language used in PPXGB, and the CKKS scheme also has Python implementations.
\subsection{Weights \& Biases}
Weights \& Biases (W\&B) \cite{wandb} is a tool which aids in training ML models, via effective logging and visualisation of training runs. Including W\&B in the training pipeline allows for logging of metrics to ensure models are training properly and to keep track of the performance of each model.
W\&B also provides a free service for doing \emph{Sweeps} of model \emph{hyperparameters}. These are the parameters associated with the structure of a model such as the size of a model layer, or the number of decision trees in a model. A Sweep is a sequence of training runs, each using different hyperparameters. The results of each training run are logged to W\&B for comparison. The choice of which hyperparameters used in each training run can be either a random choice, or using a bayesian method to try converge toward better performance on a chosen metric.
\subsection{Jupyter Notebook}
Jupyter Notebooks \cite{granger2021jupyter} are interactive Python notebooks which allow researchers to display their results in an interactive manner, by running inline code alongside standard markdown text. They are used in the simulation to aid in understanding which code would be run by each of the parties involved.
\subsection{Model Training Packages}
The XGBoost Python Package \cite{chen2016xgboost} is used for training the XGBoost models. PyTorch \cite{NEURIPS2019_9015} was found to be suitable for converting models to HE, as it uses an object oriented approach.
\subsection{TenSEAL}
TenSEAL \cite{benaissa2021tenseal} is a library which contains Python bindings for the C++ CKKS scheme implementation in Microsoft SEAL \cite{sealcrypto}. TenSEAL have also provided tutorials\footnote{https://github.com/OpenMined/TenSEAL/tree/main/tutorials} on converting plaintext NNs to HE models, which was helpful for writing this implementation.
\section{Model Training}
Each time a dataset is used for training, the first 65\% of transactions are used as the \emph{training set}, the next 15\% are used as the \emph{validation set}, and the final 20\% are used as the test set. The model is first trained on the training set with various hyperparameters. The choice of hyperparameters is then based on performance on the validation set. Final results are then taken on the test set.
The purpose of splitting the dataset in this manner is to ensure the resulting model performs well across different distributions, i.e. to prevent overfitting. If the model performs well on the validation and test sets, we can be confident that it would perform well on further new transactions.
Two scripts \verb|train_xgb.py| and \verb|train_nn.py| are provided to allow for simple training of each model. The \verb|dataset| flag can be used when running the scripts to choose the dataset to train on. The \verb|wandb| flag can be used to log the metrics from the training run to W\&B. Logging the training run is important to ensure that the loss function is converging towards a minimum on the training set. If it is not converging, there is likely a problem with either the loss function, or perhaps the learning rate is too high. The full output of an XGBoost training run on the ULB Dataset is shown in Appendix \ref{fig: sim-ulb-training}.
\subsection{Undersampling}
As the datasets are highly skewed towards legitimate transactions, improvements may be found in a model's performance by using a method called \emph{undersampling} \cite{dal2015calibrating}. In undersampling, a subset of the training set is selected which increases the total percentage of fraudulent transactions. The intention of undersampling is to achieve better \emph{recall} on the fraudulent transactions, as the model is less skewed toward labelling transactions as legitimate. A hyperparameter \verb|undersampling_num_negatives| is implemented in the training pipeline which determines the number of legitimate transactions included in the training set.
\subsection{Use of Categorical Variables}
NNs typically use learned embedding layers to handle categorical features \cite{guo2016entity}. However, little work has been done on HE implementations of embedding layers. XGBoost does not currently support any extra optimizations for the use of categorical variables. For these reasons, the categorical features in the Vesta dataset are converted to numerical data.
For conversion to numerical, the pre-processing steps given in the starter code\footnote{https://www.kaggle.com/code/inversion/ieee-simple-xgboost/notebook} provided by the organisors of the competition mentioned in Section \ref{sec: comp}. A simple label encoding converts the categorical features to a corresponding number and the NaN features are replaced with -999.
\subsection{Choosing Model Hyperparameters}
The \verb|wandb-sweep| flag can be used to perform a hyperparameter Sweep of the model. These Sweeps were performed on each model to determine the hyperparameters which performed best on the validation set.
Table \ref{tab: xgboost_hyperparams} shows the ranges of hyperparameters used in the Sweeps for XGBoost on each dataset. \verb|max_depth| controls the maximum depth of each decision tree, and \verb|num_estimators| controls the number of decision trees in the model. Note that \verb|max_depth| was hardcoded as 7 for the Vesta dataset, as it was found that using trees any deeper resulted in impractical model encryption times.
\begin{table}[H]\centering
\caption{XGBoost Hyperparameter Selection}\label{tab: xgboost_hyperparams}
\scriptsize
\begin{tabular}{lrrrr}\toprule
\textbf{Dataset} &\textbf{max\_depth} &\textbf{num\_estimators} &\textbf{undersampling\_num\_negatives} \\\midrule
ULB &6-12 &50-150 &400-2600 \\
Vesta &7 &50-150 &400-300,000 \\
\bottomrule
\end{tabular}
\end{table}
Table \ref{tab: nn_hyperparams} similarly shows the hyperparameters considered for the NN model. \verb|hidden_layer_size| controls the number of neurons in the hidden layer of the model, while \verb|pos_weight| controls an option in the training loss function (BCEWithLogitsLoss), which gives successfully labelled fraudlent transactions an extra weighting. A square activation function was used, as standard activations such as ReLU do not have direct implementations in CKKS, and it only adds a multiplicative depth of 1 per layer.
\begin{table}[H]\centering
\caption{Neural Network Hyperparameter Selection}\label{tab: nn_hyperparams}
\scriptsize
\begin{tabular}{lrrrr}\toprule
\textbf{Dataset} &\textbf{hidden\_layer\_size} &\textbf{pos\_weight} &\textbf{undersampling\_num\_negatives} \\\midrule
ULB &1-5 &400-2600 \\
Vesta &1-5 &400-300,000 \\
\bottomrule
\end{tabular}
\end{table}
\section{Conversion to Homomorphic Encryption}
\subsection{HE XGBoost}
\label{sec: he-conversion-xgboost}
PPXGB encrypts the model by iterating through each of the trees and encrypting each node in the tree recursively using OPE. Note: An additional requirement when using the PPXGB model is that the max and min values in the dataset must be known in order to use the OPE scheme. Once a leaf node is reached, the encryption scheme switches to Paillier. A HMAC is used as a pseudorandom function to obfuscate the names of the features in a transaction \cite{gavzi2014exact} for additional privacy.
An improvement was made to PPXGB to allow for parallelising the tree encryption. As each tree can be encrypted independently, the encryption can be spread across multpile CPU cores. In this implementation, the number of cores can be specified using the \verb|cores| flag. The list of trees is then evenly split into \verb|cores| lists, and encryption for each list is run on a seperate core. In theory, if the number of trees $T$ is greater than the number of cores chosen $C$, there should be a speedup by a factor of $C$ over the original PPXGB function. In practice, thread management adds a cost which reduces the speedup.
\subsection{HE Neural Network}
A layer in a NN performs a matrix mutiplication of the layer input with the layer \emph{weights}, and then adds the \emph{bias} of the layer to produce the output. This evaluate the output of all the neurons in a layer in one operation. The CKKS scheme supports ciphertext-plaintext multiplication. This means the NN layer weights and biases do not need to be encrypted to be used as HE layer weights. Converting from a plaintext NN to a HE NN involves using a CKKS matrix multiplication on the encrypted input with the plaintext layer weights, followed by adding the plaintext layer bias.
A comparison between the model classes showing the corresponding layer operations is shown in Appendix \ref{fig: he-layers-comp}. A debug option was also implemented to display the output of each layer, as shown in Appendix \ref{fig: nn-he-nn-comp}.
\section{HE Inference}
For the HE XGBoost model, Client must be supplied with the OPE key, the HMAC function, and the Paillier private key. Client hashes the transaction feature names with the HMAC, then encrypts the transaction features using the OPE key. This ciphertext is then sent to Host. When the encrypted result is received from Host, Client uses the Paillier private key to decrypt and view the result.
For the HE NN model, Client generates a CKKS public/private key pair. The transaction features are encoded as a CKKS vector and encrypted using the public key. This ciphertext is then sent to Host, along with the encryption context used, so Host can use the same context for the CKKS layer operations. The context could either be the same for every Client, or perhaps some Clients could choose higher levels of security. When the encrypted result is received from Host, Client uses the private key to decrypt and view the result.
\section{Security}
PPXGB uses the python-paillier library \cite{PythonPaillier} for its implementation of the Paillier scheme, which chose a default key length of 3078 to provide 128 bits of security. The symmetric OPE key and the seed used for the HMAC are both 128 bits in length, providing 128 bits of security.
Microsoft SEAL requires that the security context is set up with 128 bits of security by default, by ensuring their encryption contexts are in accordance with the Homomorphic Encryption Security Standard \cite{HomomorphicEncryptionSecurityStandard}. This ensures that if the HE NN runs, it will have 128 bits of security.
\section{Simulation}
A simulation is provided to demonstrate the order of events in a Client-Host use case as described in Section \ref{sec: introduction-use-cases} using the models chosen for this implementation. Additionally, a Secure Middle Server is used for encrypting the XGBoost models. The \verb|Client.ipynb|, \verb|Host.ipynb|, and \verb|Secure Middle Server.ipynb| notebooks represent the respective entities.
Host begins by training XGBoost on the UML and Vesta datasets. Then the NN model is trained on the UML dataset. In each case, the hyperparameters used are those chosen in Tables \ref{tab: xgboost_hyperparams} \& \ref{tab: nn_hyperparams}. The plaintext models are saved to the Host Files.
Next, the Secure Middle Server loads the XGBoost models from Host Files. Encryption/decryption keys are generated, and each model is encrypted. The encrypted models are saved to Host Files, and the keys are saved to Client Files. In practice, the Secure Middle Server would not have direct access to these file systems, and the transmission of the models and keys would be done over secure channels. For simplicity of the simulation, access to both file systems is granted.
Host provides Client with the specification for encoding and encrypting their transaction data, so that it is sent to Host in the correct format for input to the HE Model. Now that Host and Client have the necessary models and keys, the standard protocols for transaction inference described in Section \ref{sec: sys-arch} is done over HTTP.
Each part of the inference process is timed for comparison: encryption time, inference time, and the round trip time from the encryption of the input to decryption of output. This round-trip-time is the most accurate representation of how fast the system performs. However, as this simulation is designed to be run on the same machine, the typical latency involved with transmission over the internet is not included. Appendix \ref{fig: sim-client-host} shows an example of the Client and Host interaction in the Simulation.
\chapter{Results}
\label{sec: evaluation}
In this section the results of HE conversion testing, fraud detection, latencies and storage sizes are presented.
\section{Similarity Test}
When the chosen XGBoost and NN models are selected, a test is done to ensure that their HE versions function correctly. A random subset of the test set is selected and inference on is performed using both the plaintext and HE models. The results from the HE model are decrypted, converted to binary, and compared with the results of the plaintext model. If they are not the same, the test is deemed a failure.
A subset is used rather than the full test set, as encrypting the full test set would take a lot of compute and time. An even balance of 75 fraudulent and 75 legitimate transactions are used in the test. This style of testing for HE inference is discussed in Section \ref{sec: further-he-testing}.
The HE XGBoost passed this test with all tested combinations of hyperparameters. The HE NN failed this test whenever more than one hidden layer was used.
\section{Fraud Detection Performance}
\subsection{Metrics Used}
There is no consensus in the fraud detection literature on which metrics are best for evaluation. Using accuracy (percentage of transactions labelled correctly) is not practical due to the unbalanced data. Labelling every transaction as legitimate would result in very high accuracy.
In these results, the threshold used for converting a probability prediction into a binary value is 0.5, i.e. any transaction given at least 50\% chance of being fraudulent is labelled as fraudulent. Different Hosts/Clients may choose this threshold to be lower or higher, depending on how sensitive they want their detector to be.
Threshold-free metrics evaluate models on their output before conversion to binary, and threshold-based metrics evaluate the binary decision. There is little consensus in the fraud detection literature on which metrics are best. In Reproducible Machine Learning for Credit Card Fraud Detection - Practical Handbook a combination of both is recommended \cite{leborgne2022fraud} . In these results, the AUC ROC score and the Average Precision score are the threshold-free metrics chosen, and the percentage of each type of transaction labelled correctly are chosen as the threshold-based evaluation.
\subsection{Models Chosen}
The fraud detector to use on the test set is chosen based on how it performs on the validation set. Selecting for different metrics should provide different results on the test set. A comparison is made between the results obtained by the fraud detectors when each of the different metrics are selected for. Figure \ref{fig: ap-selection} shows selecting the NN model which had the best Average Precision score in a Sweep on the ULB dataset.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{5-evaluation/nn-ulb-ap.png}
\caption{Selecting the NN model with the best Average Precision score on the ULB validation set.}
\label{fig: ap-selection}
\end{figure}
An extra model which was selected by inspection is also included in each case, to check if a person's judgement could result in better performance than by using a metric. The model which was deemed to have the best balance of correct labels on the validation set was selected in each case. An example of the XGBoost model selected by inspection on the Vesta dataset is shown in Figure \ref{fig: inspection-selection}.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{5-evaluation/xgboost-vesta-inspection.png}
\caption{XGBoost model chosen by Inspection for Vesta dataset. Chart shows number of false negatives vs true positives detected on validation set.}
\label{fig: inspection-selection}
\end{figure}
The links to the relevant sets of training runs used for model selection, logged in W\&B, are listed in Table \ref{tab: training-run-links}.
\begin{table}[H]\centering
\caption{Links to training runs used for model selection}\label{tab: training-run-links}
\scriptsize
\begin{tabular}{lrrr}\toprule
\textbf{Model} &\textbf{Dataset} &\textbf{URL} \\\midrule
XGBoost &ULB &\url{https://wandb.ai/nuggimane/cc-ulb-xgboost/sweeps/o60jf75p} \\
&Vesta &\url{https://wandb.ai/nuggimane/cc-vesta-xgboost/sweeps/sff350wf} \\
& & \\
NN &ULB &\url{https://wandb.ai/nuggimane/cc-ulb-nn/sweeps/uyqqub3d}{} \\
&Vesta &\url{https://wandb.ai/nuggimane/cc-vesta-nn} \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Detection Results}
The results for each dataset are shown in Tables \ref{tab: ulb-results} and \ref{tab: vesta-results}. Only models which passed the Similarity Test are used, i.e. HE NN models used have one hidden layer. The Fraudulent column shows the fraction of fraudulent transactions correctly detected as fraudulent, i.e. the recall. The Legitimate shows a similar fraction for the legitimate transactions. No NN models performed well on the Vesta dataset, so only one is shown.
\begin{table}[H]\centering
\caption{Detection results on ULB Dataset. Models with overall best performance are shown in bold.}\label{tab: ulb-results}
\scriptsize
\begin{tabular}{lrrrrrr}\toprule
\textbf{Model} &\textbf{Metric Selected For} &\textbf{AUC ROC} &\textbf{Average Precision} &\textbf{Fraudulent} &\textbf{Legitimate} \\\midrule
XGBoost &AUC ROC &0.989 &0.703 &0.88 &0.9689 \\
&\textbf{Average Precision} &\textbf{0.986} &\textbf{0.774} &\textbf{0.89} &\textbf{0.981} \\
&Inspection &0.982 &0.784 &0.81 &0.9977 \\
& & & & & \\
NN &AUC ROC &0.981 &0.744 &0.68 &0.9997 \\
&\textbf{Average Precision} &\textbf{0.981} &\textbf{0.739} &\textbf{0.77} &\textbf{0.9992} \\
&Inspection &0.982 &0.724 &0.77 &0.9993 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[H]\centering
\caption{Detection results on Vesta Dataset. Models with overall best performance are shown in bold.}\label{tab: vesta-results}
\scriptsize
\begin{tabular}{lrrrrrr}\toprule
\textbf{Model} &\textbf{Metric Selected For} &\textbf{AUC ROC} &\textbf{Average Precision} &\textbf{Fraudulent} &\textbf{Legitimate} \\\midrule
XGBoost &AUC ROC &0.892 &0.498 &0.38 &0.9929 \\
&Average Precision &0.895 &0.508 &0.37 &0.9944 \\
&\textbf{Inspection} &\textbf{0.892} &\textbf{0.483} &\textbf{0.47} &\textbf{0.9789} \\
& & & & & \\
NN &\textbf{AUC ROC} &\textbf{0.725} &\textbf{0.142} &\textbf{0.72} &\textbf{0.5789} \\
\bottomrule
\end{tabular}
\end{table}
\section{Benchmarking}
Next, the results from the benchmarking of various latencies and storage sizes in the system are shown. As the NN model for the Vesta dataset failed to work as a fraud detector, it was excluded from further HE implementation benchmarking.
\subsection{Conditions for Benchmarking}
Latency benchmarking was run on an Intel Core i7-6700K CPU @ 4.00GHz with an NVIDIA GeForce GTX1080 GPU. The models used for benchmarking were those with the overall best performance, as shown in Figures \ref{tab: ulb-results} and \ref{tab: vesta-results}.
The CKKS scheme's encryption context can be tuned to achieve better performance. Using higher poly modulus degree and coefficient modulus results in better precision, but also higher latency. The HE NN for the ULB dataset had 47 neurons in its layer, which allowed for a bit precision of 30 to pass the Similarity Test. 16384 was chosen as the poly modulus degree, with a coefficient modulus of [40, 30, 30, 30, 40].
The HE XGBoost used for benchmarking on ULB had a max depth of 9 and 51 trees, while the model used for on Vesta had a max depth of 7 and 92 trees. A maximum depth of 7 used for all XGBoost models considered for Vesta, as testing showed that using a higher depth resulted in model encryption times that were impractical (over 12 minutes). Using a fixed depth of 7 kept the model encryption time around 3 minutes.
\subsection{Latency Results}
The training time for the NN depends on the number of epochs chosen, while XGBoost automatically chooses when to stop training. The number of epochs trained on the ULB dataset was 100. Although it still showed poor detection performance, more epochs (300) were used to train on the Vesta dataset. Early stopping could potentially be added to the NN when a targeted validation loss is reached, however, training time is less critical than the other latencies.
Table \ref{tab: latency-results} displays the results for the various latencies involved in training and inference. For the sub-second metrics which require extra precision, 100 runs were done each time, selecting the fastest time found.
\begin{adjustwidth}{-2.5 cm}{-2.5cm}\centering\begin{threeparttable}[H]
\caption{Latency Results}\label{tab: latency-results}
\scriptsize
\begin{tabular}{lrrrrrrr}\toprule
\textbf{Model} &\textbf{Dataset} &\textbf{Training} &\textbf{Model Encryption} &\textbf{Transaction Encryption} &\textbf{Plaintext Inference} &\textbf{Encrypted Inference} \\\midrule
XGBoost &ULB &0.5s &13s &174ms &0.7ms &6ms \\
&Vesta &6s &2mins &2s &5.5ms &22ms \\
& & & & & & \\
NN &ULB &22s &N/A &10ms &0.08ms &296ms \\
\bottomrule
\end{tabular}
\end{threeparttable}\end{adjustwidth}
\subsection{Storage}
Table \ref{tab: storage-results} compares the sizes of transactions and models in their plaintext and ciphertext forms. These are important due to storage usage and loading times. The difference in the size of plaintext transactions between the XGBoost and NN models is because they are stored as a Pandas DataFrame \cite{mckinney2011pandas} for XGBoost, whereas the transactions for the NN are stored as NumPy arrays \cite{harris2020array}.
\begin{table}[H]\centering
\caption{Storage Size Results}\label{tab: storage-results}
\scriptsize
\begin{tabular}{lrrrrrr}\toprule
\textbf{Model} &\textbf{Dataset} &\textbf{Plaintext Transaction} &\textbf{Encrypted Transaction} &\textbf{Plaintext Model} &\textbf{Encrypted Model} \\\midrule
XGBoost &ULB &1.2 kB &2.4 kB &92 kB &386 kB \\
&Vesta &10.7 kB &28.1 kB &896 kB &4512 kB \\
& & & & & \\
NN &ULB &0.387 kB &641 kB &8 kB &N/A \\
\bottomrule
\end{tabular}
\end{table}
\chapter{Discussion}
In this section the performance of the models are discussed, limitations of the system are explained, and potential further work is proposed.
\section{HE Model Comparison}
\subsection{Performance on ULB Dataset}
On the ULB dataset, the XGBoost model was successfully implemented as a CC fraud detector, with the best model correctly classifying 89\% of the fraudulent transactions and 98.1\% of legitimate transactions. The NN model also found success, with the best model achieving results of 77\% and 99.93\%. XGBoost was better at detecting fraudulent transactions, but the NN model produced less false positives.
In the latency benchmarking, HE XGBoost had an encrypted inference time of 6ms, while HE NN had a time of 296ms. When the encryption time of the transaction is added, the total latencies become 180ms and 306ms respectively. As the storage size of an encrypted HE XGBoost transaction is only 2.4kB, this would also be transmitted faster over the internet than the 641 kB transaction encrypted for HE NN.
Both models were successful on this dataset. HE XGBoost performs much faster, but the simpler deployment of HE NN may still make it the preferred choice.
\subsection{Performance on Vesta Dataset}
On the Vesta dataset, HE XGBoost found worse performance than on ULB, but the best model still correctly classified 47\% of fraudulent transactions and 97.89\% of legitimate transactions. The NN model failed to work as a fraud detector, likely due to the limited depth. Although it detected 72\% of the fraudulent transactions, it also labelled 42\% of legitimate transactions as fraudulent, which is unacceptable for a fraud detection system.
HE XGBoost had a fast inference time of 23ms, but saw a transaction encryption time of 2 seconds. Better encryption times could make HE XGBoost a decent model for this dataset.
\subsection{Metric Selected For}
No single basis for model choice between AUC ROC, Average Precision and Inspection consistently provided the best detection performance on the test set. For XGBoost on ULB, selecting for Average Precision produced the best results, while on the Vesta dataset, selecting by Inspection was the better option. The best metric for NN selection on ULB was probably Average Precision again, but very similar performance was found by Inspection. Overall, using Average Precision and the Inspection method proved to be the best option for these datasets.
\section{Limitations}
\subsection{Encrypted Outputs}
\label{sec: limitation-encrypted}
The models proposed above produce an encrypted output, which can only be decrypted using the secret key, which is held by the Client. This creates a limitation on the \hyperref[sec: use-cases-customer-bank]{Client \& Bank} use case, as the bank Host will be unable to determine if a given transaction is fraudulent, as the result is encrypted. Two possible solutions for this problem are as follows:
\begin{enumerate}
\item The Client's secret key is used in a TEE to decrypt the output. This would involve Client encrypting their secret key using a key shared with Host and transmitting it to a TEE. Inside the TEE, the secret key is decrypted, and used to decrypt the output of the HE model. The secret key is discarded, and the Host receives whether the transaction was deemed fraudulent.
\item The output of the model is not encrypted. This is possible using the XGBoost model, and other decision tree models. The inputs to the model are still encrypted, and still traverse the tree using OPE, but the leaf nodes are in plaintext, rather than using Paillier. This results in a model which takes encrypted inputs, and produces plaintext outputs.
This does however introduce further security concerns. Host could theoretically derive information about an encrypted transaction by producing the same output of the model via brute-force, using transaction data generated by Host.
\end{enumerate}
One might ask why should HE be used at all if TEEs are going to be used in the pipeline anyways. The reason is that deploying complex attested code in TEEs is difficult to implement, especially at scale. Deploying a whole model inside a TEE could be very difficult, but using them for simple encryption/decryption steps to aid in the deployment of a HE model seems feasible.
\subsection{Lack of Real Data}
Using anonymized datasets means that our evaluation is undertaken on data that is not a direct representation of what would be used for training and inference in a production environment. Solutions to this would involve applying to gain access to private datasets, or using a simulator which accurately represents the distributions found in real data.
It is possible that some of the unknown features in these datasets could contain features engineered via aggregation, i.e. a value in a given transaction may contain information about previous transactions. This would undermine the \hyperref[sec: use-cases-customer-bank]{Client \& Bank} use case, which would need transaction features to be independent from other transactions.
\subsection{NN Model Depth}
The model depth of the NN model was limited to one hidden layer by the Similarity Test. Even when using the highest CKKS precision of 60 bits, deeper NNs still failed. As each hidden layer adds another square activation function, it is likely this is the issue.
This limitation means that if potential detection performance increases are available with increasing model size, that performance is unavailable to this implementation.
\subsection{Lack of Failure Assessment}
In a plaintext fraud detection system, production datapoints which result in false-positives or true-negatives can be reviewed, and changes to the model can be made accordingly. As the Host is unable to view the transaction data, they are unable to review why failures in the system occur.
One potential solution to this is that users could be asked to share information about a specific transaction if it was a transaction that caused a failure. If the transaction was not particularly private to the Client, they may be willing to give the plaintext version to improve the system. This is similar to how software applications may ask for the local system logs after a crash.
\subsection{HE XGBoost Information Leakage}
As the tree nodes of the model use OPE to make comparisons on the input data, if the plaintext model is known to the Host, they could watch the path taken by the input data down each of the trees. Knowing the result of each of the comparisons made allows Host to determine a range of possible values for each feature in the input data. This attack could be done by Host, or by a third-party attacker if they have access to the plaintext model, the encrypted model, and encrypted transaction data. A prevention for this attack could be to also run the training in the Secure Middle Server, as it was in the original algorithm proposed \cite{meng2020privacy}. But if Host has the data used for training and the information on how the model was trained, they could potentially make a replica model.
\section{Further Work}
\label{sec: further work}
\subsection{Deeper NN}
\label{sec: further-work-deeper-ckks}
To increase the depth of the CKKS NN, an interactive bootstrapping protocol could be introduced. Sending data back for decryption after every N layers in the model, but this would likely be too slow. It is also possible that the practical multiplicative depth could be increased by using other activation functions such as a polynomial approximation of ReLU.
\subsection{HE Inference speedups}
Parallelization across cores was implemented for the encryption of the PPXGB model, but it still struggled with long transaction encryption times. Further improvements could be reached by using GPU, FPGA, or ASIC.
\subsection{Implementation on Live Data}
As the datasets used have been anonymized, a true proof that this system is viable would be to train it on a private commercial dataset, and then test it on production transactions.
\subsection{Ensemble}
An ensemble of the two models could possibly be implemented, where the output of the HE XGBoost model and the HE NN model are combined to produce the final result. The leaf nodes of the XGBoost model could be encrypted using CKKS using the same key used for input to the NN model. This would allow the result of the two models to be added together. Alternatively, the two results could be returned to Client, and they would be combined after decryption.
\subsection{Feature Engineering using HE}
Feature engineering is widely used in CC fraud detection systems. This could potentially be implemented for encrypted transactions. An implementation would probably involve aggregating features across consecutive transactions via homomorphic addition, and producing mean values using homomorphic division.
\subsection{Testing of HE Models}
\label{sec: further-he-testing}
As HE models are slower than plaintext models, testing their performance on the full test set used for testing the plaintext model may be infeasible, as that also involves encrypting the full test set. Further work to build a confidence score of how well a HE model performs, or how similarly it performs to its plaintext equivalent could be valuable across domains.
\subsection{Blockchain Use Case}
Another proposed use case of the system relates to blockchain transaction fraud detection \cite{bhowmik2021comparative}. A transaction is made and added to a blockchain, along with extra transaction features. A smart contract \cite{zheng2020overview} then runs a HE fraud detector on these encrypted features. If the transaction is predicted as fraudulent, the funds are not transferred.
\chapter{Conclusion}
In this paper, a system for private credit card fraud detection using homomorphic encryption was proposed. Two models, XGBoost and a feedforward classifier neural network, were trained as fraud detectors on plaintext data. Models were chosen based on their performance on a validation set. Choosing models based on their average precision score and by inspection produced the best performance on the test set. They were then converted to models using homomorphic encryption, for inference on encrypted transactions.
A comparison of the models found that the XGBoost model had better detection performance and faster latency on the chosen datasets than the neural network model. However, the complexity of deploying the encrypted XGBoost model may result in the neural network being preferred in some cases.
\label{sec: conclusion}
\chapter{Appendix}
\section{Plaintext-CKKS Neural Network Comparisons}
\begin{figure}[H]
\centering
\includegraphics[width=0.7\textwidth]{8-appendix/he-layers-comparison.png}
\caption{Comparison of code in plaintext and CKKS NNs. Corresponding layer operations are highlighted.}
\label{fig: he-layers-comp}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{8-appendix/nn-he-nn-comp.png}
\caption{Comparison of data flow through plaintext and CKKS NNs.}
\label{fig: nn-he-nn-comp}
\end{figure}
\section{Simulation}
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{8-appendix/sim-ulb-training.png}
\caption{Training run of XGBoost on ULB dataset.}
\label{fig: sim-ulb-training}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{8-appendix/sim-client-host.png}
\caption{Interaction in Simulation between Client and Host}
\label{fig: sim-client-host}
\end{figure}
\chapter{Figures, Tables, Referencing}
It is very important to properly refer in the text to any figures, tables or previously published work that you are discussing. Adequate and consistent referencing is one of the criteria which will be used to assess your project report.
\section{Figures}
Graphs, pictures and other images should be included in your report as a numbered, captioned figure. An example is given in Figure \ref{veldis}.
\begin{figure}[h]
\centering
\includegraphics{background/5e1-1.pdf}
\caption{Velocity distribution on the mid-plane for an inlet velocity for case 1.}
\label{veldis}
\end{figure}
The figure and caption should be centred. The figure numbering starts at 1 at the beginning of each chapter. The caption should provide a brief description of what is being shown. The figure should appear in the document after it is referred to in the text. No figure should be included which is not referred to in the text. Ensure that the size and resolution of images imported from software are sufficient to read any text.
\section{Tables}
Tables are an important way of displaying your results. Table \ref{tab:treatments} is a sample table, adapted from the Master/Doctoral Thesis template at \url{http://www.latextemplates.com/cat/theses}, which was generated with this code:
{\footnotesize
\begin{verbatim}
\begin{table}[b]
\caption{The effects of treatments X and Y on the four groups studied.}
\label{tab:treatments}
\centering
\begin{tabular}{l l l}
\toprule
\textbf{Groups} & \textbf{Treatment X} & \textbf{Treatment Y} \\\midrule
1 & 0.2 & 0.8\\
2 & 0.17 & 0.7\\
3 & 0.24 & 0.75\\
4 & 0.68 & 0.3\\
\bottomrule\\
\end{tabular}
\end{table}
\end{verbatim}
}
\begin{table}[b]
\caption{The effects of treatments X and Y on the four groups studied.}
\label{tab:treatments}
\centering
\begin{tabular}{l l l}
\toprule
\textbf{Groups} & \textbf{Treatment X} & \textbf{Treatment Y} \\
\midrule
1 & 0.2 & 0.8\\
2 & 0.17 & 0.7\\
3 & 0.24 & 0.75\\
4 & 0.68 & 0.3\\
\bottomrule\\
\end{tabular}
\end{table}
Tables are numbered in the same way as figures. Typically tables also have a short caption, but this is not universally true. The number and caption appear above the table, not below as with figures. Again, no table should appear in the report which has not been referred to in the text. Tables should come after they are discussed in the text. The exact formatting of the table depends somewhat on the content of the table, but in general, the text in the table should be the same font and size as the main text.
\section{Equations}
All equations should be numbered sequentially. Do not restart the numbering at the beginning of each chapter. Unlike figures and tables, you may not need to refer to every equation in the text. You should take care to format equations properly. Do no simply try to use plain text. Use the equation layout facilities. An example of how equations should appear is shown in Equation \ref{sampleequation}. Here is the code for it:
{\footnotesize
\begin{verbatim}
\begin{equation}
\textrm{div}(\underline{u}) = \frac{\delta u}{\delta x} + \frac{\delta v}{\delta y} +
\frac{\delta w}{\delta z} = 0
\label{sampleequation}
\end{equation}
\end{verbatim}
}
\begin{equation}
\textrm{div}(\underline{u}) = \frac{\delta u}{\delta x} + \frac{\delta v}{\delta y} + \frac{\delta w}{\delta z} = 0
\label{sampleequation}
\end{equation}
\section{Referencing published work}
It is important to give appropriate credit to other people for the work that they have shared through publications. In fact, you must sign a declaration in your report stating that you understand the nature of plagiarism. As well as avoiding plagiarism, citing results or data from the literature can strengthen your argument, provide a favourable comparison for your results, or even demonstrate how superior your work is.
There are many styles to reference published work. For example, the parenthetical style (which is also called the \emph{Harvard style}) uses the author and date of publication (e.g. ``Smith and Jones, 2001''). There is also the Vancouver style (or the \emph{citation sequence style}), which is used in this document. In the Vancouver style, the publications are cited using bracketed numbers which refer to the list in the References section at the end of the report. The references are listed in the order that they are cited in the report. A variant is \emph{name sequence style}, in which the publications are referenced by number, but the list is arranged alphabetically. The following paragraph shows the use of the Vancouver style:
\begin{quote}
Several studies have examined the sound field around tandem cylinders generated by flow\cite{fitzpatrick2003flow,finnegan2010experimental}, while other investigations have focused on the effect of an applied sound field on the flow\cite{hall2003vortex}. Papers from conference proceedings\cite{jordan2001array}, books\cite{paidoussis2010fluid} and technical reports\cite{reyes2007power} can be dealt with in the same style.
\end{quote}
The Vancouver style has the advantage that it is a little more compact in the text and does not distract from the flow of the sentence if there are a lot of citations. However, it has the disadvantage that it is not immediately clear to the reader what particular work has been referenced.
It actually does not matter which particular referencing style is used as long as three important considerations are observed:
\begin{itemize}
\item the referencing style used throughout the document is consistent;
\item all material used or discussed in the text is properly cited;
\item nothing is included in the reference list that has not been cited.
\end{itemize}
This template has a suitable referencing style already set up -- you should use it and use the built-in BibTeX system to manage your references. See above for examples of how to cite a reference and look in the \texttt{sample.bib} file to see BibTeX references. Remember \href{http://scholar.google.com}{Google Scholar} and other search engines will give you BibTeX references for lots of academic publications. Otherwise, you can easily make up your own based on the examples in that file.
\section*{\Huge{Declaration}}
\chapter*{Abstract}
Credit card fraud is a problem continuously faced by financial institutions and their customers, which is mitigated by fraud detection systems. However, these systems require the use of sensitive customer transaction data, which introduces both a lack of privacy for the customer and a data breach vulnerability to the card provider. This paper proposes a system for private fraud detection on encrypted transactions using homomorphic encryption. Two models, XGBoost and a feedforward classifier neural network, are trained as fraud detectors on plaintext data. They are then converted to models which use homomorphic encryption for private inference. Latency, storage, and detection results are discussed, along with use cases and feasibility of deployment. The XGBoost model has better performance, with an encrypted inference as low as 6ms, compared to 296ms for the neural network. However, the neural network implementation may still be preferred, as it is simpler to deploy securely. A codebase for the system is also provided, for simulation and further development.
\newpage
\onehalfspacing\raggedright
\section*{\Huge{Acknowledgements}}
Firstly, I would like to thank my supervisor Dr. Ciarán Mc Goldrick for his guidance and valuable feedback during the course of my research. Also, thank you to the Evervault team for introducing me to the concept of homomorphic encryption. I would also like to express my gratitude to my parents, and to my lab partner Katherine, for their continuous support and encouragement.
\tableofcontents
\listoffigures
\listoftables
\newpage
\mainmatter
\input{1-introduction/introduction.tex}
\input{2-background/background.tex}
\input{3-system-design/system-design.tex}
\input{4-implementation/technical-implementation}
\input{5-evaluation/evaluation}
\input{6-discussion/discussion}
\input{7-conclusion/conclusion.tex}
\bibliographystyle{unsrtnat}
|
1,116,691,497,720 | arxiv | \section{Numerical estimates}\label{sec:cestimates}
Summarizing the previous section the control~$u$ and solution~$\theta$ on~$[\tau,\tau+R']$ are given by the infinite series
\begin{IEEEeqnarray}{rCl}
\label{eq:u}u(t) &=& \sum_{i\ge1}\frac{y^{(i)}(t)}{(2i-1)!}\\
\theta(t,x) &=& \sum_{i\ge 0}y^{(i)}(t)\frac{x^{2i} }{(2i)!}\\
y(t) &=& \phi_s\Bigl(\frac{t-\tau}{R'}\Bigr)\sum_{k\ge0}y_k\frac{(t-\tau)^k}{k!}\\
\label{eq:y}y_k &=& \sqrt{2}\Bigl(\sum_{n\ge 0} c_{n} e^{-n^2\pi ^2 \tau } n^{2k} \Bigr) (-\pi^2)^k;
\end{IEEEeqnarray}
moreover $y$ hence $u$ and~$\theta$ are identically zero on~$[\tau+R',T]$.
The aim of this section is to show that the partial sums
\begin{IEEEeqnarray}{rCl}
\label{eq:ubar}\overline{u}(t) &:=& \sum_{1\le i\le \oi}\frac{y^{(i)}(t)}{(2i-1)!}\\
\label{eq:thetabar}\overline{\theta}(t,x) &:=& \sum_{0\le i\le \oi}y^{(i)}(t)\frac{x^{2i} }{(2i)!}\\
\label{eq:ybar}\overline{y}(t) &:=& \phi_s\Bigl(\frac{t-\tau}{R'}\Bigr)\sum_{0\le k\le\ok}y_k\frac{(t-\tau)^k}{k!}\\
\label{eq:ykbar}\overline{y_k} &:=& \sqrt{2}\Bigl(\sum_{0\le n\le \on} c_{n} e^{-n^2\pi ^2 \tau } n^{2k} \Bigr) (-\pi^2)^k.
\end{IEEEeqnarray}
for given $\oi,\ok,\on\in\N$ provide very good approximations of the above series, and to give explicit error estimates.
\begin{thm}
\label{thm4}
There exist positive constants $C,C_1,C_2,C_3$ such that for all
$\theta_0\in L^2$, $\oi,\ok,\on\in\N$, and~$t\in[\tau,T]$
\bes
\norm{\theta(t)-\overline{\theta} (t)}_{L^\infty}
\le C\left( e^{-C_1 \, \oi \ln \oi} + e^{-C_2\, \ok} + e^{-C_3\, \on ^2} \right)
\norm{\theta_0}_{L^2}
\ees
\end{thm}
\begin{pf}
First notice that for $(t,x)\in [\tau,T]\times[0,1]$
\bes
\label{delta}
\abs{\theta (t,x) - \overline{\theta} (t,x)} \le \Delta_1 + \Delta_2 + \Delta_3,
\ees
where
\begin{IEEEeqnarray*}{rCl}
\Delta_1 &:=& \Bigl\vert \sum_{i>\oi}y^{(i)}(t)\frac{x^{2i}}{(2i)!} \Bigr\vert\\
\Delta_2 &:=& \Bigl\vert \sum_{0\le i\le \oi } \partial_t ^i \Big[ \phi (t)
\sum_{k > \ok } y_{k} \frac{(t-\tau )^k}{k!} \Big] \frac{x^{2i}}{(2i)!} \Bigr\vert\\
\Delta_3 &:=& \Bigl\vert \sum_{0\le i\le \oi } \partial _t ^i \Big[ \phi (t)
\sum_{0\le k \le \ok }
\\&&\quad
\sqrt{2} \big(\sum_{n> \on} c_n e^{-n^2\pi ^2 \tau } n^{2k}\big) (-\pi^2)^k \frac{(t-\tau )^k}{k!} \Big] \frac{x^{2i}}{(2i)!} \Bigr\vert .
\end{IEEEeqnarray*}
By lemma~\ref{lem:steering} $y$ is Gevrey of order~$s$ on~$[\tau,T]$ with some~$M_1\norm{\theta_0}_{L^2},R_1>0$ hence
\begin{IEEEeqnarray*}{rCl}
\Delta_1 &\leq& \sum_{i>\oi}\frac{\abs{y^{(i)}(t)}}{(2i)!}\\
&\leq& M_1\norm{\theta_0}_{L^2}\sum_{i>\oi}\frac{i!^s}{(2i)!R_1^i}\\
&\leq& M_1'\norm{\theta_0}_{L^2}\sum_{i>\oi}\frac{\sqrt i}{(4R_1)^i\sqrt i^{2-s}} \Bigl(\frac{i}{e}\Bigr)^{(s-2)i}\\
&\leq& M_1'\norm{\theta_0}_{L^2}\sum_{i>\oi}\frac{ i^{\frac{s-1}{2}}e^{(2-s)(1-\ln i)i} }{(4R_1)^i},
\end{IEEEeqnarray*}
where we have used Stirling's formula. Pick $C_1<2-s$ and $\sigma\in (C_1, 2-s)$. Then for $i > \oi$
\bes
(4R_1)^{-i} i^{\frac{s-1}{2}} e^{(2-s) (1-\ln i )i } \le K_1e^{-\sigma i (\ln i -1 )},
\ees
where $K_1=K_1(s, \sigma , R_1)$. But
\begin{IEEEeqnarray*}{rCl}
\sum_{i > \oi} e^{-\sigma i(\ln i -1 ) } &\le& \int_{\oi} ^\infty e ^{-\sigma x (\ln x -1 ) } dx\\
&\leq& K_1' \int _{\oi (\ln \oi -1 )} ^\infty e ^{-\sigma x} dx\\
&\leq& K_1' e^{-C_1 \oi \ln \oi},
\end{IEEEeqnarray*}
so that eventually
\bes
\Delta_1 \le K_1''\norm{\theta_0}_{L^2} e^{-C_1\oi\ln \oi}.
\ees
For $\Delta_2$ we first notice that for $t\in\{ z\in \C; |z-\tau |\le \rho'\tau \}$, where~$\rho'$ satisfies
$R'/\tau<\rho'<1$,
\begin{IEEEeqnarray*}{rCl}
\big\vert \sum_{k>\ok} y_k \frac{( z -\tau ) ^k}{k!} \big\vert
&\le& \sum_{k>\ok} |y_k| \frac{{\rho '} ^k \tau ^k }{k!}\\
&\leq& M_1\norm{\theta_0}_{L^2}\sum_{k>\ok}{\rho'}^k\\
&\leq& M_1''\norm{\theta_0}_{L^2}{\rho'}^{\ok+1}.
\end{IEEEeqnarray*}
By the Cauchy estimates we thus have for $\tau\le t\le\tau+R'$
\bes
\Big\vert \partial_t^i\big[\sum_{k> \ok } y_{j,k} \frac{(t-\tau ) ^k }{k!}\big] \Big\vert
\le M_1''\norm{\theta_0}_{L^2}{\rho'}^{\ok + 1} \frac{i!}{R_1^i},
\ees
hence
\bes
\Big\vert\partial _t ^i \big[\phi (t) \sum_{k>\ok} y_k\frac{(t-\tau )^k}{k!}\big] \Big\vert
\le M_2\norm{\theta_0}_{L^2}{\rho'}^{\ok + 1} \frac{i! ^s}{R_2^i}.
\ees
It follows that
\bes
\Delta_2
\le \sum_{0\le i\le \oi} M_2\norm{\theta_0}_{L^2}{\rho '}^{\ok + 1}
\frac{i!^s}{(2i)! R_3 ^i}
\le K_2\norm{\theta_0}_{L^2}e^{-C_2\ok}
\ees
where $0<C_2 <\ln\frac{\tau}{R'}$.
To estimate $\Delta_3$ first notice that for~$\alpha>0$
\bes
\sum_{n>\on}e^{-\alpha n^2} \leq \int_{\on}^\infty e^{-\alpha x^2}dx
= \int_{\on^2}^\infty\frac{e^{-\alpha y}}{2\sqrt y}dy
\leq
\frac{e^{-\alpha n^2}}{2\alpha\on}.
\ees
Pick $\rho '$ and $\rho ''$ with $R\/\tau < \rho ' <\rho '' <1$. Then
\begin{IEEEeqnarray*}{rCl}
\Big\vert\sum_{n>\on}c_n e^{-n^2\pi ^2 \tau } n^{2k}\Big\vert
&\leq& \norm{\theta_0}_{L^2}\frac{e^{-k}k^k}{(\pi^2\rho''\tau)^k}
\sum_{n>\on}e^{-n^2\pi^2(1-\rho'')\tau}\\
&\leq& K_3\norm{\theta_0}_{L^2}\frac{k!}{(\pi^2\rho''\tau)^k}
\frac{e^{-\pi^2(1-\rho'')\tau\on^2}}{\on},
\end{IEEEeqnarray*}
where we have used first that $x\to e^{-x}x^k$ is maximum for $x=k$, and then that $e^{-k}k^k\le Ck!$ by Stirling's formula.
Hence for $z\in\{ z\in\C;\ \ |z- \tau | \le \rho'\tau \}$
\begin{IEEEeqnarray*}{rCl}
\IEEEeqnarraymulticol{3}{l}{
\Big\vert\sum_{0\le k\le \ok}\sqrt{2}
\Big( \sum_{n>\on} c_{j,n} e^{-n^2\pi ^2 \tau} n^{2k}\Big) (-\pi^2)^k\frac{( z -\tau )^k}{k!} \Big\vert}\\
\qquad
&\le& K_3'\norm{\theta_0}_{L^2}\sum_{0\le k\le\ok}
\frac{e^{-\pi^2(1-\rho'')\tau\on^2}}{\on} \Big(\frac{\rho'}{\rho''}\Big)^k\\
&\le& K_3''\norm{\theta_0}_{L^2}\frac{e^{-\pi^2(1-\rho'')\tau\on^2}}{\on}.
\end{IEEEeqnarray*}
Setting $C_3:=\pi^2(1-\rho'')\tau<\pi^2(\tau-R')$, this yields for $\tau\le t\le \tau+R'$
\begin{IEEEeqnarray*}{rCl}
\IEEEeqnarraymulticol{3}{l}{
\Big\vert
\partial _t^i \big[ \sum_{0\le k\le \ok} \sqrt{2} \big( \sum_{n> \on} c_n e^{-n^2\pi ^2 \tau} n^{2k} \big) (-\pi^2)^k\frac{(t-\tau )^k}{k!}\big]
\Big\vert}\\
\qquad
&\le& M_3\norm{\theta_0}_{L^2} e^{-C_3\on ^2 } \frac{i!}{R_1^i}
\end{IEEEeqnarray*}
and
\begin{IEEEeqnarray*}{rCl}
\IEEEeqnarraymulticol{3}{l}{
\Big\vert
\partial _t^i \big[\phi (t) \sum_{0\le k\le \ok} \sqrt{2} \big( \sum_{n> \on} c_{j,n} e^{-n^2\pi ^2 \tau} n^{2k} \big) (-\pi^2)^k\frac{(t-\tau )^k}{k!} \big]
\Big\vert}\\
\qquad
&\le& M_3'\norm{\theta_0}_{L^2} e^{-C_3\on ^2 } \frac{i! ^s}{R_2^i}.
\end{IEEEeqnarray*}
We then conclude
\bes
\Delta_3 \le \sum_{0\le i\le \oi} M_3'\norm{\theta_0}_{L^2} e^{-C_3\on^2} \frac{i!^s}{(2i)! R_2^i}.
\le M_3''\norm{\theta_0}_{L^2}e^{-C_3\on ^2}
\label{delta4}
\ees
Collecting the inequalities for $\Delta_1,\Delta_2,\Delta_3$ eventually gives the statement of the theorem.\qed
\end{pf}
Let $\hat\theta$ denote the solution of \eqref{B1}--\eqref{B3} where the ``exact'' control~$u$~\eqref{eq:u} is replaced by the truncated control~$\overline{u}$~\eqref{eq:ubar}, still starting from the initial condition~$\theta_0$. Notice $\hat\theta$ is the ``exact'' solution obtained when applying the truncated control, while $\overline{\theta}$ is the truncated solution obtained when applying the ``exact'' control. Since in ``real life'' only the truncated control can be actually applied it is important to know how well $\hat\theta$ approximates~$\theta$, the ``exact'' solution when applying the ``exact'' control. It turns that $\hat\theta$ satisfies the same relation as~$\overline\theta$ in Theorem~\ref{thm4}; the proof is omitted for lack of space but follows from the proof of Theorem~\ref{thm4}.
\begin{cor}\label{cor3}
With the same notations as in Theorem \ref{thm4},
\bes
\norm{\theta(t)-\hat{\theta} (t)}_{L^\infty}
\le C\left( e^{-C_1 \, \oi \ln \oi} + e^{-C_2\, \ok} + e^{-C_3\, \on ^2} \right)
\norm{\theta_0}_{L^2}
\ees
\end{cor}
\section{The heat equation is flat}\label{sec:flatness}
We claim the system~\eqref{B1}--\eqref{B3} is ``flat'' with $y(t):=\theta(0,t)$ as a flat output, which means there is (in appropriate spaces of smooth functions) a $1-1$ correspondence between arbitrary functions $t\mapsto y(t)$ and solutions of~\eqref{B1}--\eqref{B3}.
We first seek a formal solution in the form
\bes
\theta(t,x):=\sum_{i\ge0}\frac{x^i}{i!}a_i(t)
\ees
where the $a_i$'s are functions yet to define. Plugging this expression into~\eqref{B1} yields
\bes
\sum_{i\ge 0}\frac{x^i}{i!}[a_{i+2}-a_i']=0,
\ees
hence $a_{i+2}=a_i'$ for all~$i\ge0$. On the other hand $y(t)=\theta(0,t)=a_0(t)$, and \eqref{B2} implies~$a_1(t)=0$. As a consequence $a_{2i}=y^{(i)}$ and $a_{2i+1}=0$ for all~$i\ge0$. The formal solution thus reads
\be
\label{AA10}
\theta(t,x)=\sum_{i\ge 0}\frac{x^{2i}}{(2i)!}y^{(i)}(t)
\ee
while the formal control is given by
\be
\label{AA10bis}
u(t)=\theta_x(1,t)=\sum_{i\ge 1}\frac{y^{(i)}(t)}{(2i-1)!}.
\ee
We now give a meaning to this formal solution by restricting $t\mapsto~y(t)$
to be Gevrey of order~$s\in [0,2)$.
\begin{prop}
\label{prop1}
Let $s\in [0,2)$, $-\infty <t_1<t_2<\infty$, and $y\in C^\infty ([t_1,t_2] )$ satisfying for some constants $M,R>0$
\be
\label{AA11}
\abs{y^{(i)}(t)} \le M \frac{i!^s}{R^i}, \qquad \forall i\ge 0,\ \forall t\in [t_1,t_2].
\ee
Then the function $\theta$ defined by \eqref{AA10} is Gevrey of order $s$ in $t$ and $s/2$ in $x$ on
$[t_1,t_2]\times [0,1]$; hence the control~$u$ defined by \eqref{AA10bis} is also Gevrey of order $s$ on $[t_1,t_2]$.
\end{prop}
\begin{pf}
We must prove the formal series
\be
\label{AA12}
\partial _t ^m\partial _x ^n \theta (t,x) = \sum_{2i\ge n} \frac{ x^{2i-n } }{ (2i-n)! } y^{(i+n)}(t)
\ee
is uniformly convergent on $[t_1,t_2]\times [0,1]$ with growth estimates of the form
\be
\label{AA12bis}
\abs{\partial _t ^m\partial _x ^n \theta (t,x)} \le C
\frac{m! ^s}{R_1^m}\, \frac{n! ^\frac{s}{2}}{R_2^n} \cdot
\ee
By \eqref{AA11}, we have for all $(t,x)\in [t_1,t_2]\times [0,1]$
\begin{eqnarray*}
\left\vert \frac{x^{2i-n}}{(2i-n)!} y^{(i+m)} (t) \right\vert &\le& \frac{M}{R^{i+m}} \, \frac{(i+m)! ^s}{(2i-n)!} \\
&\le& \frac{M}{R^{i+m} }\, \frac{(2^{i+m} i! \, m! )^s}{(2i-n)!} \\
&\le& \frac{M}{R^{i+m}}\, \frac{2^{si} \bigl(2^{-2i} \sqrt{\pi i} \, (2i)! \bigr)^\frac{s}{2} }{(2i-n)!}\, \frac{m!^s}{2^{-sm}} \\
&\le& M \frac{(\pi i) ^{\frac{s}{4}} }{R_1 ^i (2i-n)! ^{1-\frac{s}{2}} } n! ^{\frac{s}{2}} \frac{m!^s}{R_1 ^m},
\end{eqnarray*}
where we have set $R_1 = 2^{-s}R$; we have used Stirling's formula for $(2i)!$ and twice $(i+j)! \le 2^{i+j} i!j!$. Since $\sum_{2i\ge n} \frac{(\pi i)^\frac{s}{4} }{R_1 ^i (2i-n)! ^{1-\frac{s}{2} } } <\infty$ the series in~\eqref{AA12} are uniformly convergent for all $m,n\ge 0$, hence $\theta \in C^\infty ([t_1,t_2]\times [0,1])$.
Finally, since
\bes
\sum_{2i\ge n} \frac{ M(\pi i)^\frac{s}{4} }{R_1 ^i (2i-n)! ^{1-\frac{s}{2} } }
\le M\Bigl(\frac{\pi}{2}\Bigr)^\frac{s}{4} R_1 ^{-\frac{n}{2}} \sum_{j\ge 0} \frac{ j^\frac{s}{4} + n^\frac{s}{4} }{R_1 ^\frac{j}{2} j! ^{1-\frac{s}{2} } }
\le C R_2^{-n}
\ees
where $R_2\in (0,\sqrt{R_1})$ and $C>0$ is some constant independent of~$n$, we have the desired estimates~\eqref{AA12bis}.\qed
\end{pf}
\section{Introduction}
The controllability of the heat equation was first considered in the 1-D case, \cite{FattoR1971ARMA,Jones1977JMAA,Littm1978ASNSPCS}), and very precise results were obtained by
the classical moment approach. Next using Carleman estimates and duality arguments the null controllability was proved in \cite{FursiI1996book,LebeaR1995CPDE} for any bounded domain in~$\R^N$, any control time $T$, and any control region. This Carleman approach proves very efficient also with semilinear parabolic equations, \cite{FursiI1996book}.
By contrast the numerical control of the heat equation (or of parabolic equations) is in its early stage, see e.g.~\cite{MunchZ2010IP,BoyerHL2011NM,MicuZ2011SCL}. A natural candidate for the control input is the control of minimal $L^2-$norm, which may be obtained as a trace of the solution of the (backward) adjoint problem whose terminal state is the minimizer of a suitable quadratic cost. Unfortunately its computation is a hard task~\cite{MicuZ2011SCL}; indeed the terminal state of the adjoint problem associated with some regular initial state of the control problem may be highly irregular, which leads to severe troubles in the numerical computation of the control function.
All the above results rely on some observability inequalities for the adjoint system. A direct approach which does not involve the adjoint problem was proposed in~\cite{Jones1977JMAA,Littm1978ASNSPCS,LinL1995AMO}. In \cite{Jones1977JMAA} a fundamental solution for the heat equation with compact support in time was introduced and used to prove null controllability. The results in \cite{Jones1977JMAA,Rosie2002CAM} can be used to derive control results on a bounded interval with one boundary control in some Gevrey class. An extension of those results to the semilinear heat equation in 1D was obtained in \cite{LinL1995AMO} in a more explicit way through the resolution of an ill-posed problem with data of Gevrey order 2 in~$t$.
In this paper we derive in a straightforward way the null controllability of the 1-D heat equation
\begin{IEEEeqnarray}{rCl'l}
\theta_t(t,x) -\theta_{xx}(t,x) &=& 0, &(t,x)\in(0,T)\times(0,1) \label{B1}\\
\theta_x(t,0) &=& 0, &t\in(0,T) \label{B2}\\
\theta_x(t,1) &=& u(t), &t\in(0,T) \label{B3}
\end{IEEEeqnarray}
with initial condition
\begin{IEEEeqnarray*}{rCl'l}
\theta(0,x) &=&\theta_0(x), &x\in (0,1).
\label{eq:ic}
\end{IEEEeqnarray*}
This system describes the dynamics of the temperature~$\theta$ in an insulated metal rod where the control~$u$
is the heat flux at one end. More precisely given any final time~$T>0$ and any initial state $\theta_0\in L^2(0,1)$
we provide an explicit control input $u\in L^2(0,T)$ such that the state reached at time~$T$ is zero, i.e.
\begin{IEEEeqnarray*}{rCl'l}
\theta(T,x) &=&0, &x\in(0,1).
\end{IEEEeqnarray*}
We use the so-called {\em flatness approach}, \cite{FliesLMR1995IJoC}, which consists in parameterizing the solution~$\theta$ and the control~$u$ by the derivatives of a ``flat output''~$y$ (section~\ref{sec:flatness}); this notion was initially introduced for finite-dimensional (nonlinear) systems, and later extended to in particular parabolic PDEs, \cite{LarocMR2000IJRNC,LynchR2002IJC,MeureZ2008MCMDS,Meure2011A}. Choosing a suitable trajectory for this flat output~$y$ then yields an explicit series for a control achieving the exact steering to zero (section~\ref{sec:controllability}). This generalizes~\cite{LarocMR2000IJRNC}, where only approximate controllability was achieved through a similar construction. We then give accurate error estimates when the various series involved are replaced by their partial sums, which is paramount for an actual numerical scheme (section~\ref{sec:cestimates}). Numerical experiments demonstrate the relevance of the approach (section~\ref{sec:numerics}).
In the sequel we will consider series with infinitely many derivatives of some functions. The notion of Gevrey order is a way of estimating the growth of these derivatives: we say that a function $y\in C^\infty([0,T])$ is {\em Gevrey of order $s\geq0$ on~$[0,T]$} if there exist positive constants~$M,R$ such that
\bes
\abs{y^{(p)}(t)} \leq M\frac{p!^s}{R^p} \qquad \forall t\in [0,T],\ \forall p\ge 0.
\ees
More generally if $K\subset\R^N$ is a compact set and $y$ is a function of class~$C^\infty$ on~$K$ (i.e. $y$ is the restriction to $K$ of a function of class~$C^\infty$ on some open neighbourhood $\Omega$ of~$K$), we say $y$ is {\em Gevrey of order $s_1$ in $x_1$, $s_2$ in $x_2$,\ldots,$s_N$ in $x_N$ on~$K$} if there exist positive constants $M,R_1,...,R_N$ such that
\bes
\abs{\partial_{x_1}^{p_1}\partial_{x_2}^{p_2}\cdots\partial_{x_N}^{p_N}y(x)} \le
M\frac{\prod_{i=1}^N (p_i !)^{s_i}}{\prod_{i=1}^N R_i^{p_i} }, \quad \forall x\in K,\ \forall p\in \N ^N.
\ees
By definition, a Gevrey function of order $s$ is also of order $r$ for~$r\geq s$. Gevrey functions of order~1 are analytic (entire if $s<1$). Gevrey functions of order~$s>1$ have a divergent Taylor expansion; the larger~$s$, the ``more divergent'' the Taylor expansion. Important properties of analytic functions generalize to Gevrey functions of order $s>1$: the scaling, addition, multiplication and derivation of Gevrey functions of order $s>1$ is of order~$s$, see~\cite{Ramis1978,Rudin1987book}. But contrary to analytic functions, functions of order $s>1$ may be constant on an open set without being constant everywhere. For example the ``step function''
\bes
\phi_s(t):=\begin{cases}
1 & \text{if $t\leq0$}\\
0 & \text{if $t\geq1$}\\
\dfrac{ e^{-(1-t)^{-k}} }{ e^{-(1-t)^{-k}} + e^{-t^{-k}} }
&\text{if $t\in]0,1[$},
\end{cases}
\ees
where $k = (s-1)^{-1}$ is Gevrey of order~$s$ on $[0,1]$ (and in fact on~$\R$); notice $\phi_s(0)=1$, $\phi_s(1)=0$ and $\phi_s^{(i)}(0)=\phi_s^{(i)}(1)=0$ for all~$i\geq1$.
In conjunction with growth estimates we will repeatedly use Stirling's formula $n!\sim(n/e)^n\sqrt{2\pi n}$.
\section{#1}\setcounter{equation}{0}}
\section{Null controllability}\label{sec:controllability}
In this section we derive an explicit control steering the system from any initial state~$\theta_0\in L^2(0,1)$ at time~$0$ to the final state~$0$ at time~$T>0$. Two ideas are involved: on the one hand thanks to the flatness property it is easy to find a control achieving the steering to zero starting from a certain set of initial conditions (lemma~\ref{lem:steering}); on the other hand thanks to the regularizing property of the heat equation this set is reached from any~$\theta_0\in L^2(0,1)$ when applying first a zero control for some time (lemma~\ref{lem:zerocontrol}).
\begin{lem}\label{lem:steering}
Let $(y_i)_{i\ge0}$ be a sequence of real numbers such that for some constants~$M,R>0$
\be
\label{AA20}
\abs{y_i}\le M\frac{i!}{R^i}\qquad \forall i\ge 0.
\ee
Then the function defined on~$[t_1,t_1+R']$, $R'<R$, by
\bes
y(t):=\phi_s\Bigl(\frac{t-t_1}{R'}\Bigr)\sum_{i\ge0}y_i\frac{(t-t_1)^i}{i!},
\ees
is Gevrey of order~$s>1$ on $[t_1,t_1+R']$ and satisfies for all $i\geq0$
\begin{IEEEeqnarray}{rCl}
y^{(i)}(t_1) &=& y_i\label{eq:yt1}\\
y^{(i)}(t_1+R') &=& 0\label{eq:yt2}.
\end{IEEEeqnarray}
Moreover the control defined on~$[t_1,t_1+R']$ by
\be\label{AA31}
u(t):=\sum_{i\ge1}\frac{y^{(i)}(t)}{(2i-1)!}
\ee
is also Gevrey of order $s$ on $[t_1,t_1+R']$ and steers the system from the initial state $\sum_{i\ge0}y_i\frac{x^{2i}}{(2i)!}$ at time~$t_1$ to the final state~$0$ at time~$t_1+R'$.
\end{lem}
\begin{pf}
Let $\overline{y}(z):=\sum_{i\geq0}y_i\frac{z^i}{i!}$. The growth property~\eqref{AA20} implies $\overline{y}$ is analytic on the disc $\{z\in\C;\ \abs z<R\}$, the convergence being moreover uniform for $\abs z\leq R'<R$. Therefore $\overline{y}$ is Gevrey of order~$1$, hence of order~$s>1$, on~$[t_1,t_1+R']$. On the other hand $\phi(t):=\phi_s\Bigl(\frac{t-t_1}{R'}\Bigr)$ is also Gevrey of order~$s$ on $[t_1,t_1+R']$, hence so is the product~$y$ of $\overline y$ and~$\phi$. The boundary values~\eqref{eq:yt1}-\eqref{eq:yt2} follow at once from the definition of~$\phi_s$.
The control~$u$ in~\eqref{AA31} achieves the steering to zero; indeed
\be
\theta(t,x):=\sum_{i\ge 0}\frac{ x^{2i}}{(2i)!}y^{(i)}(t),
\ees
as well as~$u$, is by proposition~\ref{prop1} Gevrey of order~$s$ in~$t$ and $s/2$ in~$x$, and obviously satisfies~$\theta(t_1+R',x)=0$. \qed
\end{pf}
\begin{lem}\label{lem:zerocontrol}
Let $\theta_0\in L^2(0,1)$ and $\tau>0$. Consider the final state $\theta_\tau(x):=\theta(\tau,x)$ reached when applying the control $u(t):=0$, $t\in[0,\tau]$, starting from the initial state~$\theta_0$.
Then $\theta_\tau$ is analytic in $\C$ and can be expanded as
\bes
\theta_\tau(x) = \sum_{i\ge0}y_i\frac{x^{2i}}{(2i)!}, \quad x\in\C,
\ees
with
\be
\abs{y_i} \le C\Bigl(1+\frac{1}{\sqrt\tau}\Bigr)\frac{i!}{\tau^i}
\ees
where $C$ is some positive constant depending only on~$\theta_0$.
\end{lem}
\begin{pf}
Decompose $\theta _0$ as the Fourier series of cosines
\bes
\theta_0(x)=\sum_{n\ge 0}c_n\sqrt{2}\cos(n\pi x)
\ees
where the convergence holds in $L^2(0,1)$ and
\bes
2|c_0|^2+\sum_{n\ge 1} |c_n|^2 =\int_0^1 |\theta _0(x)|^2 dx <\infty.
\ees
The solution starting from~$\theta_0$ then reads
\be
\theta (t,x)=\sum_{n\ge 0}c_ne^{-n^2\pi^2t}\sqrt{2}\cos(n\pi x)
\label{C100}
\ee
and in particular
\bes
\theta_\tau(x)=\sum_{n\ge 0}c_n e^{-n^2\pi^2\tau}\sqrt{2}\cos(n\pi x)
\ees
The series for $\theta_\tau$ is analytic in~$\C$ since for all~$\abs{x}\leq r$
\bes
\abs{c_ne^{-n^2\pi^2\tau}\sqrt{2}\cos(n\pi x)} \le C_1\left(\sup_{k\ge0}\abs{c_k}\right)e^{-n^2\pi^2\tau+n\pi r}
\ees
where $C_1$ is some positive constant; this ensures the uniform convergence of the series in every open disk of radius $r>0$.
Moreover
\begin{IEEEeqnarray*}{rCl}
\theta_\tau(x) &=& \sqrt{2}\sum_{n\ge 0}c_ne^{-n^2\pi^2\tau}\sum_{i\ge0}(-1)^i\frac{(n\pi x)^{2i}}{(2i)!}\\
&=& \sum_{i\ge0} \frac{x^{2i}}{(2i)!} \underbrace{\left(\sqrt{2}(-1)^i\sum_{n\ge0}c_n e^{-n^2\pi^2\tau}(n\pi)^{2i}\right)}_{=:y_i}
\end{IEEEeqnarray*}
The change in the order of summation will be justified once we have proved that $y_i$, $i\geq0$, is absolutely convergent and
\bes
\sum_{i\ge0}\abs{y_i}\frac{x^{2i}}{(2i)!} <\infty, \quad\forall x\geq0.
\ees
For $i\ge 0$ let $h_i(x):=e^{-\tau\pi^2x^2}(\pi x)^{2i}$ and $N_i:=\left[\bigl(\frac{i}{\pi ^2\tau}\bigr)^{\frac{1}{2}}\right]$. The map $h_i$ is increasing on $\bigl[0,\left(\frac{i}{\pi ^2\tau}\right)^{\frac{1}{2}}\bigr]$ and decreasing on $\bigl[\left(\frac{i}{\pi ^2\tau}\right)^{\frac{1}{2}},+\infty\bigr)$ hence
\begin{IEEEeqnarray*}{rCl}
\sum_{n\ge0}h_i(n) &\le& \int_0^{N_i}h_i(x)dx + h_i(N_i)\\
&& +\: h_i (N_i + 1) + \int_{N_{i+1} }^\infty h_i(x)dx\\
&\le& 2h_i\left(\Bigl(\frac{i}{\pi^2\tau}\Bigr)^{\frac{1}{2}}\right) + \int_0^\infty h_i(x)dx\\
&\le& C_2\frac{i!}{\tau^i\sqrt{i}} + \int_0^\infty h_i(x)dx;
\end{IEEEeqnarray*}
$C_2$ is some positive constant and we have used Stirling's formula. On the other hand integrating by parts yields
\begin{IEEEeqnarray*}{rCl}
\int_0^\infty h_i(x)dx &=& \frac{2i-1}{2\tau}\int_0^\infty h_{i-1}(x)dx\\
&=& \frac{(2i-1)\cdots 3\cdot1}{(2\tau)^i}\int_0^\infty e^{-\tau\pi^2x^2}dx\\
&=& \frac{(2i)!}{2^ii!(2\tau)^i}\cdot\frac{1}{\pi\sqrt\tau} \int_0^\infty e^{-x^2}dx\\
&\leq& C_3\frac{i!}{\tau^i\sqrt{i\tau}},
\end{IEEEeqnarray*}
where $C_3$ is some positive constant and we have again used Stirling's formula. As a consequence
\be\label{Z7}
\abs{y_i}\le\sqrt{2}\sup_{n\ge 0}\abs{c_n}\sum_{n\ge0}h_i(n) \le C\Bigl(1+\frac{1}{\sqrt\tau}\Bigr)\frac{i!}{\tau^i}
\ee
where $C$ is some positive constant. Finally
\bes
\sum_{i\ge0}\abs{y_i}\frac{x^{2i}}{(2i)!}
\leq C\Bigl(1+\frac{1}{\sqrt\tau}\Bigr)\sum_{i\ge0}\underbrace{\frac{i!}{(2i)!}\Bigl(\frac{x^2}{\tau}\Bigr)^i}_{=:v_i}<\infty
\ees
since $\frac{v_{i+1}}{v_i}\sim\frac{1}{4i}\frac{x^2}{\tau}$.\qed
\end{pf}
With the two previous lemma at hand we can now state our main controllability result.
\begin{thm}\label{thm1}
Let $\theta_0\in L^2(0,1)$ and~$T>0$. Pick any $\tau\in(0,T)$ and $s\in (1,2)$. Then there exists a function~$y$
Gevrey of order~$s$ on~$[\tau,T]$ such that the control
\bes
u(t):=\begin{cases}
0 & \text{if $0\le t\le\tau$}\\
\sum_{i\ge1}\frac{y^{(i)}(t)}{(2i-1)!} &\text{if $\tau <t\le T$}.
\end{cases}
\ees
steers the system from the initial state $\theta_0$ at time~$0$ to the final state~$0$ at time~$T$.
Moreover $u$ is Gevrey of order~$s$ on~$[0,T]$; $t\mapsto\theta(t,\cdot)$ is in $C\bigl([0,T],L^2(0,1)\bigr)$; $\theta$ is Gevrey of order~$s$ in~$t$ and~$s/2$ in~$x$ on~$[\varepsilon,T]\times[0,1]$ for all $\varepsilon\in(0,T)$.
\end{thm}
\begin{pf}
By lemma~\ref{lem:zerocontrol} the state reached at time~$\tau$ reads $\sum_{i\ge0}y_i\frac{x^{2i}}{(2i)!}$ with the sequence
$(y_i)_{i\ge0}$ satisfying the growth property of lemma~\ref{lem:steering} with $M:=1+\frac{1}{\sqrt\tau}$ and~$R:=\tau$. Hence the desired function is given by
\bes
y(t):=\phi_s\Bigl(\frac{t-\tau}{R'}\Bigr)\sum_{i\ge0}y_i\frac{(t-\tau)^i}{i!},
\ees
on~$[\tau,\tau+R']$, where $R'<\tau$ and $R'\leq T-\tau$; and by $y(t):=0$ on~$[\tau+R',T]$. Moreover $y$ and~$u$ are Gevrey of order~$s$ on~$[\tau,T]$, and by construction $\theta(\tau,x)=\theta(\tau^+,x)=\sum_{i\ge0}y_i\frac{x^{2i}}{(2i)!}$ and $u(\tau)=u(\tau^+)=0$.
By Proposition~\ref{prop1} the solution~$\theta$ on~$[\tau,T]\times[0,1]$ is well-defined and Gevrey of order~$s$ in~$t$ and $s/2$ in~$x$ hence $t\mapsto\theta(t,\cdot)\in C((0,T];C^1([0,1]))$ and $u\in C([0,T])$.
On the other hand it is easily seen that $\theta$ is Gevrey of order~$1$ in~$t$ and $1/2$ in~$x$ on~$[\varepsilon,\tau]\times[0,1]$ for all $\varepsilon\in (0,\tau )$.
Thus the solution $\theta$ is Gevrey of order $1$ in $t$ and $1/2$ in $x$ on $[\varepsilon , \tau ]\times [0,1]$ while it is Gevrey of order $s$ in $t$ and $s/2$ in $x$ on $[\tau ,T]\times [0,1]$.
To prove $\theta$ is Gevrey of order $s$ in $t$ and $s/2$ in $x$ on $[\varepsilon , T]\times [0,1]$ it is then sufficient to check
$\partial _t^k \theta (\tau, x ) = \partial _t ^k \theta (\tau ^+, x)$ for $k\geq0$ and~$x\in [0,1]$. But
\begin{IEEEeqnarray*}{rCl}
\partial _t^k \theta (\tau ^+, x ) &=& \sum_{i\ge 0} \frac{x^{2i}}{(2i)!} y^{(i+k)} (\tau ) \\
&=& \sum_{i\ge 0} \frac{x^{2i}}{(2i)!} y_{i+k} \\
&=& \sqrt{2} \sum_{i\ge 0} \frac{x^{2i}}{(2i)!}
\left( \sum_{n\ge 0} c_n e^{-n^2\pi ^2 \tau } n^{2(i+k)} \right) (-\pi^2)^{i+k} \\
&=& \sum_{n\ge 0} c_n (-n^2\pi ^2) ^k e^{-n^2\pi ^2 \tau } \sqrt{2} \cos (n\pi x) \\
&=& \partial _t ^k \theta (\tau, x).
\end{IEEEeqnarray*}
As a consequence $u$ is also Gevrey of order~$s$ on~$[0,T]$. \qed
\end{pf}
\section{Numerical experiments}\label{sec:numerics}
We hare conducted a number of numerical experiments to demonstrate the relevance of our approach, and to investigate the influence of the main parameters. We have focused on the control effort, which is probably the most important quantity. The results are summarized in the two tables below, where $\norm{\overline u}_{L^2}$ (top table) and $\norm{\overline u}_{L^\infty}$ (bottom table) are given for different values of the parameters $s$ and~$R'$ in~\eqref{eq:ybar}. Figures~\ref{fig:surf1130} and~\ref{fig:control1130} give moreover the complete temperature and control evolution for the case $R'=0.2$ and~$s=1.6$. For all experiments the ``regularization time'' $\tau$ is~$0.3$, and the initial condition $\theta_0$ is a step function with $\theta_0(x)=-1$ on~$[0,1/2)$ and $\theta_0(x)=1$ on~$(1/2,1)$ hence Fourier coefficients $c_{2p}=0$ and $c_{2p+1}=\frac{(-1)^{p+1}}{2p+1}\frac{2\sqrt2}{\pi}$ for $p\geq0$; $\oi,\ok,\on$ are ``large enough'' for a good accuracy of the truncated series\eqref{eq:ubar}~--\eqref{eq:ykbar}.
\begin{center}
\begin{tabular}{|r|c|c|c|c|c|}\hline
$\mathbf{s,R'}$& \textbf{0.15}& \textbf{0.20} & \textbf{0.25} & \textbf{0.30}\\ \hline
\textbf{1.5}& 693& 63.3& 12.7& 3.82\\
\hline
\textbf{1.6}& 35.3& 6.41& 1.95& 0.78\\
\hline
\textbf{1.7}& 7.49& 1.95& 0.74& 0.34\\
\hline
\textbf{1.8}& 5.53& 1.24& 0.48& 0.23\\
\hline
\textbf{1.9}& 5.71& 1.29& 0.47& 0.22\\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|r|c|c|c|c|c|}\hline
$\mathbf{s,R'}$& \textbf{0.15}& \textbf{0.20} & \textbf{0.25} & \textbf{0.30}\\ \hline
\textbf{1.5}& 3666& 330& 55.2& 18.1\\
\hline
\textbf{1.6}& 118& 23.6& 7.17& 2.76\\
\hline
\textbf{1.7}& 18.6& 4.78& 1.73& 0.76\\
\hline
\textbf{1.8}& 36.4& 4.59& 1.44& 0.65\\
\hline
\textbf{1.9}& 47.8& 9.66& 2.13& 0.82\\
\hline
\end{tabular}
\end{center}
A few conclusions can be drawn from these experiments:
\begin{itemize}
\item when $R'$ gets small there seems to be an ``optimal'' value of~$s$ which minimizes the control effort (the values are different for $\norm{\overline u}_{L^2}$ and $\norm{\overline u}_{L^\infty}$)
\item when $s$ it too small or too large the derivatives $\overline y^{(i)}$ tend to ``crowd'' near the extremities of~$[\tau,\tau+R']$ with large amplitudes
\item the smaller $R'$ the more noticeable this phenomenon, with of course an increased control effort.
\end{itemize}
An interesting question is the tradeoff between $\tau$ and~$R'$ to reach the zero state at time~$T:=\tau+R'$ with the smallest control effort: longer regularization~$\tau$ or longer duration~$R'$ of the active control? With the present construction of the function~$y$~\eqref{eq:y} we have by design $R'\leq\tau$, which is probably an important restriction. Other, but more complicated, constructions without this limitation are possible and will be studied in the future.
\begin{figure}[ht!]
\centering
\includegraphics[width=1.0\columnwidth]{surf1130cropped}
\caption{$\overline\theta(t,x)$ for $R'=0.2$ and $s=1.6$.}
\label{fig:surf1130}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[width=1.0\columnwidth]{control1130cropped}
\caption{$\overline u(t)$ and $\norm{\overline u}_{L^2(0,t)}$ for $R'=0.2$ and $s=1.6$.}
\label{fig:control1130}
\end{figure}
\section{Conclusion}
|
1,116,691,497,721 | arxiv | \section{Introduction}\label{IntroductionSection}
The treatment of missing data in different areas of science, statistics, economics, or pattern recognition has been and still is a wide subject
of interest, since a variety of situations may lead to incomplete data in a set of information.
In particle and nuclear physics there are several situations where a certain fraction of data may be missing.
Typical cases are the set of space points which contribute to the tracking of charged particles, or the different values of
the energy loss of particles in a multilayer detector.
There are several reasons why data may be missing.
In the simplest case, they may be missing completely at random (MCAR), e.g. the probability that a data is missing does not depend on the value
of the variable. In most cases however such probability either depends on the other variables in the data set or on the value of the variable under consideration.
In the former situation the data are said to be missing at random (MAR), while in the latter situation the missing data mechanism is denoted as non-ignorable or
missing not at random (MNAR).
Examples of the MCAR case are the passage of minimum ionizing charged particles
through a multilayer detector, in the limit of very small thickness (thus limiting the amount of energy loss and multiple scattering in each layer),
with detection efficiency smaller than 100\%, due to dead areas or other inefficiencies.
Examples of the MNAR case are given by a detector which has a finite energy threshold, modelled for instance as a sigmoid function.\\
The simplest approach in case of a certain fraction of missing information from a set of variables is to disregard the event where at least one variable is
missing. Such approach, although retaining only complete events, may lead to a substantial loss of events either when the elementary fraction of missing
events is large or when the number of variables is large. As an example, in a process with $d$=100 variables (such as the number of
space points in a large tracking detector),
even an elementary fraction $\eta$= 0.1\% in each variable leads to a net loss of 1-(1-$\eta$)$^{d}$ = 1-(0.999)$^{100}\sim$10\%.
Figure \ref{OverallMissingRateVSIneffVSNDim} shows the contour lines corresponding to different overall fractions of missing events
(10\%, 20\%, 30\% and 40\%), as a function of the dimensionality $d$ of the problem and of the elementary fraction of missing events $\eta$ in each variable.
The importance of using all collected events is of special concern in case of rare events, where disregarding the event due to its incompleteness may
lead to a substantial fraction of potentially interesting events being lost. This is the reason why different methods have been employed
to impute the missing values of a variable according to the statistical properties of the variables of interest which define the
event \cite{Graham2009,Horton2007,Luengo2010}. Most of such methods rely on normal distributions of the variables, which however is not a good
representation in many problems. The energy loss of a charged particle in a thin detector is a typical example of a variable whose distribution
has an asymmetric shape (Landau tail). For such reason, it is of interest to develop and test methods which are not biased by the assumption
on normal distributions. \\
In this paper an approach based on multivariate skew-normal (MSN) distribution was developed. The method was then applied to a test case, where
simulated energy losses of pions, kaons and protons in a multilayer Silicon detector were considered. The performance of a neural network was then
evaluated under missing data and after recovering them with the approach described here. Sect. \ref{MissingDataSection} describes in some more detail the problem of
missing data reconstruction and the methods adopted throughout the paper, namely Multiple Imputation (MI) and Maximum Likelihood (ML) methods,
while the application to the test case is discussed in Sect. \ref{MethodApplicationSection}. Various methods
of imputation of the missing data were then considered and applied to the same set of simulated data, comparing their performance in terms of
identification efficiency. The results of such comparison are reported in Sect. \ref{ResultsSection}. Some detail of the algorithms being employed are described
in \ref{SkewNormalAlgorithmDerivation} .
\begin{figure}[!t]
\centering
\includegraphics[scale=0.4]{MissingFractionVSIneffVSNDim.pdf}%
\vspace{0.1cm}
\caption{Overall missing fraction $f$ as a function of the variable detection inefficiency $\eta$ and the number of
observables $d$ in the analysis. The contour lines are relative to $f$=10\%, 20\%, 30\%, 40\%.}
\label{OverallMissingRateVSIneffVSNDim}
\end{figure}
\section{Missing Data Reconstruction}\label{MissingDataSection}
The most common strategy of dealing with missing data is list-wise deletion (LD), that consists in rejecting all events with missing observables.
Such approach decreases the effective sample size, specially with a large number of observables in analysis, and correspondingly the
power of any statistical tests to be performed with that sample, which in presence of rare events it is not desirable.
If the data are not missing at random, such procedure may also lead to a selection bias. In cases when the missing data mechanism
is MCAR or MAR list-wise deletion does not add any bias and if the event sample size is not a critical issue it should be the preferred approach.\\
The alternative approach than removing events with missing data, is to fill in or impute the missing values,
with the aim of recovering the relevant features of the data set as a whole (mean, variances, parameters and so forth), rather than
obtaining precise estimates of single missing values. Indeed, imputed values should not be trusted nor directly used to draw inferences.\\
The simplest way is to replace each missing value with the mean of the observed values for that variable. This strategy significantly alters the
distribution for that observable and all derived summary indicators, for example the variance which is typically underestimated.
If more than one group is present in the data sample, the mean estimation tend to be pulled towards the most abundant group.
For that reason such method is not recommended.\\
A wide number of modern imputation methods exists in literature also used in the context of neural network analysis,
among them regression-based single imputation, multiple imputation (MI), maximum likelihood (ML) methods, methods based on K-Nearest Neighbor (KNN)
or K-means clustering, Singular Value Decomposition (SVD), Support Vector Machines (SVM). A review of some of these methods is available in
\cite{Graham2009,Horton2007,Luengo2010}.\\Neural networks themselves could be also used to impute missing data.
This is typically achieved by designing a set of neural network classifiers, with each specialized network performing classification on a different
set of input attributes. However when the dimension of the input vector is large and missing data are present in many variables, such approach lead to a
large number of combinations of networks to be implemented.\\In this work we will adopt maximum likelihood methods based on multivariate normal (MN) mixtures
as compared to the popular multiple imputation approach.
Both methods assume a normal model, which is not appropriate to the energy deposit data under consideration. For that reason we designed in this work
a ML method based on multivariate skew-normal (MSN) mixture models, presented in paragraph \ref{SkewNormalImputationSection}.
\subsection{Multiple Imputation}\label{MultipleImputationSection}
Single imputation methods, for example those based on linear regression, are intrinsically limited.
They infact proceed by calculating the regression of the incomplete variable on the other complete variables and
substituting the predicted mean for each missing observation. Since imputed values always lie on the regression line, the actual dispersion of the data is
ignored and therefore the variance is underestimated, leading to a bias in the parameter estimates.\\The multiple imputation approaches proceed instead
by introducing a random variation in the process, i.e. a random normal error into the regression equation, and by generating several data sets, each
with different imputed values, to partially restore the lost variance. To account for the fact that only a single draw from the data population is taken,
multiple random draws from the posterior distribution of the population, each imputed multiple times, are also introduced to completely restore the
variance of the data. Multiple imputation algorithms available in literature typically differs for how this latest step is performed.
Some uses bootstrap procedures to generate random draws, others adopt Data Augmentation (DA) techniques or the Markov chain Monte Carlo (MCMC) algorithm and so forth.
If the MAR assumption holds the method leads to almost unbiased estimators.
\subsection{Maximum Likelihood Imputation}\label{LikelihoodImputationSection}
In the likelihood approach a model is assumed to describe the observed data. If $p$ variables are considered for the analysis, we denote a sample of $N$ data
observations with $\mathbf{x}$= ($\mathbf{x}_{1}$, $\mathbf{x}_{2}$, \dots, $\mathbf{x}_{N}$), being $\mathbf{x}_i$ a $p$-dimensional data vector.
For a given observation $i$ we may have missing patterns. We indicate with $\mathbf{x}_{i,o}$ the observed patterns and with $\mathbf{x}_{i,m}$ the missing patterns.
A widely used approach consists in assuming a mixture model F($\mathbf{x}$;$\mbox{\boldmath $\Theta$}$) with $K$ components with probability density functions \emph{pdf} f($\mathbf{x}$;$\mbox{\boldmath $\theta$}_k$)
and parameters $\mbox{\boldmath $\theta$}_{k}$:
\begin{equation}
F(\mathbf{x};\mbox{\boldmath $\Theta$})= \sum_{k=1}^{K}\pi_{k}f_{k}(\mathbf{x};\mbox{\boldmath $\theta$}_{k})%
\end{equation}
where $\pi_{k}$ are the component weights, constrained as $\sum_{k=1}^{K}\pi_{k}=1$. $\mbox{\boldmath $\Theta$}$ denotes the overall parameters of the mixture,
estimated by maximizing the log-likelihood function:
\begin{equation}
\mathcal{L}(\mbox{\boldmath $\Theta$}; \mathbf{x})= \log\prod_{i=1}^{N}F(\mathbf{x}_i;\mbox{\boldmath $\Theta$}) = \sum_{i=1}^{N}\log \sum_{k=1}^{K}\pi_{k}f_{k}(\mathbf{x}_i;\mbox{\boldmath $\theta$}_{k}) \, . \label{loglik}
\end{equation}
Usually the maximization has to be carried out numerically. However, for some particular choice of the components $f_k$, a closed form solution
may be derived by using the \emph{Expectation-Maximization} (EM) algorithm. Notable examples are the multivariate normal and t-distribution and
corresponding skew distributions, skew-normal and skew-t. In the EM framework the $N$ observed data are considered incomplete and a set of variable
indicators $\mathbf{Z}=(\mathbf{z}_1,\ldots,\mathbf{z}_N)$ is introduced, such that $\mathbf{z}_i=(z_{1i},\ldots,z_{Ki})$ (i=1,\dots N) with $z_{ki}$=1 if $\mathbf{x}_i$
comes from the $k$-th component and $z_{ki}$=0 otherwise.\\
Provided that the missing data are ignorable (MAR or MCAR), the missing features $\mathbf{x}_{i,m}$ can be incorporated in the
model as additional latent parameters together with the $z_{i}$.
In the following paragraphs we report the results obtained for normal and skew-normal mixtures.
\subsubsection{Normal Mixture Models}\label{NormalImputationSection}
Consider a multivariate normal (MN) distribution $f_{k}(\mathbf{x};\mbox{\boldmath $\theta$}_{k})$ as $k$-th component:
\begin{equation*}
f_{k}(\mathbf{x};\mbox{\boldmath $\theta$}_{k})= \frac{1}{2\pi |\mbox{\boldmath $\Sigma$}_{k}|^{1/2}}\exp\left[-\frac{1}{2}(\mathbf{x}-\mbox{\boldmath $\mu$}_{k})^{T}\mbox{\boldmath $\Sigma$}_{k}^{-1}(\mathbf{x}-\mbox{\boldmath $\mu$}_{k})\right]
\end{equation*}
with mean vector $\mbox{\boldmath $\mu$}_{k}$ and covariance matrix $\mbox{\boldmath $\Sigma$}_{k}$ for the $k$-th component.\\
For the $i$-th observation the mixture parameters can be partitioned into an observed and missing part: ($\mbox{\boldmath $\mu$}_{ik,o}$, $\mbox{\boldmath $\mu$}_{ik,m}$),
($\mbox{\boldmath $\Sigma$}_{ik,oo}$, $\mbox{\boldmath $\Sigma$}_{ik,mm}$, $\mbox{\boldmath $\Sigma$}_{ik,om}$).
Following \cite{Zoubin1994,Delalleau2012} the iterative solution provided by
the EM algorithm is given by:
\begin{align}
\tau_{ik}^{(t+1)}&= \frac{\pi_{k}^{(t)}f_{k}(\mathbf{x}_{i,o},\mbox{\boldmath $\theta$}_{k,o}^{(t)})}{\sum_{k=1}^{K}\pi_{k}^{(t)}f_{k}(\mathbf{x}_{i,o},\mbox{\boldmath $\theta$}_{k,o}^{(t)})}\\
\pi_{k}^{(t+1)}&= \frac{1}{N}\sum_{i=1}^{N}\tau_{ik}^{(t+1)}\\
\mbox{\boldmath $\mu$}_{k}^{(t+1)}&= \frac{\sum_{i=1}^{N}\tau_{ik}^{(t+1)}\hat{\mathbf{x}}_{ik}^{(t+1)}}{\sum_{i=1}^{N}\tau_{ik}^{(t+1)}}\\%
\mbox{\boldmath $\Sigma$}_{k}^{(t+1)}&= \frac{\sum_{i=1}^{N}\tau_{ik}^{(t+1)}(\hat{\mathbf{x}}_{ik}^{(t+1)}-\mbox{\boldmath $\mu$}_{k}^{(t+1)})^{T}(\hat{\mathbf{x}}_{ik}^{(t+1)}-
\mbox{\boldmath $\mu$}_{k}^{(t+1)})}{\sum_{i=1}^{N}\tau_{ik}^{(t+1)}}+\frac{\tau_{ik}^{(t+1)}\hat{\mathbf{C}}_{ik}^{(t+1)}}{\sum_{i=1}^{N}\tau_{ik}^{(t+1)}}
\end{align}
with $\hat{\mathbf{x}}_{ik}$ and $\hat{\mathbf{C}}_{ik}$ respectively given by:
\begin{equation}
\hat{\mathbf{x}}_{ik}=
\left\{
\begin{array}{l}
\hat{\mathbf{x}}_{ik,o}= \mathbf{x}_{i,o}\\%
\hat{\mathbf{x}}_{ik,m}= \mbox{\boldmath $\mu$}_{k,m}+\mbox{\boldmath $\Sigma$}_{k,mo}\mbox{\boldmath $\Sigma$}_{k,oo}^{-1}(\mathbf{x}_{i,o}-\mbox{\boldmath $\mu$}_{k,o})
\end{array}
\right.\\
\end{equation}
\begin{equation}
\hat{\mathbf{C}}_{ik}=
\left.
\begin{pmatrix}
\mbox{\boldmath $0$} & \mbox{\boldmath $0$}\\%
\mbox{\boldmath $0$} & \hat{\mathbf{C}}_{ik,mm}
\end{pmatrix}
\right.
,\;\hat{\mathbf{C}}_{ik,mm}= \mbox{\boldmath $\Sigma$}_{k,mm}-\mbox{\boldmath $\Sigma$}_{k,mo}\mbox{\boldmath $\Sigma$}_{k,oo}^{-1}\mbox{\boldmath $\Sigma$}_{k,mo}^{T}
\end{equation}
In the above expressions, $\hat{\mathbf{x}}_{ik,m}=\mathbb{E}[z_{ik}\mathbf{x}_{i,m}|\mathbf{x}_{i,o},\mbox{\boldmath $\theta$}_{k}]$ is the expectation of the missing
values $\mathbf{x}_{i,m}$ given the observed data $\mathbf{x}_{i,o}$, assuming that component $k$ generated $\mathbf{x}_{i}$, while $\hat{\mathbf{C}}_{ik,mm}$
represents the covariance of the missing values $\mathbf{x}_{i,m}$.\\
At convergence, the missing values $\mathbf{x}_{i,m}$ are estimated by the weighted sum of $\hat{\mathbf{x}}_{ik,m}$:
\begin{equation}
\mathbf{x}_{i,m}= \sum_{k=1}^{K}\tau_{ik}\hat{\mathbf{x}}_{ik,m}
\end{equation}
We note that if the $p$ variables $\mathbf{x}$ are independent, the covariance matrix is diagonal and therefore $\mbox{\boldmath $\Sigma$}_{k,mo}$=$\mbox{\boldmath $0$}$. In this
case the imputation rule yields $\mathbf{x}_{i,m}$= $\sum_{k=1}^{K}\tau_{ik}\mbox{\boldmath $\mu$}_{k,m}$.
\subsubsection{Skew-Normal Mixture Models}\label{SkewNormalImputationSection}
Consider a multivariate skew-normal (MSN) distribution $f_{k}(\mathbf{x},\mbox{\boldmath $\theta$}_{k})$ as $k$-th component:
\begin{equation*}
f_{k}(\mathbf{x};\mbox{\boldmath $\theta$}_{k})= 2\phi_{p}(\mathbf{x};\mbox{\boldmath $\xi$}_{k},\mbox{\boldmath $\Omega$}_{k})\Phi(\mbox{\boldmath $\delta$}_{k}^{T}\mbox{\boldmath $\Omega$}_{k}^{-1}(\mathbf{x}-\mbox{\boldmath $\xi$}_{k});0,1-\mbox{\boldmath $\delta$}_{k}^{T}\mbox{\boldmath $\Omega$}_{k}^{-1}\mbox{\boldmath $\delta$}_{k})
\end{equation*}\label{MSNDefinition}
with location vector $\mbox{\boldmath $\xi$}_{k}$, scale matrix $\mbox{\boldmath $\Sigma$}_{k}$, skewness vector $\mbox{\boldmath $\delta$}_{k}$ and $\mbox{\boldmath $\Omega$}_{k}=\mbox{\boldmath $\Sigma$}_{k}+\mbox{\boldmath $\delta$}_{k}\mbox{\boldmath $\delta$}_{k}^{T}$ for the $k$-th
component \footnote{Note that $\mbox{\boldmath $\xi$}$ and $\mbox{\boldmath $\Sigma$}$ do not represent the mean and covariance of the distribution. These are given by:
\begin{eqnarray*}
&\mathbb{E}(\mathbf{X})=\mbox{\boldmath $\xi$}+\sqrt{\frac{2}{\pi}}\\
&\mbox{cov}(\mathbf{X})=\mbox{\boldmath $\Sigma$}+\biggl(1-\frac{2}{\pi}\biggl)\mbox{\boldmath $\delta$}\mbox{\boldmath $\delta^{T}$}
\end{eqnarray*}\label{SkewNormalHierarchical}
}.
$\phi_{p}(\cdot)$ denotes the p-variate normal \emph{pdf} and $\Phi(\cdot)$ the \emph{cdf} of the univariate normal.
It is worth noting that several forms of the skew-normal distribution exists in literature, both restricted and unrestricted. A nice
comparison between different definitions is reported in \cite{LeeMcLachlan2012}. In this work we assumed the
restricted form provided by Pyne et al \cite{Pyne2009}.\\
For the $i$-th observation the mixture parameters can be divided into an observed and missing part: ($\mbox{\boldmath $\xi$}_{ik,o}$, $\mbox{\boldmath $\xi$}_{ik,m}$),
($\mbox{\boldmath $\Sigma$}_{ik,oo}$, $\mbox{\boldmath $\Sigma$}_{ik,mm}$, $\mbox{\boldmath $\Sigma$}_{ik,om}$), ($\mbox{\boldmath $\delta$}_{ik,o}$,$\mbox{\boldmath $\delta$}_{ik,m}$).
A closed form solution to the problem has been provided by Lin et al \cite{Lin2009,Lin2010}
for the unrestricted skew-normal model. In this work we explicitly derived the EM solution for the restricted form, given by:
\begin{align*}
&\tau_{ik}^{(t+1)}= \frac{\pi_{k}^{(t)}f(\mathbf{x}_{i,o},\mbox{\boldmath $\theta$}_{k,o}^{(t)})}{\sum_{k=1}^{K}\pi_{k}^{(t)}f_{k}(\mathbf{x}_{i,o},\mbox{\boldmath $\theta$}_{k,o}^{(t)})}\\%
&\pi_{k}^{(t+1)}= \frac{1}{N}\sum_{i=1}^{N}\tau_{ik}^{(t+1)}\\%
&\mbox{\boldmath $\xi$}_{k}^{(t+1)}= \frac{\sum_{i=1}^{N}\tau_{ik}^{(t+1)}(\hat{\mathbf{x}}_{ik}-\mbox{\boldmath $\delta$}_{k}^{(t)}e_{1,ik}^{(t)})}{\sum_{i=1}^{N}\tau_{ik}^{(t+1)}}\\%
&\mbox{\boldmath $\delta$}_{k}^{(t+1)}= \frac{\sum_{i=1}^{N}\tau_{ik}^{(t+1)}e_{1,ik}^{(t)}(\tilde{\mathbf{x}}_{ik}-\mbox{\boldmath $\xi$}_{k}^{(t+1)})}{\sum_{i=1}^{N}\tau_{ik}^{(t+1)}e_{2,ik}^{(t)}}\\%
&\mbox{\boldmath $\Sigma$}_{k}^{(t+1)}= \frac{\sum_{i=1}^{N}\tau_{ik}^{(t+1)} (\hat{\mathbf{x}}_{ik}-\mbox{\boldmath $\xi$}_{k}^{(t+1)}) (\hat{\mathbf{x}}_{ik}-\mbox{\boldmath $\xi$}_{k}^{(t+1)})^{T} }
{\sum_{i=1}^{N}\tau_{ik}^{(t+1)}} -
\frac{\sum_{i=1}^{N}\tau_{ik}^{(t+1)} e_{1,ik}^{(t)}[\mbox{\boldmath $\delta$}_{k}^{(t)}(\tilde{\mathbf{x}}_{ik}-\mbox{\boldmath $\xi$}_{k}^{(t+1)})^{T}+(\tilde{\mathbf{x}}_{ik}-\mbox{\boldmath $\xi$}_{k}^{(t+1)})
\mbox{\boldmath $\delta$}_{k}^{T (t+1)}]}
{\sum_{i=1}^{N}\tau_{ik}^{(t+1)}}+\\%
+& \frac{\sum_{i=1}^{N}\tau_{ik}^{(t+1)} e_{2,ik}\mbox{\boldmath $\delta$}_{k}^{(t+1)}\mbox{\boldmath $\delta$}_{k}^{T (t+1)}]}
{\sum_{i=1}^{N}\tau_{ik}^{(t+1)}}+
\frac{\sum_{i=1}^{N}\tau_{ik}^{(t+1)}\hat{\mathbf{C}}_{ik}}{\sum_{i=1}^{N}\tau_{ik}^{(t+1)}}
\end{align*}
in which the needed expectation terms are given by:
\begin{align*}
&e_{1,ik}= \mu_{ik,o}+\sigma_{ik,o}\frac{\phi(\mu_{ik,o}/\sigma_{ik,o})}{\Phi(\mu_{ik,o}/\sigma_{ik,o})}\\%
&e_{2,ik}= \mu_{ik,o}^{2}+\sigma_{ik,o}^{2}+\mu_{ik,o}\sigma_{ik,o}
\frac{\phi(\mu_{ik,o}/\sigma_{ik,o})}{\Phi(\mu_{ik,o}/\sigma_{ik,o})}\\%
&\hat{\mathbf{x}}_{ik}=
\left\{
\begin{array}{l}
\hat{\mathbf{x}}_{ik,o}= \mathbf{x}_{i,o}\\%
\hat{\mathbf{x}}_{ik,m}= \mbox{\boldmath $\xi$}_{ik,m}+\mbox{\boldmath $\delta$}_{ik,m}e_{1,ik}+\mbox{\boldmath $\Sigma$}_{ik,mo}\mbox{\boldmath $\Sigma$}_{ik,oo}^{-1}(\mathbf{x}_{i,o}-\mbox{\boldmath $\xi$}_{ik,o}-
\mbox{\boldmath $\delta$}_{ik,o}e_{1,ik})
\end{array}
\right.\\
&\tilde{\mathbf{x}}_{ik}=
\left\{
\begin{array}{l}
\tilde{\mathbf{x}}_{ik,o}= \mathbf{x}_{i,o}\\%
\tilde{\mathbf{x}}_{ik,m}= \mbox{\boldmath $\xi$}_{ik,m}+\mbox{\boldmath $\delta$}_{ik,m}e_{2,ik}/e_{1,ik}+
\mbox{\boldmath $\Sigma$}_{ik,mo}\mbox{\boldmath $\Sigma$}_{ik,oo}^{-1}(\mathbf{x}_{i,o}-\mbox{\boldmath $\xi$}_{ik,o}-\mbox{\boldmath $\delta$}_{ik,o}e_{2,ik}/e_{1,ik})
\end{array}
\right.\\
&\hat{\mathbf{C}}_{ik}=
\left.
\begin{pmatrix}
\mbox{\boldmath $0$} & \mbox{\boldmath $0$}\\%
\mbox{\boldmath $0$} & \hat{\mathbf{C}}_{ik,mm}
\end{pmatrix}
\right.
,\;\hat{\mathbf{C}}_{ik,mm}=\mbox{\boldmath $\Sigma$}_{ik,mm}-\mbox{\boldmath $\Sigma$}_{ik,mo}\mbox{\boldmath $\Sigma$}_{ik,oo}^{-1}\mbox{\boldmath $\Sigma$}_{ik,om} +
(e_{2,ik}-e_{1,ik}^2)(\mbox{\boldmath $\delta$}_{ik,m}-\mbox{\boldmath $\Sigma$}_{ik,mo}\mbox{\boldmath $\Sigma$}_{ik,oo}^{-1}\mbox{\boldmath $\delta$}_{ik,o})(\mbox{\boldmath $\delta$}_{ik,m}-\mbox{\boldmath $\Sigma$}_{ik,mo}\mbox{\boldmath $\Sigma$}_{ik,oo}^{-1}\mbox{\boldmath $\delta$}_{ik,o})^{T}
\end{align*}
with $\mu_{ik,o}=\mbox{\boldmath $\delta$}_{ik,o}^T\mbox{\boldmath $\Omega$}_{ik,oo}^{-1}(\mathbf{x}_{i,o}-\mbox{\boldmath $\xi$}_{ik,o})$ and $\sigma_{ik,o}^{2}=1-\mbox{\boldmath $\delta$}_{ik,o}^{T}\mbox{\boldmath $\Omega$}_{ik,oo}^{-1}\mbox{\boldmath $\delta$}_{ik,o}$.\\
In the above expressions, $\hat{\mathbf{x}}_{ik,m}$ is the expectation of the missing
values $\mathbf{x}_{i,m}$ given the observed data $\mathbf{x}_{i,o}$, assuming that component $k$ generated $\mathbf{x}_{i}$, while $\hat{\mathbf{C}}_{ik,mm}$
represents the covariance of the missing values $\mathbf{x}_{i,m}$.\\
At convergence, the missing values $\mathbf{x}_{i,m}$ are estimated by the weighted sum of $\hat{\mathbf{x}}_{ik,m}$:
\begin{equation}
\mathbf{x}_{i,m}= \sum_{k=1}^{K}\tau_{ik}\hat{\mathbf{x}}_{ik,m}
\end{equation}
More details on the derivation are reported in the \ref{SkewNormalAlgorithmDerivation} .
\section{Method Application}\label{MethodApplicationSection}
For our application we considered a test case, namely the interaction of charged particles (pions, kaons and protons) on a multilayer Silicon detector, considering
the energy losses in each layer as relevant observables. This is a well known technique in particle and nuclear physics
to separate and identify different particles according to their
specific energy loss as a function of the particle momentum, usually provided by their curvature in a magnetic field or by the time-of-flight
measurement.
Missing data may arise in this scenario either from detector inefficiencies, i.e. inactive or damaged detector areas, or
from faulty electronic channels.
Even though the number of assumed variables in this example is not large (six layers were considered, as is the case of a typical inner
tracking detector, such as the present configuration of the ALICE detector at LHC\cite{ALICE}), the case
may be usefully employed to demonstrate the
capabilities of the various approaches and compare their performance.
For the classification of charged particles we adopted a neural network approach, comparing its performance according to the way
in which missing data were previously imputed by one of the methods
described in Section \ref{MissingDataSection}.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.4]{TruncatedEnergyLossVSMomentum.pdf}%
\vspace{0.1cm}
\caption{Specific energy loss $dE/dX$ for pions, kaons and protons as a function of their particle momentum, as obtained from a GEANT
simulation. The average of the two smallest energy deposits in the six layers is considered in the plot.}
\label{EnergyLossVSMomentum}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[scale=0.13]{AnalysisMethodSketch_last_light.pdf}%
\vspace{0.1cm}
\caption{Schema of the analysis method.}
\label{AnalysisMethodSketchFig}
\end{figure}
\subsection{Simulated data set}\label{SimulationSection}
To generate the simulated data set, the interaction of $\pi^+$, $K^+$ and protons was implemented in a GEANT software replica of a six-layers
Silicon plane detector. Each layer had 500 $\mu$m thickness.
Particles were generated with a relative abundance of 0.80/0.15/0.05 for pions/kaons/protons, with a uniform momentum distribution in the
interval 0-1 GeV/c.
The energy loss deposited in each layer was stored event-by-event for subsequent analysis. About 10$^{6}$ events were produced.
Since the identification performances are expected to strongly depends on the particle momentum,
the data set was divided into individual momentum bins of
width 50 MeV/c. For each bin, half of the data sample was used as training and cross validation for the
neural network learning stage.
\begin{figure}[!t]
\centering
\subtable[MN imputation]{\includegraphics[scale=0.4]{SampleMGImputationPlot_PaperFig.pdf}\label{SampleImputationPlot1}}%
\subtable[MSN imputation]{\includegraphics[scale=0.4]{SampleMSNImputationPlot_PaperFig.pdf}\label{SampleImputationPlot2}}%
\vspace{0.1cm}
\caption{Normal (left panel) and skew-normal (right panel) imputation results for a sample data set with 10\% missing data.
True (simulated) data are shown in black filled dots, while imputed data with open black dots. The colored area represents the
contour levels of the fitted normal/skew-normal mixture model.}
\label{SampleImputationPlot}
\end{figure}
The remaining part of the data set available in the momentum bin was used to draw $N_{\textsf{s}}$=100 random test
samples with $N$=1000 events. We generated randomly missing data in each test sample according to a prespecified missing probability,
assumed equal for
each detector layer. Such samples were finally used to test the considered imputation methods.\\
Fig. \ref{EnergyLossVSMomentum} shows a typical scatter plot of the specific energy loss
$dE/dX$ as a function of the particle momentum $p$. In particular we reported in the plot the truncated energy loss, computed by taking the average of the
two smallest values in the six detector layers.
\begin{figure}[!t]
\centering
\subtable[$p$=(0.25-0.30) GeV/c]{\includegraphics[scale=0.3]{NNOutput_p0_25-0_3.pdf}\label{NNOutputPlot1}}%
\hspace{-0.9cm}
\subtable[$p$=(0.55-0.60) GeV/c]{\includegraphics[scale=0.3]{NNOutput_p0_55-0_6.pdf}\label{NNOutputPlot2}}%
\hspace{-0.9cm}
\subtable[$p$=(0.85-0.90) GeV/c]{\includegraphics[scale=0.3]{NNOutput_p0_85-0_9.pdf}\label{NNOutputPlot3}}%
\vspace{0.1cm}
\caption{Distribution of the neural network outputs of pions(red histogram), kaons (green histogram) and protons (blue histogram) events
obtained when presenting a sample data set to the designed network for three different momentum bins:
(0.25-0.30) GeV/c (left panel), (0.55-0.60) GeV/c (central panel) and (0.85-0.90) GeV/c (right panel).}
\label{NNOutputPlot}
\end{figure}
The individual energy loss distribution for a given momentum bin has the typical Landau shape, with an asymmetric tail at high energy loss.
This aspect is important, since most of the available algorithms to impute missing data are most suited for Gaussian distributions.\\
To reduce the impact of Landau tails and get rid of the momentum dependence we normalized the simulated energy losses obtained in the
six layers to the logarithm of the average energy loss for kaons, e.g. we considered $e\equiv\ln(dE/dX)-\ln(dE/dX_{kaons}(p))$ as variable
for the classification analysis.
\subsection{Neural network approach to particle identification}\label{NeuralNetworkSection}
Neural network approaches to particle identification in nuclear and particle physics have been extensively used in the past. In this work we considered
a feed-forward neural network. The strategy is sketched in Fig. \ref{AnalysisMethodSketchFig}.
We train a neural network for each momentum bin
with a set of complete simulated data. After a few trials, a 6:6:6:1 network topology was chosen for the entire momentum range,
where the input neurons were the
six energy losses in the different layers and the output layer is the particle identification (PID) code (1=pion, 2=kaon, 3=proton).
Two hidden layers with six neurons each were introduced to improve the particle separation capabilities at high momenta. Hyperbolic tangent
activation functions
were used for the neurons in the hidden layer. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm was employed as learning method and a number of
iterations in the order of 1000 provides both optimal classification results and network generalization capabilities.\\The trained network is then used to
identify the test data samples in which missing data were previously imputed separately with each of the methods presented in the
previous Section: Mean,
Maximum Likelihood for MN and MSN mixtures and Multiple Imputation. The maximum-likelihood algorithms were completely developed in C++ employing
the implementation of multivariate distributions of the R statistical tool \cite{RProject} via the RInside/Rcpp interface \cite{RInside}.
Several implementation to perform multiple imputation are available in current statistical tools.
To compare with other algorithms we made use for this work of the \textsc{Amelia II} package \cite{AmeliaPackage} present in the R tool.
\begin{figure}[!t]
\centering
\subtable[$p$=(0.25-0.30) GeV/c]{\includegraphics[scale=0.27]{ImpPerf_MSE_p0_25-0_3_last.pdf}\label{DiffPlot1}}%
\hspace{-0.25cm}
\subtable[$p$=(0.55-0.60) GeV/c]{\includegraphics[scale=0.27]{ImpPerf_MSE_p0_55-0_6_last.pdf}\label{DiffPlot2}}%
\hspace{-0.25cm}
\subtable[$p$=(0.85-0.90) GeV/c]{\includegraphics[scale=0.27]{ImpPerf_MSE_p0_85-0_9_last.pdf}\label{DiffPlot3}}%
\vspace{0.1cm}
\caption{Average differences between the true data and the imputed data,
for three different momentum bins:
(0.25-0.30) GeV/c (left panel), (0.55-0.60) GeV/c (central panel) and (0.85-0.90) GeV/c (right panel).}
\label{Differences}
\end{figure}
\begin{figure}[!th]
\centering
\subtable[$p$=(0.25-0.30) GeV/c - pion]{\includegraphics[scale=0.28]{PionEffVSMissFraction_p0_25-0_3_last.pdf}\label{EfficiencyLowMomentumPlot1}}%
\hspace{-0.2cm}%
\subtable[$p$=(0.25-0.30) GeV/c - kaon]{\includegraphics[scale=0.28]{KaonEffVSMissFraction_p0_25-0_3_last.pdf}\label{EfficiencyLowMomentumPlot2}}%
\hspace{-0.2cm}%
\subtable[$p$=(0.25-0.30) GeV/c - proton]{\includegraphics[scale=0.28]{ProtonEffVSMissFraction_p0_25-0_3_last.pdf}\label{EfficiencyLowMomentumPlot3}}%
\\%
\vspace{-0.3cm}
\subtable[$p$=(0.55-0.60) GeV/c - pion]{\includegraphics[scale=0.28]{PionEffVSMissFraction_p0_55-0_6_last.pdf}\label{EfficiencyMediumMomentumPlot1}}%
\hspace{-0.2cm}%
\subtable[$p$=(0.55-0.60) GeV/c - kaon]{\includegraphics[scale=0.28]{KaonEffVSMissFraction_p0_55-0_6_last.pdf}\label{EfficiencyMediumMomentumPlot2}}%
\hspace{-0.2cm}%
\subtable[$p$=(0.55-0.60) GeV/c - proton]{\includegraphics[scale=0.28]{ProtonEffVSMissFraction_p0_55-0_6_last.pdf}\label{EfficiencyMediumMomentumPlot3}}%
\\%
\vspace{-0.3cm}
\subtable[$p$=(0.85-0.90) GeV/c - pion]{\includegraphics[scale=0.28]{PionEffVSMissFraction_p0_85-0_9_last.pdf}\label{EfficiencyHighMomentumPlot1}}%
\hspace{-0.2cm}%
\subtable[$p$=(0.85-0.90) GeV/c - kaon]{\includegraphics[scale=0.28]{KaonEffVSMissFraction_p0_85-0_9_last.pdf}\label{EfficiencyHighMomentumPlot2}}%
\hspace{-0.2cm}%
\subtable[$p$=(0.85-0.90) GeV/c - proton]{\includegraphics[scale=0.28]{ProtonEffVSMissFraction_p0_85-0_9_last.pdf}\label{EfficiencyHighMomentumPlot3}}
\vspace{-0.2cm}
\caption{Classification efficiency for pions, kaons and protons, respectively shown from left to right obtained with different imputation methods
for three momentum bins: (0.25-0.3) GeV/c (top panels), (0.55-0.60) GeV/c (middle panels) and (0.85-0.90) GeV/c (bottom panels). Multiple imputation (MI) results
are shown with blue triangle markers, mean imputation with black markers, maximum likelihood with multivariate normal model (ML-MN) with green dots
and maximum likelihood with multivariate skew-normal model (ML-MSN) with red squares.}
\label{EfficiencyPlot}
\end{figure}
\begin{figure}[!th]
\centering
\subtable[$p$=(0.25-0.30) GeV/c - pion]{\includegraphics[scale=0.28]{PionPurityVSMissFraction_p0_25-0_3_last.pdf}\label{PurityLowMomentumPlot1}}%
\hspace{-0.2cm}%
\subtable[$p$=(0.25-0.30) GeV/c - kaon]{\includegraphics[scale=0.28]{KaonPurityVSMissFraction_p0_25-0_3_last.pdf}\label{PurityLowMomentumPlot2}}%
\hspace{-0.2cm}%
\subtable[$p$=(0.25-0.30) GeV/c - proton]{\includegraphics[scale=0.28]{ProtonPurityVSMissFraction_p0_25-0_3_last.pdf}\label{PurityLowMomentumPlot3}}%
\\%
\vspace{-0.3cm}
\subtable[$p$=(0.55-0.60) GeV/c - pion]{\includegraphics[scale=0.28]{PionPurityVSMissFraction_p0_55-0_6_last.pdf}\label{PurityMediumMomentumPlot1}}%
\hspace{-0.2cm}%
\subtable[$p$=(0.55-0.60) GeV/c - kaon]{\includegraphics[scale=0.28]{KaonPurityVSMissFraction_p0_55-0_6_last.pdf}\label{PurityMediumMomentumPlot2}}%
\hspace{-0.2cm}%
\subtable[$p$=(0.55-0.60) GeV/c - proton]{\includegraphics[scale=0.28]{ProtonPurityVSMissFraction_p0_55-0_6_last.pdf}\label{PurityMediumMomentumPlot3}}%
\\%
\vspace{-0.3cm}
\subtable[$p$=(0.85-0.90) GeV/c - pion]{\includegraphics[scale=0.28]{PionPurityVSMissFraction_p0_85-0_9_last.pdf}\label{PurityHighMomentumPlot1}}%
\hspace{-0.2cm}%
\subtable[$p$=(0.85-0.90) GeV/c - kaon]{\includegraphics[scale=0.28]{KaonPurityVSMissFraction_p0_85-0_9_last.pdf}\label{PurityHighMomentumPlot2}}%
\hspace{-0.2cm}%
\subtable[$p$=(0.85-0.90) GeV/c - proton]{\includegraphics[scale=0.28]{ProtonPurityVSMissFraction_p0_85-0_9_last.pdf}\label{PurityHighMomentumPlot3}}
\vspace{-0.2cm}
\caption{Classification purity for pions, kaons and protons, respectively shown from left to right obtained with different imputation methods
for three momentum bins: (0.25-0.3) GeV/c (top panels), (0.55-0.60) GeV/c (middle panels) and (0.85-0.90) GeV/c (bottom panels).
Multiple imputation (MI) results
are shown with blue triangle markers, mean imputation with black markers, maximum likelihood with multivariate normal model (ML-MN) with green dots
and maximum likelihood with multivariate skew-normal model (ML-MSN) with red squares.}
\label{PurityPlot}
\end{figure}
\section{Results}\label{ResultsSection}
In Fig. \ref{SampleImputationPlot} we report the scatter plot of the normalized energy losses obtained in the first two detector layers
for a sample test data set relative to a low particle momentum bin, namely (0.25-0.3 GeV/c). As specified before, instead of plotting the energy loss,
the normalized (to that of kaons) logarithmic energy loss was reported. The full black dots indicate the complete (simulated) data, while the empty
dots represent the imputed data obtained with the ML method for MN (Fig. \ref{SampleImputationPlot1}) and MSN (Fig. \ref{SampleImputationPlot2}) mixtures.
The colored areas represent the contour levels of the MN and MSN bivariate model fitted to the data. Due to its asymmetry, the MSN model slightly
better reproduces the data outside the region with the highest density with respect to the Gaussian model.
We performed the reconstruction analysis on the entire set of simulated data, assuming the energy losses from all detectors
as relevant observables
for the analysis. Fig. \ref{NNOutputPlot} shows the distributions of the neural network output for three different momentum bins, namely
(0.25-0.30 GeV/c), where a good separation between the different species may be expected on the basis of their energy losses,
and (0.55-0.60), (0.85-0.90) GeV/c, where a partial
overlapping of the three species is expected, with a larger contamination (i.e. smaller purity) in each sample.
Proper cuts on such distributions have been applied and optimized for each momentum interval to classify the three species.
An overall comparison among the various imputation methods may be carried out observing the differences between true (simulated) and
imputed data. Fig. \ref{Differences} shows the overall average quadratic differences (all the particle species and all six variables included)
as a function of the missing fraction of data and for three
particle momentum bins. The differences originating from the four
imputation methods are shown with different symbols. As it is clearly seen, at low momenta (where the three species are well separated)
the largest differences are observed with the mean imputation method,
whereas the multiple imputation (MI) gives smaller values and the two ML methods give even smaller (and comparable) differences.
The similarity between the normal and skew-normal ML imputation results is likely a consequence of the transformation of the original
variables, which results in a more symmetric distribution with respect to the original Landau distribution.
In the results shown above, the mean is largely dominated by the most abundant species, in this case the pions, which explains the corresponding
large differences observed in the small momentum bin, where the values of the energy losses for the three species at a given momentum are
sensibly different.
In the intermediate momentum bin, the
trend is still visible, even if the effect is reduced. Finally, at higher momenta, all the methods give comparable results in terms of
differences between missing values and imputed values.
The results reported in Fig.\ref{Differences} are an average over the various species.
However, the effect of the various imputation methods may be expected to be strongly different when a specific species is selected,
due to their relative abundances. When missing data are replaced by
data imputed by the various methods to build the neural network, this results in a different efficiency and purity of the different species.
In Fig. \ref{EfficiencyPlot} we report the average classification efficiencies for the three species (left panels: pions,
middle panels: kaons, right panels: protons),
achieved by imputing the missing
values according to the different imputation methods,
in three different momentum bins: (0.25-0.30) GeV/c (top panels), (0.55-0.60) GeV/c (middle panels) and (0.85-0.90) GeV/c (bottom panels).
Multiple imputation results are shown in blue triangles,
while simple mean imputation in black triangles. Maximum likelihood results are shown with red squares for the skew-normal model and with green dots for
the normal model.
As expected, the simple mean imputation provides a good identification only for the most abundant species, the pions,
whereas for kaons and protons, the classification
capabilities are significantly deteriorated as the fraction of missing data increases, especially in the low momentum bin.
The other imputation methods provides compatible results for low particle
momenta, where the three species are perfectly separated. No significant loss of efficiency is observed even for a large fraction of missing data.
At intermediate and higher momenta the various imputation algorithms provide comparable results in case of the most abundant species (pions), while the
multivariate approach, either in the normal model (ML-MN) or in the skew-normal model (ML-MSN) provide the best results - by a large factor -
in case of protons, which are the less abundant species.
In the high momentum bin, where an overlap occurs in the NN output between the various species, the kaon efficiency may exhibit a
different behaviour, with an increasing efficiency as a function of the missing data fraction, even with the mean imputation method.
This is explained as due to the increasing number of kaons passing the cut which discriminates kaons from pions. An increasing deterioration of the kaon
purity is also observed, as shown in Fig. \ref{PurityPlot}, which reports the corresponding results in terms of classification purity
for the three species and the same momentum bins.
\section{Summary}\label{SummarySection}
We have quantitatively discussed different imputation methods to handle missing values in a data set where these values had to be used as input
neurons in an artificial neural network. Apart from the simple imputation method making use of the mean value, the Multiple Imputation
and the Maximum Likelihood Imputation, either as Normal or Skew-Normal models, were considered and tested against a simple test case, dealing
with the identification of different particle species through a multi-layer telescope detector in different momentum bins,
corresponding to various degree of difficulty in the separation between the different species.
While it is easy to understand that missing values cannot be recovered by the simple mean imputation method, especially for the species which
are less abundant, since the imputed values are strongly influenced by the most abundant one, the detailed application of more sophisticated
imputation methods is not trivial, and requires different aspects to be taken into account. Methods based on the ML are generally believed to
provide more realistic results over the simple mean imputation or even the Multiple Imputation method. However, depending on the specific
application where the missing values need to be replaced by a realistic guess, different choices are possible, and in general a careful study
needs to be carried out before trusting the final results.
The test case considered in the present paper, more than providing a unique solution to this complex problem, was rather intended as a possibility
to discuss and implement the basic steps which need to be addressed in most cases. The capability of any imputation method had to be checked
against simulated data first of all looking at the average differences between real and imputed data. However, dealing with the problem of
recognizing different species in an artificial neural network under various scenarios of increasing difficulty, it became clear that the
capabilities of the various methods are to be checked in each specific condition and in general the same method may behave slightly differently
when applied to different circumstances. Only after taking into account such aspects, may a specific imputation method be considered as a
reliable way to impute missing values in a data set.
Concerning the application of the ML method to problems which exhibit intrinsically asymmetric distributions, as it is for example the Landau
distribution describing the specific energy loss of charged particles in thin layers, the ML with a skew-normal model should be derived for the
particular case of interest. This was done in this paper, and the detailed results are shown as an example in \ref{SkewNormalAlgorithmDerivation}.
However, the possibility to proceed to a suitable transformation of variables in order to make the distributions of interest more similar to a normal
distribution, has also to be considered. It was shown in this paper that in such a case, the performance of the ML normal and ML skew normal
should be expected to be similar, as it was observed in our case.
Many of the considerations discussed in the present paper may also be applied to other physical problems of interest in which
event statistics is a major concern. One of this is the mass identification of high energy primary cosmic rays in a hybrid detector,
where missing values in a data set may arise from MCAR (i.e. inactive/unavailable areas for observables measured with surface stations),
MAR (i.e. selection cuts on atmospheric parameters for observables recorded with fluorescence detectors)
and MNAR (e.g. quality and fiducial cuts on same composition observables or on different shower variables directly correlated with them) mechanisms.
\section{Acknowledgements}\label{AcknowledgementsSection}
We warmly thank Prof. S.Ingrassia for useful discussions and previous collaboration works on the use of statistical methods applied to
real data sets.
|
1,116,691,497,722 | arxiv | \section{Introduction}
GRBs, amongst the farthest and the most powerful objects ever observed in the Universe, are still a mystery after
50 years from their
discovery time by the Vela Satellites \citep{klebesadel73}.
Phenomenologically, GRBs are traditionally classified in short SGRBs ($T_{90}<2$s) and long LGRBs
($T_{90}>2$s) \citep{mazets81,kouveliotou93}, depending on their duration, where $T_{90}$ is the time
in which the 90\% (between 5\% and 95\%) of radiation is emitted in the prompt emission.
However, \cite{norris2006} discovered the existence
of an intermediate class (IC), or SGRBs with Extended Emission
(SGRBsEE), that shows mixed properties between SGRBs and LGRBs. Another relevant classification related to the spectral
features distinguishing normal GRBs from X-ray Flashes (XRFs) appears.
The XRFs \citep{Heise2001,Kippen2001} are extra-galactic transient X-ray sources with spatial distribution,
spectral and temporal characteristics similar to LGRBs. The remarkable property that distinguishes XRFs from GRBs is
that their $\nu F_{\nu}$ prompt emission spectrum peaks at energies
typically one order of magnitude lower than the
observed peak energies of GRBs. XRFs are empirically defined by a greater fluence (time-integrated flux) in the X-ray
band
($2-30$ keV) than in the gamma-ray band ($30-400$ keV). This classification is also relevant for
the investigation
of GRB relations since some of them become stronger or weaker by introducing different GRB categories, see sec.
\ref{Dainotti}.\\
One of the historical models used to explain the GRB phenomenon is the ``fireball" model \citep{wijers97,meszaros1998,meszaros2006} in
which a compact
central engine (either the collapsed core of a massive star or the merger product of a neutron star binary) launches a highly
relativistic, and jetted electron/positron/baryon plasma. Interactions of blobs within the jet are believed to produce the prompt emission,
which consists of high photon energies such as gamma rays and hard X-rays. Instead, the interaction of the jet with the ambient
material causes the afterglow phase, namely a long lasting multi-wavelength emission (X-ray, optical and sometimes
also radio), which follows the prompt.
However, problems in explaining the light curves within this model have been shown by \cite{Willingale2007}, hereafter W07.
More specifically, for $\sim 50\%$ of GRBs, the observed afterglow is in agreement with the model, but for the rest, the temporal and
spectral indices do not conform and are suggestive of continued late energy injection. The difficulty of the standard fireball models
appeared when Swift\footnote{The Swift satellite was launched in 2004.
With the instruments on board, the Burst Alert Telescope (BAT, divided in four standard channels 15-25; 25-50; 50-100; 100-150 keV),
the X-Ray Telescope (XRT, 0.3-10 keV), and
the Ultra-Violet/Optical Telescope (UVOT, 170-650 nm), Swift provides a rapid follow-up of the afterglows in several
wavelengths with better coverage than previous missions.} observations had revealed a more complex behaviour of the light curves
\citep{Obrien06,sakamoto07,zhang07c} than in the past and pointed out that GRBs often follow
``canonical" light curves \citep{Nousek2006}. In fact, the light curves can be divided into two, three and even more
segments. The second segment, when it is flat, is called plateau emission.
X-ray plateaus can be interpreted as occurring due to
an accreting black hole (BH) \citep{Cannizzo2009,cannizzo2011,Kumar2008}
or a top-heavy jet evolution \citep{duffell15}. In addition, the fact that a
newly born magnetar could be formed either via the collapse
of a massive star or during the merger of two neutron stars
motivated the interpretation of the X-ray plateaus as
resulting from the delayed injection of rotational energy
($\dot{E}_{rot}\sim 10^{50}-10^{51}$ erg s$^{-1}$) from a fast spinning
magnetar \citep{usov92,zhang2001,dallosso2011,metzger11,rowlinson12,rowlinson14,rea15}.
These models are summarized in sec. \ref{Dainotti2008interpretation}.
Therefore, in this context, the discovery of relations amongst relevant
physical parameters between prompt and plateau phases is very important so as to use them
as possible model discriminators.
In fact, many theoretical models have
been presented in the literature to explain the wide variety of observations, but each
model has some advantages and drawbacks. The use of the phenomenological relations corrected for selection biases
can boost the understanding of the mechanism responsible for such emissions.
Moreover, being observed at much larger redshift range than the SNe, it has
long been tempting to consider GRBs as useful cosmological probes, extending the redshift range by almost
an order of a magnitude further than the available SNe Ia, observed up to $z=2.26$ \citep{rodney15}.
Indeed, GRBs are observed up to redshift $z=9.4$ \citep{cucchiara11},
which is much more distant than SNe Ia, and, therefore, they can help to understand the nature of the dark energy (DE),
which is the main goal of modern cosmology, and determine the evolution of the equation of state (EoS), $w$, at very
high $z$.
So far, the most robust standard candles are the SNe Ia which, by being excellent distance indicators, provide a unique probe for measuring
the expansion history of the Universe whose discovery has been awarded the Nobel Prize in 2011 \citep{riess98,perlmutter98}.
Up-to-date, $w$ has been measured to be \textcolor{red}{$-1$} within $5\%$ of the Einstein's cosmological constant, $\Omega_{\Lambda}$, the pure vacuum
energy. Measurement of the Hubble constant, $H_0$, provides another constraint on $w$ when combined
with Cosmic Microwave Background Radiation (CMBR) and Baryon Acoustic Oscillation (BAO) measurements \citep{weinberg2013}.
Therefore, the use of other estimates provided by GRBs would be helpful to confirm further and/or constrain the ranges of values of
$H_0$.
However, different from the SNe Ia, which originate from white dwarves reaching
the Chandrasekhar limit and always releasing the same amount of energy,
GRBs cannot yet be considered standard candles with their isotropic energies spanning over $8$ orders of magnitude. Therefore, finding out
universal relations among observable properties can help to standardize their energetics and/or luminosities.
It is for this reason that the study of GRB relations is relevant for both understanding the GRB emission mechanism, for finding a good distance
indicator and for estimating the cosmological parameters at high $z$.\\
Until now, for cosmological purposes, the most used relations are the prompt emission relations:
Amati \citep{AmatiEtal02} and Ghirlanda relations \citep{Ghirlanda2004}. The scatter of these relations is
significantly reduced providing constraints on the cosmological parameters, see \cite{ghirlanda06} and
\cite{ghirlanda09b} for details.
By adopting a maximum likelihood approach which allows for correct quantification of
the extrinsic scatter of the relation, \cite{Amati2008} constrained the matter density
$\Omega_M$ (for a flat Universe)
to 0.04-0.40 (68\% confidence level, CL), with a best-fit value of
$\Omega_M \sim 0.15$, and exclude $\Omega_M=1$ at $> 99.9$\% CL. Releasing the assumption
of a flat Universe, they found evidence for a low value of $\Omega_M$ (0.04-0.50
at 68\% CL) as well as a weak dependence of the dispersion of the relation between the prompt peak energy in
the $\nu F_\nu$ spectrum and the total gamma isotropic energy, $\log E_{\gamma,peak}-\log E_{\gamma,iso}$,
on $\Omega_{\Lambda}$ (with an upper limit of $\Omega_{\Lambda} \sim 1.15$ at 90\% CL). This
approach makes no assumptions about the $\log E_{\gamma,peak}-\log E_{\gamma,iso}$ relation and it does not use
other calibrators to set the normalization of the relation. Therefore, the treatment of the
data is not affected by the so-called circularity problem (to calibrate the GRB luminosity relations
for constraining cosmological models a particular cosmological model
has to be assumed a priori) and the results are independent of those derived
via SNe Ia (or other cosmological probes).
Nowadays, the values of the cosmological parameters confirmed by measurements from the Planck Collaboration
for the $\Lambda$CDM model are $\Omega_M=0.3089\pm0.0062$, $\Omega_{\Lambda}=0.6911\pm 0.0062$, and
$H_0=67.74\pm0.46$ Km s$^{-1}$ Mpc$^{-1}$.
For the investigation of the properties of DE, \cite{amati13} showed the 68\% CL
contours in the $\Omega_M-\Omega_{\Lambda}$ plane obtained by assuming a sample of 250 GRBs
expected shortly compared to those from other cosmological probes such as SNe Ia, CMB and Galaxy Clusters.\\
They obtained the simulated data sets via Monte Carlo techniques
by taking into account the slope, normalization, and dispersion of the observed
$\log E_{\gamma,peak}-\log E_{\gamma,iso}$ relation,
the observed $z$ distribution of GRBs and the distribution
of the uncertainties in the measured values of $\log E_{\gamma,peak}$ and $\log E_{\gamma,iso}$.
These simulations indicated that with a sample of 250 GRBs, the
accuracy in measuring $\Omega_M$ would be comparable to that currently provided by SNe
data. In addition, they reported the estimates of $\Omega_M$ and the parameter of the DE EoS, $w_0$, derived from
the present and expected future samples. They assumed that the $\log E_{\gamma,peak}-\log E_{\gamma,iso}$
relation
is calibrated with a 10\% accuracy by using, e.g., the luminosity distances provided
by SNe Ia and the self-calibration of the relation with a large enough number of GRBs
lying within a narrow range of z ($\Delta z \sim 0.1-0.2$).
Generally speaking, as the number of GRBs in each redshift bin increases,
also the feasibility and accuracy of the self-calibration of GRB relations will improve.
\textcolor{red}{For a review on GRB prompt relations, see \cite{dainotti2016b}}.\\
\textcolor{red}{Even though the errors on $\Omega_M$ obtained in \cite{amati13} may lead to GRBs as promising
standard candles, because they are
almost comparable with SNe (0.06 for GRBs versus 0.04 for SNe, as provided for the SNe sample by \citealt{betoule14} and \citealt{calcino17}), these
results show that $\Omega_M$ has an error which is 20 times larger then the value obtained by Planck. Thus, GRBs in a
near future can be comparable with SNe Ia, but not likely with Planck. On the other hand, there is discrepancy among
the values of $H_0$ computed by CMB and
SNe \citep{Planck2015} and thus adding a new effective cosmological probe as GRBs can help to cast light on
this discrepancy and break the degeneracy among several cosmological parameters.}\\
It is clear from this context that selection biases play a major and crucial role even for the close-by probes such
as SNe Ia in determining the correct cosmological parameters.
This problem is more relevant for GRBs, which are particularly affected by the Malmquist bias effect
(Malmquist 1920, Eddington 1940) that favours the brightest objects against faint ones at large distances.
Therefore, it is necessary to investigate carefully the problem of selection effects and how to overcome them before
using GRB relations as distance estimators, as cosmological probes, and as model discriminators. This is indeed the
major aim of this review.
Besides, this work is useful, especially for those embarking on
the study of GRB relations, because it aims at constituting a brief, but a complete compendium of afterglow and
prompt-afterglow relations.\\
The review is organized as follows: in section \ref{notations}, we explain the nomenclature and definitions in all
review, in sections \ref{Afterglow correlations} and \ref{promptaftcor}, we analyze the relations between the afterglow
parameters and between parameters of both the prompt and afterglow phases. In section \ref{Selection effects}, we
describe how these relations can be affected by selection biases. In section \ref{redshiftestimator}, we present how to obtain a
redshift estimator and in section \ref{cosmology}, we report the use of the Dainotti relation as an example of GRB application as a cosmological tool.
Finally, \textcolor{red}{in section \ref{discussion}, we briefly summarize some findings about the physical models and the
cosmological usage of the analyzed relations, while in the last section} we draw our conclusions.
\section{Notations}\label{notations}
For clarity, we report a summary of the nomenclature adopted in the review.
\begin{itemize}
\item $L$, $E$, $F$, $S$, and $T$ indicate the luminosity, the energy, the flux, the fluence and the time
which can be observed in several
wavelengths, denoted with the first subscript, and at different times or part of the light curve,
denoted instead with the second subscript. In addition, with
$\alpha$, $\beta$ and $\nu$, we represent the temporal and spectral decay indices and the frequencies.\\
More specifically:
\item $T_{X,a}$ and $T_{O,a}$ denote the time in the X-ray at the end of the plateau and the same time, but in the optical
wavelength respectively. $F_{X,a}$ are $F_{O,a}$ are their respective fluxes, while $L_{X,a}$ and $L_{O,a}$ are their
respective luminosities. An approximation of the energy of the plateau is $E_{X,plateau}=(L_{X,a}\times T^*_{X,a})$,
see the left panel of Fig. \ref{fig:notation}.
\item $T_{O,peak}$ and $T_{X,f}$ are the peak time in the optical and the
time since ejection of the pulse.
$L_{O,peak}$ and $L_{X,f}$ are their respective luminosities. $F_{O,peak}$ is the respective flux of $T_{O,peak}$.
\item $T_{X,peak}$ is the peak time in the X-ray and $F_{X,peak}$ and $L_{X,peak}$ are its flux and luminosity
respectively.
\item $T_{X,p}$ and $T_{X,t}$ are the time at the end of the prompt emission
within the W07 model and the time at which the flat and the step decay behaviours of
the light curves join respectively.
\item $T_{90}$ and $T_{45}$ are the times in which the
90\% (between 5\% and 95\%) and 45\% (between 5\%-50\%) of radiation is emitted in the prompt emission respectively.
\item $\tau_{lag}$ and $\tau_{RT}$ are the differences in arrival time to the observer of the high
energy photons and low energy photons and the shortest time over which the light curve increases by the $50\%$ of the
peak flux of the pulse.
\item $L_{X,200\rm{s}}$, $L_{X,10}$, $L_{X,11}$, $L_{X,12}$, $L_{X,1\rm{d}}$ and $L_{O,200\rm{s}}$, $L_{O,10}$, $L_{O,11}$,
$L_{O,12}$, $L_{O,1\rm{d}}$
are the X-ray and optical luminosities at 200 s, at 10, 11, 12 hours and at 1 day respectively;
$L_{O,100s}$, $L_{O,1000s}$, $L_{O,10000s}$, $L_{O,7}$ are the optical luminosity at 100 s, 1000 s,
10000 s and 7 hours; $L_{\gamma,iso}$ and $L_L(\nu,T_{X,a})$ are the isotropic prompt emission mean
luminosity and the optical
or X-ray luminosity of the late prompt emission at the time $T_{X,a}$.
\item $F_{X,11}$, $F_{X,1\rm{d}}$ and $F_{O,11}$, $F_{O,1\rm{d}}$ are the X-ray and optical fluxes at 11 hours and at 1 day respectively;
$F_{\gamma,prompt}$, $F_{X,afterglow}$ are the gamma-ray flux in the prompt and the X-ray flux in the afterglow
respectively.
$E_{\gamma,prompt}$ and $E_{X,afterglow}$ are their respective isotropic energies and $L_{\gamma,prompt}$ and $L_{X,afterglow}$
are the respective luminosities.
$S_{\gamma,prompt}$ indicates
the prompt fluence in the gamma band correspondent to the rest frame isotropic prompt energy $E_{\gamma,prompt}$.
\item $E_{O,afterglow}$, $E_{\gamma,iso}$ and $E_{X,f}$ are the optical isotropic energy in the afterglow phase,
the total gamma isotropic energy and the prompt emission energy of the pulse.
\item $E_{k,aft}$, $E_{\gamma,peak}$ and $E_{\gamma,cor}$ are the isotropic kinetic afterglow energy in X-ray,
the prompt peak energy in
the $\nu F_\nu$ spectrum and the isotropic energy corrected for the beaming factor.
\item $\alpha_{X,a}$, $\alpha_{O,>200\rm{s}}$, $\alpha_{X,>200\rm{s}}$,
$\alpha_{\nu,fl}$ and $\alpha_{\nu,st}$ are the
X-ray temporal decay index in the afterglow phase, in the optical after $200$ s,
in the X-ray after $200$ s and the optical or X-ray flat and steep temporal decay indices respectively.
\item $\beta_{X,a}$, $\beta_{OX,a}$ and $\beta_{O,>200\rm{s}}$ are the spectral index of the
plateau emission in X-ray, the optical-to-X-ray spectral index for
the end time of the plateau and the optical spectral index after $200$ s.
\item $\nu_X$, $\nu_O$, $\nu_c$, $\nu_m$ are the X-ray and optical frequencies, and the cooling and the peak frequencies
of the synchrotron radiation.
\end{itemize}
All the time quantities described above are given in the observer frame, while with the upper index $*$
we denote in the text the observables in the GRB rest frame. The rest frame times are the observed times
divided by the cosmic time expansion, for example, $T^*_{X,a}=T_{X,a}/(1+z)$ denotes the rest frame time
at the end of the plateau emission.
\pagebreak
\begin{figure}[htbp]
\centering
\includegraphics[width=0.37\hsize,angle=0]{will.pdf}
\includegraphics[width=0.37\hsize,angle=0]{notation.pdf}
\caption{\footnotesize Left panel: the functional form of the fitting model from \cite{Willingale2007}. Right panel:
the observed light curve for GRB 061121 with the best-fit W07
model superimposed from \cite{dainotti16a}. The red dot marks the end of the flat plateau phase in the
X-ray afterglow ($T_{X,a}$, $F_{X,a}$). A similar configuration appears in the optical range.}
\label{fig:notation}
\end{figure}
In the following table we will give a list of the abbreviations/acronyms used through the text:
\begin{table}[htbp]
\footnotesize
\begin{center}{
\begin{tabular}{|c|c|}
\hline
Abbreviation& Meaning\\
\hline
DE& Dark Energy\\
EoS& Equation of State\\
CL& Confidence Level\\
IC& Intermediate Class GRB\\
SGRB& Short GRB\\
LGRB& Long GRBs\\
SGRBsEE& Short GRBs with extended emission\\
XRFs& X-ray Flashes\\
SNe&Supernovae\\
BH& Black Hole\\
z& redshift\\
FS& Forward Shock\\
RS& Reverse Shock\\
$H_0$&Hubble constant\\
$\Omega_M$& Matter density in $\Lambda$CDM model\\
$\Omega_{\Lambda}$ & Dark Energy density in $\Lambda$CDM model\\
$\Omega_k$ & curvature in $\Lambda$CDM model\\
$\sigma_{\log L_{X,a}}$ & error on the luminosity\\
$\sigma_{\log T^{*}_{X,a}}$& error on the time\\
E4& sample with $\sigma_E=(\sigma_{\log L_{X,a}}^2+\sigma_{\log T^{*}_{X,a}}^2)^{1/2}<4$\\
E0095& sample with $\sigma_E=(\sigma_{\log L_{X,a}}^2+\sigma_{\log T^{*}_{X,a}}^2)^{1/2}<0.095$ \\
W07& Willingale et al. (2007)\\
$\Gamma$& Lorentz Factor\\
$V$&Variability of the GRB light curve\\
$h$& Hubble constant divided by 100\\
$w_0$, $w_a$& coefficients of the DE EoS $w(z)=w_0+w_a z(1+z)^{-1}$ \\
HD & Hubble Diagram\\
a & normalization of the relation\\
b & slope of the relation\\
$\sigma_{int}$ & intrinsic scatter of the relation\\
$b_{int}$ & intrinsic slope of the relation\\
\hline
\end{tabular}}
\caption{\footnotesize Table with abbreviations.}
\label{abbreviations}
\end{center}
\end{table}
\section{The Afterglow Relations}
\label{Afterglow correlations}
Several relations appeared in literature relating only parameters in the afterglow, such as the $L_X(T_{a})-T^{*}_{X,a}$ relation \citep{Dainotti2008}
and similar ones in the optical and X-ray bands such as the unified $L_X(T_{a})$-$T_{X,a}^*$ and $L_{O,a}$\,-\,$T^*_{O,a}$ \citep{ghisellini09} and the $L_{O,200\rm{s}}$\,-\,$\alpha_{O,>200\rm{s}}$ relations \citep{oates2012}.
\subsection{The Dainotti relation (\texorpdfstring{$L_X(T_{a})$\,-\,$T^{*}_{X,a}$}{Lg})} \label{Dainotti}
The first relation to shed light on the plateau properties has been the $L_X(T_a)$\,-\,$T^{*}_{X,a}$
one, hereafter also referred as LT. The phenomenon is an anti-relation between the X-ray luminosity
at the end of the plateau, $L_X(T_a)$, and the time in the X-ray at the end of the plateau, $T^{*}_{X,a}$,
for simplicity of notation we will refer to $L_X(T_a)$ as $L_{X,a}$.\\
It was discovered by \cite{Dainotti2008} using
33 LGRBs detected by the Swift satellite in the X-ray energy band observed by XRT.
Among the 107 GRBs fitted by W07 phenomenological model, shown in the left panel of Fig. \ref{fig:notation}, only the GRBs that
have a good spectral
fitting of the plateau and firm determination of $z$ have been chosen. The functional form of the LT
relation obtained is the following:
\begin{equation}
\log L_{X,a} = a + b \times \log T^{*}_{X,a},
\end{equation}
with a normalization $a=48.54$, a slope $b=-0.74 ^{+0.20}_{-0.19}$, an intrinsic scatter, $\sigma_{int}=0.43$ and a
Spearman correlation coefficient\footnote{A computation of statistical dependence
between two variables stating how good the relation
between these variables can be represented employing a monotonic function. It assumes a value between $-1$ and $+1$.}
$\rho=-0.74$.
$L_{X,a}$ in the Swift XRT passband, $(E_{min}, E_{max})=(0.3,10)$ keV, has been
computed from the following equation:
\begin{equation}
L_{X,a} (z)= 4 \pi D_L^2(z, \Omega_M, h) \, F_{X,a} \times K
\label{eq: lx}
\end{equation}
where $D_L(z, \Omega_M, h)$ represents the GRB luminosity distance for a given $z$, $F_{X,a}$ indicates the flux
in the X-ray at the end of the plateau,
and $K=\frac{1}{(1+z)^{(1-\beta_{X,a})}}$ denotes the K-correction for cosmic expansion \citep{Bloom2001}.
This anti-relation shows that the shorter the plateau duration, the more luminous the plateau.
Since the ratio between the errors on both variables is close to unity, it means that both errors need to be considered
and the Marquardt Levenberg algorithm is not the best fitting method to be applied in this circumstance.
Therefore, a Bayesian
approach \citep{Dagostini2005} needs to be considered. This method takes into account the errors of both variables and an intrinsic scatter, $\sigma_{int}$, of
unknown nature. However, the results of both the D'Agostini method and the Marquardt Levenberg algorithm
are comparable. Due to the higher accuracy of the first method from now on the authors prefer this technique in their papers.
Evidently, the tighter the relation, the better the chances to constrain
the cosmological parameters. With this specific challenge in mind,
a subsample of bursts has been chosen with particular selection criteria both
on luminosity and time, namely $\log L_{X,a} > 45$ and $1 \le \log T_{X,a}^{*} \le 5$.
After this selection has been applied, a subsample of 28 LGRBs was obtained with
$(a, b, \sigma_{int})=(48.09, -0.58 \pm 0.18, 0.33)$, thus reducing considerably the scatter.
\begin{figure}[htbp]
\centering
\includegraphics[width=8.1cm,height=7.4cm,angle=0]{upperlimitcopy.pdf}
\includegraphics[width=8.1cm,height=7.3cm,angle=0]{GRB-SNE-LONG.pdf}
\caption{\footnotesize Left panel: $\log L_{X,a}$ (equivalent to $\log L_X^*$ in this plot) vs. $\log T^{*}_{X,a}$ for $62$
long afterglows with the error energy parameter $\sigma_E < 4$, and the best
fitted relation line in black, from \cite{Dainotti2010}. The red line fitted to the 8 lowest error
(red) points produces an upper envelope of the full data set.
The upper envelope points with the best fitted line are separately presented in an inset panel.
\textcolor{red}{Right panel: LONG-NO-SNe 128 GRBs (blue points fitted with
a solid blue line) and the 19 events from LONG-SNe (red
empty triangles) fitted with a red dashed line from \cite{dainotti16c}}}
\label{fig:Dainotti2010}
\end{figure}
In agreement with these results, through the analysis of the late prompt phase in optical and X-ray light curves
of 33 LGRBs,
also \cite{ghisellini09} found a common observational model for optical and X-ray light curves
with the same value for the slope, $b=-0.58^{+0.18}_{-0.18}$, obtained by \cite{Dainotti2008}
when the time is limited between $1 \le \log T_{X,a}^{*} \le 5$.\\
Instead, \cite{Dainotti2010} from a sample of 62 LGRBs found $b=-1.06^{+0.27}_{-0.28}$, while for
the 8 IC GRBs pointed out a much steeper relation ($b=-1.72^{+0.22}_{-0.21}$). Finally, taking into
account the errors on luminosity ($\sigma_{\log L_{X,a}}$) and time ($\sigma_{\log T^{*}_{X,a}}$), the
8 GRBs with the smallest errors were defined as the ones with $\sigma_E=(\sigma_{\log L_{X,a}}^2+\sigma_{\log T^{*}_{X,a}}^2)^{1/2}<0.095$.
For this subsample, \cite{Dainotti2010} found a slope $-1.05^{+0.19}_{-0.20}$, see Fig. \ref{fig:Dainotti2010}, the right panel of Fig.
\ref{fig:Dainotti2010b} and Table \ref{tbl7}.\\
Similar to \cite{Dainotti2010}, also \cite{bernardini2012} and \cite{Sultana2012}, with a sample of 64 and 14 LGRBs
respectively, found a slope $b\approx -1$, for details see Table \ref{tbl7}.\\
Expanding the sample again to 77 LGRBs, \cite{dainotti11a} discovered a relation with $b=-1.20^{+0.27}_{-0.30}$.
Later, \cite{mangano12}, considering in their sample of 50 LGRBs those GRBs with no visible plateau phase and employing
a broken power law as a fitting model, found a steeper slope ($b=-1.38^{+0.16}_{-0.16}$).
Thus, from all these analyses it is clear that a steepening of the slope has been
observed when the sample size is increased.\\
Therefore, before going further with additional analysis,
\cite{Dainotti2013a} decided to show how selection biases can influence the slope of the relation. They showed that
the steepening of the relation results from selection biases, while the intrinsic slope of the relation is
$b=-1.07^{+0.09}_{-0.14}$, see section \ref{Selection effects}. Summarizing, \cite{Dainotti2013a} with a sample of
101 GRBs, confirmed the previous results from \cite{Dainotti2010}, as well as \cite{rowlinson14}, with a data set
of 159 GRBs.
\begin{figure}[htbp]
\centering
\includegraphics[width=7cm,height=6cm,angle=0]{f7arowlinson13copy.pdf}
\includegraphics[width=8.2cm,height=6.2cm,angle=0]{lumflux.pdf}
\caption{\footnotesize Left panel:
the plateau flux versus the plateau duration for a sample of 22 SGRBs from \cite{rowlinson2013}. Blue stars are GRBs with two or more
breaks in their light curves, while green circles have one break.
Right panel:
$\log L_{X,a}$ versus $\log F_{X,a}$ for the full GRB sample from \cite{Dainotti2010}. The 8 upper envelope
points are shown as red squares, while the IC GRBs are represented by green triangles.}
\label{fig:Dainotti2010b}
\end{figure}
\cite{dainotti15} also confirmed previous results of \cite{Dainotti2013a} but with a larger sample of 123 LGRBs.
All the samples discussed are observed by SWIFT/XRT.
\begin{table}[htbp]
\footnotesize
\begin{center}{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Author & N& Type& Slope& Norm & Corr.coeff. & P \\
\hline
Dainotti et al. (2008)&28 & $1<T^{*}_{X,a}<5$&$-0.58^{+0.18}_{-0.18}$&48.09& -0.80&$1.6\times10^{-7}$\\
Dainotti et al. (2008)&33 &All GRBs&$-0.74^{+0.20}_{-0.19}$&48.54& -0.74&$10^{-9}$\\
Cardone et al. (2009)&28&L&$-0.58^{+0.18}_{-0.18}$& 48.09&-0.74&$10^{-9}$\\
Ghisellini et al. (2009)&33&L&$-0.58^{+0.18}_{-0.18}$&48.09& -0.74&$10^{-9}$\\
Cardone et al. (2010)&66&L&$-1.04^{+0.23}_{-0.22}$&$50.22^{+0.77}_{-0.76}$&-0.68&$7.6\times10^{-9}$\\
Dainotti et al. (2010) & 62 &L&$-1.06^{+0.27}_{-0.28}$&$51.06^{+1.02}_{-1.02}$&-0.76&$1.85\times10^{-11}$\\
Dainotti et al. (2010) & 8 &high luminosity&$-1.05^{+0.19}_{-0.20}$&$51.39^{+0.90}_{-0.90}$&-0.93&$1.7\times10^{-2}$\\
Dainotti et al. (2010) & 8 & IC&$-1.72^{+0.22}_{-0.21}$&$52.57^{+1.04}_{-1.04}$&-0.66&$7.4\times10^{-2}$\\
Dainotti et al. (2011a) &77 &L &$-1.20^{+0.27}_{-0.30}$&$51.04^{+0.27}_{-0.30}$&-0.69& $7.7\times10^{-8}$\\
Sultana et al. (2012)&14&L&$-1.10^{+0.03}_{-0.03}$&$51.57^{+0.10}_{-0.10}$&-0.88&$10^{-5}$\\
Bernardini et al. (2012)& 64 & L&$-1.06^{+0.06}_{-0.06}$&51.06&-0.68&$7.6\times10^{-9}$\\
Mangano et al. (2012) & 50 & L& $-1.38^{+0.16}_{-0.16}$&$52.2^{+0.06}_{-0.06}$&-0.81&$2.4\times10^{-10}$\\
Dainotti et al. (2013a)& 101&ALL intrinsic &$-1.07^{+0.09}_{-0.14}$&52.94&-0.74&$10^{-18}$ \\
Dainotti et al. (2013b) & 101&All GRBs&$-1.32^{+0.18}_{-0.17}$&$52.8^{+0.9}_{-0.3}$&-0.74&$10^{-18}$ \\
Dainotti et al. (2013b)& 101&without short&$-1.27^{+0.18}_{-0.26}$&52.94&-0.74&$10^{-18}$ \\
Dainotti et al. (2013b) & 101&simulated&$-1.52^{+0.04}_{-0.24}$&$53.27_{-0.48}^{+0.54}$&-0.74&$10^{-18}$ \\
Postnikov et al. (2014)&101&L ($z<1.4$)&$-1.51^{+0.26}_{-0.27}$&$53.27_{-0.48}^{+0.54}$&-0.74&$10^{-18}$\\
Rowlinson et al. (2014)&159 &intrinsic &$-1.07^{+0.09}_{-0.14}$&52.94&-0.74&$10^{-18}$ \\
Rowlinson et al. (2014)&159 &observed &$-1.40^{+0.19}_{-0.19}$&$52.73^{+0.52}_{-0.52}$&-0.74&$10^{-18}$\\
Rowlinson et al. (2014)&159 &simulated&$-1.30^{+0.03}_{-0.03}$&$52.73^{+0.52}_{-0.52}$&-0.74&$10^{-18}$\\
Dainotti et al (2015)&123&L&$-0.90^{+0.19}_{-0.17}$&$51.14^{+0.58}_{-0.58}$&-0.74&$10^{-15}$\\
\textcolor{red}{Dainotti et al. (2016c)}&\textcolor{red}{19}& \textcolor{red}{L-SNe}&\textcolor{red}{$-1.5^{+0.3}_{-0.3}$}&\textcolor{red}{$51.85^{+0.94}_{-0.94}$}&\textcolor{red}{-0.83}&\textcolor{red}{$5\times10^{-6}$}\\
\hline
\end{tabular}}
\caption{\footnotesize Summary of the LT relation. All the measurements are performed by the Swift XRT Telescope.
The first column represents the authors, the second one the number of GRBs in the used sample,
the third one the GRB type (S=Short, L=Long, IC=Intermediate), the fourth and the fifth ones are
the slope and normalization of the relation and the last two columns are the correlation coefficient and the chance
probability, P.}
\label{tbl7}
\end{center}
\end{table}
In the context of reducing the scatter of the LT relation, \cite{Delvecchio16} investigated the temporal decay indices
$\alpha_{X,a}$ after the plateau phase for a sample of $176$ GRBs detected by Swift within two different models:
a simple power law, considering the decaying phase after the plateau phase, and the W07 one. It is pointed out that the results
are independent of the chosen model. It was checked if there are some common characteristics in GRBs phenomena that can
allow them to be used as standardizable candles like SNe Ia and to obtain some constraints revealing which is the best physical
interpretation describing the plateau emission.
The interesting result is that the LT relation for the low and high luminosity GRBs seems to depend differently on the $\alpha_{X,a}$
parameter, thus possibly implying a diverse density medium.\\
\textcolor{red}{Continuing the search for a standard set of GRBs, \cite{dainotti16c} analyzed 176 GRB afterglow plateaus observed by
Swift with known redshifts which revealed that the subsample of LGRBs associated with SNe (LONG-SNe) presents a very
high correlation coefficient for the LT relation. They investigated the category of LONG GRBs associated
spectroscopically with SNe in order to compare the LT correlation for this sample with the one for LGRBs for which
no associated SN has been observed (hereafter LONG-NO-SNe, 128 GRBs). They checked if there is a difference among these
subsamples. They adopted first a non-parametric statistical method, the \cite{Efron1992} one, to take into
account redshift evolution and check if and how this effect may steepen the slope for the
LONG-NO-SNe sample. This procedure is necessary due to the fact that this sample is observed at much higher redshift
than the GRB-SNe sample. Therefore, removing selection bias is the first step before any comparison among samples
observed at different redshifts could be properly performed. They have demonstrated that there is no evolution for the
slope of the LONG-NO-SNe sample and no evolution is expected for the LONG-SNe sample. The difference among the slopes
is statistically significant with the probability P=0.005 for LONG-SNe. This possibly suggests that the LONG-SNe with
firm spectroscopic features of the SNe associated might not require a standard energy reservoir in the plateau phase
unlike the LONG-NO-SNe. Therefore, this analysis may open new perspectives in future theoretical investigations of the
GRBs with plateau emission and associated with SNe. They also discuss how much this difference can be due to the jet
opening angle effect. The difference between the SNe-LONG (A+B) and LONG-NO-SNe sample is only statistically significant
at the 10\% level when we consider the beaming correction. Thus, one can argue that the difference in slopes can be
partially due to the effect of the presence of low luminosity GRBs in the LONG-SNe sample that are not corrected for beaming.
However, the beaming corrections are not very accurate due to indeterminate jet opening angles, so the debate remains
open and it can only be resolved when we will gather more data.}\\
In Table \ref{tbl7}, we report a summary of the parameters $a$ and $b$ with $\rho$ and $P$ for the LT relation.
In conclusion, the most reliable parameters for this relation are those from
\cite{Dainotti2013a}, because they have demonstrated that the intrinsic slope not affected by selection biases
is determined to be $-1$ as computed through the Efron and Petrosian (EP) method.
\subsubsection{Physical interpretation of the Dainotti relation \texorpdfstring{($L_X(T_a)$\,-\,$T^{*}_{X,a}$)}{Lg}}\label{Dainotti2008interpretation}
Here, we revise the theoretical interpretation of the LT relation, which is based mainly
on the accretion \citep{Cannizzo2009,cannizzo2011} and the magnetar models \citep{zhang2001,dallosso2011,rowlinson12,rowlinson2013,rowlinson14}.\\
The first one states that an accretion disc is created from the motion of
the material around the GRB progenitor star collapsing towards its progenitor core. After it is compressed by
the gravitational forces, the GRB emission takes place. For LGRBs, the early rate of decline in the initial steep
decay phase of the light curve
may provide information about the radial density distribution within the progenitor \citep{Kumar2008}.\\
\cite{Cannizzo2009} predicted a steeper relation slope (\textcolor{red}{-3/2}) than the observed one
(\textcolor{red}{$\sim -1$}), which on the other
hand is in good agreement with the prior emission model of \cite{yamazaki09}.\\
Later, \cite{cannizzo2011}, using a sample of $62$ LGRBs and few SGRBs simulated the fall-back disks
surrounding the BH. They found that a circularization radius of the mass around the BH with value $10^{10}-10^{11}$ cm can give an
estimate for the plateau
duration of around $10^4$s for LGRBs maintaining the initial fall back mass at $10^{-4}$ solar masses ($M_{\odot}$),
see the left panel
of Fig. \ref{fig:cannizzo11}. For SGRBs the radius is estimated to be $10^8$ cm.
The LT relation provides a lower limit for the accreting mass estimates $\Delta M \approx 10^{-4}$ to $10^{-3} M_{\odot}$\footnote{This value can be
derived considering the total inferred accretion mass $\Delta M/M=\Delta E_X/f^{-1}*\epsilon_{acc}*c^2$ where c
is the light speed, f is the X-ray afterglow beaming factor, $\epsilon_{acc}$ is the efficiency of the accretion onto the BH and $E_X$
is the observed total energy of the plateau + later decay phases (the integral over time between $T_{X,t}$ and the end of afterglow,
see Eq. 2 of W07).}. From their results, it was claimed that the LT relation could be obtained if a typical energy reservoir
in the fall-back mass is assumed, see the right panel of Fig. \ref{fig:cannizzo11}. However, in their analysis the very steep
initial decay following the prompt emission, which have been modelled by \cite{lindner10} as fall-back of the
progenitor core, is not considered.
\begin{figure}[htbp]
\includegraphics[width=8cm,height=6cm,angle=0]{f6cannizzo11.pdf}
\includegraphics[width=8cm,height=6cm,angle=0]{f1cannizzo11.pdf}
\caption{\footnotesize Left panel: model light curves for LGRB parameters from \cite{cannizzo2011}, keeping
the starting fall-back disk mass constant at $10^{-4} M_{\odot}$ but changing the initial radius and normalization.
Right panel: total accretion mass for the plateau $+$ later decay phases
of GRBs from \cite{cannizzo2011}, considering 62 LGRBs from \cite{Dainotti2010}. The region in red
represents a limiting XRT detection flux level $f_{\rm II} = 10^{-12}$ erg cm$^{-1}$ s$^{-1}$
(assuming a plateau duration $t_{\rm II} = 10^4$ s) in order to study a plateau to sufficient accuracy.
A beaming factor $f=1/300$ and a net efficiency for powering the X-ray flux
$\epsilon_{\rm net}=\epsilon_{\rm acc}\epsilon_X=0.03$ were assumed.}
\label{fig:cannizzo11}
\end{figure}
Regarding the magnetar model, \cite {zhang2001} studied the effects of an injecting central
engine on the GRB afterglow radiation, concentrating on a strongly magnetized millisecond pulsar.
For specific starting values of rotation period and magnetic field of the pulsar, the afterglow light curves should
exhibit an achromatic bump lasting from minutes to months, and the observation of such characteristics could
set some limits on the progenitor models.
More recently, \cite{dallosso2011} investigated the energy evolution in a relativistic
shock from a spinning down magnetar in spherical symmetry. With their fit of few observed Swift XRT light curves and
the parameters of this model, namely a spin period of ($1-3$ ms), and high values of magnetic fields ($B\sim 10^{14}-10^{15}$ G),
they managed to well reproduce the properties of the shallow decay phase and the LT relation, see the left panel of
Fig. \ref{fig:rowlinson2014}.\\
Afterward, \cite{bernardini2012} with a sample of 64 LGRBs confirmed, as previously founded by \cite{dallosso2011}, that
the shallow decay phase of the GRB light curves and the LT relation can be well explained.\\
Then, \cite{rowlinson12} and \cite{rowlinson2013} pointed out that energy injection is a fundamental mechanism
for describing the plateau emission of both LGRBs and SGRBs. In fact, the remnant of NS-NS mergers can form a magnetar,
and indeed
the origin of the majority of SGRBs is well explained through the energy injection by a magnetar.
\begin{figure}[htbp]
\includegraphics[width=0.495\hsize,angle=0]{dallosso11a.pdf}
\includegraphics[width=0.495\hsize,angle=0]{param_space.pdf}
\caption{\footnotesize Left panel: five theoretical light curves obtained by \cite{dallosso2011}, changing the initial energy of the afterglow,
the dipole magnetic field, B, and the initial spin
period of the NS, P. Right panel: the grey shaded areas are the homogeneous distribution of B and P employed to simulate the observable magnetar plateaus from \cite{rowlinson14}. The upper and lower limits on B and the upper
limit on P are computed considering the sample of GRBs fitted with the magnetar model
\citep{lyons2010,dallosso2011,bernardini2012,gompertz2013,rowlinson2013,yi2014,lu2014}.
The dashed black vertical line (1) at 0.66 ms is the minimum P allowed. The dotted black line (2) indicates a limit on P and B strengths imposed by the fastest slew time of XRT in their sample in the rest frame of the highest $z$ GRB, as plateaus with durations shorter than the
slew time are unobservable. The black dash-dot lines (3-6) are the observational constraints for the dimmest XRT plateau
observable assuming the lowest $z$ in the GRB sample. These limits vary as a function of the beaming and efficiency of
the magnetar emission: (3) Minimum beaming angle and efficiency (1 degree and 1\% respectively), (4) Minimum efficiency (1\%)
and maximum beaming angle (isotropic), (5) Maximum efficiency (100\%) and minimum beaming angle, (6) Maximum efficiency and beaming
angle. The observed distributions indicate that the samples have low efficiencies and
small beaming angles.}
\label{fig:rowlinson2014}
\end{figure}
Later, \cite{rowlinson14}, using 159 GRBs from Swift catalogue, analytically demonstrated that the central
engine model accounts for the LT relation assuming that the compact object is
injecting energy into the forward shock (FS), a shock driven out
into the surrounding circumstellar medium. The luminosity and plateau duration can be computed
as follows:
\begin{equation}
\log L_{X,a} \sim \log (B^2_{p}P^{-4}_{0}R^6)
\label{eq10}
\end{equation}
and
\begin{equation}
\log T^*_{X,a} = \log (2.05 \times IB^{-2}_{p}P^2_{0}R^{-6}),
\label{eq11}
\end{equation}
where $T^*_{X,a}$ is in units of $10^3$ s, $L_{X,a}$ is in units of $10^{49}$ erg s$^{-1}$, $I$ is the moment of inertia in
units of $10^{45}$ g cm$^2$, $B_{p}$ is the magnetic field strength at the poles
in units of $10^{15}$ G, $R$ is the radius of the NS in units of $10^6$ cm and
$P_{0}$ is the initial period of the compact object in milliseconds.
Then, substituting the radius from eq. \ref{eq11} into eq. \ref{eq10}, it was derived that:
\begin{equation}
\log \ (L_{X,a}) \sim \log \ (10^{52}I^{-1}P^{-2}_{0})-\log \ (T^*_{X,a}).
\end{equation}
Therefore, an intrinsic relation $\log L_{X,a} \sim - \log T_{X,a}^{{*}}$ is confirmed directly from this formulation.
Although some magnetar plateaus are inconsistent with energy injection into the FS,
\cite{rowlinson14} showed that this emission is narrowly beamed
and has $\leq 20$\% efficiency in conversion of rotational energy from the compact object into the
observed plateau luminosity. In addition, the intrinsic LT relation slope, namely the one where
the selection biases are
appropriately removed, is explained within the spin-down of a newly formed magnetar at 1 $\sigma$ level,
see right panel of Fig. \ref{fig:rowlinson2014}.
The scatter in the relation is mainly due to the range of the initial spin periods.\\
After several papers discussing the origin of the LT relation within the context of the magnetar model,
very recently a debate has been opened by \cite{rea15} on the reliability of this model as the correct
interpretation for the X-ray plateaus.
Using GRBs with known $z$ detected by Swift from its launch to August 2014, \cite{rea15} concluded that the initial
magnetic field distribution,
used to interpret the GRB X-ray plateaus within the magnetar model does not match the features
of GRB-magnetars with the Galactic magnetar
population. Therefore, even though there are large uncertainties in these estimates due to GRB rates, metallicity and
star
formation, the GRB-magnetar model in its present
form is safe only if two kinds of magnetar progenitors are allowed. Namely, the GRB should be different from Galactic
magnetar ones
(for example for different metallicities) and should be considered supermagnetars (magnetars with an initial magnetic
field significantly large).
Finally, they set a limit of about $ \leq 16$ on the number of stable magnetars produced in the Milky Way via a GRB in
the past Myr. \textcolor{red}{However, it can be argued that since the rates of Galactic magnetars and GRBs are
really different, the number of Galactic magnetars cannot fully describe the origin of GRBs.
In fact the Galactic magnetar rate is likely to be greater than 10\% than the core collapse SNe rate, while GRB rate is
much lower than that. In addition, the number of magnetars in the Milky Way may not be used as a constraint on the GRB
rate because the spin-down rates of GRB magnetars should be very rapid. Due to the low GRB rate it would not be easy to
detect these supermagnetars.}
Thus, it can be claimed that no conflict stands among this paper and the previous ones.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.495\hsize,angle=0]{f9vaneerten14.pdf}
\includegraphics[width=0.495\hsize,angle=0]{f9bvaneerten14.pdf}
\caption{\footnotesize Optical and X-ray light curves for wind (left panel) and ISM (right plot)
scenario's from \cite{vaneerten14a}.
Thick light grey curves represent the analytical solutions for
prolonged and impulsive energy injection. Thick dashed light
grey and the thick dotted light grey curves indicate the forward shock region emission only and the reverse shock region only respectively. The grey vertical lines show
(1) the arrival time of emission
from the jet back and (2) the arrival time of emission
from the jet front. The solid vertical lines indicate arrival times of emission along the jet axis
for these two events; the dashed vertical lines express the arrival
times of emission from an angle $\theta = 1/\gamma$.}
\label{fig:vaneerten14a}
\end{figure}
Still in the context of the energy injection models, \cite{vaneerten14a} found a relation between the optical flux
at the end of the plateau and the time at the end of the plateau itself $F_{O,a} \sim T_{O,a}^{-0.78 \pm 0.08}$
\citep{Panaitescu2011,Li2012} for which observed frame variables were
considered. The range of all parameters describing the emission
($E_{\gamma,iso}$, the fraction of the magnetic energy, $\epsilon_B$, the initial density, $n_0$) is the principal cause of the scatter in the
relation, but it does not affect the slope. Finally, it was claimed that both the wind ($\propto A/r^2$, where A is a constant)
and the interstellar medium can reproduce the observed relation within both the reverse shock (RS, a shock driven back into the
expanding bubble of the ejecta) and FS scenarios, see Fig. \ref{fig:vaneerten14a}.
\begin{figure}[htbp]
\centering
\includegraphics[width=8cm,height=6cm,angle=0]{f1leventis14.pdf}
\includegraphics[width=8cm,height=6.2cm,angle=0]{f5leventis14.pdf}
\caption{\footnotesize Left panel: optical and X-ray light curves before and after
the injection break from \cite{Leventis2014}. The contributions
of the FS (dotted line) and RS (dashed line) are shown for both. The considered parameters are
$E = 10^{51}$ erg, $n_1=50$ cm$^{-3}$, $\Delta t=5 \times 10^3$ s, $\eta=600$,
q=0, $\epsilon_e=\epsilon_B = 0.1$, p=2.3, $\theta_j=90^o$, $d=10^{28}$ cm and $z=0.56$.
Right panel: index of the $F_{X,a}-T_{X,a}$ relation as a function of the electron distribution index, p, for the FS
and the RS from \cite{Leventis2014}. The lightly shaded region includes values allowed
by the scaling from \cite{Panaitescu2011}, while the darker region
indicates the scaling from \cite{Li2012}. The five dashed lines show
the five possible indices for the FS, while the three solid lines display the three possible
(independent of p) indices for the RS.}
\label{fig:leventis14}
\end{figure}
Considering alternative models explaining the LT relation, \cite{sultana2013} studied the evolution of
the Lorentz gamma factor, $\Gamma=1/ \sqrt{1-v^2/c^2}$ (where $v$ is the relative velocity between the inertial reference
frames and $c$ is the light speed), during the whole duration of the light curves within the context of the Supercritical Pile Model.
This model provides an explanation for both the steep-decline and the plateau or the steep-decline and the power law decay phase
of the GRB afterglow, as observed in a large number of light curves, and for the
LT relation. One of their most important results is that the duration of the
plateau in the evolution of $\Gamma$ becomes shorter with a decreasing value of $M_0c^2$, where $M_0$ is the initial rest mass of
the flow. This occurrence means that the more luminous the plateau, the shorter its duration and the
smaller the $M_0c^2$, namely the energy.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.495\hsize,angle=0]{f1avaneerten14b.pdf}
\includegraphics[width=0.495\hsize,angle=0]{f1cvaneerten14b.pdf}
\caption{\footnotesize Comparison of the slopes for 1000 thin shell data set runs and slopes of the observed LT relation in optical
(horizontal direction) and the LT relation in X-ray (vertical direction) from \cite{vaneerten14b}
for the FS (left panel) and the RS (right panel) cases.
Grey band expresses 1 $\sigma$ errors on the relations, while green dots represent runs consistent at 1 $\sigma$ error bars
for both, orange dots are compatible at 3 $\sigma$, but not at 1 $\sigma$ and red dots pass neither test.
Vertical grey lines show more scattered LT in optical error bars from \cite{Panaitescu2011}.}
\label{fig:vaneerten14b}
\end{figure}
Instead, in the context of the RS and FS emissions, \cite{Leventis2014}, investigating the synchrotron radiation
in the thick shell scenario (i.e. when the RS is relativistic), found that this radiation
is compatible with the presence of the plateau phase, see the left panel of Fig. \ref{fig:leventis14}.
In addition, analyzing the $\log F_{X,a}$\,-\,$\log T_{X,a}$
relation in the framework of this model, they arrived at the conclusion that smooth energy injection
through the RS is favoured respect to the FS, see the right panel of Fig. \ref{fig:leventis14}.\\
\cite{vaneerten14b}, with a simulated sample of GRBs, found out that the observed LT relation
rules out basic thin shell models, but not basic thick ones. In fact, in the thick model, the
plateau phase comes from the late central
source activity or from additional kinetic energy transfer from slower ejecta
which catches up with the blast wave. As a drawback, in this
context, it is difficult to distinguish between FS and RS
emissions, or homogeneous and stellar wind-type environments.\\
In the thin shell case, the plateau phase is given by the pre-deceleration emission from a
slower component in a two-component or jet-type model, but this scenario is not in agreement with the observed LT
relation, see Fig. \ref{fig:vaneerten14b}.
This, however, does not imply that acceptable fits using a thin shell model are not possible, but
further analysis is needed to exclude without any doubts thin shell models.
\textcolor{red}{Another model which has not been tested yet on this correlation is the photospheric emission model
from stratified jets \citep{Ito2014}.}
\pagebreak
\subsection{The unified \texorpdfstring{$L_X(T_a)$-$T_{X,a}^*$ and $L_{O,a}$\,-\,$T^*_{O,a}$}{Lg} relations}\label{ghisellini}
In order to describe the unified picture of the X-ray and optical afterglow, it is necessary to introduce relevant
features regarding optical luminosities. To this end, \cite{boer2000} studied
the afterglow decay index in 8 GRBs in both X-ray and optical wavelengths. In the X-ray, the brightest GRBs had decay
indices around $1.6$ and the dimmest GRBs had decay indices around $1.11$.
Instead, they didn't observe this trend for the optical light curves,
probably due to the host galaxy absorption.\\
Later, \cite{nardini06a} discovered that the
monochromatic optical luminosities at 12 hours, $L_{O,12}$, of 24 LGRBs cluster at
$\log L_{O,12} = 30.65$ erg s$^{-1}$ Hz$^{-1}$,
with $\sigma_{int} = 0.28$. The distribution of $L_{O,12}$ is less scattered
than the one of $L_{X,12}$, the luminosity at 12 hours in the X-ray, and the one of the
ratio $L_{O,12}/E_{\gamma,prompt}$, where $E_{\gamma,prompt}$ is the rest frame isotropic prompt
energy. Three bursts are outliers because they have luminosity which is smaller by a
factor $\sim 15$. This result suggests the existence of a family of intrinsically optically underluminous dark
GRBs, namely GRBs where the optical-to-X-ray
spectral index, $\beta_{OX,a}$, is shallower than the X-ray spectral index minus 0.5, $\beta_{X,a}-0.5$.\\
\cite{Liang2006} confirmed these results.
They found a bimodal distribution of $L_{O,1\rm{d}}$ using 44 GRBs.
\cite{nardini08} also confirmed these findings. They
analyzed selection effects present in their observations extending
the sample to 55 LGRBs
with known $z$ and rest-frame optical extinction detected by the Swift satellite.\\
In contrast, \cite{melandri2008}, \cite{oates2009}, \cite{zaninoni13} and \cite{melandri14} found no bimodality in
the distributions of $L_{O,12}$, $L_{O,1\rm{d}}$ and $L_{O,11}$, investigating samples
of 44, 24, 40 and 47 GRBs respectively.\\
Instead, with the aim of finding a unifying representation of the GRB afterglow phase,
\cite{ghisellini09} fitted the light curves assuming this functional form:
\begin{equation}
L_L(\nu,t)=L_L(\nu,T_{X,a})\frac{(t/T_{X,t})^{-\alpha_{\nu,fl}}}{1+(t/T_{X,t})^{\alpha_{\nu,st}-\alpha_{\nu,fl}}}.
\end{equation}
They used a data sample of 33 LGRBs detected by Swift in X-ray (0.3-10 keV) and optical R
bands (see the left and middle panels of Fig. \ref{fig:ghisellini}).
Within this approximation, the agreement with data is reasonably good, and they confirmed the X-ray
LT relation.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.3\hsize,angle=0]{total_lcx.pdf}
\includegraphics[width=0.3\hsize,angle=0]{total_lco.pdf}
\includegraphics[width=0.385\hsize,angle=0]{liangfig04.pdf}
\caption{\footnotesize The light curves of the full sample from \cite{ghisellini09} in the X-rays (left
panel) and optical (middle panel). The vertical lines represent $\log L_{X,12}$ and $\log L_{O,12}$ in the rest frame time
respectively. Instead, the dashed lines indicate the $\log t^{-5/4}$ (blue) and the $\log t^{-5/3}$ (red) behaviours.
Right panel: relation between $ L_{O,peak}$ (equivalent to $L_{R,p}$ in the picture) and $T^*_{O,peak}$ of the data set from \cite{Liang2010}. Line represents the best fit.}
\label{fig:ghisellini}
\end{figure}
Through their analysis using a data sample of 32 Swift GRBs, \cite{Liang2010} found that the
optical peak luminosity, $L_{O,peak}$, in the R band
in units of $10^{47}$ erg s$^{-1}$ and the optical peak time, $T^*_{O,peak}$, are anti-correlated, see
the right panel of Fig. \ref{fig:ghisellini}, with a slope $b=-2.49\pm 0.39$ and $\rho=-0.90$.
They deduced that a fainter bump has its maximum later than brighter ones and it also presents a
longer duration.\\
\cite{Panaitescu2011} showed a similar relation to the one presented in \cite{Liang2010}.
They found a $\log F_{O,a} \sim \log T^{-1}_{O,a}$ anti-relation using 37 Swift GRBs.
This result may indicate a unique mechanism for the optical
afterglow even though the optical energy has a quite large scatter.
\begin{figure}[htbp]
\centering
\includegraphics[width=5.35cm,angle=0]{f9acopy.pdf}
\includegraphics[width=5.4cm,angle=0]{f8copy.pdf}
\includegraphics[width=5.25cm,angle=0]{f8bcopy.pdf}
\caption{\footnotesize Left panel: $ L^{S}_{O,a}$ (equivalent to $L^S_{R,p}$ in the picture) as a
function of $T^{S,*}_{O,a}$
(equivalent to $t_{b}$ in the picture) from \cite{Li2012}. The grey circles represent
the X-ray data from \cite{Dainotti2010}. Lines correspond to the best fit lines. Middle and Right panels:
$L^{S}_{O,a}$ and $T^{S}_{O,a}$ distributions for the full GRB data set from \cite{Li2012}.}
\label{fig:li2012a}
\end{figure}
Afterwards, \cite{Li2012} found a relation (see the left panel of Fig. \ref{fig:li2012a})
similar to the LT relation, but in the R band. They used 39 GRBs with optical data available in the literature.
This relation is between the optical luminosity at the end of the plateau, $L_{O,a}$, in units of
$10^{48}$ erg s$^{-1}$ and the optical end of the plateau time,
$\log T^*_{O,a}$, in the shallow decay phase of the GRB light
curves, denoted with the index S. They found a slope $b=-0.78\pm 0.08$, $\rho=0.86$ and $P<10^{-4}$.
\begin{table}[htbp]
\footnotesize
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Correlations & Author & N& Slope& Corr.coeff.& P \\
\hline
$L_{O,peak}-T_{O,peak}$ & Liang et al. (2010) &32 &$-2.49\pm 0.39$&-0.90& \\
$L_{O,a}-T_{O,a}$&Panaitescu \& Vestrand (2011)&37&$-1$&&\\
$L^{S}_{O,a}$-$T^{S}_{O,a}$&Li et al. (2012)&39&$-0.78\pm 0.08$&$0.86$&$<10^{-4}$ \\
\hline
\end{tabular}
\caption{\footnotesize Summary of the relations in this section. The first column represents the relation in
log scale,
the second one the authors, and the third one the number of GRBs in the used sample. Afterwards, the fourth column is
the slope of the relation and the last two columns are the correlation coefficient and the chance
probability, P.}
\label{tblopt}
\end{center}
\end{table}
$L^{S}_{O,a}$ varies from $10^{43}$ to $10^{47}$ erg s$^{-1}$,
and in some GRBs with an early break reaches $\sim 10^{49}$ erg s$^{-1}$, see the middle panel of Fig.
\ref{fig:li2012a}. $T^S_{O,a}$ spans from tens of seconds to several days after the GRB trigger,
with a typical shallow peak
time $T^{S}_{O,a}$ of $\sim 10^4$ seconds, see the right panel of Fig. \ref{fig:li2012a}.
By plotting $L_{O,a}$ in units of $10^{48}$ erg s$^{-1}$ as a function of
$T^*_{O,a}$ in the burst frame, they observed that optical data have a similar trend to the X-ray data.
In fact, this power law relation, presented in the left panel of Fig. \ref{fig:li2012a},
with an index of $-0.78\pm 0.08$ is similar to that derived for the X-ray
flares (see sec. \ref{lisotp}). XRF phenomena are described in sec. 1. As a consequence,
they recovered the LT relation. In Table \ref{tblopt} a summary of the relations described in this section is
displayed.
\subsubsection{Physical interpretation of the unified \texorpdfstring{$L_X(T_a)$-$T_{X,a}^*$ and $L_{O,a}$\,-\,$T^*_{O,a}$}{Lg} relations}
In the unified $L_X(T_a)$-$T_{X,a}^*$ and $L_{O,a}$\,-\,$T^*_{O,a}$ relations
\cite{ghisellini09} considered the flux as the sum of synchrotron radiation caused by
the standard FS due to the fireball impacting the
circumburst medium and of another component may be produced by a long-lived central engine, which resembles mechanisms
attributed to a ``late prompt".
Even if based in part on a phenomenological model, \cite{ghisellini09} explained situations in which achromatic and
chromatic
jet break are either present or not in the observed light curves.\\
In addition, from their analysis, the decay slope of the late prompt emission results to be
$\alpha_{X,a}=-5/4$ (see blue dashed line for X-ray and optical emission
in the left and middle panels of Fig. \ref{fig:ghisellini} respectively), really close within the errors
to the value of the temporal accretion rate
of fall-back material (i.e. $\sim \log t^{-5/3}$,
see red dashed line for X-ray and for optical emission in the left and middle panels of
Fig. \ref{fig:ghisellini} respectively). This explains the activity of the central engine for such a long duration.
For a similar interpretation within the context of the accretion onto the BH related to LT relation
see sec. \ref{Dainotti2008interpretation}.\\
\cite{Liang2010} claimed that the external shock model explains well the anti-relation between
$L_{O,peak}$ and $T_{O,peak}$, because later deceleration time is equivalent to slower ejecta and thus to a less
luminous emission.\\
Furthermore, \cite{panaitescu08} from the analysis of the $\log L_{O,a}-\log T^*_{O,a}$ relation explained the peaky
afterglows (those with $L_{O,a}\propto T^{-1}_{O,a}$) as being a bit outside the cone of view, while the plateau as
off-axis events and due to the angular structure of the jet.
Later, \cite{Panaitescu2011} asserted that
the double-jet structure of the ejecta is problematic. To overcome this issue, they suggested a
model in which both the peaky and plateau afterglows depend on how much time the central engine allows for the energy
injection.
More specifically, impulsive ejecta with a narrow range of $\Gamma$ are responsible for the peaky afterglows, while the plateau
afterglows are produced by a distribution of initial $\Gamma$ which keeps the energy injection till $10^5$ s.\\
Later, \cite{Li2012} pointed out that late
GRB central engine activities can account for both optical flares
and the optical shallow-decay segments. These activities
can be either erratic (for flares) or steady (for internal plateaus).
A normal decay follows the external plateaus with $\alpha_{X,a}$ typically around $-1$, thus possibly originated by an
external shock with the shallow decay segment
caused by continuous energy injection into the blast wave \citep{rees98,dai98,sari2000,zhang2001}. Instead, the internal
plateaus, found first by \cite{troja07} in GRB 070110 and later studied statistically by \cite{liang2007}, are
followed by a much steeper decay ($\alpha_{X,a}$ steeper than -3), which
needs to be powered by internal dissipation of a late outflow. In summary, the afterglow can be interpreted as a mix
of internal and external components.
\subsection{The \texorpdfstring{$L_{O,200\rm{s}}$\,-\,$\alpha_{O,>200\rm{s}}$ relation and its physical interpretation}{Lg}} \label{Oates}
\cite{oates2012} discovered a relation between the optical luminosity at 200 s,
$\log L_{O,200\rm{s}}$, and the optical temporal decay index after 200 s, $\alpha_{O,>200\rm{s}}$,
see the right panel of Fig. \ref{fig:14}. They used a sample of 48 LGRB afterglow light curves at $1600$ \AA{}
detected by UVOT on board of the Swift satellite, see the left panel of Fig. \ref{fig:14}.
The best fit line for this relation is given by:
\begin{equation}
\log \, L_{O,200 \rm{s}} = (28.08 \pm 0.13)-(3.636 \pm 0.004) \times \alpha_{O,>200\rm{s}},
\end{equation}
with $\rho=-0.58$ and a significance of 99.998\% (4.2 $\sigma$).
This relation means that the brightest GRBs decay faster than the dimmest ones. To obtain
the light curves employed for building the relation, they used the criteria
from \cite{oates2009} in order to guarantee that the entire UVOT light curve is not noisy,
namely with a high signal to noise (S/N) ratio\textcolor{red}{: the optical/UV light curves must
be observed in the V filter of the UVOT with a magnitude $\le 17.8$, UVOT observations must
have begun within the first 400 s after the BAT trigger and the afterglow must have been observed until at
least $10^5$ s after the trigger.}
Their results pointed out the dependence of this relation is on the differences in the observing angle and on
the rate of the energy release from the central engine.\\
As a further step, \cite{oates15}, using the same data set, investigated the same
relation both in optical and in X-ray wavelengths in order to make a comparison, and they confirmed previous
optical results finding a similar slope for both relations.
In addition, they analyzed the connection between
the temporal decay indices after $200$ s (in X-ray and optical) obtaining as best fit relation
$\alpha_{X,>200\rm{s}} = \alpha_{O,>200\rm{s}} - 0.25$, see the left panel of Fig. \ref{fig:oates15}.
They yielded some similarities between optical and
X-ray components of GRBs from these studies. \textcolor{red}{Their results were in disagreement
with those previously found by \cite{urata07}, who investigated the relation between the optical and X-ray temporal
decay
indices in the normal decay phase derived from the external shock model.
In fact, a good fraction of outliers was found in this previous work.}\\
\cite{racusin16} studied a similar relation using 237 Swift LGRBs,
but in X-ray. For this relation, it was found that slope \textcolor{red}{$b=-0.27$}$\pm0.04$ and solid evidence
for a strong connection between optical and X-ray components of GRBs was discovered as well.
In conclusion, the Monte Carlo simulations and the statistical tests validated
the relation between $\log L_{O,200\rm{s}}$ and $\alpha_{O,>200\rm{s}}$
by \cite{oates2012}. In addition,
it shows a possible connection with its equivalent,
the LT relation in X-ray, implying a common physical mechanism.
In Table \ref{tbloates} a summary of the relations described in this section is reported.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.49\hsize,clip]{Lum_light_curves.pdf}
\includegraphics[width=0.49\hsize,angle=0,clip]{oates.pdf}
\caption{\footnotesize Left panel: ``optical light curves of 56 GRBs from \cite{oates2012}".
Right panel: ``$\log L_{O,200\rm{s}}$ vs. $\alpha_{O,>200\rm{s}}$ from \cite{oates2012}.
The red solid line indicates the best fit line
and the blue dashed lines show the 3 $\sigma$ variance".}
\label{fig:14}
\end{figure}
Regarding the physical interpretation of the $\log L_{O,200\rm{s}}$\,-\,$\alpha_{O,>200\rm{s}}$ relation,
\cite{oates2012} explored several scenarios. The first one implies that the relation can be due to the interaction
of the jet
with the external medium. In a
straightforward scenario $\alpha_{O,>200\rm{s}}$ is not a fixed value and all optical afterglows stem from
only one closure relation where $\alpha_{O,>200\rm{s}}$ and $\beta_{O,>200\rm{s}}$ are related linearly.
Thus a relation between
$\log L_{O,200\rm{s}}$ and $\beta_{O,>200\rm{s}}$ should naturally appear.
Contrary to this expectation, $\alpha_{O,>200\rm{s}}$ and $\beta_{O,>200\rm{s}}$ are poorly correlated, see the right
panel of Fig. \ref{fig:oates15}, and there
is no evidence for the existence of a relation between $\beta_{O,>200\rm{s}}$ and $\log L_{O,200\rm{s}}$. Therefore, this scenario cannot be
ascribed as the cause of the $\log L_{O,200\rm{s}}$\,-\,$\alpha_{O,>200\rm{s}}$ relation.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.495\hsize,angle=0,clip]{OXdecay_at200s.pdf}
\includegraphics[width=0.495\hsize,angle=0,clip]{Decay_200s_vs_beta.pdf}
\caption{\footnotesize Left panel: ``$\alpha_{O,>200\rm{s}}$ and $\alpha_{X,>200\rm{s}}$
from \cite{oates15}.
The red solid line represents the best fit regression and the blue dashed lines
represent 3 times the root mean square (RMS) deviation. The relationships expected between the optical/UV
X-ray light curves from the GRB closure relations are also shown.
The pink dotted line represents $\alpha_{O,>200\rm{s}}=\alpha_{X,>200\rm{s}}$. The light blue dotted-dashed
lines represent $\alpha_{X,>200\rm{s}}=\alpha_{O,>200\rm{s}}\pm0.25$. In the top right corner it is given
the coefficient $\rho$ with $P$, and it is
provided the best fit slope and constant determined by linear regression". Right panel: ``$\alpha_{O,>200\rm{s}}$ and $\beta_{O,>200\rm{s}}$
for the sample of 48 LGRBs from \cite{oates2012}. The lines represent 3 closure relations and a colour scale is used to
display the range in $\log L_{O,200\rm{s}}$".}
\label{fig:oates15}
\end{figure}
In the second scenario, they assumed that the $\log L_{O,200\rm{s}}$\,-\,$\alpha_{O,>200\rm{s}}$
relation is produced by few closure relations indicated by lines in the right panel of Fig. \ref{fig:oates15}.
However, from this picture, the $\alpha_{O,>200\rm{s}}$ and $\beta_{O,>200\rm{s}}$ values with similar luminosities do not gather around a particular
closure relation, thus also the basic standard model is not a good explanation of the
$\log L_{O,200\rm{s}}-\alpha_{O,>200\rm{s}}$ relation.
As a conclusion, the afterglow model is more complex than it was
considered in the past. It is highly likely that there are physical properties that control the emission mechanism and the decay
rate in the afterglow that still need to be investigated.\\
Therefore, \cite{oates2012} proposed two additional alternatives.
The first is related to some properties of the central engine
influencing the rate of energy release so that for fainter
afterglows, the energy is released more slowly. Otherwise, the relation can be due to different observing angles where
observers at smaller viewing angles see brighter and faster decaying light curves.
\begin{table}[htbp]
\footnotesize
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Correlations & Author & N& Slope& Norm & Corr.coeff.& P \\
\hline
$L_{O,200\rm{s}}$\,-\,$\alpha_{O,>200\rm{s}}$ & Oates et al. (2012) &48 &$-3.636 \pm 0.004$&$28.08 \pm 0.13$&$-0.58$&$2\times10^{-4}$ \\
&Oates et al. (2015)&48 &$-3.636 \pm 0.004$&$28.08 \pm 0.13$&$-0.58$&$2\times10^{-4}$ \\
$L_{X,200\rm{s}}$\,-\,$\alpha_{X,>200\rm{s}}$&Racusin et al. (2016)&237&\textcolor{red}{$-0.27$}$\pm0.04$&$-6.99\pm1.11$&$0.59$&$10^{-6}$\\
\hline
\end{tabular}
\caption{\footnotesize Summary of the relations in this section. The first column represents the relation in
log scale,
the second one the authors, and the third one the number of GRBs in the used sample. Afterwards, the fourth and
fifth columns are
the slope and normalization of the relation and the last two columns are the correlation coefficient and the chance
probability, P.}
\label{tbloates}
\end{center}
\end{table}
As pointed out by \cite{Dainotti2013a}, the $\log L_{O,200\rm{s}}$\,-\,$\alpha_{O,>200\rm{s}}$
relation is related to the LT one since both show an anti-relation between luminosity and decay rate of the
light curve or time. The key point would be to understand how they relate to each other and the possible
common physics that eventually drives both of them.
To this end, \cite{oates15} compared the observed relations with the ones obtained with the simulated sample.
The luminosity-decay relationship in the optical/UV is in agreement with that in the X-ray, inferring a common
mechanism.
\section{The Prompt-Afterglow Relations}\label{promptaftcor}
As we have discussed in the previous paragraphs, the nature of the plateau and the relations
(e.g. the optical one) based on similar physics and directly related to the plateau are still under investigation.
For this reason, several models have been proposed. To further enhance its theoretical understanding, it is necessary
to evaluate
the connection between plateaus and prompt phases. To this end, we hereby review the prompt-afterglow relations,
thus helping to establish a more complete picture of the plateau GRB phenomenon.
\subsection{The \texorpdfstring{$E_{\gamma,afterglow}-E_{X,prompt}$ relation}{Lg} and its physical interpretation}
W07 analyzed the relation between the gamma flux in the prompt phase, $F_{\gamma,prompt}$, and the X-ray flux
in the afterglow, $F_{X,afterglow}$ using 107 Swift GRBs, see the upper left panel of Fig. \ref{ee2}.
They calculated $F_{X,afterglow}$ in the XRT band (0.3-10 keV), while
$F_{\gamma,prompt}$ in the BAT (15-150 keV) plus the XRT (0.3-10 keV) energy band.
For GRBs with known redshift, as shown in the upper right panel of Fig. \ref{ee2}, they investigated
the prompt isotropic energy, $E_{\gamma,prompt}$, and the afterglow isotropic energy, $E_{X,afterglow}$,
assuming a cosmology with $H_0 = 71$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\Lambda}= 0.73$ and $\Omega_M= 0.27$.
\begin{figure}[htbp]
\includegraphics[width=0.495\hsize,angle=0]{wil3.pdf}
\includegraphics[width=0.495\hsize,angle=0]{epreaft.pdf}
\includegraphics[width=0.495\hsize,angle=0]{fig4liang07.pdf}
\includegraphics[width=0.41\hsize,angle=0]{ee.pdf}
\caption{\footnotesize Upper left panel: the $F_{X,afterglow}$ in the XRT
band (0.3-10 keV) vs. $F_{\gamma,prompt}$ computed from the BAT $T_{90}$ flux (15-150 keV)
plus the XRT flux (0.3-10 keV)
from \cite{Willingale2007}. The dotted line represents where $F_{X,afterglow}$ and $F_{\gamma,prompt}$ are identical.
Upper right panel: $\log E_{\gamma,prompt}$ vs. $\log E_{X,afterglow}$
from \cite{Willingale2007}. Symbols show the position of the afterglow in the $\beta_{X,a}$-$\alpha_{X,a}$
plane. GRBs that fall in the pre-jet-break region are plotted as dots, those that fall above this in the
post-jet-break region are plotted as stars, and those below the pre-jet-break band are plotted as squares.
The dotted line represents equality between $\log E_{\gamma,prompt}$ and $\log E_{X,afterglow}$.
Bottom left panel: the $\log E_{\gamma,prompt}-\log E_{X,afterglow}$ relation ($E_{\gamma,iso}$ and
$E_{X,iso}$ respectively in the picture) from \cite{liang2007}.
The solid line is the best fit. The dashed line indicates the 2 $\sigma$ area.
Bottom right panel: $\log E_{X,plateau} $ vs. $\log E_{\gamma,prompt}$ from \cite{ghisellini09}.
The dashed line represents the least square fit with $ \log E_{X,plateau}$ ($T_{a}L_{T_a}$ in the picture)
$ \sim 0.86\times \log E_{\gamma,prompt}$
($E_{\gamma,iso}$ in the picture) ($P= 2\times 10^{-7}$, without the outlier GRB 070125).}
\label{ee2}
\end{figure}
At the same time, \cite{liang2007} focused on the relation between
$E_{\gamma,prompt}$ and $E_{X,afterglow}$ using a sample of 53 LGRBs. They pointed out a good relation with
$b=1\pm0.16$, see the bottom left panel of Fig. \ref{ee2}.\\
In agreement with these results, \cite{Liang2010} and \cite{Panaitescu2011} analyzed this relation,
using respectively 32 and 37 GRBs, but considering energy bands different from that used in \cite{liang2007};
they obtained the slopes $b=0.76\pm0.14$ and $b=1.18$ respectively (see the left and middle panels of Fig. \ref{ee3}).
\begin{figure}[htbp]
\includegraphics[width=5.2cm,height=4.7cm,angle=0]{liangfig053.pdf}
\includegraphics[width=5.2cm,height=4.8cm,angle=0]{fig2bpanaitescu.pdf}
\includegraphics[width=5.8cm,height=4.8cm,angle=0]{fig10racusin.pdf}
\caption{\footnotesize Left panel: ``relation between
$ E_{\gamma,prompt}$ and $ E_{O,afterglow}$ ($E_{\gamma,iso}$ and
$E_{R,iso}$ respectively in the picture), for the optically selected sample, from \cite{Liang2010}.
Line is the best fit".
Middle panel: ``relation between
$\log E_{\gamma,prompt}$
and $\log E_{O,afterglow}$ ($E_{\gamma,iso}$ and
$L_p \times t_p$ respectively in the picture) from \cite{Panaitescu2011}. Black symbols are for afterglows with optical
peaks,
red symbols
for optical plateaus, open circles for afterglows of uncertain type. $r(\log E_{X,afterglow},\log E_{\gamma,prompt}) = 0.66$
for all 37 afterglows.
This linear correlation coefficients correspond to a probability $P=10^{-5.3}$".
Right panel: ``$E_{k,aft}$ as a function of $E_{\gamma,prompt}$ from \cite{racusin11}. The dashed lines
indicated different values of $\eta$. The bursts detected by LAT on board of Fermi
tend to have high $E_{\gamma,prompt}$ ($E_{\gamma,iso}$ in the picture), but average $E_{k,aft}$, and therefore
higher values of $\eta$ than the samples from BAT on board of Swift or GBM on board of Fermi".}
\label{ee3}
\end{figure}
\cite{rowlinson2013} and \cite{grupe2013} confirmed these results, see the left and middle panels of Fig.
\ref{fig:grupe13}. In fact, they obtained a $E_{\gamma,prompt}-E_{X,afterglow}$ relation with slope
$b\sim 1$ using 43 SGRBs and 232 GRBs with spectroscopic redshifts detected by Swift respectively.\\
Finally, \cite{dainotti15} analyzed this relation to find some constraints on the ratio of
$E_{X,afterglow}$ to $E_{\gamma,prompt}$, considering a sample of 123 LGRBs, see the right panel of Fig. \ref{fig:grupe13}.\\
Instead, \cite{ghisellini09}, with a sample of 33 LGRBs, considered a similar relation, but assuming the X-ray plateau
energy, $E_{X,plateau}$, as an estimation of $E_{X,afterglow}$, see the bottom right panel of Fig. \ref{ee2}; they found
a slope $b=0.86$. \\
In addition, \cite{ghisellini09} also investigated the relation between $E_{\gamma,prompt}$
and the kinetic isotropic energy in the afterglow, $E_{k,aft}$, with the same sample,
finding a relation with $b=0.42$.
Similarly, \cite{racusin11} studied the same relation, using 69 GRBs and assuming different efficiencies to find some
limits between $E_{k,aft}$ and $E_{\gamma,prompt}$, see the right panel of Fig. \ref{ee3}.\\
This relation was most likely used to study the differences in detection
of several instruments and to analyze the transferring process of kinetic energy into the prompt emission in GRBs,
making the relation by \cite{racusin11} the most reliable one.\\
\textcolor{red}{To summarize, for comparing the energies in the prompt and the afterglow phases,
a $E_{\gamma,prompt}-E_{X,afterglow}$ relation was studied by \cite{liang2007} and
confirmed by \cite{rowlinson2013}, \cite{grupe2013} and \cite{dainotti15}. The last study found also some limitations
on the ratio among the prompt and the afterglow energies. Furthermore, instead of $E_{X,afterglow}$, $E_{X,plateau}$
was considered for the investigation, although this quantity provided similar results to the previous ones
\citep{ghisellini09}. Finally, the relation between
$E_{\gamma,prompt}$ and $E_{k,aft}$ was studied by \cite{ghisellini09} and confirmed by \cite{racusin11}, who examined
the energy transfer in the prompt phase. These relations are relevant because of their usefulness for investigating
the efficiency of the emission processes during the transition from the prompt phase to the afterglow one, and for
explaining which the connection between these two emission phases is. As a main result, \cite{ghisellini09}
and \cite{racusin11} claimed that the fraction of kinetic energy transferred from the prompt phase to the afterglow one
is around 10\%. In particular, \cite{racusin11} yielded that this value of the transferred kinetic
energy, for BAT-detected GRBs, is in agreement with the analysis by \cite{zhang07b} for which the internal shock model
well describes this value in the case of a late energy transfer from the fireball to the surrounding medium
\citep{zhang2005}.}\\
In Table \ref{tbl3}, a summary of the relations described in this section is presented.\\
As regards the physical interpretation of the $E_{X,afterglow}-E_{\gamma,prompt}$ relation,
\cite{racusin11}, estimating the efficiency parameter $\eta$ for the BAT sample, confirmed
the \cite{zhang07b} result for which $\sim57\%$ of BAT bursts have $\eta<10\%$.
However, for the samples from the Gamma Burst Monitor (GBM)
and the Large Area Telescope (LAT), on board the Fermi satellite\footnote{The Fermi Gamma ray
Space Telescope (FGST), launched in 2008 and still running, is a space observatory being
used to perform gamma ray astronomy observations from low Earth orbit. Its main instrument is the Large Area Telescope (LAT), an
imaging gamma ray detector, (a pair-conversion instrument) which detects photons with energy from about 20 MeV to 300 GeV with a
field of view of about 20\% of the sky; it is a sequel to the EGRET instrument on the Compton gamma ray observatory (CGRO). Another
instrument aboard Fermi is the Gamma Ray Burst Monitor (GBM), which is used to study prompt GRBs from $8$ keV to $30$ MeV.},
they found that only 25\% of the GBM bursts and none of the LAT bursts have
$\eta<10\%$. This implies that Fermi GRBs are more efficient at transferring kinetic energy
into prompt radiation.
\begin{figure}[htbp]
\includegraphics[width=4cm,height=3.85cm,angle=0]{f5rowlinson13.pdf}
\includegraphics[width=6.3cm,angle=0]{fig11agrupe13.pdf}
\includegraphics[width=6cm,angle=0]{energypromptvsafterglowcopy.pdf}
\caption{\footnotesize Left panel: the prompt BAT 15-150 keV fluence vs. the X-ray fluence in the 15-150
keV energy band from \cite{rowlinson2013}. Blue stars are GRBs with 2 or more breaks in their light curves,
green circles have 1 break and red triangles have
no significant breaks in their light curves. The black line
indicates where the shallow decay phase fluence is equal to the prompt fluence. Middle panel: relations of the 0.3-10 keV XRT
fluence with fluence in the 15-150 keV BAT band from \cite{grupe2013}. Short bursts are represented with
triangles and high-redshift (z $> 3.5$) bursts with circles.
Right panel: $<\log E_{\gamma,prompt}>$ vs. $\log E_{X,afterglow}$ relation from \cite{dainotti15} for 123 LGRBs.
The solid line for equal $\log E_{\gamma,prompt}$ and $\log E_{X,afterglow}$ is given for reference.}
\label{fig:grupe13}
\end{figure}
\begin{table}[htbp]
\footnotesize
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Correlations & Author & N& Slope& Norm & Corr.coeff.& P \\
\hline
$E_{X,afterglow}-E_{\gamma,prompt}$ & Liang et al. (2007) & 53 &$1.00^{+0.16}_{-0.16}$&$-0.50^{+8.10}_{-8.10}$&0.79&$<10^{-4}$ \\
$E_{O,afterglow}-E_{\gamma,prompt}$ &Liang et al. (2010)&32&$0.76^{+0.14}_{-0.14}$&$1.30^{+0.14}_{-0.14}$&0.82&$<10^{-4}$\\
&Panaitescu \& Vestrand (2011)&37&1.18&&0.66& $10^{-5.3}$\\
\hline
$E_{X,plateau}-E_{\gamma,prompt}$ & Ghisellini et al. (2009)&33&0.86&&&$2\times10^{-7}$ \\
\hline
$ E_{k,aft}- E_{\gamma,prompt}$ & Ghisellini et al. (2009)&33&0.42&&&$10^{-3}$ \\
\hline
\end{tabular}
\caption{\footnotesize Summary of the relations in this section. The first column represents the relation in log scale,
the second one the authors, and the third one the number of GRBs in the used sample. Afterwards, the fourth and fifth columns are
the slope and normalization of the relation and the last two columns are the correlation coefficient and the chance probability, P.}
\label{tbl3}
\end{center}
\end{table}
\subsection{The \texorpdfstring{$L_{X,afterglow}-E_{\gamma,prompt}$}{Lg} relation and its physical interpretation}
\cite{berger07} investigated the prompt and afterglow energies in the observed frame of 16 SGRBs.
A large fraction of them (80\%) follows a linear relation between
the prompt fluence in the gamma band, $S_{\gamma,prompt}$, in the BAT range and
the X-ray flux at 1 day, $F_{X,1\rm{d}}$, in the XRT band given by:
\begin{equation}
\log F_{X,1\rm{d}} \sim (1.01 \pm 0.09) \times \log S_{\gamma,prompt},
\end{equation}
with $\rho=0.86$ and $P=5.3\times10^{-5}$.
\cite{gehrels08} confirmed his results investigating the same relation, but with X-ray
fluxes at 11 hours, $F_{X,11}$, see Fig. \ref{fig:gehrels08b}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.85\hsize,clip]{f2gehrels08.pdf}
\caption{\footnotesize $F_{X,11}$-$S_{\gamma,prompt}$ ($F_{X-ray}$ and
$S_{\gamma-ray}$ respectively in the picture) relation for Swift
SGRBs (in red) and LGRBs (in blue) from \cite{gehrels08} .
The XRT $F_{X,11}$ are computed at 3 keV and the BAT
$S_{\gamma,prompt}$ are detected between 15 and 150 keV \citep{sakamoto08}.}
\label{fig:gehrels08b}
\end{figure}
Later, \cite{nysewander09} considered the relation between $F_{X,11}$ or the optical flux at 11 hours, $F_{O,11}$,
and $E_{\gamma,prompt}$, finding an almost linear relation, see Fig. \ref{fig:nysawander}. They used a data set
of 37 SGRBs and 421 LGRBs detected by Swift.
\cite{Panaitescu2011} confirmed, in part, these results. They found a similar relation between
$E_{\gamma,prompt}$ and $F_{O,a}$ using 37 GRBs, but with a higher slope ($b=1.67$), see the left panel of Fig.
\ref{fig:gehrels08}.\\
\cite{kaneko07} showed a linear relation $L_{X,10}\propto E_{\gamma,prompt}$, where $L_{X,10}$ is the X-ray
luminosity at 10 hours calculated in the
2-10 keV energy range, while $E_{\gamma,prompt}$ in the 20-2000 keV energy range, see the left panel of
Fig. \ref{fig:kaneko}.
This relation compares four long events spectroscopically associated with SNe with ``regular" energetic LGRBs
($E_{\gamma,prompt}\sim 10^{52}-10^{54}$ erg). The results possibly indicate a common
efficiency $\eta$ for transforming kinetic energy into gamma rays
in the prompt phase for both these four events and for ``regular" energetic LGRBs.\\
The same relation has been studied in the context of the low luminosity versus normal luminosity GRBs.
Indeed, \cite{amati07} found that the relation between $L_{X,10}$, in the 2-10 keV band,
and $E_{\gamma,prompt}$, in the 1-10000 keV band, becomes stronger ($P\sim 10^{-11}$) including sub-energetic GRBs
as GRB 060218, GRB 980425
and GRB 031203, see the middle panel of Fig. \ref{fig:kaneko}. Therefore, it is claimed
that sub-energetic GRBs are intrinsically faint and are considered to some extent normal cosmological GRBs.\\
Finally, \cite{berger07} also analyzed the relation between the X-ray luminosity at one day,
$L_{X,1\rm{d}}$, and $E_{\gamma,prompt}$, using 13 SGRBs with measured $z$. They found a slope $b=1.13\pm0.16$
(see the right panel of Fig. \ref{fig:gehrels08}).
\begin{figure}[htbp]
\centering
\includegraphics[width=0.495\hsize,angle=0,clip]{f3nysawander09.pdf}
\includegraphics[width=0.495\hsize,angle=0,clip]{f4nysawander09.pdf}
\includegraphics[width=0.495\hsize,angle=0,clip]{f5nysawander09.pdf}
\includegraphics[width=0.495\hsize,angle=0,clip]{f6nysawander09.pdf}
\caption{\footnotesize Upper left panel: ``a plot of $F_{O,11}$ (corrected for Galactic
extinction) vs. 15-150 keV $S_{\gamma,prompt}$
for both LGRBs (grey) and SGRBs (red) from \cite{nysewander09}. Note that
below a fluence of 10$^{-7}$ erg cm$^{-2}$, no optical afterglow of
an SGRB has been discovered, while above 10$^{-7}$, all reasonably
deep observing campaigns, but one (GRB 061210) have detected an
optical afterglow". Upper right panel: ``a plot of $F_{X,11}$
vs. 15-150 keV $S_{\gamma,prompt}$ ($E_{\gamma,iso}$ in the picture) for both
LGRBs (grey) and SGRBs (red) from \cite{nysewander09}".
Bottom left panel: ``a plot of $L_{O,11}$ (corrected for
Galactic extinction) vs. $E_{\gamma,prompt}$ from \cite{nysewander09}. Dashed upper limits represent SGRBs with a host
galaxy determined by XRT error circle only. The classification of
GRB 060614 and GRB 060505 is uncertain, therefore, they are labelled
as ``possibly short" ". Bottom right panel: ``a plot of $L_{X,11}$ vs. $E_{\gamma,prompt}$ from \cite{nysewander09}. The open circles
represent SGRBs with a host galaxy determined by XRT error circle
only. The classification of GRB 060614 and GRB 060505 is uncertain,
therefore, they are labelled as ``possibly short" ".}
\label{fig:nysawander}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=8.1cm,height=6.3cm,angle=0,clip]{fig2apanaitescu.pdf}
\includegraphics[width=8.1cm,height=6.3cm,angle=0,clip]{fig2berger.pdf}
\caption{\footnotesize Left panel: $E_{\gamma,prompt}$-$F_{O,a}$ ($E_{\gamma,iso}$ and
$F_p$ respectively in the picture) relation from \cite{Panaitescu2011}.
Black symbols are for afterglows with optical peaks, red symbols for optical plateaus,
open circles for afterglows of unknown kind. Right panel: $L_{X,1\rm{d}}$ vs. $E_{\gamma,prompt}$ ($E_{\gamma,iso}$
in the picture) for the SGRBs with a known $z$ (solid black circles),
redshift constraints (open black circles) and without any redshift
information (grey symbols connected by dotted lines) from \cite{berger07}.}
\label{fig:gehrels08}
\end{figure}
\cite{Liang2010} confirmed his results in the optical range using a sample of 32 Swift GRBs
($E_{\gamma,prompt}-L_{O,peak}$ with $b=1.40 \pm 0.08$, see the right panel of Fig.
\ref{fig:kaneko}). In addition, \cite{kann10} also confirmed his results with a sample of 76 LGRBs
($E_{\gamma,prompt}-L_{O,1\rm{d}}$ with $b=0.36$, see the left panel of Fig. \ref{fig:berger}).
\begin{figure}[htbp]
\centering
\includegraphics[width=5.5cm,height=5cm,angle=0,clip]{f20kaneko06.pdf}
\includegraphics[width=5cm,height=5cm,angle=0,clip]{ELamati07.pdf}
\includegraphics[width=5.75cm,height=5cm,angle=0,clip]{liangfig052.pdf}
\caption{\footnotesize Left panel: ``$L_{\rm X,10}$ of SN-GRBs (source frame: 2$-$10 keV) as a
function of their $E_{\gamma,prompt}$, $E_{\gamma,iso}$ in the picture, (20$-$2000 keV)
from \cite{kaneko07}.
$z$ for each event is also shown in colour". Middle panel: ``$L_{X,10}$ (in 2-10 keV range)
vs. $E_{\gamma,prompt}$ ($E_{iso}$ in the picture) for the events included in the sample of \cite{Nousek2006} (triangles) plus the 3 sub-energetic GRB\,980425,
GRB\,031203, GRB\,060218, the other GRB/SN event GRB\,030329 (circles), and
3 GRBs with known $z$ and deep limits to the peak magnitude
of associated SN, XRF\,040701, GRB\,060505 and GRB\,060614 (diamonds) from \cite{amati07}. Empty triangles
indicate those GRBs for which the 1-10000 keV $E_{\gamma,prompt}$ was computed based on
the 100-500 keV $E_{\gamma,prompt}$ reported by \cite{Nousek2006} by assuming
an average spectral index.
The plotted lines are the best-fit power laws obtained
without (dotted) and with (dashed) sub-energetic GRBs and GRB\,030329". Right panel: ``relation between $E_{\gamma,prompt}$
and $L_{O,peak}$ ($E_{\gamma,iso}$ and
$L_{R,p}$ respectively in the picture) for the optically selected sample from \cite{Liang2010}. Line is the best fit".}
\label{fig:kaneko}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=5.5cm,height=4.4cm]{f8kann10.pdf}
\includegraphics[width=0.325\hsize,angle=0,clip]{figure14berger14.pdf}
\includegraphics[width=0.325\hsize,angle=0,clip]{figure15berger14.pdf}
\caption{\footnotesize Left panel: ``$F_{O,1\rm{d}}$ in the $R$ band
plotted against the bolometric $E_{\gamma,prompt}$ ($E_{iso,bol}$ in the picture) for all GRBs in the optically selected sample from \cite{kann10}
(except GRB 991208, which was only discovered after several days, and GRBs 060210, 060607A, 060906 and 080319C, where the follow-up
does not extend to one day). While no tight relation is visible, there is a trend of increasing optical luminosity with increasing
prompt energy release. This is confirmed by a linear fit (in log-log space), using a Monte Carlo analysis to account for the
asymmetric errors. The dashed line shows the best fit, while the dotted line marks the 3 $\sigma$ error region. Several special
GRBs are marked". Middle panel: ``$L_{X,11}$ vs. $E_{\gamma,prompt}$ for
SGRBs (blue) and LGRBs (grey) from \cite{berger14}. Open symbols for SGRBs
indicate events without a known $z$, for which a fiducial value
of $z=0.75$ is assumed. The dashed blue and red lines are the
best-fit power law relations to the trends for SGRBs and LGRBs,
respectively, while the dotted black line is the expected
relation based on the afterglow synchrotron model with
$\nu_X>\nu_c$ and $p=2.4$ ($\log L_{X,11}\sim1.1\times \log E_{\gamma,prompt}$).
The inset shows the distribution of the ratio $\log (L_{X,11} \times
(11\,{\rm hr})^{1.3}/ E_{\gamma,prompt}^{1.1})$, for the full samples
(thick lines) and for the region where SGRBs and LGRBs have equal $E_{\gamma,prompt}$ values (thin lines).
The lower level of $L_{X,11}$ relative to
$E_{\gamma,prompt}$ for SGRBs is evident from these various
comparisons". Right panel: same as in the middle panel, ``but for the
isotropic-equivalent afterglow optical luminosity at a rest frame
time of 7 hours ($L_{O,7}$), still from \cite{berger14}. The dotted black line is the
expected relation based on the afterglow model for
$\nu_m<\nu_{O}<\nu_c$ and $p=2.4$ ($\log L_{O,7}\sim 1.35\times \log E_{\gamma,prompt}$). The inset
shows the distribution of the
ratio $\log (L_{O,7} \times (7\,{\rm hr})^{1.05}/
E_{\gamma,prompt}^{1.35})$, for the full samples (thick lines) and for the region where SGRBs and LGRBs
have equal $E_{\gamma,prompt}$ values (thin
lines). The lower level of $L_{O,7}$ relative to
$E_{\gamma,prompt}$ for SGRBs is evident from these various
comparisons".}
\label{fig:berger}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=8.1cm,angle=0,clip]{Eiso_OLum.pdf}
\includegraphics[width=8.1cm,angle=0,clip]{Eiso_XLum.pdf}
\caption{\footnotesize Left panel: $\log L_{O,200\rm{s}}$-$\log E_{\gamma,prompt}$ ($E_{iso}$ in the picture) relation from \cite{oates15}. Right
panel: $\log L_{X,200\rm{s}}$-$\log E_{\gamma,prompt}$ ($E_{iso}$ in the picture) relation from \cite{oates15}.}
\label{fig:liang2}
\end{figure}
Similarly, \cite{Dainotti11b} analyzed the relation between
$\log L_{X,a}$ and $\log E_{\gamma,prompt}$
using the light curves of 66 LGRBs from the Swift BAT+XRT repository, http://www.swift.\\ac.uk/burst$_{-}$analyser/.
Their sample has been divided into two subsamples: E4 formed of 62 LGRBs and E0095 consisting of 8 LGRBs, assuming
$\sigma_E$ as a parameter representing the goodness of the fit.
For the E4 subsample it was found:
\begin{equation}
\log L_{X,a}=28.03^{+2.98}_{-2.97}+0.52_{-0.06}^{+0.07}\times \log E_{\gamma,prompt},
\end{equation}
with $\rho=0.43$ and $P=1.4\times10^{-5}$, while for the E0095 subsample
\begin{equation}
\log L_{X,a}=29.82^{+7.11}_{-7.82}+0.49_{-0.16}^{+0.21}\times \log E_{\gamma,prompt},
\end{equation}
with $\rho=0.83$ and $P=3.2 \times 10^{-2}$.
Thus, it was concluded that the small error energy sample led to a higher relation
and to the existence of a subset of GRBs which can yield a ``standardizable candle".
Furthermore, since $\log L_{X,a}$ and $\log T_{X,a}^*$ are strongly correlated, and the slope is roughly -1, the energy
reservoir of the plateau is roughly constant. Since $\log E_{\gamma,peak}$ and $\log E_{\gamma,prompt}$ are both linked with
$\log L_{X,a}$,
then the $\log E_{\gamma,peak}-\log E_{\gamma,prompt}-\log E_{X,plateau}$ relation is straightforward.
For its modification
taking into account $\log E_{\gamma,iso}$ of the whole X-ray light curves see \cite{Bernardini13}.
As further confirmations of the $L_{X,a}-E_{\gamma,prompt}$ relation, \cite{davanzo12} and \cite{margutti13}
found a relation between $\log L_{X,a}$ and $E_{\gamma,prompt}$ with slope $b\sim1$ and $\rho\approx 0.70$,
using 58 and 297 Swift LGRBs respectively. \\
Furthermore, \cite{berger14} studied the relation between the X-ray luminosities at 11 hours, $L_{X,11}$, and $E_{\gamma,peak}$, and the relation between
the optical luminosity at 7 hours $L_{O,7}$ and $E_{\gamma,peak}$ for a sample of 70 SGRBs
and 73 LGRBs detected mostly by Swift. He found that the observed relations are flatter
than the ones simulated by \cite{kann10}, see the middle and right panels of Fig. \ref{fig:berger}. \\
Regarding the relation between $E_{\gamma,prompt}$ and the optical luminosities, \cite{oates15} analyzed the relation
between $L_{O,200\rm{s}}$ or $L_{X,200\rm{s}}$ and $\log E_{\gamma,prompt}$ with a sample of 48 LGRBs. They claimed
a strong connection between prompt and afterglow phases, see Fig. \ref{fig:liang2} and Table \ref{tbl4}. This relation permits to
study some important spectral characteristics of GRBs, the optical and X-ray components of the radiation process and
the standard afterglow model. In Table \ref{tbl4}, a summary of the relations described in this section is shown.\\
Regarding the physical interpretation of the $L_{X,afterglow}-E_{\gamma,prompt}$ relation,
\cite{gehrels08} underlined that the optical
and X-ray radiation are characterized by $\beta_{OX,a}\approx0.75$. This value matches the slow cooling case,
important at 11 hours, when the electron distribution power law index is $p=2.5$ for $\nu_m < \nu_O < \nu_X < \nu_c$.\\
\cite{oates15} pointed out that within the standard afterglow model, the $\log E_{\gamma,prompt}-(\log L_{O,200\rm{s}}, \log L_{X,200\rm{s}})$
relations are expected.
However, the slopes of the simulated and observed relations are inconsistent at $>$ 3 $\sigma$ due to values set for the $\eta$
parameter. If the distribution of the efficiencies is not sufficiently narrow the relation will be more disperse. Thus, the
simulations repeated with $\eta=0.1$ and $\eta=0.9$ gave, anyway, incompatible results between the simulated and observed slopes at
$>$ 3 $\sigma$.
\begin{table}[htbp]
\footnotesize
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Correlations & Author & N & Slope& Norm & Corr.coeff. & P \\
\hline
$ F_{X,1\rm{d}}- S_{\gamma,prompt}$ & Berger (2007)& 16& $1.01^{+0.09}_{-0.09}$&& 0.86 &$5.3\times10^{-5}$\\
\hline
$F_{X,11}-S_{\gamma,prompt}$ & Gehrels et al. (2008) & 111 & $0.63^{+0.04}_{-0.04}$&$2.11^{+0.21}_{-0.21}$&0.53&$4\times10^{-9}$ \\
&Gehrels et al. (2008) & 10 &$0.36^{+0.17}_{-0.17}$&$0.06^{+1.07}_{-1.07}$&0.35&0.31\\
\hline
$ F_{O,11}-E_{\gamma,prompt}$ & Nysewander et al. (2009)&421&$\sim 1$&&& \\
$ F_{O,11}-E_{\gamma,prompt}$ & Nysewander et al. (2009)&37&$\sim 1$&&& \\
\hline
$F_{X,11}- E_{\gamma,prompt}$ & Nysewander et al. (2009)&421&$\sim 1$&&&\\
$F_{X,11}- E_{\gamma,prompt}$ & Nysewander et al. (2009)&37&$\sim 1$&&&\\
\hline
$ F_{O,a}-E_{\gamma,prompt}$ & Panaitescu\&Vestrand (2011)&37&1.67&&0.75&$10^{-7.3}$ \\
\hline
$L_{X,1\rm{d}}-E_{\gamma,prompt}$ & Berger (2007) &13& $1.13^{+0.16}_{-0.16}$&$43.43^{+0.20}_{-0.20}$& 0.94 &$3.2\times10^{-6}$\\
\hline
$ L_{O,peak}-E_{\gamma,prompt}$ & Liang et al. (2010) &32& $1.40^{+0.08}_{-0.08}$&$0.83^{+0.15}_{-0.15}$& 0.87 &$10^{-4}$\\
\hline
$L_{O,1\rm{d}}- E_{\gamma,prompt}$ & Kann et al. (2010) &76&0.36&&&\\
\hline
$ L_{X,a}- E_{\gamma,prompt}$ & Dainotti et al. (2011b) & 62& $0.52_{-0.06}^{+0.07}$&$28.03^{+2.98}_{-2.97}$&0.43&$1.4\times10^{-5}$\\
& Dainotti et al. (2011b) & 8 & $0.49_{-0.16}^{+0.21}$& $29.82^{+7.11}_{-7.82}$&0.83&$3.2 \times10^{-2}$\\
& D'Avanzo et al. (2012)& 58 & $\sim 1$& &$\approx 0.70$&\\
& Margutti et al. (2013)& 297 & $\sim 1$& &$\approx 0.70$ &\\
\hline
$L_{X,11}-E_{\gamma,prompt}$ & Berger (2014)&73&0.72&44.75&&\\
& Berger (2014)&70&0.83&43.93&&\\
\hline
$ L_{O,7}-E_{\gamma,prompt}$ & Berger (2014)&73&0.73&43.70&&\\
& Berger (2014)&70&0.74& 42.84&&\\
\hline
$ L_{X,200\rm{s}}- E_{\gamma,prompt}$ & Oates et al. (2015)&48&$1.10^{+0.15}_{-0.15}$&$-27.81^{+7.89}_{-7.89}$&0.83&$5.04\times10^{-13}$\\
\hline
$L_{O,200\rm{s}}-E_{\gamma,prompt}$ & Oates et al. (2015)&48&$1.09^{+0.13}_{-0.13}$&$-25.27^{+6.92}_{-6.92}$&0.76&$4.51\times10^{-10}$\\
\hline
\end{tabular}
\caption{\footnotesize Summary of the relations in this section. The first column represents the relation in log scale,
the second one the authors, and the third one the number of GRBs in the used sample. Afterwards, the fourth and fifth columns are
the slope and normalization of the relation and the last two columns are the correlation coefficient and the chance probability, P.}
\label{tbl4}
\end{center}
\end{table}
\subsection{The \texorpdfstring{$L_{X,a}-L_{O,a}$}{Lg} relation and its physical interpretation}
In the observed frame, \cite{jakobsson04} studied the $\log F_{O,11}$ versus $\log F_{X,11}$ distribution, in
the optical R band and in the $2-10$ keV band respectively, using all known GRBs with a detected X-ray
afterglow, see the left panel of Fig. \ref{OX_comp}.
Different from the previous definition of dark bursts (where dark bursts were simply defined as
those bursts in which the optical transient is not observed), they defined these bursts as
GRBs where the optical-to-X-ray
spectral index, $\beta_{OX,a}$, is shallower than the X-ray spectral index minus 0.5, $\beta_{X,a}-0.5$.
They found out 5 dark bursts among 52 observed by Beppo-SAX\footnote{Beppo-SAX, (1996-2003), was an Italian-Dutch
satellite
capable of simultaneously observing targets over more than 3 decades of energy,
from $0.1$ to $600$ keV with relatively large area, good (for that time) energy resolution and imaging capabilities (with a spatial
resolution of $1$ arcminute between $0.1$ and $10$ keV). The instruments on board Beppo-SAX are Low Energy Concentrator Spectrometer
(LECS), Medium Energy Concentrator Spectrometer (MECS), High Pressure Gas Scintillation Proportional Counter (HPGSPC), Phoswich
Detector System (PDS) and Wide Field Camera (WFC, from $2-30$ keV and from $100-600$ keV). The first four instruments point to the
same direction allowing observations in the broad energy range (0.1-300 keV). With the WFC it was possible to model the afterglow
as a simple power law, mainly due to the lack of observations during a certain period in the GRB light curve.}.
This analysis aimed at distinguishing dark GRBs through Swift.
\cite{gehrels07} and \cite{gehrels08} confirmed the results
using a data sample of 19 SGRBs and 37 LGRBs$+$6 SGRBs respectively, see the middle and right panels of
Fig. \ref{OX_comp}. In particular, \cite{gehrels08} obtained a slope $b=0.38 \pm 0.03$
for LGRBs and $b=0.14\pm0.45$ for SGRBs.
\begin{figure}[htbp]
\includegraphics[width=5.5cm,height=4.9cm]{figjakobsson1copy.pdf}
\includegraphics[width=5.5cm,height=5.2cm]{fig1gehrels07.pdf}
\includegraphics[width=5.2cm,height=5.2cm,angle=0]{f1gehrels08.pdf}
\caption{\footnotesize Left panel: $\log F_{O,11}$-$\log F_{X,11}$ ($F_{opt}$ and $F_X$ respectively in the
plot) distribution for the data set from \cite{jakobsson04}.
Filled symbols show optical detections while open symbols represent upper limits. Lines of constant $\beta_{OX,a}$
are displayed with the corresponding value. Dark bursts are those which have $\beta_{OX,a}< 0.5$.
Middle panel: $F_{X,11}$-$F_{O,11}$ relation for Swift SGRBs and LGRBs from Gehrels (2007). Comparison is made to pre-Swift GRBs
and to lines of optical to X-ray spectral index from \cite{jakobsson04}. The grey points indicate LGRBs,
the black points represent SGRBs and the small black points without error bars
are the pre-Swift GRBs. Right panel: $F_{O,11}$-$F_{X,11}$ relation for Swift SGRBs (shown in red) and LGRBs (shown in blue)
from \cite{gehrels08}. The three circled bursts are those
for which $z>3.9$. The pre-Swift GRBs taken from \cite{jakobsson04} are presented in green.
Also the dark burst separation line
$\beta_{OX,a} = 0.5$ \citep{jakobsson04} and a line showing $\beta_{OX,a} = 1.0$ are represented.}
\label{OX_comp}
\end{figure}
Instead in the rest-rest frame, \cite{berger14} studied the relation between $L_{O,7}$ and $L_{X,11}$ on 70 SGRBs
and 73 LGRBs, finding some similarities between SGRBs and LGRBs and a central value
$<L_{O,7}/L_{X,11}> \approx 0.08$, see the left panel of Fig. \ref{fig:berger14}.\\
\cite{oates15} improved their study. They analyzed a similar relation with a sample of 48 LGRBs,
but using $L_{O,200\rm{s}}$ and $L_{X,200\rm{s}}$, see the right panel of Fig. \ref{fig:berger14}.
The slope obtained has a value $b=0.91 \pm 0.22$.\\
This relation helps to explore the synchrotron spectrum of GRBs and to obtain some constraints on the circumburst
medium for both LGRBs and SGRBs. In Table \ref{tblgehrels} a summary of the relations
described in this section is displayed.
\begin{figure}[htbp]
\includegraphics[width=0.47\hsize]{figure16berger14.pdf}
\includegraphics[width=0.5\hsize]{OXLum.pdf}
\caption{\footnotesize Left panel: ``$L_{O,7}$ vs.
$L_{X,11}$ from \cite{berger14}. The dotted black line marks a linear relation, expected for
$\nu_X\sim \nu_c$. The inset shows the distribution of the ratio
$L_{O,7}/L_{X,11}$, indicating that both SGRBs and LGRBs
exhibit a similar ratio, and that in general $L_{O,7}/
L_{X,11}\sim 1$, indicative of $\nu_X\sim \nu_c$ for SGRBs".
Right panel: ``$\log L_{O,200\rm{s}}$ vs. $\log L_{X,200\rm{s}}$ from \cite{oates15}.
The red solid line represents the best fit regression and the blue dashed line
represents 3 times the RMS deviation. In the top right corner, it is given
$\rho$ and $P$ and it is provided the best-fit slope and constant determined by linear regression".}
\label{fig:berger14}
\end{figure}
\begin{table}[htbp]
\footnotesize
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Correlations & Author & N& Slope& Norm & Corr.coeff.& P \\
\hline
$F_{X,11}-F_{O,11}$ &Gehrels et al. (2008) & 6 &$0.14\pm0.45$&$0.72 \pm 0.94$&$0.06$&$0.68$ \\
&&37&$0.38 \pm 0.03$&$1.62\pm0.04$&$0.44$&$0.006$\\
\hline
$L_{X,11}-L_{O,7}$ & Berger (2014) &70&0.08&&& \\
&&73&0.08&&&\\
\hline
$L_{X,200\rm{s}}-L_{O,200\rm{s}}$ & Oates et al. (2015)&48&$0.91 \pm 0.22$&$1.04\pm6.94$&0.81&$5.26\times10^{-12}$ \\
\hline
\end{tabular}
\caption{\footnotesize Summary of the relations in this section. The first column represents the relation in log scale,
the second one the authors, and the third one the number of GRBs in the used sample. Afterwards, the fourth and fifth columns are
the slope and normalization of the relation and the last two columns are the correlation coefficient and the chance probability, P.}
\label{tblgehrels}
\end{center}
\end{table}
Regarding the physical interpretation of the $L_{X,a}-L_{O,a}$ relation, \cite{berger14} showed that,
in the context of the synchrotron model, the comparison of $L_{O,7}$ and $L_{X,11}$ indicated that usually $\nu_c$ is
near or higher than the X-ray band. Indeed, LGRBs have often
greater circumburst medium densities (about $50$ times greater than SGRBs) and therefore $\nu_c \sim \nu_X$.
\subsection{The \texorpdfstring{$L_X(T_a)-L_{\gamma,iso}$}{Lg} relation}\label{Dainotti2011b}
In \cite{Dainotti11b} the connections between the physical properties of the prompt emission and $\log L_{X,a}$ were analyzed
using a sample of 62 Swift LGRBs. A relation was found between $\log L_{X,a}$ in the XRT band and the isotropic prompt luminosity,
$\log < L_{\gamma,iso}>_{45}\equiv \log (E_{\gamma,prompt}/T_{45})$, in the BAT energy band,
see the left panel of Fig. \ref{fig:16}. This relationship can be fitted with the following equation:
\begin{equation}
\log\ L_{X,a} = 20.58^{+6.66}_{-6.73} + 0.67^{+0.14}_{-0.15} \times \log\ <L_{\gamma,iso}>_{45},
\end{equation}
obtaining $\rho=0.59$ and $P=7.7\times10^{-8}$.
In this paper $\log L_{X,a}$ was related to several prompt luminosities defined using different time scales,
such as $T_{90}$, $T_{45}$ (the time in which the
45\% between 5\%-50\% of radiation is emitted in the prompt emission), and $T_{X,p}$ (the time at
the end of the prompt emission within the W07 model).
Furthermore, the E4 (defined in Table \ref{abbreviations}) subsample of $62$ LGRBs with known $z$ from a sample of
77 Swift LGRBs and the E0095 subsample of 8 GRBs with smooth light curves were used, see black and
red points in the left panel of Fig. \ref{fig:16}.\\
Therefore, it has been shown that the GRB subsample with the strongest correlation coefficient
for the LT relation also implies the tightest prompt-afterglow relations.
This subsample can be used as a standard one for astrophysical and cosmological studies.\\
In the middle panel of Fig. \ref{fig:16}, the correlation coefficients $\rho$ are shown for the following distributions: $(\log <L_{\gamma,iso}>_{45},\log <L_{\gamma,iso}>_{90},
\log <L_{\gamma,iso}>_{T_{X,p}},\log E_{\gamma,prompt})-\log L_{X,a}$, represented by different colours,
namely red, black, green and blue respectively.\\
No significant relations for the IC bursts have been found out. However, the paucity of the data does not allow a
definitive statement.
From this analysis, it is clear that the inclusion of the IC GRB class does not strengthen the existing relations.
\begin{table}[htbp]
\footnotesize
\begin{center}
\begin{tabular}{|c|l c|c c|}
\hline
& \multicolumn{2}{c|}{E4} & \multicolumn{2}{c|}{E0095} \\
\hline
Correlations & \multicolumn{2}{l|}{\hspace{20pt}$\rho$ \hspace{60pt} (b, a)} & \multicolumn{2}{l|}{\hspace{20pt}$\rho$ \hspace{60pt} (b, a)} \\
\cline{2-5} & \multicolumn{2}{l|}{\hspace{95pt}{P}} & \multicolumn{2}{l|}{\hspace{95pt}{P}} \\
\hline
$ L_{X,a}- <L_{\gamma,iso}>_{45}$ & 0.59 & $(0.67_{-0.15}^{+0.14}, 20.58 _{-6.73}^{+6.66})$ & 0.95 & $(0.73 _{-0.11}^{+0.16}, 17.90_{-6.0}^{+5.29})$ \\
& 0.62 & $7.7 \times 10^{-8}$ & 0.90& $2.3 \times 10^{-3}$ \\
$ L_{X,a}- <L_{\gamma,iso}>_{90}$ & 0.60 & $(0.63 _{-0.16}^{+0.15}, 22.05_{-7.31}^{+7.14})$ & 0.93 & $(0.84 _{-0.12}^{+0.11}, 11.86_{-3.44}^{+3.43})$ \\
& 0.62 & $7.7 \times 10^{-8}$ &0.94 & $2.7 \times 10^{-3}$\\
$ L_{X,a}- <L_{\gamma,iso}>_{T_{X,p}}$ & 0.46 & $(0.73_{-0.14}^{+0.09}, 16.61_{-4.35}^{+4.35})$ & 0.95 & $(0.93_{-0.23}^{+0.20},7.70_{-3.46}^{+3.47})$\\
& 0.56 & $2.21 \times 10^{-6}$ &0.90 & $2.3 \times 10^{-3}$\\
$ L_{X,a}- E_{\gamma,prompt}$ & 0.43 & $(0.52 _{-0.06}^{+0.07}, 28.03_{-2.97}^{+2.98})$ & 0.83 & $(0.49 _{-0.16}^{+0.21}, 29.82 _{-7.82}^{+7.11})$ \\
&0.52 & $1.4 \times 10^{-5}$ & 0.75 & $3.2\times 10^{-2}$ \\
$ T^*_{X,a}- E_{\gamma,prompt}$ & -0.19 & $(-0.49_{-0.08}^{+0.09},54.51_{-0.30}^{+0.37})$ & -0.81 &$(-0.96_{-0.22}^{+0.21}, 54.67_{-0.69}^{+0.69})$\\
&-0.21 & $1.0 \times 10^{-1}$ & -0.69 & $5.8 \times 10^{-2}$\\
$ L_{X,a}- E_{\gamma,peak}$ & 0.54 & $(1.06_{-0.23}^{+0.53}, 43.88_{-1.00}^{+0.61})$ & $0.74$ & $(1.5_{-0.94}^{+0.65}, 43.10_{-2.26}^{+2.53})$\\
& 0.51& $2.2 \times 10^{-5}$ &0.80 & $1.7 \times 10^{-2}$\\
$ T^*_{X,a}- E_{\gamma,peak}$ & -0.36 & $(-0.66_{-0.29}^{+0.20}, 4.96_{-0.80}^{+0.81})$ & -0.74 & $(-1.40_{-0.65}^{+0.66}, 7.04_{-1.77}^{+1.79})$\\
& -0.35& $5.2 \times 10^{-3}$ & -0.77 & $2.5 \times 10^{-2}$ \\
$ <L_{\gamma,iso}>_{45}- E_{\gamma,peak}$ & 0.81 & $(1.14_{-0.25}^{+0.22}, 49.27_{-0.60}^{+0.61})$ & 0.76 & $(1.45_{-0.54}^{+0.26}, 48.48_{-1.04}^{+1.05})$\\
&0.67& $2.6 \times 10^{-9}$ & 0.92& $1.2 \times 10^{-3}$\\
\hline
\end{tabular}
\caption{\footnotesize Correlation coefficients $\rho$, the respective relation fit line parameters (a, b), and the
correlation coefficient $r$ with the respective random occurrence probability $P$, for the considered prompt-afterglow and
prompt-prompt distributions in log scale from \cite{Dainotti11b}.}
\label{tbl9}
\end{center}
\end{table}
In general, this study pointed out that the plateau phase results connected to the inner engine.
In addition, also relations between $\log L_{X,a}$ and several other prompt
emission parameters were analyzed, including $\log E_{\gamma,peak}$ and the variability, $\log V$.
As a result, relevant relations are found between these quantities, except for the variability parameter, see
Table \ref{tbl9}.
\begin{figure}[htbp]
\centering
\includegraphics[width=5.6cm,height=5.85cm,angle=0,clip]{llrelationcopy.pdf}
\includegraphics[width=5.05cm,height=4.85cm,angle=0,clip]{rho1copy.pdf}
\includegraphics[width=5.6cm,height=5cm,angle=0,clip]{fig11bgrupe13.pdf}
\caption{\footnotesize Left panel: $\log L_{X,a}$ vs. $\log <L_{\gamma,iso}>_{45}$ relation ($\log L^*_{X,a}$ and
$\log L^*_p$ respectively) for the E4 data set from \cite{Dainotti11b},
with the fitted relation dashed line in black. The red line is fitted to the 8 lowest error (red) points of the
E0095 subset.
Middle panel: correlation coefficients $\rho$ for $\log\ L_{X,a}-\log\ <L_{\gamma,iso}>_{45}$ (red squares),
$\log\ L_{X,a}-\log\ <L_{\gamma,iso}>_{90}$ (black circles), $\log\ L_{X,a}-\log\ <L_{\gamma,iso}>_{T_{X,p}}$ (green asterisks) and
$\log\ L_{X,a}-\log\ E_{\gamma,prompt}$
(blue squares) relations, obtained using the LGRB subset with the maximum \textcolor{red}{u $=$ }$\sigma_E$ from \cite{Dainotti11b}.
Right panel: Luminosity in the 0.3-10 keV XRT band ($L_{X,a}$) vs. luminosity in the
15-150 keV BAT band ($L_{\gamma,iso}$) from \cite{grupe2013}.
Short bursts are represented by triangles and high-redshift (z $> 3.5$) bursts by circles.}
\label{fig:16}
\end{figure}
Finally, as shown in Table \ref{tbl9}, only
a very small relationship exists between $\log T^{*}_{X,a}-\log E_{\gamma,prompt}$ with $\rho=-0.19$.
Also \cite{grupe2013} claimed the existence of relations between $\log <L_{\gamma,iso}>_{90}$
and $\log L_{X,a}$ (see the right panel of Fig. \ref{fig:16}) and between $\log <L_{\gamma,iso}>_{90}$ and
$\log T^*_{X,a}$ using a sample of 232 GRBs. The latter can be derived straightforwardly from the
$\log T^*_{X,a}-\log E_{\gamma,prompt}$ relation,
being $\log <L_{\gamma,iso}>_{90}$ computed as $\log (E_{\gamma,prompt}/T_{90})$.
\subsection{The \texorpdfstring{$L_{X,peak}-L_X(T_a)$}{Lg} relation}\label{lpromptla}
\cite{dainotti15} further investigated the prompt-afterglow relations presenting an updated analysis of
123 Swift BAT+XRT light curves of LGRBs with known $z$ and afterglow plateau phase. The relation between the peak
luminosity of the prompt phase in the X-ray, $\log L_{X,peak}$, and $\log L_{X,a}$ can be written as follows:
\begin{equation}
\log L_{X,a}=A+B\times \log L_{X,peak},
\end{equation}
with $A=-14.67 \pm 3.46$, $B=1.21^{+0.14}_{-0.13}$ and with $\rho=0.79$ and $P <0.05$, see the left panel of
Fig. \ref{fig:lpeakla}. In the literature $L_{X,peak}$ is denoted as:
\begin{equation}
L_{X,peak}=4\pi\times D_L(z, \Omega_M, h)^2\times F_{X,peak}\times K.
\end{equation}
\begin{figure}[htbp]
\includegraphics[width=0.495\hsize,angle=0]{LxvsLisoredshiftevolutionwithoutlowluminousevent_FINAL.pdf}
\includegraphics[width=0.495\hsize,angle=0]{PeakFluxvsFluxatTaredshiftbinscopy.pdf}
\caption{\footnotesize Left panel: ``GRB distributions in redshift bins on the $\log L_{X,a}$-$\log L_{X,peak}$
plane from \cite{dainotti15}, where $\log L_{X,peak}$ is computed using the approach used in the Second BAT Catalogue. The
sample is split into 4 different equipopulated redshift bins: $z \le 0.84$
(blue), $0.84 \leq z < 1.8$ (magenta), $1.8 \leq z < 2.9$ (green) and $z \geq
2.9$ (red). The dashed line is the fitting relation line". Right panel: ``GRB distributions in redshift bins on the
$\log F_{X,a}-\log F_{X,peak}$ plane from \cite{dainotti15}, where $\log F_{X,peak}$
is computed following the approach used in the Second BAT Catalogue. The
sample is split into 4 different equipopulated redshift bins: $z \le 0.84$
(blue), $0.84 \leq z < 1.8$ (magenta), $1.8 \leq z < 2.9$ (green) and $z \geq
2.9$ (red)".}
\label{fig:lpeakla}
\end{figure}
The relation $<\log L_{\gamma,iso}>-\log L_{X,a}$ \citep{Dainotti11b}
for the same GRB sample presented a correlation coefficient, $\rho=0.60$, smaller than the one of
the $\log L_{X,peak}-\log L_{X,a}$ relation, see sec. \ref{Dainotti2011b}.
This implied that a better definition of the luminosity
or energy parameters improves $\rho$ by 24\%.
In the left panel of Fig. \ref{fig:lpeakla} $\log L_{X,peak}$ is calculated directly from the peak
flux in X-ray, $F_{X,peak}$, considering a broken or a simple power law as the best fit of the spectral model.
Thus, the error propagation due to time and energy
is not involved, differently from the earlier considered luminosities. In addition, \cite{dainotti15} preferred the $\log L_{X,peak}-\log L_{X,a}$ to the
relations presented in \cite{Dainotti11b}, namely the $(\log E_{\gamma,prompt}, \log E_{\gamma,peak})-\log L_{X,a}$,
due to the fact that $\log E_{\gamma,prompt}$ and $\log E_{\gamma,peak}$ can
undergo double bias truncation due to high and low energy detector threshold. Instead, this problem does not appear for $\log L_{X,peak}$ \citep{Lloyd99}.
Furthermore, to show that the $\log L_{X,peak}-\log L_{X,a}$ relation is robust,
the redshift dependence induced by the distance
luminosity was eliminated employing fluxes rather than luminosities. A relation
between $\log F_{X,a}$ and $\log F_{X,peak}$ was obtained with $\rho=0.63$, see the right panel of
Fig. \ref{fig:lpeakla}.
\begin{figure}[htbp]
\includegraphics[width=0.495\hsize,angle=0]{Fig2leftpanelFINAL.pdf}
\includegraphics[width=0.495\hsize,angle=0]{Fig2rightpanel5Dicember.pdf}
\caption{\footnotesize Left panel: $\log L$-$\log T^*$ relation for all the pulses in the prompt (black symbols) and in the
afterglow (red symbols) emissions from \cite{dainotti15}. $\log L$ is the same as $\log L_{X,f}$ for the prompt
emission pulses, while indicates $\log L_{X,a}$
for the afterglow emission pulses, and, the time $\log T^*$ indicates $\log T^*_{X,f}$ for the prompt emission pulses and $\log T^*_{X,a}$
for the afterglow phase. The green points show the maximum luminosity prompt emission pulses ($\log T_{Lmax}$, $\log L_{max}$).
Right panel: $\log E$ vs. $\log T^*$ relation for all the pulses in the prompt (black symbols) and in the
afterglow (red symbols) emissions from \cite{dainotti15}. $\log E$ represents $\log E_{X,f}$
for the prompt emission pulses, while it represents $\log E_{X,plateau}$ for the afterglow emission pulses, and
the time $\log T^*$ indicates $\log T^*_{X,f}$ for the prompt emission pulses and $\log T^*_{X,a}$ for the
afterglow phase. The yellow points display the maximum energy prompt emission pulses ($\log T_{Emax}, \log E_{max}$).}
\label{fig:energylpeakla}
\end{figure}
However, for further details about a quantitative analysis of the selection effects see sec.
\ref{Selection effects}.\\
Finally, \cite{dainotti15} showed that the LT relation has a different slope,
at more than 2 $\sigma$, from the one of the prompt phase relation between the
time since ejection of the pulse and the respective luminosity, $\log L_{X,f}-\log T^{*}_{X,f}$
\citep{willingale2010}, see the left panel of
Fig. \ref{fig:energylpeakla}.
This difference also implied a discrepancy in the distributions of the energy and time, see the right panel of
Fig. \ref{fig:energylpeakla}.
The interpretation of this discrepancy between the slopes opens a new perspective in the theoretical understanding of
these observational facts, see the next section for details.\\
As a further step, \cite{dainotti16a} analyzed this relation
adding as a third parameter $T_{X,a}$ with a sample of 122 LGRBs (without XRFs and GRBs associated to SNe). They found
a tight relation:
\begin{equation}
\log L_{X,a}=(15.69 \pm 3.8) + (0.67 \pm 0.07) \times \log L_{X,peak}-(0.80 \pm 0.07) \log T_{X,a},
\end{equation}
with $\rho=0.93$, $P\leq 2.2\times 10^{-16}$, and $\sigma_{int}= 0.44 \pm 0.03$.
Additionally, the scatter
could be further reduced considering the subsample of 40 LGRBs having light curves with good data coverage and flat
plateaus:
\begin{equation}
\log L_{X,a}=(15.75 \pm 5.3) + (0.67 \pm 0.1) \times \log L_{X,peak}-(0.77 \pm 0.1) \log T_{X,a},
\end{equation}
with $\rho=0.90$, $P=4.41 \times 10^{-15}$, and $\sigma_{int}= 0.27 \pm 0.04$.
These results may suggest the use of this plane as a ``fundamental" plane for GRBs and for
further cosmological studies.
\subsubsection{Physical interpretation of the \texorpdfstring{$L_X(T_a)-L_{\gamma,iso}$ and the $L_{X,peak}-L_X(T_a)$}{Lg} relations}
In \cite{dainotti15}, the two distinct slopes of the luminosity-duration and
the energy-duration distributions of prompt and plateau pulses could reveal that these two are different characteristics
of the radiation: the former may be generated by internal
shocks and the latter by the external shocks. Indeed, if the plateau is produced by synchrotron radiation from the
external shock, then all the pulses have analogous physical conditions (e.g. the power
law index of the electron distribution). In addition, the prompt-afterglow
connections were analyzed in order to better explain the existing physical
models of GRB emission predicting the $\log L_{X,a}-\log L_{\gamma,iso}$ and the $\log L_{X,peak}-\log L_{X,a}$
relations together with the
LT one in the prompt and afterglow phases. They claimed that the model better explaining these
relationships is the one by \cite{hascoet2014}.
In this work they considered two scenarios: one in the standard FS model assuming a modification
of the microphysics parameters to decrease the efficiency at initial stages of the GRB evolution; in the
latter the early afterglow stems from a long-lived RS in the FS scenario. In the FS
scenario a wind external medium is assumed together with a microphysics parameter $\epsilon_e \propto n^{-\nu}$, the amount of the internal energy going into electrons
(or positrons), where n is the density medium. In the case of $\nu \approx 1$ is possible to reproduce a flat plateau. Thus, even operating on
just one parameter can lead to the formation of a
plateau that also reproduces the $\log L_{X,a}-\log L_{X,iso}$ and the $\log L_{X,peak}-\log L_{X,a}$ relations.
Alternatively, in the RS scenario, in order to obtain the observed prompt-afterglow relationships, the typical
$\Gamma$ of the ejecta should rise with the burst energy.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.95\hsize,angle=0]{fig3kazanas15.pdf}
\caption{\footnotesize ``The histogram of the BAT to XRT flux ratio for a number of Swift GRB from \cite{kazanas15}.
The distribution shows clearly a
preferred value for this ratio of order $\sim 10^3-10^4$. The vertical line shows also the proton to electron mass
ratio $m_p/m_e$".}
\label{fig:kazanas15}
\end{figure}
In addition, \cite{ruffini14} pointed out that the induced gravitational collapse paradigm
can recover the $\log L_{X,a}-\log L_{\gamma,iso}$ and the $\log L_{X,peak}-\log L_{X,a}$ relations. This model
considers the very energetic ($10^{52}-10^{54}$ erg) LGRBs for which the SNe has been seen.
The light curves were built assuming for the external medium either a radial structure for the wind
\citep{Guida2008,bernardini06,bernardini07,caito09} or a partition of the
shell \citep{Dainotti2007}, therefore well matching the afterglow plateau and the prompt emission.\\
Recently, \cite{kazanas15} within the context of the Supercritical Pile GRB Model claimed that they can reproduce
the $\log L_{X,a}-\log L_{\gamma,iso}$
and the $\log L_{X,peak}-\log L_{X,a}$ relations, because the ratio, R, between the luminosities appears consistent with the
one between the mean prompt energy flux from BAT and the afterglow plateau fluxes detected by XRT. In particular,
$R \approx 2000$ is close to the proton to electron mass ratio, see Fig. \ref{fig:kazanas15}.\\
Indeed, this is a new challenge for theoretical modelling that would need to consider, simultaneously, the several
prompt-afterglow connections in order to better reproduce the phenomenology of the relations from a statistical point of view.
\subsection{The \texorpdfstring{$L^F_{O,peak}-T^{*F}_{O,peak}$}{Lg} relation and its physical interpretation}\label{lisotp}
\cite{Liang2010} studied the relation between the width of the light curve
flares, $w$, and $T_{O,peak}$ of the flares, denoted with the index F, using a sample of 32 Swift GRBs, see the left panel of Fig. \ref{fig:liang3}.
This relation reads as follows:
\begin{equation}
\log w^F = (0.05 \pm 0.27) + (1.16 \pm 0.10) \times \log T_{O,peak}^{F},
\end{equation}
with $\rho=0.94$.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\hsize,angle=0]{figwtpliang.pdf}
\includegraphics[width=0.47\hsize,angle=0]{f5a.pdf}
\caption{\footnotesize Left panel: $\log w^F$-$\log T_{O,peak}^F$ distribution from \cite{Liang2010}.
Right panel: $\log w^F$-$\log T_{O,peak}^F$ distribution from \cite{Li2012}. In both panels lines represent the best fit.}
\label{fig:liang3}
\end{figure}
Later, \cite{Li2012} found the same relation as \cite{Liang2010}, but with smaller values of normalization
and slope, using 24 flares from 19 single-pulse GRBs observed with CGRO/BATSE\footnote{Among the instruments of the
Compton Gamma Ray Observatory (CGRO) satellite, running from 1991 to 2001, the
Burst and Transient Source Experiment (BATSE) played a
fundamental role in the measurements of GRB spectral features in the range from $20$ keV to $8$ MeV. Bursts were typically
detected at rates of roughly one per day over the 9-year CGRO mission within a time interval ranging from $\sim0.1$ s up to
about $100$ s. Therefore, this satellite enabled careful analysis of the spectral properties of the GRB prompt emission.},
see the right panel of Fig. \ref{fig:liang3}.
However, for these 19 GRBs only in 14 GRBs a flare activity is distinctly visible.
The relationship was given by:
\begin{equation}
\log w^F = -0.32 + 1.01 \times \log T^F_{O,peak}.
\end{equation}
They claimed that earlier flares are brighter and narrower than later ones.
They compared the $w^F-T^F_{O,peak}$ distribution for the
X-ray flares detected by Swift/XRT with the one for the
optical flares in the R band. As a conclusion, they seemed to have a similar behaviour
\citep{chincarini07,margutti2010}, see the right panel of Fig. \ref{fig:liang3}.\\
Furthermore, in the rest frame band, they found a relation between the
$L_{O,peak}$ of the flares in the R energy in units of $10^{48}$ erg s$^{-1}$ and $T^*_{O,peak}$ of the flares
using 19 GRBs, see Fig. \ref{fig:li2012}.
Both prompt pulses and X-ray and optical flares are correlated and present a visible temporal evolution, as seen in
Fig. \ref{fig:li2012}. This relation is given by:
\begin{equation}
\log L^{F}_{O,peak}=(1.89\pm 0.52)-(1.15\pm0.15)\times \log T^{*F}_{O,peak},
\end{equation}
with $\rho=0.85$ and $P<10^{-4}$.
$T^{F}_{O,peak}$ spans from $\sim$ tens of seconds to $\sim 10^6$ seconds, instead the $L^{F}_{O,peak}$ varies from $10^{43}$ to
$10^{49}$ erg s$^{-1}$, with an average value of $10^{46}$ erg s$^{-1}$.
In addition, considering only the most luminous GRBs, they found that $T^{*F}_{O,peak}$ was strongly anti-correlated to
$E_{\gamma,prompt}$ in the $1-10^4$ keV energy band:
\begin{equation}
\log T^{*F}_{O,peak}=(5.38\pm 0.30)-(0.78\pm 0.09)\times \log E_{\gamma,prompt}/10^{50},
\end{equation}
with $\rho=0.92$.
These outcomes revealed that the GRB flares in the optical wavelength with higher $E_{\gamma,prompt}$ peak earlier and
are much more luminous. In Table \ref{tbl2} a summary of the relations described in this section is displayed.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.95\hsize,angle=0]{f5bcopy.pdf}
\caption{\footnotesize $\log L^F_{O,peak}$-$\log T_{O,peak}^{*F}$ relation from \cite{Li2012}. Lines represent
the best fits, black dots indicate optical flares, and the grey circles with errors show X-ray flares associated
with the optical flares.}
\label{fig:li2012}
\end{figure}
\begin{table}[htbp]
\footnotesize
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Correlations & Author & N & Slope& Norm & Corr.coeff. & P \\
\hline
$w^F-T^F_{O,peak}$ & Liang et al. (2010) & 32 & $1.16^{+0.10}_{-0.10}$&$0.05^{+0.27}_{-0.27}$&0.94& $<10^{-4}$\\
&Li et al. (2012)&19&1.01&-0.32&&\\
\hline
$L^{F}_{O,peak}-T^{*F}_{O,peak}$ & Li et al. (2012)&19&$-1.15^{+0.15}_{-0.15}$& $1.89^{+0.52}_{-0.52}$&0.85&$<10^{-4}$ \\
\hline
$T^{*F}_{O,peak}-E_{\gamma,prompt}$ &Li et al. (2012)&19&$-0.78^{+0.09}_{-0.09}$&$5.38^{+0.30}_{-0.30}$&0.92&$<10^{-4}$ \\
\hline
\end{tabular}
\caption{\footnotesize Summary of the relations in this section. The first column represents the relation in log scale,
the second one the authors, the third one the number of GRBs in the used sample, and the fourth and the fifth columns are the slope and
normalization of the relation. The last two columns are the correlation coefficient and the chance
probability, P.}
\label{tbl2}
\end{center}
\end{table}
As regards the physical interpretation of the $L^F_{O,peak}-T^{*F}_{O,peak}$ relationship,
\cite{Li2012} found that the flares are separated
components superimposed to the afterglow phase. The coupling between $L^{F}_{O,peak}$ and $T^{*F}_{O,peak}$ suggested that the prompt $\gamma$-ray and
late optical flare emission may arise from the same mechanism, namely from a central
engine that can periodically eject a number of shells during the emission. Impacts of these shells could
create internal shocks or magnetic turbulent reconnections,
which would emerge from the variability \citep{kobayashi97,zhang2011}. \cite{fenimore95} revealed
no relevant pattern in the width and intensity distributions using gamma ray data only.
In addition, the usual tendency of the $w^F-T^F_{O,peak}$ relation
cannot be due to hydrodynamical diffusion of the shells emitted at recent times, but it is necessary that the central engine radiates
thicker and fainter shells at late stages \citep{maxham09}. This could be explained as flares generated by clumps,
such that the diffusion during the accretion mechanism would extend the accretion duration onto the BH \citep{perna06,proga06}.
\section{Selection Effects} \label{Selection effects}
Selection effects are distortions or biases that usually occur when the sample observed is not representative of the
``true'' population itself. This kind of biases usually affects GRB relations.
\cite{Efron1992}, \cite{Lloyd99}, \cite{Dainotti2013a,Dainotti2015b} and \cite{petrosian14} emphasized that when
dealing with a multivariate data set, it is imperative to determine first the true
relations among the variables, not those introduced by the observational selection effects,
before obtaining the individual distributions of the variables themselves.
This study is needed for claiming the existence of the intrinsic relations. A relation can be called
intrinsic only if it is carefully tested and corrected for these biases.\\
The selection effects present in the relations discussed above are mostly due to the dependence of
the parameters on the redshift,
like in the case of the time and the luminosity evolution, or due to the threshold of the detector used.\\
In this section, we describe several different methods to deal with selection biases.\\
In paragraph \ref{redind}, we discuss the redshift induced relation
through a qualitative method, while in \ref{efronpetmethod} we present a more quantitative approach through the EP method.
In \ref{intrinsicorrelation}, we describe how to obtain the intrinsic relations corrected by selection biases, and
in \ref{ghisellinisel} we report the selection effects for the optical and X-ray luminosities. Lastly, in \ref{montecarlosimulation}
we show the evaluation of the intrinsic relation through Monte Carlo simulations.
\subsection{Redshift induced relations}\label{redind}
An important source of possible selection effects is the dependence of the variables on the redshift. To this
end, \cite{dainotti11a} investigated the redshift evolution of the parameters of the LT relation, because a change of
the relation slope has been observed when comparing several analyses \citep{Dainotti2008,Dainotti2010}. Namely,
in the first paper, it was found $b=-0.74^{+0.20}_{-0.19}$ and in the latter $b=-1.06^{+0.27}_{-0.28}$. Therefore, it became crucial to
understand the reason of this change, even if the two slopes are still comparable at the 1-$\sigma$ level.
The distribution of the 62 LGRBs in the sample is not uniform within the range $(z_{min}, z_{max}) = (0.08, 8.26)$ with
few data points at large redshifts. Even if this sample is sparse, it was important to investigate whether the calibration
coefficients $(a, b, \sigma_{int})$ were in agreement within the error bars over this large redshift interval, see the left panel
of Fig. \ref{fig:redshiftbin}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.495\hsize,angle=0]{plotbinzed.pdf}
\includegraphics[width=0.495\hsize,angle=0]{fig1Bcopy.pdf}
\caption{\footnotesize Left panel: ``$\log L_{X,a}-\log T^*_{X,a}$ relation divided in the three redshift
bins $Z1=(0.08, 1.56)$, $Z2=(1.71, 3.08)$ and $Z3=(3.21, 8.26)$ from \cite{dainotti11a}. With the blue points it is
represented the $Z1$ sample, with the green ones the $Z2$ sample and with the red points the $Z3$ sample. The respective fitted
lines are
in the same colours". Right panel: ``$\log L_{X,a}-\log T^*_{X,a}$ distribution from \cite{Dainotti2013a}
for the sample of 101 GRB afterglows divided in $5$ equipopulated redshift bins shown by different colours: black for
$z < 0.89$, magenta for $0.89 \leq z \leq 1.68$,
blue for $1.68 < z \leq 2.45$, green $2.45 < z \leq 3.45$, red for $ z \geq 3.45$. Solid
lines show the fitted relations".}
\label{fig:redshiftbin}
\end{figure}
For this reason, the data set was separated in three redshift bins with the same number of elements,
$Z1=(0.08, 1.56)$, $Z2=(1.71, 3.08)$ and $Z3=(3.21, 8.26)$ presented as blue, green and red points respectively in the left panel
of Fig. \ref{fig:redshiftbin}. The results are presented in Table \ref{tbl8}.
\begin{table}[htbp]
\footnotesize
\begin{center}{
\begin{tabular}{|c|c|c|c|c|}
\hline
Id & $\rho$ & $(b, a, \sigma_{int})_{bf}$ & $b_{median}$ & $(\sigma_{int})_{median}$ \\
\hline
Z1 & -0.69 & (-1.20, 51.04, 0.98) & $-1.08_{-0.30}^{+0.27}$ & $1.01_{-0.16}^{+0.20}$ \\
Z2 & -0.83 & (-0.90, 50.82, 0.43) & $-0.86_{-0.16}^{+0.18}$ & $0.45_{-0.08}^{+0.09}$ \\
Z3 & -0.63 & (-0.61, 50.14, 0.26) & $-0.58_{-0.15}^{+0.14}$ & $0.26_{-0.06}^{+0.07}$ \\
\hline
\end{tabular}}
\caption{\footnotesize Results of the calibration procedure for GRBs divided in three
equally populated redshift bins with $(z_{min}, z_{max}) = (0.08, 1.56)$,
$(1.71, 3.08)$, $(3.21, 8.26)$ for bins Z1, Z2, Z3 from \cite{dainotti11a}. The subscript $bf$ displays
the best fit values, while the $median$ subscript shows the median values.}
\label{tbl8}
\end{center}
\end{table}
The correlation coefficient $\rho$ was found quite high in each redshift bins, supporting the
independence of the LT relation on $z$. The slopes $b$ for
bins $Z1$ and $Z2$ are comparable within the $68\%$ CL, while the slopes in bins $Z1$ and $Z3$ only
within the $95\%$ CL, see Table \ref{tbl8}. On the contrary, the normalization $a$ is comparable in all the bins.
From this analysis, it is not possible to confirm that the LT relation is shallower for larger
$z$ GRBs, due to the low number of data points and the presence of high $\sigma_E$ GRBs. Finally,
bigger samples with small $\sigma_E$ values and a more uniform $z$ binning are required to overcome this problem.\\
For this reason, \cite{Dainotti2013a} performed a similar analysis, but with a larger sample consisting of $101$ GRBs.
Specifically, this updated sample was split in $5$ redshift ranges with the same number of elements,
thus having $20$ GRBs in each subgroup, represented in the right panel of Fig. \ref{fig:redshiftbin} by different
colours: black for $z < 0.89$, magenta for $0.89 \leq z \leq 1.68$, blue for $1.68 < z \leq 2.45$, green for
$2.45 < z \leq 3.45$ and red for $ z \geq 3.45$. The fitted lines for each
redshift bin are also shown in the same colour code. The distribution of the subsamples presented different power law
slopes when
the whole sample was divided into bins. The objects in the different bins exhibited some separation into different regions
of the LT plane.
Moreover, the slope of the relation for each redshift bin versus the averaged redshift range has also been presented,
see the left panel of Fig. \ref{fig:slopevsz}.\\
In addition, in \cite{Dainotti2015b}, the updated sample of $176$ GRBs was divided into $5$ redshift bins consisting
of about 35 GRBs for each group, as shown in the right panel of Fig. \ref{fig:slopevsz}. A small evolution in $z$ has
been confirmed
with the following linear function $b(z)=0.10z-1.38$.
\begin{figure}[htbp]
\includegraphics[width=0.495\hsize,angle=0]{fig1Ccopy.pdf}
\includegraphics[width=0.485\hsize,angle=0]{slopevszlin.pdf}
\caption{\footnotesize Left panel: ``the variation of $b$ (and its error range) with the mean value of the
redshift bins from \cite{Dainotti2013a}". Right panel: ``$\alpha_{\tau}$, which is equivalent to the slope $b$,
vs. z using a linear function $\alpha_{\tau}=0.10 z-1.38$ from \cite{Dainotti2015b}".}
\label{fig:slopevsz}
\end{figure}
Regarding the $\log L_{X,peak}-\log L_{X,a}$ relation,
\cite{dainotti15} showed that it is not produced by the dependence on the redshift of its variables.
To estimate the redshift
evolution, the sample was separated into $4$ redshift bins as shown in the left panel of Fig. \ref{fig:lpeakla}.
The GRB distribution in each bin is not grouped
or constrained within a specific region, therefore indicating no strong redshift evolution.
For $\log L_{X,a}$ it was found that there was negligible
redshift evolution of the afterglow X-ray luminosity \citep{Dainotti2013a}, while for $\log L_{X,peak}$ has been
demonstrated that there is significant
redshift evolution \citep{yonetoku04,petrosian14,dainotti15}.
For more details, see sec. \ref{luminosityev} and \ref{timeev}.
\subsection{Redshift induced relations through Efron and Petrosian method}\label{efronpetmethod}
For a quantitative study of the redshift evolution, which is the dependence of the variables on the redshift, we here refer to the
EP method which is specifically designed to overcome the biases resulting from incomplete data.
The Efron \& Petrosian technique, applied to GRBs \citep{petrosian09,Lloyd99,Lloyd2000}, allows to compute the intrinsic
slope of the relation by creating new bias-free observables, called local variables and denoted with the symbol ${'}$.
For these quantities,
the redshift evolution and the selection effects due to instrumental thresholds are removed.
The EP method uses a modification of the Kendall tau test\footnote{The Kendall $\tau$ is a non-parametric statistical test used to
measure the association between two measured quantities. It is a measure of rank relation: the similarity of the
orderings of the data when ranked by each of the quantities.} $\tau$ to compute the best fit values of the parameters which represent
the luminosity and time evolutionary functions. For details about the definition of $\tau$ see \cite{Efron1992}.
\subsubsection{Luminosity evolution}\label{luminosityev}
The relation between luminosity and $z$ is called luminosity evolution.
We discuss the luminosity evolution for both prompt and plateau phases. Before applying the EP method to the
plateau phase, the limiting plateau flux, $F_{\rm{lim}}$, which gives the minimum observed luminosity for a given
$z$ needs to be parameterized.
The XRT sensitivity, $F_{lim,XRT}=10^{-14}$ erg cm$^{-2}$ s$^{-1}$, is not high enough to represent
the truncation of the data set.
Hence, as claimed by \cite{cannizzo2011}, a better choice for the flux threshold is $10^{-12}$ erg cm$^{-2}$ s$^{-1}$.
Several threshold fluxes were analyzed \citep{Dainotti2013a}, finally
$F_{lim,XRT} = 1.5 \times $10$^{-12}$ erg cm$^{-2}$ s$^{-1}$, which leaves 90 out of 101 GRBs, was selected (see the left panel of
Fig. \ref{fig:19}).
Regarding instead the prompt limiting flux, \cite{dainotti15} chose a BAT flux limit
$F_{lim,BAT}=4\times10^{-8}$ erg cm$^{-2}$ s$^{-1}$, which also allows 90\% of GRBs in the sample, see the right panel of Fig. \ref{fig:19}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.49\hsize,angle=0,clip]{selection.pdf}
\includegraphics[width=0.48\hsize,angle=0,clip]{sellumprompt.pdf}
\caption{\footnotesize Left panel: ``the bivariate distribution of $\log L_{X,a}$ and $z$ with two different flux limits
from \cite{Dainotti2013a}. The instrumental XRT flux limit, $1.0\times10^{-14}$ erg cm$^{-2}$ s$^{-1}$ (dashed
green line), is too low to be representative of the flux limit, $1.5 \times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$ (solid
red line) represents better the limit of the sample". Right panel: ``the bivariate distribution of $\log L_{X,peak}$ and $z$ with the
flux limit assuming the K correction $K = 1$ from \cite{dainotti15}. The BAT flux limit, $4.0 \times 10^{-8}$ erg cm$^{-2}$ s$^{-1}$
(solid red line), better represents the limit of the sample".}
\label{fig:19}
\end{figure}
In \cite{Dainotti2013a}, the relation function, g(z), is defined when determining the evolution of $L_{X,a}$
so that the local variable $L'_{X,a} \equiv L_{X,a}/g(z)$ is not dependent anymore from $z$.
The evolutionary function is parameterized by a simple relation function:
\begin{equation}
g(z)=(1+z)^{k_{L_{X,a}}}.
\label{lxev}
\end{equation}
More complex evolution functions lead to comparable results, see Dainotti et al. (2013a, 2015b).
\begin{figure}[htbp]
\centering
\includegraphics[width=5.05cm,height=4.9cm,angle=0,clip]{ltevolution.pdf}
\includegraphics[width=5.6cm,height=5.2cm,angle=0,clip]{ltevlumprompt.pdf}
\includegraphics[width=5.6cm,height=5.3cm,angle=0,clip]{complexluminosity.pdf}
\caption{\footnotesize Left panel: $\tau$ vs. $k_{L_{X,a}}$ from \cite{Dainotti2013a}.
The red line indicates the full sample, while the green dotted line indicates the sample of 47 GRBs
in common with the 77 LGRBs in \cite{dainotti11a}. Middle panel: $\tau$ vs. $k_{L_{X,peak}}$, using the
eq. \ref{lxev}, from \cite{dainotti15}. Right panel: $\tau$ vs. $k_{L_{X,peak}}$, using the eq. \ref{compl1}, from
\cite{dainotti15}.}
\label{fig:18bis}
\end{figure}
With this modified version of $\tau$, the value of $k_{L_{X,a}}$ for which
$\tau_{L_{X,a}} = 0$ is the one that best represents the luminosity evolution at the 1 $\sigma$ level.
$k_{L_{X,a}}=-0.05_{-0.55}^{+0.35}$ means that this evolution is negligible, see the left panel of Fig. \ref{fig:18bis}.
In the same panel, this distribution is also plotted
for a smaller sample of 47 GRBs (green dotted line) in common with the previous one of 77 LGRBs presented in \cite{dainotti11a}.\\
The results of the afterglow luminosity evolution among the two samples are compatible at 2 $\sigma$.
Instead, regarding the study of the evolution of $L_{X,peak}$, the simple relation function
(see eq. \ref{lxev}) was compared to a more complex function \citep{dainotti15} given by:
\begin{equation}
g(z)=\frac{Z^{k_L}(1+Z_{cr}^{k_L})}{Z^{k_L}+Z_{cr}^{k_L}},
\label{compl1}
\end{equation}
where $Z = 1 + z$ and $Z_{cr} = 3.5$. A relevant luminosity evolution was obtained in the prompt,
$k_{L_{X,peak}} = 2.13^{+0.33}_{-0.37}$, using the simple relation, while $k_{L_{X,peak}} = 3.09^{+0.40}_{-0.35}$
for the more complex function, see the middle and right panels of Fig. \ref{fig:18bis} respectively.
The results of the prompt luminosity evolution among the two different functions are compatible at 2 $\sigma$.
\subsubsection{Time Evolution}\label{timeev}
Similarly to the treatment of the luminosity evolution, one has also to determine the limit of the plateau end time,
$T^{*}_{X,a,{lim}}= 242/(1+z)$ s \citep{Dainotti2013a}, and of the prompt peak time
$T^*_{X,prompt,lim} = 1.74/(1+z)$ s \citep{dainotti15}, see the left and right panels of Fig. \ref{fig:19bis}
and Fig. \ref{fig:grupetime} respectively.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.495\hsize,angle=0,clip]{selection1.pdf}
\includegraphics[width=0.495\hsize,angle=0,clip]{selectiontimeprompt.pdf}
\caption{\footnotesize Left panel: ``the bivariate distribution of the rest frame time $\log T^{*}_{X,a}$ and $z$ from
\cite{Dainotti2013a}. The red line is the limiting rest frame time, $\log (T_{X,a,lim}/(1 + z))$ where the chosen limiting value of the
observed end-time of the plateau in the sample is $T_{X,a,lim} = 242$ s". Right panel: ``the bivariate distribution of the rest frame
time $\log T^*_{X,prompt}$ and $z$ from \cite{dainotti15}, where with $\log T^*_{X,prompt}$ they denoted the sum of the
peak pulses width of each single
pulse in each GRB. The chosen limiting value of the observed pulse width in the sample is $\log T_{X,prompt,lim} = 0.24$ s.
The red line is the limiting rest frame time, $\log (T_{X,prompt,lim}/(1 + z))$".}
\label{fig:19bis}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1\hsize,angle=0,clip]{f8grupe13.pdf}
\caption{\footnotesize Distributions between redshift and the observed (left panel) and rest-frame (right panel)
$T_{90}$ in the BAT energy range from \cite{grupe2013}.}
\label{fig:grupetime}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=5.05cm,height=4.75cm,angle=0,clip]{ltevolution1.pdf}
\includegraphics[width=5.6cm,height=4.65cm,angle=0,clip]{ltevtimepromptcopy.pdf}
\includegraphics[width=5.6cm,height=4.65cm,angle=0,clip]{complextimecopy.pdf}
\caption{\footnotesize Left panel: $\tau$ vs. $k_{T^{*}_{X,a}}$ from \cite{Dainotti2013a}. The red line indicates the full sample, while the
green dotted line indicates the 47 GRBs in common with the sample presented in \cite{dainotti11a}. Middle panel: $\tau$
vs. $k_{T^*_{X,prompt}}$, using the eq. \ref{taev}, from \cite{dainotti15}. Right panel: $\tau$ vs. $k_{T^*_{X,prompt}}$, using the eq. \ref{compl2}, from \cite{dainotti15}.}
\label{fig:18}
\end{figure}
To determine the evolution of $T^*_{X,a}$, so that the de-evolved variable $T'_{X,a} \equiv T^{*}_{X,a}/f(z)$ is not correlated
with z, the relation function $f(z)$ \citep{Dainotti2013a} was specified:
\begin{equation}
f(z)=(1+z)^{k_{T^{*}_{X,a}}}.
\label{taev}
\end{equation}
The values of $k_{T^{*}_{X,a}}$ for which $\tau_{T^{*}_{X,a}} = 0$ is the one that best matches the plateau end time evolution
at the 1 $\sigma$ uncertainty. $\tau_{T^{*}_{X,a}}$ versus $k_{T^{*}_{X,a}}$ distribution
shows a consistent evolution in $T^{*}_{X,a}$, as seen in the left panel of Fig. \ref{fig:18}, namely
$k_{T^{*}_{X,a}}=-0.85_{-0.30}^{+0.30}$. In the same panel this distribution is also displayed for a smaller
sample of 47 GRBs (green dotted line) in common with the previous one of 77 GRBs presented
in \cite{dainotti11a}.
The results of the afterglow time evolution among the two samples are compatible at 1.5 $\sigma$.\\
Regarding the prompt time evolution,
a more complex function was also used in addition to the simple relation function \citep{dainotti15}:
\begin{equation}
f(z)=\frac{Z^{k^*_T}(1+Z_{cr}^{k^*_T})}{Z^{k^*_T}+Z_{cr}^{k^*_T}},
\label{compl2}
\end{equation}
where $Z = 1 + z$ and $Z_{cr} = 3.5$.
As a conclusion, a not relevant time evolution in the prompt was found for both the simple function,
$k_{T^*_{X,prompt}}= -0.62^{+0.38}_{-0.38}$, and for the more complex one $k_{T^*_{X,prompt}}=-0.17^{+0.24}_{-0.27}$,
see the middle and right panels of Fig. \ref{fig:18} respectively.
The results of the prompt time evolution among the two different functions are compatible at 1 $\sigma$.
\subsection{Evaluation of the intrinsic slope}\label{intrinsicorrelation}
The last step to determine if a relation is intrinsic is to evaluate its ``true" slope.
To this end, the EP method was used in the local time ($T'_{X,a}$) and luminosity ($L'_{X,a}$) space
obtaining an intrinsic slope for the LT relation $b_{int}=1/\alpha=-1.07^{+0.09}_{-0.14}$.
The significance of this relation is at 12 $\sigma$ level. It
can be derived directly from the left panel of Fig. \ref{fig:17} \citep{Dainotti2013a}, because
if there was no relation it would have been that $\tau=0$ for $b_{int}=0$ at 1 $\sigma$.\\
Instead, regarding the evaluation of the intrinsic slope in the $\log L_{X,peak}-\log L_{X,a}$ relation,
\cite{dainotti15} used a different method, namely the partial correlation coefficient.
This is the degree of association between two random variables calculated as a function of $b_{int}$ in
the following way:
\begin{equation}
r_{L^{'}_{X,peak} L^{'}_{X,a}, D_L}=\frac{r_{L^{'}_{X,peak},L^{'}_{X,a}}-r_{L^{'}_{X,peak},D_L}*r_{L^{'}_{X,a},D_L}}{(1-r^{2}_{L^{'}_{X,peak},D_L})*{(1-r^{2}_{L^{'}_{X,a},D_L})}},
\end{equation}
where $\log L^{'}_{X,a}=L^{'}_{X,a}$ and $\log L^{'}_{X,peak}=L^{'}_{X,peak}$.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.43\hsize,angle=0,clip]{evolutionalphacopy.pdf}
\includegraphics[width=0.54\hsize,angle=0,clip]{Pearsonintrinsicslopeluminosities_FINAL.pdf}
\caption{\footnotesize Left panel: $\tau$ vs. $b_{int}$ (indicated with $\alpha$ in the picture) from \cite{Dainotti2013a}.
Right panel: r vs. $b_{int}$ from \cite{dainotti15} with the
best value where $\log L_{X,peak}$ and $\log L_{X,a}$ are strongly correlated (the central thick line). The two thinner
lines indicate the $0.05\%$ probability that the sample is drawn by chance.}
\label{fig:17}
\end{figure}
As displayed in the right panel of Fig. \ref{fig:17}, the relation is
highly significant when $b_{int}= 1.14^{+0.83}_{-0.32}$, which is at 1 $\sigma$ of the observed slope.\\
In addition, following an analysis similar to the one of \cite{Butler2010}, \cite{Dainotti2015b} simulated a sample for
which biases on both time and luminosity are considered. Particularly, they assumed the biases to be roughly the same whichever
monotonic efficiency function for the luminosity detection is taken. This method presented how an unknown
efficiency function could affect the slope
of any relation and the GRB density rate. Then, biases in slope or normalization
caused by the truncations were analyzed. This gave distinct fit values that allow for studying the scatter of the relation and its
selection effects. This analysis has shown, together with the one in \cite{Dainotti2013a}, that the LT
relation can be corrected by selection effects and therefore can be used in principle as redshift estimator
(see sec. \ref{redshiftestimator}) and as a valuable cosmological tool (see sec. \ref{cosmology}).
As regards other relations, \cite{davanzo12} for the $L_{X,a}-E_{\gamma,prompt}$ relation,
\cite{oates15} for the $L_{O,200\rm{s}}-\alpha_{O,>200\rm{s}}$ relation,
and \cite{racusin16} for the $L_{X,200\rm{s}}$\,-\,$\alpha_{X,>200\rm{s}}$ relation, also used the partial correlation
coefficient method to show that the redshift dependence does not induce these relations.
\subsection{Selection effects for the optical and X-ray luminosities}\label{ghisellinisel}
In this section we discuss the selection effects due to the limiting optical and X-ray luminosities
relevant for the relations mentioned above.
\cite{nardini08a} investigated
if the observed luminosity distribution can be the result of selection effects by studying the optically dark afterglows.
By simulating the $\log L_{O,12}$, $z$, host galaxy dust
absorption, $A^{host}_V$, and telescope limiting magnitude for each of the 30000 GRBs, the observed optical luminosity
distribution was contrasted to
the simulated one.
From this simulated distribution regarding the intrinsic one, it is necessary
to take only GRBs with a flux which is larger than the threshold flux of the associated detector. This corresponds
with a lower luminosity truncation, which is around $\log \ L_{O,12} \approx 31.2$ (erg s$^{-1}$ Hz$^{-1}$). Therefore, the
fact that we do not
observe GRBs with such a luminosity puts a limit to the luminosity function.\\
They also checked statistically the presence of a low luminosity category of events which are at $3.6$
$\sigma$ off the central value of the distribution.
They pointed out that if the absorption is chromatic, the observed luminosity distribution does not match with any unimodal
one. If many GRBs are absorbed by
``grey" achromatic dust, then a unimodal luminosity distribution
can be obtained.
In summary, dark bursts could belong to an optically subluminous group or to a category of bursts for which
a high achromatic absorption is present.\\
As regards the evaluation of the selection effects of $L_{O,peak}$, the biases
in the detection of $F_{O,peak}$ need to be considered.
As found from \cite{panaitescu08}, for a typical optical afterglow spectrum ($F_{O,a}\propto T_{O,a}^{-1}$),
variations in
the observer offset angle induce a $\log F_{O,peak}-\log T_{O,peak}$ anti-relation that is flatter than
what is measured. In fact, an observational selection
effect could steepen the slope of the anti-relation between $\log F_{O,peak}$ and $\log T_{O,peak}$.\\
In addition, SGRBs observed by Swift seem to be fluence-limited, while LGRBs detected with the same telescope
are flux-limited \citep{gehrels08} due to the instrument trigger.\\
\cite{nysewander09} pointed out that the ratio $F_{O,11}/F_{X,11}$ may be influenced
by absorption of photons in the host galaxy. Furthermore, they showed that $F_{X,11}$ should
be precise, because the LGRBs observed in the XRT passband do not present X-ray column absorptions, differently
from the majority of LGRBs.
The computed optical absorption of LGRB afterglows indicates smaller column densities ($N_H$)
than in the X-ray, with optical absorptions ($A_V$) about one-tenth to one magnitude
\citep{shady07,cenko09}. Regarding the SGRBs, they have more luminous optical emission relative to the X-ray
than what is assumed by the standard model.
Later, \cite{kann10} claimed that the grouping of the optical luminosity at the time of 1 day,
$L_{O,1\rm{d}}$, is less remarkable than the one described by \cite{liang06} and \cite{nardini06a}
for GRBs observed by Swift.
This suggested that the grouping pointed out in pre-Swift data can
be due to selection effects only. Finally, \cite{berger14} claimed that the optical afterglow detection
can influence the luminosity distribution towards places with larger densities medium.
\subsection{Selection effects in the \texorpdfstring{$L_{O,200\rm{s}}-\alpha_{O,>200\rm{s}}$}{Lg} relation}\label{montecarlosimulation}
\cite{oates2012} ensured that a high S/N light curve, covering both early and
late times, can be constructed from the UVOT multi-filter observations using the criteria from \cite{oates2009}.
If the faintest optical/UV afterglows decay more slowly than the brightest ones,
then at late time the luminosity distribution is less dispersed and the correlation coefficient of the
$\log L_{O,200\rm{s}}-\alpha_{O,>200\rm{s}}$ relation must become smaller and/or
negligible. Indeed, both of these effects were observed in their sample.
Furthermore, the $\log L_{O,200\rm{s}}-\alpha_{O,>200\rm{s}}$ relation may arise, by chance, from the way in which the
sample is chosen. Thus, to verify if this is not the case, they computed Monte Carlo simulations.
Among the $10^6$ trials, 34 have a correlation
coefficient indicating a more significant relation than the original one. This points out that, at 4.2 $\sigma$ confidence,
the $\log L_{O,200\rm{s}}-\alpha_{O,>200\rm{s}}$ relation is not caused by the selection criteria nor does it happens by chance, and
thus it is intrinsic.
\section{Redshift Estimator}\label{redshiftestimator}
As we have pointed out in the introduction, the study of GRBs as possible distance estimators is relevant,
since for many of them
$z$ is unknown. Therefore, having a relation which is able to infer the distance from known quantities observed
independently of $z$ would allow a better investigation of the GRB population.
Moreover, in the cases in which $z$ is uncertain, the estimator can give hints on the
upper and lower limits of the distance at which the GRB is placed. Some examples of redshift estimators for the
prompt relations \citep{atteia03,yonetoku04,tsutsui13} have been reported. In these papers, a method
is developed for inverting GRB luminosity relations in respect to the redshift to have an expression of the distance
as a function of z.
The methodology used for the prompt emission relations can be then applied also to the afterglow or prompt-afterglow
phase relations.
In this respect, \cite{dainotti11a} investigated the LT relation as a redshift estimator.
From this relation, the best fit parameters of the slope and normalization are derived, while parameters such as
$\log F_{X,a}$, $\log T_{X,a}$ and $\beta_{X,a}$ are known, because they are measured.
Therefore, the LT relation can be inverted to obtain
an estimate of $z$ as it has been done for the prompt relations by \cite{yonetoku04}.
With this intention, let us return to the eq. \ref{eq: lx} and write it in another form:
\begin{eqnarray}
\log{L_{X,a}} & = & \log{(4 \pi F_{X,a})} + 2 \log{D_L(z, \Omega_M, h)} - (1 - \beta_{X,a}) \log{(1 + z)} \nonumber \\
~ & = & \log{(4 \pi F_{X,a})} + (1 + \beta_{X,a}) \log{(1 + z)} + 2 \log{r(z)} + 2 \log{(c/H_0)} \nonumber \\
~ & = & a \log{\left ( \frac{T_{X,a}}{1 + z} \right )} + b
\end{eqnarray}
where $r(z)=D_L(z, \Omega_M, h)\times(H_0/c)$. Solving respect to $z$, it was obtained:
\begin{equation}
(1 + \beta_{X,a} + a) \log{(1 + z)} + 2 \log{r(z)} = a \log T_{X,a} + b - \log{(4 \pi F_{X,a})} - 2 \log{(c/H_0)}.
\label{eq: zedest}
\end{equation}
The numerical solution of this equation may encounter some problems that must be taken into account:
$(\log T_{X,a}, \log \\F_{X,a}, \beta_{X,a})$ and the LT calibration parameters $(a, b)$ are influenced by their
own errors. Furthermore, the errors on $(a, b)$ are not symmetric and $\sigma_{int}$ is summed to the total
error in a nonlinear way. For details about possible solutions on how to consider the errors see \cite{dainotti11a}.
The above solution was employed for the E4 and the E0095 samples, pointing out that the LT
relation can still not be considered as a precise redshift estimator, see Fig. \ref{fig:theorobservredshift}.
\begin{figure}[htbp]
\centering
\includegraphics[width=1\hsize,angle=0]{zobservedvsestimatednomeancopy.pdf}
\caption{\footnotesize $z_{obs}$-$z_{est}$ distribution for the 62 LGRBs divided in three $\sigma_E$ ranges from
\cite{dainotti11a}: $\sigma_E \le 0.095$ is represented by red points, $0.095 \le \sigma_E \le 0.3$ is represented
by blue points, $0.3 \le \sigma_E \le 4$ is represented by black points.}
\label{fig:theorobservredshift}
\end{figure}
Assuming $\Delta z = z_{obs} - z_{est}$, where $z_{obs}$ and $z_{est}$ are the observed and the estimated redshifts respectively,
it has been shown that $\sim 20\%$ of GRBs in the E4 sample (black, $0.3 \le \sigma_E \le 4$, and blue,
$0.095 \le \sigma_E \le 0.3$, points in Fig. \ref{fig:theorobservredshift}) has
$|\Delta z/\sigma(z_{est})| \le 1$. While for the E0095 subsample $28\%$ has $|\Delta z/\sigma(z_{est})| \le 1$,
red dots in Fig. \ref{fig:theorobservredshift}.
The percentage of successful solutions rises
to $\sim 53\%$ ($\sim 57\%$) for the E4 (E0095) sample if $|\Delta z/\sigma(z_{est})| \le 3$ is considered.
The comparison of the results for both the E4 and
E0095 samples is proof that $\sigma_E$ has no strong influence on the redshift estimate.
The reason why the redshift indicator has not yet given successful results depends on the intrinsic scatter of the LT
relation.
Thus, it is useful to check whether better results can be achieved by increasing the
data sample size. For this reason, an E0095 subsample was simulated creating $(\log T_{X,a}, \beta_{X,a}, z)$ values from a distribution
similar to the observed one for the E4 sample.
Then, $\log L_{X,a}$ was selected from a Gaussian distribution with mean
value obtained by the LT relation and with $\sigma_{Gauss}=\sigma_{int}$.
These values were employed to compute $\log F_{X,a}$ and to reproduce the noise for all the quantities so that the relative errors
resembled the observations. Then, using Markov chains as input to the
redshift estimate formula, it is concluded that only enlarging the sample is not an appropriate methodology to increase the
success of the LT relation as a redshift estimator.\\
In fact, with ${\cal{N}} \simeq 50$, the number of GRBs with
$|\Delta z/\sigma(z_{est})| \le 1$ first rises to $\sim 34\%$ and then
diminish to $\sim 20\%$ for ${\cal{N}} \simeq 200$, while $\langle \Delta z/z_{obs} \rangle \simeq -17\%$ for both
${\cal{N}} \simeq 50$ and ${\cal{N}} \simeq 200$.
The fact that enlarging the sample does not improve the result could be expected. Indeed,
a bigger sample conducts
to tighter constraints on the $(a, b, \sigma_{int})$ values, but does not affect $\sigma_{int}$ which is the principal cause of
inconsistencies between the observed and the estimated $z$.\\
Therefore, an alternative way was explored: $\sigma_{int}$ was decreased and the best fit $(a, b)$ parameters of the E0095 subsample
were chosen. In fact, fixing $\sigma_{int} = 0.10$ gives $f(|\Delta z/z_{obs}| \le 1) \simeq 66\%$.
These outcomes suggested that the LT relation could be employed as a redshift estimator only in the case that a subsample of
GRBs could be determined with $\sigma_{int} = 0.10-0.20$. If such a sample is achievable
is not clear yet due to the paucity of the E0095 subsample. In fact, it is difficult to find out
some useful indicators that can help to define GRBs close to the best fit line of the LT relation. To obtain $\sim 50$ GRBs to calibrate
the LT relation with $\sigma_{int} \sim 0.20$ it has been estimated that a sample of $\sim 600$ GRBs with
observed $(\log T_{X,a}, \log F_{X,a}, \beta_{X,a}, z)$ values is needed. However, even if this is a challenging goal, it may be possible to find out
properties of GRB afterglows which enable us to reduce the $\sigma_{int}$ of the LT relation with a much smaller sample.
Finally, an interesting feature would be to correct for the selection effects all the physical quantities of
the relations mentioned above.
In this manner, it would be possible to average them in order to create a more precise redshift
estimator.
\section{Cosmology}\label{cosmology}
The study of the Hubble Diagram (HD), namely the distribution of the distance modulus $\mu(z)$\footnote{The difference between
the apparent magnitude m, ideally corrected from the effects of interstellar
absorption, and the absolute magnitude M of an astronomical object.} versus $z$ of SNe Ia, opened the way to the investigation of
the nature of DE.
As it is known from the literature, $\mu(z)$ is proportional to the logarithm of the luminosity distance
$D_L(z, \Omega_M, h)$ through the following equation:
\begin{equation}
\mu(z) = 25 + 5 \times \log D_L(z, \Omega_M, h).
\end{equation}
In addition, $D_L(z, \Omega_M, h)$ is related to different DE EoSs.
\subsection{The problem of the calibration}
One of the most important issues presented in the use of GRB relations for cosmological studies is the so-called
circularity problem. Namely, a cosmological model needs to be assumed to compute
$D_L(z, \Omega_M, h)$. This is due to the fact that local GRBs are not available apart from the case of GRB 980425.
Indeed, this kind of GRBs would be observed at $z < 0.01$ and their measure would be independent of a particular
cosmological setting.
This issue could be overcome in three ways: a) through the calibration of these relations by several low $z$ GRBs
(in fact, at $z\le 0.1$ the luminosity distance is not sensitive to the balance of $\Omega_M$ and $\Omega_{\Lambda}$
for a given $H_0$, where $H_0$ is between 65 and 72); b)
through a solid theoretical model in order to explain the observed 2D relations. Namely, this would fix their slopes and
normalizations independently of cosmology, but this task still has to be achieved; c) through the calibration of the standard
candles using GRBs in a narrow redshift range ($\Delta z$) near a fiducial redshift, $z_c$. We here describe some examples on how to overcome the problem of circularity using prompt
relations.
\begin{figure}[htbp]
\centering
\includegraphics[width=16.5cm, height=8cm,angle=0]{fig3liang06copy.pdf}
\caption{\footnotesize ``Distribution of $\log P$ in the (N, $\Delta z$) plane from \cite{liang06}. The grey contours
mark the areas where the dependencies of $b_1$ and $b_2$ on $\Omega_M$ are statistically significant (P $< 10^{-4}$). The white region is suitable for the calibration purpose".}
\label{fig:liang06}
\end{figure}
The treatment of this problem will be the same once we consider afterglow or prompt-afterglow relations.
\cite{liang06} suggested a new GRB luminosity indicator, $E_{\gamma,iso} = aE^{b_1}_{\gamma,peak}T^{b_2}_{O,a}$,
different from the previous GRB luminosity indicators that are generally
written in the form of $L=a\prod x^{b_i}_i$, where a is the normalization, $x_i$
is the i-th observable, and $b_i$ is its corresponding power law index. It was demonstrated
that while $a$ relies on the cosmological parameters, this is not the case for $b_1$ and $b_2$ until $\Delta z$ is sufficiently
little, see Fig. \ref{fig:liang06}. The choice of $\Delta z$ for a given GRB sample could be evaluated
depending on its dimension and the errors on the variables.
The most suitable approach would be to assemble GRBs within a small redshift range
around a central $z_c$ ($z_c \sim 1$ or $z_c \sim 2$), because the GRB $z$ distribution peaks in this interval
\textcolor{red}{(see also \citealt{wang2011} and \citealt{wang15})}.\\
In addition, also \cite{ghirlanda06} defined the
luminosity indicator $E_{\gamma,peak}= a \times E^b_{\gamma,cor}$ using the $\log E_{\gamma,peak}-\log E_{\gamma,cor}$ relation \citep{Ghirlanda2004},
where
\begin{equation}
E_{\gamma,cor}=(1-\cos\theta_{jet})\times 4\pi \times D_L^2(z, \Omega_M, h)\times S_{\gamma,prompt}/(1+z)^2
\end{equation}
is the energy corrected for the beaming factor and $\theta_{jet}$ is the opening angle of the jet.
They calculated the minimum number of GRBs (N), within $\Delta z$ around a certain $z_c$, needed to
calibrate the relation, considering a sample of 19 GRBs detected mostly by Beppo-SAX and Swift.
Particularly, they fitted the relation for each value of $\Omega_M$ and $\Omega_{\Lambda}$ using
a set of N GRBs distributed in the interval $\Delta z$ (centered around $z_c$). If the variation
of the slope, b, is less than $1\%$ the relation is assumed calibrated.
$N$, $\Delta z$ and $z_c$ are free parameters. They checked several $z_c$ and distinct $z$ dispersions
$\Delta z \in (0.05, 0.5)$ by Monte Carlo simulations.
At every $z$ the smaller the N the bigger the variation of the slope, $\Delta b$ (for the same $\Delta z$),
because the relation is more scattered. On the other hand, for greater $z_c$ a
tinier $\Delta z$ is necessary to maintain $\Delta b$ in its little state. Finally, they found that 12 GRBs with
$z \in (0.9, 1.1)$ can be sufficient to calibrate the slope of the $\log E_{\gamma,peak}-\log E_{\gamma,cor}$
relation.
Instead, at $z_c = 2$ a narrower redshift bin is needed, for example $z \in (1.95, 2.05)$.\\
However, this method might becomes unsuccessful, because the sample size of the observed GRBs is not sufficiently big.
Another method for a model-independent calibration may be obtained employing SNe Ia as distance
indicators. This method is based on
the assumption that a GRB at redshift z must have the same distance modulus $\mu(z)$ of a SNe Ia at the same redshift.
In this way, GRBs should be considered as complementary to SNe Ia at very high z, thus allowing for the construction of
a very long distance
ladder. Therefore, interpolating the SNe Ia HD provides the value of $\mu(z)$ for a subsample
of GRBs with $z \le 1.4$, which can be employed for the calibration of the 2D relations \citep{kodama2008,liang08,wei09}.
This value is given by the formula:
\begin{eqnarray}
\mu(z) & = & 25 + (5/2) (\log \ y - k) \nonumber \\ ~ & = & 25 + (5/2) (a + b \log\ x -k),
\label{modulus}
\end{eqnarray}
where $y = k D_L^2(z, \Omega_M, h)$ is a given quantity with $k$ a redshift independent constant,
and $a$ and $b$ are the relation parameters. Presuming that this calibration is redshift independent,
the HD at higher $z$ can be constructed using the calibrated relations for the
other GRBs in the data set.\\
Finally, \cite{li2014} analyzed the light curves of 8 LGRBs associated with SNe finding a relation
between the peak magnitude and the decline rate at 5, 10 and 15 days as in SNe Ia. However, from the
comparison with the well-known relation for SNe Ia \citep{phillips93}, it was pointed out that these two objects have
two different progenitors. More importantly, this discovery allowed
to use GRBs associated with SNe as possible standard candles. In addition, \cite{cano2014} investigated the optical
light curves of 8 LGRBs associated with SNe discovering evidence of a relation between their luminosity
and the width of the GRB light curves relative to the template of the well-known SN 1998bw. This result also confirmed
the possibility of using GRBs associated with SNe as standard candles.
\subsection{Applications of GRB afterglow relations}
In this section, we describe some applications to cosmology only for the LT relation, because this is the only
afterglow relation that has been used so far as a cosmological probe. However, the method is very general and it can
be employed for all the other relations presented in the review.
The idea to use afterglow GRBs phase as cosmological rulers was proposed for the first time in 2009, when the LT relation
was used to derive a new HD \citep{cardone09,cardone10}.\\
More specifically, \cite{cardone09} revised the data set
used in \cite{schaefer2007} appending the LT relation. They used a Bayesian fitting method, similar to that used
in \cite{firmani06} for the $\log E_{\gamma,peak}$-$\log E_{\gamma,cor}$ relation, to calibrate the
different GRB relations known at that time assuming a fiducial $\Lambda$CDM model compatible with the data provided
by the Wilkinson Microwave Anisotropy Probe, WMAP5.\\
A new HD including $83$ objects was obtained (69 from
\cite{schaefer2007} plus 14 new GRBs obtained by the LT relation) computing the mean performed over six relations
($\log E_{\gamma,cor}-\log E_{\gamma,peak}$, $\log L_{\gamma,iso}-\log V$, with $V$ the variability which measures
the difference between the observed light curve and a smoothed version of that light curve,
$\log L_{X,a}-\log T_{X,a}^*$,
$\log L_{\gamma,iso}-\log \tau_{lag}$, with $\tau_{lag}$ the difference in arrival time to the
observer of the high energy photons and low energy photons, $\log L_{\gamma,iso}-\log \tau_{RT}$,
with $\tau_{RT}$ the shortest time over which the light curve increases by the $50\%$ of the
peak flux of the pulse, and $\log L_{\gamma,iso}-\log E_{\gamma,peak}$).\\
To elude the circularity problem, local regression was run to calculate $\mu(z)$ from
the newest SNe Ia sample containing 307 SNe Ia in the range
$0.015 \le z \le 1.55$. Indeed,
the GRB relations mentioned before were calibrated while considering only GRBs with $z \le 1.4$ in order to cover the same
redshift range spanned by the SNe Ia data. This SNe Ia sample is the input for the local
regression estimate of $\mu(z)$.\\
The basic idea of the local regression analysis consists of several stages described in \cite{cardone09}.
To find out which are the optimal parameters of this procedure, a large sample of simulations was carried out.
They set the value of the model parameters
$(\Omega_M, w_0, w_a, h)$, with $w_0$ and $w_a$ given by the coefficient of the DE EoS $w(z)=w_0+w_a z(1+z)^{-1}$
\citep{schaefer2007}, in the ranges $0.15 \le \Omega_M \le 0.45$, $-1.5 \le w_0 \le -0.5$, $-2.0 \le w_a \le 2.0$
and $0.60 \le h \le 0.80$. For each $z$ value, $\mu(z_i)$ was selected from a Gaussian distribution centered on
the predicted value and with $\sigma_{int}= 0.15$, consistent with the
$\sigma_{int}$ of the SNe Ia absolute magnitude. This way, a mock
catalogue with the same $z$ and error distribution of the SNe sample was built. Each
$\mu(z)$ value derived from this procedure is compared to the input one.
The local regression method correctly produces the underlying $\mu(z)$ at each $z$ from the SNe Ia sample, whichever
is the cosmological model.\\
Furthermore, comparing their HD to the one derived by
\cite{schaefer2007}, referred as the Schaefer HD, they have updated the Schaefer HD in three ways, namely
updating the $\Lambda$CDM model parameters, using a Bayesian fitting procedure and adding the LT relation.
To analyze the influence of these changes, the sample of 69 GRBs adopted by
\cite{schaefer2007} was also used and the distance moduli were computed with the new calibration,
but without considering the LT relation. It was found that $\mu_{new}/\mu_{old}$ is close to 1 within 5\%.
Thus, this calibration procedure has not modified the results.\\
In conclusion, it was pointed out that the $\mu(z)$ for each of the GRBs in common to \cite{schaefer2007}
and \cite{Dainotti2008} samples is compatible with the one computed using the set of \cite{schaefer2007} relations.
Therefore, no systematic bias is added by also considering the LT relation.
On the other hand, the addition of the LT relation to the pre-existing ones not only decreases the errors on
$\mu(z)$ by $\sim 14$\%, but also expands the data set from $69$ to $83$ GRBs.\\
While \cite{cardone09} added the LT relation to a set of other $5$ known prompt emission relations, \cite{cardone10}
used instead the LT relation alone (66 LGRBs) or in combination with other cosmological tools in order to find
some constraints on the cosmological parameters at large $z$.
The GRBs were divided in E0095 and E4 samples, indicating that the introduction of
the LT relation alone also provides constraints compatible with previous outcomes, since the HD spans over a
large redshift range $(0.033, 8.2)$.\\
Furthermore, considering three different cosmological models, namely the $\Lambda$CDM, the CPL \citep{Chevallier2001}
and the quintessence (QCDM), it was discovered that the $\Lambda$CDM model is preferred. To better show the
impact of GRBs, the fit was repeated only with other probes, such as SNe Ia or Baryon acoustic Oscillations,
excluding the GRBs. The addition of GRBs does not significantly narrow the parameters
confidence ranges, but GRBs drive the constraints on $w_a$ to $0$. This result indicates
that the consideration of a big sample of E0095 GRBs may lead to a constant
EoS DE model.\\
In addition, we may note that, different from what was done in the literature at the time of their publication, the HD for the
E0095 and E4 samples is the only GRB HD built with a single relation in the afterglow containing a statistically significant
sample.\\
Furthermore, the LT relation does not request the mix of several relations to rise the number of
GRBs with a known $\mu(z)$. In fact, each relation is influenced by its own biases and intrinsic scatter; therefore,
using all of them in the same HD can affect the evaluation of the cosmological parameters.
The $\sigma_{int}$ of the LT relation may be considerably decreased if only the E0095 subsample
is analyzed. However, considering the whole sample of 66 LGRBs, \cite{cardone10}
constrained $\Omega_M$ and $H_0$ obtaining values compatible with the ones presented in the literature.\\
This analysis clearly claimed that the LT relation can be considered for building a GRB HD without adding
any bias in the study of the cosmological parameters.
Equivalent findings were achieved considering E0095 GRBs even if they are just $12\%$ of the whole sample.
Therefore, a further
investigation of E0095 GRBs can boost their use as standard sample for studying the DE mystery.\\
As a further development, \cite{Dainotti2013b} pointed out
to what extent a separation of 5 $\sigma$ above and below the intrinsic value, $b_{int}=-1.07_{-0.14}^{+0.09}$, of
the slope of the LT relation can influence the cosmological results.\\
For this study, a simulated data set of 101 GRBs obtained through a Monte Carlo simulation was
collected assuming
$b=-1.52$, $\sigma_{int}=0.93$ (larger than the scatter computed from the original data set, namely $\sigma_{int}=0.66$),
and the fiducial $\Lambda$CDM flat cosmological model with $\Omega_M = 0.291$ and $H_0 = 71$ Km s$^{-1}$ Mpc$^{-1}$.
They investigated how much the scatter in the cosmological parameters can be diminished if, instead of the total
sample (hereafter Full), a highly luminous subsample (hereafter High Luminosity)
is considered, constrained by the condition that $\log L_{X,a} \geq 48.7$.
The choice of this selection cut at a given luminosity is explained in \cite{Dainotti2013a}, who
showed that the local luminosity function is similar to the observed luminosity one
for $\log L_{X,a} \geq 48$.\\
The methodology is similar to what has been done by \cite{Amati2008} for the $\log E_{\gamma,peak}-\log E_{\gamma,iso}$ relation, namely
the fit has been performed varying simultaneously both the calibration parameters, $p_{GRB}=(a, b, \sigma_{int})$, and the
cosmological parameters, $p_c=(\Omega_M , \Omega_{\Lambda}, w_0, w_a, h)$, each time for a
given model in order to correctly take this issue into account.\\
In order to have stronger limits on the cosmological parameters two samples were added to the data set, the $H(z)$
sample ($H(z)=H_0 \times \sqrt{\Omega_M(1+z)^3+\Omega_k(1+z)^2+\Omega_{\Lambda}}$) over the
redshift range $0.10 \le z \le 1.75$ \citep{stern2010} and the Union $2.1$ SNe Ia
sample containing 580 objects over the redshift range $0.015 \le z \le 1.414$ \citep{suzuki2012}.\\
A Markov Chain Monte Carlo (MCMC) method was used, running three parallel chains and applying
the Gelman-Rubin test\footnote{The Gelman-Rubin diagnostics relies on parallel chains
to test whether they all converge to the same posterior distribution.} in order to analyze the convergence for an assumed
cosmological model characterized by a given set of cosmological parameters $p_c$ to be determined.\\
From this statistical analysis results regarding the Full GRB sample, $b$, $a$ and $\sigma_{int}$ of the LT
relation are independent of the chosen cosmological model and the presence of the
SNe Ia and $H(z)$ data in the sample.
In addition, even if a 5 $\sigma$ scatter in $b_{int}$ is assumed, the results for the Full sample
are in agreement with earlier outcomes \citep{Dainotti2008,dainotti11a} where exclusively flat models were assumed.\\
On the other hand, due to the wide errors on the simulated data,
the cosmological parameters are not emerging in the calibration procedure. However, the signature of the cosmology
will appear considering a greater data set with low errors on $(\log \ T_{X,a}^*, \log \ L_{X,a})$.\\
Furthermore, for the Full sample, it was studied how much the deviation from the $b_{int}$ of the LT relation influences
the cosmological parameters. To analyze this
problem, a model parameterized in terms of the present day values of $\Omega_M$, $\Omega_{\Lambda}$ and $H_0$
was considered.
\begin{figure}[htbp]
\includegraphics[width=0.95\hsize]{jointprobfullcopy.pdf}
\includegraphics[width=0.95\hsize]{jointprobhlcopy.pdf}
\caption{\footnotesize Upper left panel: ``regions of confidence for the marginalized likelihood function $ {\mathcal L}(b,\sigma)$ from
\cite{Dainotti2013b}, obtained
marginalizing over $a$ and the cosmological parameters using the Full sample. The bright brown regions indicate the
1 $\sigma$ (full zone) and 2 $\sigma$ (bright grey) regions of confidence respectively. On the axes are plotted the box-and-whisker
diagrams relatively to the $b$ and
$\sigma_{int}$ parameters: the bottom and top of the diagrams are the 25th and 75th percentile (the lower and upper quartiles,
respectively), and the band near the middle of the box is the 50th percentile (the median)". Upper right panel: ``regions of confidence
for the marginalized likelihood function ${ \mathcal L}(\Omega_M, h)$, obtained using the Full sample, from \cite{Dainotti2013b}".
Bottom left panel: ``regions of
confidence for the marginalized likelihood function $ {\mathcal L}(b,\sigma)$ from \cite{Dainotti2013b},
obtained marginalizing over $a$ and the cosmological parameters for the High Luminosity sample. The bright brown regions indicate the
1 $\sigma$ (full zone) and 2 $\sigma$ (bright grey) regions of confidence respectively. On the axes are plotted the box-and-whisker
diagrams relatively to the $b$ and
$\sigma_{int}$ parameters: the bottom and top of the diagrams are the 25th and 75th percentile (the lower and upper quartiles,
respectively), and the band near the middle of the box is the 50th percentile (the median)". Bottom right panel: ``regions of confidence
for the marginalized likelihood function ${ \mathcal L}(\Omega_M, h)$, obtained using the High Luminosity sample, from
\cite{Dainotti2013b}".}
\label{conreg}
\end{figure}
Although $h$ is comparable with the values from both the local distance estimators \citep{riess2009} and CMBR
data \citep{komatsu2011}, the median values for
$(\Omega_M, \Omega_{\Lambda})$ are broader if compared to a fiducial $\Omega_M \sim 0.27$ recovered in earlier works
\citep{davis07}.
For this reason, considering for the Full sample, a distinct $b_{int}$ will lead to a disagreement of $13\%$ with the
best value of the $\Omega_M$ parameter (see the upper panels of Fig. \ref{conreg}).
Even if the median values of the fit for the sample that also has SNe Ia and $H(z)$ data do not conduct towards
flat models, a spatially flat Universe accords with, for example,
the WMAP7 cosmological parameters within $95\%$ giving $\Omega_k = -0.080_{-0.093}^{+0.071}$.
This difference can be deduced, because in this case it is not possible to distinguish among flat and not
flat models and this distinction is still not possible when SNe Ia data are present in the
fit. Thus, constraining the model to be spatially flat, but shaping the DE EoS with $w(z)$,
leads to a couple $(w_0, w_a)$ completely different irrespective of whether SNe Ia and $H(z)$ data are considered or not in the
sample. Regarding instead the High Luminosity subsample, the limits on the calibration parameters mostly do not
depend on either the used cosmological model or if SNe Ia and $H(z)$ data are considered in the sample.
Furthermore, for the High Luminosity subsample it is shown that adding the SNe Ia and
$H(z)$ data does not ameliorate the constraints on the calibration parameters.\\
Finally, the Full sample outcomes are comparable to those of the flat cosmological model for the SNe Ia sample, while the
High Luminosity subsample diverges by $5\%$ in the value of $H_0$ as computed in \cite{Petersen2010}, and the scatter
in $\Omega_M$ is underestimated by $13\%$, see the bottom panels of Fig. \ref{conreg}.
In conclusion, an optimal procedure is to consider a High luminosity subsample provided by a cut exactly at
$\log L_{X,a}=48$; otherwise, the
luminosity and time evolutions should be added in the computation of the cosmological parameters.\\
Later, another application of GRBs to cosmology is presented in \cite{postnikov14} where the DE EoS was analyzed as a
function of $z$ without assuming any a priori $w(z)$ functional form.\\
To build a GRB $(\mu, z)$ diagram, 580 SNe Ia from the Union 2.1 compendium \citep{suzuki2012} were used together with
54 LGRBs in the overlapping
redshift ($z \leq 1.4$ see the left panel of Fig. \ref{fig:hubblediagram})
region between GRBs and SNe Ia. In addition, a standard $w=-1$ cosmological model was assumed.
\begin{figure}[htbp]
\centering
\includegraphics[width=8.1cm,height=5.4cm,angle=0]{hubbledia.pdf}
\includegraphics[width=8.1cm,height=5.5cm]{muGRBs_eoscopy.pdf}
\caption{\footnotesize Left panel: ``($z_j$, $\mu_j \pm \Delta \mu_j$) for SNe Ia from \cite{postnikov14}. GRBs are inferred
from the relation assuming a flat $w = -1$ cosmology and stand out only from their larger error bars, no
discontinuity is evident, implying a first order consistency of a $w = -1$ model out to very high $z$.
The SNe Ia data were taken from the Union 2.1 compendium \citep{suzuki2012}". Right panel: ``distance ladder from
\cite{postnikov14}. GRBs in the SNe Ia overlap redshift range, where cosmology is well
constrained, are used to calculate the GRB
intrinsic correlation coefficient. This relation is then used to calculate $D_L(z, \Omega_M, h)$ for high $z$ GRBs from
their
X-ray afterglow luminosity curves. Standard constant $w$ solutions are shown for reference. Vertical dashed line marks farthest SNe Ia
event. Inset to the right shows a histogram of the GRB sample distribution in $z$. Inset to the left shows resulting most probable
EoS, together with a small sample of models probed, confidence intervals are so large, that only extreme variations with respect to
$w=-1$ can be excluded".}
\label{fig:hubblediagram}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=1\hsize]{centeredDATAtwo4copy.pdf}
\caption{\footnotesize ``Tree of $w(z>1)$ curves inferred from synthetic GRB samples constructed for $w(z)=-1$ cosmologies
in \cite{postnikov14}, showing to what extent correlated GRB errors constrain EoS at high $z$ ($z>1$).
In the left side plot GRB errors taken from
actual data are used, while in the right side plot GRB errors reduced by a factor of $4$ are considered".}
\label{fig:hubblediagram2}
\end{figure}
One order of magnitude expansion
in redshift interval is supplied by the GRB data set considering the correlation coefficients obtained for the SNe Ia.
This detail allows for the enlargement of the cosmological model out to $z=8.2$.
In fact, a relation was found given by:
\begin{equation}
\log L_{X,a}=53.27_{-0.48}^{+0.54}-1.51^{+0.26}_{-0.27} \times \log T^*_{X,a},
\end{equation}
with $\rho=-0.74$ and $P=10^{-18}$.\\
\cite{postnikov14} used a Bayesian statistical analysis, similarly to
\cite{firmani06} and \cite{cardone10}, in which the hypothesis is related to a particular $w(z)$ function with the
selection of $H_0$ and the present DE density parameter, $\Omega_{\Lambda0}$.
The assumption of isotropy for the
cosmological model, reliable limits on the EoS and also a
fixed value for $w(z)$ in the $z \leq 0.01$ redshift interval were employed.
In addition, a huge number of randomly chosen $w(z)$ models were used.\\
To test the procedure, their pattern is verified through the simulated data sets obtained from several input cosmological
models with relative errors and $z$ distribution equal to the real data. Through this procedure, employing the LT
relation, a data set of GRBs detected by the
Swift satellite, with $z$ from $0.033$ to $9.44$, was adopted (see inset in the right panel of
Fig. \ref{fig:hubblediagram}). Thus, it is possible to investigate the history of the Universe out to $z \approx 10$.
(However, an additional analysis would be beneficial if we would consider the sample without the GRB at
$z=9.4$. We note that indeed in \cite{cardone10} a sample of canonical GRBs was used in which this burst has not
been included).\\
In order to do that, they simulated $2000$ constant EoSs uniformly spaced between $-4 \leq w_{\Lambda} \leq 2$,
with $w_{\Lambda}$ the DE EoS. Beginning from SNe Ia data sample, a precise solution was found to be
in agreement with the cosmological constant and a small confidence interval, $w=-0.99 \pm 0.2$, see the right panel of Fig. \ref{fig:hubblediagram}.
Furthermore, it is shown that assuming also that the BAO limits do not differ from the solution of the EoS, but it
considerably decreases the
confidence interval ($w=-0.99 \pm 0.06$). In fact, the insertion of the BAO notably constrained the confidence
region of the solutions, especially for the present DE density parameter, giving $\Omega_{\Lambda0}=0.723 \pm 0.025$.\\
As a further step, the $w(z)$ model which leads to the best evaluation of $D_L(z, \Omega_M, h)$, $z$ of
the SNe Ia sample and the BAO constraints needs to be selected.
The confidence region of the allowed $w(z)$ curves is significantly constrained taking into account
also the BAO data.\\
Afterwards, also considering that GRB data should constrain the cosmological parameters, apart from obtaining
one order of magnitude expansion in the redshift range, it was extremely difficult to
constrain the high $z$ $w(z)$ functional form, considered the paucity of points over a broad
redshift interval and the error bars related to these data.
This is visible in the left panel of Fig. \ref{fig:hubblediagram2}, where a simulated GRB data set having the same $z$
distribution and error bars as the real data, but with assumed $w=-1$ Universe, is provided.
It is noted that only strong $w(z)$ fluctuations are not allowed. Then, decreasing the errors by a
factor of $4$ led to more intriguing high $z$ DE constraints, see the right panel of Fig. \ref{fig:hubblediagram2}.\\
In addition, the small number of elements in the SNe Ia overlapping region indicated broad error bars on the GRB correlation
coefficients. Meanwhile, the broad error bars for high $z$ GRBs generated a very flat probability
distribution (represented by the uniform black shading area in the left panel of Fig. \ref{fig:hubblediagram2})
for the several EoSs checked. Therefore, there will be great interest for the $1<z<4$ region of the GRB HD as soon as
the GRB data set is enlarged and the quality of data is upgraded.
\section{Summary and discussion}\label{discussion}
\textcolor{red}{From the analysis of the relations mentioned in previous sections, it is visible that:}
\begin{enumerate}
\item \textcolor{red}{The accretion model \citep{Cannizzo2009,cannizzo2011} and the magnetar model
\citep{usov92,dallosso2011,rowlinson12} seem to give the best explanation of the Dainotti relation
(giving best fit slopes -3/2 and -1 respectively). The magnetar model seems to be favoured compared to the accretion
one, because the intrinsic slope computed in \cite{Dainotti2013a} is exactly $-1.07_{-0.9}^{+0.14}$.}
\item \textcolor{red}{A more complex jet structure is needed for interpreting the $\log L_{O,200\rm{s}}$\,-\,$\alpha_{O,>200\rm{s}}$
relation \citep{oates2012}. Indeed, \cite{oates2012} showed that the standard afterglow model cannot explain this
relation, especially taking into account the closure relations \citep{Sari98}, which relate temporal decay and spectral
indices. Therefore, in order to interpret their results, they claimed either the presence of some features of the central engine
which dominate the energy release or that the observations were made by observers at different angular distances from the source's axis.
\cite{Dainotti2013a} pointed out a similarity between the $\log L_{O,200\rm{s}}$\,-\,$\alpha_{O,>200\rm{s}}$
relation and the $L_X-T^{*}_a$ relation, making worthy of investigating the possibility of a single physical mechanism inducing both of them.}
\item \textcolor{red}{In the external shock model the $L_X(T_a)-L_{\gamma,iso}$ and the $L_{X,peak}-L_X(T_a)$ relations
cannot lead to a net distinction among constant or wind type density media, but they are able to
exclude so far the thin shell models and to favour the thick shell ones. Among the models that very well
describe the $L_X(T_a)-L_{\gamma,iso}$ and the $L_{X,peak}-L_X(T_a)$ relations there is the one
by \cite{hascoet2014}. They investigated the standard FS model with
a wind external medium and a microphysics parameter $\epsilon_e \propto n^{-\nu}$, and they found out
that for values $\nu \approx 1$ is possible to reproduce a flat plateau phase, and consequently the relations
mentioned above.
This shows how important the study of correlations especially with the aim of discriminating among models.}
\item \textcolor{red}{In regard to the prompt-afterglow relations, mentioned in section \ref{promptaftcor},
involving the energies and the luminosities for the prompt and the afterglow phases, it is pointed out that they help
to interpret the connection between these two GRB phases. For example, \cite{racusin11} pointed out that
the fraction of kinetic energy transferred from the prompt phase to the afterglow one, for BAT-detected GRBs,
is around 10\%, in agreement with the analysis by \cite{zhang07b}.
However, from the investigation of these relations, the synchrotron radiation process seems to not explain completely the observations, and also the scatter present in these
relations is significant. Therefore, further analysis will be useful.}
\item \textcolor{red}{The study of the $L^F_{O,peak}-T^{*F}_{O,peak}$ relation sheds light on the nature of the flares in
the GRB light curves. From the analysis carried on by \cite{Li2012}, it was found out that the flares are additional and
distinct components of the afterglow phase. They also claimed that a periodically-emitting energy central engine can explain the optical
and $\gamma$-ray flares in the afterglow phase.}
\item \textcolor{red}{One of the greatest issues that may undermine the GRB relations as model discriminators and as cosmological tools are selection bias and the evolution with the redshift of the physical quantities involved in these relations.
An example of selection biases is given by \cite{Dainotti2013a}, who used the \cite{Efron1992} method to deal with
the redshift evolution of the X-ray luminosity and the time, to evaluate the intrinsic $L_X-T^{*}_a$ relation.
Furthermore, \cite{Dainotti2015b} assumed an unknown efficiency function for the detector and investigated the biases due
to the detector's threshold and how they affect the X-ray luminosity and the time measurements. The methods described
can be also useful to deal with the selection effects for the optical luminosity and in the $\log L_{O,200\rm{s}}$\,-\,$\alpha_{O,>200\rm{s}}$ relation and any other relation.}
\item \textcolor{red}{Regarding the use of correlations as cosmological tools, we still have to further reduce the scatter of the GRB
measurements and the dispersion of the relations themselves to allow GRBs to be complementary with the measurement
of SNe Ia. Indeed, the redshift evolution effect and the threshold of the detector can generate
relevant selection biases on the physical quantities which however we know how to treat analytically with robust
statistical techniques as we have shown in several sections. Nevertheless, more precise calibration methods, with the help of
other cosmological objects, and more space missions dedicated to detect faint GRBs and GRBs at high redshift
(for example the future SVOM mission) can shed new light on the use of GRBs as cosmological tools.
Lastly, other open questions are concerned with how much cosmological parameters can reduce their degeneracy adding
GRBs into the set of cosmological standard candles. For example, different results of the value of $w$ can lead to
scenarios which can be compatible with a non-flat cosmological model.}
\end{enumerate}
\section{Conclusions}\label{conclusion}
In this work, we have summarized the bivariate relations among the GRB afterglow parameters and their characteristics in order
to discuss
their intrinsic nature and the possibility to use them as standardizable candles.
It has been shown with different methodologies
that some of the relations presented are intrinsic. However, the intrinsic slope has been determined
only for a few relations.
For the other relations, we are not aware of their intrinsic slopes and
consequently how far the use of the observed relations can influence the evaluation of the theoretical models and the ``best"
cosmological settings \citep{Dainotti2013b}.
Therefore, the estimate of the intrinsic relations is crucial for the determination of the most plausible model that can explain
the plateau phase and the afterglow emission.\\
In fact, though there are several theoretical interpretations describing each relation, as we
have shown, in many cases, more than one is viable. This result indicates
that the emission processes that rule the GRBs still have to be further investigated.
To this end, it is necessary to use the intrinsic relations and not the observed ones affected by
selection biases to test the theoretical models. Moreover, the pure afterglow relations have the advantage
of not presenting the double truncation in the flux limit, thus facilitating the correction for selection
effects and their use as redshift estimators and cosmological tools.\\
A very challenging future step would be to use the corrected relations as a reliable redshift estimator and to determine a
further estimate of $H_0$, $\Omega_{\Lambda}$ and $w$. In particular, it is advisable to
use all the afterglow relations which are not yet employed for cosmological studies as new probes, after they
are corrected for selection biases, in order to reduce the
intrinsic scatter as it has been done in \cite{schaefer2007} for the prompt relations.
\section{Acknowledgments}
This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. We thank S. Capozziello
for fruitful comments. M.G.D is grateful to the Marie Curie Program, because the research leading to these results has received funding from the
European Union Seventh FrameWork Program (FP7-2007/2013) under grant agreement N 626267. R.D.V. is grateful to the
Polish National Science Centre through the grant DEC-2012/04/A/ST9/00083.
|
1,116,691,497,723 | arxiv | \section*{Introduction}
Let $f: {\mathbb P}^2 \dashrightarrow {\mathbb P}^2$ be a dominant rational self-map of the complex
projective plane~${\mathbb P}^2$. A great deal of effort has gone into understanding how the
ergodic theory of $f$ is governed by two numerical invariants, the first and
second dynamical degrees $\lambda_1(f)$ and $\lambda_2(f)$, of $f$. The two
cases $\lambda_2(f) > \lambda_1(f)$ and $\lambda_1(f) > \lambda_2(f)$ are well-studied, the former corresponding to ``predominantly repelling dynamics'' and the latter to ``predominantly saddle-type
dynamics.'' The borderline case $\lambda_1 = \lambda_2$ has received much less attention.
In this paper we provide many examples of rational maps $f: {\mathbb P}^2 \dashrightarrow
{\mathbb P}^2$ with $\lambda_1(f) = \lambda_2(f)$ that exhibit both repelling and
saddle-type dynamics equally.
In order to situate and state our main results, let us discuss what is and is not known in a bit more detail.
If we write $f$ in homogeneous coordinates as $f = [f_1:f_2:f_3]$, with the $f_i$ having no common factors of positive degree, then the common degree of the $f_i$ is called the {\em algebraic degree} $d(f)$. The {\em first dynamical degree} is then the asymptotic growth rate
\begin{equation}
\label{eqn:firstdegree}
\lambda_1(f) := \lim_{n \rightarrow \infty} d(f^n)^{1/n}
\end{equation}
of the algebraic degrees. If $d(f^n) = d(f)^n$ for all $n$ then $f$ is said to be
{\em algebraically stable} and we have $\lambda_1(f) = d(f)$.
The {\em topological degree} $\lambda_2(f)$ is the number of preimages $\# f^{-1}(p)$ of a generic point $p \in {\mathbb P}^2$. It is well-behaved under
iteration, satisfying $\lambda_2(f^n) = \lambda_2(f)^n$, and thus is also called the {\em second dynamical degree} of $f$.
The basic ergodic theoretic invariant of $f$ is its topological entropy $h_{top}(f)$. Gromov \cite{gro} and Dinh-Sibony \cite{DS_ENTROPY} showed that entropy is controlled by the dynamical degrees. Specifically,
\begin{equation}
\label{eqn:gromovbd}
h_{top}(f) \leq \log\max\{\lambda_1(f),\lambda_2(f)\}.
\end{equation}
Guedj \cite{Guedj} and Dinh-Nguyen-Truong \cite{DNT}, following Briend and Duval
\cite{BD2}, showed further that when $\lambda_2>\lambda_1$, the map $f$ has a
unique invariant measure $\mu$ of maximal entropy $\log \lambda_2$, that $\mu$
is mixing and repelling (both Lyapunov exponents are positive), that repelling
periodic points for $f$ are asymptotically equidistributed with respect to
$\mu$, and that $\mu$ does not charge small (i.e. pluripolar) sets.
On the other hand, work of Diller-Dujardin-Guedj \cite{ddg2,ddg3} gives that when $\lambda_1~>~\lambda_2$ and certain
additional technical hypotheses are satisfied, then $f$ has (again) a mixing invariant measure $\nu$ of maximal entropy $\log \lambda_1$. But in this case $\nu$ is of \emph{saddle} type (one Lyapunov exponent is positive and the other is negative), and saddle periodic points for $f$ are asymptotically equidistributed with respect to~$\nu$. Uniqueness of the measure of maximal entropy when $\lambda_1>\lambda_2$ has been established only in special cases, notably for polynomial automorphisms of ${\mathbb C}^2$ \cite{BLS} and for surface automorphisms \cite{CANTAT_K3}.
Almost nothing beyond the bound on topological entropy is known about the ergodic theory of rational maps with equal dynamical degrees $\lambda_1 = \lambda_2$. Products $f:=(f_1,f_2)$ of one dimensional maps $f_j:{\mathbb P}^1\to{\mathbb P}^1$ in which one of the factors is linear furnish simple examples of plane rational maps with $\lambda_1=\lambda_2$, and these suggest some possibilities for the ergodic theory.
Consider for example the rational maps $f,g,h: {\mathbb P}^2 \dashrightarrow {\mathbb P}^2$, given on ${\mathbb C}^2$ by
\begin{align*}
f(x,y) = (x+1,y^2), \,\, g(x,y) = (2x,y^2), \,\, \mbox{and} \,\, h(x,y) = (x,y^2).
\end{align*}
Each of these has $\lambda_1 = \lambda_2 = 2$. The map $f: {\mathbb P}^2
\dashrightarrow {\mathbb P}^2$ has topological entropy~$0$; see \cite[Example 1.4]{GUEDJ_ETDS}.
Meanwhile, $g : {\mathbb P}^2 \dashrightarrow {\mathbb P}^2$ has a unique measure $\mu$ of maximal entropy
$\log 2$ which is normalized Lebesgue measure on the unit circle
$\{x=0\} \times \{|y| = 1\}$. However, if one changes surface, compactifying ${\mathbb C}^2$ by ${\mathbb P}^1\times {\mathbb P}^1$ instead of ${\mathbb P}^2$, the resulting map $g: {\mathbb P}^1\times {\mathbb P}^1 \rightarrow {\mathbb P}^1 \times {\mathbb P}^1$ acquires a second measure of maximal entropy $\nu$ given by
normalized Lebesgue measure on $\{x = \infty\} \times \{|y| = 1\}$. The measure $\mu$ is repelling, while $\nu$ is of saddle type.
Finally, for each $x_0 \in {\mathbb C}$ the map $h$ has normalized Lebesgue measure on the unit circle $\{|y| = 1\}$
as measure of maximal entropy within each vertical complex line
$\{x=x_0\}$.
Of course, not all maps with equal dynamical degrees are products, but many
non-product examples $f:{\mathbb P}^2\dashrightarrow{\mathbb P}^2$ are still semiconjugate to rational maps
$\check f~:~{\mathbb P}^1~\to~{\mathbb P}^1$ via a some rational map ${\mathbb P}^2\dashrightarrow{\mathbb P}^1$. Such maps are
said to have `invariant fibrations'. They include \cite{df} all maps with
$\lambda_1=\lambda_2 = 1$ (i.e. all examples where $f$ is birational) and
examples \cite{BG,BGN,GZ,SABOT} that arise in connection with spectral theory
for operators on self-similar spaces. Based on this evidence, Guedj asked
whether any rational map with equal (maximal) dynamical degrees must preserve a
fibration \cite[p.103]{GUEDJ_HABILITATION}. However, examples were recently
found by Bedford-Cantat-Kim \cite{BCK} and Kaschner-P\'erez-Roeder \cite{KPR}
showing that this is not the case.
The product examples $f,g,h$ defined above are each degenerate in another way. The ergodic measures of maximal entropy for these three maps are all supported in algebraic curves (lines, in fact). Since birational changes of coordinate can contract curves, such measures are not very robust. The measures obtained in \cite{Guedj} and \cite{ddg2} for `cohomologically hyperbolic' cases $\lambda_1\neq \lambda_2$ are much more diffuse; in particular they do not charge algebraic curves.
The main results of this paper show that there are large families of rational maps with equal dynamical degrees $\lambda_1 = \lambda_2$ that admit exactly two measures of maximal entropy without any of the above problems.
\begin{thma}
Let $f:{\mathbb P}^2\dashrightarrow{\mathbb P}^2$ be a rational map with equal algebraic and topological degrees $d(f) = \lambda_2(f)\geq 2$. Then there is an open subset of linear maps $T\in {\mathop{\mathrm{Aut}}}({\mathbb P}^2)$ for which the map $f_T := T\circ f$ satisfies the following.
\begin{itemize}
\item $f_T$ is algebraically stable and therefore has $d(f) = \lambda_1(f_T) = \lambda_2(f_T)$.
\item There is no $f_T$ invariant foliation; in particular, $f_T$ is not rationally semiconjugate to a holomorphic self-map of a Riemann surface.
\item There are exactly two $f_T$-invariant and ergodic measures $\mu$ and $\nu$ of maximal entropy $\log d(f)$.
\item $f_T$ is uniformly expanding on $\mathop{\rm supp}\mu$.
\item $f_T$ is uniformly hyperbolic of saddle type on $\mathop{\rm supp}\nu$.
\item Neither $\mathop{\rm supp}\nu$ nor $\mathop{\rm supp}\mu$ are contained in any algebraic curve.
\item Any point $p\in {\mathbb P}^2$ whose forward images are not indeterminate for $f_T$ has a forward orbit asymptotic to either $\mathop{\rm supp}\mu$, $\mathop{\rm supp}\nu$ or one of finitely many attracting periodic points.
\end{itemize}
\end{thma}
\begin{remark*}
It is easy to find mappings $f$ satisfying the hypothesis of Theorem~A. For
example, suppose $g: {\mathbb P}^2 \rightarrow {\mathbb P}^2$ is an endomorphism and $h: {\mathbb P}^2
\dashrightarrow {\mathbb P}^2$ is a birational map, both of algebraic degree $a$. Then,
$f: = g\circ h$ has $d(f)~=~\lambda_2(f)~=~a^2$.
\end{remark*}
\vspace{-0.01in}
The subset of ${\mathop{\mathrm{Aut}}}({\mathbb P}^2)$ from Theorem A is open with respect to the metric topology on ${\mathop{\mathrm{Aut}}}({\mathbb P}^2)$, not the Zariski topology. (In fact, we will never use the Zariski topology in this paper.)
Therefore, we do not know whether there might be different, but also fairly robust possibilities for the ergodic theory of a rational map with equal degrees. Are there for instance, large families of such maps with exactly one measure of maximal entropy? More than two? However each of the first two conclusions of Theorem {{A} } are obtained by (essentially) verifying that $T$ can be chosen to avoid countably many algebraic coincidences. So once the conclusions hold for one $T$, they apply `generically.' This is the content of the next result, which answers a question posed to us by Charles Favre.
\vspace{-0.01in}
\begin{thmb}
Let $f:{\mathbb P}^2\dashrightarrow{\mathbb P}^2$ be a rational map with equal algebraic and topological degrees $d(f) = \lambda_2(f)\geq 2$. Then for all $T\in{\mathop{\mathrm{Aut}}}({\mathbb P}^2)$ outside a countable union of proper algebraic subsets,
\begin{itemize}
\item $f_T$ is algebraically stable, with $\lambda_1(f_T) = \lambda_2(f_T)$.
\item There is no $f_T$ invariant foliation.
\end{itemize}
\end{thmb}
\noindent
The connection between dynamical degrees and ergodic theory
extends to the much more general context of meromorphic self-maps on
compact K\"ahler manifolds of any dimension. The reader may consult~\cite{GUEDJ_ETDS} to gain a good sense of the larger picture, much of which is still conjectural. However, a recent paper
of Vigny~\cite{VIGNY} validates this picture for ``generic cohomologically
hyperbolic rational maps'' of ${\mathbb P}^k$.
\vspace{0.05in}
Some necessary background on the dynamics of rational maps is presented in~\S \ref{SEC_REF}. \S \ref{SEC_PROOFS} gives the proofs of Theorems {{A} } and {B}. The main techniques that we use are hyperbolic dynamics, elementary geometry of algebraic curves, and an application of Theorem 4.1' from \cite{KPR}. One can obtain some (though not nearly all) of the conclusions of Theorem {{A} } with less exertion by appealing to work of De Th{\'e}lin and Vigny \cite[Theorem 1]{DE_THELIN_VIGNY}. In order to keep the discussion more self-contained, we do not take this route here.
Given a rational map $f:{\mathbb P}^2\dashrightarrow{\mathbb P}^2$ with equal degrees $d(f) = \lambda_2(f)$,
one can in principle use our arguments for Theorem {{A} } to identify a
specific linear transformation $T\in{\mathop{\mathrm{Aut}}}({\mathbb P}^2)$ for which $f_T$ satisfies all
conclusions of the Theorem. In practice however, it seems a bit daunting to
arrange everything we need from $T$ to make the theorem work. Hence in the
final section \S \ref{SEC_PERTURBED_CHEBYSHEV} of this paper, we consider a
family of rational maps $f_t:{\mathbb P}^2\dashrightarrow{\mathbb P}^2$ that depend on a single real
parameter $t\in(0,1]$ and for which symmetry considerations allow us to verify
the conclusions of Theorem {{A} } more directly. The initial map $f =
f_1$ is closely related to the one variable quadratic Chebyshev map. We show
that the conclusions of Theorem {{A} } hold for all $t\in (0,1]$ close
enough to $0$. An interesting additional outcome is that for all such~$t$, the
measure $\mu$ is real, supported on ${\mathbb R}{\mathbb P}^2\subset{\mathbb P}^2$, whereas the measure
$\nu$ is not.
\section{Rational maps, dynamical degrees and entropy}\label{SEC_REF}
Throughout this paper ${\mathbb P}^2$ will denote the complex projective plane. Unless otherwise specified, we will measure distance between points with the usual Fubini-Study metric, normalized so that ${\mathbb P}^2$ has unit volume. For any set $X\subset {\mathbb P}^2$ and $r>0$, we let $B_r(X) := \{p\in{\mathbb P}^2:{\rm dist}(p,X)<r\}$ be the $r$-neighborhood of $X$.
Henceforth $f:{\mathbb P}^2\dashrightarrow{\mathbb P}^2$ will denote a rational map of the complex
projective plane ${\mathbb P}^2$. Let us recall some definitions, facts and
conventions concerning such maps. In homogeneous coordinates, $f$ is given by
$f = [f_1:f_2:f_3]$, where the components $f_j[x_1:x_2:x_3]$ are homogeneous
polynomials, all of the same degree $d$ and with no non-constant common
factors. The common zeroes of the components $f_j$ correspond to a finite set
${\mathcal I} = {\mathcal I}(f)\subset{\mathbb P}^2$ on which $f$ is not definable as a continuous map. On ${\mathbb P}^2 \setminus {\mathcal I}$, the map
$f$ is well-defined and holomorphic. We define the ``image" of a point $p\in
{\mathcal I}$ to be the set $f(p)$ of all possible limits $\lim_{p_j\to p} f(p_j)$. This
is always a non-trivial algebraic curve. Note that here and in the rest of the paper, algebraic curves are allowed to be reducible unless otherwise noted.
We assume throughout that $f$ is {\em dominant}, meaning that $f({\mathbb P}^2 \setminus {\mathcal I})$ is not contained in an algebraic curve.
We define the `image' of an algebraic curve $V\subset{\mathbb P}^2$ under $f$ to be its (set-theoretic) proper transform
\begin{align}\label{EQN_IMAGE_CURVE}
f(V) := \overline{f(V\setminus {\mathcal I})}.
\end{align}
If $V$ is irreducible, then $f(V)$ is either another irreducible algebraic
curve or a point. For a rational map of ${\mathbb P}^2$,
the latter can only happen if $V \cap {\mathcal I} \neq \emptyset$; see \cite[Proposition 1.2]{FS2}.
If $V$ is an irreducible curve with $f(V)$ a point, we call $V$
\emph{exceptional} for $f$, letting ${\mathcal E}$ denote the union of all (finitely
many) exceptional curves.
We define the preimage $f^{-1}(V)$ of a
curve $V$ to be the union of all non-exceptional irreducible curves $C$ such
that $f(C) \subset V$. A curve $V\subset X$ is \emph{invariant} if $f(V) = V$
and \emph{totally invariant} if, $f^{-1}(V) = V$.
We say that an irreducible curve $V$ is \emph{ramified} for $f$ if $V$ is not exceptional, but $f$ is not locally one-to-one near any point in $V$. The collection of all exceptional and ramified curves forms the critical set $\mathop{\mathrm{crit}}(f)$ of $f$. These may be assembled with multiplicities defined by the order of vanishing of the Jacobian determinant into the \emph{critical divisor} $\mathop{\mathrm{crit}}(f) \in \div({\mathbb P}^2)$ of $f$.
As in the introduction, we let $d=d(f)$, $\lambda_1 = \lambda_1(f)$ and $\lambda_2 = \lambda_2(f)$ denote the algebraic and first and second dynamical degrees of $f$. The integer $\lambda_2$ is (alternatively) the multiplier for the induced action $f^*$ on $H^4({\mathbb P}^2,{\mathbb R}) \cong {\mathbb R}$. Likewise, the algebraic degree $d$ of $f$ is the multiplier of the pullback action $f^*$ on $H^2({\mathbb P}^2,{\mathbb R})\cong {\mathbb R}$. Indeed any divisor $D\in\div({\mathbb P}^2)$ admits (see e.g. \cite[\S 1.2]{ddg1}) a natural pullback $f^*D$ and pushforward $f_*D$ by $f$, and the resulting divisors satisfy $\mathop{\rm deg} f_*D = \mathop{\rm deg} f^*D = d\cdot \mathop{\rm deg} D$.
We point out that if $C$ is an irreducible curve, then the support of $f_*C$ is equal to the \emph{total transform} of $C$ by $f$, i.e. to the proper transform $f(C)$ together with the images of all points in ${\mathcal I}(f)\cap C$. In particular $\mathop{\rm supp} f_*C$ will typically be reducible when $C$ meets ${\mathcal I}(f)$. On the other hand, when $C$ is disjoint from ${\mathcal I}(f)$, we have $\mathop{\rm supp} f_*C = f(C)$ is irreducible and more precisely, $f_*C = \lambda f(C)$ where $\lambda$ is the topological degree of the restriction $f|_C:C\to f(C)$.
Likewise, if $C$ is irreducible and does not contain the image of any exceptional curve, then
$$
f^*C = \sum_{f(C') = C} \nu' C',
$$
where the sum is over irreducible curves $C'$, and $f$ is locally $\nu'$-to-1 near a general point in $C'$.
One always has that $\lambda_2(f^n) = \lambda_2(f)^n$, and typically also $d(f^n) = d^n$. However, when ${\mathcal I}\neq \emptyset$ the latter equality can fail. There is a useful geometric characterization of (the opposite of) this possibility.
\begin{proposition}[See \cite{FS2, df}]
\label{prop:ascriterion}
The following are equivalent.
\begin{itemize}
\item $d(f^n) = d^n$ for all $n\in{\mathbb N}$;
\item $\lambda_1(f) = d$;
\item No exceptional curve has forward orbit containing a point of indeterminacy.
\end{itemize}
\end{proposition}
\noindent As in the introduction, we will call $f$ \emph{algebraically stable} if one/all of the conditions in this proposition hold.
The upper bound \eqref{eqn:gromovbd} for the topological entropy\footnote{We remark that there is some subtlety defining $h_{top}(f)$ if $\mathcal{I} \neq \emptyset$. This issue is addressed in~\cite{DS_ENTROPY}.} shows the dynamical significance of dynamical degrees.
It is expected that this bound is actually an equality in most situations, and more precisely, that there exists an $f$-invariant measure $\mu$ whose metric entropy satisfies $h_{\mu}(f) = \log\max\{\lambda_1,\lambda_2\}$. Theorem {{A} } of this article concerns the nature of such measures when $\lambda_1=\lambda_2$.
\section{Proof of Theorems {{A} } and {B}}\label{SEC_PROOFS}
\subsection{Linear perturbations of plane rational maps}
For any linear map $T\in{\mathop{\mathrm{Aut}}}({\mathbb P}^2)$, we will let $f_T$ denote the `perturbed' map $T\circ f$. Note that the algebraic and topological degrees of $f_T$ are the same as those of $f$. So are the indeterminacy, exceptional and critical sets of $f_T$.
We will be interested in choosing $T$ so that most points of ${\mathbb P}^2$ map by $f_T$ into a small neighborhood of some fixed line. More specifically, for the remainder of this section we choose a line $L_0\subset{\mathbb P}^2$, a point ${p_0}\in {\mathbb P}^2$ not contained in $L_0$ and a surjective linear map $T_0:{\mathbb P}^2\setminus\{{p_0}\}\to L_0$. As the discussion proceeds we will need to impose further conditions on these choices. Let us list all of these now.
\begin{itemize}
\item[(A)] ${p_0}\notin f({\mathcal I})\cup f(L_0)$;
\item[(B)] $L_0\cap {\mathcal I} = \emptyset$;
\item[(C)] ${p_0}$ is a regular value of $f$, with $\lambda_2$ distinct preimages;
\item[(D)] $\mathop{\rm deg} f(L_0) \geq 2$. In particular, $L_0$ is not exceptional.
\end{itemize}
Property (B) implies that we can choose
$\epsilon > 0$ sufficiently small so that
\begin{align}\label{EQN_NO_INDET_IN_TUBE}
B_\epsilon(L_0) \cap {\mathcal I} = \emptyset.
\end{align}
Since any exceptional curve for $f$ must intersect ${\mathcal I}$,
this also implies no exceptional curve is contained in $B_\epsilon(L_0)$. For the remainder of this section, we will suppose
$\epsilon > 0$ is sufficiently small that (\ref{EQN_NO_INDET_IN_TUBE}) holds.
Note that for any $\delta>0$ the set
$$
{\mathcal T}(T_0,\delta) := \{T\in{\mathop{\mathrm{Aut}}}({\mathbb P}^2): {\rm dist}(T(p),T_0(p))< \delta \text{ for all } p\notin B_\delta({p_0})\}
$$
is non-empty and open in ${\mathop{\mathrm{Aut}}}({\mathbb P}^2)$.
\begin{proposition}
\label{prop:twotraps}
Suppose that conditions (A) and (B) hold and $\epsilon > 0$ is small enough that $B_\epsilon(L_0)$ and $B_\epsilon(f^{-1}(p_0))$ are disjoint from each other and from ${\mathcal I}$. Then there exists $\delta >0$ such that for all $T\in {\mathcal T}(T_0,\delta)$
\begin{itemize}
\item $f_T({\mathbb P}^2\setminus B_\epsilon(f^{-1}(p_0))) \subset B_\epsilon(L_0)$;
\item $f_T^{-1}({\mathbb P}^2\setminus B_\epsilon(L_0)) \subset B_\epsilon(f^{-1}(p_0))$.
\end{itemize}
In particular, $f_T({\mathcal I})\subset B_\epsilon(L_0)$ and any ergodic $f_T$-invariant measure is supported entirely in $B_\epsilon(L_0)$ or in $B_\epsilon(f_T^{-1}(p_0))$.
If in addition, condition (C) holds, then we may further arrange that all points in $B_\epsilon(f^{-1}(p_0))$ are regular for $f_T$ and (therefore) $B_\epsilon(L_0)$ contains all critical values of $f_T$.
\end{proposition}
\begin{proof} Straightforward check.
\end{proof}
\begin{proposition}\label{PROP_AS}
If conditions (A) and (B) hold and $\delta>0$ is small enough, then for all $T\in{\mathcal T}(T_0,\delta)$, the map $f_T:{\mathbb P}^2\dashrightarrow{\mathbb P}^2$ is algebraically stable.
\end{proposition}
\begin{proof}
Choose $\epsilon$ and then $\delta$ as in Proposition \ref{prop:twotraps}. Since each exceptional curve $C$ for $f$ meets $L_0\subset B_\epsilon(L_0)$, we have $f_T^n(C)\in B_\epsilon(L_0)$ for all $n\geq 1$. In particular, the forward orbit of $C$ is disjoint from ${\mathcal I}$, i.e. $f_T$ is algebraically stable.
\end{proof}
\begin{corollary}\label{COR_AS}
The map $f_T$ is algebraically stable for all $T\in{\mathop{\mathrm{Aut}}}({\mathbb P}^2)$ outside a countable union of proper algebraic subsets.
\end{corollary}
\begin{proof}
Given $n\in{\mathbb N}$ and a point $p\in {\mathbb P}^2$, the condition $p\in{\mathcal I}(f_T^n)$ amounts to an algebraic constraint on $T$. By Proposition \ref{prop:ascriterion}, $f_T$ fails to be algebraically stable precisely when there exists some (smallest) $n\in{\mathbb N}$ and a point $p$ in the finite set $f_T({\mathcal E})$ such that $p\in{\mathcal I}(f_T^n)$. Accounting for all possible $n\in{\mathbb N}$, we see that $f_T$ is algebraically stable for all $T$ outside a countable union of algebraic subsets of ${\mathop{\mathrm{Aut}}}({\mathbb P}^2)$. Proposition \ref{PROP_AS} guarantees that these subsets are all proper.
\end{proof}
\subsection{(No) invariant foliations for maps with equal degrees}
The map $f:{\mathbb P}^2~\dashrightarrow~{\mathbb P}^2$ is said to `preserve a rational fibration' if it is
rationally semiconjugate to a one dimensional map, i.e. if there is a rational
map $\phi:{\mathbb P}^2\dashrightarrow {\mathbb P}^1$ and a rational function $\check f:{\mathbb P}^1 \to {\mathbb P}^1$ such
that $\phi\circ f = \check f\circ \phi$ (at all points where both compositions
are defined). If either the base map $\check f$, or the fiber map
$f_x:\phi^{-1}(x) \to \phi^{-1}(f(x))$ ($x\in{\mathbb P}^1$ a general point) has degree
one, then we have (see e.g. Lemma 4.1 in \cite{ddg1}) equality $\lambda_1(f) =
\lambda_2(f)$ of the dynamical degrees of $f$.
It was demonstrated recently \cite{KPR} that $\lambda_1(f) =
\lambda_2(f)$ does not imply $f$ preserves a rational fibration, nor indeed even a singular holomorphic foliation.
Specifically, the following criterion appears in that paper:
\begin{theorem}[Theorem 4.1' from \cite{KPR}]\label{THM_CONDITIONS_FOR_NO_FOLIATION}
Assume that $f$ has an indeterminate point $q$ such that
\begin{enumerate}
\item there are irreducible curves $C_n\subset f^n(q)$ with $\limsup \, \mathop{\rm deg}(C_n) = \infty$; and
\item $q$ has an infinite preorbit along which $f$ is a finite holomorphic map.
\end{enumerate}
Then no iterate of $f$ preserves a singular holomorphic foliation.
\end{theorem}
\noindent
We refer the reader to \cite{FP,KPR}, and the references therein for more details
about the transformation of singular holomorphic foliations by rational maps.
The following lemma shows that the condition (D) can be readily satisfied:
\begin{lemma}
Suppose that $f$ has equal algebraic and topological degrees $d=\lambda_2\geq 2$. Let $p\in{\mathbb P}^2$ be a point whose image $f(p)$ is a regular value of $f$, with $\lambda_2$ distinct $f$-preimages, all outside ${\mathcal I}$. Let $L,L'$ be lines through $p$ that do not meet ${\mathcal I}$ and have distinct images $f(L)\neq f(L')$. Then one of the two irreducible curves $f(L)$ or $f(L')$ is not a line. In particular, for almost all lines $L\subset{\mathbb P}^2$, we have that $\mathop{\rm deg} f(L)~\geq~2$.
\end{lemma}
\begin{proof}
Suppose on the contrary that both images are lines. Then the facts that $L\cap {\mathcal I} = \emptyset$ and that $f_*L = dL$ as divisors imply that $f:L\to f(L)$ is $d$-to-1. Since $\lambda_2 = d$, it follows that $f^{-1}(f(L)) = L$. Likewise $f^{-1}(f(L')) = L'$. But then $f^{-1}(f(p)) = f^{-1}(f(L)\cap f(L')) \subset L\cap L' = \{p\}$ which contradicts the hypothesis that $f(p)$ has $\lambda_2\geq 2$
distinct preimages.
\end{proof}
\begin{lemma}
\label{lem:curvegrows}
Suppose that conditions (A), (B) and (D) hold. For any sufficiently small $\epsilon>0$ there exists $\delta>0$ such that for any $T\in {\mathcal T}(T_0,\delta)$, any irreducible algebraic curve $C\subset B_\epsilon(L_0)$, and any $n\geq 0$, the image $f_T^n(C)$ is irreducible, lies in $B_\epsilon(L_0)$ and has degree at least $2^n\mathop{\rm deg} C$.
\end{lemma}
\begin{proof}
Recall that $\epsilon$ is sufficiently small that $B_\epsilon(L_0)
\cap {\mathcal I} = \emptyset$.
Choose $\delta>0$ small enough that $T\in{\mathcal T}(T_0,\delta)$
guarantees that $f_T(\overline{B_\epsilon(L_0)}) \subset B_\epsilon(L_0)$.
This assures that $f_T^n|_{B_\epsilon(L_0)}: B_\epsilon(L_0) \rightarrow
B_\epsilon(L_0)$ is holomorphic for every $n \geq 1$. Therefore, for any
irreducible curve $C \subset B_\epsilon(L_0)$ the irreducible curve $f_T^n(C)$ is just the usual set-theoretic image
of $C$ under $f_T^n$.
Since there are no exceptional curves for $f_T$ contained in $B_\epsilon(L_0)$, the image $f_T^n(C)$ is non-trivial.
It therefore suffices to show that $\mathop{\rm deg} f_T(C) \geq 2 \mathop{\rm deg} C$. Since $\mathop{\rm deg} f(L_0)\geq 2$, we can find a line $L$ through $p_0$ such that $L\cap f(L_0)$ contains at least two points, and all of them are transverse intersections. Shrinking $\epsilon$ (and therefore $\delta$) if necessary, we may assume that $f_T^{-1}(L)\cap B_\epsilon(L_0)$ is a union of $k\geq 2$ disks $\mathcal{D}_1,\dots,\mathcal{D}_k$ such that each is properly embedded in $B_\epsilon(L_0)$ and the images $f_T(\mathcal{D}_j)\subset L$ are mutually disjoint. Since $C\subset B_\epsilon(L_0)$, we have on topological grounds that $C$ meets each disk $\mathcal{D}_j$ in at least $\mathop{\rm deg} C$ points (counted with multiplicity). Hence $f(C)$ meets $L$ in at least $k\mathop{\rm deg} C$ points counted with multiplicity. Thus $\mathop{\rm deg} f(C) \geq k\mathop{\rm deg} C \geq 2\mathop{\rm deg} C$.
\end{proof}
\begin{theorem} \label{THM_NO_FOLIATION}
Suppose that ${\mathcal I}\neq \emptyset$ and that conditions (A)-(D) hold. Then for any $\delta>0$ small enough and any $T\in{\mathcal T}(T_0,\delta)$, there is no $f_T$-invariant singular holomorphic foliation of ${\mathbb P}^2$.
\end{theorem}
Note that ${\mathcal I}\neq \emptyset$ as soon as $\lambda_2 < d^2$, e.g. in the case of interest here $\lambda_2 = d\geq 2$.
\begin{proof}
Choose $\epsilon>0$ and then $\delta>0$ small enough so that all conclusions of
Proposition \ref{prop:twotraps} and Lemma \ref{lem:curvegrows} apply. Fix any
$q\in {\mathcal I}$, any $T\in \mathcal{T}(T_0,\delta)$, and any non-trivial irreducible algebraic curve $C \subset f_T(q) \subset B_\epsilon(L_0)$.
Lemma
\ref{lem:curvegrows} tells us that the forward images $f_T^{n-1}(C) \subset f_T^n(q)$
have degree tending to infinity with $n$. Meanwhile, Proposition \ref{prop:twotraps}
implies that $f_T^{-n}(q)$ consists of exactly $\lambda_2^n$ regular points for~$f_T$. Theorem \ref{THM_CONDITIONS_FOR_NO_FOLIATION} therefore guarantees that
there is no $f_T$-invariant foliation.
\end{proof}
\begin{corollary}
\label{COR_GENERIC_NIF}
Let $f:{\mathbb P}^2\dashrightarrow{\mathbb P}^2$ be a rational map with equal algebraic and topological
degrees $\lambda:= d(f) = \lambda_2(f)\geq 2$. Then, the map $f_T:= T \circ f:
{\mathbb P}^2~\dashrightarrow~{\mathbb P}^2$ admits no invariant foliation for any $T\in{\mathop{\mathrm{Aut}}}({\mathbb P}^2)$ outside
a countable union of proper algebraic subsets.
\end{corollary}
\begin{proof}
Let $q \in {\mathcal I}$ be any indeterminate point and let $C \subset f(q)$ be some chosen irreducible curve.
This induces a choice $C_{T} := T(C)$ of an irreducible component of $f_T(q)$ for each $T\in{\mathop{\mathrm{Aut}}}({\mathbb P}^2)$.
In the proof of Theorem \ref{THM_NO_FOLIATION} we saw that there exists $S \in {\mathop{\mathrm{Aut}}}({\mathbb P}^2)$
so that
\begin{enumerate}
\item
for each $n \geq 1$, $f_S^{-n}(q)$ consists of $\lambda^n$ preimages each of which is a regular point for $f_S^n$, and
\item for each $n \geq 1$, $f_S^{n-1}(C_{S}) \subset f_S^n(q)$ is an irreducible algebraic curve with $\mathop{\rm deg}(f_S^{n-1}(C_{S})) \rightarrow~\infty$, as $n \rightarrow \infty$.
\end{enumerate}
These two conditions imply the (weaker) criteria laid out in Theorem \ref{THM_CONDITIONS_FOR_NO_FOLIATION}. To establish the present corollary, it will suffice to fix both $q\in{\mathcal I}$ and $n\in{\mathbb N}$ and show that the two conditions continue to hold when $S$ is replaced by any $T\in{\mathop{\mathrm{Aut}}}({\mathbb P}^2)$ outside of some algebraic subset. Existence of $S$ implies we may take the subset to be proper.
Note that for any subvariety $V\subset{\mathbb P}^2$, the condition that $q\in f_T^n(V)$ amounts to an algebraic constraint on $T$. Taking $V = \mathop{\mathrm{crit}}(f) \cup f_T({\mathcal I})$, we obtain that
$f_T^{-n}(q)$ consists of $\lambda_2^n$ distinct preimages, each of which is
a regular point for $f_T^n$, for all $T\in{\mathop{\mathrm{Aut}}}({\mathbb P}^2)$ outside some algebraic
subset.
For fixed $n$, the set of $T$ for which $f_T^{n-1}(C_T)$ is reducible is again
an algebraic subset of ${\mathop{\mathrm{Aut}}}({\mathbb P}^2)$. The degree of $f_S^{n-1}(C_{S})$ is equal
to the number of set theoretic intersections $\#(f_S^{n-1}(C_{S})\cap L)$ of
$f_S^{n-1}(C_{S})$ with some line $L\subset{\mathbb P}^2$. Therefore, for $T$ outside
of some proper algebraic set $\#(f_T^{n-1}(C_{T})\cap L)$ will be finite.
Moreover, it follows from the Weierstrass Preparation Theorem that as a
function of $T$, the quantity $\#(f_T^{n-1}(C_{T}) \cap L)$ is finite,
constant and maximal (in particular at least $\#(f_S^{n-1}(C_{S}) \cap L)$),
off another proper algebraic subset of ${\mathop{\mathrm{Aut}}}({\mathbb P}^2)$. We conclude that $\mathop{\rm deg}
f_T^{n-1}(C_{T}) \to\infty$ for all $T$ outside a countable union of such
sets. \end{proof}
\subsection{The saddle measure}
Recall that a rational map $f_0:{\mathbb P}^1\to{\mathbb P}^1$ is \emph{hyperbolic}, i.e. uniformly expanding on its Julia set, if all critical points of $f_0$ lie in basins of attracting periodic points.
\begin{proposition}
\label{prop:hypmap}
Let $L_0\subset {\mathbb P}^2$ be any non-exceptional line disjoint from ${\mathcal I}$ and ${p_0}\in{\mathbb P}^2$ be any point outside $L_0$ and $f(L_0)$. Then one can choose a linear map $T_0:{\mathbb P}^2\setminus\{{p_0}\}\to L_0$ so that $T_0\circ f|_{L_0}$ is a degree $d$ hyperbolic rational self-map of ${\mathbb P}^1$ (i.e. of $L_0$).
\end{proposition}
\begin{proof}
Since we assume $L_0$ is disjoint from ${\mathcal I}$, the total
transform of $L_0$ under $f$ coincides with the proper transform $f(L_0)$.
Hence, the push forward divisor $f_*L_0$ has irreducible support equal to $f(L_0)$.
Because we have equivalence of divisors $f_*L_0 \sim dL_0$, it follows that
$f:L_0\to f(L_0)$ is $m$-to-$1$ where $m\mathop{\rm deg} f(L_0) = d$. Now choose ${p_0}$
outside $L_0\cup f(L_0)$. Let $\Pi:{\mathbb P}^2\setminus\{{p_0}\}\to~L_0$ denote the
central projection sending each line through ${p_0}$ to its intersection with $L$.
The restriction $\Pi:f(L_0)\to L_0$ is $\mathop{\rm deg} f(L_0)$-to-1. Hence the
restriction $\tilde f_0 := (\Pi\circ f)|_{L_0}$ is a degree $d$ rational map of
${\mathbb P}^1$. We choose the identification $L_0\cong {\mathbb P}^1$ so that $\infty$ is not
equal to $\tilde f_0(0)$ or to any critical value of $\tilde f_0$. Let $f_0(z)
= \alpha \tilde f_0(z)$ for some $\alpha\in {\mathbb C}^*$. Taking $|\alpha|$ small
enough, we obtain that all critical values of $f_0$ lie in a small disk
$\Delta\subset{\mathbb C}$ about $z=0$ and that $f_0(\overline{\Delta}) \subset \Delta$.
It follows that $\Delta$ is in the Fatou set of $f_0$ and therefore $f_0$ is
hyperbolic. Let $A:L_0\to L_0$ denote the contraction $z\mapsto \alpha z$.
Then the (linear) map $T_0 := A\circ \Pi:{\mathbb P}^2\setminus\{{p_0}\}\to L_0$ satisfies
the conclusion of the proposition.
\end{proof}
With $T_0$ as in the Proposition \ref{prop:hypmap}, we let $f_0$ denote the restriction $(T_0\circ f)|_{L_0}$
and ${\mathcal{J}}$ denote the Julia set of $f_0$. Recall (see e.g. \cite{Jon2}) that a
\emph{history} of a point $z\in{\mathcal{J}}$ is a sequence $(z_{-n})_{n\geq 0}$
terminating with $z_0 = z$ and satisfying $f(z_{-(n+1)})~=~z_{-n}$ for all
$n\in{\mathbb N}$. We let $\hat {\mathcal{J}}$ denote the set of all histories of points in ${\mathcal{J}}$
and endow $\hat {\mathcal{J}}$ with the product topology. The \emph{natural extension} of
$f_0:{\mathcal{J}}\to{\mathcal{J}}$ is the homeomorphism $\hat f:\hat{\mathcal{J}}~\to~\hat{\mathcal{J}}$ given by $\hat
f(\hat z) = (f_0(z_{-n}))_{n\geq 0}$. The canonical projection $\pi:~\hat {\mathcal{J}}~\to~{\mathcal{J}}$, given by $\pi(\hat z) = z_0$ is a semiconjugacy of $\hat f$ onto $f_0$
that induces a one-to-one entropy-preserving correspondence $\pi_*$ between
$\hat f$ invariant probability measures on $\hat {\mathcal{J}}$ and $f_0$ invariant
probability measures on ${\mathcal{J}}$. The natural extension has the universal property
that any other semiconjugacy of an invertible system onto $f_0:{\mathcal{J}}\to{\mathcal{J}}$ factors
through $\pi$. (See \cite[Ch. 1]{PU} for more details on the natural extension.)
\begin{theorem}
\label{thm:nuexists}
Suppose that conditions (A),(B), and (D) hold and that \\ $T_0:{\mathbb P}^2~\setminus~\{{p_0}\}~\to~L_0$ is chosen so that the induced one dimensional map $f_0 = T_0\circ f:{\mathbb P}^1\to {\mathbb P}^1$ is hyperbolic with Julia set ${\mathcal{J}}$. Then for any $\epsilon>0$ small enough, there exists $\delta>0$ such that $T\in{\mathcal T}(T_0,\delta)$ implies that there is a unique $f_T$-invariant measure $\nu$ on $B_\epsilon(L_0)$ with entropy $\log d$. Moreover
\begin{itemize}
\item[(a)] $f_T$ is uniformly hyperbolic of saddle type on $\mathop{\rm supp}\nu$.
\item[(b)] The canonical projection $\pi:\hat {\mathcal{J}}\to {\mathcal{J}}$ factors $\pi = \pi_T\circ \hat\pi$ into semiconjugacies $\hat\pi_T:\hat{\mathcal{J}} \to \mathop{\rm supp}\nu$ of $\hat f$ onto $f_T$, and $\pi_T:\mathop{\rm supp}\nu\to {\mathcal{J}}$ of $f_T$ onto~$f_0$.
\item[(c)] $\mathop{\rm supp}\nu$ is not contained in an algebraic curve.
\item[(d)] Any point $p\in B_\epsilon(L_0)$ has forward orbit $(f_T^n(p))_{n\geq 0}$ asymptotic to either $\mathop{\rm supp}\nu$ or to one of finitely many attracting periodic cycles.
\end{itemize}
\end{theorem}
This theorem is closely related to papers on attractors and attracting sets for
endomorphisms of ${\mathbb P}^2$ by Forn{\ae}ss-Weickert \cite{FW_ATTRACTORS},
Jonsson-Weickert \cite{JW_ATTRACTORS}, Forn{\ae}ss-Sibony \cite{FS_EXAMPLES},
Dinh \cite{DINH_ATTRACTORS} and Daurat \cite{DAURAT}. It is also closely related to
work on H\'enon and H\'enon-Like maps by Hubbard and Oberste-Vorth \cite{HOV2} and Dujardin \cite{DUJARDIN_HENON_LIKE}. Regardless, we present a more or less self-contained proof here.
Recall that a sequence $(z_n)_{n\geq 0}\subset L_0$ is a $\delta$ pseudo-orbit for $f_0$ if ${\rm dist}(f(z_n),z_{n+1})~<~\delta$ for all $n\geq 0$.
\begin{proof}
Let $\epsilon>0$ be small enough that Proposition \ref{prop:twotraps} applies. Because $f_0$ is hyperbolic, we may shrink $\epsilon$ still further if necessary and choose an open cover $\{{\mathcal U}^-,{\mathcal U}^+\}$ of $L_0$ and $\delta>0$ with the following properties:
\begin{itemize}
\item all conclusions of Proposition \ref{prop:twotraps} apply;
\item ${\mathcal{J}} \subset {\mathcal U}^- \subset \overline{{\mathcal U}^-}\subset f_0({\mathcal U}^-)$;
\item $f_0(\overline{{\mathcal U}^+}) \subset {\mathcal U}^+\subset L_0\setminus {\mathcal{J}}$;
\item $f_0$ is uniformly expanding on ${\mathcal U}^-$ and uniformly contracting on ${\mathcal U}^+$;
\item for any $z\in L_0$, we have $f_0^n(z)\in {\mathcal U}^-$ for all $n\in {\mathbb N}$ if and only if $z\in{\mathcal{J}}$;
\item any $\delta$ pseudo-orbit in ${\mathcal U}^-$ is $\epsilon$-shadowed by a unique orbit in ${\mathcal{J}}$;
\item any $\delta$ pseudo-orbit $(z_n)_{n\geq 0}$ with $z_0\in{\mathcal U}^+$ is completely contained in ${\mathcal U}^+$ and eventually contained in an $\epsilon$ neighborhood of some attracting cycle.
\end{itemize}
Strictly speaking, one must replace the Fubini-Study metric on ${\mathcal U}^\pm$ with some other metric to get the uniform expansion/contraction in the fourth item. Alternatively, one can replace $f_0$ with a high enough iterate. These things do not affect our arguments.
Let $\Pi~:~{\mathbb P}^2\setminus\{p_0\}~\to~L_0$ be the central projection. We extend ${\mathcal U}^-,{\mathcal U}^+$ to open subsets of ${\mathbb P}^2$, setting ${\mathcal U}^\pm_\epsilon := \Pi^{-1}({\mathcal U}^\pm)\cap B_\epsilon(L_0)$. Fixing $T\in {\mathcal T}(T_0,\delta)$, we let
$$
{\mathcal L}_T := \{p\in B_\epsilon(L_0):f_T^n(p) \in {\mathcal U}^-_\epsilon \text{ for all } n\geq 0\}.
$$
For any point $p\in {\mathcal L}_T$, the sequence $(\Pi\circ f_T^n(p))_{n\geq
0}$ is a $\delta$ pseudo-orbit for $f_0$ that is completely contained in
${\mathcal U}^-$, so there is a unique point $\pi_T(p) \in {\mathcal{J}}$ whose $f_0$-orbit $\delta$
shadows that of $p$. Clearly $\pi_T\circ f_T = f_0\circ \pi_T$. The fact that
$f_0$ is uniformly expanding on ${\mathcal U}^-$ guarantees that the semiconjugacy
$\pi_T:{\mathcal L}_T\to{\mathcal{J}}$ is continuous.
Similarly, for any point $p\notin{\mathcal L}_T$, the sequence $(\Pi\circ f_T^n(p))_{n\geq 0}$ is a $\delta$ pseudo-orbit for $f_0$ eventually contained ${\mathcal U}^+$ and therefore eventually contained in an $\epsilon$ neighborhood of some attracting periodic cycle $(z_n)_{n=0}^{N-1}$. Since $f_0$ is contracting on ${\mathcal U}^+$, $f_T$ is contracting on ${\mathcal U}^+_\epsilon$. It follows that there is a unique attracting periodic cycle $(p_n)_{n=0}^{N-1}$ for $f_T$ with $d(p_n,z_n) < \epsilon$ for all $0\leq n\leq N-1$ and that the orbit of $p$ is asymptotic to this cycle. That is, every point $p\in B_\epsilon(L_0)\setminus {\mathcal L}_T$ has a forward orbit asymptotic to one of finitely many attracting cycles.
\begin{lemma}
\label{lem:lam} For small enough $\delta>0$ and $T\in{\mathcal T}(\pi,\delta)$, the map $\pi_T:{\mathcal L}_T\to {\mathcal{J}}$ is a fibration of ${\mathcal L}_T$ by complex disks that are properly embedded in $B_\epsilon(L_0)$, and the maps $f_T:\pi_T^{-1}(z) \to \pi_T^{-1}(f_0(z))$ are uniformly contracting. In particular $f_T$ is uniformly hyperbolic of saddle type on ${\mathcal L}_T$.
\end{lemma}
The proof is a small variation on that of the Hadamard-Perron Theorem (see e.g. \cite[Thm. 6.2.8]{KH}), helped along a bit by the holomorphic context. We sketch it for the sake of completeness.
\begin{proof}
Replacing $f_T$ and $f_0$ with iterates for the moment, we may assume that
$f_0$ expands the Fubini-Study metric on ${\mathcal U}^-$ by a uniform factor of $4$.
Let $\Pi:{\mathbb P}^2\setminus\{{p_0}\}\to L_0$ be the central projection. Given $z\in {\mathcal{J}}$,
let $D_n, D_n' \subset L$ be the disks of radius $\delta$, $\delta/2$ about
$f_0^n(z)$. Thus $f_0$ maps $D_{n-1}'$ biholomorphically onto a
neighborhood of $\overline{D_n}$.
Setting $P_n = \Pi^{-1}(D_n)\cap B_\epsilon(L_0)$, we call a disk $\Delta$ in $P_n$ \emph{vertical} if $\Delta$ meets $L_0$ transversely in a single point and $\partial\Delta\subset \partial B_\epsilon(L_0)$. Provided $\delta > 0$ is small enough, $f_T^{-1}(\Delta)$ meets $P_{n-1}$ in a unique vertical disk $f_T^\sharp(\Delta)$. If, e.g. by linear projection from a point in $L_0$ well outside ${\mathcal{J}}$, we regard all vertical disks in $P_n$ as graphs over the `central disk' $\Pi^{-1}(f_0^n(z))$, then $f_T^\sharp$ uniformly contracts distances between graphs. It follows that if we choose vertical disks $\Delta_n\subset P_n$ for each $n\in{\mathbb N}$, then $(f_T^n)^\sharp(\Delta_n)$ converges uniformly to a vertical disk $\Delta_z \subset P_0$. Necessarily $f^n(\Delta_z)\subset P_n$ for all $n\in{\mathbb N}$ so that $\pi_T(\Delta_z) = z$. Since the $P_n$ are foliated by vertical disks, it follows that in fact $\Delta_z = \pi_T^{-1}(z)$, i.e. any point $p\in P_0\setminus \Delta_z$ satisfies $f_T^n(p)\notin P_n$ for some $n$ large enough.
Since $f_{T_0}$ contracts all vertical disks to $L_0$, it follows (for small enough $\delta$) that $f_T(\Delta_z) \subset B_{\epsilon/2}(L_0)\cap \Delta_{f(z)}$. So by the Schwarz Lemma $f_T:\Delta_z\to \Delta_{f(z)}$ is uniformly contracting. Since $f_0$ is expanding on ${\mathcal{J}}$, $f_T$ must uniformly expand the distance between distinct `stable' disks $\Delta_z,\Delta_{z'}$. Hence $f_T$ is hyperbolic of saddle type on ${\mathcal L}_T$.
\end{proof}
Let $\Omega_T := \bigcap f_T^n({\mathcal L}_T)$. Since ${\mathcal L}_T$ is closed in $B_\epsilon(L_0)$ and $f(\overline{B_\epsilon(L_0)}) \subset B_\epsilon(L_0)$, it follows that $\Omega_T$ is compact and non-empty and satisfies $f_T(\Omega_T) = \Omega_T$. The previous lemma and $f_0^{-1}({\mathcal{J}}) = {\mathcal{J}}$ guarantee that $\pi_T:\Omega_T \to {\mathcal{J}}$ is surjective. Since $f_T$ is uniformly contracting along fibers of $\pi_T$, we have for each history $\hat z\in\hat{\mathcal{J}}$ that the intersection $\bigcap_{n\geq 0} \overline{f_T^n(\pi_T^{-1}(z_{-n}))}$ is a single point $\hat\pi_T(\hat z)$. More or less by construction, the resulting map $\hat\pi_T:\hat{\mathcal{J}} \to \Omega_T$ semiconjugates $\hat f$ to $f_T$ and satisfies $\pi = \pi_T\circ\hat\pi_T(\hat z)$. It is continuous because $f_T$ is continuous and surjective because $f_T(\Omega_T) = \Omega_T$.
Thus, by the universal property of the natural extension we have that $\hat f:\hat J\to\hat J$ is also the natural extension of $f_T:\Omega_T\to\Omega_T$. The unique invariant probability measure $\check \nu$ of maximal entropy for $f_0:{\mathcal{J}} \to {\mathcal{J}}$ lifts to a unique measure of maximal entropy $\hat \nu$ for $\hat f:\hat J\to\hat J$. Pushing forward to $\nu := \hat\pi_{T*}\hat \nu$ gives the unique measure of maximal entropy $\log d$ for $f_T:\Omega_T\to\Omega_T$.
To complete the proof of Theorem \ref{thm:nuexists} it remains to show that $\mathop{\rm supp}\nu$ is not contained in an algebraic curve. Suppose, in order to reach a contradiction, that $\mathop{\rm supp}\nu$ is contained in an algebraic curve $C$. Since $\nu$ is ergodic with infinite support, we can suppose each irreducible component of $C$ intersects $\mathop{\rm supp} \nu$ in an infinite set with positive measure, and that these components are permuted cyclically by $f_T$.
It follows from Lemma \ref{lem:curvegrows} that no irreducible component of $C$ is contained in $B_\epsilon(L_0)$.
Slightly shrinking $\epsilon$ if necessary, we may assume that $B_\epsilon(L_0)$ omits at least three points of each such component. So $C\cap B_\epsilon(L_0)$ is forward invariant by $f_T$, and its normalization $S$ is a Riemann surface whose connected components are all hyperbolic. Hence the restriction of $f_T$ to $C\cap B_\epsilon(L_0)$ lifts to a
map on $S$ which is distance non-increasing in the hyperbolic metric. In particular the restriction of
$f_T$ to $C\cap B_\epsilon(L_0)$ has topological entropy zero. This
contradicts the fact that $\nu$ is an $f_T$-invariant measure of positive
entropy supported on $C\cap B_\epsilon(L_0)$.
\end{proof}
\subsection{The repelling measure}
Recall \cite{KITCHENS} that for any positive integer $k$, the \emph{one-sided $k$-shift} is the set $\Sigma_k$ of all sequences $(i_n)_{n\geq 0}$ that take values $i_n\in\{0,\dots,k~-~1\}$, together with the map $\sigma:\Sigma_k\to\Sigma_k$, $\sigma:(i_n) \mapsto (i_{n+1})$. The one-sided $k$-shift has topological entropy $\log k$ and admits a unique invariant measure that achieves this entropy.
\begin{theorem}
\label{thm:muexists}
Suppose that conditions (A),(B) and (C) hold. Then for $\epsilon>0$ small enough there exists $\delta>0$ such that for any $T\in {\mathcal T}(T_0,\delta)$
\begin{itemize}
\item ${\mathcal A} := \bigcap_{n\geq 0} f_T^{-n}(B_\epsilon(f^{-1}(p_0)))$ is a Cantor set totally invariant by $f_T$;
\item $f_T$ is uniformly expanding on ${\mathcal A}$;
\item $f_T|_{\mathcal A}$ is topologically conjugate to the one-sided $\lambda_2$-shift;
\item in particular, there is a unique $f_T$-invariant measure $\mu$ with $h_\mu(f_T) =
\log\lambda_2$ having
$\mathop{\rm supp}\mu \subset B_\epsilon(f^{-1}(p_0))$.
\end{itemize}
\end{theorem}
\begin{proof}
Choose $\epsilon>0$ and then $\delta>0$ small enough that all conclusions of Proposition~\ref{prop:twotraps} apply. Since $p_0$ is a regular value of $f$, we may further assume that $B_\epsilon(f^{-1}(p_0))$ is a disjoint union of $\lambda_2$ open balls, each centered at a preimage $p_j$ of $p_0$ and mapped biholomorphically by $f$ onto a neighborhood of $p_0$. In particular, $B_\epsilon(f^{-1}(p_0))$ is Kobayashi complete hyperbolic. (See \cite{KRANTZ} for a gentle introduction to the Kobayashi metric and \cite{LANG} for more details.)
Shrinking $\delta$ if necessary, we may assume that $T$ maps the complement of
$f(B_\epsilon(p_j))$ into $B_\epsilon(L_0)$, so that $f_T(B_\epsilon(p_j))$
contains $\overline{B_\epsilon(f^{-1}(p_0))}$. It follows that $f$ uniformly
expands the Kobayashi distance on $B_\epsilon(f^{-1}(p_0))$.
Standard arguments now tell us that if we assign to each point $p\in {\mathcal A}$ its `itinerary' $\iota(p) = (i_n)$, where $f_T^n(p) \in B_\epsilon(p_{i_n})$, then the resulting map $\iota:{\mathcal A} \to \Sigma_{\lambda_2}$ is a homeomorphism that conjugates $f_T|{\mathcal A}$ to the shift map $\sigma$. Pulling back the unique measure of maximal entropy for $\sigma$ gives us the measure $\mu$ in the final conclusion of the theorem.
\end{proof}
In order to show that the repelling measure $\mu$ is not contained in an algebraic curve, we first indicate some useful refinements of Proposition \ref{prop:hypmap}.
\begin{proposition}
\label{prop:hypmapplus}
Suppose $p_0$ and $L_0$ satisfy $p_0 \not \in f(L_0)$ and Properties (B-D) and let $L_1$ be some chosen line
through two distinct points of $f^{-1}(p_0)$.
One can choose a linear map
$T_0:{\mathbb P}^2\setminus\{{p_0}\}\to L_0$ so that $T_0\circ f|_{L_0}$ is a degree $d$ hyperbolic rational self-map of $L_0$ and, additionally, there exists a disk
$\Delta\subset L_0$ satisfying
\begin{itemize}
\item $\Delta$ is disjoint from the critical set of $f$ and from $L_1 \cap L_0$,
\item $\Delta$ contains
$T_0\circ f(\overline{\Delta})$, $T_0\circ f({\mathcal E})$, and $T_0\circ f(L_1 \cap L_0)$, and
\item each irreducible component of ${\mathcal E}$ has infinite forward orbit under $T_0 \circ f$.
\end{itemize}
\end{proposition}
\begin{proof}
In the proof of Proposition \ref{prop:hypmap} we can (additionally)
identify $0$ and $\infty$ with regular points of $f$ different from $L_1\cap L_0$.
All conclusions of Proposition \ref{prop:hypmap}
and the first two conclusions above follow by taking both $|\alpha|$ and $\Delta$
small enough.
Let $T_0$ be the map obtained in the previous paragraph.
Concerning the third conclusion, note that $T_0 \circ f$ is injective on $\Delta$ and has a unique fixed point $\eta$,
so it suffices to arrange that no irreducible component $E$ of ${\mathcal E}$ has image $T_0 \circ f(E)~=~\eta$.
This can be done without affecting any of the previous properties by an arbitrarily small perturbation of $T_0$.
\end{proof}
\begin{theorem}
\label{THM_MU_NOT_1D}
Suppose in Theorem \ref{thm:muexists} that the rational map $f$ has equal degrees $d = \lambda_2$. If $\epsilon>0$ and $\delta>0$ are small enough and $T_0$ is chosen to satisfy the hypotheses of Propositions \ref{prop:hypmap} and \ref{prop:hypmapplus}, then there is no algebraic curve in ${\mathbb P}^2$ that contains $\mathop{\rm supp}\mu$.
\end{theorem}
We begin proving this by drastically narrowing down the possibilities for an algebraic curve that contains $\mathop{\rm supp}\mu$.
\begin{lemma}
\label{LEM_SMOOTH_CURVE}
Suppose under the hypotheses of Theorem \ref{THM_MU_NOT_1D} that $T\in{\mathcal T}(T_0,\delta)$ and $C\subset{\mathbb P}^2$ is an algebraic curve containing $\mathop{\rm supp}\mu$, where $\mu$ is the repelling measure for $f_T$. Then ${\mathcal E}(f) = {\mathcal E}(f_T) \neq \emptyset$ and we may assume that
\begin{itemize}
\item $C$ is smooth and rational, in particular irreducible.
\item $C = f_T^{-1}(C)$ is totally invariant.
\item ${\mathcal I}(f_T^n) \cap C = \emptyset$ for all $n\geq 1$.
\item The forward orbit $f_T^n({\mathcal E}(f))$ is contained in $C$.
\end{itemize}
\end{lemma}
\begin{proof}
The measure of maximal entropy on the $\lambda_2$-shift is mixing, hence so is $\mu$. We may therefore assume that $C$ is irreducible and invariant, i.e. $f_T(C) = C$. Since $\mu$ has entropy $\log\lambda_2$ and totally disconnected support, it further follows that $C$ is rational and that $f_T|_C$ is (at least and therefore exactly) $\lambda_2$-to-1.
In particular, $C$ is \emph{totally} invariant, i.e. $f_{T}^{-1}(C) = C$. Furthermore,
\begin{align*}
\lambda_2\mathop{\rm deg} C\leq \mathop{\rm deg} f_{T*}C = d\mathop{\rm deg} C = \lambda_2\mathop{\rm deg} C
\end{align*}
implies that $f_{T*} C = \lambda_2 C$ and therefore that $C$ contains no indeterminate points for $f_T$. Invariance of $C$ implies that it contains no indeterminate points for $f_T^n$ for any $n\geq 1$.
Since $\mathop{\rm supp}\mu$ is disjoint from the critical set of $f_T$, we see that $f_T$ is not critical along $C$. Hence $f_T^* C = C + E_C$ where $E_C$ is a non-trivial effective divisor supported on the exceptional set ${\mathcal E}$. In particular ${\mathcal E}\neq \emptyset$. Each irreducible component $E$ of ${\mathcal E}$ must meet $C$ at some non-indeterminate point $p$, so it follows that $f_T^n(E) = f_T^n(p) \subset C$ for all $n\geq 1$.
It remains to prove that $C$ is smooth. Suppose that $p$ is a singular point of $C$ and that $q\in{\mathbb P}^2$ is any $f_T$-preimage of $p$. Any local defining function $\psi$ for $C$ vanishes to order at least $2$ at $p$. Hence the local defining function $\psi\circ f_T$ for $f_T^*C$ vanishes to order at least $2$ at $q$, i.e $f_T^*C$ is singular at $q$. Since $f_T^*C = C + E_C$, we infer that either $C$ is singular at $q$ or $q\in{\mathcal E}$. In the first case, we repeat the argument with $q$ in place of $p$, etc. Since $C$ has only finitely many singular points, we will eventually find ourselves in the second case. In short, \emph{any} backward orbit of $p$ contains a point in $f_T({\mathcal E})\cap C$.
It follows from Proposition \ref{prop:hypmapplus} that for $T$ close enough to
$T_0$ any irreducible component of $E$ of ${\mathcal E}$ has $f_T(E)$
in a forward invariant neighborhood $W$ of the
disk $\Delta$. Moreover, $f_T|_W$ is a biholomorphism onto its
image with a unique fixed point. By the third part of Proposition \ref{prop:hypmapplus}, this fixed point is
different from $f_T(E)$.
Therefore, $p$ has an infinite forward orbit along which $C$ would be singular, giving a contradiction.
We conclude that $C$ is smooth.
\end{proof}
\begin{proof}[Proof of Theorem \ref{THM_MU_NOT_1D}]
If the theorem fails, then there is a sequence $T_i \in {\mathop{\mathrm{Aut}}}({\mathbb P}^2)$, with
$T_i \rightarrow T_0$, for which the repelling invariant measure $\mu_i$
associated to $f_{T_i}$ is contained in an algebraic curve $C_i$. By
Lemma \ref{LEM_SMOOTH_CURVE}, each $C_i$ is smooth and rational. In particular each has degree one or two.
Refining, we may assume that $\mathop{\rm deg} C_i$ is independent of $i$ and that $C_i \to C_\infty$, where $C_\infty$ is a (possibly reducible) divisor of the same degree, and the convergence can be understood to take place with respect to coefficients of the homogeneous defining polynomials for $C_i$.
We claim that $L_0$ is an irreducible component of $C_\infty$. Indeed, $C_i$ contains the forward orbit of the exceptional set ${\mathcal E}$ by $f_{T_i}$. As $i\to\infty$ this forward orbit converges to that of ${\mathcal E}\cap L_0$ by $T_0\circ f$.
By Proposition \ref{prop:hypmapplus}, the latter is an infinite subset of $L_0$, so that $L_0$ must be contained in $C_\infty$. So the claim follows from the fact two algebraic curves with infinitely many points in common must share an irreducible component.
Since $\mathop{\rm supp}\mu\subset C_i$ converges to $f^{-1}(p_0)$ as $i\to\infty$, it further follows that $f^{-1}(p_0)\subset C_\infty$. By hypothesis $f^{-1}(p_0)$ contains at least two points, all distinct from $L_0$.
Hence the only possibility here is that $C_\infty = L_0\cup L_1$, where $L_1\neq L_0$ contains the entire preimage $f^{-1}(p_0)$ of $p_0$. Proposition \ref{prop:hypmapplus} implies that $L_1\cap L_0$ is a point in the Fatou set of the one dimensional map $f_0 := T_0\circ f:L_0 \to L_0$. As a rational map of degree $d>1$, the map $f_0$ has at least one repelling fixed point $s\in L_0$. Since it lies in the Julia set ${\mathcal{J}}$ of $f_0$, the point $s$ differs from $L_0\cap L_1$.
Let $\pi_{T_i}:{\mathcal L}_{T_i}\to {\mathcal{J}}$ be as in the proof of Theorem \ref{thm:nuexists}. By Lemma \ref{lem:lam}, the sets $D_i = \pi_{T_i}^{-1}(s)$ are complex disks properly embedded in $B(L_0,\epsilon)$, forward invariant and contracted by $f_{T_i}$. The point $s_i := \bigcap f_{T_i}^n(D_i)$ is fixed of saddle type for $f_{T_i}$.
As $i\to\infty$, we have $s_i\to s$ and $D_i \to B(L_0,\epsilon)\cap \Pi^{-1}(s)$ uniformly. Since $C_i\to L_0\cup L_1$, we also have for large $i$ that the central projection $\Pi:{\mathbb P}^2\setminus\{p_0\}$ onto $L_0$ restricts to an injective map of $C_i\cap B(L_0,\epsilon)$ onto $L_0$ minus a small neighborhood of $L_0\cap L_1$. In particular $C_i\cap D_i\neq \emptyset$ for $i$ large enough.
Since $C_i\cap D_i$ is forward invariant and closed, it follows that $C_i$ contains the saddle point $s_i$. Since $C_i$ also contains a repelling fixed point near each preimage of $p_0$ and the unique attracting point in $B(L_0,\epsilon)$, we see that $f|_{C_i}$ has at least $d + 2$ fixed points. However, a degree $d$ map of ${\mathbb P}^1$ has only $d+1$ fixed points. This contradiction concludes the proof.
\end{proof}
We have now established the main results stated at the beginning of this paper.
That is, if equality $\lambda_2 = d$ holds for topological and algebraic degrees of
$f$, then Proposition \ref{PROP_AS} and Theorems \ref{THM_NO_FOLIATION},
\ref{thm:nuexists}, \ref{thm:muexists} and \ref{THM_MU_NOT_1D} together give Theorem {A}; and Theorem {{B} } follows from Corollaries \ref{COR_AS} and \ref{COR_GENERIC_NIF}.
\section{A specific example}\label{SEC_PERTURBED_CHEBYSHEV}
\label{section:unperturbed}
In the remainder of this article, we consider some rather restricted linear perturbations of a particular rational map $f: {\mathbb P}^2 \dashrightarrow {\mathbb P}^2$ that is closely related to the one variable quadratic Chebyshev map and was studied in detail by Han Liu in his PhD thesis \cite{han}. Specifically $f = g \circ h$ where
\begin{equation}
\begin{array}{rcl}
g[x_1:x_2:x_3] & = & [x_1^2:x_2^2:x_3^2], \quad \mbox{and} \\
h[x_1:x_2:x_3] & = & [x_1(-x_1+x_2+x_3):x_2(x_1-x_2+x_3):x_3(x_1+x_2-x_3)].
\end{array}
\end{equation}
Note that $h$ is a birational involution, linearly conjugate to the standard Cremona involution $[x_1,x_2,x_3]\mapsto [x_2x_3,x_3x_1,x_1x_2]$ via the (unique) automorphism of ${\mathbb P}^2$ that sends the points $[1,0,0],[0,1,0],[0,0,1],[1,1,1]$ to $[0,1,1], [1,0,1], [1,1,0],$\\ $[1,1,1]$, respectively. Hence the indeterminacy set of $h$ is
$$
{\mathcal I}(h) = \{a_1,a_2,a_3\} := \{[0,1,1],[1,0,1],[0,1,1]\},
$$
and the exceptional set ${\mathcal E}(h)$ consists of the three lines $A_1,A_2,A_3$ joining these points. Specifically $h(A_i) = a_i$ and vice versa, where $A_i$ is the line joining the pair ${\mathcal I}(h)\setminus\{a_i\}$.
Both maps $g$ and $h$ preserve the rational two form $\eta$ given in the affine
coordinates $(x,y)\mapsto [x,y,1]$ by $\eta = \frac{dx\wedge dy}{xy}$. That
is, $g^*\eta = 4\eta$ and $h^*\eta = \eta$. Hence $f^*\eta = (4\cdot 1)\eta$.
Plane rational maps that preserve meromorphic two forms (up to a multiplicative factor) are considered at length in
\cite{DL}.
The map $f$ considered here is one of the simplest
instances we know of a non-invertible rational map that preserves a two form
and has non-obvious dynamics. These dynamics are completely described in
\cite{han}, but here we pay attention only to aspects relevant to the theme of
this article.
Several elementary properties of $f$ are readily inferred from those of $g$ and $h$.
\begin{proposition}\label{PROP:BASIC_CHEB}
$f$ is a dominant rational map with topological and algebraic degrees $\lambda_2(f) = d(f) = 4$. Moreover,
\begin{enumerate}
\item $f$ is symmetric in the homogeneous coordinates $x_1,x_2,x_3$ and preserves (modulo indeterminate points) the real slice ${\mathbb{RP}}^2 := \{[x_1,x_2,x_3]\in{\mathbb P}^2: x_j\in{\mathbb R}\}$.
\item The poles $X_j := \{x_j=0\}$ of $\eta$ are each totally invariant by $f$. These lines are also the ramification locus of $f$, and the restriction $f:X_j\to X_j$ to any one of them, when expressed in the two non-vanishing homogeneous coordinates, is the one variable map $z\mapsto z^2$.
\item The points $\{e_1,e_2,e_3\} := [1,0,0],[0,1,0],[0,0,1]$ are each fixed and superattracting.
\item The only other (non-indeterminate) fixed point of $f$ is $[1,1,1]$, which is repelling.
\item The exceptional and indeterminacy loci of $f$ coincide with those of $h$, and $f(A_j) = a_j$. In particular, $f$ is not algebraically stable on ${\mathbb P}^2$.
\item The critical divisor of $f$ is reduced with degree six. Specifically $\mathop{\mathrm{crit}}(f) = \sum_i A_i + \sum_j X_j$.
\end{enumerate}
\end{proposition}
In order to understand the real dynamics of $f$, it is convenient to employ affine coordinates that are adapted to emphasize the symmetry of $f$ with respect to the homogeneous variables. Specifically, in what follows \emph{adapted affine coordinates} will mean affine coordinates $(x',y')$ on ${\mathbb P}^2\setminus L_0$, where $L_0=\{x_1+x_2+x_3=0\}$, that identify the repelling fixed point $[1,1,1]$ with $(0,0)$ and the superattracting fixed points $[1:0:0],[0:1:0],[0:0:1]$ with vertices of an equilateral triangle centered at $(0,0)$. In these coordinates, Figure \ref{fig:basins} shows the real points in the basins associated to each superattracting point $e_1,e_2,e_3$. This picture reveals some interesting aspects of the dynamics of $f$. The complement of the (closures of the) basins is (apparently) the open disk ${\mathcal U}$ inscribed in the triangle $\{x_1x_2x_3=0\}$. One can check that $\partial{\mathcal U} = {\mathbb R}{\mathbb P}^2\cap Q$, where $Q$ is the algebraic curve defined by $\rho(x_1,x_2,x_3) = x_1^2 + x_2^2+x_3^2 - 2(x_1x_2 + x_2x_3 + x_3x_1) = 0$) and that $Q\cap\{x_1x_2x_3=0\} = {\mathcal I}$, each intersection being a tangency.
\begin{figure}
\centering
\includegraphics[width=90mm]{figure1.jpg}
\caption{Real parts of the three superattracting basins shown in adapted affine coordinates. Alternating white/colored bands indicate the amount of time it takes for a point to get close to a superattracting point. The disk ${\mathcal U}$ in the center is the complement of the basin closures. Note also that the common boundary of each pair of basins is contained in an exceptional line $A_j$, and that ${\mathcal I}\subset\partial{\mathcal U}$ is the set where all three basins meet. \label{fig:basins}}
\end{figure}
\begin{proposition}
\label{prop:Q}
The conic curve $Q$ and the region ${\mathcal U}$ have the following properties.
\begin{enumerate}
\item $f(a_j) = Q$ for each $a_j\in \mathcal{I}(f)$.
\item $f(p) = p$ for every $p\in Q\setminus \mathcal{I}(f)$.
\item ${\mathcal U}$ is totally invariant modulo the lines $A_j$, i.e. for every $p\notin A_1\cup A_2\cup A_3$, we have $p\in {\mathcal U}$ if and only if $f(p) \in {\mathcal U}$.
\end{enumerate}
\end{proposition}
\begin{proof}
We have that $f(a_j) = g(h(a_j)) = g(A_j)$ is a conic curve tangent to each coordinate axis $X_i$ at the point $g(A_j\cap X_i) = \mathcal{I}(f)\cap X_i$, independent of $j$. Counting the number of conditions that the tangencies impose on the defining polynomial, one finds there is only one such conic, so we must have $f(a_j) = Q$.
To see that $Q$ is totally invariant by $f$, note that $g^*Q$ is a divisor of degree 4 with defining polynomial that is symmetric in the homogeneous coordinates $x_1,x_2,x_3$ and that the support of $g^*Q$ includes (by the previous paragraph) the lines $A_1,A_2,A_3$. It follows that $g^*Q = A_1+A_2 + A_3 + L_0$, where $L_0 = \{x_0+x_1+x_1=0\}$ is the `line at infinity' in Figure \ref{fig:basins}. Now $h^{-1} = h$ collapses each line $A_j$ to a point, and $h^*L_0$ is an effective divisor of degree two passing through each point in $\mathcal{I}(f)$ and (again) symmetric in $x_1,x_2,x_3$. It follows that $f^{-1}(Q) = h^{-1}(L_0) = \mathop{\mathop{\rm supp}} h^*L_0 = Q.$
Now we argue that $f|_Q = \mathrm{id}$. Since $h$ is an involution mapping $L_0$ to $Q$, we have that $h$ maps $Q$ bijectively onto $L_0$. Thus, $g(L_0) = Q$, and $g_*L_0 = k Q$ where $k$ is the topological degree of the restriction $g|_{L_0}$. Since $g(L_0)$ is a divisor of degree two, we have $k=1$, i.e. $g:L_0 \to Q$ is also injective. Hence $f|_Q$ is an automorphism of a rational curve. Symmetry with respect to homogeneous coordinates dictates that $f$ fixes each of the three points on $Q$ where two of the three homogeneous coordinates agree. Hence the restricted map, a linear fractional transformation fixing three points, must be the identity.
Using the fact that $(1,1)\in {\mathcal U}$ one sees that
\begin{align*}
{\mathcal U} = \{(x,y)\in {\mathbb R}^2:\rho_\mathrm{aff}(x,y) < 0\},
\end{align*}
where $(x,y) = [x,y,1]$ are (non-adapted) affine coordinates and $\rho_\mathrm{aff}(x,y) = \rho(x,y,1)$. Continuing the computation of the pullback of $Q$ begun above, we arrive at
$$
f^*Q = h^*(A_1+A_2+A_3+L_0) = 2(A_1+A_2+A_3) + Q.
$$
Hence $\rho_\mathrm{aff}\circ f = c \rho_\mathrm{aff} \rho_1^2\rho_2^2\rho_3^2$ where $\rho_j$ is an affine defining function for the line $A_j$ and $c$ is a constant. Applying this formula to $(1,1) = f(1,1)$, we see that $c > 0$. Hence for any point $p\notin A_1\cup A_2\cup A_3$, we see that $\rho_\mathrm{aff}(p)$ has the same sign as $\rho_\mathrm{aff}(f(p))$. This proves the final assertion in the proposition.
\end{proof}
\begin{corollary}
\label{cor:eqmeasure} $f$ has topological entropy equal to $\log 4$, and there is a unique measure of maximal entropy for $f$ given by $\mu = \frac{1}{2\pi^2} {\bm 1}_{{\mathcal U}} \eta$. In particular, $h_{top}(f|_{{\mathbb{RP}}^2}) = h_{top}(f) = \log 4$ and repelling periodic points of $f$ are dense in ${\mathcal U}$.
\end{corollary}
\begin{proof}
Note that the two form $\eta$ naturally defines a positive measure on ${\mathcal U}$. A routine computation shows that this measure has finite mass equal to $2\pi^2$. Since ${\mathcal U}$ is totally invariant by $f$ and $f^*\eta = 4\eta$, it follows that $f^*\mu = 4\mu = \lambda_2(f)\mu$. Hence by \cite{PARRY} $h_\mu(f) = \log 4$. But the variational principal and Gromov's bound \eqref{eqn:gromovbd} for the entropy of a rational map tell us that
$$
h_{\mu}(f) \leq h_{top}(f) \leq \log\max\{\lambda_1(f),\lambda_2(f)\} \leq \log\max\{d(f),\lambda_2(f)\} = \log 4
$$
so that $h_{top}(f) =\log 4$, too. In fact, $\lambda_1(f) < d(f)$ by Part (5) of Proposition \ref{PROP:BASIC_CHEB}, so uniqueness of the measure of maximal entropy and density of repelling cycles follows from the main results of \cite{Guedj}.
\end{proof}
\begin{remark} It is shown in \cite{han} that $\lambda_1(f) = 2$.
\end{remark}
The lines $A_j$ partition the disk ${\mathcal U}$ into four simply connected open sets ${\mathcal U}_i$, $i~=~0,1,2,3$. We index these so that ${\mathcal U}_0$ denotes the center triangle, and ${\mathcal U}_i$, $i=1,2,3$, denotes the set bounded by $A_i$ and $Q$. The partition $\left\{\overline{{\mathcal U}_i}\right\}_{i=0}^3$ maps forward well.
\begin{figure}
\centering
\includegraphics[width=90mm]{figure2.jpg}
\caption{Dynamics on ${\mathcal U}$. Each point $p$ is colored according to the connected component of ${\mathcal U}\setminus \{A_1,A_2,A_3\}$ that contains $f^4(p)$. \label{fig:partition}}
\end{figure}
\begin{proposition}\label{prop:partition}
$f$ maps each region ${\mathcal U}_j$ homeomorphically onto ${\mathcal U}$.
\end{proposition}
\begin{proof}
The last conclusion of Proposition \ref{prop:Q} tells us that $f({\mathcal U}_j) \subset {\mathcal U}$. Since $f$ has topological degree $4$, it will suffice to show that ${\mathcal U}\subset f({\mathcal U}_j)$ for $j=0,1,2,3$.
Consider first the image of ${\mathcal U}_0$. From the fact that $(x,y)\mapsto (1/x,1/y)$ preserves the first quadrant, it follows that $h({\mathcal U}_0) = {\mathcal U}_0$. One then sees that $g$ maps $\partial {\mathcal U}_0$ homeomorphically to $Q$. Indeed $A_3 \cap \partial {\mathcal U}_0$ is the line segment in ${\mathbb R}^2$ joining $(1,0)$ to $(0,1)$), and one verifies easily that $s$ maps this segment homeomorphically onto the portion of $Q$ joining $(1,0)$ to $(0,1)$. Similar observations apply to the the other two sides of ${\mathcal U}_0$. It follows that $f({\mathcal U}_0) = s({\mathcal U}_0) = {\mathcal U}$.
One shows similarly, that $f({\mathcal U}_j) = {\mathcal U}$ for $j=1,2,3$. For instance $h({\mathcal U}_3) = \{(x,y)\in{\mathbb R}^2:(-x,-y)\in {\mathcal U}_0\}$. Hence $f({\mathcal U}_3) = g(h({\mathcal U}_3)) = g({\mathcal U}_0) = {\mathcal U}$.
\end{proof}
Proposition \ref{prop:partition} suggests that ${\mathcal U}_j$, $j=0,1,2,3$ might be a Markov partition for the dynamics of $f$ on ${\mathcal U}$. The presence of points of indeterminacy in $\partial{\mathcal U}$ makes this idea a bit tricky to verify, but it is nevertheless carried out in detail in \cite{han}, which gives a complete account of both the real and complex dynamics of $f$ on all of ${\mathbb P}^2$. Our purpose here is to consider the dynamics of a family $f_t$, $t\in[0,1]$ of perturbations of $f$, so we turn now to these.
For each $t\in[0,1]$ we set $f_t:=T_t\circ f$ where $T_t$ is the linear map given in homogeneous coordinates by
$$
T_t[x_1:x_2:x_3] = [x_1:x_2:x_3]-\frac{1-t}{3}(x_1+x_2+x_3)[1:1:1].
$$
Then $T_0$ is the central projection from ${p_0}:=[1:1:1]$ to the line $L_0 := \{x_1+x_2+x_3=0\}$, and in adapted affine coordinates $T_t$, $t\neq 0$ is the scaling map $(x',y')\mapsto t^{-1}(x',y')$. In particular, for $t\in (0,1)$, we have that ${\mathcal U}_t := T_t({\mathcal U}) \supset\overline{{\mathcal U}}$. Since ${\mathcal U}$ is totally invariant by $f$, it follows that
\begin{equation}
\label{EQN_BACKWARD}
f_t^{-n}\left(\overline{{\mathcal U}}\right) \subset {\mathcal U} \text{ for all } n\in{\mathbb N}.
\end{equation}
As in \S \ref{SEC_PROOFS}, the indeterminacy set ${\mathcal I} := {\mathcal I}(f) = {\mathcal I}(f_t)$ is independent of the perturbation.
Our goal in the remainder of this section is to establish the following two results about dynamics of $f_t$.
\begin{theorem}\label{THM_REAL_MEASURE}
For all $t \in (0,1)$, the following are true for $f_t$.
\begin{itemize}
\item[(a)] $f_t$ is algebraically stable with dynamical degrees $\lambda_1=\lambda_2 = 4$.
\item[(b)] There is no $f_t$-invariant foliation.
\item[(c)] The topological entropy of $f_t$ as a real (and complex) map is $\log 4$.
\item[(d)] More precisely, $f_t$ admits an ergodic invariant measure $\mu$ of (maximal) entropy $\log 4$ with $\mathop{\rm supp}\mu\subset{\mathcal U}$
\item[(e)] $\mathop{\rm supp}\mu$ is not contained in any algebraic curve.
\end{itemize}
\end{theorem}
\begin{theorem}\label{THM_TWO_MEASURES}
For all positive $t$ close enough to $0$, $f_t$ admits exactly two ergodic measures $\mu$ (the measure in Theorem \ref{THM_REAL_MEASURE}) and $\nu$ of maximal entropy $\log 4$. Moreover,
\begin{itemize}
\item[(f)] $f_t$ is hyperbolic of repelling type on ${\mathcal A} := \mathop{\rm supp}\mu$.
\item[(g)] $f_t$ is hyperbolic of saddle type on $\Omega:= \mathop{\rm supp}\nu$.
\item[(h)] Periodic points are dense in ${\mathcal A}$ and $\Omega$
\item[(i)] Neither measure is supported on an algebraic curve.
\item[(j)] ${\mathcal A}$ is real whereas $\Omega$ is not.
\item[(k)] $f_t$ has three periodic points ${e}_1,{e}_2,$ and ${e}_3$ not in ${\mathcal A}\cup\Omega$, all fixed and attracting.
\item[(l)] For any $z\in{\mathbb P}^2$, exactly one of the following occurs.
\begin{itemize}
\item $f_t^n(z)\in {\mathcal A}$ for all $n\geq 0$.
\item $f_t^n(z) \in {\mathcal I}$ for some $n\geq 0$.
\item $f_t^n(z) \to \Omega$ as $n\to\infty$.
\item $f_t^n(z)$ tends to one of the attracting point ${e}_j$ as $n\to\infty$.
\end{itemize}
\end{itemize}
\end{theorem}
\begin{remark} If $t\in (0,1)$ is not close to $0$, then we do not know whether $f_t$ admits measures (real or complex) of maximal entropy different from $\mu$.
\end{remark}
\subsection{Dynamical properties of $f_t$ that hold for any $t\in (0,1)$.}
Allowing for now that $t$ is any parameter in $(0,1)$, we will prove the assertions in Theorem \ref{THM_REAL_MEASURE} more or less in order.
\begin{proof}[Proof of (a)] We already have $d(f_t) = d(f) = 4 = \lambda_2(f) = \lambda_2(f_t)$.
Recall that ${\mathcal I} = {\mathcal I}(f) \subset \partial{{\mathcal U}}$. Hence by \eqref{EQN_BACKWARD}, the entire backward orbit of ${\mathcal I}$ is contained in ${\mathcal U}$. On the other hand, $f_t({\mathcal E}) = T_t({\mathcal I}) \subset f_t(\partial{{\mathcal U}})$ does not meet ${\mathcal U}$. Hence $f_t$ is algebraically stable, and it follows that $\lambda_1(f_t) = d(f_t) = 4$.
\end{proof}
\begin{proof}[Proof of (b)]
Let $p=a_j$ be any point in ${\mathcal I}$. It suffices to show that both criteria of
Theorem \ref{THM_CONDITIONS_FOR_NO_FOLIATION} are satisfied. Since
$p\in\partial{{\mathcal U}}$ and $f_t({\mathcal E})$ lies outside $\overline{{\mathcal U}}$, it follows from
\eqref{EQN_BACKWARD} that the entire backward orbit of $p$ is contained in
${\mathcal U}\setminus{\mathcal E}$. In particular, $f_t$ is a finite map at each point of the
entire backward orbit of $p$. Meanwhile, since the branch locus of $f_t$ is
also disjoint from ${\mathcal U}$, we infer that $f$ is regular at every point in the
backward orbit of $p$. In particular, the backward orbit has infinitely many
distinct points. So the second condition of Theorem
\ref{THM_CONDITIONS_FOR_NO_FOLIATION} holds.
For the first condition, note that $f_t(p) = T_t(Q)$. In particular $f_t(p)\cap\overline{{\mathbb R}^2} = \partial{\mathcal U}_t$ is disjoint from $\overline{{\mathcal U}}$. So from \eqref{EQN_BACKWARD} again, we infer that $f_t^n(p)\cap {\mathcal I} = \emptyset$ for all $n > 0$. In particular $f_t^n(p)$ is irreducible for all $n > 0$.
It remains to show that $\mathop{\rm deg} f_t^n(p) \to \infty$. Notice that the diagonal line $L =\{[x:x:z]\}$ is invariant
under $f_t$. Moreover, $L$ is not contained in $f_t^n(p)$ for $n~>~0$, since the latter is irreducible and symmetric in the homogeneous coordinates $x_1,x_2,x_3$. Consider the
intersection between $f_t^n(p)$ for $n\geq 2$ and $L$.
Since $f_t^{n-1}(p)$ meets the exceptional line $A_3$, and $f_t(A_3) \in
\partial{\mathcal U}_t\cap L$, it follows that $f_t^n(p)$ contains orbit segment
$f_t(A_3),\dots,f_t^{n-1}(A_3)\in L$. It follows from \eqref{EQN_BACKWARD}
that the points in this segment are distinct. Hence $\mathop{\rm deg} f_t^n(p) \geq \#
f_t^n(p)\cap L \geq n-1$ for all $n\in{\mathbb N}$.
\end{proof}
\begin{proof}[Proof of (c)]
Since $T_t({\mathcal U}) \supset\overline{{\mathcal U}}$ and $I(f_t) \subset\partial{\mathcal U}$, it follows
from Proposition \ref{prop:Q} and Equation (\ref{EQN_BACKWARD})
that
$$
{\mathcal K} := \bigcap_{n \geq 0} f_t^{-n}(\overline{{\mathcal U}})
$$
is a totally invariant subset of ${\mathcal U}$. Proposition \ref{prop:partition} gives
that $f_t^{-1}(\overline{{\mathcal U}})$ is a disjoint union of four compact sets
${\mathcal U}_j\cap f_t^{-1}(\overline{{\mathcal U}}) \subsetneq {\mathcal U}_j$, $j=0,1,2,3$, and that $f_t$ maps each of
these sets homeomorphically onto $\overline{{\mathcal U}}$. Hence we can assign to each
$p\in {\mathcal K}$ its \emph{itinerary} $\iota(p) = (i_n)_{n\geq 0}$ where $i_n\in
\{0,1,2,3\}$ is chosen to satisfy $f_t^n(p) \in {\mathcal U}_{i_n}$. Standard arguments
then tell us that the map $\iota:{\mathcal K} \to \Sigma_4$ from points to itineraries is
a continuous semiconjugacy onto the full 4-shift $\sigma:\Sigma_4\to \Sigma_4$
(given by $(i_n)\mapsto (i_{n+1})$). It follows that $\log 4 \geq h_{top}(f_t)
\geq h_{top}(\sigma) = \log 4$.
\end{proof}
\begin{proof}[Proof of (d)]
The $4$-shift admits a unique measure of maximal entropy $\check \mu$, and the support of this measure is all of $\Sigma_4$. Starting with any (not necessarily invariant) measure $\mu_0$ on ${\mathcal K}$ such that $\iota_*\mu_0 = \check\mu$, any weak limit $\mu$ of the sequence
$$
\frac{1}{N+1}\sum_{n=0}^N f_{t*}^n\mu_0
$$
will be an invariant measure satisfying $\iota_*\mu = \check\mu$. Taking an extreme point of
all of the possible limits, the measure can be assumed to be ergodic. The inequalities
$
h_{\check\mu}(\sigma) \leq h_\mu(f_t)\leq h_{top}(f_t)
$
imply that that $h_\mu(f_t) = \log 4$.
\end{proof}
\begin{proof}[Proof of (e)]
Here we reuse some of the arguments for Theorem \ref{THM_MU_NOT_1D}. Suppose $\mathop{\rm supp}\mu$ is contained in an algebraic curve $C\subset{\mathbb P}^2$. We may assume that each component of $C$ intersects $\mathop{\rm supp} \mu$ in a set of positive measure, containing infinitely many points. Ergodicity of $\mu$ implies that the components of $C$ are permuted in a single cycle. The arguments for Lemma \ref{LEM_SMOOTH_CURVE} imply (again) that $f_t^{-1}(C) = C$ and that no component of $C$ is critical for $f_t$. Thus
\begin{align*}
f_t^* C = C + \sum n_i A_i
\end{align*}
with at least one coefficient, say e.g. $n_3$, positive. Hence $f_t(A_3)\in C$, and by invariance of $C$ the entire forward orbit of $A_3$ is contained in $C$. It is also, however, an infinite set contained in the forward invariant diagonal line $L = \{[x:x:z]\}$ that we considered in part (c). Since the components of $C$ are cyclically permuted by $f_t$, each of them intersects $L$ in infinitely many distinct points. We conclude that $C = L$. A simple calculation shows that $L$ is not backward invariant, and this contradiction concludes the proof.
\end{proof}
\subsection{Dynamical properties of $f_t$ that hold for $t$ near $0$}
\noindent
It turns out that the dynamics of the limiting one dimensional map $f_0:L_0\to L_0$ can be fairly easily understood. In particular, the points $[1:-1:0], [-1:0:1], [0:1:-1]$ where $L_0$ meets the $f$-invariant branch curves $X_1,X_2,X_3$ are necessarily fixed and attracting for $f_0$. Let $z$ denote the (unique) affine coordinate on $L_0$ that identifies these three points with $0,1,\infty\in{\mathbb P}^1$. One finds by direct computation that $f_0|_{L_0}$ becomes the rational function
$$
\label{EQN_1D_MAPPING_R}
r(z) = -{\frac {z \left( 5\,{z}^{3}-12\,{z}^{2}+6\,z-4 \right) }{4\,{z}^{3}-6
\,{z}^{2}+12\,z-5}}.
$$
Figure \ref{FIG_JULIA} shows a computer generated image of the basins of attraction the three fixed points of $r(z)$.
\begin{figure}
\includegraphics[scale=0.26]{figure3.png}
\caption{Julia set for $f_{0|L_0} : L_0 \rightarrow L_0$, with the basins of attraction of $0$ in
green, $1$ in red, and $\infty$ in blue. The whole figure corresponds
(approximately) to $-3.5~\leq~{\rm re}(z)~\leq~4.5$ and $-1.5 \leq {\rm im}(z)
\leq 1.5$.\label{FIG_JULIA}}
\end{figure}
\begin{lemma}\label{LEM_J1_HYPERBOLIC} The Julia set ${\mathcal{J}}$ for $f_0|_{L_0}$ is hyperbolic and connected. Moreover $L_0\setminus {\mathcal{J}}$ consists of the basins of the three attracting fixed points $[1:-1:0],[-1:0:1],[0:1:-1]\in L_0$. Consequently ${\mathcal{J}}$ is not real, i.e. not contained in $L_0\cap{\mathbb R}{\mathbb P}^2$.
\end{lemma}
\begin{proof}
In addition to the three attracting fixed points: $0, 1$ and $\infty$, the above map $r(z)$ has six
critical points, which come in three conjugate pairs.
By the Fatou-Julia Lemma, each of the attracting fixed points must have at least one
critical point in its immediate basin of attraction. However, because of the real symmetry, each attracting fixed point actually has a conjugate pair of critical points in its immediate basin,
implying that ${\mathcal{J}}$ is hyperbolic.
To see that ${\mathcal{J}}$ is connected, we will check that each Fatou component of $r$
is homeomorphic to a disc. By the threefold symmetry permuting $0$, $1$, and
$\infty$, it suffices to do this for each component in the basin of attraction
of $0$. The immediate basin of $0$ is mapped to itself by $r$ under a ramified
cover with two simple critical points. The Riemann-Hurwitz Theorem can be used to
rule out all possible degrees of this cover other than degree $3$, in which
case it implies that the immediate basin has Euler characteristic $1$. Since
$r$ has no critical points outside the immediate basins of $0,1,$ and $\infty$,
each of the other components of the basin of attraction of $0$ is mapped
conformally onto the immediate basin by an iterate of $r$. Thus, these
components are discs as well.
\end{proof}
All conclusions of Theorem \ref{THM_TWO_MEASURES} now proceed directly from Lemma \ref{LEM_J1_HYPERBOLIC}, Theorem \ref{THM_REAL_MEASURE}, and the earlier Theorems \ref{thm:muexists} and \ref{thm:nuexists}.
\subsection*{Acknowledgments}
We thank Micha{\l} Misiurewicz for helpful discussions. We also thank the referee for his or her very careful reading and for many good suggestions for improving this paper.
The first two authors were supported by NSF grant DMS-1066978 and the third author by NSF grant DMS-1348589.
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1,116,691,497,724 | arxiv | \section{Introduction}
Consider an irreducible polynomial $f(X,Y)\in {\mbox{\dbl C}} [X,Y]$.
Assume there exists a pair of operators $(P,Q)\in A_1$, where $A_1= {\mbox{\dbl C}} [x][\partial ]$ is the first Weyl algebra, such that $f(P,Q)=0$ and $\mathop{\rm ord}\nolimits (P)>0$ or $\mathop{\rm ord}\nolimits (Q)>0$,
where $\mathop{\rm ord}\nolimits (P)=k$ if $P=\sum_{i=0}^k a_i \partial^i$ with $a_i\neq 0$
(we'll call such a pair {\it a non-trivial solution} of the equation $f=0$).
The group of automorphisms of the first Weyl algebra $A_1$ acts on the set of solutions of the equation $f(X,Y)=0$, i.e. if $P,Q\in A_1$ satisfy the equation and $\varphi\in Aut(A_1)$, then $\varphi (P),\varphi (Q)$ also satisfy the equation. A natural and important problem is to describe the orbit space of the group action of $Aut(A_1)$ in the set of solutions.
Y. Berest, cf. \cite{MZh}, proposed the following interesting conjecture:
{\it If the Riemann surface corresponding to the equation $f=0$ has genus $g=1$ then the orbit space is infinite, and if $g>1$ then there are only finite number of orbits.}
This conjecture was first studied in the work \cite{MZh}, where its connection with the well known open conjecture of Dixmier was announced. Recall that the Dixmier conjecture (formulated in his seminal work \cite{Dixmier}) claims that any non-zero endomorphism of $A_{1}$ is actually an automorphism. Although the results of the work \cite{MZh} showed that the Berest conjecture is not true as it was stated, it is still interesting, as it is not clear whether there exist examples of Riemann surfaces (or algebraic curves defined by one equation, see the discussion below) with finite number of orbits.
The aim of this paper is to clarify the connection of the last question with the Dixmier conjecture. First, we can formulate a more general question.
Let $K$ be a field of characteristic zero. From now on we will denote by $A_1$ the first Weyl algebra over $K$: $A_1= K [x][\partial ]$\footnote{Although elements in $A_{1}$ can be written in many kinds of forms, we'll write them in the form $P=\sum a_{ij}x^{i}\partial^{j}$.
In Dixmier's paper this algebra was defined as $A_{1}=K[p,q]$ with the relation $[p,q]=1$, and he put $p$ on the left of $q$. When comparing, we should take $\partial$ instead of $p$, and $x$ instead of $q$. Though Dixmier used another form of elements (he put $p$ on the left of $q$), the difference is not essential for applications of his results in our paper.}. Consider any non-zero polynomial
$$
f(X,Y)=\sum_{i,j=0}^n \alpha_{ij}X^iY^j = 0, \quad \alpha_{ij}\in K.
$$
The first result of our paper (theorem \ref{T:7}) says that any solution $(P,Q)\in A_1$, or even any solution in a more general ring of differential operators, is a pair of commuting operators: $[P,Q]=0$. This is a generalization of an old result by Burchnall and Chaundy \cite{BC}, which is obtained with the help of even more "ancient" technique by Schur \cite{Schur}. Thus, the question of Berest appears to be closely connected with the well known theory of commuting ordinary differential operators (for a review of the modern state of this theory see e.g. the book \cite{Zheglov_book} and references therein).
Consider now the commutative ring $B=K[P,Q]\subset A_1$. It is well known that $B$ is an integral domain, i.e. it defines an affine plain algebraic curve. This curve admits a natural compactification. Namely,
recall that in $A_1$ there is a natural order filtration
$$
K[x]=A_1^0\subset A_1^1\subset A_1^2 \subset \ldots , \quad A_1^n=\{P\in A_1\, \mathop{\rm ord}\nolimits (P)\le n\}.
$$
This filtration induces the filtration on $B$: $B^n=B\cap A_1^n$. The compactification (the projective spectral curve) is defined as $C:=\mathop {\rm Proj} \tilde{B}$, where $\tilde{B}=\oplus_{i=0}^{\infty} B^n$ is the Rees ring. Denote by $p_a(C)$ the arithmetic genus $C$. The Berest conjecture now can be improved as a question what is the orbit space of solutions of a given irreducible equation $f(X,Y)=0$ depending on the genus $p_a(C)$.
Let's note that the question whether there exists a non-trivial solution of a given equation is rather non-trivial. First non-trivial examples appeared in the same paper of Dixmier \cite{Dixmier}:
$$
L_{4}=\left(\partial^2-x^3-\alpha\right)^2-2x,\quad
L_{6}=\left(\partial^2-x^3-\alpha\right)^3-
\frac{3}{2}\left(x\left(\partial^2-x^3-\alpha\right)+
\left(\partial^2-x^3-\alpha\right)x\right),
$$
$$f(L_4, L_6):\quad L_6^2=L_4^3-\alpha \quad p_a(C)=1.$$
Further examples were invented in connection with one question of I.M. Gelfand, who asked whether there are commuting operators from the first Weyl algebra such that their spectral curve has any given genus and their rank is a given one, where the rank of any given commutative ring $B\subset A_1$ is defined as the number $\mathop {\rm rk} B= GCD\{\mathop{\rm ord}\nolimits (Q), Q\in B\}$.
An answer to this question was given only recently, after works of A. Mironov \cite{M1}, \cite{M2}, who constructed a series of examples of rank two and three operators corresponding to a hyperelliptic curve of any given genus, and works of O. Mokhov \cite{Mokh1}, \cite{Mokh2}, who found a way to produce commuting operators of any rank $>1$ with the help of Mironov's examples. At last, in a paper \cite{ML} L. Makar-Limanov proved that any ring $B$ of rank one can be embedded into the ring $K[z]$, i.e. the spectral curve of any such ring is rational (and if one considers a non-trivial ring $B$, this curve is singular). Interested reader will also benefit by looking at recent works \cite{Vardan34}, \cite{Previato2019} related to this question.
All these results show that there are many non-trivial solutions in $A_1$, i.e. the question of Berest is meaningful. It was studied since the work \cite{MZh} in several papers: in \cite{MZh} and \cite{SMZh} it was shown that the number of orbits is infinite for spectral curves of genus one (more precisely, even the number of orbits of pairs $(L_4, L_6)$ with $\mathop{\rm ord}\nolimits (L_4)=4$, $\mathop{\rm ord}\nolimits (L_6)=6$ is infinite) and there are examples of hyperelliptic curves of any genus with infinite number of orbits. In the paper \cite{MirDav} the same result was established for almost all hyperelliptic curves of genus 2 (the field $K={\mbox{\dbl C}}$ in all these works). An interesting result for $K={\mbox{\dbl Q}}$ was obtained in a recent paper \cite{Gundareva} by A. Gundareva: for an elliptic curve over ${\mbox{\dbl Q}}$ the orbits of pairs $(L_4, L_6)$, where
$L_{4}=(\partial^2+V(x))^2+W(x),$ and $W(x)$ is a polynomial of order $2$, are parametrized by the rational points of this curve.
The main result of our paper is related to the Berest conjecture as follows: if there exists a non-zero polynomial such that the number of orbits of its non-trivial solutions in $A_1$ is finite and non-zero, then the Dixmier conjecture is true, see theorem \ref{T:main}.
In view of this statement it is reasonable to continue the study of the Berest conjecture. Since commuting differential operators admit the well known effective classification in terms of algebro-geometric spectral data, it would be interesting to understand the action of the automorphism group $\mathop {\rm Aut} (A_1)$ on these data. Let's mention that even in the simplest most studied case of genus one spectral curves (see the works \cite{KN}, \cite{G}, \cite{Mokh}, \cite{Grun}, \cite{PZ}, \cite{PW}, \cite{BZ}) this question has not yet been considered, although the methods of the last cited papers allow us to hope that it is possible.
The paper is organized as follows. In section 2 we introduce notation and recall necessary facts from the Dixmier paper \cite{Dixmier}, as the proof of the main result makes extensive use of the technique developed in \cite{Dixmier}. In section 3 we prove that any non-trivial solution is a commuting pair of operators. In section 4 we prove our main result.
{\bf Acknowledgements.} The work was supported by the School of Mathematical Sciences, Peking University and Sino-Russian Mathematics Center as well as by the Moscow Center of Fundamental and applied mathematics at Lomonosov Moscow State University.
\section{Preliminaries}
\subsection{Some Definitions and notations}
Suppose $K$ is a field of characteristic zero, with algebraic closure $\bar{K}$. Denote by $\bar{A}_1=A_1\otimes_K\bar{K}$.
Recall that the Newton Polygon of a polynomial $f(x,y)=\sum c_{ij}x^{i}y^{j}\in K[x,y]$ is the convex hull of all the points $(i,j)$ with respect to those $c_{ij}\neq 0$.
Following \cite{Dixmier}, we define $E(f)=\{(i,j)|c_{ij}\neq 0 \}$ as the points set of the Newton Polygon. Suppose $\sigma, \rho$ are two real numbers, we define the weight degree $v_{\sigma,\rho}(f)$ and the top points set $E(f,\sigma,\rho)$ as follows:
\begin{Def}
$v_{\sigma,\rho}(f)=sup\{\sigma i+\rho j|(i,j)\in E(f)\}, \\E(f,\sigma,\rho)=\{(i,j)|v_{\sigma,\rho}(f)=\sigma i +\rho j \}$.
\end{Def}
If $E(f)=E(f,\sigma,\rho)$, then we call such $f$ {\it $(\sigma,\rho)$-homogeneous}.
\begin{Def}
For any polynomial $f$, we call $f_{\sigma , \rho}=f_{\sigma , \rho} (f):= \sum_{(i,j)\in E(f,\sigma,\rho)} c_{ij}x^{i}y^{j}$ the {\it $(\sigma, \rho)$-homogeneous polynomial associated to} $f$.
The line $l$ through the points of the Newton polygon of $f_{\sigma , \rho}$ is called the {\it $(\sigma, \rho)$-top line} of the Newton polygon.
\end{Def}
We define the Newton Polygon of an operator $P\in A_1$, $P= \sum a_{ij}x^{i}\partial^{j}$ as the Newton polygon of the polynomial $f= \sum a_{ij}x^{i}y^{j}$, and we'll use the same notations $v_{\sigma,\rho}(P)$, $f_{\sigma ,\rho}(P)$, $E(P)$ for operators.
Set $\mathop{\rm ord}\nolimits_x(P):=v_{1,0}(P)$. Note that $\mathop{\rm ord}\nolimits (P)=v_{0,1}(P)$. For any differential operator (not necessarily from $A_1$) define $HT(P):= a_n$, if $P=\sum_{i=0}^na_i\partial^i$ with $a_n\neq 0$. We'll say $P$ is monic if $HT(P)=1$.
For any two polynomials $f,g$ we define the standard Poisson bracket $\{f,g\}=\frac{\partial f}{\partial x}\frac{\partial g}{\partial y}-\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}$. We denote $\{ad f\}g :=\{f,g\}$, and for any integer $n,\{ad f\}^{n}g:=\{f,\{f,\cdots g\}\}$. As usual, $(ad (P))Q:=[P,Q]$, where $P,Q\in A_1$.
Suppose $P\in A_{1}$, for any $Q\in A_1$ define he space
$$V_{Q}:=\sum_{n\geq 0}K(ad P)^{n}(Q).$$
Define the spaces $F(P):=\{Q\in A_{1}|\dim V_{Q}<+\infty\}$ and
$$ F(P,\lambda, \bar{A}_1):=\{Q\in \bar{A}_{1}|\exists n\geq 0, (ad P-\lambda)^n(Q)=0\} \quad \lambda\in \bar{K}.
$$
According to \cite[6.1]{Dixmier} we have $F(P)\otimes_K \bar{K}=\oplus_{\lambda\in \bar{K}}F(P,\lambda , \bar{A}_1) $.
We can also define the nilpotent space
$$N(P):=F(P,0,A_{1})=F(P,0,\bar{A}_{1})\cap A_1$$
and for any integer $n$, the spaces $N(P,n)=\{Q\in A_{1}|(adP)^{n}Q\neq 0, (adP)^{n+1}Q=0 \}$.
The semi-simple space is defined as
$$D(P,\lambda , \bar{A}_1):=\{Q\in \bar{A}_{1}|(ad P)Q=\lambda Q\}$$
By \cite[Prop. 6.5]{Dixmier} it is known that for $\lambda \neq 0$ $D(P,\lambda , \bar{A}_1)=F(P,\lambda , \bar{A}_1)$, and $D(P, \bar{A}_1)=\oplus_{\lambda\in \bar{K}} D(P,\lambda ,\bar{A}_1) $.
The centralizer is defined as
$$C(P)=\{Q\in A_{1}|[P,Q]=0 \}.$$
Obviously, $C(P)=D(P,0, \bar{A}_1)\cap A_1$. Set $D(P):= D(P, \bar{A}_1)\cap A_1$. By \cite[Corollary 6.6]{Dixmier} for any $P\in A_1$, we have either $D(P)=F(P)$ or $N(P)=F(P)$.
The tame automorphisms of $A_{1}$ are defined as follows: for any integer $n$, and $\lambda\in K$, define
\begin{equation}
\label{E:tame}
\Phi_{n,\lambda}:
\begin{cases}
\partial \rightarrow & \partial
\\x \rightarrow & x+\lambda \partial^{n}
\end{cases}, \quad
\Phi'_{n,\lambda}:
\begin{cases}
\partial \rightarrow & \partial+\lambda x^{n}
\\x \rightarrow & x
\end{cases},
\quad
\Phi_{a,b,c,d}:
\begin{cases}
\partial \rightarrow & a\partial+bx
\\x \rightarrow & c\partial +dx
\end{cases}
\end{equation}
with $ad-bc=1$.
\subsection{Some important theorems from Dixmier's paper}
For the convenience of the reader we collect here the most important results from the Dixmier paper which will be extensively used below.
\begin{theorem}{(\cite[Lemma 2.4, 2.7]{Dixmier})}
\label{T:Dixmier2.7}
Suppose $P,Q$ are two differential operators from $A_{1} $, $\sigma, \rho$ are real numbers, with $\sigma +\rho >0, v=v_{\sigma.\rho}(P) $ and $ w=w_{\sigma.\rho}(Q) $. If now $f_{1}$, $g_{1}$ are the $(\sigma,\rho)$-homogeneous polynomials associated to $P, Q$, then
\begin{enumerate}
\item
$\exists T,U\in A_{1}$ such that
\begin{enumerate}
\item
$[P,Q]=T+U$
\item
$v_{\sigma.\rho}(U)<v+w-\sigma-\rho$
\item
$E(T)=E(T,\sigma,\rho)$ and $v_{\sigma.\rho}(T)=v+w-\sigma-\rho$ if $T\neq 0$.
\end{enumerate}
\item
The following are equivalent
\begin{enumerate}
\item
T=0
\item
$\{f_{1},g_{1} \}=0$;
If $v,w$ are integers, then these two conditions are equivalent to
\item
$g_{1}^{v}$ is proportional to $f_{1}^{w}$
\end{enumerate}
\item Suppose $T\neq 0$, then the polynomial $(\sigma,\rho)$ associated to $[P,Q]$ is $\{f_{1},g_{1}\}$.
\item We have $f_{\sigma ,\rho} (PQ)=f_1 g_1$, $v_{\sigma ,\rho}(PQ)=v+w$.
\end{enumerate}
\end{theorem}
\begin{rem}
This theorem gives a necessary condition for two ordinary differential operators to commute. If $[P,Q]=T+U$ with $T=0$, then we'll say that $P,Q$ {\it almost commute} for short.
If $f=f_{\sigma,\rho}(P), g=g_{\sigma,\rho}(Q)$ are homogeneous polynomials associated to the pair of integers $(\sigma,\rho)$ such $f,g$ have some powers proportional to each other, we'll say $f,g$ almost commute too.
\end{rem}
With this notation we can give a defintion of "monomial type":
\begin{Def}
Suppose $(\sigma,\rho)$ are real non-negative numbers, $P$ is an ordinary differential operator. $P$ is called of monomial type, if the $(\sigma,\rho)$-homogeneous polynomial of $P$ is a monomial.
Suppose $\varphi$ is an endomorphism, and the $(\sigma,\rho)$-homogeneous polynomials of $\varphi(\partial),\varphi(x)$ are monomials which are almost commute. Then we call $\varphi$ is of monomial type associated with $(\sigma,\rho)$.
\end{Def}
\begin{theorem}{(\cite[7.3]{Dixmier})}
\label{T:Dixmier7.3}
Suppose $\sigma, \rho$ are integers, $P\in A_{1}$, $Q\in F(P)$, $v=v_{\sigma.\rho}(P)$, $w=w_{\sigma.\rho}(Q)$, $f$, $g$ are polynomials $(\sigma,\rho)$-associated to $P, Q$. If $v>\sigma +\rho $ and $f$ is not a monomial, then one of the following conditions hold:
\begin{enumerate}
\item[(a)] $g^{v}$ is proportional to $f^{w}$
\item[(b)] $\sigma>\rho$, $\rho | \sigma$, $f(Y,Z)$ is in the form of $\lambda Y^{\alpha}(Y^{\frac{\sigma}{\rho}}+\mu Z )^{\beta}$
\item[(c)] $\sigma<\rho$, $\sigma | \rho$, $f(Y,Z)$ is in the form of $\lambda Z^{\alpha}(Z^{\frac{\rho}{\sigma}}+\mu Y )^{\beta}$
\item[(d)] $\sigma=\rho$, $f(Y,Z) $ is in the form of $\lambda(Y+\mu Z)^{\alpha} (Y+\mu' Z )^{\beta}$
\\Here $\alpha, \beta$ are integers not less than 0
\end{enumerate}
\end{theorem}
\begin{theorem}{(\cite[7.4]{Dixmier})}
\label{T:Dixmier7.4}
In the notations of the previous theorem if
\begin{enumerate}
\item $v>\sigma +\rho$,
\item $f$ is not a monomial,
\item the conditions (b),(c),(d) do not hold,
\end{enumerate}
then $F(P)=C(P)$.
\end{theorem}
\begin{theorem}{(\cite[8.5]{Dixmier})}
\label{T:Dixmier8.5}
Suppose $P=a_{00}+a_{10}\partial+a_{20}\partial^{2}+\cdots +a_{ro}\partial^{r}+a_{01}x+a_{11}x\partial$, with at least one of $a_{01},a_{11}$ is not 0. Then $\exists \Phi \in Aut(A_{1})$ such that $\Phi(P)=a\partial +bx+cx\partial $.
\end{theorem}
\begin{theorem}{(\cite[8.6]{Dixmier})}
\label{T:Dixmier8.6}
Suppose $P=a\partial^{2}+2bx\partial +cx^{2}+d\partial +ex +g\in A_1$, then
\begin{enumerate}
\item if $b^{2}-ac=0$ then $\exists \Phi \in Aut(A_{1})$ such that $\Phi(P)\in K[\partial]$
\item if $b^{2}-ac\neq 0$, then $\exists \Phi \in Aut(A_{1})$ such that $\Phi(P)=\lambda \partial^{2} +\mu x^{2}+ \nu$ with $\lambda, \mu, \nu \in K$, $\lambda \neq 0$, $\mu \neq 0$.
\end{enumerate}
\end{theorem}
\begin{theorem}{(\cite[8.10]{Dixmier})}
\label{T:Dixmier8.10}
$Aut(A_{1})$ is generated by all $\Phi_{n,\lambda}, \Phi'_{n,\lambda}$ and $\Phi_{a,b,c,d}$.
\end{theorem}
\section{Commutativity of polynomial solutions}
\subsection{Some preperations}
In this section we collect several preliminary statements necessary for the proof of our first theorem. The proof is based on the well known Schur theory from \cite{Schur}.
\begin{lemma}
\label{L:Alexander Lemma 1}
\begin{enumerate}
\item Assume that $Q\in A_{1}$, $Q\notin K$ is an arbitrary differential operator, then there exists the automorphism $\varphi$ of $A_{1}$, such that $HT (\varphi (Q))\in K$.
\item If $P$ is an another differential operator, $\varphi \in End(A_{1})-\{0\}$, then $[P,Q]=0 $ iff $[\varphi(P),\varphi(Q) ]=0 $
\end{enumerate}
\end{lemma}
\begin{proof}
1. Let $k=\mathop{\rm ord}\nolimits_x (Q)$. If $k>0$, then applying the tame automorphism $\Phi_{n,\lambda}$ with $n\gg 0$ and appropriate $\lambda$ we obtain the statement. If $k=0$, we can apply first the automorphism $\Phi_{0,1,-1,0}$ and then $\Phi_{n,\lambda}$ as above.
2. Since $A_1$ is a simple algebra, any endomorphism is either zero or injective.
\end{proof}
\begin{lemma}
\label{L:Alexander Lemma 2}
If $P$ and $Q$ are two differential operators (not necessarily from $A_1$, but with an integral coefficient ring) with $HT(P)\notin K$ and $HT(Q)\in K$, then
\begin{enumerate}
\item $[P,Q]\neq 0$;
\item There is no non-zero polynomial $f(X,Y)=\sum c_{ij}X^{i}Y^{j}$, such that $f(P,Q)=0$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item Suppose $P=a_{p}\partial^{p}+a_{p-1}\partial^{p-1}+\dots$ and $Q=b_q\partial^{q}+b_{q-1}\partial^{q-1}+\dots$ with $a_{p}\notin K$, $b_q\in K$, then $[P,Q]=a_{p}'b_q\partial^{p+q-1}+\dots \neq 0$
\item For any polynomial $f$ denote $N_{f}:=max\{ip+jq \}$. Denote by $R_{f}(l)$ the coefficient of $\partial^{l} $ in $f(P,Q)$. Without loss of generality we can assume $Q$ is monic. Given any $f\in K[X,Y]$ let's write $f(P,Q)$ in the form of
$$k_{1}P^{u_{1}}Q^{v_{1}}+k_{2}P^{u_{2}}Q^{v_{2}}+\dots +k_{m}P^{u_{m}}Q^{v_{m}}+O(N_{f}),$$
where $u_{i}p+v_{i}q=N_{f}$, $u_{1}>u_{2}>\dots >u_{m}$ and monomials in $O(N_{f})$ like $k_{j}P^{u_{j}}Q^{v_{j}}$ satisfy $u_{j}p+v_{j}q<N_{f}$.
If $f(P,Q)=0$, then $R_{f}(N_{f})=k_{1}a_{p}^{u_{1}}+k_{2}a_{p}^{u_{2}}+\dots +k_{m}a_{p}^{u_{m}}=0$. Looking at the degree w.r.t. $x$ in such equation, we'll have $k_{1}=k_{2}=\dots =k_{m}=0$, which means $f=0$.
\end{enumerate}
\end{proof}
\begin{lemma}
\label{L:Alexander Lemma 3}
Suppose $P,Q$ are two differential operators (not necessarily from $A_1$, but with an integral coefficient ring) with $HT(P), HT(Q)\in K$, with $ord(P)=p, ord(Q)=q$. Then
\begin{enumerate}
\item there exists a pseudo-differential operator $S$, such that $S^{-1}QS=c\partial^{q}$, $c\in K$;
\item $[P,Q]=0$ iff $[S^{-1}PS,S^{-1}QS]=0$;
\item $f(P,Q)=0$ iff $f(S^{-1}PS,S^{-1}QS)=0$;
\item $[P,Q]=0$ iff $S^{-1}PS\in K((\partial^{-1}))$;
\item $f(P,Q)=0$ iff \quad $\forall l>0\quad P^{l}f(P,Q)=0$.
\end{enumerate}
\end{lemma}
\begin{proof} Items 1, 2, 3, 4 are proved e.g. in \cite[Chapter 4]{Zheglov_book}. Item 5 follows from the fact that the ring of differential operators is an integral domain.
\end{proof}
\subsection{Commutativity of solutions}
Now we are ready to prove the first theorem.
\begin{theorem}\label{T:7}
Let $f \in K[X,Y]$, $f(X,Y)=\sum_{i+j<N}c_{ij}X^{i}Y^{j}$ be a non-zero polynomial. Assume that $f(P,Q)=\sum c_{ij}P^{i}Q^{j}=0$ where $P,Q$ are two differential operators (not necessarily from $A_1$, but with an integral coefficient ring) with $HT(Q)\in K$\footnote{By applying an appropriate change of coordinates, this condition is reachable for many coefficient rings, cf. \cite[Rem. 1.6]{BZ}}.
Then $[P,Q]=0$.
In particular, in view of lemma \ref{L:Alexander Lemma 1}, this is true for any solution $(P,Q)\in A_1^2$.
\end{theorem}
\begin{proof}
Assume the converse: suppose $[P,Q]\ne 0$. Then by lemma \ref{L:Alexander Lemma 2} we have $HT(P)\in K$.
Using lemma \ref{L:Alexander Lemma 3}, we can replace the pair $(P,Q)$ with the pair $(S^{-1}PS, S^{-1}QS)$ with $S^{-1}QS =c\partial^q$. Assume that $S^{-1}PS=a_p\partial^{p}+a_{p-1}\partial ^{p-1}+\ldots,$ $a_p\in K$. Suppose that $a_{z}$ is the first coefficient which is not a constant, i.e. $\forall j>z, a_{j}\in K$.
Since $f(P,Q)=0$ then for any $l\in {\mbox{\dbl Z}}$ $R_f(l)=0$. It's easy to observe that first non-constant terms in the expansion of the expression $f(P,Q)$ can appear only in $R_{f}(N_{f}-p+z)$.
Suppose $f(P,Q)$ is of the form
\begin{center}
$k_{1}P^{u_{1}}Q^{v_{1}}+k_{2}P^{u_{2}}Q^{v_{2}}+\dots +k_{m}P^{u_{m}}Q^{v_{m}}+O(N_{f})$
\end{center}
with the same notation as in lemma \ref{L:Alexander Lemma 2}, and here we can assume $u_{m}>0$ due to lemma \ref{L:Alexander Lemma 3}, item 5. Then
\begin{equation}
\label{E:(i)}
R_{f}(N_{f}-p+z)=k_{1}a_{z}u_{1}+k_{2}a_{z}u_{2}+\dots +k_{m}a_{z}u_{m} =\theta_{1}
\end{equation}
for some constant $\theta_{1}\in K$.
Next, consider $g(P,Q):=Pf(P,Q)=0$ and do the same for $R_g(N_{g}-p+z)=R_g(N_{f}+z)$. Then
\begin{equation}
\label{E:(ii)}
R_{g}(N_{g}-p+z)=k_{1}a_{z}(u_{1}+1)+k_{2}a_{z}(u_{2}+1)+\dots +k_{m}a_{z}(u_{m}+1) =\theta_{2}
\end{equation}
for some constant $\theta_{2}$.
Since $a_{z}\notin K$, from \eqref{E:(i)}, \eqref{E:(ii)} we obtain that
$$
\begin{cases}
k_{1}u_{1}+k_{2}u_{2}+\dots +k_{m}u_{m}=0
\\k_{1}(u_{1}+1)+k_{2}(u_{2}+1)+\dots +k_{m}(u_{m}+1)=0.
\end{cases}
$$
Hence
\begin{equation}
\label{E:(iii)}
\begin{cases}
k_{1}+k_{2}+\dots +k_{m}=0
\\k_{1}(u_{1}-u_{m})+k_{2}(u_{2}-u_{m})+\dots +k_{m-1}(u_{1}-u_{m})=0
\end{cases}
\end{equation}
Notice that the expression
$$k_{1}P^{u_{1}-u_{m}}Q^{v_{1}}+k_{2}P^{u_{2}-u_{m}}Q^{v_{2}}+\dots +k_{m-1}P^{u_{m-1}-u_{m}}Q^{v_{m-1}}+k_{m}Q^{v_{m}}$$
has coefficient at $\partial^{N_{f}-(u_{m}+1)p+z}$ equal to
$$[k_{1}(u_{1}-u_{m})+k_{2}(u_{2}-u_{m})+\dots +k_{m-1}(u_{1}-u_{m})]a_{z}-\theta_{3}$$
for some constant $\theta_{3}$. Consider now the equality
\begin{multline*}
k_{1}P^{u_{1}}Q^{v_{1}}+k_{2}P^{u_{2}}Q^{v_{2}}+\dots +k_{m}P^{u_{m}}Q^{v_{m}}=\\
P^{u_{m}}(k_{1}P^{u_{1}-u_{m}}Q^{v_{1}}+k_{2}P^{u_{2}-u_{m}}Q^{v_{2}}+\dots +k_{m-1}P^{u_{m-1}-u_{m}}Q^{v_{m-1}}+k_{m}Q^{v_{m}})
\end{multline*}
Comparing the coefficients at $\partial^{N_{f}-p+z}$ on both sides of this equality, we get
\begin{multline*}
k_{1}a_{z}u_{1}+k_{2}a_{z}u_{2}+\dots +k_{m}a_{z}u_{m}=\\
[k_{1}(u_{1}-u_{m})+k_{2}(u_{2}-u_{m})+\dots +k_{m-1}(u_{1}-u_{m})]a_{z}-\theta_{3} +u_{m}a_{z}+\theta_{4}\end{multline*}
for some constant $\theta_{4}$. Using \eqref{E:(i)} and \eqref{E:(iii)}, we get $\theta_{1}=-\theta_{3}+u_{m}a_{z}+\theta_{4}$, which contradicts to the assumption that $a_{z}\notin K $.
\end{proof}
\section{The main result}
In this section we'll prove our main result:
\begin{theorem}
\label{T:main}
Assume there exists a non-zero polynomial $f(X,Y)\in K[X,Y]$, which has a non-trivial solution $(P,Q)\in A_{1}^{2}$, and the number of orbits under the group action of $Aut(A_1)$ on solutions of $f$ in $A_{1}^{2}$ is finite. Then the Dixmier conjecture holds, i.e $\forall \varphi\in End(A_{1})-\{0\}$, $\varphi$ is an automorphism.
\end{theorem}
The proof will be divided into several steps.
\subsection{$\varphi$ is of rectangular type}
Since $\varphi [\partial, x]=[\varphi(\partial),\varphi(x)]=1$, it's easy to see that $F(\varphi(\partial))\neq C(\varphi(\partial))$.
\begin{Def}
Suppose $\varphi$ is an endomorphism of $A_{1}$. We'll say $\varphi(\partial)$ has mixture term if $\exists (i,j)\in E(\varphi(\partial))$ with $i\neq 0$, $j\neq 0$. We'll say $\varphi(\partial)$ doesn't have mixture terms if $\forall (i,j)\in E(\varphi(\partial))$ it holds either $i=0$ or $j=0$.
\end{Def}
\begin{lemma}\label{L:4}
Suppose $\varphi$ is an endomorphism of $A_{1}$. If $\varphi(\partial)$ doesn't have mixture terms or has only one mixture term with $(i,j)=(1,1)$, then $\varphi$ is an automorphism.
\end{lemma}
\begin{proof}
It's a very easy case when $\varphi(\partial)$ doesn't have mixture terms, i.e $\varphi(\partial)=f(\partial)+g(x)$, with some polynomials $f,g$. Suppose $f(\partial)=a_{n}\partial^{n}+\cdots +a_{0}$, and $g(x)=b_{m}x^{m}+\cdots +b_{0}$. Take $(\sigma,\rho)=(m,n)$, then $v_{\sigma,\rho}(\varphi(\partial))=mn$, the polynomial $(\sigma,\rho)$ associated to $\varphi(\partial)$ is $f(x,\partial)=a_{n}\partial^{n}+b_{m}x^{m} $ and obviously it can't be written as the power of another polynomial.
If one of $m,n$ is not more than 1, we can easily verify that $\varphi$ is an automorphism. If $n>1, m>1$, then $(m-1)(n-1)>0 \Rightarrow mn-m-n\geq 0 \Rightarrow v_{\sigma,\rho}(\varphi(\partial))-\sigma-\rho \geq 0 \Rightarrow v_{\sigma,\rho}(\varphi(\partial))+v_{\sigma,\rho}(\varphi(x))-\sigma-\rho > 0$. By theorem \ref{T:Dixmier2.7} we get that $f_{\sigma ,\rho} (\varphi (x))$ must be some power of $f$, then by taking composition with the tame automorphism
\begin{equation}
\label{E:iv}
\varphi':\begin{cases}
\partial \rightarrow & \varphi(\partial)
\\x \rightarrow &\varphi(x)-\beta\varphi (\partial)^{\frac{w}{v}}
\end{cases}
\end{equation}
for appropriate $w,v, \beta$, we 'll still have $[\varphi'(\partial),\varphi'(x)]=1$, but $v_{\sigma,\rho}(\varphi'(x))<v_{\sigma,\rho}(\varphi(x))$. We can continue until $v_{\sigma,\rho}(\varphi(x))<v_{\sigma,\rho}(\varphi(\partial))$, this means $\varphi(x)=0$, a contradiction.
Also, if $\varphi(\partial)$ has only one mixture term $x\partial$, $\varphi(\partial)=cx\partial +f(\partial)+g(x)$, with $f(\partial)=a_{n}\partial^{n}+\cdots +a_{0}$, and $g(x)=b_{m}x^{m}+\cdots +b_{0}$, take $(\sigma,\rho)=(m,n)$, then $v_{\sigma,\rho}(\varphi(\partial))=mn$ or $m+n$.
\begin{enumerate}
\item[I] If $mn>m+n$, then $f(x,\partial)=a_{n}\partial^{n}+b_{m}x^{m}$, and we can use the same trick as before like \eqref{E:iv}.
\item[II] If $mn<m+n$, then $(m-1)(n-1)<1$ and one of $m,n$ is not more than 1. By theorem \ref{T:Dixmier8.5}, $\exists \Phi \in A_{1}$ such that $\varphi'=\Phi \circ \varphi (\partial)=ax+b\partial +cx\partial$. We can use the trick in \eqref{E:iv} again, and assume that $\varphi'(x)=dx+e\partial +fx\partial$. Direct calculation shows that this is impossible when $c\neq 0$.
\item[III] If $mn=m+n$, then $(m-1)(n-1)=1 \Rightarrow m=1,n=1$, now the polynomial $(\sigma,\rho)=(1,1)$-associated to $\varphi(\partial)$ is $f(x,\partial)=a\partial^{2}+bx\partial +cx^{2}$. By theorem \ref{T:Dixmier8.6} $\exists \Phi$ such that $\Phi \circ \varphi(\partial)\in K[\partial]$ or $=\lambda \partial^{2}+ \mu x^{2} +\nu $, the first case is clear and the latter case has been discussed before.
\end{enumerate}
\end{proof}
From now on, we can always assume that $\varphi(\partial)$ has mixture terms $x^{i}\partial^{j}$ such that one of $i,j$ is more than 1. Hence $\forall \sigma,\rho >0$, $v_{\sigma,\rho}(\varphi(\partial))-\sigma-\rho>0$.
\begin{lemma} \label{L:5}
Assume $\varphi(\partial)=\sum a_{ij}x^{i}\partial^{j}$ and there exists $a_{ij}\neq 0$ such that one of $i,j$ is more than 1. Set
$$
l=sup\{i|\quad \exists j, a_{ij}\neq 0\}, \quad k=sup\{j|\quad \exists i, a_{ij}\neq 0 \}.
$$
Then at least one of $a_{l0}, a_{0k}, a_{lk}$ is not equal to 0.
\end{lemma}
\begin{proof} If not, then we can assume $I=sup\{i|a_{ik}\neq 0 \}$ and $J=sup\{j|a_{jl}\neq 0 \}$, then the Newton Polygon of $\varphi(\partial)$ is as follows.
\begin{figure}[!h]
\label{P:Figure 1}
\centering
\begin{tikzpicture}
\draw[line width=1pt, ->](0,0) -- (6,0) node[right] {$x$};
\draw[line width=1pt, ->](0,0) -- (0,5) node[above] {$\partial$};
\draw[dashed] (-0.5,3)node[below]{$k$} --(5.5,3);
\draw[dashed] (5,-0.5)node[right]{$l$}--(5,3.5);
\filldraw[blue](3,3) node[above]{$(I,k)$} circle(2pt);
\filldraw[blue](5,1) node[right]{$(l,J)$} circle(2pt);
\filldraw[blue](4.3,1.7)circle(2pt);
\filldraw[black](2,1)circle(2pt);
\filldraw[black](1,2)circle(2pt);
\filldraw[blue](4.5,2.5)circle(2pt);
\draw[dashed,purple] (5.5,-0.5) --(4,4);
\filldraw[blue](3.7,2.3)circle(2pt);
\draw[line width=1pt,red](2.5,3.5)--(5.5,0.5);
\end{tikzpicture}
\caption{Newton Polygon of $\varphi(\partial)$ }
\end{figure}
Choose the line $l:\sigma x+\rho y=\theta$ for some $\theta\in K$ such that $(I,k),(l,J)$ lie on this line (for example, $\sigma=k-J, \rho=l-I, \theta=lk-IJ$). Then we have two possible conditions:
\begin{enumerate}
\item This line is a $(\sigma,\rho)$-top line of the Newton Polygon of $\varphi(\partial)$:
From our assumption, there are no points on the axis in this top line. Hence the polynomial $(\sigma,\rho)$-associated to $\varphi(\partial)$ is $f(x,\partial)=a_{Ik}x^{I}\partial ^{k}+a_{lJ}x^{l}\partial ^{J}+\cdots$.
Now we already have $v_{\sigma,\rho}(\varphi(\partial))>\sigma+\rho$, and $f$ is not a monomial. Then use theorem \ref{T:Dixmier7.4}, we are in the case (b) or (c) or (d) in theorem \ref{T:Dixmier7.3}. It's impossible, since there are no point on the axis.
\item This line is not the $(\sigma,\rho)$-top line of the Newton Polygon, i.e there is another point in the Newton Polygon of $\varphi(\partial)$ over this line. By our assumptions, this point lies strictly inside the triangle on the picture.
Then choose the top line going through the point $(l,J)$ and, say, $(I',k')$ ($l:\sigma x+\rho y=\theta'$) with coefficients $\sigma=k'-J, \rho=l-I'$. Again the line has no points on the axis, so it's impossible, too by the same arguments.
\end{enumerate}
\end{proof}
The case when $a_{lk}\neq 0$ is definitely what we want, this motivates the following definition:
\begin{Def}
The Newton Polygon $N$ is called of rectangular type, if $N$ is contained in the rectangle constructed by $\{(0,0),(0,k),(l,0),(l,k)\}$, and $(l,k)$ is the vertex of $N$ with $l,k \ge 1$.
If the Newton Polygons of $\varphi(\partial),\varphi(x)$ are of rectangular type, then we call such $\varphi$ of rectangular type (a specially condition of monomial type).
\end{Def}
\begin{lemma} \label{L:6}
Suppose the Newton polygon of $\varphi(\partial)$ is of the rectangular type, then the Newton polygon of $\varphi(x)$ is also of the rectangular type.
\begin{proof}
For any real pair $(\sigma,\rho)$, suppose $f_{\sigma,\rho}$ is the polynomial $(\sigma,\rho)$ associated to $\varphi(\partial)$ and $g_{\sigma,\rho}$ is the polynomial $(\sigma,\rho)$ associated to $\varphi(x)$.
According to theorem \ref{T:Dixmier2.7}, $[\varphi(\partial),\varphi(x)]=1=T+U$, with
$$
v_{\sigma,\rho}(T)=v_{\sigma,\rho}(\varphi(\partial))+v_{\sigma,\rho}(\varphi(x))-\sigma-\rho .
$$
Since $\varphi(\partial)$ is of rectangular type, i.e $\varphi(\partial)=x^{l}\partial^{k}+\cdots$ with $l,k\ge 1$, for any positive real pair $(\sigma,\rho)$ we have $f_{\sigma,\rho}(\varphi(\partial))=x^{l}\partial^{k}$, $v_{\sigma,\rho}(\varphi(\partial))\ge \sigma + \rho $, on the other hand $v_{\sigma,\rho}(\varphi(x))>0$. Thus we get that $T=0$, this means $\varphi(\partial)$ is almost commute with $\varphi(x)$, and $\varphi(x)$ is of monomial type associated to $(\sigma,\rho)$.
Suppose $g_{1,1}=x^{l'}\partial^{k'}$. Since $\varphi(x)$ is of monomial type associated to arbitrary real positive pair $(\sigma,\rho)$, it's easy to see now the Newton Polygon of $\varphi(x)$ is contained in the rectangle constructed by $\{(0,0),(0,k'),(l',0),(l',k')\}$, with $(l',k')\in E(\varphi(x))$, $l',k'\ge 1$. Hence $\varphi(x)$ is also of rectangular type.
\end{proof}
\end{lemma}
Looking back on lemma \ref{L:5}, if $a_{lk}=0$, then one of $a_{l0},a_{0k}$ is not equal to 0, in such situation $\varphi$ is not of rectangular type. However, we have
\begin{lemma} \label{L:7}
Suppose $\varphi$ is an endomorphism of $A_{1}$. Then either $\varphi \in Aut(A_{1}) $ or $\exists \Phi \in Aut(A_{1})$ such that $\Phi \circ \varphi $ is of rectangular type.
\begin{proof}
We keep the same notations as in lemma \ref{L:5}.
\begin{enumerate}
\item[(1)]$a_{l0}=0$, the Newton Polygon of $\varphi(\partial)$ looks like in Figure \ref{F:Figure 2}.
\begin{figure}[!h]
\label{F:Figure 2}
\centering
\begin{tikzpicture}
\draw[line width=1pt, ->](0,0) -- (6,0) node[right] {$x$};
\draw[line width=1pt, ->](0,0) -- (0,5) node[above] {$\partial$};
\filldraw[black](3,1) node[right]{$(l,J)$} circle(2pt);
\filldraw[blue](0,4) node[left]{$(0,k)$} circle(2pt);
\filldraw[blue](2,2) circle(2pt);
\filldraw[blue] (1,3) circle(2pt);
\draw[dashed, blue] (-0.5,4.5) --(4.5,-0.5);
\draw[dashed] (-0.5,1)node[below]{$J$} --(3,1);
\draw[dashed] (3,-0.5)node[right]{$l$}--(3,1);
\filldraw[red](0,2.5) node[left]{$(0,k')$} circle(2pt);
\draw[line width=1pt , red](0,2.5) -- (5,0);
\end{tikzpicture}
\caption{Changing variable of $\varphi(\partial)$ in case (1)}
\end{figure}
Choose proper $(\sigma,\rho)$, such that the top line go across $(0,k)$ and $(l,J)$, for example $\sigma=k-J,\rho=l,\theta=kl$, the top line equation: $\sigma x+\rho y =\theta$.
Hence the polynomial $(\sigma,\rho)$-associated to $\varphi(\partial)$ is
$$
f_{\sigma,\rho}(\varphi(\partial))=a_{0k}\partial^{k}+a_{lJ}x^{l}\partial^{J}+\cdots
$$
Now $f_{\sigma,\rho}(\varphi(\partial)),v_{\sigma,\rho}(\varphi(\partial))$ satisfy the condition of theorem \ref{T:Dixmier7.3}. Since $F(\varphi(\partial))\neq C(\varphi(\partial))$, we are in the case (b),(c) or (d) of theorem \ref{T:Dixmier7.3}. Since only $a_{0k}\neq 0$, we see that (c) can not happen, only (b) or (d).
Now change the variable: choose $\Phi\in Aut(A_{1})$
$$
\Phi :\begin{cases}
\partial \rightarrow &\partial
\\x \rightarrow & x+\mu \partial^{\frac{\sigma}{\rho}}
\end{cases}
$$
for an appropriate $\mu$.
Then $\Phi \circ \varphi(\partial)$ has only one point left on the original top line (only $(l,J)$ survives). And now the new Newton Polygon of $\Phi\circ \varphi(\partial)$ becomes smaller than before and the top line becomes flatter. We can continue on this procedure until the top line is strictly flat, after that the Newton Polygon of the new $\tilde{\varphi}=\Phi \circ \varphi$ is of the rectangular type.
\item[(2)]$a_{0k}=0$, with the same discussion of (1), we can be in the case (c) or (d) of theorem \ref{T:Dixmier7.3}. The Newton Polygon of $\varphi(\partial)$ looks like Figure \ref{P:Figure 3}.
\begin{figure}[!h]
\label{P:Figure 3}
\centering
\begin{tikzpicture}
\draw[line width=1pt, ->](0,0) -- (6,0) node[right] {$x$};
\draw[line width=1pt, ->](0,0) -- (0,5) node[above] {$\partial$};
\filldraw[black](1,3) node[right]{$(I,k)$} circle(2pt);
\filldraw[blue](4,0) node[below]{$(l,0)$} circle(2pt);
\filldraw[blue](2,2) circle(2pt);
\filldraw[blue] (3,1) circle(2pt);
\draw[dashed, blue] (-0.5,4.5) --(4.5,-0.5);
\draw[dashed] (-0.5,3)node[below]{$k$} --(1,3);
\draw[dashed] (1,-0.5)node[right]{$I$}--(1,3);
\filldraw[red](2.5,0) node[below]{$(l',0)$} circle(2pt);
\draw[line width=1pt , red](2.5,0) -- (0.5,4);
\end{tikzpicture}
\caption{Changing variable of $\varphi(\partial)$ in case (2)}
\end{figure}
Now take
$$
\Phi:\begin{cases}
\partial \rightarrow &\partial +\mu x^{\frac{\rho}{\sigma}}
\\x \rightarrow & x.
\end{cases}
$$
Then $\Phi \circ \varphi(\partial)$ has only one point left on the original top line (only $(I,k)$ survives). We can continue on this procedure until the top line is strictly vertical, after that the Newton Polygon of the new $\tilde{\varphi}=\Phi \circ \varphi$ is of rectangular type.
\item[(3)] $a_{l0}\neq 0, a_{0k}\neq 0$, there exists point over the line across these two points. Choose the proper $(\sigma,\rho)$ such that the top line go across $a_{l0}$ or $a_{0k}$ (but can't be two of them), the discussion will be the same as in (1) and (2).
\begin{figure}[!h]
\centering
\begin{tikzpicture}
\draw[line width=1pt, ->](0,0) -- (6,0) node[right] {$x$};
\draw[line width=1pt, ->](0,0) -- (0,6) node[above] {$\partial$};
\filldraw[black](3,3) node[right]{$(I,J)$} circle(2pt);
\filldraw[blue](5,0) node[below]{$(l,0)$} circle(2pt);
\filldraw[purple](0,5) node[left]{$(0,k)$} circle(2pt);
\filldraw[blue] (4,1.5) circle(2pt);
\filldraw[purple] (1.5,4) circle(2pt);
\draw[dashed, blue] (5,0) --(3,3);
\draw[dashed, purple] (0,5) --(3,3);
\draw[line width=1pt , red](4,0) -- (3,3);
\draw[line width=1pt , orange](0,4) -- (3,3);
\filldraw[red](4,0) node[below]{$(l',0)$} circle(2pt);
\filldraw[orange](0,4) node[left]{$(0,k')$} circle(2pt);
\end{tikzpicture}
\caption{Changing variable of $\varphi(\partial)$ in case (3)}
\end{figure}
\item[(4)]$a_{l0}\neq 0, a_{0k}\neq 0$, there doesn't exist point over the line across these two points, i.e
there can be three conditions, (b) or (c) or (d). Choose the proper $(\sigma,\rho)$ such that the top line just go across $(0,k)$ and $(l,0)$, for example $\sigma=k,\rho=l,\theta=kl$, the top line equation: $\sigma x+\rho\partial =\theta$.
\\In case (b), just do as in (1), the survived vertex is on the axis when changing variable, after that the Newton Polygon becomes smaller and flatter than before.
\\In case (c), just do as in (2), the survived vertex is on the axis when changing variable, after that the Newton Polygon becomes smaller and more vertical than before.
\\In case (d) $f_{\sigma,\rho}(\varphi (\partial))$ is of the form of $\lambda (\partial+\mu x)^{\alpha}(\partial+\mu'x)^{\beta}$.
\item[(4i)]$\mu=\mu'$ or one of $\alpha, \beta$ is equal to zero, then take $\Phi \in \mathop {\rm Aut} (A_{1}):$
$$
\Phi :\begin{cases}
\partial \rightarrow &\partial- \mu x
\\x \rightarrow & x.
\end{cases}
$$
Then the new Newton Polygon becomes more vertical than before, since the only survived vertex is $(0,\alpha+\beta)$. And this procedure can be continued.
\item[(4ii)] $\mu \neq \mu'$ and $\alpha, \beta \neq 0$, then take $\Phi_1, \Phi_2 \in \mathop {\rm Aut} (A_{1})$:
$$
\Phi_1 :\begin{cases}
\partial \rightarrow &\partial-\mu x
\\x \rightarrow & x,
\end{cases}
\quad
\Phi_2 :\begin{cases}
\partial \rightarrow &\partial
\\x \rightarrow & x-(\mu'-\mu)^{-1}\partial.
\end{cases}
$$
After applying these automorphisms the only survived vertex on the top line is not on the axis, so the situation of $\Phi_2\circ\Phi_1 \circ \varphi(\partial)$ comes into (3).
\begin{figure}[!h]
\centering
\begin{tikzpicture}
\draw[line width=1pt, ->](0,0) -- (6,0) node[right] {$x$};
\draw[line width=1pt, ->](0,0) -- (0,6) node[above] {$\partial$};
\filldraw[black](2.5,2.5) node[right]{$(I,J)$} circle(2pt);
\filldraw[blue](5,0) node[below]{$(l,0)$} circle(2pt);
\filldraw[blue](0,5) node[left]{$(0,k)$} circle(2pt);
\filldraw[blue] (3.75,1.25) circle(2pt);
\filldraw[blue] (1.25,3.75) circle(2pt);
\draw[dashed, blue] (5,0) --(0,5);
\draw[line width=1pt , red](4,0) -- (2.5,2.5);
\draw[line width=1pt , orange](0,4) -- (2.5,2.5);
\filldraw[red](4,0) node[below]{$(l',0)$} circle(2pt);
\filldraw[orange](0,4) node[left]{$(0,k')$} circle(2pt);
\end{tikzpicture}
\caption{Changing variable of $\varphi(\partial)$ in case (4ii)}
\end{figure}
In view of lemma \ref{L:6} we can continue on this trick until $\varphi$ becomes either a tame automorphism or an endomorphism of rectangular type.
\end{enumerate}
\end{proof}
\end{lemma}
From now on, we can always assume that $\varphi$ is of the rectangular type. It's easy to see that now for any positive pair $(\sigma,\rho)$, $f_{\sigma,\rho}(\varphi(\partial))$ is a monomial.
\begin{rem}
\label{R:4.1}
\begin{enumerate}
\item [(a)] Suppose $\varphi$ is of rectangular type. We can see for any real positive pair $(\sigma,\rho)$, $\varphi$ is always of monomial type and moreover, the monomials $f_{\sigma,\rho}(\varphi(\partial , x))$ are the same for different pairs, this is why we note it as the special case for monomial type at the beginning.
\item [(b)] Using the procedure from the proof of lemma \ref{L:5}, we can easily draw the picture of the Newton Polygon of any automorphism. Besides, the following lemma holds:
\end{enumerate}
\end{rem}
\begin{lemma} \label{L:8}
Any automorphism of $A_{1}$ can't be of rectangular type. More precisely: if $\Phi \in Aut(A_{1})$, with $n=sup\{i|\quad \exists j, a_{ij}\neq 0\}$, $m=sup\{j|\quad \exists i, a_{ij}\neq 0 \}$ (in the notation of lemma \ref{L:5}) then
\begin{enumerate}
\item $(0,n),(m,0)\in E(\Phi(\partial))$;
\item $\forall i,j>0,(i,n),(m,j)\notin E(\Phi(\partial)) $;
\item one of the rate $\frac{m}{n}$ or $\frac{n}{m}$ must be an integer.
\end{enumerate}
\begin{proof}
Just use induction, take the decomposition of $\Phi$ into the tame automorphism by theorem \ref{T:Dixmier8.10}. It's easy to see that any tame automorphism satisfies such condition and the composition of them also satisfy the condition.
\end{proof}
\end{lemma}
\begin{figure}[!h]
\centering
\begin{tikzpicture}
\draw[line width=1pt, ->](0,0) -- (6,0) node[right] {$x$};
\draw[line width=1pt, ->](0,0) -- (0,6) node[above] {$\partial$};
\draw[line width=1pt , red](1,0) -- (0,1);
\draw[line width=1pt , orange](1,0) -- (0,2);
\draw[line width=1pt , blue](4,0) -- (0,2);
\draw[line width=1pt , purple](4,0) -- (0,4);
\filldraw[black](4,0) node[below]{$m$} circle(2pt);
\filldraw[black](0,4) node[left]{$n$} circle(2pt);
\end{tikzpicture}
\caption{Decompostion of automorphism $\Phi(\partial)$}
\end{figure}
\subsection{$\varphi$ has a fixed point $P$}
\begin{Def}
Suppose $\varphi$ is an endomorphism of rectangular type. Let $(\sigma,\rho)$ be any two positive integer numbers.
Assume $f_{\sigma,\rho}(\varphi(\partial))=c_1 x^{l}\partial^{k}$, $f_{\sigma,\rho}(\varphi(x))=c_2 x^{l'}\partial^{k'}$, where $l,k,l',k'$ are non-zero integers, $c_1, c_2\in K$. By theorem \ref{T:Dixmier2.7}, $f_{\sigma,\rho}(\varphi(\partial))$ is proportional with $f_{\sigma,\rho}(\varphi(x))$. Define $\epsilon(\varphi)\in \mathbf{Q}$ as
$$
(l',k')=\epsilon(\varphi)(l,k)
$$
We'll call this $\epsilon(\varphi)$ as the rate of $\varphi(\partial)$ with $\varphi(x)$. Note that $\epsilon(\varphi)$ does not depend on $(\sigma,\rho)$, cf. remark \ref{R:4.1}.
\end{Def}
Before we continue our discussion, we propose a Lemma first:
\begin{lemma}\label{L:9}
Suppose $\varphi$ is an endomorphism of rectangular type.
\begin{enumerate}
\item For any $q\in{\mbox{\dbl N}}$ the endomorphism $\varphi^q$ is of rectangular type with $\epsilon(\varphi^{q})=\epsilon(\varphi)$.
\item Choose $(\sigma,\rho):=(p\epsilon,p) \in {\mbox{\dbl N}}^2$ (for some $p\in{\mbox{\dbl N}}$), where $\epsilon \in \mathbf{Q}$, $\epsilon=\epsilon(\varphi)$.
Assume $P\in A_{1}$, $P\notin K$ is such that $f_{\sigma,\rho}(P)$ is a monomial. Then $f_{\sigma,\rho}(\varphi(P))$ is a monomial with
\begin{equation} \label{E:vi}
v_{\sigma,\rho}(\varphi(P))=(k+\epsilon l)v_{\sigma,\rho}(P)
\end{equation}
Here $f_{\sigma,\rho}(\varphi(\partial))=c_1x^{l}\partial^{k}$, $f_{\sigma,\rho}(\varphi(x))=c_2x^{l\epsilon}\partial^{k\epsilon}$.
\end{enumerate}
\end{lemma}
\begin{proof} 1. Note that for any monomial $x^{u}\partial^{v}$ we have $\varphi(x^{u}\partial^{v})=\varphi(x)^{u}\varphi(\partial)^{v}$ and by theorem \ref{T:Dixmier2.7}, item 4 we have therefore
$$
f_{\sigma ,\rho}(\varphi(x^{u}\partial^{v}))=cx^{l(u\epsilon+v)}\partial^{k(u\epsilon +v)}
$$
for any positive $\sigma ,\rho$ (cf. remark \ref{R:4.1}) and some $0\neq c\in K$. In particular, we can choose $\sigma ,\rho$ in such a way that the corresponding $(\sigma ,\rho)$-line $\sigma x+\rho y=\theta$ is almost flat or almost vertical. More precisely, we need to choose such $\sigma ,\rho$ that the lines through the vertex $(l,k)$ and through the vertex $(l\epsilon ,k\epsilon)$ intersect the vertical axe at the point $(0,a)$ and correspondingly at the point $(0,a\epsilon)$ with $a<k+1$ and $a\epsilon <k\epsilon +1 $ (and choose such $\sigma ,\rho$ that this lines intersect the flat axe at the point $(b,0)$ and correspondingly at the point $(b\epsilon,0)$ with $b<l+1$ and $b\epsilon <l\epsilon +1$). With this choice it just suffices to prove that the monomials $f_{\sigma ,\rho}(\varphi(x^{u}\partial^{v}))$, where $(u,v)$ are the vertices inside the rectangle $\{(0,0),(0,k),(l,0),(l,k)\}$, belong to the rectangle $\{(0,0),(0,k(l\epsilon +k)),(l(l\epsilon +k),0),(l(l\epsilon +k),k(l\epsilon +k))\}$ (and correspondingly, the monomials $f_{\sigma ,\rho}(\varphi(x^{u}\partial^{v}))$, where $(u,v)$ are the vertices inside the rectangle $\{(0,0),(0,k\epsilon),(l\epsilon,0),(l\epsilon,k\epsilon)\}$, belong to the rectangle $\{(0,0),(0,\epsilon k(l\epsilon +k)),(\epsilon l(l\epsilon +k),0),(\epsilon l(l\epsilon +k), \epsilon k(l\epsilon +k))\}$). Indeed, if it is true, then all other monomials of $\varphi(x^{u}\partial^{v})$ will belong to the same rectangles $\{(0,0),(0,k(l\epsilon +k)),(l(l\epsilon +k),0),(l(l\epsilon +k),k(l\epsilon +k))\}$, $\{(0,0),(0,\epsilon k(l\epsilon +k)),(\epsilon l(l\epsilon +k),0),(\epsilon l(l\epsilon +k), \epsilon k(l\epsilon +k))\}$, i.e. $\varphi$ will be of rectangular type.
But this is easy: $(u,v)$ is a vertex inside the rectangle $\{(0,0),(0,k),(l,0),(l,k)\}$ iff $u\le l$ and $v\le k$. Clearly, if these inequalities hold, then $l(u\epsilon+v)\le l(l\epsilon +k)$ and $k(u\epsilon +v)\le k(l\epsilon +k)$, i.e. $f_{\sigma ,\rho}(\varphi(x^{u}\partial^{v}))$ belongs to the rectangle $\{(0,0),(0,k(l\epsilon +k)),(l(l\epsilon +k),0),(l(l\epsilon +k),k(l\epsilon +k))\}$.
Thus, $\varphi^2$ is of rectangular type, and the same arguments show that $\varphi^q$ is of rectangular type for any $q\in {\mbox{\dbl N}}$. A simple calculation now shows that: for any $q\in {\mbox{\dbl N}}$,
\begin{equation}
\label{E:cases}
\begin{cases}
f_{\sigma ,\rho}(\varphi^{q}(\partial))=\tilde{c}_1x^{l(k+\epsilon l)^{q-1}}\partial^{k(k+\epsilon l)^{q-1}}
\\f_{\sigma ,\rho}(\varphi^{q}(x))=\tilde{c}_2x^{l\epsilon(k+\epsilon l)^{q-1}}\partial^{k\epsilon(k+\epsilon l)^{q-1}}
\end{cases}
\end{equation}
so that $\epsilon(\varphi^{q})=\epsilon(\varphi)$, too.
2. Suppose $P=cx^{u}\partial^{v} +f(x,\partial)$, with $f_{\sigma,\rho}(P)=cx^{u}\partial^{v}$. Then $\varphi(P)=\varphi(cx^{u}\partial^{v})+\varphi(f)$. Note that for any monomial $x^{u}\partial^{v}$ we have and
$$v_{\sigma,\rho}(\varphi(x^{u'}\partial^{v'}))= (\rho k+\sigma l)(u'\epsilon+v')=(k+\epsilon l)v_{\sigma,\rho}(x^{u'}\partial^{v'}).
$$
Since the weights of all monomials in $f$ are less than $(u\sigma+v\rho )$, we get from this
$$v_{\sigma,\rho}(\varphi(P))= (k+\epsilon l)v_{\sigma,\rho}(P), \quad
v_{\sigma,\rho}(\varphi(f))<(k+\epsilon l)v_{\sigma,\rho}(P),
$$
and $f_{\sigma,\rho}(\varphi(P))=f_{\sigma,\rho}(cx^{u}\partial^{v})$ is a monomial.
\end{proof}
Let $\varphi$ be an endomorphism of rectangular type, and assume it is not an automorphism. If $\epsilon (\varphi )$ is an integer, we can choose $\Phi_{\epsilon ,c} \in \mathop {\rm Aut}(A_{1})$ and consider a new endomorphism $\varphi':=\varphi \circ \Phi_{\epsilon ,c}$. Since $\varphi' (\partial )=\varphi (\partial)$, we have by lemma \ref{L:6} that $\varphi'$ is again of rectangular type. Besides, $\epsilon (\varphi')< \epsilon (\varphi )$ for an appropriate $c\in K$. We can continue this process until $\epsilon (\varphi )$ becomes not an integer. Doing the same with other types of tame automorphisms, without loss of generality we can assume that neither of $\epsilon (\varphi ),\frac{1}{\epsilon (\varphi )}$ is integer (since $\varphi$ is assumed to be not an automorphism).
Suppose the conditions of theorem \ref{T:main} holds, choose a non-trivial equation $F$ such that the number of solution orbits is finite. Then choose a non-trivial solution $(P,Q)$ in a solution orbit, choose $(\sigma,\rho)=(p\epsilon (\varphi ),p)$ such that $\sigma, \rho \in {\mbox{\dbl N}}$.
W.l.o.g we can suppose $f_{\sigma,\rho}(P),f_{\sigma,\rho}(Q)$ are monomials. If not, according to theorem \ref{T:7} we know that $[P,Q]=0$ (so they are almost commute, too). Take automorphism $\Phi_{N,1}$ (or $\Phi_{N,1}'$) with $N\gg 0$ an integer big enough. Then $(\Phi(P),\Phi(Q))$ is also a non-trivial solution to the equation $F=0$, and $f_{\sigma,\rho}(\Phi(P)),f_{\sigma,\rho}(\Phi(Q))$ are monomials.
Since the number of orbits of solutions is finite, so the set $\{\mathcal{O}(\varphi^{n}(P,Q)), n\in{\mbox{\dbl N}} \}$ is a finite set, too (here $\mathcal{O}()$ denotes the orbit of the group $\mathop {\rm Aut} (A_1)$). Hence there exist $p,q\in {\mbox{\dbl N}}$ such that $\varphi^{p+q}(P,Q)$ is on the same orbit with $\varphi^{q}(P,Q)$. According to the definition of the orbit, we know $\exists \psi \in \mathop {\rm Aut}( A_{1})$, such that
\begin{equation} \label{E:v}
\varphi^{p}(\varphi^{q}(P,Q))=\psi\circ \varphi ^{q}(P,Q)
\end{equation}
By lemma \ref{L:9} we have:
$$f_{\sigma,\rho}(\varphi^{q}(\partial))=\tilde{c}_1x^{l(k+\epsilon l)^{q-1}}\partial^{k(k+\epsilon l)^{q-1}},\quad f_{\sigma,\rho}(\varphi^{q}(x))=\tilde{c}_2x^{l\epsilon(k+\epsilon l)^{q-1}}\partial^{k\epsilon(k+\epsilon l)^{q-1}}$$
are non-trivial monomials for any $q\in{\mbox{\dbl N}}$ and $f_{\sigma,\rho}(\varphi^{q}(P)),f_{\sigma,\rho}(\varphi^{q}(Q))$ are monomials for arbitrary $q\in{\mbox{\dbl N}}$, too.
The Newton Polygon of $\psi^{-1}(\partial )$ must be in the shape of Figure 6 we draw in the section 4.1. Observe that the $(\sigma,\rho)$-top line can only go across one point of $\psi^{-1}(\partial)$, say $(0,n)$ or $(m,0)$, since one of $\frac{n}{m},\frac{m}{n}$ must be an integer and neither of $\epsilon,\frac{1}{\epsilon}$ is an integer. So the $(\sigma,\rho)$-homogeneous polynomial associated to $\psi^{-1}(\partial)$ is a monomial, hence the $(\sigma,\rho)$-homogeneous polynomial associated to $\psi^{-1}(x)$ is a monomial too by theorem \ref{T:Dixmier2.7}. Using lemma \ref{L:9} again we know that $f_{\sigma, \rho}(\varphi^{p}\circ\psi^{-1}(x,\partial))$ are monomials for any $p\in{\mbox{\dbl N}}$.
Now let $p,q$ be as in \eqref{E:v}. Put $\tilde{\varphi}=\varphi^{p}\circ \psi^{-1},(\tilde{P},\tilde{Q})=\varphi^{p}(\varphi^{q}(P,Q))=\psi\circ \varphi ^{q}(P,Q)$. Simple calculation shows that $\tilde{\varphi}(\tilde{P},\tilde{Q})=(\tilde{P},\tilde{Q})$. Now take the new $\tilde{\varphi}$ instead of $\varphi$, and $\tilde{P}$ instead of $P$, we know that the new $\tilde{\varphi}$ has a fixed point $\tilde{P}$.
Note that the new $\tilde{\varphi}$ may not be of rectangular type any more. But it is of monomial type associated to the old $(\sigma,\rho)=(p\epsilon,p)$ and simple observation shows that the vertex in the top line is not on the axis. The new rate of $\tilde{\varphi}$ may change, too (actually, it must be changed), i.e $\epsilon(\tilde{\varphi})\neq \epsilon (\varphi )$.
\begin{rem}
\begin{enumerate}
\item [(a)]The reason why $\tilde{\epsilon}=\epsilon(\tilde{\varphi})$ must be changed: if not, $f_{\sigma,\rho}(P)$ can't be a monomial any more. This is because $\varphi(P)=P$, so $v_{\sigma,\rho}(\varphi(P))=v_{\sigma,\rho}(P)$. If $f_{\sigma,\rho}(P)$ is a monomial, then $v_{\sigma,\rho}(\varphi(P))=(k+\epsilon l)v_{\sigma,\rho}(P)$ due to lemma \ref{L:9} -- a contradiction, since $P$ is not a constant and therefore $v_{\sigma,\rho}(P)>0$.
\item[(b)] Since here we assume $\tilde{\varphi}=\varphi^{p}\circ \psi^{-1}$, if we get $\tilde{\varphi}$ is an automorphism, we still need to show that $\varphi$ is an automorphism. But this is clear: any endomorphism is an automorphism iff it is surjective. If $\tilde{\varphi}$ is an automorphism, it is surjective, so is $\varphi$.
\end{enumerate}
\end{rem}
For convenience we still denote $(\tilde{\varphi},\tilde{P},\tilde{l},\tilde{k},\tilde{\epsilon})$ as $(\varphi,P,l,k,\epsilon)$. In particular, $\varphi(\partial)=c_1x^{l}\partial^{k}+\cdots, \varphi(x)=c_2x^{l\epsilon}\partial^{k\epsilon} +\cdots$ again, but now $\epsilon$ or $1/\epsilon$ can be integer. Denote by $(\sigma_{0},\rho_{0})$ the old $(\sigma,\rho)$. We have seen above that $\varphi$ is of monomial type associated to $(\sigma_{0},\rho_{0})$, and the Newton polygon of $\varphi (\partial )$ looks like on Figure 7 (the Newton polygon of $\varphi (x )$ looks similarly in view of theorem \ref{T:Dixmier2.7}).
\begin{figure}[!h]
\label{pic}
\centering
\begin{tikzpicture}
\draw[line width=1pt, ->](0,0) -- (6,0) node[right] {$x$};
\draw[line width=1pt, ->](0,0) -- (0,6) node[above] {$\partial$};
\filldraw[black](2.5,2.5) node[right]{$(l,k)$} circle(2pt);
\filldraw[blue](5,0) circle(2pt);
\filldraw[blue](0,5) circle(2pt);
\draw[dashed, blue] (5,0) --(0,5);
\draw[line width=2pt , black](4,0) -- (2.5,2.5);
\draw[line width=2pt , black](0,4) -- (2.5,2.5);
\draw[line width=2pt , black](4,0) -- (0,0);
\draw[line width=2pt , black](0,4) -- (0,0);
\filldraw[black](4,0) circle(2pt);
\filldraw[black](0,4) circle(2pt);
\filldraw[black](0,0) circle(2pt);
\end{tikzpicture}
\caption{$\varphi$ is of monomial type associated to $(\sigma_{0},\rho_{0})$}
\end{figure}
Next step in our proof is to change $\varphi$ into rectangular type again. The difference between the situation now and lemma \ref{L:7} is:
\begin{enumerate}
\item[(a)] We are only in the case like (3) in the proof of lemma \ref{L:7}.
\item[(b)] We have to keep $P$ invariant when changing $\varphi$.
\end{enumerate}
\begin{lemma}\label{L:10}
Suppose $\varphi$ is an endomorphism of $A_{1}$, $\varphi$ is of monomial type associated to $(\sigma_{0},\rho_{0})$ and the Newton polygon of $\varphi (\partial )$ looks like on Figure 7, $P\in A_1$, $P\notin K$, with $\varphi(P)=P$. Then there exists an automorphism $\Phi$ such that $\tilde{\varphi}:=\Phi\circ \varphi\circ \Phi^{-1}, \tilde{P}=\Phi(P)$ satisfy
\begin{enumerate}
\item [(1)]$\tilde{\varphi}$ is of rectangular type,
\item[(2)] $\tilde{\varphi}(\tilde{P})=\tilde{P}$, $\tilde{P}\notin K$.
\end{enumerate}
\begin{proof} Obviously, if the condition holds, then (2) is satisfied, so we pay our attentions to (1). Like in the proof of lemma \ref{L:7}, we are in case (3), i.e. we only need to deal with the situations (1) and (2) there.
\begin{enumerate}
\item [(I)]For the upper case, we are in the conditions (c) or (d) of theorem \ref{T:Dixmier7.3}. Take $\Phi\in \mathop {\rm Aut}(A_{1})$
$$\Phi :\begin{cases}
\partial \rightarrow &\partial
\\x \rightarrow & x+\mu \partial^{\frac{\sigma}{\rho}}
\end{cases}$$
for an appropriate $\mu\in K$.
Then put $\tilde{\varphi}=\Phi\circ \varphi\circ \Phi^{-1}, \tilde{P}=\Phi(P)$. Note that $\Phi^{-1}(\partial)=\partial$, so that the Newton Polygon of $\tilde{\varphi}(\partial)$ is totally equal to $\Phi \circ \varphi(\partial)$. We know the Newton Polygon of $\tilde{\varphi}(\partial)$ becomes flatter than before, and we can continue on this procedure until it is strictly flat. We may denote the result as $\varphi,P$ again.
\begin{figure}[!h]
\centering
\begin{tikzpicture}
\draw[line width=1pt, ->](0,0) -- (6,0) node[right] {$x$};
\draw[line width=1pt, ->](0,0) -- (0,6) node[above] {$\partial$};
\filldraw[black](3,3) node[right]{$(l,k)$} circle(2pt);
\filldraw[black](5,0) circle(2pt);
\filldraw[black](0,3) circle(2pt);
\filldraw[black](0,0) circle(2pt);
\draw[line width=2pt , black](3,3) -- (5,0);
\draw[line width=2pt , black](0,0) -- (5,0);
\draw[line width=2pt , black](0,3) -- (3,3);
\draw[line width=2pt , black](0,0) -- (0,3);
\end{tikzpicture}
\caption{The Newton Polygon of $\tilde{\varphi}(\partial)$ after changing variable}
\end{figure}
\item[(II)]For the right case, we are in the conditions (b) or (d).
\\Now take
$$\Phi:\begin{cases}
\partial \rightarrow &\partial +\mu x^{\frac{\rho}{\sigma}}
\\x \rightarrow & x
\end{cases}$$
for an appropriate $\mu\in K$
and $\tilde{\varphi}=\Phi\circ \varphi\circ \Phi^{-1}, \tilde{P}=\Phi(P)$.
\\We should pay our attention to the Newton Polygons of $\varphi(x)$ and $\tilde{\varphi}(x)$ this time. Since the Newton polygon of $\varphi(\partial)$ has $(0,1)$-top line going across $(l,k)$ and $\varphi(\partial)$ is almost commute with $\varphi(x)$, we know by theorem \ref{T:Dixmier2.7} that $\varphi(x)$ has Newton Polygon with the $(0,1)$-top line going across $(l',k')$ too. More specifically, it is $k'=sup\{j|(i,j)\in E(\varphi(x))\}$ and $l'=sup\{i|(i,k')\in E(\varphi(x))\}$.
\begin{figure}[!h]
\centering
\begin{tikzpicture}
\draw[line width=1pt, ->](0,0) -- (6,0) node[right] {$x$};
\draw[line width=1pt, ->](0,0) -- (0,6) node[above] {$\partial$};
\filldraw[black](3,3) node[right]{$(l',k')$} circle(2pt);
\filldraw[black](5,0) circle(2pt);
\filldraw[black](0,3) circle(2pt);
\filldraw[black](0,0) circle(2pt);
\draw[line width=2pt , black](3,3) -- (5,0);
\draw[line width=2pt , black](0,0) -- (5,0);
\draw[line width=2pt , black](0,3) -- (3,3);
\draw[line width=2pt , black](0,0) -- (0,3);
\draw[dashed, red] (3,3) --(4,0);
\filldraw[red](4,0) circle(2pt);
\end{tikzpicture}
\caption{The Newton Polygon of $\varphi(x)$ when changing variable}
\end{figure}
As in (1), we can continue this procedure until the right top line of $\tilde{\varphi}(x)$ becomes strictly vertical, after that we get $\tilde{\varphi}$ of rectangular type.
\end{enumerate}
\end{proof}
\end{lemma}
For convenience, we'll still denote the new $(\tilde{\varphi},\tilde{\epsilon},\tilde{P})$ from this lemma as $(\varphi,\epsilon,P)$.
\begin{rem}
We have to point out here $\tilde{\epsilon}$ may be an integer or the inverse of an integer now. But it's not important any more, the purpose of making $\epsilon$ not an integer or the inverse of an integer was to make sure that $\exists (\sigma_{0},\rho_{0})$ and $\varphi$ of monomial type associated to $ (\sigma_{0},\rho_{0})$, which is definitely the condition in lemma \ref{L:10}.
\end{rem}
\subsection{$v_{\sigma,\rho}(P)=m(\sigma+\rho)$}
\label{S:ooo}
Suppose $\varphi$ is an endomorphism of rectangular type with a fixed point $P\notin K$ (i.e $\varphi(P)=P$), $\epsilon :=\epsilon (\varphi )$, $(\sigma ,\rho)=(p\epsilon,p)\in {\mbox{\dbl N}}^2$ and suppose $f_{\sigma ,\rho}(\varphi(\partial))=c_1x^{l}\partial^{k}$, $f_{\sigma ,\rho}(\varphi(x))=c_2x^{l'}\partial^{k'}$, with $(l',k')=\epsilon(l,k)$.
Since $\mathop{\rm ord}\nolimits_{x}(P)<\infty $, we know $\exists N$, such that $(ad\partial)^{N}P=0$, and since $\varphi$ is an endomorphism, then for any integer $q>0, (ad\varphi^{q}(\partial))^{N}P=0 $. Then due to theorem \ref{T:Dixmier2.7}, we know that $\exists m<N, (ad\varphi^{q}(\partial))^{m}P$ is almost commute with $\varphi^{q}(\partial)$(then almost commute with $\varphi(\partial)$).
Define $m(q)$ as {\it the first positive integer} such that $(ad\varphi^{q}(\partial))^{m(q)}P$ is almost commute with $\varphi(\partial)$ for any $q\in {\mbox{\dbl N}}$ (it is well-defined as we discuss above).
In the same way we define $n(q)$ to be {\it the first positive integer} such that \\
$f_{\sigma ,\rho}((ad\varphi^{q}(\partial))^{n(q)}P)$ is a monomial. Obviously, it's well defined, with $n(q)\leq m(q)$ for any $q\in {\mbox{\dbl N}}$.
Since ${\mbox{\dbl N}}$ is an infinite set and the number of possible pairs $(n(q),m(q))$ is finite, there exist two different positive numbers $p,q$, $p<q$ such that $n:=n(p)=n(q)$, $m:=m(p)=m(q)$.
\begin{lemma}
\label{L:calculations}
Let $\varphi$, $P$, $(\sigma ,\rho)$, $p<q$, $n,m$ be as above. Then $n=m$ and
$$v_{\sigma,\rho}(P)=m(\sigma+\rho).$$
\end{lemma}
\begin{proof}
Assume $\theta_{0}=v_{\sigma,\rho}(P)$ (note that $\theta_{0}>0$ since $P\notin K$), and set
\\$\theta_{1}=v_{\sigma,\rho}(ad\varphi^{q}(\partial)P)$
\\$\cdots$
\\$\theta_{m}=v_{\sigma,\rho}([ad\varphi^{q}(\partial)]^{m}P)$
\\From our assumption on $m$ and theorem \ref{T:Dixmier2.7} it follows that the equations
$$
\theta_{i}=\theta_{i-1}+v_{\sigma,\rho}(\varphi^{q}(\partial))-\sigma-\rho
$$
hold for $1\leq i\leq m$. However, we have (cf. formulae \eqref{E:cases})
\begin{equation}\label{E:vii}
\theta_{m}=mk(k+\epsilon l)^{q}+\theta_{0}-m(\sigma+\rho).
\end{equation}
On another side, set $\eta_{0}=\theta_{0}=v_{\sigma,\rho}(P)$,
\\$\eta_{1}=v_{\sigma,\rho}(ad\varphi^{p}(\partial)P)$
\\$\cdots$
\\$\eta_{m}=v_{\sigma,\rho}([ad\varphi^{p}(\partial)]^{m}P)$
\\ Then we have
\begin{equation}\label{E:viii}
\eta_{m}=mk(k+\epsilon l)^{p}+\eta_{0}-m(\sigma+\rho)
\end{equation}
Since $f_{\sigma,\rho}([ad\varphi^{p}(\partial)]^{m}(P))$ and $f_{\sigma,\rho}([ad\varphi^{q}(\partial)]^{m}(P))$ are both monomials according to our assumptions, then condition in Lemma \ref{L:5} is satisfied. Applying $\varphi$ $(q-p)$ times and using \eqref{E:vi}, we get
\begin{equation}\label{E:ix}
\theta_{m}=(k+\epsilon l)^{q-p}\eta_{m}.
\end{equation}
Comparing with \eqref{E:vii}, \eqref{E:viii}, \eqref{E:ix}, we get
\begin{equation}\label{E:x}
\theta_{0}=m(\sigma+\rho).
\end{equation}
We can rewrite this procedure for $n$ instead of $m$ in the same way, but since the result \eqref{E:x} has no relation with $p,q$, we immediately get $n=m$, which means, the first time $f_{\sigma,\rho}([ad\varphi^{p}(\partial)]^{n}P)$ becomes a monomial, it's almost commute with $f_{\sigma,\rho}(\varphi(\partial))$, i.e. is proportional to it. In other words, $[ad\varphi^{q}(\partial)]^{m-1}(P)$ is not of monomial type associated to $(\sigma,\rho)=(p\epsilon,p)$.
\end{proof}
\subsection{Proof of the main theorem: the last step}
Again, suppose $\varphi$ is an endomorphism of rectangular type with a fixed point $P\notin K$ (i.e $\varphi(P)=P$), and suppose $\varphi(\partial)=c_1x^{l}\partial^{k}+\cdots, \varphi(x)=c_2x^{l'}\partial^{k'} +\cdots$, with $(l',k')=\epsilon(l,k)$, $v_{\sigma,\rho}(P)=m(\sigma+\rho)$ as in the previous section. We are now going to show it's impossible.
First we should notice that $f_{\sigma,\rho}(P)$ can't be a monomial. If not, since $\varphi(P)=P$, we have $v_{\sigma,\rho}(\varphi(P))=v_{\sigma,\rho}(P)$, on the other hand, by lemma \ref{L:6} we know that $v_{\sigma,\rho}(\varphi(P))=(k+\epsilon l)v_{\sigma,\rho}(P)$, it's impossible as $v_{\sigma,\rho}(P)>0$.
Now assume $f_{\sigma,\rho}(P)=c_{1}x^{n_{1}}\partial^{m_{1}}+c_{2}x^{n_{2}}\partial^{m_{2}}+\cdots$, (here $n_{i},m_{i}$ has no relation with $n,m$). According to the definition of weight degree $v_{\sigma,\rho}$ and by lemma \ref{L:calculations}, we know that for any integer $i$ such that $c_{i}\neq 0$
\begin{equation}\label{E:xi}
\sigma n_{i}+\rho m_{i}=m(\sigma+\rho).
\end{equation}
\begin{figure}[!h]
\centering
\begin{tikzpicture}
\draw[line width=1pt, ->](0,0) -- (6,0) node[right] {$x$};
\draw[line width=1pt, ->](0,0) -- (0,6) node[above] {$\partial$};
\filldraw[black](1.5,1) node[right] {$(n_{3},m_{3})$} circle(2pt);
\filldraw[black](1,2) node[right] {$(n_{2},m_{2})$} circle(2pt);
\filldraw[black](0.5,3) node[right] {$(n_{1},m_{1})$} circle(2pt);
\draw[line width=2pt , black](0,4) -- (2,0);
\end{tikzpicture}
\caption{$P$ can't be in monomial type associated to $(p\epsilon,p)$}
\end{figure}
We have calculate before (see \eqref{E:cases}):
$$\begin{cases}
\varphi^{q}(\partial)=\tilde{c}_1x^{l(k+\epsilon l)^{q-1}}\partial^{k(k+\epsilon l)^{q-1}}+\cdots
\\\varphi^{q}(x)=\tilde{c}_2x^{l\epsilon(k+\epsilon l)^{q-1}}\partial^{k\epsilon(k+\epsilon l)^{q-1}}+\cdots
\end{cases}$$
Denote $u=l(k+\epsilon l)^{q-1}, v=k(k+\epsilon l)^{q-1}$ for convenience. According to the definition of $m$, we know $m+1$ is the first number such that $\{ad\varphi^{q}(\partial) \}^{m+1}(P)=0$.
Consider $$\{x^{u}\partial^{v},x^{n_{i}}\partial^{m_{i}} \}=(vn_{i}-um_{i})x^{n_{i}+u-1}\partial ^{m_{i}+v-1} $$
and for any integer $r,\exists \lambda=\lambda(r,i) \in K$, such that
$$ \{adx^{u}\partial^{v} \}^{r}(x^{n_{i}}\partial^{m_{i}})=\lambda x^{n_{i}+r(u-1)}\partial ^{m_{i}+r(v-1)}$$
Since $f_{\sigma,\rho}([ad\varphi^{q}(\partial)]^{m}(P))$ is a monomial, the homogeneous terms in $P$ must be eliminated by taking multiple Poisson bracket until there is only one term left.
Notice that if $i\neq j$, then
$$\lambda(i,r) x^{n_{i}+r(u-1)}\partial ^{m_{i}+r(v-1)}\neq \lambda(j,r) x^{n_{j}+r(u-1)}\partial ^{m_{j}+r(v-1)}$$
This means the terms will be eliminated only if $\lambda(i,r)$ for some $r$.
\begin{figure}[!h]
\centering
\begin{tikzpicture}
\draw[line width=1pt, ->](0,0) -- (6,0) node[right] {$x$};
\draw[line width=1pt, ->](0,0) -- (0,8) node[above] {$\partial$};
\filldraw[black](1.125,0.75) circle(2pt);
\filldraw[black](0.75,1.5) circle(2pt);
\filldraw[black](0.375,2.25) circle(2pt);
\draw[line width=2pt , black](0,6) node[left] {$\{ad\varphi^{q}(\partial) \}^{r+1}(P)$} -- (3,0);
\filldraw[black](1.875,2.25) circle(2pt);
\filldraw[black](1.5,3) circle(2pt);
\filldraw[black](1.125,3.75) circle(2pt);
\draw[line width=2pt , black](0,3) node[left] {$\{ad\varphi^{q}(\partial) \}^{r}(P)$} -- (1.5,0);
\draw[dashed, red,->] (1.125,0.75) --(1.875,2.25);
\draw[dashed, red,->] (0.75,1.5) --(1.5,3);
\draw[dashed, red,->] (0.375,2.25) --(1.125,3.75);
\end{tikzpicture}
\caption{$i,j$ can not meat with each other after applying the Lie-bracket}
\end{figure}
Now suppose $\xi =\xi (i)$ is the last integer that $\lambda=\lambda(\xi,i )$ is not equal to 0. This means the coefficient of the Poisson bracket $\{x^{u}\partial^{v},x^{n_{i}+\xi(u-1)}\partial ^{m_{i}+\xi(v-1)}\}$ is equal to 0. Thus, we have
$$v(n_{i}+\xi(u-1))-u(m_{i}+\xi(v-1))=0. $$
Simplifying the equation, we have $v n_{i}-um_{i}-v\xi+u\xi=0$. Since $u=l(k+\epsilon l)^{q-1}, v=k(k+\epsilon l)^{q-1}$, we can eliminate the common factor $(k+\epsilon l)^{q-1}$, thus we have
$$k n_{i}-l m_{i}-k\xi+l\xi=0. $$
Eliminate $m_{i}$ by \eqref{E:xi}, we have
$$k n_{i}-l(\frac{m(\sigma+\rho)-\sigma n_{i}}{\rho}) -k\xi+l\xi=0. $$
From this we can get a representation of $n_{i}$:
\begin{equation}\label{xii}
n_{i}=\frac{lm(\sigma+\rho)+\xi\rho(k-l)}{k\rho+l\sigma}
\end{equation}
If $\xi(i)=m$, we can solve the equation and get $n_{i}=m_{i}=m$ (this means $x^m\partial^n= cf_{\sigma ,\rho}([ad\varphi^{q}(\partial)]^{m}(P))$ is the last monomial surviving after applying the Poisson bracket).
But we can see if $\xi(j)=m-1$, then $n_{j}$ can not be an integer since $ \rho(k-l)<k\rho+l\sigma$. This means there are no monomials $x^{m_{j}}\partial^{n_{j}}$ eliminated after applying the Poisson bracket $\xi=m-1$ times, and this means $[ad\varphi^{q}(\partial)]^{m-1}(P)$ is of monomial type associated to $(\sigma,\rho)$. This contradicts to the result we get in lemma \ref{L:calculations} in section \ref{S:ooo} -- that $m=n$, thus $f_{\sigma,\rho}(P)$ must be a monomial - a contradiction.
Thus we get that $\forall \varphi\in End(A_{1})-\{0\}$, $\varphi$ must be an automorphism, as there are no endomorphisms of the rectangular type with a fixed point. And this means the Dixmier conjecture holds.
|
1,116,691,497,725 | arxiv | \section{Introduction}
Recent studies in mathematical psychology \cite{martinez2016quantum}, \cite{kvam2021temporal}, \cite{busemeyer2020application}, show that the Lindbladian equations from quantum mechanics facilitate modeling peculiar aspects of human decision making. Such quantum decision models capture {\em order effects}
(humans perceive $P(H|A\cap B) $ and $P(H|B \cap A)$ differently in decision making)
and violation of the {\em sure thing principle} (human perception of probabilities in decision making violates the total probability rule). Motivated by the design of human-machine interaction systems, this paper addresses the following question: {\em Given a sequence of human decisions over time, how can a controller (machine) adapt the Lindbladian dynamics (of the human decision maker) so as to converge to a specific decision?} To investigate this, we develop a novel generalization of recent results involving finite-step stochastic Lyapunov functions. Thus at an abstract level, we study the stochastic stability of a switched controlled Lindbladian dynamic system where the switching occurs due to the interaction of the controller (machine) and decision maker (human) at specific (possibly random) time instants.
\subsection{Decision Making Context}
Figure~\ref{fig:Model} shows our schematic setup. The finite-valued random variable $s\sim \pi_0(\cdot)$
denotes the underlying state of nature, where $\pi_0$ is a known probability mass function.
The input signals $y_k$ and $z_k$ are noisy observations of the state with conditional observation densities $p(y|s)$ and $p(z|s)$, respectively. The human's psychological state $\rho$ is represented as a density operator in Hilbert Space, which evolves via the Lindbladian equation parametrized by the observation $y_k$ and input $u_k$. The density operator $\rho$ encodes a probability distribution over actions $\{a_j\}_{j \in \{1,\dots,m\}}$, and at each time point an action is taken according to this distribution. The machine observes the actions and outputs a feedback control signal $u_k$ to the human.
\subsubsection*{Examples} Several
examples in robotics \cite{askarpour2019formal}, interactive marketing/advertising \cite{belanche2020consumer} and recommender systems \cite{lu2012recommender} exploit models for human decision making.
One example is a machine assisted healthcare system for patients with dementia \cite{HOEY2010503}, in which the patient is assisted by a machine (smart watch) to wash his hands. The machine's sensor detects whether a certain set of sequential actions are followed by the patient, and then sends those results to a controller which gives an audio/video command to the patient.
In this context the underlying states ($s$) are the tap water, soap dispenser and towel dispenser, which are partially observed by both sensor and the patient. The patient has a psychological state ($\rho$) and the resultant hand washing actions ($a_k$) are sensed by the sensor, then the controller gives the control input ($u_k$). In our work, we model the psychological state of the patient as a Lindbladian evolution as shown in Figure~\ref{fig:Model} since this accounts for a wider range of human behavior, such as irrational decisions which could be made by the dementia patient, than classical models.
\begin{figure}[h!]
\centering
\resizebox{0.5\textwidth}{!}{%
\begin{tikzpicture}[scale=1, auto, >=stealth']
\large
\matrix[ampersand replacement=\&, row sep=0mm, column sep=0cm] {
\&\&\&\&\\
\&\&\node[block] (F1) {Psychological State \[\rho_{k+1}=\mathcal{L}_{(y_k,u_k)}\,\rho_k\]};\&\node[connector] (u1) {};
\&
\&\\
\node[state] (X){State\[s\]};\&\&\&\&\\
\&\&\node[block] (F2) {Sensor};\&\&\node[block] (f1) {Controller};
\\
};
\draw [dashed] (F2) -- (f1);
\draw [connector] (F1) -- node {Decision($a_{k}$)} (F2);
\draw [connector] (f1.north) -- ++(0,0cm) |- node [right]
{Control($u_{k,T}$)} ($(F1.east) + (0cm, 0em)$);
\draw [connector] ($(X.north) - (0cm, 0cm)$) -- ++(0,0cm) |- node [left]
{Input($y_k$)} ($(F1.west) - (0cm, 0em)$);
\draw [connector] ($(X.south) - (0cm, 0em)$)-- ++(0,0cm) |- node [left]
{Input($z_k$)} ($(F2.west) - (0cm, 0em)$);
\draw[dashed,blue] (F2.north west) rectangle (f1.south east);
\end{tikzpicture}
}
\caption{Human-Machine Interaction Model}
\label{fig:Model}
\end{figure}
\subsection{Main Results and Organization}
Given the described human-machine decision making system, the question we ask is: Can the human's decision preference be guided by the input control signals such that a desired target action is eventually taken at every time step? Our results reveal that this indeed is the case. We show this by developing a novel Lyapunov stability result for a Lindbladian dynamic system.
The main results and organization of the paper is as follows:\\
(1) Sec. II introduces the open-quantum cognitive model of Martinez et. al. \cite{martinez2016quantum} and represents the discretized process in a form that will be mathematically useful for us.
\\
2) Sec. III presents Theorem (\ref{thm:convg}), which is our main result and shows the stochastic stability of our human-machine decision system. The proof uses the methodology of Amini et. al \cite{AMINI20132683}, along with Lyapunov techniques and Theorem (\ref{thm:extension}) which we provide in Sec. IV.\\
3) Sec. IV provides a generalization of a finite-step Lyapunov stability result given in Qin et. al. \cite{qin2019lyapunov} in Theorem (\ref{thm:rand}), to the case when the finite-step interval $T$ is a random variable. Also Theorem (\ref{thm:extension}) is a modified form of this result which is used to prove Theorem (\ref{thm:convg}) in Sec. III.
\subsection{Literature Review}
Generative models for human decision making are studied extensively in behavioral economics and psychology.
The classical formalisms of human decision making are the Expected Utility models of Von-Neumann and Morgenstern (1953)\cite{morgenstern1953theory} and Savage (1954) \cite{savage1951theory}. Despite the successes of these models, numerous experimental findings, most notably those of Kahneman and Tverksy \cite{kahneman1982judgment}, have demonstrated violations of the proposed decision making axioms. There have since been subsequent efforts to develop axiomatic systems which encompass wider ranges of human behavior, such as the Prospect Theory \cite{kahneman2013prospect}. However, given the complexity of human psychology and behavior it is no surprise that current models still have points of failure. The theory of Quantum Decision Making (\cite{busemeyer2012quantum}, \cite{khrennikov2010ubiquitous}, \cite{yukalov2010mathematical} and references therein) has emerged as a new paradigm which is capable of generalizing current models and accounting for certain violations of axiomatic assumptions. For example, it has been empirically shown that humans routinely violate Savage's 'Sure Thing Principle' \cite{khrennikov2009quantum}, \cite{aerts2011quantum}, which is equivalent to violation of the law of total probability, and that human decision making is affected by the order of presentation of information \cite{trueblood2011quantum} \cite{busemeyer2011quantum} ("order effects"). These violations are natural motivators for treating the decision making agent's mental state as a quantum state in Hilbert Space; The mathematics of quantum probability was developed as an explanation of observed self-interfering and non-commutative behaviors of physical systems, directly analogous to the findings which Quantum Decision Theory (QDT) aims to treat.
{\em Remark}. QDT models in psychology do not claim that the brain is acting as a quantum device in any physical sense. Instead QDT serves as a {\em parsimonious generative blackbox model} for human decision making that is backed up by extensive experimental studies \cite{kvam2021temporal}, \cite{busemeyer2012quantum}.
Within Quantum Decision Theory, several recent advances have utilized quantum dynamical systems to model time-evolving decision preferences. The classical model for this type of time-evolving mental state is a Markovian model, but in \cite{busemeyer2009empirical} an alternative formulation based on Schr\"{o}dinger's Equation is developed. This model is shown to both reconcile observed violations of the law of total probability via quantum interference effects and model choice-induced preference changes via quantum projection. This is further advanced in \cite{asano2012quantum}, and \cite{martinez2016quantum} where the mental state is modeled as an open-quantum system. This open-quantum system representation allows for a generalization of the widely used Markovian model of preference evolution, while maintaining these advantages of the quantum framework. Busemeyer et. al. \cite{kvam2021temporal} provide empirical analysis which supports the use of open-quantum models and conclude "An open system model that incorporates elements of both classical and quantum dynamics provides the
best available single system account of these three characteristics—evolution, oscillation, and choice-induced
preference change".
\subsection*{Notation}
\begin{enumerate}
\item $\ket{n} \in \mathcal{S}$ is $n^{th}$ basis vector in a Hilbert space $\mathcal{S}$.
\item $\mathcal{H}(\{\ket{s_1}, \dots, \ket{s_k}\})$: Hilbert Space spanned by the orthonormal basis vector set $\{\ket {x_1},\dots,\ket {x_k}\}$
\item $A^{\dagger}$: adjoint of $A$
\item $\ket{n}\bra{m}$: outer product of $\ket{n}$ and $\ket{m}$
\item Density operator $\rho$: $\rho=\ket{\Psi}\bra{\Psi}$ for some $\ket{\Psi}\in \mathcal{S}$
\item Trace of operator $A$: $\Tr(A)=\sum_{l=1}^{N}\bra{n}A\ket{n}$ for basis $\ket{n}\in\mathcal{S}$
\item Random events are defined on $(\Omega,\mathcal{F},\mathbb{P})$.
\end{enumerate}
\section{Quantum Probability Model for Human Decision Making}
\label{sec2}
This section presents the open-quantum system model that we will use to represent the decision preference evolution of the human decision maker.
We define the evolution of the density operator of the decision maker using the open-system Quantum Lindbladian Equation as given in \cite{martinez2016quantum}.
\subsection{Lindbladian Dynamics}
Given a state of nature which is a random variable that can take on $n$ values, the human decision maker chooses one of $m$ possible actions.
The quantum based model for human decision making is governed by the Lindbladian evolution of the psychological state. With $\mathcal{L}$ denoting the Lindblad operator, the Lindbladian ordinary differential equation for the dynamics of the psychological state $\rho_t$ over time $t \in [0,\infty)$ is specified as
\begin{equation}
\frac{d\rho_t}{dt} = \mathcal{L}_{(\alpha,\lambda,\phi)} \, \rho_t, \quad \rho_0 = \frac{1}{nm}\text{diag}(1,\cdots,1)_{mn\times mn} \label{eq:lindbladian} \end{equation}
Here $(\alpha,\lambda,\phi)$ are free parameters which determine the quantum decision maker's behavior, as discussed in Appendix~\ref{constr_lind}. We assume that the machine has full knowledge of these behavioral parameters, as methods for estimating these via training are outside the scope of this paper.
In subsequent sections we will control the evolution of $\rho_t$ and formulate Lyapunov stability conditions. We are interested in the general case when the machine can only observe the human's actions and output a control every $T$ time steps, where $T$ is a random variable. We thus define the evolution of the density operator in (\ref{M_update}), from the controller's perspective, over $T$-step iterations.
The psychological model comprises the following:
\begin{enumerate}
\item A state of nature $s \in \{1,\ldots,n\} = \mathcal{X}$, with probability mass function $\pi_0(s)$
\item An action $a_k \in \{1,\ldots,m\} = \mathcal{A}$ by the human at discrete time~$k$, for $k = 0,1,2,\ldots$.
\item Scalar control input $u_k \in [-\bar{u},\bar{u}],\, \bar{u} \in {\rm I\hspace{-.07cm}R}_+$, from the machine. This controls the parameters of the Lindbladian operator in equation (\ref{M_update}), and models a recommendation signal (for example a posterior probability of the state of nature) by the machine to the human.
\item The discrete time evolution of the psychological state
\begin{equation}
\label{M_update}
\rho_{k+T} = \mathbb{M}_{\mu_k, T}^{u_k}\,\rho_k = \frac{M_{\mu_k, T}^{u_k}\, \rho_k\, M_{\mu_k, T}^{u_k\dagger}}{\Tr(M_{\mu_k, T}^{u_k}\, \rho_k\, M_{\mu_k, T}^{u_k\dagger})}
\end{equation}
where $k = 0,1,\ldots$ denotes discrete-time.
Recall the random variable $T: \Omega \rightarrow \mathbb{N}$ specifies the time intervals over which the machine interacts with the human. $T$ has a known probability mass function $\pi_T(\cdot)$. $\mu_k$ is a $T$-length sequence $\{\mu_{k_i}\}_{i = 1}^T $of random actions $\mu_{k_i}$ taking values in $\mathcal{A}$.
See Appendix~\ref{DT_lind} for the definition of $\mathbb{M}_{\mu_k, T}^{u_k}$.
\item The action probability distribution at time $k$
\begin{equation}
\label{eq:ProbEvo}
p(a_k = \mu_k)=\Tr(M_{\mu_k}^{u_k}\, \rho_k \, M_{\mu_k}^{u_k\dagger}), \mu_k \in \{1,\cdots,m\}
\end{equation}
\end{enumerate}
See Appendix~\ref{constr_lind} and~\ref{DT_lind} for further model details.
\subsection{Practicality in Modeling Human Decision Making}
The above Lindbladian model
captures important human decision features such as the sure-thing principle and order effects, which we now describe.
These features cannot be explained by purely Markovian models without sacrificing their explanatory power.
\subsubsection*{The violation of the sure-thing principle}
The total probability law, also called the Sure Thing Principle (STP), is
\[P(A)=P(B)\,P(A|B)+P(\bar{B})\,P(A|\bar B)\]
for events $A$ and $B$. Violation occurs when $=$ is replaced by either $<$ or
$>$.
Suppose $P(A|B) = 0.6$, $P(A|\bar{B})= 0.5$. Then if the probability the human decision maker chooses action $A$ is either greater than 0.6 or less than 0.5, then the STP has been violated.
\cite{martinez2016quantum} shows that the Lindbladian model accounts for violations of the STP. Pothos and Busemeyer \cite{pothos2009quantum} (see also \cite{khrennikov2009quantum}) review empirical evidence for the violation of STP and show how quantum models can account for it by introducing quantum interference in the probability evolution.
\subsubsection*{Order Effects}
It is well-established in psychology \cite{shanteau1970additive}, \cite{furnham1986robustness}, \cite{walker1972order} that the order of presented information can affect the final judgement of a human \cite{hogarth1992order}. Order effects are not easily accounted for using classical set-theoretic probability axioms, since $P(H | A \cap B) = P(H| B \cap A)$, i.e. the order of presentation of events $A$ and $B$ does not influence the final probability judgement of $H$. Alternative models of inference have been proposed, such as the averaging model \cite{shanteau1970additive} and the belief-adjustment model \cite{hogarth1992order}, but these are only heuristic ad-hoc models which lack axiomatic foundations. Quantum probability is a natural axiomatic framework which can account for these effects, see \cite{trueblood2011quantum}, \cite{busemeyer2012quantum}, \cite{khrennikov2010ubiquitous}, \cite{busemeyer2011quantum} and references therein. Order effects naturally arise from the non-commutative structure of quantum interactions.
\section{Machine Control of Human Decision Maker}
\label{sec3}
This section exploits the Lyapunov function formulated in \cite{AMINI20132683} and a generalized finite-step convergence theorem, (that will be proved in Section~\ref{sec4}), to prove our main result, Theorem \ref{thm:convg}. This theorem states that regardless of the initial psychological state of the human, the machine is able to control the preference in such a way that the target action is eventually chosen at every time step with probability one.
We first define some notation:
With $ d =nm$, let $D$ denote the space of non-negative Hermitian matrices with trace~1:
\begin{equation}
D := \{\rho \in \mathbb{C}^{d \times d} : \rho = \rho^{\dagger}, \Tr(\rho) = 1, \rho \geq 0\}
\end{equation}
Let $\{\ket{b_r}\}_{r=1}^d$ be a set of orthonormal vectors in $\mathbb{C}^d$, where each $\ket{b_r}$ corresponds to a unique state-action pair. Let $\mathcal{S}$ be the Hilbert space formed by taking these $\{\ket{b_r}\}_{r=1}^d$ as an orthonormal basis. We consider scalar control inputs $u_k \in {\rm I\hspace{-.07cm}R}$ satisfying constraints given in Appendix \ref{ap:contr}. For our purposes it suffices that $u_k \in [-1,1]$, see \cite{AMINI20132683} for details. The term {\em 'Open-loop (super) martingale'} below denotes a (super) martingale when the control input $u_k = 0$ for $k = 0,1,\ldots$.
The following is the main result:\\
\begin{theorem}
\label{thm:convg}
Given the discrete time density operator evolution (\ref{M_update}) and any target state $\ket{\bar{b}_r}, r \in \{1,\dots,d\}$, there exists a control sequence $\{u_k\}_{k \in \mathbb{N}}$ generated by the machine such that the human psychological state $\rho_k$ converges to $\ket{\bar{b}_r}\bra{\bar{b}_r}$ with probability one for any initial $\rho_0 \in D$.\\
\end{theorem}
\begin{proof}
We will follow the formulation developed in \cite{AMINI20132683}. First
with $\beta_k = \{\mu_k, T\}$, rewrite (\ref{M_update}) as
\begin{equation}
\label{eq:dens_up}
\rho_{k+T} = \mathbb{M}_{\beta_k}^{u_k}\rho_k = \frac{M_{\beta_k}^{u_k}\, \rho_k\, M_{\beta_k}^{u_k \dagger}}{\Tr(M_{\beta_k}^{u_k}\, \rho_k\, M_{\beta_k}^{u_k \dagger})}
\end{equation}
We define the following Lyapunov function, which forms a supermartingale under both open-loop (zero-input) and closed-loop (feedback control ($u_k$)) conditions for the process (\ref{eq:dens_up}):
\begin{equation}
\label{Ly_fcn}
V_{\epsilon}(\rho):= \sum_{r=1}^d \sigma_r \bra{b_r} \rho \ket{b_r} - \frac{\epsilon}{2}\sum_{r=1}^d \bra{b_r} \rho \ket{b_r}^2
\end{equation}
where $\sigma_r$ is non-negative $\forall r \in \{1,\ldots,d\}$ and $\epsilon$ is strictly positive. The set $\{\sigma_r\}_{r=1}^d$ and $\epsilon$ are chosen according to \cite{AMINI20132683} such that the Lyapunov function $V_{\epsilon}(\rho_k)>0 \, \forall\, \rho_k \in D$ and $\rho_k$ converges to the intended subspace $\ket{\bar{b}_r} \bra{\bar{b}_r}$ with probability 1. By \cite{AMINI20132683} and \cite{6160433}, $\bra{b_r} \rho \ket{b_r}$ is an open-loop martingale given the density operator evolution (\ref{eq:dens_up}) (see Appendix~\ref{ap:martingale} for proof). $V_{\epsilon}$ is a concave function of the open-loop martingales $\bra{b_r} \rho \ket{b_r}$ and therefore is an open-loop ($u_k = 0$) supermartingale given the process (\ref{eq:dens_up}).
\begin{equation*}
\mathbb{E}[V_{\epsilon}(\rho_{k+T}) |\, \rho_k, u_k = 0] - V_{\epsilon}(\rho_k) \leq 0
\end{equation*}
The following feedback control mechanism is used
\begin{equation*}
u_k := \underset{u \in [-1,1]}{\text{argmin}}\, \mathbb{E}[V_{\epsilon}(\rho_{k+T}) | \rho_k, u]
\end{equation*}
to get $\mathbb{E}[V_{\epsilon}(\rho_{k+T}) | \rho_k, u_k] \leq \mathbb{E}[V_{\epsilon}(\rho_{k+T}) | \rho_k, u = 0]$.
Here the expectation is taken with respect to $\beta_k$,
\begin{equation*}
\begin{split}
\mathbb{E}[V_{\epsilon}(\rho_{k+T}) | \rho_k, u] &=
\mathbb{E}[V_{\epsilon}(\mathbb{M}_{\beta_k}^{u} \rho_k)]
= \int_{\Omega} V_{\epsilon} (\mathbb{M}_{\beta_k(\omega)}^u \rho_k)d\omega
\end{split}
\end{equation*}
where $\Omega$ is the sample space under which the process is induced.
Define
$
\tilde{Q}(\rho_k) := \mathbb{E}[V_{\epsilon}(\rho_{k+T}) | \rho_k, u_k] - V_{\epsilon}(\rho_k) \leq \mathbb{E}[V_{\epsilon}(\rho_{k+T}) | \rho_k, u_k = 0] - V_{\epsilon}(\rho_k) \leq 0
$.
$V_{\epsilon}(\rho)$ is a continuous function, so using \cite[Chapter 8]{kushner1971introduction}, we have the $T$-step control sequence $\{\rho_{k+iT}\}_{i \in \mathbb{N}}$ that converges to the set $\textrm{$D_{\infty}$}:= \{\rho : \tilde{Q}(\rho) = 0\}$ with probability one.\\ We will first show that the set $D_{\infty}$ \ is restricted to our desired state $\{\ket{\bar{b}_r} \bra{\bar{b}_r}\}$, then that the entire sequence $\{\rho_k\}_{k \in \mathbb{N}}$ converges to this set. The former is proved in Lemma 2 of \cite{AMINI20132683}; For any target subspace $\ket{\bar{b}_r} \bra{\bar{b}_r}$, the set $\{\sigma_r\}_{r=1}^d$ can be chosen in such a way that $D_{\infty}$ $ = \ket{\bar{b}_r} \bra{\bar{b}_r}$. The idea is the following:
A state $\rho_k$ is in the limit set $D_{\infty}$ \ iff for all $u \in [-1,1],$
\begin{equation}
\label{supm}
\mathbb{E}[V_{\epsilon}(\rho_{k+T}) | \rho_k, u] - V_{\epsilon}(\rho_k) \geq 0
\end{equation}
Also, since $V_{\epsilon}$ is an open-loop supermartingale, $\forall \rho_k \in D$:
\begin{equation}
\label{subm}
\mathbb{E}[V_{\epsilon}(\rho_{k+T}) | \rho_k, u = 0] - V_{\epsilon}(\rho_k) \leq 0
\end{equation}
\begin{comment}
and $\not\exists\,u$ which satisfies (\ref{subm})
\end{comment}
By Lemma 2 of \cite{AMINI20132683}, given any $\bar{r} \in \{1,\dots, d\}$ and fixed $\epsilon > 0$, the weights $\sigma_1, \dots, \sigma_d$ can be chosen so that $V_{\epsilon}$ satisfies the following property: $\forall r \in \{1,\dots,d\}, u \mapsto \mathbb{E}[V_{\epsilon}(\rho_{k+T}) \ | \rho_k = \ket{b_r}\bra{b_r}, u_k = u]$ has a strict local minimum at $u=0$ for $r = \bar{r}$ and a strict local maximum at $u=0$ for $r \neq \bar{r}$. This combined with (\ref{subm}) ensures that for any $r \neq \bar{r}, \exists\, u \in [-1,1]$ such that $\mathbb{E}[V_{\epsilon}(\rho_{k+T}) \ |\, \rho_k = \ket{b_r}\bra{b_r}, u_k = u] - V_{\epsilon}(\ket{b_r}\bra{b_r}) < 0$. Therefore, by (\ref{supm}), we have that $\ket{b_r}\bra{b_r}$ is in the limit set $l_{\infty}$ if and only if $r = \bar{r}$.
We now show that $\mathbb{P}[\lim_{k \to \infty}\rho_k = \ket{\bar{b_r}}\bra{\bar{b_r}}] = 1$, i.e. the entire sequence converges to the target state with probability one. This utilizes Theorem \ref{thm:extension} which is developed in Section \ref{sec4}.
We have shown that
$
\mathbb{P}[\,\lim_{i \to \infty} \rho_{k + iT} = \ket{\bar{b}_r}\bra{\bar{b}_r}] = \\
\mathbb{P}[\,\exists\,K \in \mathbb{N} : \rho_{k + iT} = \ket{\bar{b}_r}\bra{\bar{b}_r} \ \forall i \geq K]=1
$.
The set $\{\sigma_r\}$ was chosen such that $\mathbb{E}[V_{\epsilon}(\rho_{k+T}) \ | \rho_k = \ket{b_r}\bra{b_r}, u_k = u]$ has a strict local minimum at $u=0$ for $r = \bar{r}$ so
$
u_{k+iT}= \underset{u \in [-1,1]}{\text{argmin}}\, \{\mathbb{E}[V_{\epsilon}(\rho_{k+(i+1)T}) | \rho_{k+iT}, u]\} = 0 \, \forall\, i \geq K
$.
So, for $q \in \{1,\dots,T\}$,
$
\mathbb{E}[V_{\epsilon}(\rho_{k+(i+q+1)T}) | \rho_{k+(i+q)T}, u=0] = \mathbb{E}[V_{\epsilon}(\mathbb{M}_{\beta_k}^{0}\rho_{k+(i+q)T})]\leq V_{\epsilon}(\rho_{k+(i+q)T})
$
since $V_{\epsilon}$ is an open-loop supermartingale. We now know $\exists\, K \in \mathbb{N}$ such that $\mathbb{E}[V_{\epsilon}(\rho_{k+T}) | \rho_k, u_k] \leq V_{\epsilon}(\rho_k) \ \forall\, k \geq K.$ Since there exists a unique mapping from elements of $D$ to elements of ${\rm I\hspace{-.07cm}R}^{2d^2}$, we apply Theorem \ref{thm:extension} to prove $\mathbb{P}[\lim_{k \to \infty} \rho_k = \ket{\bar{b}_r}\bra{\bar{b}_r}] = 1.$
Since $\ \forall a_j \in \mathcal{A}, \exists$ a set $\{\ket{b_{j_1}},\dots,\ket{b_{j_n}}\} = a_j \otimes \mathcal{X}$, the convergence to any $\ket{\bar{b}_r}\bra{\bar{b}_r}, \ket{\bar{b}_r} \in a_j \otimes \mathcal{X}$, implies the convergence to $a_j \in \mathcal{A}.$
\end{proof}
To summarize, we showed that for the discrete time psychological state evolution of (\ref{M_update}), there exists a control policy which allows the machine to guide the human decisions such that a target decision is made asymptotically, almost surely.
\section{Finite Step Stochastic Lyapunov Stability}
\label{sec4}
The purpose of this section is two-fold. First,
we generalize Theorem 1 of Qin et. al. \cite{qin2019lyapunov} to the case when the finite step size can be a random variable. Our main result below is Theorem \ref{thm:extension}. Recall that we used this result in Section \ref{sec3} to prove stability of the human-decision system. Second, Theorem \ref{thm:rand} below is of independent interest.
Consider the discrete time stochastic system described by
\begin{equation}
\label{eq1}
x_{k+1} = f(x_k, y_{k+1}), \ \ \ k= 0,1,2,\ldots
\end{equation}
Here $x_k \in {\rm I\hspace{-.07cm}R}^n$, and $\{y_k : k =0,1,2,\ldots\}$ is a ${\rm I\hspace{-.07cm}R}^d$ valued stochastic process on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
Consider the filtration (increasing sequence of $\sigma$-fields) defined by $\mathcal{F}_0 = \{\emptyset, \Omega\}, \ \mathcal{F}_k = \sigma(y_1, \dots, y_k) \textrm{ for } k \geq 1$. We choose $x_0 \in {\rm I\hspace{-.07cm}R}^n$ as a constant with probability one. Thus $\{x_n\}$ is an ${\rm I\hspace{-.07cm}R}^n$-valued stochastic process adapted to $\mathcal{F}_k$. \\
\begin{theorem}
\label{thm:extension}
For the discrete-time stochastic system (\ref{eq1}), let $V: {\rm I\hspace{-.07cm}R}^n \rightarrow {\rm I\hspace{-.07cm}R}$ be a continuous non-negative and radially unbounded function. Suppose we have the condition:\\
(a) There exists a random variable $T: \Omega \rightarrow \mathbb{N}$ such that for any $k$, $\mathbb{E}[V(x_{k+T}) | \mathcal{F}_k] - V(x_k) \leq -\varphi(x_k)$, where $\varphi: {\rm I\hspace{-.07cm}R}^n \rightarrow {\rm I\hspace{-.07cm}R}$ is continuous and satisfies $\varphi(x) \geq 0$ for any $x \in {\rm I\hspace{-.07cm}R}^n$.
Then for any initial condition $x_0 \in {\rm I\hspace{-.07cm}R}^n$, $x_k$ converges to $\mathcal{D}_1 := \{x \in {\rm I\hspace{-.07cm}R}^n : \varphi(x) = 0\}$ with probability one.
This Theorem follows from Theorems \ref{thm:qin} and \ref{thm:rand}; proofs are given for both of these. \\
\end{theorem}
\begin{theorem}
\label{thm:qin}
For the discrete-time stochastic system (\ref{eq1}), let $V: {\rm I\hspace{-.07cm}R}^n \rightarrow {\rm I\hspace{-.07cm}R}$ be a continuous non-negative and radially unbounded function. Define the set $Q_{\lambda} = \{x : V(x) < \lambda\}$ for some positive $\lambda$, and assume that:\\
\ \ \ (a) $\mathbb{E}[V(x_{k+1}) | \mathcal{F}_k] - V(x_k) \leq 0$ for any $k$ such that $x_k \in Q_{\lambda}$ \\
\ \ \ (b) There exists an integer $T \geq 1$, independent of $\omega \in \Omega$, such that for any $k, \mathbb{E}[V(x_{k+T}) | \mathcal{F}_k] - V(x_k) \leq -\varphi(x_k)$, where $\varphi: {\rm I\hspace{-.07cm}R}^n \rightarrow {\rm I\hspace{-.07cm}R}$ is continuous and satisfies $\varphi(x) \geq 0$ for any $x \in Q_{\lambda}$ \\
Then for any initial condition $x_0 \in Q_{\lambda}$, $x_k$ converges to $\mathcal{D}_1 := \{x \in Q_{\lambda} : \varphi(x) = 0\}$ with probability at least $1 - V(x_0)/\lambda$ \\
\end{theorem}
\begin{proof}We have
\begin{equation}
\label{eq2}
\mathbb{E}[V(x_{k+T})|\mathcal{F}_k] - V(x_k) \leq -\varphi(x_k) \leq 0 , \forall x_k \in Q_{\lambda}
\end{equation} where $\varphi(x)$ is continuous. Now, Kushner \cite[p.196]{kushner1971introduction} has shown that if we start with $x_0 \in Q_{\lambda}$ then the probability of staying in $Q_{\lambda}$ during the entire resultant trajectory is at least $1 - V(x_0)/\lambda$, i.e.
\begin{equation}
\label{eq3}
\mathbb{P}[\sup_{k \in \mathbb{N}}V(x_k) \geq \lambda] \leq V(x_0)/\lambda
\end{equation}
Next construct $T$ subsequences of $\{X_k\}$ as follows: $\{X_i^{(0)}\} = \{X_0,X_T,\dots\}, \{X_i^{(1)}\} = \{X_1,X_{T+1},\dots\},\dots, \{X_i^{(T-1)}\} = \{X_{T-1},X_{2T-1},\dots\}.$ \\
Suppose $\varphi(x) \geq 0 \ \forall x:$ Then for all $k \in K := \{0,\dots,T-1\}$, $\{V(X_i^{(k)})\}$ is a non-negative supermartingale process by (\ref{eq2}), and thus by Doob's convergence theorem converges to a limit with probability 1. From (\ref{eq2}) we have for all $k \in K$ and $n \in \mathbb{N}$
$\sum_{l=1}^{n}\mathbb{E}(V(X_l^{(k)}))-\mathbb{E}(V(X_{l-1}^{(k)}))\leq-\mathbb{E}(\sum_{l=0}^{n-1}\varphi(X_l^{(k)}))
$
and $0 \leq \mathbb{E}(V(X_n^{(k)})) \leq \mathbb{E}(V(X_0^{(k)})) - \mathbb{E}(\sum_{l=0}^{n-1}\varphi(X_l^{(k)}))$.
We use Fatou's Lemma to obtain $\mathbb{E}(\sum_{l=0}^{\infty}\varphi(X_l^{(k)})) < \infty$ and by Borel-Cantelli we have $\mathbb{P}[\lim_{n \to \infty}\varphi(X_n^{(k)}) = 0] = 1\, \forall\, k \in K
$.
Now suppose $\varphi(x) \geq 0$ only for $x \in Q_{\lambda}.$ Stop $\{X_n^{(k)}\}$ on first leaving $Q_{\lambda}$. Then for $x \notin Q_{\lambda}$, $\varphi(x) = 0$ for this stopped set. This stopped process is a supermartingale and the proof is the same as above. \\
It is now apparent that $\lim_{n \to \infty}\varphi(X_n^{(k)}(\omega)) = 0 \ \forall k \in K$ and $\omega \in \bar{\Omega} = \{\omega \in \Omega : x_n(\omega) \in Q_{\lambda} \ \forall n \in \mathbb{N}\}$, so we have $\mathbb{P}[\lim_{n \to \infty}\varphi(X_n(\omega)) = 0] \geq 1- V(x_0) / \lambda$ by the analysis in Appendix \ref{pf:subseq_conv} and (\ref{eq3}). \\
\end{proof}
\begin{theorem}
\label{thm:rand}
Theorem (\ref{thm:qin}) holds when $T$ is an integer-valued random variable $T: \Omega \rightarrow \mathbb{N}$, where $\Omega$ is the underlying sample space \\
\end{theorem}
\begin{comment}
\textbf{Proof (Approach 1):} \\
Begin by inducing the same stopped process as in Proof 1 so that (5) always holds.
Now define a new random process $\{X_{k_i}\}_{i \in \mathbb{N}}$ as follows: \\
Initializing with $X_{k_0}$, define $\forall \omega \in \Omega: X_{k_{i+1}}(\omega) = X_{k_i + T(\omega)}(\omega)$, the $(k_i + T(\omega))$'th element in the original stochastic process $X_i$. Then by (a) and (b), with the condition that $T$ is independent from $\omega$ removed, we have:
\begin{multline*}
\mathbb{E}[V(X_{k_{i+1}})|\mathcal{F}_k] - V(X_{k_i}) \leq -\varphi(X_{k_i}) \leq 0 , \\ \forall X_{k_i} \in Q_{\lambda}, \ i \in \mathbb{N}.
\end{multline*}
Thus by the same method as in Proof 1, we can obtain
\begin{equation*}
0 \leq \mathbb{E}(V(X_{k_{i+1}})) \leq \mathbb{E}(V(X_{k_0})) - \mathbb{E}(\sum_{l=0}^{i}\varphi(X_{k_l}))
\end{equation*}
and so by Borel-Cantelli:
\begin{equation*}
\mathbb{P}[\lim_{i \to \infty}\varphi(X_{k_i}) = 0] = 1 \ \ \forall k_0 \in \mathbb{N}
\end{equation*}
where $\{X_{k_i}\}$ is the sequence induced by the initial condition $X_{k_0}$. \\
Now let $\Omega^{*} := \{\omega : \lim_{i \to \infty}\varphi(X_{k_i}(\omega)) = 0\}.$ Then $\ \forall \omega \in \Omega^{*},$ the set of subsequences $\{X_{k_i}(\omega)\}_{i \geq 1, k_0 = 1,\dots,T(\omega)}$ exhaust the original sequence $\{X_i(\omega)\}$ and thus $\lim_{i \to \infty}\varphi(X_i(\omega)) = 0.$ So now observe that
\begin{equation*}
\mathbb{P}[\lim_{n \to \infty}\varphi(X_n(\omega)) = 0] \geq \mathbb{P}[\Omega^{*} \cap \bar{\Omega}] = 1- V(x_0) / \lambda
\end{equation*}
where $\bar{\Omega}$ is the same as defined in Proof 1. \\
\end{comment}
\begin{proof}
This follows from the previous proof, with expectations conditioned on $T(\omega)=\tau$, the set of sequences with interval length $\tau \in \mathbb{N}$:
$
\sum_{l=1}^{n}\mathbb{E}(V(X_l^{(k)}))-V(X_{l-1}^{(k)})|T=\tau)\leq - \mathbb{E}(\sum_{l=0}^{n-1}\varphi(X_l^{(k)})|T=\tau)
$. Applying Fatou's Lemma yields $\mathbb{E}(\sum_{l=0}^{\infty}\varphi(X_l^{(k)})|T=\tau) < \infty$ and so by Borel-Cantelli we have $
\mathbb{P}[\lim_{n \to \infty}\varphi(X_n^{(k)}) = 0|T=\tau] = 1 \ \ \forall k \in K
$. The same arguments from the proof of Theorem (\ref{thm:qin}) yield
$
\mathbb{P}[\lim_{n \to \infty}\varphi(X_n(\omega)) = 0|T=\tau] \geq 1- V(x_0) / \lambda
$. So,
$
\mathbb{P}[\lim_{n \to \infty}\varphi(X_n(\omega)) = 0|T=\tau]\mathbb P(T=\tau) \geq (1- V(x_0) / \lambda)\mathbb P(T=\tau)
\implies \mathbb{P}[\lim_{n \to \infty}\varphi(X_n(\omega)) = 0] = \sum_{\tau \in \mathbb{N}}\mathbb{P}[\lim_{n \to \infty}\varphi(X_n(\omega)) = 0|T=\tau]\mathbb P(T=\tau) \geq \sum_{\tau \in \mathbb{N}}(1- V(x_0) / \lambda)\mathbb P(T=\tau) = 1- V(x_0) / \lambda
$
\end{proof}
To summarize, this section provided a generalization of the finite-step Lyapunov function result of Qin. et. al. \cite{qin2019lyapunov}. We applied this to show stability of the Lindbladian dynamics to prove almost sure convergence of the density operator (psychological state), but the generalization is of independent interest.
\section{Conclusion and Extensions}
\label{sec5}
Our main result, Theorem (\ref{thm:convg}), showed that for a human-machine decision system modeled as a controlled quantum decision system, there exists an optimal control policy under which the human's action choice can be guided to an arbitrary target action with probability one. This can be useful in tense or stressful decision situations when the human is subject to cognitive bias and irrational preferences; The machine can act as a rational Bayesian expected utility maximizer and control the human's preference to a Bayesian optimal choice. In proving this we have utilized a novel random finite-step Lyapunov function result, which we present and prove in Sec. (\ref{sec4}), and which stands as an independent result. There are several simplifying assumptions we have made, which warrant further investigating. For one, we have assumed that the machine knows the $(\alpha,\lambda,\phi)$ parametrization of the human's Lindbladian mental operator. It would be interesting to see what the analysis yields when these are only estimates with some distribution.
It is worthwhile extending our results to more general human-machine decision systems.
\section{Appendix}
\subsection{Convergence of constructed subsequences implies convergence of sequence}
\label{pf:subseq_conv}
We have $\lim_{n \to \infty}\varphi(X_n^{(k)}(\omega)) = 0 \ \forall \omega \in \bar{\Omega}$. Let $\omega \in \bar{\Omega}$ and $\varphi_n^{(k)}$ denote $\varphi(X_n^{(k)}(\omega))$ and $\varphi_n$ denote $\varphi(X_n(\omega))$. \\
We have: $\forall\, \epsilon > 0 \ \exists\, N_k$ such that $\varphi_n^{(k)} < \epsilon \ \forall n > N_k$. Take $N^{*} = \max_{k \in \{0,\dots,T-1\}}N_k$. Suppose $\lim_{n \to \infty}\varphi_n \neq 0$: Then $\exists\, \epsilon > 0$ such that $\forall \ N \in \mathbb{N} \ \exists \ n_0 > N$ with $\varphi_{n_0} > \epsilon.$ Since the subsequences are exhaustive, i.e. for any $\varphi_n \ \exists\, k, m,$ such that $\varphi_n = \varphi_m^{(k)}$, we know that for any $\epsilon > 0$, $\varphi_n < \epsilon \ \forall\, n > N^{*}$ so such a $n_0$ does not exists for $N^{*}$ and thus by contradiction we have $\lim_{n \to \infty}\varphi_n = 0.$ \\
\subsection{Lindbladian Psychological Model Construction}
Let psychological state space $\mathcal{S}:=\mathcal{H}({\{\ket{\mathcal{E}_l,\mathcal{A}_j}\}}_{\substack{{1\leq l\leq n}\\{1\leq j\leq m}}})$ be a Hilbert space. Here $\mathcal{H}$ is defined in Notation (2). Each $\ket{\mathcal{E}_l}$ is an $n$-dimensional complex vector indexed by the state $l$, and $\ket{\mathcal{A}_j}$ is an $m$-dimensional complex vector indexed by the action $j$.
\label{constr_lind}
The evolution of the density operator is given by $
\frac{d\rho}{dt} = \mathcal{L}_{(\alpha,\lambda,\phi)} \, \rho
$ where
\begin{equation}
\label{Lindblad}
\begin{split}
&\mathcal{L}_{(\alpha,\lambda,\phi)} \, \rho =-i(1-\alpha)[H,\rho]\\&+\alpha \sum_{m,n}\gamma_{(m,n)}\left(L_{(m,n)}\,\rho\, L_{(m,n)}^{\dagger}-\frac{1}{2}\{L_{(m,n)}^{\dagger}L_{(m,n)},\rho\}\right)
\end{split}
\end{equation}
Here $[A,B] = AB - BA$, $\{A,B\} = AB + BA$, $A^*$ is complex conjugate of $A$,
${H}=\text{diag}(\mathbf{1}_m,\cdots,\mathbf{1}_m)_{mn\times\,mn}$ with $\mathbf{1}_m$ an $m \times m$ matrix of ones. $L_{(m,n)} = \ket{m}\bra{n}$ is the jump operator, which represents the jump from $m^{th}$ cognitive state to $n^{th}$ cognitive state.
$\gamma_{(m,n)}:=[C(\lambda,\phi)]_{m,n}=[(1-\phi)\Pi^{T}(\lambda) + \phi K^{T}]_{m,n}$. For utility function $u: \mathcal{A} \times \mathcal{X} \rightarrow {\rm I\hspace{-.07cm}R}$,
$p(a_j|\mathcal{E}_l)=\frac{u(a_j|\mathcal{E}_l)^{\lambda}}{\sum_{j=l}^m u(a_j|\mathcal{E}_l)^{\lambda}}$
we define
$\mathbb{P}(\mathcal{E}_l):=\begin{bmatrix}
p(a_1|\mathcal{E}_l) & p(a_2|\mathcal{E}_l)&\cdots& p(a_{m}|\mathcal{E}_l) \\
\end{bmatrix}\otimes\mathbf{1}_{n\times1}$ and
$\Pi(\lambda) = \text{diag}(
P(\mathcal{E}_1),\cdots,P(\mathcal{E}_{n}))$
where $\mathbf{1}_{n\times1}$ is a vector with all 1's and, $A \otimes B$ is the kronecker product of $A$ and $B$. Define
$\eta_k(s) = p(s | u_k, y_k)$
given the noisy observation $y_k$ and input signal $u_k$, with $s \in \mathcal{X}$.
We define
$K:={\begin{bmatrix}
\eta_k(\mathcal{E}_1) & \eta_k(\mathcal{E}_2)&\cdots&\eta_k(\mathcal{E}_{n})
\end{bmatrix} }\otimes\mathbf{1}_{m\times1}\otimes \mathbb{I}_{m\times m}$.
\begin{comment}
Despite the apparent complexity of the model, the above formulation has all of its components specified by the problem formulation.
\end{comment}
$\alpha \in [0,1]$ represents the amount of quantum behavior in the evolution of the density operator.
$\lambda \in [0,\infty)$ can be thought of as the agent's ability to discriminate between the profitability of different actions.
$\phi \in (0,1)$ represents the relevance of discrimination between underlying states $\{\mathcal{E}_1, \dots, \mathcal{E}_{n}\}$.
\begin{comment}
\\ When $\alpha = 1$, the system reduces to a classical (completely dissipative) Markovian evolution. when $\alpha = 0$, the system is a purely quantum system which oscillates indefinitely and does not converge to a steady state.\\
$\lambda \in [0,\infty)$ can be thought of as the agent's ability to discriminate between the profitability of different actions. When $\lambda = 0$, each action $a_j \in \mathcal{A}$ has the same probability of being chosen, and when $\lambda \rightarrow \infty$, only the dominant action is chosen.\\
The parameter $\phi \in (0,1)$ represents the relevance of discrimination between underlying states $\{\mathcal{E}_1, \dots, \mathcal{E}_{n}\}$. There can be a wide variety of different observed preference evolutions within the three dimensional parameter space defined by ($\alpha, \lambda, \phi$).\\
In this work, we are assuming that the machine in our human-machine interaction has full knowledge of the human's parameters; The problem of parameter estimation and machine training is outside the scope of this paper, see Section \ref{sec5}.
\end{comment}
\begin{comment}
As shown in \cite{martinez2016quantum}, we solve the above matrix ordinary differential equation by vectorizing $\rho$
\begin{equation}
\label{vecLindblad}
\dfrac{d\Tilde{\rho}}{dt}=\Tilde{\mathcal{L}}\tilde{\rho}
\end{equation}
where $\tilde{\rho}=\vect({\rho})$ and
\begin{equation}
\begin{split}
\label{eq:lindvec}
\tilde{\mathcal{L}} &= -i(1-\alpha)(H \otimes \mathbb{I}_N - \mathbb{I}_N \otimes H^T) \\&+\alpha \sum_{(m,n)}\gamma_{(m,n)}(L_{(m,n)} \otimes L^{*}_{(m,n)}
\\ &-\frac{1}{2}(L^{\dagger}_{(m,n)}L_{(m,n)}\otimes \mathbb{I}_N + \mathbb{I}_N \otimes (L^{\dagger}_{(m,n)}L_{(m,n)})^{*})
\end{split}
\end{equation}
The corresponding solution for an initial $\rho(0)$ is:
\begin{equation}
\label{eq:lindvecSol}
\tilde\rho(t)=e^{\tilde\mathcal{L} t}\tilde\rho(0)
\end{equation}
and
We obtain the action probability $p(a_j)$ by taking the partial trace with respect to the state space after performing a measurement on $\rho$ w.r.t to action $a_j, j \in \{1,\dots,m\}$.
\[p(a_t=j)=\Tr_{\mathcal{X}}(M_{a_j}\,\rho\, M_{a_j}^{\dagger})\]
\end{comment}
When we consider the closed-loop feedback control mechanism, the scalar control input $u_k$ directly parametrizes the structure of the cognitive matrix, so that we can have $C_{u_k}(\lambda,\phi)$. We leave this as a parameter and do not define the specific effect of $u_k$ on $C_{u_k}(\lambda,\phi)$, as long as $u_k$ satisfies the constraints $(i) - (iv)$ of Section~\ref{ap:contr}.
\subsection{Discrete time representation of Lindblad Equation}
\label{DT_lind}
We time discretize the Lindbladian dynamics by representing it using Kraus operators to get (\ref{M_update}), see \cite{Pearle_2012}.
\begin{comment}
\begin{equation*}
\begin{split}
\rho(t+\Delta t)&=\left[\mathbbm{1}-\Delta t(iH+\frac{1}{2}\sum_{m,n}\gamma_{m,n}L_{(m,n)}^{\dagger}L_{(m,n)})\right]\,\rho(t) \\ &\left[\mathbbm{1}-\Delta t(-iH+\frac{1}{2}\sum_{m,n}\gamma_{m,n}L_{(m,n)}^{\dagger}L_{(m,n)})\right]\\ + &\Delta t\sum_{m,n}\gamma_{m,n}L_{(m,n)}\,\rho(t)\,L_{(m,n)}^{\dagger}
\end{split}
\end{equation*}
\end{comment}
We discretize the evolution in equation (\ref{Lindblad}) in multiples of $\Delta t$ in order to incorporate $T$ time steps and rewrite it as
$
\rho_{k+T}=\sum_{\mu}M_{\mu_k,T}^{u_k}\,\rho_{k}\,M_{\mu_k,T}^{u_k \dagger}
$
where $M_{0,T}^{u_k}=\left[\mathbb{I}-T\Delta t(iH+\frac{1}{2}\sum_{m,n}\gamma_{m,n}^{u_k}L_{(m,n)}^{\dagger}L_{(m,n)})\right]$
and $M_{\mu\neq0,T}^{u_k}=\sqrt{T\Delta t\gamma_{m,n}^{u_k}}L_{(m,n)}$, $T\in\mathbb{N}$. \begin{comment}
The above expression has same solution as the discrete evaluation of (\ref{eq:lindvecSol})
\[\tilde{\rho}(t+T\Delta t) = e^{\tilde{\mathcal{L}}T\Delta t}\,\tilde{\rho}(t)\] which can be rewritten as
\begin{equation}
\label{vecLind}
\tilde\rho_{k+T}=e^{\tilde{\mathcal{L}}T\Delta t}\,\tilde\rho_k
\end{equation}
after performing the inverse vectorization.
\end{comment}
We then perform a measurement to get $\rho_{k+T} = \frac{P_{\nu_k}(\sum_{\mu_k}M_{\mu_k,T}^{u_k}\,\rho_{k}\,M_{\mu_k,T}^{u_k \dagger})P_{\nu_k}^{\dagger}}{\Tr(P_{\nu_k}(\sum_{\mu_k}M_{\mu_k,T}^{u_k}\,\rho_{k}\,M_{\mu_k,T}^{u_k \dagger}) P_{\nu_k}^{\dagger})}
$
and thus obtain (\ref{M_update}).
\begin{comment}
To summarize, $\rho$ is evolved according to the Lindbladian equation until $T$ time increment , and then the resulting $\rho_{k+T}$ is projected onto an action subspace. We assume that this measurement process is random as $T$ we assume to be a random variable, so we are performing a random, number or measurements, resulting in equation (\ref{M_update})
\end{comment}
\subsection{Control Input Constraints}
\label{ap:contr}
($i$): For each $u_k, \sum_{\mu}M_{\mu}^{u_k \dagger}\, M_{\mu}^{u_k} = \mathbb{I}$ for general quantum measurements $\mu$.\\
($ii$): For $u_k = 0$, all $M_{\mu}^0$ are diagonal in the same orthonormal basis $\{\ket{n} |\, n \in \{1,\dots,d\}\}: M_{\mu}^0 = \sum_{n=1}^d c_{\mu,n}\ket{n}\bra{n}, c_{\mu,n} \in \mathbb{C}$ \\
($iii$): For all $n_1 \neq n_2$ in $\{1,\dots,d\}$, $\,\exists\, \mu \in \{1,\dots,d\}$ such that $|c_{\mu,n_1}|^2 \neq |c_{\mu,n_2}|^2.$ \\
($iv$): All $M_{\mu}^{u_k}$ are $C^2$ functions of $u_k$
\subsection{Martingale Proof}
\label{ap:martingale}
We prove that under the evolution of (\ref{M_update}), $\bra{b_r} \rho_k \ket{b_r}$ is a $T$-step martingale. Denote
$
\mathbb{M}_{\mu_k}\rho_k$ $ = \mathbb{M}_{\mu_k, T=1}^0 \rho_k=\frac{M_{\mu_k, 1}^0\, \rho_k\, M_{\mu_k, 1}^{0\dagger}}{\Tr(M_{\mu_k, 1}^0\, \rho_k\, M_{\mu_k, 1}^{0\dagger})} $.
Since $\bra{b_r} \rho_k \ket{b_r} $ $= \Tr(\ket{b_r}\bra{b_r} \rho_k)$,
$\mathbb{E}[\Tr(\ket{b_r}\bra{b_r}\, \rho_{k+1}) | \rho_k,u_k]$ $=\sum_{\mu_k = 1}^m \Tr(M_{\mu_k}\, \rho_k \, M_{\mu_k}^{\dagger}) \Tr(\ket{b_r}\bra{b_r}\, \mathbb{M}_{\mu_k}\, \rho_k)$ $=\sum_{\mu_k = 1}^m \Tr(\ket{b_r}\bra{b_r}\, M_{\mu_k}\, \rho_k\, M_{\mu_k}^{\dagger})$ $=\Tr(\sum_{\mu_k = 1}^m M_{\mu_k} M_{\mu_k}^{\dagger} \ket{b_r}\bra{b_r} \rho_k)=\Tr(\ket{b_r}\bra{b_r} \rho_k).
$
By induction we have $\mathbb{E}[\bra{b_r} \rho_{k+T} \ket{b_r}] = \bra{b_r} \rho_k \ket{b_r}$
\bibliographystyle{acm}
|
1,116,691,497,726 | arxiv | \section{Introduction}
Quantum properties of light can be used in a myriad of applications, ranging from enhanced
sensing~\cite{goda_quantum-enhanced_2008, collaboration_gravitational_2011}, to spectroscopy~\cite{polzik_spectroscopy_1992, ribeiro_sub-shot-noise_1997},
to metrology~\cite{giovannetti_quantum-enhanced_2004, marino_absolute_2011}, and quantum information processing~\cite{braunstein_quantum_2005, obrien_photonic_2009}. Especially in
the latter case, it would be desirable to have nonclassical light sources on compact platforms with very small footprint, high degree of
confinement, low power operation, and compatibility with complementary metal oxide semiconductor (CMOS) technology. In addition to the possibility of leveraging on a mature fabrication technology that is already in place, such a CMOS compatible
platform would enable the integration of microelectronics with quantum photonics on the same substrate. Many efforts have been directed at the development of on-chip single-photon
light sources~\cite{clemmen_continuous_2009, sharping_generation_2006, suhara_generation_2009, horn_monolithic_2012, davanco_telecommunications-band_2012, takesue2008generation}, detectors \cite{najafi_-chip_2015}, and logic gates~\cite{obrien_photonic_2009}. The development of brighter nonclassical light sources, such as squeezed light sources \cite{lvovsky_squeezed_2014}, on CMOS-compatible platforms has been lagging behind.
Generation of squeezed states of light requires an optical nonlinearity and was initially demonstrated in several off-chip platforms including optical parametric oscillators (OPOs) using parametric down conversion~\cite{heidmann_observation_1987}, and in atomic vapors~\cite{slusher_observation_1985} and optical fibers~\cite{shelby_broad-band_1986} using four-wave mixing. In such nonlinear parametric processes two ``twin" beams are generated, called signal and idler beams, with strong\texttt{} quantum correlations in the intensities of the two beams,
leading to noise in their intensity difference reduced below the standard quantum limit. In order to boost the effective optical nonlinearity it is customary to use a cavity. This introduces
a tradeoff between high bandwidth and high squeezing. On the one hand, the cavity should have a high finesse, in order to achieve stronger squeezing. On the other hand, typically
a high finesse implies low bandwidth, as is the case in free space OPOs. But for applications such as entanglement based quantum key distribution (QKD) in the continuous variable regime \cite{silberhorn_quantum_2002}, a large squeezing bandwidth is essential to ensure high data rates. Bright twin beam intensity difference squeezing has been recently demonstrated in whispering gallery mode crystalline resonators made of lithium niobate~\cite{furst_quantum_2011}, but the platform is not integrated on chip, and has a cavity bandwidth of 30 MHz. Recently, Ast \emph{et al.}~\cite{ast_high-bandwidth_2013} observed broadband squeezing over a GHz by using a low finesse cavity for field enhancement. However, as Ast \emph{et al.} point out~\cite{ast_continuous-wave_2012, ast_high-bandwidth_2013}, the consequences are a very high parametric oscillation threshold. Another approach to achieve
strong squeezing is to use a pulsed pump, in order to achieve very high intensities and thus strong nonlinear behavior. Using a periodically poled lithium niobate waveguide,
Eto \emph{et al.}~\cite{eto_efficient_2011} measured squeezing of -5~dB. In their work, since there is no cavity, there is no fundamental bandwidth tradeoff, the data rate limit being related to the repetition rate of the pulsed pump. However, CMOS-compatibility, and a high degree of confinement, which are desirable in making compact integrated photonic structures, is lacking.
It should be noted that Safavi-Naeini \emph{et al.}~\cite{safavi-naeini_squeezed_2013} have reported 0.2 dB squeezing of light using a silicon optomechanical resonator, which, although CMOS compatible, is intrinsically restricted to a bandwidth of a few MHz around the mechanical resonance, and uses suspended structures operating at low temperatures, making it less robust compared to sources based on all-optical nonlinearities.
\begin{figure}
\includegraphics[width = .50\textwidth]{fig1_FWM_PRA3.eps}
\caption{(a) Schematic of the generation of intensity correlated signal and idler beams via four wave mixing in the on-chip microring optical parametric oscillator (OPO). (b) Top: Optical spectrum analyzer scan of the light coupled out of the chip, at a pump power just above the parametric oscillation threshold. The Si$_3$N$_4$ ring is pumped at 1549.6 nm, generating signal and idler modes at 1540.2 nm and 1559.2 nm, respectively, which are 15 cavity modes away from the pump wavelength, as can be seen in the bottom figure. Bottom: Transmission spectrum of the ring for transverse electric (TE) polarization.}
\label{fig:FWMprocess}
\end{figure}
In this article, we report the
observation of all-optical squeezing in an on-chip monolithically integrated CMOS-compatible platform, generated in a micron-size silicon nitride oscillator~\cite{levy_cmos-compatible_2010} with gigahertz cavity linewidth. Owing to the small size of on-chip cavities, it is possible to obtain a large finesse and still have relatively wide cavity bandwidths. Large intra-cavity pump enhancement and strong squeezing can thus be obtained. Our OPO is based on the third order nonlinear process of four-wave mixing (FWM), as shown in Fig. \ref{fig:FWMprocess}.
These devices consist of microring resonators fabricated on deposited films of Si$_3$N$_4$ (inset of Fig. \ref{fig:exptsetup}). Note that the OPO can generate in principle a very large number of beams spanning more than an octave~\cite{okawachi_octave-spanning_2011, foster_silicon-based_2011}. Here we generate squeezing by using a pump power that is just above the threshold, when only two modes oscillate.
\section{Methodology}
\subsection{Design}
We fabricate the microring resonators out of silicon nitride because of its high nonlinear refractive index~\cite{levy_cmos-compatible_2010} (n$_2 = 2.5 \times 10^{-15}\, {\rm cm}^2\, {\rm W}^{-1}$, about 10 times that of silica) and very low propagation loss ($< 0.5$ dB cm$^{-1}$). It should be noted that the nonlinear refractive index of silicon is an order of magnitude higher than that of Si$_3$N$_4$, but nonlinear losses such as two photon absorption and free carrier absorption at 1550 nm prohibit parametric oscillation in silicon. The devices are compact, with a bus waveguide length of 1.5 mm and a ring circumference of 1.8 mm, corresponding to a free spectral range (FSR) of 80 GHz.
{
In order to obtain FWM gain and optical parametric oscillation, the dispersion is engineered to be anomalous at the pump wavelength to compensate for the Kerr nonlinear phase shift due to the pump \cite{turner_tailored_2006, levy_cmos-compatible_2010}. We design the ring waveguide cross section to be 820 nm high and 1700 nm wide, resulting in a slightly anomalous group velocity dispersion at the pump wavelength of 1549.6 nm.
}
In order to obtain significant squeezing, the cavity output coupling losses must be a large fraction of the overall losses. Relatively large losses are also desired for a large cavity bandwidth. On the other hand, a low pump power threshold is desirable, since it avoids detrimental thermal effects and also minimizes the influence of technical noise from the pump laser. The squeezing factor depends on the ratio between the internal ring losses, given by the ring's intrinsic quality factor ($Q_i$), and the coupling coefficient between the bus waveguide and the ring resonator, determined by the loaded quality factor ($Q_L$). The smaller the ratio of these losses, the better the squeezing obtained. {The squeezing factor relative to the shot noise level can be quantified as~\cite{brambilla_nondegenerate_1991, castelli_quantum_1994, matsko_optical_2005, chembo_quantum_2014},
\begin{equation}
S(\Omega) = 1 - \frac{\eta_c \eta_d}{1 + \Omega^2 \tau_c^2}
\end{equation}
where $\Omega$ is the sideband frequency, $\tau_c$ is the cavity photon lifetime, $\eta_d$ is the detection efficiency and $\eta_c = 1 - Q_L/Q_i$ is the ratio of the coupling losses to the total losses.}
The simultaneous requirement of large coupling losses and low threshold is a challenge, since the threshold is inversely proportional to the product of $Q_i$ and $Q_L$. We can meet these requirements by designing a ring with a very large $Q_i$, larger than 2 million, and operating in the highly overcoupled regime. The high intrinsic Q, achieved using the recently demonstrated fabrication process of thick Si$_3$N$_4$ deposition (Ref. \cite{luke_overcoming_2013}), enables the generation of the beams with a low pump power threshold of around 90 mW despite operating in this overcoupled regime. The corresponding intracavity power is estimated to be 7 W, which is an order of magnitude lower than the intracavity pump oscillation threshold of 65 W in Ref. \cite{ast_high-bandwidth_2013}. The loaded Q (200,000) is designed to be much lower than the intrinsic Q ($Q_i \approx 2$ million), facilitating a broad cavity linewidth of the order of 1 GHz.
The key to the realization of on-chip optical squeezing is the engineering of the dispersion and quality factors of the longitudinal modes of the microring resonator, to generate the twin beams at two well distinguished frequencies with low threshold pump powers. In contrast to many tabletop squeezing experiments, where the twin beams can usually be separated spatially based on their different polarizations, in our platform the beams are co-propagating in the same output waveguide with equal polarizations, and therefore in order to be distinguishable, the beams are required to have well separated frequencies. A delicate interplay between the group velocity dispersion and the quality factor of the ring, which together shape the FWM gain curve around the pump determines at which frequencies the twin beams will be generated \cite{lamont_route_2013, herr_universal_2012, chembo_quantum_2014}. {Specifically, the frequency difference between the pump and the first pair of oscillating signal and idler modes scales inversely as the square root of the second order dispersion ($D_2$) and the loaded quality factor($Q_L$): $\omega _{\rm pump} - \omega_{\rm signal} \propto \mathrm{FSR}/\sqrt{Q_L D_2}$}. Here we engineer the structure to ensure that the signal and the idler beams are generated at two well distinguished wavelengths separated by around 20 nm, at 1540.2 and 1559.2 nm when the OPO is pumped at 1549.6 nm, enabling the beams to be spatially separated using a diffraction grating with very low loss. These two modes are 15 cavity resonances away from the pump wavelength, as can be seen from the transmission spectrum of the ring shown in Fig. \ref{fig:FWMprocess}(b). The large wavelength separation between twin beams not only enables spatial separation of the twin beams, but also enables filtering of the pump wavelength, which is essential to mitigate the effects of the residual pump from affecting the twin beam squeezing measurements.
\subsection{Device Fabrication}
{
The Si$_3$N$_4$ waveguides and ring resonators were fabricated in 820 nm-thick silicon nitride films to provide low loss and high optical confinement. A 4 $\rm \mu$m thick silicon dioxide layer was first thermally grown on a virgin silicon wafer as an under cladding. The nitride layer was grown using low pressure chemical vapor deposition (LPCVD) in two steps of 400 nm each, interleaved with annealing in a nitrogen atmosphere for 3 hours at 1200 \textdegree C. The devices were patterned with electron beam lithography using MaN-2403 resist. After exposure and development, the resist was post-exposure baked for 5 min. at 115 \textdegree C, and etched in an inductively coupled plasma reactive ion etcher (ICP RIE) using CHF$_3$/O$_2$ chemistry. The devices were finally clad with 250 nm of high temperature oxide (HTO) deposited at 800 \textdegree C, followed by 2 $\mu$m of silicon dioxide using plasma enhanced chemical vapor deposition. Further details about the fabrication process can be found in \cite{luke_overcoming_2013}.
}
\begin{figure}
\includegraphics[width = .50\textwidth]{fig2_exptsetup.eps}
\caption{Experimental setup. The on-chip microring resonator is pumped by a continuous wave laser followed by an erbium doped fiber amplifier (EDFA) from JDS Uniphase and a bandpass filter (BPF). To ensure efficient coupling in and out of the chip, inverse tapers are used \cite{almeida_nanotaper_2003}. A lensed fiber is used for input coupling. The output from the waveguide is collected with a high NA objective lens. The diffraction grating spatially separates the pump and the twin beams. After blocking the pump, the beams are focused on a balanced detector. The RF output of the balanced detector is sent to an electrical spectrum analyzer. A small fraction of the beams is sent to an optical spectrum analyzer (OSA) to monitor the onset of parametric oscillation. The inset shows a microscope image of the ring resonator coupled to the bus waveguide. The shape of the fabricated ring is non-circular to fit the resonator within one field of the electron beam lithography tool and hence avoid stitching errors.
EDFA: Erbium doped fiber amplifier. BPF: Bandpass filter. OSA: Optical spectrum analyzer. VOA: Variable optical attenuator. M1, M2: mirrors. BS: Beam splitter}
\label{fig:exptsetup}
\end{figure}
\subsection{Experimental setup}
The experimental setup used to measure the squeezing is shown in Fig. \ref{fig:exptsetup}. The ring was pumped by a continuous wave, tunable laser (New Focus Velocity 6328) followed by an erbium doped fiber amplifier (JDSU OAB1552+20FA6) and a bandpass filter (BPF) with a bandwidth of 9 nm to reduce the amplified spontaneous emission noise generated by the EDFA. Light is evanescently coupled to the microring through a waveguide. The output from the waveguide was collected with a high NA (NA = 0.25) objective lens, leading to a loss of less than 1 dB. A diffraction grating with an efficiency of 85\% was used to spatially separate the pump and the two beams. After blocking the pump, the beams were focused on the two inputs of a Thorlabs PDB 150C balanced detector. The balanced detection system consists of a pair of well-matched InGaAs photodiodes with a quantum efficiency of 80\%, followed by a low noise transimpedance amplifier to amplify the difference in photocurrents between the detectors. To observe the variance of the intensity difference noise between the two balanced detectors, the RF output of the balanced detection system was sent to an electronic spectrum analyzer. The detection system had a bandwidth of 5 MHz and a transimpedance gain of 10$^5$ V/A. A small fraction (5\%) of the light coming out of the chip was sent to an optical spectrum analyzer (OSA) to observe the onset of parametric oscillation. The setup includes a variable optical attenuator (VOA) to calibrate the shot noise level and to check the degradation of squeezing with attenuation of the twin beams.
\section{Results and discussion}
We measure the squeezing in intensity difference of the two beams generated when the triply resonant OPO is pumped above threshold by comparing the noise level to the shot noise level generated from the coherent pump source.
\subsection{Shot noise calibration}
{We calibrate the shot noise level by splitting the pump beam on a 50:50 beam splitter, directing the two halves to the two detectors and monitoring the dc and ac components of the balanced detector output simultaneously. The common mode rejection ratio (CMRR) of the detectors was independently measured to be more than 27 dB. The laser is detuned far from the microring cavity resonance during shot noise calibration measurements. The electronic noise of the detection system is measured by blocking all light incident on the detectors. As seen from Fig. \ref{fig:spectrashot}, the detection system is linear over 20 dB near a sideband frequency of 3 MHz. There is excess technical noise at frequencies below 2 MHz, which cannot be rejected completely by the finite CMRR of the detection system. Focusing on the noise at 3 MHz, it can be seen from Fig. \ref{fig:shotnoise} that the shot noise level is linearly proportional to the optical power in each beam, in accordance with theory.
}
\begin{figure}
\includegraphics[width = .5\textwidth]{fig3_spectrashotnoise_automated.eps}
\caption{Linearity of the balanced detection system from 0.5 MHz to 5 MHz. The data have been dark noise corrected and normalized to the shot noise spectrum at a base local oscillator (LO) power of $P_0 = 7.8\, \mu $W, corresponding to the horizontal solid line at 0 dB. The dashed lines depict the mean optical power in the corresponding spectrum referenced to the base LO power $P_0$, in dB. There is significant technical noise at low frequencies, which is clearly evident at relative powers above 10 dB. The data have more fluctuations at high frequencies due to higher electronic noise and consequently lower dark noise clearance for the base LO power $P_0$. The dark noise varies from 5 dB below the 0 dB reference at 0.5 MHz to 1 dB below the reference at 5 MHz. Around 3 MHz, the detection system is linear over more than 20 dB, which is also substantiated by the linearity of the shot noise calibration measurement in Fig.~\ref{fig:shotnoise}.
}
\label{fig:spectrashot}
\end{figure}
\begin{figure}
\includegraphics[width = .50\textwidth]{fig4_linear_shotnoisepW_errorbar_correct_PRA3.eps}
\caption{Shot noise calibration, showing the linearity of the noise power in the photocurrent difference with optical power, as expected from theory. The data are taken at a Fourier sideband frequency of 3 MHz, using a spectrum analyzer with a resolution bandwidth of 30 kHz and a video bandwidth of 100 Hz. Data points have been corrected by subtracting the electronic noise of the balanced detection system. Error bars are extracted from the raw data.
}
\label{fig:shotnoise}
\end{figure}
\subsection{Squeezing measurements}
We observe more than 30 \% sub-shot noise quantum intensity correlations between the twin beams generated when the pump laser is tuned to an on-chip resonance at 1549.6 nm. Twin beam intensity difference squeezing measurements are presented in Fig. \ref{fig:squeezing}(a). The solid line corresponds to the signal-idler intensity correlation measurement, which is below the shot noise level, demonstrating clear intensity difference squeezing. These measurements were taken at Fourier sideband frequencies from 0.5 to 5 MHz, using a spectrum analyzer with a resolution bandwidth of 30 kHz and a video bandwidth of 100 Hz. Squeezing is not observed at very low frequencies, owing to technical noise in the pump laser. The on-chip OPOs used here could in principle act as a platform for generating large squeezing factors over broad bandwidths due to the highly overcoupled design of the rings.
{We confirm that the squeezing factor degrades linearly with increasing attenuation, and the intensity difference noise approaches the shot noise level for high attenuation, as is typical of squeezed states (Fig.~\ref{fig:squeezing}(b)).
By using the variable optical attenuator (VOA) in the experimental setup (Fig.~\ref{fig:exptsetup}), we measure the influence of decreasing transmission through the VOA on the squeezed twin beams. The dependence of the variance of the intensity difference noise, normalized to the corresponding shot noise level, can be modeled by mixing the unattenuated two-mode squeezed state with a vacuum state, on a beam splitter with transmittivity $\eta$:
\begin{equation}
\Delta^2 X_-(\eta) = \eta\, \Delta^2 X^{(0)}_- + (1-\eta)\, \Delta^2 X_v
\end{equation}
where the left hand side represents the variance after attenuation, $\Delta^2 X_-^{(0)}$ represents the unattenuated variance (i.e. before the VOA), and $X_v = 1$ is the variance of vacuum. All variances are normalized to the shot noise level at the corresponding power.}
The observed noise reduction of 1.7 $\pm$ 0.4 dB corresponds to a generated squeezing of 5 dB when corrected for detection losses and the non-ideal quantum efficiency of the detectors. This is less than the 10 dB of squeezing expected from the ratio of $Q_i$ and $Q_L$ owing to residual excess pump noise of 25 dB relative to the shot noise level, and the possible rotation of the optimally squeezed quadratures by the process of FWM \cite{mckinstrie_schmidt_2013, corzo_rotation_2013}. Furthermore, higher intrinsic quality factors of 7 million have been demonstrated in silicon nitride rings~\cite{luke_overcoming_2013}, resulting in a lower oscillation threshold, which not only helps in reducing excess noise in the pump, but also leads to a higher ratio of $Q_i$ to $Q_L$. It should thus be possible to reach much stronger noise reductions in this platform. Fundamentally the on-chip OPOs are expected to exhibit squeezing over GHz bandwidths in view of the broad linewidth of the cavity. We have demonstrated here squeezing in the MHz range, limited only by the bandwidth of our low dark noise detectors.
\begin{figure}
\includegraphics[width = .50\textwidth]{fig5_sqvsatten_sqvsfreq_line.eps}
\caption{Intensity difference squeezing. a) The variance of the signal and idler photocurrent difference, normalized to the shot noise level (SNL), on a linear scale. The dashed line at 1 represents the SNL, and the solid line represents the intensity difference fluctuations. The dark noise clearance was 16 dB at 0.5 MHz, and decreased to 5 dB at 5 MHz. The data was taken with a pump power of 93 mW. The power in the signal and idler beams is 45 $\rm \mu$W, which requires low dark noise detectors, commercially unavailable at higher frequencies. {Squeezing at low frequencies is masked by excess technical noise below $\sim$ 2 MHz, that can be clearly seen in Fig.~\ref{fig:spectrashot}}. b) Variation of squeezing with attenuation at a sideband frequency of 3 MHz. The x axis is the transmittivity of the variable optical attenuator, $\eta$. The y-axis is the intensity difference noise in shot noise units, that is, the variance of the signal and idler photocurrent difference compared to the shot noise level. Error bars are determined from the standard deviation of the measured data points.
}
\label{fig:squeezing}
\end{figure}
\section{Conclusions and outlook}
These results constitute an experimental demonstration of all-optical squeezing in an integrated CMOS compatible platform.
Our source generates bright squeezed light using a singly pumped FWM process, in contrast to other sources of above threshold squeezing which utilize parametric down-conversion. Since FWM is based on the third order nonlinearity, the technique presented here can be extended to several different material platforms, in contrast to the more restrictive second order nonlinearity only found in non-centrosymmetric materials. For example, FWM oscillation has been reported in silica~\cite{delhaye_optical_2007}, crystalline fluorides~\cite{savchenkov_tunable_2008}, hydex~\cite{razzari_cmos-compatible_2010}, aluminum nitride~\cite{jung_optical_2013} and diamond~\cite{hausmann_diamond_2014}. Our demonstration paves the way for a myriad of on-chip quantum optics experiments over broad bandwidths in a scalable, compact and robust platform. An experiment to measure phase anticorrelations of the twin beams is under way, enabling a demonstration of continuous-variable quantum entanglement~\cite{villar_generation_2005}. This will open the way to realize deterministic quantum information protocols at very high speeds. In future, the frequency separation of the twin beams can be done on chip using ring-resonator-based add-drop filters tuned to the signal, idler and pump wavelengths so that they are demultiplexed to different waveguides~\cite{barwicz_microring-resonator-based_2004}. This opens up the possibility to cascade the on-chip OPO with photonic structures to manipulate squeezed and entangled states of light generated by the OPO, further emphasizing the highly scalable nature of our platform.
The introduction of such non-classical light sources into future data communications by leveraging the mature infrastructure of microelectronics, currently being introduced into silicon photonics~\cite{jalali_silicon_2006}, is a very promising avenue to be explored.
\begin{acknowledgments}
We acknowledge fruitful discussions with Vivek Venkataraman, Alessandro S. Villar, Jaime Cardenas, Carl Poitras, Yoshitomo Okawachi, K. Saha, St\'{e}phane Clemmen and Marcelo Martinelli. The authors gratefully acknowledge support from DARPA for award \#W911NF-11-1-0202 supervised by Dr. Jamil Abo-Shaeer, and from AFOSR for award \#BAA-AFOSR-2012-02 supervised by Dr. Enrique Parra. P.N. acknowledges support from Funda\c{c}\~{a}o de Amparo \`{a} Pesquisa do Estado de S\~ao Paulo (FAPESP grant \#2011/12140-6). This work was performed in part at the Cornell NanoScale Facility, a member of the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation (Grant ECCS-0335765). This work made use of the Cornell Center for Materials Research Shared Facilities which are supported through the NSF MRSEC program (DMR-1120296).
\end{acknowledgments}
|
1,116,691,497,727 | arxiv | \section{Introduction}
\label{intro}
In a two-dimensional photonic crystal with inversion symmetry the band gap may become vanishingly small at corners of the Brillouin zone, where two bands touch as a pair of cones. Such a conical singularity is also referred to as a Dirac point, because the two-dimensional Dirac equation has the same conical dispersion relation. In a seminal work \cite{Rag06}, Raghu and Haldane investigated the effects of broken inversion symmetry and broken time reversal symmetry on the Dirac point of an infinite photonic crystal. Here we consider the transmission of radiation through an ideal but finite crystal, embedded in free space.
As we will show, the proximity to the Dirac point is associated with an unusual scaling of the transmitted photon current $I$ with the length $L$ of the photonic crystal. We assume that $L$ is large compared to the lattice constant $a$ but small compared to the transverse dimension $W$ of the crystal. For a true band gap, $I$ would be suppressed exponentially with increasing $L$ when the frequency $\omega$ lies in the gap. Instead, we find that near the Dirac point $I\propto 1/L$. The $1/L$-scaling is reminiscent of diffusion through a disordered medium, but here it appears in the absence of any disorder inside the photonic crystal.
Such ``pseudo-diffusive'' scaling was discovered in Refs.\ \cite{Kat06,Two06} for electrical conduction through graphene (a two-dimensional carbon lattice with a Dirac point in the spectrum). Both the electronic and optical problems are governed by the same Dirac equation inside the medium, but the coupling to the outside space is different. In the electronic problem, the coupling can become nearly ideal for electrical contacts made out of heavily doped graphene \cite{Kat06,Two06}, or by suitably matching the Fermi energy in metallic contacts \cite{Sch06,Bla06}. An analogous freedom does not exist in the optical case.
The major part of our analysis is therefore devoted to the question how nonideal interfaces affect the dependence of $I$ on $\omega$ and $L$. Our conclusion is that
\begin{equation}
I/I_{0}=\Gamma_{0}W/L\label{IGamma}
\end{equation}
at the Dirac point, with $I_{0}$ the incident current per mode and $\Gamma_{0}$ an effective interface transparency. The properties of the interfaces determine the proportionality constant $\Gamma_{0}$, and they also determine whether $I$ as a function of $\omega$ has a minimum or a maximum near the Dirac point, but they leave the $1/L$-scaling unaffected.
In Sec.\ \ref{waveeq} we formulate the wave equations inside and outside the medium. The Helmholtz equation in free space is matched to the Dirac equation inside the photonic crystal by means of an interface matrix in Sec.\ \ref{matching}. This matrix could be calculated numerically, for a specific model for the termination of the crystal, but to arrive at general results we work with the general form of the interface matrix (constrained by the requirement of current conservation). The mode dependent transmission probability through the crystal is derived in Sec.\ \ref{transmission}. It depends on a pair of interface parameters for each of the two interfaces. In Sec.\ \ref{totaltrans} we then show that the extremal transmission near the Dirac point scales $\propto 1/L$ regardless of the values of these parameters. We conclude in Sec.\ \ref{conclude} with suggestions for experiments.
\section{Wave equations}
\label{waveeq}
\begin{figure}[tb]
\centerline{\includegraphics[width=0.9\linewidth]{PC_layout_pores_flat}}
\caption{\label{fig_layout}
(Color online.) Photonic crystal formed by a dielectric medium perforated by parallel cylindrical holes on a triangular lattice (upper panel: front view; lower panel: top view). The dashed lines indicate the radiation incident on the armchair edge of the crystal, with the electric field polarized in the $z$-direction.
}
\end{figure}
We consider a two-dimensional photonic crystal consisting of a triangular or honeycomb lattice in the $x$-$y$ plane formed by cylindrical air-filled holes along the $z$-axis in a dielectric medium (see Fig.\ \ref{fig_layout}). The crystal has a width $W$ along the $y$-direction and a length $L$ along the $x$-direction, both dimensions being large compared to the lattice constant $a$. Monochromatic radiation (frequency $\omega$) is incident on the plane $x=0$, with the electric field $E(x,y)e^{i\omega t}$ polarized along the $z$-axis.
In the free space outside of the photonic crystal ($x<0$ and $x>L$) the Maxwell equations reduce to the Helmholtz equation
\begin{equation}
\left(\partial_{x}^{2}+\partial_{y}^{2}\right)E(x,y)+\frac{\omega^{2}}{c^{2}}E(x,y)=0.\label{Helmholtz}
\end{equation}
The mean (time averaged) photon number flux in the $x$-direction is given by \cite{Hou90}
\begin{equation}
j_{H}=\frac{\varepsilon_{0}c^{2}}{4i\hbar\omega^{2}}\left(E^{\ast}\frac{\partial E}{\partial x}-E\frac{\partial E^{\ast}}{\partial x}\right).\label{jHelmholtz}
\end{equation}
Inside the photonic crystal ($0<x<L$) the Maxwell equations reduce to the Dirac equation \cite{Rag06}
\begin{eqnarray}
\begin{pmatrix}
0&-iv_{D}(\partial_{x}-i\partial_{y})\\
-iv_{D}(\partial_{x}+i\partial_{y})&0
\end{pmatrix}
\begin{pmatrix}
\Psi_{1}\\
\Psi_{2}
\end{pmatrix}\nonumber\\
=(\omega-\omega_{D})
\begin{pmatrix}
\Psi_{1}\\
\Psi_{2}
\end{pmatrix},\label{Dirac}
\end{eqnarray}
for the amplitudes $\Psi_{1},\Psi_{2}$ of a doublet of two degenerate Bloch states at one of the corners of the hexagonal first Brillouin zone.
As explained by Raghu and Haldane \cite{Rag06,note1}, the modes at the six zone corners $\bm{K}_{p},\bm{K}'_{p}$ ($p=1,2,3$), which are degenerate for a homogeneous dielectric, are split by the periodic dielectric modulation into a pair of doublets at frequency $\omega_{D}$ and a pair of singlets at a different frequency. The first doublet and singlet have wave vectors at the first set of equivalent corners $\bm{K}_{p}$, while the second doublet and singlet are at $\bm{K}'_{p}$. Each doublet mixes and splits linearly forming a Dirac point as the wave vector is shifted by $\delta\bm{k}$ from a zone corner. The Dirac equation (\ref{Dirac}) gives the envelope field $\propto e^{i\delta\bm{k}\cdot\bm{r}}$ of one of these doublets.
The frequency $\omega_{D}$ and velocity $v_{D}$ in the Dirac equation depend on the strength of the periodic dielectric modulation, tending to $\omega_{D}=c'|\bm{K}_{p}|=c'|\bm{K}'_{p}|=4\pi c'/3a$ and $v_{D}=c'/2$ in the limit of weak modulation. (The speed of light $c'$ in the homogeneous dielectric is smaller than the free space value $c$.)
Eq.\ (\ref{Dirac}) may be written more compactly as
\begin{equation}
-iv_{D}(\nabla\cdot\bm{\sigma})\Psi=\delta\omega\Psi,\;\;\delta\omega\equiv \omega-\omega_{D},\label{Dirac2}
\end{equation}
in terms of the spinor $\Psi=(\Psi_{1},\Psi_{2})$ and the vector of Pauli matrices $\bm{\sigma}=(\sigma_{x},\sigma_{y})$. In the same notation, the
velocity operator for the Dirac equation is $v_{D}\bm{\sigma}$. The mean photon number flux $j_{D}$ in the $x$-direction is therefore given by
\begin{equation}
j_{D}=v_{D}\Psi^{\ast}\sigma_{x}\Psi=v_{D}(\Psi_{1}^{\ast}\Psi_{2}+\Psi_{2}^{\ast}\Psi_{1}).\label{jDirac}
\end{equation}
The termination of the photonic crystal in the $y$-direction introduces boundary conditions at the edges $y=0$ and $y=W$ which depend on the details of the edges, for example on edges being of zigzag,
armchair, or other type. For a wide and short crystal, $W\gg L$, these details become irrelevant and we may use periodic boundary conditions [$\Psi(x,0)=\Psi(x,W)$] for simplicity.
\section{Wave matching}
\label{matching}
\begin{figure}[tb]
\centerline{\includegraphics[width=0.9\linewidth]{Brillouin}}
\caption{\label{fig_Brillouin}
Right panels: Hexagonal first Brillouin zone of the photonic crystal (top) and dispersion relation of the doublet near one of the zone corners (bottom). Filled and open dots distinguish the two sets of equivalent zone corners, centered at $\bm{K}_{p}$ and $\bm{K}'_{p}$, respectively. The small circles centered at the zone corners are the equal-frequency contours at a frequency $\omega$ just above the frequency $\omega_{D}$ of the Dirac point. Left panels: Equal-frequency contour in free space (top) and corresponding dispersion relation (bottom). A plane wave in free space with $k_{x}$ close to $k_{0}$ (arrows in the upper left panel) excites Bloch waves in the photonic crystal with $\bm{k}$ close to $\bm{K}_{1}$ and $\bm{K}_{2}$ (arrows in the upper right panel), as dictated by conservation of $k_{y}$ and $\omega$ (dotted horizontal lines).
}
\end{figure}
The excitation of modes near a Dirac point has been discussed by Notomi \cite{Not00}, in terms of a figure similar to Fig.\ \ref{fig_Brillouin}. Because the $y$-component of the wave vector is conserved across the boundary at $x=0$, the doublet near $\bm{K}_{1}=(K_{x},K_{y})$ or $\bm{K}_{2}=(-K_{x},K_{y})$ can only be excited if the incident radiation has a wave vector $\bm{k}=(k_{x},k_{y})$ with $k_{y}$ near $K_{y}$. The conservation of $k_{y}$ holds up to translation by a reciprocal lattice vector. We will consider here the case of
$|\bm{k}|<|\bm{K}_{p}|$, where no coupling to $\bm{K}_{3}$ is
allowed. The actual radius of the equal frequency
contour in the
free space at $\omega = \omega_D$ will depend on a particular
photonic crystal realization.
The incident plane waves $E_{\rm incident}=E_{0}e^{i\bm{k}\cdot\bm{r}}$ in free space that excite Bloch waves at a frequency $\delta\omega=\omega-\omega_{D}$ have $k_{y}=K_{y}[1+{\cal O}(\delta\omega/\omega_{D})]$ and $k_{x}=k_{0}[1+{\cal O}(\delta\omega/\omega_{D})]$ with
\begin{equation}
k_{0}=\sqrt{(\omega_{D}/c)^{2}-K_{y}^{2}}.\label{k0def}
\end{equation}
For $\delta\omega\ll\omega_{D}$ we may therefore write the incident wave in the form
\begin{equation}
E_{\rm incident}(x,y)=E_{+}(x,y)e^{ik_{0}x+iK_{y}y},\label{Eindef}
\end{equation}
with $E_{+}$ a slowly varying function. Similarly, the reflected wave will have $k_{y}\approx K_{y}$ and $k_{x}\approx -k_{0}$, so that we may write it as
\begin{equation}
E_{\rm reflected}(x,y)=E_{-}(x,y)e^{-ik_{0}x+iK_{y}y},\label{Erdef}
\end{equation}
with $E_{-}$ slowly varying.
The orientation of the Brillouin zone shown in Fig.\ \ref{fig_Brillouin} corresponds to an armchair edge of the triangular lattice at $x=0$. For this orientation only one of the two inequivalent doublets is excited for a given $k_{y}$. (The other doublet at $\bm{K}'_{1}$,\ $\bm{K}'_{2}$ is excited for $-k_{y}$.) A $90^{\circ}$ rotation of the Brillouin zone would correspond to a zigzag edge. Then a linear combination of the two inequivalent doublets is excited near $k_{y}=0$. For simplicity, we will restrict ourselves here to the case shown in the figure of separately excitable doublets.
While the conservation of the wave vector component parallel to the boundary determines which modes in the photonic crystal are excited, it does not determine with what strength. For that purpose we need to match the solutions of the Helmholtz and Dirac equations at $x=0$. The matching should preserve the flux through the boundary, so it is convenient to write the flux in the same form at both sides of the boundary.
The photon number flux (\ref{jHelmholtz}) for the Helmholtz equation may be written in the same form as the flux (\ref{jDirac}) for the Dirac equation, by
\begin{subequations}
\label{jHelmholtz2}
\begin{eqnarray}
&&j_{H}=v_{H}{\cal E}^{\ast}\sigma_{x}{\cal E},\label{jHelmholtz2a}\\
&&v_{H}=\frac{\varepsilon_{0}c^{2}k_{0}}{4\hbar\omega^{2}},\;\;
{\cal E}=\begin{pmatrix}
E_{+}+E_{-}\\E_{+}-E_{-}
\end{pmatrix}.
\label{jHelmholtz2b}
\end{eqnarray}
\end{subequations}
(In the prefactor $k_{0}$ we have neglected corrections of order $\delta\omega/\omega_{D}$.) Flux conservation then requires
\begin{equation}
v_{H}{\cal E}^{\ast}\sigma_{x}{\cal E}=v_{D}\Psi^{\ast}\sigma_{x}\Psi,\;\; {\rm at}\;\; x=0.\label{fluxconserve}
\end{equation}
The matching condition has the general form \cite{And89}
\begin{equation}
\Psi=(v_{H}/v_{D})^{1/2}M{\cal E},\;\;{\rm at}\;\; x=0.\label{Mdef}
\end{equation}
The flux conservation condition (\ref{fluxconserve}) implies that the transfer matrix $M$ should satisfy a generalized unitarity condition,
\begin{equation}
M^{-1}=\sigma_{x}M^{\dagger}\sigma_{x}.\label{symplectic}
\end{equation}
Eq.\ (\ref{symplectic}) restricts $M$ to a three-parameter form
\begin{equation}
M=e^{\gamma\sigma_{z}}e^{\beta\sigma_{y}}e^{i\alpha\sigma_{x}}\label{Malpha}
\end{equation}
(ignoring an irrelevant scalar phase factor). The real parameters $\alpha,\beta,\gamma$ depend on details of the boundary at the scale of the lattice constant --- they can not be determined from the Helmholtz or Dirac equations (the latter only holds on length scales $\gg a$).
We now show that the value of $\alpha$ becomes irrelevant close to the Dirac point. At the boundary the incident and reflected waves have the form
\begin{equation}
{\cal E}_{\rm incident}=E_{0}
\begin{pmatrix}1\\ 1\end{pmatrix},\;\;
{\cal E}_{\rm reflected}=rE_{0}
\begin{pmatrix}1\\ -1\end{pmatrix},\label{Einr}
\end{equation}
with $r$ the reflection coefficient, and $E_{0}\equiv E_{+}(0,y)$ a slowly varying function. Both ``spinors'' are eigenvectors of $\sigma_{x}$, hence the action of $e^{i\alpha\sigma_{x}}$ on ${\cal E}$ is simply a phase factor:
\begin{eqnarray}
&&M{\cal E}_{\rm incident}=e^{\gamma\sigma_{z}}e^{\beta\sigma_{y}}e^{i\alpha}{\cal E}_{\rm incident},\nonumber\\
&&M{\cal E}_{\rm reflected}=e^{\gamma\sigma_{z}}e^{\beta\sigma_{y}}e^{-i\alpha}{\cal E}_{\rm reflected}.\label{ME}
\end{eqnarray}
There is no need to determine the phase factor $e^{\pm i\alpha}$, since it has no effect on the reflection probability $|r|^{2}$.
A similar reasoning applies at the boundary $x=L$, where the matching condition reads
\begin{equation}
\Psi=(v_{H}/v_{D})^{1/2}M'{\cal E},\;\;{\rm at}\;\; x=L.\label{M2def}
\end{equation}
Flux conservation requires that $M'=e^{\gamma'\sigma_{z}}e^{\beta'\sigma_{y}}e^{i\alpha'\sigma_{x}}$, with real parameters $\alpha',\beta',\gamma'$. The value of $\alpha'$ is again irrelevant close to the Dirac point, because the spinor of the transmitted wave
\begin{equation}
{\cal E}_{\rm transmitted}=tE_{0}\begin{pmatrix}1\\ 1\end{pmatrix}\label{Et}
\end{equation}
(with $t$ the transmission coefficient) is an eigenvector of $\sigma_{x}$. So
\begin{equation}
M'{\cal E}_{\rm transmitted}=e^{\gamma'\sigma_{z}}e^{\beta'\sigma_{y}}e^{i\alpha'}{\cal E}_{\rm transmitted},\label{ME2}
\end{equation}
with a phase factor $e^{i\alpha'}$ that has no effect on the transmission probability $|t|^{2}$.
\section{Transmission probability}
\label{transmission}
We consider the case $W\gg L$ of a wide and short crystal, when we may use periodic boundary conditions at $y=0,W$ for the Bloch waves $\Psi\propto e^{i\delta\bm{k}\cdot\bm{r}}$. The transverse wave vector $\delta k_{y}$ is then discretized at $\delta k_{y}=2\pi n/W\equiv q_{n}$, with mode index $n=0,\pm 1,\pm 2,\pm 3,\ldots$. We seek the transmission amplitude $t_{n}$ of the $n$-th mode.
We first determine the transfer matrix $M_{n}(x,0)$ of the $n$-th mode $\Phi_{n}(x)e^{iq_{n}y}$ through the photonic crystal, defined by
\begin{equation}
\Phi_{n}(x)=M_{n}(x,0)\Phi_{n}(0).\label{Mndef}
\end{equation}
From the Dirac equation (\ref{Dirac2}) we obtain the differential equation
\begin{equation}
\frac{d}{dx}M_{n}(x,0)=
\left(\frac{i\delta\omega}{v_{D}}\sigma_{x}+q_{n}\sigma_{z}\right)M_{n}(x,0),\label{Mequation}
\end{equation}
with solution
\begin{equation}
M_{n}(x,0)=\cos k_{n}x+\frac{\sin k_{n}x}{k_{n}}\left(\frac{i\delta\omega}{v_{D}}\sigma_{x}+q_{n}\sigma_{z}\right).\label{Mresult}
\end{equation}
We have defined the longitudinal wave vector
\begin{equation}
k_{n}=\sqrt{(\delta\omega/v_{D})^{2}-q_{n}^{2}}.\label{kndef}
\end{equation}
The total transfer matrix through the photonic crystal, including the contributions (\ref{Mdef}) and (\ref{M2def}) from the interfaces at $x=0$ and $x=L$, is
\begin{equation}
{\cal M}=M'^{-1}M_{n}(L,0)M.\label{calMdef}
\end{equation}
It determines the transmission amplitude by
\begin{eqnarray}
{\cal M}\begin{pmatrix}
1+r_{n}\\ 1-r_{n}
\end{pmatrix}=
\begin{pmatrix}t_{n}\\ t_{n}\end{pmatrix}&\Rightarrow&
\begin{pmatrix}
1-r_{n}\\ 1+r_{n}
\end{pmatrix}={\cal M}^{\dagger}\begin{pmatrix}t_{n}\\ t_{n}\end{pmatrix}\nonumber\\
&\Rightarrow&\frac{1}{t_{n}}=\frac{1}{2}
\sum_{i=1}^{2}\sum_{j=1}^{2}
{\cal M}^{\ast}_{ij},\label{tnMrelation}
\end{eqnarray}
where we have used the current conservation relation ${\cal M}^{-1}=\sigma_{x}{\cal M}^{\dagger}\sigma_{x}$.
The general expression for the transmission probability $T_{n}=|t_{n}|^{2}$ is rather lengthy, but it simplifies in the case that the two interfaces at $x=0$ and $x=L$ are related by a reflection symmetry. For a photonic crystal that has an axis of symmetry at $x=L/2$ both $\Phi(x)$ and $\sigma_{y}\Phi(L-x)$ are solutions at the same frequency. This implies for the transfer matrix the symmetry relation
\begin{eqnarray}
\sigma_{y}{\cal M}\sigma_{y}&=&{\cal M}^{-1}\Rightarrow
\sigma_{y}M'\sigma_{y}=M\nonumber\\
&\Rightarrow&\beta'=\beta,\;\;\gamma'=-\gamma,\label{MprimeMrelation}
\end{eqnarray}
and we obtain
\begin{widetext}
\begin{eqnarray}
\frac{1}{T_{n}}&=&\left(\frac{\delta\omega\sin k_{n}L}{v_{D}k_{n}}\cosh 2\beta-\cos k_{n}L\sinh 2\beta\sinh 2\gamma-\frac{q_{n}\sin k_{n}L}{k_{n}}\sinh 2\beta\cosh 2\gamma\right)^{2}\nonumber\\
&&\mbox{}+
\left(\cos k_{n}L\cosh 2\gamma+\frac{q_{n}\sin k_{n}L}{k_{n}}\sinh 2\gamma\right)^{2}.\label{Tresult}
\end{eqnarray}
\end{widetext}
For an ideal interface (when $\beta=0=\gamma$) we recover the transmission probability of Ref.\ \cite{Two06}.
At the Dirac point, where $\delta\omega=0\Rightarrow k_{n}=iq_{n}$, Eq.\ (\ref{Tresult}) reduces further to
\begin{equation}
\frac{1}{T_{n}}=\cosh^{2}(q_{n}L+2\gamma)+\sinh^{2}2\beta\sinh^{2}(q_{n}L+2\gamma).\label{TresultDirac}
\end{equation}
More generally, for two arbitrary interfaces, the transmission probability at the Dirac point takes the form
\begin{eqnarray}
&&\frac{1}{T_{n}}=\cosh^{2}(\beta-\beta')\cosh^{2}\xi_{n}+\sinh^{2}(\beta+\beta')\sinh^{2}\xi_{n},\nonumber\\
&&\xi_{n}=q_{n}L+\gamma-\gamma'.\label{TresultDiracgeneral}
\end{eqnarray}
While the individual $T_{n}$'s depend on $\gamma$ and $\gamma'$, this dependence drops out in the total transmission $\sum_{n}T_{n}$.
\section{Photon current}
\label{totaltrans}
The transmission probabilities determine the time averaged photon current $I$ at frequency $\omega_{D}+\delta\omega$ through the photonic crystal,
\begin{equation}
I(\delta\omega)=I_{0}\sum_{n=-\infty}^{\infty}T_{n}(\delta\omega),\label{barI}
\end{equation}
where $I_{0}$ is the incident photon current per mode. The sum over $n$ is effectively cut off at $|n|\sim W/L\gg 1$, because of the exponential decay of the $T_{n}$'s for larger $|n|$. This large number of transverse modes excited in the photonic crystal close to the Dirac point corresponds in free space to a narrow range $\delta\phi\simeq a/L\ll 1$ of angles of incidence. We may therefore assume that the incident radiation is isotropic over this range of angles $\delta\phi$, so that the incident current per mode $I_{0}$ does not depend on $n$.
Since $W/L\gg 1$ the sum over modes may be replaced by an integration over wave vectors, $\sum_{n=-\infty}^{\infty}\rightarrow(W/2\pi)\int_{-\infty}^{\infty}dq_{n}$. The resulting frequency dependence of the photon current around the Dirac frequency is plotted in Figs.\ \ref{fig_I} and \ref{fig_I2}, for several values of the interface parameters. As we will now discuss, the scaling with the separation $L$ of the interfaces is fundamentally different close to the Dirac point than it is away from the Dirac point.
\begin{figure}[tb]
\centerline{\includegraphics[width=0.9\linewidth]{current_omega_beta.eps}}\medskip
\centerline{\includegraphics[width=0.9\linewidth]{current_omega_gamma.eps}}
\caption{\label{fig_I}
Frequency dependence of the transmitted current, for interface parameters $\beta'=\beta$, $\gamma'=-\gamma$. In the top panel we take $\gamma=0$ and vary $\beta$, while in the bottom panel we take $\beta=0$ and vary $\gamma$. The solid curves ($\beta=\gamma=0$) correspond to maximal coupling of the photonic crystal to free space. The curves are calculated from Eqs.\ (\ref{Tresult}) and (\ref{barI}), in the regime $W/L\gg 1$ where the sum over modes may be replaced by an integration over transverse wave vectors.
}
\end{figure}
\begin{figure}[tb]
\centerline{\includegraphics[width=0.9\linewidth]{current_omega.eps}}
\caption{\label{fig_I2}
Same as Fig.\ \ref{fig_I}, for $\beta$ and $\gamma$ both nonzero.}
\end{figure}
Substitution of Eq.\ (\ref{TresultDiracgeneral}) into Eq.\ (\ref{barI}) gives the photon current at the Dirac point,
\begin{eqnarray}
&& I(\delta\omega=0)=I_{0}\Gamma_{0}\frac{W}{L},\nonumber\\
&&\Gamma_{0}=\frac{\arctan[\sinh(\beta+\beta')/\cosh(\beta-\beta')]}{\pi\sinh(\beta+\beta')\cosh(\beta-\beta')},\label{ItotalDirac}
\end{eqnarray}
independent of the parameters $\gamma,\gamma'$. For two ideal interfaces we reach the limit
\begin{equation}
\lim_{\beta,\beta'\rightarrow 0}I(\delta\omega=0)/I_{0}=\frac{1}{\pi}\frac{W}{L},\label{Itotalideal}
\end{equation}
in agreement with Refs.\ \cite{Kat06,Two06}. Eq.\ (\ref{ItotalDirac}) shows that, regardless of the transparency of the interfaces at $x=0$ and $x=L$, the photon current at the Dirac point is inversely proportional to the separation $L$ of the interfaces (as long as $a\ll L\ll W$).
As seen in Figs.\ \ref{fig_I} and \ref{fig_I2}, the photon current at the Dirac point has
an extremum (minimum or maximum) when either $\gamma$ or $\beta$ are
equal to zero. If the interface parameters $\beta,\gamma$ are both nonzero, then the extremum is displaced from the Dirac point by a frequency shift $\delta\omega_{c}$. The photon current $I(\delta\omega_{c})$ at the extremum remains inversely proportional to $L$ as in Eq.\ (\ref{ItotalDirac}), with a different proportionality constant $\Gamma_{0}$ (which now depends on both $\beta$ and $\gamma$).
The $1/L$-scaling of the photon current applies to a frequency interval $|\delta\omega|\lesssim v_{D}/L$ around the Dirac frequency $\omega_{D}$. For $|\delta\omega|\gg v_{D}/L$ the photon current approaches the $L$-independent value
\begin{equation}
I_{\infty}=I_{0}\Gamma\frac{W\delta\omega}{\pi v_{D}},\label{Itotallargeomega}
\end{equation}
with rapid oscillations around this limiting value. The effective interface transmittance $\Gamma$ is a rather complicated function of the interface parameters $\beta,\beta',\gamma,\gamma'$. It is still somewhat smaller than unity even for maximal coupling of the photonic crystal to free space ($\Gamma=\pi/4$ for $\beta=\gamma=0$).
\section{Conclusion}
\label{conclude}
While several experiments \cite{Cas97,Ye04} have studied two-dimensional photonic crystals with a honeycomb or triangular lattice, the emphasis has been on the frequency range where the band structure has a true gap, rather than on frequencies near the Dirac point. Recent experiments on electronic conduction near the Dirac point of graphene have shown that this singularity in the band structure offers a qualitatively new transport regime \cite{Gei07}. Here we have explored the simplest optical analogue, the pseudo-diffusive transmission extremum near the Dirac point of a photonic crystal. We believe that photonic crystals offer a particularly clean and controlled way to test this prediction experimentally. The experimental test in the electronic case is severely hindered by the difficulty to maintain a homogeneous electron density throughout the system \cite{Cas07}. No such difficulty exists in a photonic crystal.
If this experimental test is successful, there are other unusual effects at the Dirac point waiting to be observed. For example, disorder has been predicted to increase --- rather than decrease --- the transmission at the Dirac point \cite{Tit06,Ryc06,Ost07}. Photonic crystals could provide an ideal testing ground for these theories.
\acknowledgments
We have benefited from discussions with A. R. Akhmerov, Ya.\ M. Blanter, and M. de Dood. This research was supported by the Dutch Science Foundation NWO/FOM.
|
1,116,691,497,728 | arxiv | \section{Introduction}
Hard X-ray emission from young Supernova Remnants (SNRs) was first detected in
SN 1006 and interpreted as synchrotron emission from electrons in the energy
range of tens of TeV \citep{koya95}. Such emission meanwhile has been found in
many young shell-type objects. These electrons are also expected to produce
very high energy (VHE) $\gamma$-rays \, in the $> 100$~GeV range by inverse Compton (IC)
collisions with low energy photons from the ambient radiation field, in
particular the Cosmic Microwave Background (CMB). In regions of high gas
density this may also be accompanied by a non-negligible emission of nonthermal
Bremsstrahlung $\gamma$-rays \,\ It is therefore not {\it a priori} clear whether the VHE
$\gamma$-rays \, detected in several shell-type SNRs up to now -- Cas~A
\citep{aha01a,al07}, RX~J1713.7-3946 \, \citep{mur00,enomoto02,aha04,aha06,aha07a}, RX
J0852.0-4622 \citep{Katagiri,aha05,aha07b,enomoto06}, and RCW~86 \citep{Hoppe}
-- are indeed the result of inelastic collisions of energetic nuclei with gas
atoms, as expected theoretically if a sizeable fraction of the SN explosion
energy goes into nuclear cosmic rays (CRs) \citep{drury94,bv97}, or whether
they are merely due to IC collisions of the X-ray synchrotron emitting
electrons.
The latter inference seems at least in a rough sense possible if the effective
magnetic field strength in these objects is sufficiently low. Then the energy
spectrum of accelerated electrons, calculated from the observed synchrotron
spectrum, contains sufficiently many particles so that a strong enough IC $\gamma$-ray \,
emission results. Since magnetic fields are difficult to determine, a possible
phenomenological approach is to assume that the interior (postshock) magnetic
field in the SNR is the result of MHD-compression of a circumstellar field
whose strength is at best a few $\mu$G. For RX~J1713.7-3946 \, and RX J0852.0-4622 such a
procedure works, at least as far as the overall {\it amplitude} of the $\gamma$-ray \,
spectrum is concerned. For Cas~A this is true as well. However, all these
remnants presumably correspond to core collapse SN explosions into the wind
bubble of a massive progenitor star, with a rather complex circumstellar
magnetic field structure. In contrast, for the remnants of several type Ia SNe
which could not be detected so far in $\gamma$-rays such as Tycho's SNR, SN
1006, and Kepler's SN, this scheme leads to large overpredictions for the $\gamma$-ray \,
flux, unless the mean interior magnetic field strength $B$ is assumed to be
significantly higher than the value $\sim 10~\mu$G expected from a $\lsim
5~\mu$G circumstellar field for a strong quasi-parallel, adiabatic shock with a
compression ratio of 4 (V\"olk et al. 2007, 2008). Since electron acceleration
cannot be responsible for such an amplification of the interstellar field, this
makes a pure electron acceleration scenario generally untenable in our view.
The acceleration of nuclear particles on the other hand cannot be deduced in a
similar way from some other electromagnetic emission -- even though the
magnetic field strength and the ion injection rate are inferred from the
{\it form} of the electron synchrotron spectrum \citep{bkv02}. The energetic
nuclear component needs to be calculated theoretically, and the amplitude of
the accelerated particle distribution can only result from a nonlinear
theory. This has been done successfully in a number of cases and shows that the
overall energy in accelerated nuclei is indeed a large fraction -- about $10$\%
-- of the available hydrodynamic explosion energy, and that the local
efficiency at those parts of the shock surface, where ion injection is
effective, amounts to even about 50 \% \citep[see e.g.][for a review]{ber05}.
Given this high acceleration efficiency, at reasonable thermal gas
densities the expected rate of hadronic collisions with gas atoms and
the resulting $\gamma$-ray \, production by $\pi^0$-decay turns out to be
consistent with the observations in the $\gamma$-ray \,\-detected objects analyzed
so far: Cas~A \citep{bpv03}, RX~J1713.7-3946 \, \citep{bv06} [hereafter referred to
as BV06], \citep{bv07}, RX J0852.0-4622 \citep{bpv08}. Also for
the other objects that we have analyzed theoretically -- Tycho's SNR
\citep{vbkr02,vbk05,vbk08}, SN 1006 \citep{bkv02,kbv05}, and Kepler's
SNR \citep{bkv06} -- and for which only upper limits exist so
far, the hadronic $\gamma$-ray \, emission is expected to dominate the IC
fraction, even though for the low-density object SN 1006 only by a
modest margin.
Within the errors, the theoretically derived magnetic field strengths also
agree with those deduced from the observations of filamentary X-ray synchrotron
features, often found at the outer rims of the SNRs
\citep[e.g.][]{vl03,long,bam03,bkv03,bv04a,vbk05,par06}. The field strengths
typically are an order of magnitude greater than the compressed
circumstellar field, substantially reducing the IC
spectrum amplitude for a given synchrotron spectrum and thus tending to yield a
negligible leptonic contribution to the observed VHE $\gamma$-ray \, flux.
Although the dominant acceleration of nuclear particles is clearly favored by
what has been said up to now, leptonic scenarios have been studied in some
detail \citep[e.g.][]{Porter06,Katz08}. Especially for the experimentally
best-studied SNR RX~J1713.7-3946 \, the question is whether an IC spectrum, scaled in
amplitude from the synchrotron spectrum in the sense described above, is
compatible with the {\it form} of the observed $\gamma$-ray \, spectrum. A second question
regards the observed spatial correlations of the morphology in hard X-rays,
assumed to be the result of synchrotron radiation, and in $\gamma$-rays. Such a
correlation has been observed in RX~J1713.7-3946 \, and might at first sight be attributed to
a common leptonic population of energetic particles. It remains to be seen
whether such an inference holds upon deeper scrutiny.
Another, not quite resolved question concerns the absence of thermal X-ray
emission in the two SNRs RX~J1713.7-3946 \, and RX J0852.0-4622 that are spatially resolved
at TeV energies. Also the radio synchrotron emission is very weak in comparison
to the strong X-ray synchrotron emission in both sources. Since the earliest
observations and analyses \citep{sla99}, the explanation for this situation has
been the assumption that the SN explosion occurred into the very rarefied
stellar wind bubble of a massive progenitor star. The model of BV06 specifies
the bubble structure in detail. It demonstrates that the main gas heating and
particle acceleration occurs beyond the wind region and bubble, when the SNR
shock propagates into the swept-up, radiatively cooled shell of interstellar
matter that was generated by the forward shock driven into the ambient ISM by
the expansion of the wind bubble. Therefore the wind bubble has two main
effects: the SNR can reach a large size quickly and it is then rather quickly
decelerated in the dense shell. During propagation of the SNR shock wave
through the wind and bubble the gas density is very low, resulting in a very
low thermal X-ray emission of this material. The shock's late encounter of the
massive shell heats the nuclear particles, but leaves little time for the
subsequent relaxation in the postshock region of the SNR that heats the thermal
electrons by Coulomb collisions with the heavy ions. In this way a minimum of
thermal X-ray emission from the remnant is combined with a maximum of hard
X-ray synchrotron and hadronic $\gamma$-ray \, emission from the SNR. In addition, the
shock is nonlinearly modified. For a SNR propagating into a uniform medium with
a uniform magnetic field this implies that the larger part of the shock surface
corresponds to a quasi-perpendicular shock with a strongly reduced injection of
nuclear particles \citep{vbk03}. Suprathermal injection of ions is only
possible in the quasi-parallel shock regions. If the spatial scales of the
quasi-perpendicular regions are large enough, the cross-field diffusion of the
highest-energy particles from the neighboring quasi-parallel shock regions also
does not reach deeply into these quasi-perpendicular regions. In the
corresponding magnetic flux tubes no magnetic field amplification occurs either
and the shock remains essentially un-modified there. This means that in the
quasi-perpendicular regions the shock dissipation and therefore the gas heating
occurs in a locally unmodified shock with the overall shock speed, leading to a
rather high gas temperature. In the case where a radiatively cooling shell of a
wind bubble is the main obstacle for the SNR expansion, the situation may be
different. The MHD instabilities of such a shell probably break it into many
small regions with strongly varying field directions. Then the spatial scales
separating the quasi-perpendicular from the quasi-parallel shock regions may
become small enough that crossfield diffusion can smear out the
quasi-perpendicular regions and particle acceleration occurs practically
everywhere over the shock surface \citep{vbk07,voelk08}. In the extreme this
implies shock modification over the entire shock region and thus a reduced gas
heating due to the subshock dissipation only\footnote{The given $\gamma$-ray \, flux
requires in addition a reduction of the gas density which reduces the
thermal emission.}. Compared to an un-modified shock
of the same overall speed, the gas temperature is then diminished by a
factor $ \approx \sigma_\mathrm{s}^2 / \sigma^2 \approx 0.25$, where
$\sigma$ and $\sigma_\mathrm{s}$ denote the overall shock compression ratio and
the subshock compression ratio, respectively. This
reduces the emission of soft thermal X-rays drastically.
Such a configuration is not easily analysed in detail with standard methods of
X-ray astronomy. However, this also leaves uncertainties in the evaluation of
the model's validity. Here, as in BV06, we proceed under the assumption that
the concrete wind bubble model, or its eventual improvement, is consistent with
the fact that no thermal emission has been found up to now.
The purpose of the present paper is a discussion of the above two
questions for RX~J1713.7-3946 \,, even though we expect analogous arguments to
hold for the other objects mentioned. In section 2 we discuss
the correlations between the gas density and the magnetic field
strength that are possible, or are even to be expected, in the
circumstellar medium before the SN explosion. In an approximate way we
also show how variations in gas density lead to correlated variations
in nuclear CR pressure and in the hadronic $\gamma$-ray \, emission. However, we shall
argue that the magnetic field {\it direction} influences
particle injection into the shock acceleration process, so that
circumstellar density enhancements are only a necessary but not a
sufficient condition for enhancements of the $\gamma$-ray \, emission. Under most
circumstances electron synchrotron emission is then expected to
increase with hadronic $\gamma$-ray \, emission as well. We also briefly
discuss the alternative purely leptonic scenario. In section 3 we
compare our calculations for the hadronic $\gamma$-ray \, emission, the
synchrotron emission, and for the IC and Bremsstrahlung $\gamma$-ray \, emission
with the latest X-ray and $\gamma$-ray \, measurements for RX~J1713.7-3946 \,. The results are
finally compared with a purely leptonic scenario, calculated by
assuming ion injection and thus also magnetic field amplification to
be negligibly small. It turns out that a hadronic origin of the $\gamma$-ray \,
emission is consistent with all measurements and with our theory, whereas
the leptonic scenario runs into serious difficulties. Section 4
contains our conclusions.
Very recently an independent discussion of these questions by \citet{tua08} has
come to our attention. Their paper is more observation oriented and is partly
complementary to ours. It uses phenomenological estimates for the accelerated
particle spectra. However their conclusions are similar to ours.
\section{Expected morphological correlations}
In this section we investigate the spatial correlations that arise from
compressions or expansions of the thermal gas with a frozen-in magnetic field,
in a given radiation field. In addition we argue that variations of the nuclear
energetic particle density -- which are most likely the result of spatial
variations of the injection rate into the acceleration process -- lead to
positively correlated enhancements of the magnetic field strength as a result
of magnetic field amplification by CR streaming instabilities.
\subsection{Compressions/de-compressions of the thermal plasma}
Consider briefly quasi-static compressions / de-compressions of the gas density
by plasma motions across the magnetic field in the MHD limit.
In this case the conservation of
magnetic flux compresses the field together with the thermal gas, increasing at
the same time the target for electrons to produce synchrotron emission as well
as the target for energetic nuclei in inelastic collisions with the gas atoms.
In both cases the densities of the X-ray synchrotron emitting electrons and
of the $\gamma$-ray \, emitting nuclei vary approximately in the same way,
because the electrons and nuclei concerned have roughly the same
energy. Therefore their spatial distributions are essentially the same.
In the 1-fluid MHD-approximation the mass density $\rho$ and the
magnetic field vector $\vec{B}$ are related to the mass velocity $\vec{u}$
through the conditions of conservation of mass and magnetic flux:
\begin{equation} {\partial \rho \over \partial t} + {\vec{u} \cdot \nabla \rho}
\equiv {\mathrm{d} \rho \over \mathrm{d} t} = -\rho \nabla \cdot \vec{u}
\label{eq1}
\end{equation}
\begin{equation}
{\partial \vec{B} \over \partial t} = \nabla \times (\vec{u} \times \vec{B}).
\label{eq2}
\end{equation}
\noindent Being interested primarily in the variations of $\rho$ and
the magnitude $B$ of $\vec{B}$, it is more useful for our considerations to
reduce Eqs.(1-2) to
equations for $B^2$ and $B/\rho$ alone, using $\nabla \cdot \vec{B}=0$:
\begin{equation}
{{1 \over 2B^2} {\mathrm{d} B^2 \over \mathrm{d} t}}
= - \nabla \cdot \vec{u}- {\vec{B} \cdot (\vec{B} \cdot \nabla)\vec{u}}
\label{eq3}
\end{equation}
\begin{equation}
{\mathrm{d} ln{(B / \rho)} \over \mathrm{d} t}
= B^{-2} \vec{B} \cdot (\vec{B} \cdot \nabla) \vec{u},
\label{eq4}
\end{equation}
\noindent where the r.h.s. of Eq.(3) is
nonzero only for an expansion/compression perpendicular to the direction of
\vec{B} and Eq.(4) shows that the ratio $B / \rho$
changes only in an expansion/compression parallel to $\vec{B}$.
With these relations in mind we consider the circumstellar medium of a
single massive star before it is reached by the SNR shock. The stellar
wind produces an expanding hot, low density bubble. This dynamical
evolution compresses the ambient interstellar medium (ISM) in an outer
shock wave in an approximately spherically symmetric manner. It is
clear that the compression of this external medium will create a
shocked shell with the gas density increasing radially outwards and,
over most of the solid angle (where the interstellar B-field is not
radial), also an increasing magnetic field strength. All pre-existing
density inhomogeneities in the external medium in the form of clouds
will by themselves have generally higher B-fields, wherever the
density is high. The subsequent compression by the wind bubble will
tend to enhance these correlated density and B-field
contrasts. Azimuthal density variations will probably form upon this
compression as well, by azimuthal instabilities of the radiatively
cooling swept-up shell, regardless of pre-existing clouds. All this
implies spatially correlated variations of $\rho$ and $B$ in the
medium upstream of the SNR shock.
As the theoretical model of BV06 suggests, the remnant of the
subsequent SN explosion should be old enough that the swept-up mass of
interstellar matter dominates the mass enclosed in the wind bubble as
well as the ejected mass from the explosion. Then we have the
situation that most of the emitting medium is a shock-modified ISM
that originally had a structure as described above. Its nonthermal
emission dominates the $\gamma$-ray \, emission as well as the synchrotron
emission of the SNR, as we observe it today.
\subsection{Injection and acceleration of nuclear particles at the SNR shock}
If the magnetic field lines in front of the accelerating forward SNR blast wave
are quasi-perpendicular to the shock normal, suprathermal ions cannot escape
from the downstream region (i.e. from the remnant interior) into the upstream
region to start being diffusively accelerated. On the other hand, nuclear
particles can be readily injected in quasi-parallel shock regions where the
field lines are more or less parallel to the shock normal. And the number of
ions injected into the acceleration from each gas volume is to a first
approximation proportional to the thermal gas density. In such field regions a
high energy density of accelerated nuclear particles will be built up by the
acceleration process. This particle population will try to escape into the
upstream medium and thereby excite streaming instabilities which lead to
magnetic field amplification in the associated magnetic flux tubes. The
enhanced field amounts at the same time to an increased target strength for
synchrotron emission by electrons, while the enhanced energetic nuclear
particle intensity will increase the number of inelastic collisions per unit
volume at given gas density. As already mentioned in the Introduction an
inhibition of acceleration in quasi-perpendicular shock regions may not occur
in the majority of quasi-perpendicular regions of the shock because of
cross-field diffusion. However, in principle this effect is there and may
occur especially at pre-existing interstellar clouds encountered by the SNR
shock.
The injection of energetic electrons into the acceleration process is not very
well understood, and is possibly not correlated with the ion
injection. Electron injection might actually occur everywhere over the SNR
shock surface. It may or may not increase with the plasma density.
Let us for a moment assume that electron injection
increases strictly monotonically with the upstream gas density.
Then we would expect a {\it nonlinear} correlation between the hard X-ray
and the $\gamma$-ray \, emissions in a hadronic model (see also section 2.3), since
electrons do not play a role in the acceleration energetics
and the resulting shock modification, acting de facto as test
particles. Therefore we expect the {\it amplitude} of the accelerated
electron energy distribution to increase with $\rho$, even though its {\it
form} is entirely determined by the accelerating ion component and is in fact
equal to the {\it form} of the ion energy distribution. The reason is that
only the nuclear component -- with its dominant energy density -- determines
the shock structure, and therefore the form of the accelerated energy
spectrum of all ultrarelativistic particles whose energy is high compared to
the proton rest energy, including the electrons (as long as their radiation
loss can be neglected). In fact, the lack of saturation of the electron
acceleration will allow the energy density of nonthermal energetic electrons
to increase strictly monotonically with their injection rate, i.e. with gas
density, in contrast to the saturation behavior of the nonthermal nuclear
particles which limits their injection rate.
Even if the electrons were injected -- and thus also accelerated --
everywhere with the same intensity, then the amplification of the magnetic
field by the ions in the quasi-parallel shock regions will automatically lead
to an enhancement of the synchrotron emission in these regions, suggesting a
correlation of the synchrotron emission with the hadronic $\gamma$-ray \, emission. This
should be qualitatively similar to the case in the bipolar remnant of SN
1006, even though probably less pronounced in wind bubbles. The bipolarity of
SN 1006 has been discussed in detail by \citet{vbk03}. For an experimental
discussion of the SN 1006 magnetic field configuration, see \citet{rbd04}.
Summarizing the arguments so far, we might expect a good and perhaps even
nonlinear spatial correlation between the synchrotron emission and the
production of energetic nuclear particles, as a result of pre-existing
correlated density and magnetic field strength inhomogeneities, as well as from
field amplification in quasi-parallel shock regions. However, this last
condition on nuclear particle injection implies that not all gas density
enhancements need to show up as regions of enhanced nuclear energetic particle
density and amplified magnetic field strength \citep{Plaga}. In other words,
not all clouds in the environment of RX~J1713.7-3946 \, need to be regions of enhanced
synchrotron and hadronic $\gamma$-ray \, emission. Some might just be shielded against the
injection of ions by an unfavorable magnetic field direction relative to the
shock normal.
\subsection{Hadronic $\gamma$-ray \, emission in the Quasi-Sedov phase}
The previous arguments did not consider the explicit modulation effect of
upstream density inhomogeneities in the upstream medium on the strength of
nuclear particle acceleration and on the resulting hadronic $\gamma$-ray \, emission. To
address this question we again make use of the assumption that the evolutionary
phase of RX~J1713.7-3946 \, is dominated by the swept-up mass from the wind bubble-structured
interstellar medium. Then the SNR interior should be subsonic with a roughly
{\it uniform} total gas and particle pressure $P_{tot} \propto \rho_1
V_\mathrm{s}^2$, as known from the Sedov solution in the case without CR
acceleration. Let us call this phase the Quasi-Sedov phase. If for example, in
a situation that deviates from spherical symmetry, the shock reaches regions of
different gas density over its surface, then the local shock velocity
$V_\mathrm{s}$ there will vary with the local upstream gas density
$\rho_1$ as $V_\mathrm{s}\propto \rho_1^{-0.5}$. In the Quasi-Sedov phase the
nuclear energetic particle pressure $P_\mathrm{c}$ is an only slowly varying
function of time, locally equal to several 10 percent of
$P_{tot}$ if ion injection is efficient
\citep[e.g.][]{kbv05}. And this nonthermal energy density is
primarily concentrated in the highest-energy particles of the spectrum, i.e. in
the VHE range. This implies an essentially uniform VHE hadronic $\gamma$-ray \, emissivity
$q_{\gamma} \propto P_\mathrm{c}$ in these regions and therefore a local
hadronic $\gamma$-ray \, production rate $Q_{\gamma}= q_{\gamma} \rho_\mathrm{d} \propto
\rho_1$, since the downstream gas density $\rho_\mathrm{d}$ is a fixed fraction
of the upstream density $\rho_1$ for a strong shock.
Therefore the hadronic $\gamma$-ray \, emission is approximately proportional to the local
gas density. Unless the electron acceleration is anticorrelated with the
upstream gas density, the X-ray synchrotron emission will be correlated with
the hadronic $\gamma$-ray \, emission, because $B$ and $\rho$ are almost always spatially
correlated.
\subsection{Spatial correlations in the leptonic scenario}
In the highly likely case that the leptonic $\gamma$-ray \, emission is dominated by the IC
emission, the $\gamma$-ray \, morphology is basically determined by the spatial
distribution of the radiating electrons. On the other hand,
the synchrotron emission is proportional to the product of the energetic
electron density and the local magnetic field energy. In the hypothetical case
that the hadronic $\gamma$-ray \, emission is negligible, the observations would of course
require a good spatial correlation between X-ray and IC $\gamma$-ray \, emission. And, the
better this correlation in such a leptonic scenario, the more the magnetic
field strength must therefore be spatially uniform.
This is a most unlikely situation for RX~J1713.7-3946 \,\, given the fact that the
gas density appears quite variable around the SNR shell and the
remnant appears to be interacting with at least some of these
molecular clouds, as CO-observations suggest \citep{Fukui,mor05}. This
makes a purely leptonic scenario already empirically highly unlikely,
independently of any theoretical arguments.
\section{Comparison of the latest $\gamma$-ray \, spectrum with the previous theoretical
model of BV06}
The latest version of the H.E.S.S. $\gamma$-ray \, spectrum \citep{aha07a}
presented in Fig.1 has not only increased statistical accuracy, but
also a flat (hard) spectral shape at the lowest energies of $~
250$~GeV, and a smooth extension and fall-off towards the 100 TeV
region. For the evaluation of this spectrum the reflectivity changes
of the H.E.S.S. mirrors have been taken into account, leading to a
roughly 15 percent increase in flux compared to the 2005
spectrum.
Note that the theoretical $\gamma$-ray \, spectra, calculated in BV06 and also presented in
Fig.2, correspond to the assumptions that RX~J1713.7-3946 \, was a core collapse supernova SN
of type II/Ib with a massive progenitor and explosion energy
$E_{sn}=1.8\times10^{51}$~erg, that it has an age of $\approx 1600$~yr and is
located at a distance of $\approx 1$~kpc. Although this general scenario
corresponds to the conclusions of most other authors, significantly larger
distances also have been considered in the literature \citep{sla99}. For
details, see the discussion in BV06.
Since the theoretical model cannot well determine the spectral amplitude, for
the reasons given in BV06, the above-mentioned flux increase is not relevant in
a comparison of theoretical and observational spectra\footnote{This is
approximately also true if cross-field diffusion of the highest-energy nuclear
particles modifies the shock almost everywhere, as discussed in the
Introduction.}.
However, the {\it forms} of the spectra agree remarkably well. We note that the
inferred leptonic IC and Nonthermal Bremsstrahlung spectra are depressed by
more than an order of magnitude relative to the observed spectrum. They cannot
explain the observations, if the magnetic field is indeed amplified to the
degree assumed in the theory and supported by the upper limit for the thickness
of the synchrotron filaments which one can derive (BV06) from the
XMM-observations of this remnant by \citet{Hiraga}. The resulting lower limit
for the magnetic field strength of $65~\mu$G has recently been supported by
Chandra observation of fast variations of the hard X-ray emission in some spots
in the remnant, possibly showing the localized existence of even mG magnetic
field strengths \citep{uat08} (see, however also \citet{butt08}).
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{0444fig1.eps}
\caption{Spatially integrated, $\gamma$-ray \, spectral energy distribution of RX~J1713.7-3946 \, . The
latest H.E.S.S.~$\gamma$-ray \, data
\citep{aha07a} ({\it blue color}) are shown together with the EGRET upper limit
for the RX~J1713.7-3946 \, position \citep{aha06} ({\it green color}). The {\it
solid} curve at energies above $10^7$~eV corresponds to $\pi^0$-decay $\gamma$-ray \,
emission, whereas the {\it dashed} and {\it dash-dotted} curves indicate the
Inverse Compton (IC) and Nonthermal Bremsstrahlung (NB) emissions,
respectively, from the theoretical model of BV06. See also \citet{bv07} .}
\label{f1}
\end{figure*}
\section{Comparison of the latest overall nonthermal spectrum with the BV06
spectrum}
We present in Fig.2 along with new HESS data the latest hard X-ray spectrum,
obtained with the Suzaku instrument \citep{ttu08}, which is given in the form
of an energy flux density by \citet{uat08}, and compare it with the theoretical
spectrum of BV06 \citep[see also][]{za07}. The Suzaku measurement does not
cover the entire SNR, and therefore it needs to be suitably normalized by the
requirement of optimum agreement with the ASCA spectrum, cf. \citet{aha06},
over the latter instrument's range between 0.5 and 10 keV. The result is shown
in Fig.2. The good agreement basically stems from the fact that the amplified
downstream field of $\approx 100~\mu$G, used to calculate the theoretical
spectrum in BV06, already leads to electron synchrotron cooling above an
electron momentum of $\approx 500~m_\mathrm{p} c$, and thus to a {\it
hardening} of the synchrotron spectrum compared to a spectrum calculated
without electron cooling (see Fig.3 below).
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{0444fig2.eps}
\caption{Spatially integrated, overall nonthermal spectral energy distribution
of RX~J1713.7-3946 \,\. The ATCA radio data \citep[cf.][]{aha06}[violet color], ASCA X-ray
data \citep[cf.][]{aha06}, Suzaku X-ray data \citep{uat08}[red color], and
2006 H.E.S.S. $\gamma$-ray \, data \citep{aha07a}[blue color] are shown. The EGRET upper
limit for the RX~J1713.7-3946 \, position \citep{aha06} [green color] is shown as well. The
{\it solid} curve at energies above $10^7$~eV corresponds to $\pi^0$-decay
$\gamma$-ray \, emission, whereas the {\it dashed} and {\it dash-dotted} curves indicate
the Inverse Compton (IC) and Nonthermal Bremsstrahlung (NB) emissions,
respectively, from the theoretical model of \citet{bv06}.}
\label{f2}
\end{figure*}
The same observed spectrum can also be compared with a theoretical spectrum
(Fig.3) in which a very low ion injection rate ($\eta = 10^{-5}$) and a rather
low downstream magnetic field strength of $20~\mu$G was assumed (see
BV06). This corresponds to a dominantly leptonic $\gamma$-ray \, test particle spectrum
without field amplification\footnote{In fact, the strength of the downstream
magnetic field might be even smaller by a factor of two or more, wherever the
shock is not locally parallel to the external field. However, adopting such a
small field would imply that even the gross amplitude of the maximum of the
observed $\gamma$-ray \, energy flux could not be fitted to the observations.}. The
IC-scattered diffuse radiation field is the CMB plus interstellar far infrared
and optical radiation fields as given in \citet{bpv03}. This corresponds to
typical values used for the Solar neighborhood
\citep[e.g.][]{drury94,gps98,Porter06}. We note that for nearby objects at
distances $d\sim 1$~kpc the CMB contribution dominates in the IC emission
\citep[see also][]{Porter06}.
The electron injection strength was fitted such that an optimum fit to the
observations in the radio and X-ray ranges is achieved, cf. Fig.3. We note that
the {\it form} of the X-ray spectrum is only very poorly fitted in this
leptonic scenario, especially when one uses the recent Suzaku
measurements. Also the $\gamma$-ray \, spectrum has a maximum which is much too sharp in
comparison with the observed H.E.S.S. spectrum. We note that, compared to
earlier measurements \citep{aha05}, the latest version of the H.E.S.S. $\gamma$-ray \,
spectrum \citep{aha07a} deviates more clearly from the IC-type spectrum with a
relatively sharp peak at $\epsilon_{\gamma}\sim 1$~TeV. Note also that our
spectrum of the nonthermal emission, that corresponds to the leptonic (or
inefficient) scenario, is almost identical to the spectrum presented by
\citet{Porter06}, even though they approximate the electron spectrum
analytically, whereas we calculate it numerically. Therefore it is also clear
that the quality of the fit achieved by \citet{Porter06} with a leptonic model
will be considerably lower if one uses the Suzaku X-ray spectrum and the new
HESS $\gamma$-ray \, spectrum instead of older, less accurate data.
At $\gamma$-ray \, energies of 1 GeV the spectral energy flux density is a factor of about
30 below the value in the hadronic scenario. It might be possible to construct
a more or less plausible form of the diffuse radiation field spectrum to obtain
a reasonable fit in the TeV region. However, it remains very doubtful in our
view whether this can avoid the sharp decline towards the GeV energy range
indicated in Fig.3. In any case such a construction cannot improve the
unacceptable fit in the hard X-ray range.
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{0444fig3.eps}
\caption{The same as in Fig.2, except that a leptonically dominated scenario
was assumed (see text).}
\label{f3}
\end{figure*}
\section{Conclusions}
We conclude that a theoretical acceleration model which takes into account
magnetic field amplification and a consistent nuclear energetic particle
production is consistent with the latest H.E.S.S. $\gamma$-ray \, and Suzaku hard X-ray
observations. It is also expected to be consistent with the observed good
correlation between X-ray synchrotron emission and VHE $\gamma$-ray \, emission. In the
face of existing gas density variations in or near the SNR, a purely leptonic
interpretation of the spatial correlation becomes poorer as this correlation
becomes closer empirically. The attempt to explain the $\gamma$-ray \, emission by leptonic
processes cannot be made consistent with the observed synchrotron spectrum. The
availability of higher-quality hard X-ray and $\gamma$-ray \, measurements leads to a
better consistency with the kinetic nonlinear theory prediction, whereas the
consistency of the leptonic model becomes much poorer.
Whether a leptonic model can lead to an acceptable fit of the $\gamma$-ray \, spectrum
through a more detailed evaluation of the local diffuse radiation field is an
open question. In our view, even such a modification would encounter enormous
difficulties in attempting to fit the $\gamma$-ray \, spectrum over the additional two
orders in $\gamma$-ray \, energy down from the VHE range to the GeV range. The leptonic
scenario seems also to be inconsistent with the filamentary X-ray morphology
which suggests substantial field amplification at least over part of the
remnant.
\begin{acknowledgements}
The authors would like to thank V.S. Ptuskin, F.A. Aharonian and
V.N. Zirakashvili for discussions on the general topic. EGB
acknowledges the partial support by the Presidium of RAS (program
No.16) and by the Russian Foundation for Basic Research (grant
07-02-00221) and the hospitality of the Max-Planck-Institut f\"ur
Kernphysik, where part of this work was carried out.
\end{acknowledgements}
|
1,116,691,497,729 | arxiv | \subsection{Adem Relations}
\paragraph{0.2.1} Now we will return to the discussion of our results on Adem relations.
Of course $B\Sigma_2 \simeq RP^\infty$, the real projective space.
One of our main results is a cochain level computation of the $\smallsmile_n$ products in $N^*(B\Sigma_2)$.
First it is well-known, and easy, that in a standard model the cochain algebra $N^*(B\Sigma_2) = {\mathbb F}_2[t]$, where $t \in N^1(B\Sigma_2)$ is dual to $x_1$, and $t^p \in N^p(B\Sigma_2)$ is dual to $x_p$.
The class $x_p$ is homologically represented by the real projective space $RP^p$.
Our first main result is this:
\begin{thm} \label{binom1}
$t^i \smallsmile_n t^j = \binom{i}{n}\binom{j}{n} t^{i+j-n} \in N^*(B\Sigma_2).$
\end{thm}
We derive this result purely combinatorially from Steenrod's definition of the $\smallsmile_i$.
In fact, what we really evaluate is Steenrod's equivariant chain map $\widetilde{AW}_{\!\!\Delta} \colon N_*(E\Sigma_2) \otimes N_*(B\Sigma_2) \to N_*(B\Sigma_2) \otimes N_*(B\Sigma_2)$, and the cochain statement follows by duality.
Our result includes the evaluation of the Squares in projective space, $Sq^k(t^i) = \binom{i}{k}t^{i+k}$, a result that is usually deduced as a consequence of the Cartan formula for the evaluation of Squares on cup products of cohomology classes.
We do not use the Cartan formula, and we obtain more, namely the evaluation of all $\smallsmile_n$ products in a specific cochain complex $N^*(B\Sigma_2)$.\\
The binomial coefficients in Theorem~\ref{binom1} correspond to rather simple binomial coefficient formulae for counting various kinds of partitions of integers, either exactly or mod 2.
There is no need for strenuous binomial coefficient manipulations that one often finds in proofs of Adem relations.\footnote{An analogue of Theorem~\ref{binom1} for an equivariant map $\widetilde{AW}_{\!\!\Delta} \colon N_*(EC_p; {\mathbb F}_p) \otimes N_*(BC_p; {\mathbb F}_p) \to N_*(BC_p; {\mathbb F}_p)^{\otimes p}$, where $C_p$ is the cyclic group of order $p$, seems to be the main obstacle to extending our results to odd primes $p$.}
\paragraph{0.2.2} Following Adem \cite{1adem}, we study classifying spaces $BV_4 \subset BD_8 \subset B\Sigma_4$ at the simplicial set level, where $\Sigma_4$ denotes the symmetric group, $D_8$ is the dihedral group, and $V_4$ is the normal Klein 4-group $V_4 \subset \Sigma_4$ containing the three products of disjoint 2-cycles.
Of course $V_4 \simeq \Sigma_2 \times \Sigma_2$, generated by any two non-identity elements.
The dihedral group, of which there are three conjugate versions in $\Sigma_4$, contains other copies of $\Sigma_2 \times \Sigma_2$.
The $D_8$ we work with contains the commuting 2-cycles $b =(12)$ and $c = (34)$, and is generated by these elements along with $a = (13)(24) \in V_4$.
We will work with the generators $\{a, bc\} \in V_4 \simeq \Sigma_2 \times \Sigma_2$.
Note that conjugation in $\Sigma_4$ by the element $(23)$ interchanges the generators $a = (13)(24)$ and $bc = (12)(34)$ of $V_4.$\footnote{In the first few sections of the paper we use the disjoint cycle notation to name permutations.
Later on, it will be more natural to name a permutation $\sigma \in \Sigma_n$ as a function, written as a sequence $(\sigma(1) \sigma(2) \cdots \sigma(n))$.}\\
The homology $H_*(BV_4) \simeq H_*(B\Sigma_2 \times B\Sigma_2)$ is generated by products of projective spaces $ RP^q \times RP^p$.
In specific simplicial chain complexes, these generators are given by $x_q \times x_p = EZ(x_q \otimes x_p)$, where $EZ$ is the Eilenberg-Zilber map $$EZ\colon N_*(B\Sigma_2) \otimes N_*(B\Sigma_2) \to N_*(B\Sigma_2 \times B\Sigma_2) \simeq N_*(BV_4).$$ It is important to point out that the map $EZ$ is pretty complicated, so $x_q \times x_p$ is actually a sum of quite a large number of basic elements.
Essentially, $EZ$ amounts to triangulating prisms.\\
With the models of contractible $G$ spaces $EG$, and classifying spaces $BG$, that we use, inclusions of groups $H \subset G$ yield inclusions of simplicial sets $EH \subset EG$ and $BH \subset BG$, hence inclusions of normalized chain complexes, $N_*(EH) \subset N_*(EG)$ and $N_*(BH) \subset N_*(BG)$.
We thus have many such inclusions corresponding to our chosen subgroups $\Sigma_2 $'s $\subset V_4 \subset D_8 \subset \Sigma_4$.
If we name an element in some set associated to one of these groups, we will generally use the same name for that element viewed in the similar set associated to a larger group.\footnote{It does not seem all that controversial to call by only one name an element of a subset of various other sets, and it saves substantially on notation.}
\paragraph{0.2.3} The cycle $x_q \times x_p \in N_{q+p}(BV_4) \subset N_{q+p}(BD_8) \subset N_{q+p}(B\Sigma_4)$ corresponds, following \S0.1.4, to a cocycle operation.
We construct an explicit $V_4$-equivariant chain homotopy $J_\Psi \colon N_*(EV_4) \to N_{*+1}(ED_8)$, which induces on coinvariants a chain homotopy $\overline{J}_\Psi \colon N_*(BV_4) \to N_{*+1}(BD_8)$, so that the cycle $x_q \times x_p + \partial \overline{J}_\Psi(x_q \times x_p) \in N_{p+q}(BD_8)$ homologous to $x_q \times x_p$ corresponds to a very specific operation expressed in terms of iterated Squares and $\smallsmile_i$ products of Squares.
The chain homotopy $J_\Psi$ is constructed using the preferred choices of $AW$ and $EZ$ maps, along with Steenrod's definition of the $\smallsmile_i$, and our explicit calculation of $\smallsmile_i$ products in $N^*(B\Sigma_2)$.
Although the main result concerning cocycle operations is first expressed entirely in terms of $\smallsmile_i$ products, it will be more familiar to express it in terms of Square operations on a cocycle $\alpha \in N^n(X)$.
For fixed cocycle degree $n$, the cocycle operation associated to $x_q \times x_p$ has degree $3n - (q+p)$.
Here is a precise statement.
\begin{thm} \label{Adem1}
For a cocycle $\alpha \in N^n(X)$ we have
\begin{equation*}
(x_q \times x_p)( \alpha) + d(\overline{J}_\Psi (x_q \times x_p)(\alpha))\, =\
\sum_\ell\ \binom{p-\ell}{p-2\ell} Sq^{2n-q-\ell}Sq^{n - p +\ell} (\alpha) + d (N_{q,p,n}(\alpha)),
\end{equation*}
where
\begin{equation*}
N_{q,p,n}(\alpha) \ =
\sum_{\substack{0< a\le \ell \in \mathbb{Z}[1/2];\ \\ a\equiv \ell\ mod\ \mathbb{Z}}}\binom{p-\ell-a}{p-2\ell} \binom{p-\ell +a}{p-2\ell}Sq^{n -p+\ell +a}(\alpha)\ \smallsmile_{q-p+2\ell+1}\ Sq^{n -p + \ell -a}(\alpha).
\end{equation*}
By symmetry, we also have an explicit formula
\begin{equation*}
(x_p \times x_q)(\alpha) + d (\overline{J}_\Psi(x_p \times x_q)(\alpha))\, =\
\sum_\ell\ \binom{q-\ell}{q-2\ell} Sq^{2n-p-\ell}Sq^{n - q +\ell} (\alpha) \ + \
d (N_{p,q,n}(\alpha)).
\end{equation*}
\end{thm}
In these formulae, $d$ is the cochain coboundary in $N^*(X)$.
The existence of the homotopies $J_\Psi$ and $\overline{J}_\Psi$ is a special case of a general result, Theorem~\ref{t:homotopy for classifying spaces}, proved in \S1.3.1.
A general formula for these homotopies is given in Theorem~\ref{t:homotopy for iota and Psi}.
Study of the underlying $V_4$-equivariant map in our special case, $\Psi \colon N_*(EV_4) \to N_*(ED_8)$, is carried out in the subsections of Section 3.
This is where our explicit calculation of Steenrod's map $\widetilde{AW}_{\!\!\Delta} \colon N_*(E\Sigma_2) \otimes N_*(B\Sigma_2) \to N_*(B\Sigma_2) \otimes N_*(B\Sigma_2)$ gets used.
\paragraph{0.2.4} The purely cohomological part of Theorem~\ref{Adem1}, that is, the terms involving compositions of Squares, was established by Adem.
But the proof was not easy, and was rather hidden by his use of chain homotopies that were not made explicit.
Our proof, even at the cochain level with coboundary terms, is quite transparent once the $\smallsmile_n$ products in $B\Sigma_2$ are computed.
The binomial coefficients in Theorem~\ref{Adem1} are essentially just repetitions of the binomial coefficients in Theorem~\ref{binom1} with different values of $i,j,n$.
\paragraph{0.2.5} We have observed in \S0.2.2 that conjugation by $(23) \in \Sigma_4$ interchanges our two chosen $\Sigma_2$'s in $V_4 \simeq \Sigma_2 \times \Sigma_2$.
But an inner automorphism $c_g$ of a group $G$ gives rise to a very specific equivariant homotopy $J_g$ on $EG$ that induces a homotopy $\overline{J}_g$ between $\operatorname{Id} \colon BG \to BG$ and $Bc_g \colon BG \to BG$.
In our case, we then have a formula at the cycle level$$x_q \times x_p + x_p \times x_q = \partial \overline{J}_{(23)} (x_q \times x_p) \in N_*(B\Sigma_4).$$
For $g\in G$, we give the general formula for $ J_g$ in \S1.3.3.\\
We apply this last equation to a cocycle $\alpha$ and add that equation to the two equations in Theorem~\ref{Adem1}.
The result is the following cochain level formula:
\begin{thm} \label{Adem2}
For a cocycle $\alpha \in N^n(X)$ we have
\begin{multline*}
\sum_\ell \binom{q-\ell}{q-2\ell} Sq^{2n-p-\ell} Sq^{n-q+\ell} (\alpha) \ + \
\sum_\ell \binom{p-\ell}{p-2\ell} Sq^{2n-q-\ell} Sq^{n-p+\ell} (\alpha) \\ = \
d \Big( (\overline{J}_\Psi (x_p \times x_q)(\alpha) +
N_{p, q,n}(\alpha) +
\overline{J}_\Psi (x_q \times x_p)(\alpha) +
N_{q, p, n}(\alpha) +
\overline{J}_{(23)} (x_q \times x_p)(\alpha) \Big).
\end{multline*}
\end{thm}
And there you have it, a very specific cocycle/coboundary formulation of Adem relations.\\
One interesting thing about the equivariant chain homotopies $J_\Psi$ and $J_{(23)}$ is that they are essentially given by exactly the same kind of formula, arising from a rather general common underlying situation.
Specifically, we mentioned above that any two equivariant maps between contractible free $G$ complexes are equivariantly homotopic.
But with preferred models there is even a canonical way to choose equivariant homotopies.
This will all be explained in due course.\\
Of course to fully explain Theorems~\ref{Adem1} and~\ref{Adem2}, we need to clarify exactly how elements of $N_*(B\Sigma_4)$ act on cocycles.
This will be accomplished using operad methods, which we begin discussing in Subsection~\ref{Operads}.
\paragraph{0.2.6} We want to emphasize that the overall structure of our proof of the Adem relations follows very closely the original proof of Adem \cite{1adem}.
As mentioned previously, Adem proved the cohomological part of Theorem~\ref{Adem1}.
Our combinatorial result Theorem~\ref{binom1} provides cochain details Adem lacked.
In addition, in his proof of the cohomological part of Theorem~\ref{Adem1}, Adem did not need an explicit $V_4$-equivariant chain homotopy like our $J_\Psi$, only that such a thing existed.
Finally, Adem also used the inner automorphism by the element $(23) \in \Sigma_4$ to show that the elements $x_q \times x_p$ and $x_p \times x_q$ that are distinct in the homology of $V_4$ map to the same element in the homology of $\Sigma_4$, hence differ by a coboundary and define the same cohomology operation.
He did not have the structured method using operads to generate specific coboundary formulae, but just the fact that a cohomology operation is determined by a homology class in the symmetric group surely corresponded to some theoretically possible behind the scenes explicit coboundary computations.
\paragraph{0.2.7} The relations in Theorem~\ref{Adem2} do not look like the usual Adem relations that express inadmissible compositions $Sq^aSq^b$ of Steenrod Squares, with $ a < 2b$, as sums of admissible compositions.
The easiest way to recover the usual Adem relations is to use the fact that the Steenrod Squares are stable cohomology operations, that is, they commute with cohomology suspension.
One looks at a high suspension $s^N[\alpha] \in H^{n+N}(S^N \wedge X)$ and identifies one of the relations for degree $n+N$ cocycles in Theorem~\ref{Adem2} so that the binomial coefficients give exactly the usual Adem relation for $Sq^aSq^b(s^N[\alpha)].$\footnote{We found this trick in some course notes of J. Lurie \cite{13lurie}, although probably goes back to Adem.} We include this argument in an appendix.
\paragraph{0.2.8} Theorem~\ref{Adem2} produces, for each cocycle degree $n$ and pair $(p, q)$, a relation between Steenrod operations of total degree $3n - (p+q)$.
With $n$ and the sum $p+q$ fixed, these relations are highly redundant.
We do not see how to use linear combinations of these relations to single out a preferred coboundary formula for a given Adem relation on degree $n$ cocycles.
The desuspension trick of \S0.2.7 also does not single out preferred coboundary formulae for Adem relations.
Of course two different coboundary expressions for a fixed relation differ by a cocycle, related to the indeterminacy of secondary cohomology operations, and that might be something interesting to pursue.
\subsection{Operads} \label{Operads}
\paragraph{0.3.1} In the early 2000's the Surj operad was introduced, along with actions on tensors of cochains generalizing Steenrod's $\smallsmile_n$ operations.
The Surj operations are multivariable cochain operations, computed as sums of products of evaluations of different cochains on faces of simplices (coface operations).
The sums are parameterized by certain diagrams.
We review the sum over diagrams formulae for the $\smallsmile_n$ in Subsection 2.2 and for the more general Surj operations in \S4.3.2.\\
The results of this paper lead to computer algorithms for producing explicit Surj operad formulae that can be used to define cochains whose boundaries are Adem relations, in the form stated in Theorem~\ref{Adem2}.
The formulae quickly involve sums of a very large number of terms as the degrees of the relations and the cocycles increase.
We do believe that it is a theoretical advance to have the better understanding of Adem relations that we establish in this paper.
But it has limited practical use.
\paragraph{0.3.2} We will briefly review how operad methods give rise to an action of $N_*(E\Sigma_n)$ as multivariable cochain operations.
We take this up in greater detail in Section~\ref{sect4}.
Given a space $X$, we make use of three operads and operad morphisms\footnote{Strictly speaking we only use the ${\mathbb F}_2$ coefficient versions of these operads.}
$${\mathcal E} \xrightarrow{TR} {\mathcal S} \xrightarrow{Eval} End(N^*(X)).$$
The first is the Barratt-Eccles operad, with ${\mathcal E}_n = N_*(E\Sigma_n)$, the normalized chains on the classical MacLane model of a contractible $\Sigma_n$ simplicial set.
The second operad is the Surj operad, also called the Step operad or the Sequence operad, whose construction and action on tensors of cochains directly generalizes Steenrod's original construction of the $\smallsmile_i$ products.
The third operad $End(C^*)$ for a cochain complex $C^*$ has operad components $End(C^*)_n = \ \mathrm{Hom}((C^*)^{\otimes n}, C^*)$, the multilinear cochain operations.
We remark that our constructions are applicable to any Barratt-Eccles algebra, i.e., a chain complex $A$ together with an operad morphism ${\mathcal E} \to End(A)$.
The Adem cochains described in this paper are in this general case only as constructive as the morphism defining the algebra structure.
\paragraph{0.3.3} The operad ${\mathcal E}$ was introduced in \cite{2barratteccles}, and studied in great detail by Berger and Fresse \cite{3bergerfresse}, \cite{4bergerfresse}.
The Surj operad ${\mathcal S}$ was more or less simultaneously introduced by Berger and Fresse and by McClure and Smith \cite{18mccluresmith}.
The operad components ${\mathcal S}_n$ are free acyclic $\Sigma_n$ chain complexes with ${\mathbb F}_2$ basis in degree $k$ named by surjections $s \colon\{1, 2, \ldots, n+k\} \to \{1, 2, \ldots, n\}$ with $s(i) \not= s(i+1)$, all $i$.
Each such surjection gives rise to a multivariable cochain operation $N^*(X)^{\otimes n} \to N^*(X)$ that lowers total degree by $k$, (when cochains are graded positively), and these operations define the operad morphism $Eval$.
When $n = 2$, the surjection named $(1212 \ldots)$ with $2+k$ entries corresponds to Steenrod's operation $\smallsmile_k$.
The operad morphism $TR$ is called table reduction and was introduced by Berger and Fresse as a way to explain how simplices in $E\Sigma_n$, which can be regarded as tables consisting of some number of permutations, act on tensors of cochains, via the Surj operad.
The important thing for us is that our chain homotopies $J_\Psi \colon N_*(EV_4) \to N_{*+1}(E\Sigma_4)$ and $J_{(23)} \colon N_*(E\Sigma_4) \to N_{*+1}(E\Sigma_4)$ take values in ${\mathcal E}$.
But by $TR$ they are pushed over to ${\mathcal S}$, and then to cochain operations.
We cannot see the needed chain homotopies directly in ${\mathcal S}$.
\paragraph{0.3.4} In the paper \cite{19medinacartan}, the second author carried out a program, somewhat similar to the joint program here, for finding an explicit coboundary formula implying the Cartan product formula for Steenrod Squares.
In fact, both that work and the work in this paper originated when we were working on the paper \cite{7brumfielmorgan} and needed a specific coboundary formula for the relation $Sq^2([a]^2) = (Sq^1[a])^2$ for a cocycle $a$ of degree 2, which is simultaneously a Cartan relation and an Adem relation.
Since that time, we have figured out our much more structured explanations of both the Cartan formula and the Adem relations in general for the mod 2 Steenrod algebra.
\paragraph{0.3.5}
We did apply our computer algorithm to Theorem~\ref{Adem2} with $(q, p) = (4,1)$ to find a formula $dx(\alpha) = Sq^2Sq^2(\alpha) + Sq^3Sq^1(\alpha)$ for a degree 3 cocycle $\alpha$.
The class $x(\alpha)$ desuspends to the class we used in \cite{7brumfielmorgan} for the corresponding Adem relation in one lower dimension.\\
In general, Theorem~\ref{Adem2} with $(q, p) = (4,1)$ ultimately produces, for each cocycle $\alpha$ of degree $n$, a formula writing a certain sum of Steenrod operations of total degree $3n - 5$ applied to $\alpha$ as the coboundary of a cochain $x(\alpha)$ of degree $4n-6$.
The Surj formula used to define the cochain $x(\alpha)$ is a sum of 116 surjections $\{1, 2, \ldots, 10\} \to \{1,2,3,4\}$ applied to the multi-tensor $\alpha^{\otimes 4}$, via the operad morphism $Eval$.
Each such surjection term gives a sum of coface operations parameterized by certain diagrams.
The number of diagrams, but not the surjections themselves, depends on $n$.\\
When $n = 3$, the diagrams for the Surj formula for the degree 6 cochain $x(\alpha)$ all yield 0, except for 26 of the 116 surjections $\{1, 2, \ldots, 10\} \to \{1,2,3,4\}$.
The number of diagrams, or coface expressions, associated to these 26 surjections is around 100.
So the coface operation formula for $x(\alpha)$ with $dx(\alpha) = Sq^2Sq^2(\alpha) + Sq^3Sq^1(\alpha)$ is quite complicated, even for cocycles $\alpha$ of degree 3.
\paragraph{0.3.6} The paper \cite{9chataurlivernet} also makes use of operad methods to study actions of the mod 2 Steenrod algebra.
Those authors' goal was not to produce explicit coboundary formulae for the Cartan and Adem relations, but rather to develop a universal operadic treatment of contexts in which the Steenrod algebra acts, or equivalently, in which the Cartan and Adem relations hold for some family of operations, somewhat similar to what May carried out in his paper \cite{17maygeneeral}.\\
We believe our work is related to results that connect operads and higher structures on cochain algebras to homotopy theory, as in \cite{14mandellpadic}, \cite{15mandellinteger}, \cite{16mayoperads}, \cite{22smith}.
It is clear that our proofs of Theorems \ref{Adem1} and \ref{Adem2}, especially the construction of the equivariant chain map $\Psi\colon N_*(EV_4) \to N_*(E\Sigma_4)$ and the equivariant chain homotopy $J_\Psi\colon N_*(EV_4) \to N_{*+1}(E\Sigma_4)$ between $\Psi$ and the inclusion map, bring into the open certain useful structure inside the Barratt-Eccles operad that has perhaps not previously been noticed.
\section*{Acknowledgment}
We thank the referee for their careful reading of this work and for many valuable suggestions improving its exposition.
\section{Simplicial Sets and Classifying Spaces}
In this section we review some basic facts about simplicial sets, their chain and cochain complexes, and some facts about classifying spaces $BG$ for groups arising from acyclic spaces $EG$ with free $G$ actions.
We also explain how using the join operation, the Eilenberg-Zilber and Alexander-Whitney maps leads to explicit chain homotopies between maps whose target is the chains on $EG$.
\subsection{Simplicial Sets and Normalized Chain Complexes}
\paragraph{1.1.1} Recall that a simplicial set $X$ consists of a collection of sets $\{X_n\}_{n \geq 0}$ indexed by the natural numbers, together with face and degeneracy operators.
If $X$ is a simplicial set and $\sigma$ is in $X_n$, the set of ``$n$-simplices" of $X$, then there is a unique simplicial map $s \colon \Delta^n \to X$ sending the top-dimensional simplex to $\sigma$, where $\Delta^n$ is the $n$-simplex, regarded as a simplicial set with standard face and degeneracy operators.
In particular, the vertices of $\Delta^n$ are $\{0 < 1 < \cdots < n\}$ and the elements of $\Delta^n_k$ are non-decreasing sequences $(i_0, i_1, \ldots, i_k)$ of vertices.
The degenerate simplices are those that repeat at least one vertex.
The degenerate $n$-simplices $\sigma \in X_n$ are images of degenerate $n$-simplices of $\Delta^n$ under the associated maps $s \colon \Delta^n \to X$.
Products of simplicial sets are simplicial sets with $(X \times Y)_n = X_n \times Y_n$, with the product face and degeneracy operators.
We sometimes use the word space for a simplicial set.
Maps of spaces always means maps of simplicial sets.
\paragraph{1.1.2} Naturally associated to simplicial sets are various chain and cochain complexes, which have homology and cohomology groups.
For any simplicial set $X$, let $N_*(X)$ denote the chain complex of normalized chains with ${\mathbb F}_2$ coefficients.
Thus in degree $n$, $N_n(X)$ is the ${\mathbb F}_2$ vector space with basis the set of $n$-simplices $X_n$, modulo the subspace generated by the degenerate simplices.
\paragraph{1.1.3} The boundary operator $\partial$ on $N_*(X)$ is defined on a basic $n$-simplex to be the sum of the codimension one faces.
Given $\sigma \in X_n$ there is the unique simplicial map $s\colon \Delta^n \to X$ with $s(\Delta^n) = \sigma$.
We can thus define the boundary formula universally by $$\partial(i_0,i_1, \ldots, i_n) = \sum_{j=0}^n\ \partial_j(i_0, i_1, \ldots, i_n) = \sum_{j=0}^n\ (i_0, \ldots \widehat{i_j} \ldots, i_n),$$ where $\widehat{i_j} $ means $i_j$ is deleted.
It is easy to check that if $\sigma$ is a degenerate simplex then $\partial(\sigma) = 0$.
\paragraph{1.1.4} The normalized cochains are defined as the dual chain complex $N^*(X) = Hom(N_*(X), {\mathbb F}_2)$, where we regard ${\mathbb F}_2$ as a chain complex concentrated in degree 0.
Cochains thus lower degree, so the natural grading then is to regard $N^*(X)$ as a chain complex concentrated in negative degrees, and this will be our convention.\footnote{Negative means $\leq 0$ and strictly negative is $< 0$.
The negative grading of cochains, which is the correct way to do it, does require some extra thought at times by those accustomed to positive grading.} Since cochains are in negative degrees, the adjoint coboundary operator $d$ on $N^*(X)$, also lowers degree by one.
So everything is a chain complex, and cochain complex just means a chain complex concentrated in negative degrees.
\paragraph{1.1.5} The Eilenberg-Zilber functorial chain map $EZ \colon N_*(X) \otimes N_*(Y) \to N_*(X \times Y)$ with ${\mathbb F}_2$ coefficients will play a prominent role in the paper.
By naturality, the general definition follows from the case $X= \Delta^n,\ Y = \Delta^m$.
In this case, we simply triangulate the prism $\Delta^n \times \Delta^m$ as the simplicial complex underlying the product of posets $(0 < 1< \cdots < n) \times (0 < 1< \cdots < m)$.
The map $EZ$ takes the tensor product of the universal simplices of dimensions $n$ and $m$ to the sum of the maximal dimension simplices of the product space.
These $n+m$ dimensional simplices correspond to strictly increasing vertex sequences in the product poset order, $(00 = i_0j_0 < i_1j_1 < \cdots < i_{n+m}j_{n+m} = nm)$, where at each step one of the indices increases by 1 and the other is unchanged.
One can prove that $EZ$ is a chain map either by working directly with these simplices or by thinking geometrically and using $\partial (\Delta^n \times \Delta^m) = \partial(\Delta^n) \times \Delta^m \cup \Delta^n \times \partial(\Delta^m)$.
We are working with ${\mathbb F}_2$ coefficients throughout, so orientation signs are irrelevant.\\
In the case $m = 1$, denote by $EZ_h$ the composition $EZ_h(z) = EZ(z \otimes (0,1))$ displayed below,
\begin{equation*}
N_*(Z) \xrightarrow{\operatorname{Id} \otimes (0,1)} N_*(Z ) \otimes N_1(\Delta^1) \xrightarrow{EZ} N_{*+1}(Z \times \Delta^1).
\end{equation*}
Then $EZ_h$ is a (classical) universal chain homotopy between top and bottom faces of cylinders,
\begin{equation*}
(EZ_h \circ \partial + \partial \circ EZ_h) (z) = (z, (1, \ldots, 1)) + (z, (0, \ldots, 0)) \in N_*(Z \times \Delta^1).
\end{equation*}
By naturality, it suffices to verify this when $Z = \Delta^n$ and $z = (0, 1, \ldots, n)$.
Since $EZ$ is a chain map, we have
\begin{equation*}
\partial \circ EZ(z \otimes (0,1)) = EZ(\partial z \otimes (0,1)+ z \otimes 1 + z \otimes 0))
\end{equation*}
and the result is clear.\\
On the other hand, to make this more precise, one can write out sums for $EZ_h(z)$ and $\partial \circ EZ_h(z)$ in the chain homotopy formula.
The $n+1$ dimensional simplices of the convex prism $\Delta^n \times \Delta^1$ are joins of simplices on the bottom and top faces,
\begin{equation*}
(00,10, \ldots, j0) * (j1, \ldots, n1) = (00, \ldots, j0, j1, \ldots, n1).
\end{equation*}
Then $EZ_h(z ) = \sum_j (00,10, \ldots, j0) * (j1, \ldots, n1)$ and $\partial EZ_h(z) = \sum_j \partial \big( (00,10, \ldots, j0) * (j1, \ldots, n1) \big)$.
Since the join operation on simplices just writes the vertices of one simplex after those of another it is easy to understand the boundary of the join of two simplices,
\begin{equation*}
\partial (a * b) = \partial a * b + a * \partial b + \epsilon(a) \, b + a \, \epsilon(b),
\end{equation*}
where on the right the join operation is extended bilinearly and where $\epsilon \colon N_*(\Delta^n)\to {\mathbb F}_2$ is the linear map defined by $\epsilon(v) =1$ for vertices and is $0$ for positive degree simplices.\\
The join operation on simplices extends to a linear map of degree one $* \colon N_*(\Delta^n) \otimes N_*(\Delta^n) \to N_*(\Delta^n)$, and the boundary formula above holds for all $a \otimes b$.
It follows immediately that
\begin{equation*}
(\partial * + * \partial )(a \otimes b) = \epsilon(a) b + a \epsilon (b).
\end{equation*}
The functorial Alexander-Whitney diagonal chain map $AW_{\Delta} \colon N_*(Z) \to N_*(Z) \otimes N_*(Z)$ will also play a prominent role.
Again by naturality it suffices to consider $Z = \Delta^n$ and $z = (0, \ldots, n)$.
The formula is
\begin{equation*}
AW_{\Delta}(0, \ldots n) = \sum_j \ (0, \ldots j) \otimes (j, \ldots n).
\end{equation*}
For arbitrary simplicial sets $Z$ this gives the well-known sum of front faces of simplices tensored with back faces.
A direct computation shows that $AW_{\Delta}$ is a chain map.
\paragraph{1.1.6} Consider a connected simplicial set $X$ for which a degree one join operation $* \colon N_*(X) \otimes N_*(X) \to N_*(X)$ has meaning, and satisfies the boundary formula $\partial (a \ast b) = \partial a \ast b + a \ast \partial b + \epsilon(a)\, b + a\, \epsilon(b)$.
Suppose we have two degree zero chain complex morphisms $\phi_0, \phi_1 \colon N_*(Z) \to N_*(X)$ for some simplicial set $Z$.
Given a simplex generator $z \in Z_n$, let $^j \partial z$ mean front faces and let $\partial^{n-j}z$ mean back faces, of degrees $j$ and $n-j$.
Consider
\begin{equation*}
J_\Phi (z) = \sum_j \ \phi_0(^j\partial z) * \phi_1 (\partial^{n-j} z) \in N_{*+1}(X).
\end{equation*}
\begin{lem} \label{l:join for homotopies}
If $Z$ is connected and if both $\phi_0$ and $\phi_1$ induce non-zero maps in degree 0 homology, then
\begin{equation*}
(J_\Phi \circ \partial + \partial \circ J_\Phi) (z) = \phi_1(z) + \phi_0(z).
\end{equation*}
\end{lem}
\begin{proof}
We will give two proofs, one exploiting $AW_{\Delta}$ and the other $EZ_h$.
The assumption that both $\phi_0$ and $\phi_1$ induce non-zero maps in degree 0 homology implies that for any vertex $v$ of $Z$ both $\phi_0(v)$ and $\phi_1(v)$ are sums of an odd number of vertices of $X$, hence $\epsilon \phi_0(v) = \epsilon \phi_1 (v) = 1$.\\
For the first proof we notice that the degree one map $J_\Phi$ is the composition
\begin{equation*}
N_*(Z) \xrightarrow{AW_{\Delta}} N_*(Z) \otimes N_*(Z) \xrightarrow {\phi_0 \otimes \phi_1} N_*(X) \otimes N_*(X) \xrightarrow{*} N_*(X).
\end{equation*}
Since the first two maps are chain maps, the boundary formula for $*$ implies
\begin{align*}
(J_\Phi \circ \partial + \partial \circ J_\Phi) (z) & =
(\partial * + * \partial)\circ \phi_0 \otimes \phi_1 \circ AW_{\Delta}(z) \\ & =
\phi_1(z) + \phi_0(z),
\end{align*}
where, denoting $z_0 = \, ^0\partial z$ and $z_n = \partial^0 z$, we used for the last equality the fact that $\epsilon \phi_0(z_0) = \epsilon \phi_1(z_n) = 1$.\\
For the second proof we first define a degree zero chain map $$H_\Phi \colon N_*(Z \times \Delta^1) \to N_*(X)$$ that agrees for any chain maps $\phi_0$ and $\phi_1$ with $(\epsilon \phi_0(z_0)) \phi_1(z)$ on the top copy $Z \times 1$ of the cylinder and with $\phi_0(z) (\epsilon \phi_1(z_n))$ on the bottom copy $Z \times 0$.
Consider an $n$-simplex in $Z \times \Delta^1$, not on the top or bottom, $$(t,u) = (t_0, \ldots, t_\ell, t_{\ell+1}, \ldots, t_n), (0 \ldots, 0, 1, \ldots, 1),$$ where there are $\ell+1$ zeros and $n-\ell$ ones in the $\Delta^1$ factor $u$.
Set
\begin{equation*}
H_\Phi(t,u) = \phi_0 (t_0, \ldots, t_\ell) * \phi_1(t_{\ell+1}, \ldots, t_n) \in N_n(X).
\end{equation*}
Using the join boundary formula, it is an exercise to prove that $H_\Phi$ is indeed a chain map.\\
We then observe that the degree one map $J_\Phi$ is the composition
\begin{equation*}
N_n(Z) \xrightarrow{EZ_h} N_{n+1}(Z \times \Delta^1) \xrightarrow{H_\Phi} N_{n+1}(X).
\end{equation*}
Since $H_\Phi$ is a chain map, with the assumption that $\epsilon \phi_0(z_0) = \epsilon \phi_1(z_n) = 1$ we have again
\begin{align*}
(J_\Phi \circ \partial + \partial \circ J_\Phi) (z) & =
H_\Phi \circ ( EZ_h \circ \partial + \partial \circ EZ_h ) (z) \\ & =
H_\Phi (z, (1, \ldots, 1)) + (z, (0, \ldots, 0)) \\ & =
\phi_1(z) + \phi_0(z)
\end{align*}
as claimed.
\end{proof}
The existence of a join operation on $N_*(X)$ implies $H_*(X; {\mathbb F}_2) \simeq H_*(point; {\mathbb F}_2)$.
We remark that Lemma \ref{l:join for homotopies} can be proven with this much weaker hypothesis, using a different chain homotopy $J_\Phi$.
But the chain homotopy obtained using a join operation leads to the construction of equivariant chain homotopies in the presence of group actions, which will be very important in later sections of the paper.
Another reason the join is important is, as was shown in \cite{medina2020prop1}, that the join together with the Alexander-Whitney diagonal and the augmentation $\epsilon$ define an $E_\infty$ structure on chain complexes of simplicial sets, generalizing the one we describe in Section 4.3 below.
\subsection{Classifying Spaces for Discrete Groups}
Let $G$ be a discrete group.
We review here the MacLane model for a contractible left $G$ simplicial set $EG$ and a classifying simplicial set $BG = G \backslash EG$.
\paragraph{1.2.1} The MacLane model for $EG$ has $(EG)_n$ equal to the set of ordered $(n+1)$-tuples $(g_0, \ldots, g_n)$ of elements of $G$.
The $i^{th}$-face map is given by deleting the entry $g_i$ and the $i^{th}$ degeneracy map is given by repeating $g_i$.
In particular, the degenerate simplices of $EG$ are those sequences that have the same group element as two successive entries.\\
There is a left action $G\times EG\to EG$ given by
\begin{equation*}
g(g_0, \ldots ,g_n)=(gg_0, \ldots, gg_n).
\end{equation*}
This is a free action and the quotient is $BG$, the MacLane model for the \textit{classifying space} for $G$.
We can identify $BG_n$ with $n$-tuples of elements in $G$ by identifying the orbit of $(g_0, \ldots ,g_n)$ with $[g_0^{-1}g_1, \ldots ,g_{n-1}^{-1}g_n]$.
With this representation of elements in $BG_n$ the face maps are given by
\begin{align*}
\partial_0 [h_1, \ldots, h_n] & = [h_2, \ldots, h_n], \\
\partial_i [h_1, \ldots, h_n] & = [h_1, \ldots, h_{i-1}, h_ih_{i+1}, \ldots,h_n], \\
\partial_n [h_1, \ldots, h_n] & = [h_1, \ldots, h_{n-1}].
\end{align*}
The degenerate simplices in $BG$ are those sequences in which at least one entry is the identity element.
\paragraph{1.2.2} This definition of $EG$ is a special case of the classifying space of a category, the category which has the elements of group $G$ as objects and for every pair $g,g'\in G$ a unique morphism between them.
A simple proof that $EG$ is contractible is given by the observation that every element $g \in G$ is a terminal object of the category underlying $EG$.\\
$BG$ is the classifying space of the quotient category of $EG$, the quotient having a single object $*$ with ${\rm Hom}(*,*) = G$ and with the composition operation being the product in $G$.
One can think of the map $EG \to BG$ as the map of classifying spaces associated to the functor between the underlying categories that takes the morphism $(g_0, g_1)$ to the morphism $g_0^{-1}g_1$.
This assignment does preserve compositions since $g_0^{-1}g_2 = (g_0^{-1}g_1)(g_1^{-1}g_2)$.
\subsection{Equivariant Maps at the $EG$ Level}
\paragraph{1.3.1} We will next prove a rather general fact stating that certain pairs of equivariant chain maps are equivariantly chain homotopic.
Actually, results like this are well-known, using acyclic model methods.
But we want explicit equivariant chain homotopies.
Our result will be crucial for the ultimate goal of producing coboundary formulae for Adem relations.
\begin{thm} \label{t:homotopy for classifying spaces}
Consider a group homomorphism $\iota \colon H \to G$ between finite groups, inducing a simplicial map $\iota \colon EH \to EG$, and a chain map $\iota_* \colon N_*(EH) \to N_*(EG)$.
Note that $\iota_*$ is $\iota$-equivariant for the free actions of $H$ on the domain and of $\iota H \subset G$ on the range.
Suppose $\Psi \colon N_*(EH) \to N_*(EG)$ is any $\iota$-equivariant chain map that induces the identity on
\begin{equation*}
H_0(EH) = {\mathbb F}_2 = H_0(EG).
\end{equation*}
Then $\iota_*$ and $\Psi$ are chain homotopic, by an $\iota$-equivariant chain homotopy $J_\Psi \colon N_*(EH) \to N_{*+1}(EG)$.
\end{thm}
\begin{proof}
First we indicate a reason this should be true somewhat different from the usual acyclic model argument.
The equivariant chain map $\Psi$ can be regarded as a zero-cycle in the chain complex
$$\mathrm{Hom}_{{\mathbb F}_2[H]}(N_*(EH), N_*(EG)).$$
Since $N_*(EH)$ is free over the group ring ${\mathbb F}_2[H]$ and $EG$ is contractible,
$$H_0(\mathrm{Hom}_{{\mathbb F}_2[H]}(N_*(EH), N_*(EG))) =
H_0(\mathrm{Hom}(N_*(BH), {\mathbb F}_2)) = {\mathbb F}_2.$$
Thus, $ \mathrm{Hom}(N_*(EH), N_*(EG))$ contains only one non-trivial equivariant homotopy class.
\end{proof}
\paragraph{1.3.2} The explicit equivariant chain homotopy between $\Psi$ and $\iota_*$ that we write down in the next theorem is a special case of the chain homotopies constructed in Lemma~\ref{l:join for homotopies} of \S1.1.6.
Note the join map of simplices extends to a multilinear map $N_j(EG) \otimes N_k(EG) \xrightarrow{*} N_{j+k+1}(EG)$ satisfying $\partial(x*y) = \partial x *y + x * \partial y + \epsilon(x) \, y + x \, \epsilon(y)$.
\begin{thm} \label{t:homotopy for iota and Psi}
With $\iota$ and $\Psi$ as in Theorem~\ref{t:homotopy for classifying spaces}, a canonical equivariant chain homotopy between $\iota_*$ and $\Psi$ is given by the formula
\begin{equation*}
J_\Psi(h_0, h_1, \ldots, h_n) = \sum_j \iota_* ( h_0, h_1, \ldots, h_j) * \Psi(h_j, h_{j+1}, \ldots, h_n).
\end{equation*}
\end{thm}
\begin{proof}
The $\iota$ equivariance of $J_\Psi$ is obvious from the formula defining it.
That $J_\Psi$ defines a chain homotopy is a direct consequence of Lemma~\ref{l:join for homotopies} after noticing that $^j \partial (h_0, \dots, h_n) = (h_0, \dots, h_j)$ and $\partial^j (h_0, \dots, h_n) = (h_j, \dots, h_n)$.
\footnote{We discussed aspects of this proof of Theorem~\ref{t:homotopy for iota and Psi}, perhaps prematurely, in \S1.1.6, because we wanted to pave the way for Theorems~\ref{t:homotopy for classifying spaces} and \ref{t:homotopy for iota and Psi}. One can also prove Theorem~\ref{t:homotopy for iota and Psi} by a lengthy but straightforward direct computation.}
\end{proof}
\paragraph{1.3.3} A special case of Theorem~\ref{t:homotopy for classifying spaces} is the map on chains induced by the right translation map of simplicial sets $EG \to EG$ given by $x \mapsto xg^{-1}$, with $\iota = \operatorname{Id} \colon G \to G$.
In this case, one actually obtains an equivariant homotopy $J_g \colon EG \times \Delta^1 \to EG$ between the identity and the map $(g_0, \ldots, g_n) \mapsto (g_0g^{-1}, \ldots, g_ng^{-1}) = g^{-1}(c_g(g_0), \ldots, c_g(g_n))$, where $c_g$ is conjugation by $g$.
Applying equivariance, these constructions project to a homotopy $\overline{J}_g\colon BG \times \Delta^1 \to BG$ between the identity and the map induced by inner automorphism $c_g$.
The chain homotopy $N_*(EG) \to N_{*+1}(EG)$ produced by Theorem~\ref{t:homotopy for iota and Psi} coincides with the chain homotopy produced by the space level homotopy.
The formula is
\begin{equation*}
J_g(g_0, \ldots, g_{n}) = \sum_j (g_0, \ldots, g_{j}, g_{j}g^{-1}, \ldots, g_{n}g^{-1}).
\end{equation*}
\subsection{The Spaces $E\Sigma_2$ and $B\Sigma_2$}
\paragraph{1.4.1} Of particular importance for us is the group $G = \Sigma_2 = \{1, T\}$.
In $E\Sigma_2$ there are just two non-degenerate simplices in each dimension, which are $\widetilde{x}_p = (T^0, T^1, \ldots, T^p) = (1, T, 1, \ldots, T^p)$ and $T\tilde{x}_p = (T, 1, T, \ldots, TT^p)$.
The (equivariant) boundary in the normalized chain complex is determined by $\partial \tilde{x}_p = T\tilde{x}_{p-1} + \tilde{x}_{p-1}$, since all codimension one faces except the first and last are degenerate.
In the classifying space $B\Sigma_2$ there is a single non-degenerate simplex in each dimension, namely $x_p = [ T, T, \ldots , T]$, which is a cycle.
For normalized cochains, we have the dual basis elements $t_p = x_p^* \in N^{-p}(B\Sigma_2)$ and $\widetilde{t}_p = \tilde{x}_p^*,\ T\widetilde{t}_p = T\tilde{x}_p^* \in N^{-p}(E\Sigma_2)$.
\paragraph{1.4.2} \textbf{REMARK.} We make an observation about the simplices of $E\Sigma_2$ and $B\Sigma_2$ that will be quite important later.
This observation extends the above observation about the chain boundary formula.
In $E\Sigma_2$, if we delete an interior interval, consisting of an odd number of adjacent vertices of a non-degenerate simplex, the result is a degenerate simplex.
If we delete an interior interval consisting of an even number of vertices, or \textit{any} initial or terminal interval of vertices, the result remains a non-degenerate simplex.
\\
It is slightly trickier understanding compositions of face operations in $B\Sigma_2$.
Of course one can always just apply the observations in the paragraph above about degenerate and non-degenerate simplices in $E\Sigma_2$ to the projection $E\Sigma_2 \to B\Sigma_2$.
But one can also reason directly in $B\Sigma_2$.
From the general face operator formulae for $BG$ in \S1.2.1, the first or last basic face operator $\partial_0$ or $\partial_p$ in $B\Sigma_2$ applied to $x_p = [T, T, \ldots, T]$ just deletes a first or last $T$, leaving the non-degenerate $x_{p-1}$.
But an interior basic face operator multiplies two adjacent $T$'s, resulting in a $1$ entry and a degenerate simplex.
If now another adjacent interior face operator is applied, the 1 and a $T$ are multiplied, eliminating the 1 and resulting in the non-degenerate $x_{p-2}$.
Thus a composition of adjacent interior face operators applied to $x_p$ results in a non-degenerate simplex if and only if the number of adjacent interior face operators composed is even.
\section{Cup$_n$ Products}
In the first two subsections of this section we explain Steenrod's explicit enhanced Alexander-Whitney chain map $\widetilde{AW}_{\!\!\Delta} \colon N_*(E\Sigma_2) \otimes N_*(X) \to N_*(X) \otimes N_*(X)$ that is used to define $\smallsmile_n$ products.
Then in Subsection 2.3 we compute the $\smallsmile_n$ products in $N^*(B\Sigma_2)$ and $N^*(E\Sigma_2)$, using some simple combinatorial facts about counting certain kinds of partitions of integers.\footnote{The proofs of the combinatorial facts are deferred to an appendix.} In particular, we prove Theorem~\ref{binom1} of the Introduction.
\subsection{Alexander-Whitney and Steenrod Maps}
\paragraph{2.1.1} We will make heavy use of the classical Alexander-Whitney map
\begin{equation*}
AW \colon N_*(X \times Y) \to N_*(X) \otimes N_*(Y).
\end{equation*}
Simplices of dimension $n$ in a product simplicial set $X \times Y$ are given by pairs $ (X \times Y)_n = X_n \times Y_n$.
The map $AW$ is defined universally on a basic product of simplices $\Delta^n \times \Delta^n$ by
$$AW \big( (0, 1, \ldots, n), (0, 1, \ldots, n) \big) = \sum_{i=0}^n (0, 1, \ldots, i) \otimes (i, i+1, \ldots, n).$$
On a general pair of $n$-simplices $(u,v)$, this yields by naturality the usual sum of front faces of $u$ tensor back faces of $v$.\\
We will call by the name $AW_{\Delta} \colon N_*(X) \to N_*(X) \otimes N_*(X)$ the map which is a chain approximation of the diagonal given by the composition $AW \circ \Delta_*$, where $\Delta_*$ is the chain map associated to the diagonal map of simplicial sets $\Delta \colon X \to X \times X$, $\Delta u = (u, u)$.
The cochain dual of $AW_{\Delta}$ defines the cochain cup product $N^*(X) \otimes N^*(X) \to N^*(X)$.\\
The map $AW$ for products of spaces is associative, hence unambiguously defied for any number of factors.
It is also natural in any number of factors.
The diagonal approximation map $AW_{\Delta}$ for single spaces is natural and (co)associative.
\paragraph{2.1.2} For any space $X$ there is an immensely important enhanced $AW$ diagonal approximation chain map $\widetilde{AW}_{\!\!\Delta} \colon N_*(E\Sigma_2) \otimes N_*(X) \to N_*(X) \otimes N_*(X)$, which is a $\Sigma_2$-equivariant chain map of degree 0.
Here $T \in \Sigma_2$ acts on $N_*(E\Sigma_2)$ in the obvious way, fixes the copy of $N_*(X)$ in the domain, and switches the two copies of $N_*(X) $ in the range.
The map $\widetilde{AW}_{\!\!\Delta}$ was constructed by Steenrod using higher homotopies between the diagonal approximations $AW_{\Delta}$ and $T AW_{\Delta}$, \cite{23steenrodproducts}.
The map $(\widetilde{AW}_{\!\!\Delta})_0 \colon \tilde{x}_0 \otimes N_*(X) \to N_*(X) \otimes N_*(X)$ is a chain map, (since $\tilde{x}_0$ is a cycle), and identifies with $AW_{\Delta}$.
The enhanced diagonal $\widetilde{AW}_{\!\!\Delta}$ is natural for maps $X \to Y$.
A precise construction of $\widetilde{AW}_{\!\!\Delta}$ is given in Subsection 2.2 below.
\paragraph{2.1.3} For $n > 0$, the cochain dual of the map $(\widetilde{AW}_{\!\!\Delta})_n \colon \tilde{x}_n \otimes N_*(X) \to N_*(X) \otimes N_*(X)$ encodes the higher $\smallsmile_n$ product.
To be precise, given cochains $\alpha, \beta$ of degrees $-p, -q$ and a simplex $u$ of dimension $p+q -n$, one has
\begin{equation*}
\langle \alpha \smallsmile_n \beta, \, u) \rangle =
\langle \alpha \otimes \beta, \ \widetilde{AW}_{\!\!\Delta}(\tilde{x}_n \otimes u) \rangle.
\end{equation*}
Since $\widetilde{AW}_{\!\!\Delta}$ is a chain map, it is a cycle of degree 0 in the complex
\begin{equation*}
\mathrm{Hom}(N_*(E\Sigma_2) \otimes N_*(X),\ N_*(X) \otimes N_*(X)).
\end{equation*}
This means $0 = \widetilde{AW}_{\!\!\Delta} \circ \partial + \partial \circ \widetilde{AW}_{\!\!\Delta}$.
Pursuing this, one obtains the coboundary formula for the $\smallsmile_n$ operations,
\begin{equation*}
d (\alpha \smallsmile_n \beta) = d\alpha \smallsmile_n \beta + \alpha \smallsmile_n d \beta + \alpha \smallsmile_{n-1} \beta + \beta \smallsmile_{n-1} \alpha.
\end{equation*}
We can ignore signs since we have ${\mathbb F}_2$ coefficients.
\paragraph{2.1.4} From the coboundary formula, if $\alpha$ and $\beta$ are cocycles then $$d(\alpha \smallsmile_n \beta) = \alpha \smallsmile_{n-1} \beta + \beta \smallsmile_{n-1} \alpha.$$ If $\alpha$ is a cocycle of degree $-i$ then for $0 \leq n \leq i$ the Steenrod Square $Sq^n \alpha = \alpha \smallsmile_{i -n} \alpha$ is a cocycle of degree $-(i + n)$.
Then $Sq^i(\alpha) = \alpha^2$.
If $\beta$ is another cocycle of degree $-i$ then $$Sq^n(\alpha + \beta) = (\alpha + \beta) \smallsmile_{i-n} (\alpha + \beta) = Sq^n(\alpha) + Sq^n(\beta) + d (\alpha \smallsmile_{i - n + 1} \beta),$$ so the $Sq^n$ are linear operations on cohomology classes.
It is easy to see from the direct combinatorial construction of $\widetilde{AW}_{\!\!\Delta}$ given in the next section that $Sq^0(\alpha) = \alpha \smallsmile_i \alpha = \alpha$.\\
Another property of Steenrod Squares that follows easily from Steenrod's direct definition is the commutativity of Squares with cohomology suspension.
The proof of this result seems almost awkward when expressed in terms of cohomology, \cite{13lurie}, \cite{21moshertangora}.
In the papers \cite{6brumfielmorgan} and \cite{7brumfielmorgan} we discovered and exploited the following cochain level formula for the integral version of $\smallsmile_n$ products:
\begin{equation*}
s(x \smallsmile_i y) = (-1)^{deg(x)+i+1}sx \smallsmile_{i+1} sy,
\end{equation*}
where $s$ is cochain suspension.
This obviously implies quite a bit more than just the fact that Steenrod Squares commute with cohomology suspension.\footnote{We believe this to be an unnoticed or under-appreciated formula.}
\paragraph{2.1.5} It is easy to compute cup products in $H^*(B\Sigma_2)$ and $H^*(E\Sigma_2)$.
For any $EG$, we have the $AW$ maps, which are $G$-equivariant,
\begin{align*}
AW( (g_0, \ldots, g_n) \times (h_0, \ldots , h_n) ) & = \sum_{i = 0} ^ n (g_o, \ldots, g_i) \otimes (h_i, \ldots, h_n), \\
AW_{\Delta}(g_0, \ldots, g_n) & = \sum_{i = 0}^n (g_0, \ldots, g_i) \otimes (g_i, \ldots, g_n).
\end{align*}
Reducing mod the $G$ action gives the $AW$ maps for $BG$.\\
In the case $G = \Sigma_2$ one can reason directly with the cells $[T, T, \ldots, T]$ of $B\Sigma_2$.
The result is easily seen to be $AW_{\Delta}(x_p) = \sum_{i+j = p} \, x_i \otimes x_j$.
Applying these formulae to dual cochains in $B\Sigma_2$, we get the cup product formula
$$t_i \smallsmile t_j = t_{i+j} \in N^*(B\Sigma_2).$$
Thus the ring $N^*(B\Sigma_2) = {\mathbb F}_2[t]$, a polynomial ring on one generator $t = t_1$ of degree $-1$.
This ring is also the cohomology ring $H^*(B\Sigma_2)$, since the differential is 0.\\
The $AW$ formulae also reveal the cup products in $N^*(E\Sigma_2)$.
Explicitly,
\begin{align*}
\widetilde{t}_p \smallsmile \widetilde{t}_q & = \widetilde{t}_{p+q} \text{ and } \widetilde{t}_p \smallsmile T \widetilde{t}_q = 0 \text{ if } p \text{ is even}, \\
\widetilde{t}_p \smallsmile \widetilde{t}_q & = 0 \text{ and } \widetilde{t}_p \smallsmile T\widetilde{t}_q = \widetilde{t}_{p+q} \text{ if } p \text{ is odd}.
\end{align*}
The other products in $N^*(E\Sigma_2)$ are determined by $T$-equivariance.
\subsection{Explicit Definition of $\widetilde{AW}_{\!\!\Delta}$}
\paragraph{2.2.1} We now recall Steenrod's explicit cochain formulae for $\smallsmile_n$ products, \cite{23steenrodproducts}.
In fact, we will define Steenrod's map $\widetilde{AW}_{\!\!\Delta} \colon N_*(E\Sigma_2) \otimes N_*(X) \to N_*(X) \otimes N_*(X)$.
A simplex $u \in X_N$ can be viewed as a simplicial map $\Delta^N \to X$, so by naturality it suffices to work on a simplex, $\Delta^N = (0, 1, \ldots, N)$.
Then $\widetilde{AW}_{\!\!\Delta}(\tilde{x}_n \otimes \Delta^N) \in N_*(\Delta^N) \otimes N_*(\Delta^N)$ is a sum of tensor products of various faces of $\Delta^N$.
Subsets $I \subset (0, 1, \ldots , N)$ name the faces of $\Delta^N$.
The sum we want is indexed by a set, ${\rm Diagrams }(N)$, of diagrams consisting of two rows of non-empty intervals of the vertices of $\Delta^N$.
The total number of intervals is $n+2$, thus $I_1, I_2, \ldots, I_{n+2}$.
Every vertex of $\Delta^N$ is in at least one interval.
For each $1 \leq j \leq n+1$, the final vertex of $I_j$ is the initial vertex of $I_{j+1}$, and this describes the only overlaps of the intervals.
We require each interior interval $I_2, I_3, \ldots I_{n+1}$ to contain more than one vertex.\\
We alternate these intervals, with $I_1, I_3, I_5, \ldots$ on the first row and $I_2, I_4, \ldots$ on the second row.
Visualize the intervals by inserting $n+1$ separating bars between vertices of $\Delta^N$, then repeating the vertex after each bar.
Thus $$(I_1 | I_2 | \ldots | I_{n+2}) = ((0, \ldots, k_1) | (k_1, \ldots, k_2 ) | \ldots | (k_{n+1}, \ldots, N)).$$
\paragraph{2.2.2} Here is Steenrod's formula.
$$\widetilde{AW}_{\!\!\Delta}(\widetilde x_n \otimes \Delta^N)\ =\!\! \sum_{{\rm Diagrams }(N)} (I_{odd} \otimes I_{even}), \text{ where } I_{odd} = \bigsqcup_j I_{2j-1} \text{ and } I_{even} = \bigsqcup_j I_{2j}.$$
In terms of a simplex $u \colon \Delta^N \to X$, with faces denoted $u(I)$, the formula is $$\widetilde{AW}_{\!\!\Delta} (\widetilde x_n \otimes u) \ =\!\! \sum_{{\rm Diagrams }(N)} u(I_{odd}) \otimes u(I_{even}) \in N_*(X) \otimes N_*(X).$$
We extend $\widetilde{AW}_{\!\!\Delta}$ equivariantly, defining $\widetilde{AW}_{\!\!\Delta}(T \widetilde x_n \otimes u)$ by switching the order of the tensor product factors in $ \widetilde{AW}_{\!\!\Delta} (\widetilde x_n \otimes u)$.
Given the intervals $I_j$, we could also form a diagram by putting the $I_1, I_3, \ldots$ on the second row and the $I_2, I_4, \ldots$ on the first row.
So the equivariance amounts to a sum over diagrams vs a sum over inverted diagrams.
Of course, it is by no means obvious that $\widetilde{AW}_{\!\!\Delta}$ is a chain map.
But Steenrod proved that it is.\\
Now, given cochains $\alpha, \beta \in N^*(X)$ of degrees $-i, -j$ with $i+j-n = N$, and a simplex $u$ of dimension $N$, Steenrod's definition of the $\smallsmile_n$ product becomes, by duality, the following coface formula:
\begin{equation*}
\langle \alpha \smallsmile_n \beta, u \rangle \ = \!\!
\sum_{{\rm Diagrams }(N)} \langle \alpha, u(I_{odd})\rangle \langle \beta, u(I_{even})\rangle.
\end{equation*}
\paragraph{2.2.3} \textbf{REMARK.} For $1 \leq k \leq n+2$, define $|I_k|$ to be the number of vertices in $I_k$.
For $2 \leq k \leq n+1$ we have $|I_k| \geq 2$.
Possibly $|I_1| = 1$ and/or $|I_{n+2}| = 1$.
Of course a diagram contributes 0 for a pair of cochains of degrees $-i, -j$ unless $i+1 = |I_1| + |I_3| + \cdots$ and $j+1 = |I_2| + |I_4| +\cdots$.
We also notice that if $n = 2m$ is even, then the intervals are $I_1, I_3, \ldots, I_{2m+1}$ and $I_2, I_4, \ldots, I_{2m+2}$.
So there are $m+1$ intervals on the first row and $m+1$ intervals on the second row.
If $n = 2m+1$ is odd then the intervals are $I_1, I_3, \ldots, I_{2m+3}$ and $I_2, I_4, \ldots, I_{2m+2}$.
So there are $m+2$ intervals on the first row and $m+1$ intervals on the second row.
\subsection{Cup$_n$ Products in $E\Sigma_2$ and $B\Sigma_2$}
\paragraph{2.3.1} We will now embark on a calculation of {\it all} $\smallsmile_n$ products in $N^*(B\Sigma_2)$ and $N^*(E\Sigma_2)$.
The computations will make use of some standard combinatorial formulae for counting ordered partitions of positive integers.\\
We fix $i,j,n$ and $N = i+j-n$.
The only non-zero cochains and chains are the dual pairs $t^i = x_i^*$ and $ t^j = x_j^*$.
We distinguish the cases $n = 2m$ even and $n = 2m+1$ odd.
By Remarks 1.4.2 and 2.2.3, in the even case $n = 2m$ the only diagrams that give non-zero evaluations on $t^i \otimes t^j$ are the diagrams for which $i + 1 = |I_1| + |I_3| + \cdots + |I_{2m+1}|$ is a positive partition, with all terms other than $|I_1|$ even, and also for which $j+1 = |I_2| + \cdots + |I_{2m+2}|$ is a positive partition with all terms other than $|I_{2m+2}|$ even.\\
The point here is that on the second row, where $t^j$ will be evaluated, the number of vertices deleted between intervals $I_{2k}$ and $I_{2k+2}$, $k \geq 1$, is $|I_{2k+1}| - 2$.
Thus by Remark 1.4.2, the $j$-face of $x_N$ named $x_N(I_{even})$ is degenerate unless all $|I_{2k+1}|$ are even.
Similarly, on the first row, the $i$-face of $x_N$ named $x_N(I_{odd})$ is degenerate unless all $|I_{2k}|$, $k \leq m$, are even.\\
In the same way by Remarks 1.4.2 and 2.2.3, in the odd case $n = 2m+1$ the only diagrams that give non-zero evaluations are the diagrams for which $i + 1 = |I_1| + |I_3| + \cdots + |I_{2m+3}|$ is a positive partition for which all but the first and last terms are even, and also for which $j+1 = |I_2| + \cdots + |I_{2m+2}|$ is a positive partition with all terms even.
\paragraph{2.3.2} We prove the following facts in an appendix.\\
\textbf{COMBINATORIAL FACT 1.} Let $n = 2m+1$.
The number mod 2 of positive ordered partitions of $i+1$ consisting of $m+2$ terms with all but the first and last even, is the binomial coefficient $\binom {i} {n}$ mod 2.
The number mod 2 of positive ordered partitions of $j+1$ consisting of $m+1$ even terms is the binomial coefficient $\binom {j} {n}$ mod 2.\\
\textbf{COMBINATORIAL FACT 2.} Let $n = 2m$.
The number mod 2 of positive ordered partitions of $i+1$ consisting of $m+1$ summands, all but the first even, is the binomial coefficient $\binom {i} {n}$ mod 2.
Thus also the number mod 2 of positive ordered partitions of $j+1$ consisting of $m+1$ summands, all but the last even, is the binomial coefficient $\binom{j} {n}$ mod 2.
\paragraph{2.3.3} The key consequence of the combinatorial facts and the discussion preceding those statements is the following, which repeats Theorem~\ref{binom1} of the introduction.
\begin{thm} \label{t:binomial repeated}
For $\widetilde{AW}_{\!\!\Delta} \colon N_*(E\Sigma_2) \otimes N_*(B\Sigma_2) \to N_*(B\Sigma_2) \otimes N_*(B\Sigma_2)$, we have the chain formula $$\widetilde{AW}_{\!\!\Delta} (\tilde{x}_n \otimes x_k) \, = \!\! \sum_{i+j = k+n} \binom{i}{n} \binom{j}{n} x_i \otimes x_j.$$ The equivalent cochain formula is $$ t^i \smallsmile_n t^j = \binom{i}{n} \binom {j} {n} t^{i+j- n}.$$
\end{thm}
\paragraph{2.3.4} \textbf{REMARK.} The Steenrod Squares are defined for cocycles $\alpha$ of degree $-i$ by $Sq^m(\alpha) = \alpha \smallsmile_{i-m} \alpha$.
Thus we have computed the Steenrod Squares in $N^*(B\Sigma_2)$ by a direct combinatorial method.
The result is $Sq^m(t^i) = \binom{i} {m} t^{i+m},$ using that $\binom{i}{m} = \binom{i} {i-m}$ and $\binom{i}{m}^2 \equiv \binom{i}{m}$ mod 2.
The usual proof of the formula for Steenrod Squares in real projective space uses the Cartan formula.
We have avoided the Cartan formula, and, moreover we have computed all $\smallsmile_n$ products in a model of real projective space.
\paragraph{2.3.5} We can also calculate the $\smallsmile_n$ products in $N^*(E\Sigma_2)$.
The method is the same, based on Remarks 1.4.2 and 2.2.3, and the combinatorics of counting partitions.
A new wrinkle arises in the $E\Sigma_2$ case dealing with the first interval $I_1$ in the diagrams for computing $\smallsmile_n$.
In evaluating a $\smallsmile_n$ product diagram on a cell $(1, T, 1 , \dots)$, the parity of $|I_1|$ determines whether the face to be evaluated on the second row begins with $1$ or $T$.
If $|I_1|$ is even, the second row begins with $T$.
But we know how to count the appropriate partitions mod 2 when $|I_1|$ is even, and also when $|I_1|$ is arbitrary.
So the case $|I_1|$ odd will be the difference, or sum, of those numbers.
Here is the result.
\begin{thm} \label{t:2.2}
For $\widetilde{AW}_{\!\!\Delta} \colon N_*(E\Sigma_2) \otimes N_*(E\Sigma_2) \to N_*(E\Sigma_2) \otimes N_*(E\Sigma_2)$ we have the chain formula
\begin{equation*}
\widetilde{AW}_{\!\!\Delta}(\tilde{x}_n \otimes \tilde{x}_k)\ =\!
\sum_{i+j = k+n} \binom{i+1}{n+1} \binom{j}{n} \ \tilde{x}_i \otimes \tilde{x}_j \ + \
\binom{i}{n+1}\binom{j}{n} \ \tilde{x}_i \otimes T \tilde{x}_j.
\end{equation*}
A formula that includes all four evaluation cases for $N_*(E\Sigma_2)$ is given for $b, a \in \{0, 1\}$ by
\begin{equation*}
\widetilde{AW}_{\!\!\Delta}(T^b \tilde{x}_n \otimes T^a\tilde{x}_k)\ = \sum_{\epsilon = 0, 1}\ \ \sum_{i+j = k+n} c_{n,k,i}^\epsilon \ S^b(T^{a} \tilde{x}_i \otimes T^{a+\epsilon}\tilde{x}_j ),
\end{equation*}
where
\begin{equation*}
c_{n,k,i}^0 = \binom{i+1}{n+1}\binom{j}{n}
\quad \text{and} \quad
c^1_{n,k,i} = \binom{i}{n+1}\binom{j}{n},
\end{equation*}
and where $S$ switches the tensor factors.
\end{thm}
\begin{proof}
The first statement will be proved in the combinatorial appendix.
The remaining evaluations are determined by equivariance.\\
For example, applying $T$ to the second variable $\tilde{x}_k$, one uses naturality of $\widetilde{AW}_{\!\!\Delta} \colon N_*(E\Sigma_2) \otimes N_*(X) \to N_*(X) \otimes N_*(X)$ in $X$.
Applying $T$ to the first variable $\widetilde x_n$, one uses the equivariance that applies the operator $S$ interchanging the two factors in the range.
\end{proof}
Note that we have
\begin{equation*}
\binom{i}{n+1} + \binom{i+1}{n+1} = \binom{i}{n+1} + \binom{i}{n} + \binom{i}{n+1} \equiv \binom{i}{n} \pmod{2},
\end{equation*}
in agreement with Theorem~\ref{t:binomial repeated}.
\section{The Chain Maps Phi and Psi}
The main results Theorem~\ref{Adem1} and Theorem~\ref{Adem2} are proved in this section, modulo explaining the operad method for extending Steenrod's $\smallsmile_n$ operations to multi-variable cochain operations.\footnote{The operad discussion is carried out in Section 4.} The key steps in the proofs of Theorem~\ref{Adem1} and Theorem~\ref{Adem2} amount to constructing some chain homotopies with target various complexes $N_*(EG)$, arising from certain equivariant chain maps.
The needed equivariant chain maps are constructed in Subsections 3.1 and 3.2.
The maps they induce on chain complexes are computed in Subsections 3.3 and 3.4.
The formulae for these chain maps in Corollary~\ref{cor3.2} and Corollary~\ref{cor3.3} look somewhat complicated, but do not really involve anything more than the classical Alexander-Whitney diagonal map and the computation of the $\smallsmile_n$ products in $N_*(E\Sigma_2)$.\\
From Subsection 1.3, certain pairs of equivariant chain maps are connected by canonical chain homotopies.
Theorem~\ref{t:3.4}, Corollary~\ref{cor3.5} and Corollary~\ref{cor3.6} in Subsection 3.4 summarize these chain homotopy formulae in our special cases.
Then in Subsection 3.5 we summarize how these results imply Theorem~\ref{Adem1} and Theorem~\ref{Adem2}, modulo explaining how operad methods produce actions of various complexes $N_*(EG)$ on multi-tensors of cochains.
\subsection{Some Dihedral Group Actions}\label{sect3.1}
\paragraph{3.1.1} Let $D_8$ be the dihedral group of order 8, with generators $a, b, c$.
These generators are all of order 2; $[b,c] = 1$; $ab = ca$; and $ac = ba$.
These are the relations that hold for the inclusion $D_8 \subset \Sigma_4$ given by $b = (12)$, $c = (34)$, $a = (13)(24)$.
Note that $V_4 \subset D_8$ is the subgroup generated by $\{a, bc\}$.\\
We define a left action of $D_8$ on $ N_*(E\Sigma_2)\otimes N_*(E\Sigma_2)\otimes N_*(E\Sigma_2)$ as follows.
Let $T$ be the generator of $\Sigma_2$ with its natural action on $ N_*(E\Sigma_2)$.
Then
\begin{align*}
a(x \otimes y \otimes z) & = Tx \otimes z \otimes y, \\
b(x \otimes y \otimes z) & = x \otimes Ty \otimes z, \\
c(x \otimes y \otimes z) & = x \otimes y \otimes Tz.
\end{align*}
One checks easily that these equations define a group action.
Since $N_*(E\Sigma_2)$ is an acyclic chain complex of free ${\mathbb F}_2[\Sigma_2]$-modules, it is easy to see that this action makes $N_*(E\Sigma_2) \otimes N_*(E\Sigma_2) \otimes N_*(E\Sigma_2)$ an acyclic chain complex of free ${\mathbb F}_2[D_8]$-modules.
\paragraph{3.1.2} Next, since $EZ$ is an associative operation on products of spaces, there is a well-defined Eilenberg-Zilber map
\begin{equation*}
EZ\colon N_*(E\Sigma_2)\otimes N_*(E\Sigma_2)\otimes N_*(E\Sigma_2) \to N_*(E\Sigma_2 \times E\Sigma_2 \times E\Sigma_2).
\end{equation*}
We also give the range of this map a $D_8$ action.
In fact, $D_8$ acts on the space $E\Sigma_2\times E\Sigma_2\times E\Sigma_2$ by the same formulae on cell triples $(x, y, z)$ of the same degree as the above formulae on basic tensor triples.
The space $E\Sigma_2 \times E\Sigma_2 \times E\Sigma_2$ is then a contractible, free $D_8$ space.
The map $EZ$ is equivariant with respect to the two actions.
The Alexander-Whitney map
\begin{equation*}
AW \colon N_*(E\Sigma_2\times E\Sigma_2\times E\Sigma_2) \to N_*(E\Sigma_2)\otimes N_*(E\Sigma_2)\otimes N_*(E\Sigma_2)
\end{equation*}
is equivariant for the actions of $b$ and $c$, but not for the action of $a$.
\paragraph{3.1.3} The action of $D_8$ on the product space $E\Sigma_2\times E\Sigma_2\times E\Sigma_2$ is determined by the action on the vertices $\Sigma_2 \times \Sigma_2 \times \Sigma_2$, extended coordinate-wise to
\begin{equation*}
(E\Sigma_2\times E\Sigma_2\times E\Sigma_2)_n \simeq (\Sigma_2\times \Sigma_2\times \Sigma_2)^{n+1}.
\end{equation*}
The free left action of $D_8$ on vertices identifies the group $D_8$ with the set $\Sigma_2 \times \Sigma_2 \times \Sigma_2$, viewed as the $D_8$-orbit of $(1, 1, 1)$.
This identification determines a multiplication on $\Sigma_2 \times \Sigma_2 \times \Sigma_2$, making this set of triples into a group isomorphic to $D_8$.
Moreover, with this group structure and the coordinatewise action of $D_8$ on $n$-simplices of $E(\Sigma_2 \times \Sigma_2 \times \Sigma_2) \simeq E\Sigma_2\times E\Sigma_2\times E\Sigma_2$, we see that in fact we have defined an isomorphism of simplicial sets $E\Sigma_2 \times E\Sigma_2 \times E\Sigma_2 \simeq ED_8$, as free $D_8$ complexes.\\
The actual formula for the induced group product on triples $\Sigma_2 \times \Sigma_2 \times \Sigma_2$ is somewhat tricky.
We can name the triples $(T^{\epsilon_1}, T^{\epsilon_2}, T^{\epsilon_3})$, with $\epsilon_j \in \{0, 1\}$.
Such a triple is identified with the element $c^{\epsilon_3} b^{\epsilon_2}a^{\epsilon_1} \in D_8$.
This is true because evaluating that $D_8$ element on $(1,1,1)$ indeed yields $(T^{\epsilon_1}, T^{\epsilon_2}, T^{\epsilon_3})$.\\
To multiply triples, one simply computes products $c^{\delta_3}b^{\delta_2}a^{\delta_1} c^{\epsilon_3} b^{\epsilon_2} a^{\epsilon_1} \in D_8$, using the relations $ac = ba, ab = ca, bc = cb$ to move $a^{\delta_1}$ across $c^{\epsilon_3}b^{\epsilon_2}$ and to commute $b$'s and $c$'s.
The result will then have the form $c^{\gamma_3}b^{\gamma_2}a^{\gamma_1},$ which translates to the product of triples.\footnote{This product on triples can be viewed as a semi-direct product multiplication on $D_8 = \Sigma_2 \ltimes (\Sigma_2 \times \Sigma_2)$, with the subgroup $\langle a \rangle = \Sigma_2$ acting by conjugation on the normal subgroup $\langle b, c\rangle = \Sigma_2 \times \Sigma_2$.}
\paragraph{3.1.4} A much simpler discussion applies to $\Sigma_2 \times \Sigma_2$ acting on $N_*(E\Sigma_2) \otimes N_*(E\Sigma_2)$ by the tensor product of the natural action of the group factors on the tensor factors, and on $ N_*(E\Sigma_2 \times E\Sigma_2)$ by the product action on the product space.
Both the $EZ$ and $AW$ maps are equivariant in this case.\\
Notice that the product $E\Sigma_2\times E\Sigma_2$ is naturally identified with $E(\Sigma_2\times \Sigma_2)$.
Thus, we can view the Alexander-Whitney map as a map
\begin{equation*}
AW\colon N_*(E(\Sigma_2\times\Sigma_2)) \to N_*(E\Sigma_2)\otimes N_*(E\Sigma_2).
\end{equation*}
It is equivariant with respect to the natural $\Sigma_2 \times \Sigma_2$-actions.
\subsection{The Map Phi} \label{sect3.2}
\paragraph{3.2.1} We have the Alexander-Whitney diagonal map
\begin{equation*}
AW_{\Delta} \colon N_*(E\Sigma_2)\to N_*(E\Sigma_2)\otimes N_*(E\Sigma_2)
\end{equation*}
and the map of Steenrod
\begin{equation*}
\widetilde{AW}_{\!\!\Delta}\colon N_*(E\Sigma_2)\otimes N_*(E\Sigma_2)\to N_*(E\Sigma_2)\otimes N_*(E\Sigma_2).
\end{equation*}
We form the composition below, $\Phi = (\operatorname{Id} \otimes \, \widetilde{AW}_{\!\!\Delta}) \circ (AW_{\Delta} \otimes \operatorname{Id})$,
\begin{equation*}
\begin{tikzcd}
N_*(E\Sigma_2) \otimes N_*(E\Sigma_2) \arrow[r, "AW_{\Delta} \otimes \operatorname{Id}"] &[10pt]
N_*(E\Sigma_2 )\otimes N_*(E\Sigma_2) \otimes N_*(E\Sigma_2) \arrow[r, "\operatorname{Id} \otimes \widetilde{AW}_{\!\!\Delta}"] &[6pt]
N_*(E\Sigma_2)\otimes N_*(E\Sigma_2) \otimes N_*(E\Sigma_2).
\end{tikzcd}
\end{equation*}
\paragraph{3.2.2} The following result will be important for the later construction of certain chain homotopies.
\begin{thm} \label{t:3.1}
The composition $\Phi$ is equivariant with respect to the embedding
\begin{equation*}
\Sigma_2\times \Sigma_2\to D_8
\end{equation*}
that sends the generator of the first factor to $a$ and the generator of the second factor to $bc$.
(This embedding coincides with the inclusion $V_4 \subset D_8$ as subgroups of $\Sigma_4$.)
\end{thm}
\begin{proof}
This follows for the involution in the first factor immediately from the fact that $AW_{\Delta} (Tx)= (T \otimes T)(AW_{\Delta} (x))$ and $\widetilde{AW}_{\!\!\Delta}(Ty \otimes z) = S(\widetilde{AW}_{\!\!\Delta}(y \otimes z))$, where $S$ denotes the switch of tensor factors.
It follows for the involution in the second factor from the fact that $\widetilde{AW}_{\!\!\Delta}(x \otimes Ty) = (T \otimes T)(\widetilde{AW}_{\!\!\Delta}(x \otimes y))$.
\end{proof}
\paragraph{3.2.3} As an immediate consequence from the definition $\widetilde x_q = (1, T, 1, \ldots,T^q)$ and the definition of $AW_{\Delta}$ we have the following.
\begin{lem*}
The Alexander-Whitney diagonal approximation for the simplicial set $N_*(E\Sigma_2)$ is given by
\begin{equation*}
AW_{\Delta} (\widetilde x_q)=\sum_{i=0}^q\widetilde x_i\otimes T^i\widetilde x_{q-i}
\end{equation*}
and
\begin{equation*}
AW_{\Delta} (T\widetilde x_q)=\sum_{i=0}^qT\widetilde x_i\otimes T^{i+1}\widetilde x_{q-i}.
\end{equation*}
\end{lem*}
\paragraph{3.2.4} We recall from Theorem~\ref{t:2.2} that the map
\begin{equation*}
\widetilde{AW}_{\!\!\Delta} \colon N_*(E\Sigma_2) \otimes N_*(E\Sigma_2) \to N_*(E\Sigma_2)\otimes N_*(E\Sigma_2)
\end{equation*}
is determined by $\Sigma_2 \times \Sigma_2$ equivariance and the formula
\begin{equation*}
\widetilde{AW}_{\!\!\Delta}(\widetilde x_n \otimes \widetilde x_k) =
\sum_{\epsilon = 0,1} \sum_{i+j = k+n} c_{n,k,i}^\epsilon \, \widetilde x_i \otimes T^\epsilon \widetilde x_j,
\end{equation*}
where the coefficients $c_{n,k,i}^\epsilon$ are given in Theorem~\ref{t:2.2}.
Specifically, with $j = n+k-i$,
\begin{equation*}
c^0_{n,k,i} = \binom{i+1}{n+1} \binom{j}{n}\ \ \ \ \text{and}\ \ \ c_{n,k,i}^1 = \binom{i}{n+1}\binom{j}{n}.
\end{equation*}
Let us write out explicitly the composition $\Phi =(\operatorname{Id} \otimes \widetilde{AW}_{\!\!\Delta}) \circ (AW_{\Delta} \otimes \operatorname{Id})$.
Thus, for $a \in \{0, 1\}$, we have from Lemma 3.2.3 and the full version of Theorem~\ref{t:2.2} that includes the equivariance
\begin{equation} \label{Phi1}
\begin{split}
\Phi(\widetilde x_q\otimes T^a\widetilde x_p) & = \sum_{i=0}^q \widetilde x_i \otimes \widetilde{AW}_{\!\!\Delta}(T^i \widetilde x_{q-i} \otimes T^a \widetilde x_p) \\ & =
\sum_{i=0}^q \widetilde x_i \otimes \left(\sum_{\epsilon=0,1} \sum_{j=0}^{p+q-i}c_{q-i,p,j}^\epsilon S^i(T^a\widetilde x_j\otimes T^{a+\epsilon} \widetilde x_{p+q-i-j})\right),
\end{split}
\end{equation}
where $S$ is the switch of factors.
We also have
\begin{equation} \label{Phi2}
\begin{split}
\Phi(T \widetilde x_q \otimes T^a \widetilde x_p) & =
\sum_{i=0}^q T \widetilde x_i \otimes \widetilde{AW}_{\!\!\Delta}(T^{i+1} \widetilde x_{q-i} \otimes T^a \widetilde x_p) \\ &=
\sum_{i=0}^q T\widetilde x_i \otimes \left(\sum_{\epsilon = 0,1} \sum_{j=0}^{p+q-i}c_{q-i,p,j}^\epsilon \ S^{i+1}(T^a\widetilde x_j\otimes T^{a+\epsilon}\widetilde x_{p+q-i-j})\right).
\end{split}
\end{equation}
In parsing these formulae, the subscripts $n, k, i, j$ in the expression for $\widetilde{AW}_{\!\!\Delta}(T^b \widetilde x_n \otimes T^a \widetilde x_k)$ of Theorem~\ref{t:2.2} become subscripts $q-i, p, j, p+q-i-j$ in the expressions for $\widetilde{AW}_{\!\!\Delta}(T^b\widetilde x_{q-i}\otimes T^a\widetilde x_p)$ that occur in identities \eqref{Phi1} and \eqref{Phi2}.
\subsection{Explicit computation of $\overline\Phi$} \label{sect3.3}
\paragraph{3.3.1} From the formulae in the previous subsection, along with equivariance, we can deduce formulae for actions of $\Phi$ on certain quotients of the domain and range of $\Phi$.
Consider the map formed from $\Phi$ by first dividing the domain of $\Phi$ by the involution on the second factor of $N_*(E\Sigma_2) \times N_*(E\Sigma_2)$, and dividing the range of $\Phi$ by the corresponding action of $bc$.
Follow that by dividing the range of $\Phi$ by the full subgroup $\{b,c\}$,
\begin{equation*}
\overline \Phi \colon N_*(E\Sigma_2)\otimes N_*(B\Sigma_2) \to N_*(E\Sigma_2) \otimes N_*(B\Sigma_2) \otimes N_*(B\Sigma_2).
\end{equation*}
From Formula~\eqref{Phi1} of \S{3.2.4}, and since $c^0_{q-i,p,j} + c^1_{q-i,p,j} = \binom{j}{q-i} \binom{p+q-i-j}{q-i}$, we have
\begin{equation} \label{Phi3}
\overline \Phi (\widetilde x_q \otimes x_p)\ = \
\sum_{i=0}^q \widetilde x_i \otimes \left( \sum_{j=0}^{p+q-i}\begin{pmatrix}j \\ q-i \end{pmatrix} \begin{pmatrix} p+q-i-j \\ q-i \end{pmatrix} S^i(x_j \otimes x_{p+q-i-j}) \right).
\end{equation}
In this summation, the binomial coefficient product is 0 unless $0 \leq q-i \leq j \leq p$.
We assume these inequalities going forward.\footnote{The binomial coefficients arose when we were counting diagrams in Section 2.2 related to partitions of integers that were used to compute $\smallsmile_n$ products of cochains. Sometimes there are no diagrams of certain shape that evaluate non-trivially on a tensor product of cochains of given dimensions. You do not need a binomial coefficient formula to count the number of positive partitions of $N$ into $M > N$ summands.}
\paragraph{3.3.2} We will rewrite the sum (3.3) so as to easily distinguish the symmetric and non-symmetric terms in the second two tensor factors of the expression.\\
We set $\ell =(p-q+i)/2$, an element of ${\mathbb Z}[\frac{1}{2}].$ We set $a=p-\ell-j$ so that $a$ is congruent to $\ell$ modulo $\mathbb{Z}$.
Also, $0 \leq p - j = \ell + a$ and $0 \leq j - q + i = \ell - a$.
Thus $-\ell \leq a \leq\ell$.\\
We have $i = q-p+2\ell, \ j = p-\ell-a, \ q-i = p-2\ell$, and $p+q-i-j = p-\ell+a$. Substitute these values into the sum in Formula~\eqref{Phi3}.
\begin{equation} \label{Phi4}
\overline \Phi(\widetilde x_q\otimes x_p) \ = \!\!\!
\sum_{\substack{-\ell \le a\le \ell;\ \\ a\equiv \ell \pmod \mathbb{Z}}}
\begin{pmatrix} p-\ell-a \\ p-2\ell \end{pmatrix}
\begin{pmatrix} p-\ell+a \\ p-2\ell \end{pmatrix}
\widetilde x_{q-p+2\ell} \otimes S^{q-p+2\ell} (x_{p-\ell-a} \otimes x_{p-\ell+a}).
\end{equation}
The sum is finite since $\ell \in {\mathbb Z}[\frac{1}{2}]$ and $0 \leq \ell \leq p/2$.
We also must have $0 \leq 2\ell+q-p$.
We can remove the powers of the switching operator $S$ in Formula~\eqref{Phi4} because for fixed $\ell$ there is an $S$-invariant term with $a = 0$ and the other terms occur in pairs with indices $\ell, a$ and $\ell, -a$ whose sum is invariant under $S^{q-p+2\ell}$.\\
\begin{cor} \label{cor3.2}
In $ N_*(E\Sigma_2)\otimes N_*(B\Sigma_2)\otimes N_*(B\Sigma_2)$ we have
$$\overline\Phi(\widetilde x_q\otimes x_p) = S_{q,p}+NS_{q,p},$$
where the symmetric terms are
$$S_{q,p}=\sum_\ell \begin{pmatrix}p-\ell \\ p-2\ell\end{pmatrix}\widetilde x_{q-p+2\ell}\otimes x_{p-\ell}\otimes x_{p-\ell},$$
and the non-symmetric terms are
\begin{align*}
NS_{q,p}\ = \!\! \sum_{\substack{-\ell \le a\le \ell;\ \ a\not=0; \\ a\equiv \ell\pmod \mathbb{Z}}}\begin{pmatrix}p-\ell-a \\ p-2\ell\end{pmatrix}\begin{pmatrix}p-\ell+a \\ p-2\ell\end{pmatrix}\widetilde x_{q-p+2\ell}\otimes x_{p-\ell-a}\otimes x_{p-\ell+a}.
\end{align*}
By symmetry, we also have a formula $\overline\Phi(\widetilde x_p\otimes x_q) = S_{p,q} + NS_{p,q}$.
\end{cor}
\paragraph{3.3.3} The map we have constructed
$$\overline\Phi\colon N_*(E\Sigma_2)\otimes N_*(B\Sigma_2) \to N_*(E\Sigma_2)\otimes N_*(B\Sigma_2)\otimes N_*(B\Sigma_2)$$
is equivariant with respect to the remaining $\Sigma_2$ actions on both sides.
On the range, this involution is $T \otimes S$, where $S$ switches the second and third factors.
$\overline\Phi$ passes to the quotient to give a map that we will also call $\overline \Phi$,
$$\overline \Phi \colon N_*(B\Sigma_2)\otimes N_*(B\Sigma_2) \to \bigl( N_*(E\Sigma_2)\otimes N_*(B\Sigma_2)\otimes N_*(B\Sigma_2)\bigr)_{\Sigma_2.}$$
This last complex can also be written as $\bigl( N_*(E\Sigma_2)\otimes N_*(E\Sigma_2)\otimes N_*(E\Sigma_2)\bigr)_{D_8},$ where $D_8$ is the dihedral group acting freely as described in \S3.1.1.
As such, the homology of this coinvariant complex is the homology of $BD_8$.
\begin{cor} \label{cor3.3}
We have
\begin{equation*}
\overline\Phi( x_q\otimes x_p) = \widehat{S}_{q,p}+\partial \widehat{NS}_{q,p} \in \bigl( N_*(E\Sigma_2)\otimes N_*(B\Sigma_2)\otimes N_*(B\Sigma_2)\bigr)_{\Sigma_2},
\end{equation*}
where
\begin{align*}
\widehat{NS}_{q,p} \ = \!\!
\sum_{\substack{0 < a\le \ell; \\ a\equiv \ell\pmod \mathbb{Z}}}\begin{pmatrix}p-\ell-a \\ p-2\ell\end{pmatrix}\begin{pmatrix}p-\ell+a \\ p-2\ell\end{pmatrix} [\widetilde x_{q-p+2\ell+1} \otimes x_{p-\ell -a} \otimes x_{p-\ell+a}]
\end{align*}
and where $$\widehat{S}_{q,p} = \sum_\ell \begin{pmatrix}p-\ell \\ p-2\ell\end{pmatrix}\ [\widetilde{x}_{q-p+2\ell}\otimes x_{p-\ell}\otimes x_{p-\ell}].$$
By symmetry, we also have a formula $\overline\Phi( x_p\otimes x_q) = \widehat{S}_{p,q} + \partial \widehat{NS}_{p,q}$.
\end{cor}
\begin{proof}
After dividing by the last $\Sigma_2$ action, we have in the coinvariant complex $[T\widetilde{x} \otimes y \otimes z] = [\widetilde{x} \otimes z \otimes y]$.
So we can combine pairs of terms in Corollary~\ref{cor3.2}.
Thus with $0 < a$ we have
\begin{align*}
\Big[ \widetilde{x}_{q-p +2\ell} \otimes (x_{p-\ell-a}\otimes x_{p-\ell+a} + x_{p-\ell+a}\otimes x_{p-\ell-a}) \Big] & =
\Big[ (\widetilde{x}_{q-p +2\ell} +T\widetilde{x}_{q-p+2l})\otimes x_{p-\ell-a}\otimes x_{p-\ell+a} \Big] \\ & =
\partial \Big[ \widetilde{x}_{q-p+2\ell+1}\otimes x_{p-\ell-a}\otimes x_{p-\ell+a} \Big].
\end{align*}
In parsing this formula and the statement of Corollary~\ref{cor3.3} it is useful to observe that basis elements in the coinvariant complex
\begin{equation*}
\Big( N_*(E\Sigma_2)\otimes N_*(B\Sigma_2)\otimes N_*(B\Sigma_2) \Big)_{\Sigma_2}
\end{equation*}
which are non-symmetric in the second two factors have unique names in the form $[T^a \widetilde x_r \otimes x_s \otimes x_t]$, with $a \in \{ 0, 1\}$ and $s < t$.
For symmetric elements, $[\widetilde{x}_r \otimes x_s \otimes x_s] = [T\widetilde{x}_r \otimes x_s \otimes x_s]$.
The result is then clear.
\end{proof}
\subsection{The Map Psi}
\paragraph{3.4.1} In \S3.1.3 we identified set-wise $D_8 = \Sigma_2 \times \Sigma_2 \times \Sigma_2$.
We also implicitly translated the product in $D_8$ to a corresponding product of triples.
The group action of $D_8$ on the left of simplices in $E(\Sigma_2 \times \Sigma_2 \times \Sigma_2)$ described in \S3.1.2 then gives an identification of free $D_8$ simplicial sets $ED_8 = E(\Sigma_2 \times \Sigma_2 \times \Sigma_2) .$ We can also identify $E(\Sigma_2 \times \Sigma_2 \times \Sigma_2) $ with $ E\Sigma_2 \times E\Sigma_2 \times E\Sigma_2$.
\paragraph{3.4.2} We form the composition below, $ \Psi = EZ \circ \Phi \circ AW $,
$$N_*(E\Sigma_2 \times E\Sigma_2) \xrightarrow{AW} N_*(E\Sigma_2)\otimes N_*(E\Sigma_2) \buildrel \Phi \over\longrightarrow N_*(E\Sigma_2)\otimes N_*(E\Sigma_2)\otimes N_*(E\Sigma_2)$$
$$ \xrightarrow{EZ} N_*(E\Sigma_2\times E\Sigma_2\times E\Sigma_2) \simeq N_*(ED_8).$$
The last equivalence is from our identification of the simplicial set $ED_8$ with $E\Sigma_2 \times E\Sigma_2 \times E\Sigma_2$ discussed just above.\\
The map $\Psi$ is equivariant with respect to the inclusion $\Sigma_2\times \Sigma_2 \simeq V_4 \subset D_8$ that sends the first generator to $a$ and the second to $bc$.
This holds because $\Phi$ has this equivariance property by Theorem~\ref{t:3.1}, and the $AW$ and $EZ$ maps are also suitably equivariant.
\begin{thm} \label{t:3.4}
Let $$\iota_* \colon N_*(E(\Sigma_2\times \Sigma_2))\to N_*(ED_8)$$
be the inclusion induced by the inclusion $\iota \colon \Sigma_2\times \Sigma_2 \simeq V_4 \subset D_8$.
Then there is an explicit $\iota$-equivariant chain homotopy
$J_\Psi$ between $\Psi$ and $\iota_*$, given by the formula
\begin{align*} J_\Psi(g_0, g_1, \ldots, g_n) & = \sum_j \iota_*(g_0, \ldots, g_j) * \Psi(g_j, \ldots, g_n) \\
& = \sum_j (g_0, \ldots, g_j, \Psi(g_j, \ldots, g_n)).
\end{align*}
\end{thm}
\begin{proof}
The statement is immediate from the equivariance we have established for $\Psi = EZ\circ\Phi\circ AW$ and Theorems~\ref{t:homotopy for classifying spaces} and \ref{t:homotopy for iota and Psi} in \S1.3.1 and \S1.3.2.
In degree 0, the map $\Psi$ is just the inclusion ${\mathbb F}_2[\Sigma_2 \times \Sigma_2] \to {\mathbb F}_2[D_8]$, hence the induced map on $H_0$ is the identity, so Theorems~\ref{t:homotopy for classifying spaces} and \ref{t:homotopy for iota and Psi} do apply.
\end{proof}
\paragraph{3.4.3} Now we set
\begin{align*}
\widetilde x_q \times \widetilde x_p =EZ(\widetilde x_q\otimes \widetilde x_p)\in N_*(E\Sigma_2\times E\Sigma_2) \nonumber \\
x_q \times x_p = EZ(x_q \otimes x_p) \in N_*(B\Sigma_2 \times B\Sigma_2).
\end{align*}
Since $AW \circ EZ = \operatorname{Id}$, we see that $AW(\widetilde x_q \times \widetilde x_p)=\widetilde x_q\otimes \widetilde x_p$, and hence
\begin{equation*}
\Psi(\widetilde x_q \times \widetilde x_p) = EZ(\Phi(\widetilde x_q\otimes \widetilde x_p)).
\end{equation*}
An immediate consequence of Theorem~\ref{t:3.4} is the following:
\begin{cor} \label{cor3.5}
$$(\partial \circ J_\Psi+ J_\Psi \circ \partial)(\widetilde x_q \times \widetilde x_p) = EZ(\Phi(\widetilde x_q\otimes \widetilde x_p))+ (\widetilde x_q \times \widetilde x_p) \in N_*(ED_8).$$
By symmetry we have
$$(\partial \circ J_\Psi+ J_\Psi \circ \partial)(\widetilde x_p \times \widetilde x_q) = EZ(\Phi(\widetilde x_p\otimes \widetilde x_q))+ (\widetilde x_p \times \widetilde x_q) \in N_*(ED_8).$$
\end{cor}
Let $\pi_*\colon N_*(EG)\to N_*(BG)$ be the map induced by the projection to the quotient for $G = V_4 \simeq \Sigma_2 \times \Sigma_2$ and $G = D_8$.
The $V_4$-equivariant homotopy $J_\Psi \colon N_*(EV_4) \to N_{*+1}(ED_8)$ yields a commutative diagram
\begin{equation}
\begin{tikzcd}
N_*(EV_4) \arrow[r, "J_\Psi"] \arrow[d, "\pi_*"] & N_{*+1}(ED_8) \arrow[d, "\pi_*"] \\
N_*(BV_4) \arrow[r, "J_\Psi"] & N_{*+1}(BD_8).\\
\end{tikzcd}
\end{equation}
\begin{cor} \label{cor3.6}
With
$\overline{J}_\Psi(x_q \times x_p) = \pi_*J_\Psi(\widetilde x_q \times \widetilde x_p)$ we have
$$\partial \overline{J}_\Psi(x_q \times x_p) = EZ(\overline \Phi(x_q \otimes x_p)) + x_q \times x_p \in N_*(BD_8).$$
By symmetry we have
$$\partial \overline{J}_\Psi(x_p \times x_q) = EZ(\overline \Phi(x_p \otimes x_q)) + x_p \times x_q \in N_*(BD_8).$$
\end{cor}
\begin{proof}
Follows from Corollary~\ref{cor3.6} since $\pi_* \partial(\widetilde x_q \times \widetilde x_p) = \partial (x_q \times x_p) = 0.$
\end{proof}
\subsection{A Brief Summary}
\paragraph{3.5.1} In \S1.3.3 we gave the general formula for a chain homotopy associated to an inner automorphism of a group.
Specializing to $\Sigma_4$, and referring to the discussion in \S0.2.5, we get $\partial \overline{J}_{(23)}(x_q \times x_p) = x_p \times x_q + x_q \times x_p \in N_*(B\Sigma_4)$.\\
Combining with Corollary~\ref{cor3.6} just above, we then have an equality
\begin{equation} \label{e:3.6}
\partial \overline{J}_\Psi(x_q \times x_p) + \partial \overline{J}_\Psi(x_p \times x_q) + \partial \overline{J}_{(23)}(x_q \times x_p)\ = \
EZ(\overline \Phi(x_q \otimes x_p)) + EZ(\overline \Phi(x_p \otimes x_q)) \in N_*(B\Sigma_4).
\end{equation}
Four of the five terms here are actually in $N_*(BD_8)$.
But recall the point made in \S0.2.2 that we use the same names in $N_*(BD_8) \subset N_*(B\Sigma_4)$.\\
To complete the proofs of Theorems \ref{Adem1} and \ref{Adem2}, which were stated in \S0.2.3 and \S0.2.5 and which the reader should review, we will explain how to `evaluate' every term in Formula \eqref{e:3.6} on a cocycle $\alpha \in N^{-n}(X)$, using an action of elements in $N_*(B\Sigma_4)$ on symmetric tensors of the form $\alpha \otimes \alpha \otimes \alpha \otimes \alpha$.
We will explain such an action in Section 4 in terms of operads.
The sum of the five evaluations of the terms in Formula \eqref{e:3.6} will be 0, which will be seen to be equivalent to Theorem~\ref{Adem2}.
The proof of Theorem~\ref{Adem1} will be a more direct application of Corollary~\ref{cor3.6}.
\paragraph{3.5.2} It turns out that the operad method allows us to suppress the $EZ$ step in the two terms on the right-hand side of Formula \eqref{e:3.6} above, and directly apply $\overline \Phi(x_q \otimes x_p)$ and $\overline \Phi(x_p \otimes x_q)$, which lie in $\bigl( N_*(E\Sigma_2)\otimes N_*(B\Sigma_2)\otimes N_*(B\Sigma_2)\bigr)$, to a symmetric tensor.
These two terms were computed in Corollary~\ref{cor3.2} in \S3.3.2, as sums of triple tensors.
The direct evaluation of $\widetilde{x}_r \otimes x_s \otimes x_t$ on a symmetric tensor will follow from Lemma~\ref{l:4.1} in \S4.3.4.
The result is
\begin{equation*}
(\widetilde{x}_r \otimes x_s \otimes x_t)(\alpha^{\otimes 4}) = (\alpha \smallsmile_s \alpha) \smallsmile_r (\alpha \smallsmile_t \alpha) = Sq^{n-s}(\alpha) \smallsmile_r Sq^{n-t}(\alpha),
\end{equation*}
where $deg(\alpha) = -n$.
Note that if $s = t$ then $Sq^{n-s}(\alpha) \smallsmile_r Sq^{n-s}(\alpha) = Sq^{2n-s-r}Sq^{n-s}(\alpha)$.\\
Bringing in the formulae of Corollary~\ref{cor3.3}, which are explicit sums of triple tensors with various coefficients and subscripts (which the reader should review and compare with Theorem~\ref{Adem1}), we find that Corollary 3.6 and Lemma~\ref{l:4.1} imply the formulae stated originally in the introduction as Theorem~\ref{Adem1}.
Specifically, from Corollary~\ref{cor3.3}, the evaluation of $\overline{\Phi}(x_q \otimes x_p)$ on a symmetric tensor $\alpha^{\otimes 4}$ consists of evaluating the symmetric part $\widehat{S}_{q,p}$ of $\overline{\Phi}(x_q \otimes x_p)$, which yields by Lemma~\ref{l:4.1} the sum of iterated Steenrod Squares appearing in Theorem~\ref{Adem1}, and evaluating the non-symmetric part $\partial \widehat{NS}_{q,p}$, which yields the coboundary of the sum of $\smallsmile_r$ products of Squares in Theorem~\ref{Adem1}.
Similarly for $\overline{\Phi}(x_p \otimes x_q)$.
Thus, evaluating the terms in Corollary~\ref{cor3.6} on $\alpha^{\otimes 4}$ completes the proof of Theorem~\ref{Adem1}.
\paragraph{3.5.3} The meaning of the evaluation of the three boundary terms on the left-hand side of Formula \eqref{e:3.6} on a symmetric tensor $\alpha^{\otimes 4}$, producing three coboundary terms in $N^*(X)$, will be explained in \S4.4.4.
Then the vanishing of the sum of the five evaluations of the terms in \eqref{e:3.6}, combined with the discussion of Theorem~\ref{Adem1} in \S3.5.2 just above, very easily translates to the statement of Theorem~\ref{Adem2}.
\paragraph{3.5.4} In a strong sense, our main theorem is really Formula \eqref{e:3.6} in \S3.5.1.
For each pair $q > p$, Formula \eqref{e:3.6} can be rewritten as a relation $R(q,p) = 0 \in N_{q+p}(B\Sigma_4)$, where $R(q,p)$ is a sum of five terms.
Explicit formulae for all five of these terms have been given at various points in our paper.
In fact, before applying equivariance to the chain homotopies $J_\Psi$ and $J_{(23)}$, we actually gave formulae for all five terms in the sum
\begin{align}
\widetilde{R}(q,p)
& = \partial J_\Psi(\widetilde x_q \times \widetilde x_p) + \partial J_\Psi(\widetilde x_p \times \widetilde x_q) + \partial J_{(23)}(\widetilde x_q \times \widetilde x_p) \nonumber \\
& + EZ(\Phi(\widetilde x_q\otimes \widetilde x_p)) + EZ(\Phi(\widetilde x_p\otimes \widetilde x_q)) \in N_{q+p}(E\Sigma_4).
\end{align}
The sum $\widetilde{R}(q,p) \in N_{q+p}(E\Sigma_4)$ projects to the sum $R(q,p) \in N_{q+p}(B\Sigma_4)$, and in the operad method to be described in Section 4 it is actually $\widetilde{R}(q,p)$ that is directly evaluated on symmetric tensors $\alpha^{\otimes 4}$, with the result depending only on $R(q,p)$.
Conceptually it is not so difficult to write a computer program for calculating the $\widetilde{R}(q,p) \in N_{q+p}(E\Sigma_4)$, and we have done so, but the output gets extremely large, even for $q+p \approx 10$.
A positive way to look at this is that each $R(q,p)$ corresponds to {\it infinitely many} different (unstable) Adem relations, obtained as in Theorem~\ref{Adem2} by evaluating $R(q,p)$ on a symmetric tensor $\alpha^{\otimes 4}$, where the cocycle $\alpha$ can have any degree.
\section{Operads} \label{sect4}
At the beginning of this section we introduce some basic terminology about operads.
We describe the symmetric group operad in the category of sets and the endomorphism operads in the category of chain complexes as examples.
In Subsections 4.2 and 4.3 we introduce the Surj operad, ${\mathcal S} = \{{\mathcal S}_n\}_{n \geq 1}$, also an operad of chain complexes, and describe how each chain complex ${\mathcal S}_n$ acts on $n$-fold tensors of cochains on spaces $X$.
The action of ${\mathcal S}_2$ coincides with Steenrod's two-variable $\smallsmile_n$ products.
Then we explain in Lemma~\ref{l:4.1} how the operad structure exhibits iterations of $\smallsmile_n$ products as part of the action of ${\mathcal S}_4$.\\
In Subsection 4.4 we introduce the Barratt-Eccles operad ${\mathcal E}$ with ${\mathcal E}_n = N_*(E\Sigma_n)$.
The operad ${\mathcal E}$ acts on multi-tensors of cochains via an operad morphism $TR\colon {\mathcal E} \to {\mathcal S}$.
The operad morphism $TR$ is very important in our paper because our key chain homotopies all have values in ${\mathcal E}$, and we need to push those chain homotopies to ${\mathcal S}$ to obtain explicit cochain operations and complete the proofs of the main theorems.
\subsection{Introduction} \label{sect4.1}
\paragraph{4.1.1} Recall that a {\em symmetric operad}, \cite{16mayoperads}, ${\mathcal P}$ in a monoidal category is a collection of objects
${\mathcal P} = \{{\mathcal P}_n\}_{n\ge 1}$ of the category together with structure maps
$${\mathcal P}_r\times ({\mathcal P_{s_1}\times\cdots\times{\mathcal P}_{s_r}}) \to {\mathcal P}_{s_1+\cdots+s_r},$$
satisfying composition and symmetry rules for $\Sigma_n$ actions on ${\mathcal P}_n$, and a unit rule for ${\mathcal P}_1$.
\paragraph{4.1.2} Let us define the operad in the category of sets determined by the symmetric groups.
We denote an element $\sigma \in \Sigma_n$ by the sequence $(\sigma(1)\ldots \sigma(n))$,
which is the reordering of $(1\ldots n)$ given by applying the permutation.\footnote{For the remainder of the paper we will write permutations in this form, rather than as products of disjoint cycles.}\\
The symmetric groups $\{\Sigma_n\}_{n\ge 1}$ form an operad where the operad structure
$$\Sigma_r\times(\Sigma_{s_1}\times \cdots \times \Sigma_{s_r})\to \Sigma_{s_1+\cdots+s_r}$$
is given by dividing the interval $[1,s_1+\ldots+s_r]$ into disjoint blocks of lengths $s_1, \ldots , s_r$, starting from the left, and then first permuting the elements of the $i^{th}$ block among themselves by applying the conjugation of the element of $\Sigma_{s_i}$ by the unique order preserving map from the $i^{th}$ block to $\{1, \ldots ,s_i\}$.
This produces an automorphism of each block.
The blocks with their new internal orderings are then permuted among themselves, according to the element of $\Sigma_r$.\footnote{One can also first permute the $r$ blocks using $\sigma \in \Sigma_r$, keeping the entries of each original block in their consecutive order. Then apply the $\sigma_i \in \Sigma_{s_i}$.
To keep straight how each permutation is applied, it helps to pretend that $r, s_1, \ldots, s_r$ are distinct integers.}
\paragraph{4.1.3} As a simple example, consider the operad structure map $\Sigma_2\times (\Sigma_2\times \Sigma_2) \to \Sigma_4$.
We have the embedding $D_8 \subset \Sigma_4$, with $a \mapsto (3412),\ b \mapsto (2134),\ c \mapsto (1243)$.
It is easy to see from the block description of the operad structure map that the triple $(T^{\epsilon_1}, (T^{\epsilon_2}, T^{\epsilon_3})) \in \Sigma_2\times (\Sigma_2\times \Sigma_2)$ maps to the permutation $c^{\epsilon_3}b^{\epsilon_2}a^{\epsilon_1} \in D_8 \subset \Sigma_4$.
Here, $\epsilon_i \in \{0,1\}$.\\
Thus, the operad structure map is a bijection $\Sigma_2\times (\Sigma_2\times \Sigma_2) \simeq D_8$, and, moreover, this bijection coincides with the bijection $D_8 \simeq \Sigma_2\times (\Sigma_2\times \Sigma_2)$ studied in \S3.1.3.
The induced product on the triples $(T^{\epsilon_1}, (T^{\epsilon_2}, T^{\epsilon_3})) \in \Sigma_2\times (\Sigma_2\times \Sigma_2)$ thus also coincides with the product on triples from \S3.1.3.
\paragraph{4.1.4} For a vector space or a chain complex $V$ there is the Endomorphism operad, with
$$End(V)_n={\ \rm Hom}(V^{\otimes n}, V),$$
with the obvious operad structure and action of the symmetric groups.
To give $V$ the structure of an {\em algebra} over a symmetric operad ${\mathcal P}$ is to give a map of symmetric operads ${\mathcal P}\to End(V)$.
\subsection{The Surj Operad ${\mathcal S}$}
In this subsection and the next we follow McClure and Smith \cite{18mccluresmith}.
\paragraph{4.2.1} We view a function $ \{1, \ldots , r+d\}\to \{1, \ldots ,r\}$ as a sequence of integers $A = (a(1) \ldots a(r+d))$, each in the interval $[1,r]$.
Fix $r$ and consider the $\mathbb{F}_2$-vector space with basis the set of all maps $\{1, \ldots ,r+d\}\to \{1, \ldots ,r\}$.
We form the quotient vector space, denoted ${\mathcal S}_r(d)$, by setting equal to zero all sequences $(a(1) \ldots a(r+d))$ that are either not surjective functions or have
$a(i)=a(i+1)$ for some $i < r+d$.
We define a chain complex structure on ${\mathcal S}_r$.
The boundary of a basis element of ${\mathcal S}_r(d)$ is
$$\partial \big(a(1) \ldots a(r+d)) = \sum_i(a(1) \ldots \widehat{a(i)} \ldots a(r+d)\big) \in {\mathcal S}_r(d-1).$$
Of course, this means that any terms in the sum that are either non-surjective functions or
have the property that two successive entries are equal are set to zero.
One sees easily that this defines a chain complex denoted ${\mathcal S}_r$.
\paragraph{4.2.2} There is the obvious left action of $\Sigma_r$ on ${\mathcal S}_r$ given by post-composition of a function with a permutation of $\{1, \ldots ,r\}$.
The operad structure on ${\mathcal S}$, which we will not define explicitly, is compatible with these actions.
This means that ${\mathcal S}$ is a {\em symmetric operad}.
For each $r\ge 1$, the action of $\Sigma_r$ on ${\mathcal S}_r$ is a free action and it turns out, \cite{18mccluresmith}, that ${\mathcal S}_r$ is an acyclic resolution of $\mathbb{F}_2$ over $\mathbb{F}_2[\Sigma_r]$.
Hence, the homology of the coinvariant complex $({\mathcal S}_r)_{\Sigma_r}$ is identified with $H_*(B\Sigma_r)$.
\paragraph{4.2.3} The basis of ${\mathcal S}_2(n)$ consisting of alternating sequences of 1's and 2's of length $n+2$ matches the basis of $N_n(E\Sigma_2)$ consisting of alternating sequences of 1's and $T$'s of length $n+1$, by dropping the last entry of a $1, 2 $ sequence.
The $\Sigma_2$ actions and the boundary formulae also coincide.
Thus we can identify ${\mathcal S}_2$ and $N_*(E\Sigma_2)$.
The full operad operations in the Surj operad ${\mathcal S}$ are quite complicated.
However, on degree zero chains, $ {\mathcal S}_2 \otimes ({\mathcal S}_2 \otimes {\mathcal S}_2) \to {\mathcal S}_4$ coincides with the map ${\mathbb F}_2[D_8] \to {\mathbb F}_2[\Sigma_4]$ induced by the dihedral group inclusion $D_8 \to \Sigma_4$ for the symmetric group operad, as described in \S4.1.3.
In all degrees, the axioms for permutation group actions on symmetric operads imply that $$N_*(E\Sigma_2) \otimes (N_*(E\Sigma_2) \otimes N_*(E\Sigma_2)) = {\mathcal S}_2 \otimes ({\mathcal S}_2 \otimes {\mathcal S}_2) \to {\mathcal S}_4$$ is equivariant for the inclusion $D_8 \to \Sigma_4$.
\subsection{Action of the Surj Operad on Normalized Cochains}
\paragraph{4.3.1} The normalized cochain complex of a simplicial set $X$, $ N^*(X)$, is an algebra over the Surj operad.
That is to say there are chain maps that we will describe in the next subsection
$${\mathcal O}_X\colon {\mathcal S}_r\otimes \underbrace{ N^*(X)\otimes\cdots\otimes N^*(X)}_{r-{\ \rm times}}\to N^*(X).$$
This means that (i) the degree of ${\mathcal O}_X(\zeta\otimes\alpha_1\otimes \cdots\otimes \alpha_r)$
is the sum of the degrees of the $\alpha_i$ plus the degree of $\zeta$, and (ii)
\begin{align*}
d({\mathcal O}_X(\zeta\otimes\alpha_1\otimes\cdots\otimes\alpha_r))=
{\mathcal O}_X \Big\{ \partial(\zeta) \otimes (\alpha_1 \otimes \cdots \otimes \alpha_r) + \sum_i \zeta \otimes (\alpha_1 \otimes \cdots \otimes d\alpha_i \otimes \cdots \otimes \alpha_r) \Big\}.
\end{align*}
Also, the $\mathcal{O}_X$ satisfy certain associativity and equivariance properties.\\
Recall our convention that $N^*(X)$ is negatively graded.
This means that the operation ${\mathcal O}_X(\zeta)$ decreases the sum of the absolute values of the degrees of the $\alpha_i$ by deg$(\zeta)$.\\
The operations ${\mathcal O}_X$, natural in $X$, implicitly determine the operad structure maps for ${\mathcal S}$.
In fact, any finite part of ${\mathcal S}$ injects into $End(N^*(\Delta^k))$ for large $k$.
\paragraph{4.3.2} We will describe the action of a function $ A = (a(1)\ldots a(n+r)) \in {\mathcal S}_r(n)$ on multi-tensors of cochains.
For each $k$, we consider a set of $r$-step diagrams, ${\rm Diagrams}(A, k)$, associated to certain collections of $n+r$ subintervals of $[0,k]$.
By this we mean a division of $[0,k]$ into non-empty intervals $I_1,I_2, \ldots ,I_{n+r}$ so that the final point of each interval agrees with the initial point of the next interval.
The interval $I_j$ is said to be at level $a(j)$.
There is one extra condition, which is that for any $j\not= j'$, if the intervals $I_j$ and $I_{j'}$ are at the same level, then they are disjoint.
For a given level $1 \leq \ell \leq r$, set $I(\ell) = \bigsqcup_{a(j) = \ell} I_j $, which we interpret as a face of the simplex $\Delta^k$.\\
The generator $A = (a(1) \dots a(n+r)) \in {\mathcal S}_r(n)$ acts in the following manner.
Let $\alpha_1 \otimes \cdots \otimes \alpha_r$ be a multi-tensor of cochains of total degree $-(n + k)$.
The operad algebra structure ${\mathcal O}_X$ will assign to $A \otimes (\alpha_1 \otimes \cdots \otimes \alpha_r)$ a cochain of degree $-k$.
Let $u \in N_k(X)$ be a simplex of dimension $k$, regarded as a simplicial map $u \colon \Delta^k \to X$.
Then
\begin{equation*}
\left \langle {\mathcal O}_X(A \otimes (\alpha_1 \otimes \cdots \otimes \alpha_r)), \ u \right \rangle \ = \!
\sum_{{\rm Diagrams}(A, k)} \prod_\ell \ \langle \alpha_\ell, \ u(I(\ell)) \rangle.
\end{equation*}
With this definition the element $ (1212 \ldots)$ in ${\mathcal S}_2(n)$ acts by
$$\alpha\otimes\beta\mapsto \alpha\smallsmile_n\beta.\footnote{We defined $\smallsmile_n$ as a sum over two-step diagrams in \S\S 2.2.3, 2.2.4, using the terminology {\it odd} and {\it even} for levels 1 and 2.}$$
\paragraph{4.3.3} The coinvariant chain complex of the $\Sigma_r$ action on ${\mathcal S}_r$ acts on $\Sigma_r$-symmetric cochains in $N^*(X)^ {\otimes r} $ as follows.
Let $\zeta \in {\mathcal S}_r(n)_{\Sigma_r}$.
Lift $\zeta$ to an element $\widetilde\zeta \in {\mathcal S}_r(n)$.
The element $\widetilde\zeta$ determines a map
$$ N^*(X)^{\otimes r}\to N^*(X)$$
raising degree by $n$.\footnote{Again recall cochains are negatively graded.}
Another lift will give a different map, but elements in the same $\Sigma_r$ orbit have the same restriction to the $\Sigma_r$-invariant elements in $ N^*(X)^{\otimes r}$.\\
Thus, we have a well-defined chain map
$$({\mathcal O}_X)_{\Sigma_r}\colon ({\mathcal S}_r)_{\Sigma_r}\otimes \bigl(\underbrace{ N^*(X)\otimes\cdots\otimes N^*(X)}_{r-{\ \rm times}}\bigr)^{\Sigma_r}\to N^*(X).$$
Restricting even further to symmetric cocycles of the form $\alpha\otimes \alpha\otimes\cdots\otimes \alpha$,
a cycle in ${\mathcal S}_r(n)_{\Sigma_r}$ determines a map
from cocycles of degree $k \leq 0$ to cocycles of degree $rk+n$.
The induced operation on cohomology depends only on the homology class of $\zeta$.
In this way $H_n(B\Sigma_r)$ acts as cohomology operations
$$H^k(X)\to H^{rk+n}(X).$$
In the special case when $r=2$ the operation associated to
$$[(\underbrace{1212\ldots}_{(n+2)})] \equiv [(\underbrace{1T1T\ldots}_{(n+1)})] = x_n \in N_n(B\Sigma_2)$$
sends a symmetric cochain $(\alpha\otimes \alpha)\in N^k(X)\otimes N^k(X)$ to $\alpha\smallsmile_n \alpha \in N^{2k+n}(X)$.
In the case when $\alpha$ is a cocycle, the result of this operation is a cocycle representing the cohomology class
$Sq^{|k|-n}([\alpha])$.
\paragraph{4.3.4} In particular, we are interested in the operad product
$${\mathcal S}_2\otimes ({\mathcal S}_2\otimes {\mathcal S}_2)\to {\mathcal S}_4.$$
This is a map of an acyclic complex with a free action of the dihedral group $D_8$ to an acyclic complex with a free $\Sigma_4$-action.
Of course, ${\mathcal S}_2$ is identified with $ N_*(E\Sigma_2)$.
This allows us to consider $$\Phi(\widetilde x_q\otimes \widetilde x_p) \in N_*(E\Sigma_2) \otimes (N_*(E\Sigma_2) \otimes N_*(E\Sigma_2))$$ as an element of the domain of this operad product, where the map $\Phi$ was defined in \S3.2.1.
From \S4.2.3, the operad product is equivariant with respect to the inclusion of $D_8 \subset \Sigma_4$ that sends the elements $b$ and $c$ in the dihedral group to the permutations $(2134)$ and $(1243)$ respectively, and sends the element $a$ in the dihedral group to $(3412)$.
The following is an important observation.
\begin{lem} \label{l:4.1}
Under the operad composition ${\mathcal S}_2\otimes ({\mathcal S}_2\otimes {\mathcal S}_2) \to {\mathcal S}_4$, the element
$\widetilde x_r \otimes (\widetilde x_s\otimes \widetilde x_t)$ maps to an element of ${\mathcal S}_4$ that acts on a cochain $\alpha_1\otimes\alpha_2\otimes\alpha_3\otimes\alpha_4$ to produce
$$(\alpha_1\smallsmile_s\alpha_2)\smallsmile_r(\alpha_3\smallsmile_t\alpha_4).$$
\end{lem}
\begin{proof}
We will not prove this by directly passing through ${\mathcal S}_4$, but rather by using the operad morphism ${\mathcal S} \to End(N^*(X))$.
We have the identification ${\mathcal S}_2 \simeq N_*(E\Sigma_2)$ of \S4.2.3, with $\smallsmile_q = (12121...) \leftrightarrow \widetilde x_q$.
The operad morphism sends $\widetilde x_r \otimes (\widetilde x_s\otimes \widetilde x_t)$ to the endomorphism operad element $\smallsmile_r \otimes (\smallsmile_s \otimes \smallsmile_t)$.
Acting on $N^*(X)^{\otimes 4}$, this gives the composition $$N^*(X)^{\otimes 4} = N^*(X)^{\otimes 2} \otimes N^*(X)^{\otimes 2} \xrightarrow{ \smallsmile_s \otimes \smallsmile_t} N^*(X) \otimes N^*(X) \xrightarrow{\smallsmile_r} N^*(X),$$ which is exactly the claim of the lemma.
\end{proof}
\subsection{The Barratt-Eccles Operad ${\mathcal E}$}
Here we follow the presentation of Berger-Fresse \cite{3bergerfresse}.
\paragraph{4.4.1} The Barratt-Eccles operad is an operad in the category of chain complexes with ${\mathcal E}_r = N_*(E\Sigma_r)$.
The ${\mathcal E}$ operad structure map $${\mathcal E}_r \otimes ({\mathcal E}_{s_1} \otimes \cdots \otimes {\mathcal E}_{s_r}) \to {\mathcal E}_{s_1 + \cdots + s_r}$$ is a composition of the operad structure on symmetric groups and the Eilenberg-Zilber map $$EZ \colon N_*(E\Sigma_r) \otimes (N_*(E\Sigma_{s_1} )\otimes \cdots \otimes N_*(E\Sigma_{s_r})) \to N_*(E\Sigma_r \times (E\Sigma_{s_1} \times \cdots \times E\Sigma_{s_r})).$$ Specifically, the ${\mathcal E}$ operad structure map is post-composition of $EZ$ with the map of normalized chain complexes induced by the set theoretic map $$\Sigma_r \times (\Sigma_{s_1} \times \cdots \times \Sigma_{s_r}) \to \Sigma_{s_1 + \cdots + s_r},$$ which is the structure map of the symmetric group operad described in \S4.1.2. \\
As is the case with the Surj operad, the Barratt-Eccles operad is a symmetric operad:
the natural actions of $\Sigma_r$ on ${\mathcal E}_r$ are compatible with the operad structures.
Furthermore, ${\mathcal E}_r$ is a free ${\mathbb F}_2[\Sigma_r]$ resolution of $\mathbb{F}_2$.
\paragraph{4.4.2}
Berger-Fresse define an operad morphism $TR \colon {\mathcal E} \to {\mathcal S}$, which they call {\em Table Reduction}.
Since Table Reduction is an operad map, for any simplicial set $X$ the normalized cochains
$N^*(X)$ form an algebra over the Barratt-Eccles operad.\\
For completeness in this paper, we include here a definition of the Table Reduction morphism.
Given a basis element $(\sigma_0, \dots, \sigma_n) \in \mathcal E_r(n)$ we define
\begin{equation*}
TR(\sigma_0, \dots, \sigma_n) = \sum_{a} s_{a} \in {\mathcal S}_r(n)
\end{equation*}
as a sum of surjections
\begin{equation*}
s_{a} \colon \{1, \dots, n+r \} \to \{1, \dots, r\}
\end{equation*}
parametrized by all tuples of integers $a = (a_0, \dots, a_n)$ with each $a_i \geq 1$ and such that $a_0 + \cdots + a_n = n + r$.
For one such tuple $a$ we now describe its associated surjection $s_a$ as a sequence $s_a = (s_a(1), s_a(2), \ldots, s_a(n+r))$.\\
Consider the following ``table" representation of the generator of $\mathcal {E}_r(n)$:
\begin{center}
\begin{tabular} {c | c c c}
& $\sigma_0(1)$ & $\cdots$ & $\sigma_0(r)$ \phantom{.} \\
& $\sigma_1(1)$ & $\cdots$ & $\sigma_1(r)$ \phantom{.} \\
& $\vdots$ & $\ddots$ & $\vdots$ \\
& $\sigma_n(1)$ & $\cdots$ & $\sigma_n(r)$ .
\end{tabular}
\end{center}
Each line of the table is a permutation of $\{1,2, \ldots, r\}$.
The first $a_0$ entries of the sequence $s_a$ are the first $a_0$ entries of the permutation on the first line of the table.
The first $a_0 -1$ of these values, that is, all except the last one, are then removed from all lower permutation lines of the table.\\
The next $a_1$ entries in the sequence $s_a$ are the first $a_1$ entries remaining in the second line of the table.
The values of all except the last of these are then removed from all lower lines in the table.
The process forming the sequence $s_a$ continues in this way.
The final $a_n$ entries of the sequence will consist of all remaining entries on the last line of the table after the first $n$ steps of the process.\\
For more details on this map we refer the reader to the original treatment in \cite{3bergerfresse}.\\
The component ${\mathcal E}_2$ in the Barratt-Eccles operad is
$ N_*(E\Sigma_2) = {\mathcal S}_2$.
On these components of the operads, Table Reduction is the identification of ${\mathcal E}_2$ with ${\mathcal S}_2$ given above.\\
We shall use the following commutative diagram
\begin{equation*}
\begin{tikzcd}
N_*(E\Sigma_2) \otimes (N_*(E\Sigma_2) \otimes N_*(E\Sigma_2)) \arrow[r] \arrow[d, "{TR \, \otimes (TR \, \otimes TR)}"] & N_*(E\Sigma_4) \arrow[d, "TR"]\\
{\mathcal S}_2\otimes ({\mathcal S}_2\otimes {\mathcal S}_2 ) \arrow[r] & {\mathcal S}_4,
\end{tikzcd}
\end{equation*}
where the horizontal arrows are the Barratt-Eccles and Surj operad structure maps.
These maps are equivariant with respect to the inclusion $D_8 \subset \Sigma_4$.\\
The Barratt-Eccles operad map
$$N_*(E\Sigma_2)\otimes ( N_*(E\Sigma_2)\otimes N_*(E\Sigma_2)) \to N_*(E\Sigma_4)$$
is the composition of
$$EZ\colon N_*(E\Sigma_2)\otimes N_*(E\Sigma_2)\otimes N_*(E\Sigma_2) \to N_*(E(\Sigma_2\times\Sigma_2\times \Sigma_2))$$
with the map
$$N_*(E(\Sigma_2\times\Sigma_2\times \Sigma_2))\to N_*(E\Sigma_4)$$
induced by the group homomorphism $(T,1,1)\mapsto a = (3412);\ \ (1, T, 1)\mapsto b = (2134);\ \ (1, 1, T) \mapsto c = (1243)$, which defines our chosen group inclusion $D_8 \subset \Sigma_4$.\footnote{It is here that we use the identification of the group operation on triples with $D_8$ that we discussed in \S3.1.3 and \S4.1.3.}
\paragraph{4.4.3} Recall from \S3.3.3 that $\overline{\Phi}(x_q \otimes x_p) \in ({\mathcal S}_2\otimes {\mathcal S}_2\otimes {\mathcal S}_2)_{D_8}$.
It is exactly the commutativity of the above diagram, and the fact that $EZ$ is part of the ${\mathcal E}$ operad structure, that explains our remark in \S3.5.2 that we can evaluate $EZ \overline{\Phi}(x_q \otimes x_p) \in N_*(BD_8) \subset N_*(B\Sigma_4) = ({\mathcal E}_4)_{\Sigma_4}$ on a cocycle $\alpha \in N^*(X)$ by directly evaluating $\overline{\Phi}(x_q \otimes x_p) \in ({\mathcal S}_4)_{\Sigma_4}$ on $\alpha^{\otimes 4}$.
The actual formula in terms of Steenrod Squares and $\smallsmile_r$ products, then comes from Corollary~\ref{cor3.3} and Lemma~\ref{l:4.1}.
We have now completed one of the last steps in the proofs of the main Theorems \ref{Adem1} and \ref{Adem2}, as summarized in Subsection 3.5.
\paragraph{4.4.4} All that remains in the proofs of Theorems \ref{Adem1} and \ref{Adem2} from the summary in Subsection 3.5 is to explain how Table Reduction is used to evaluate the three boundary terms $$ \partial \overline{J}_\Psi(x_q \times x_p),\ \partial \overline{J}_\Psi(x_p \times x_q),\ \partial \overline{J}_{(23)}(x_q \times x_p) \in N_*(B\Sigma_4)$$ from Formula \eqref{e:3.6} in \S3.5.1 on a cocycle $\alpha.$ Since Table Reduction is a $\Sigma_4$-equivariant chain map, it induces a chain map of coinvariant complexes $\overline{TR}\colon N_*(B\Sigma_4) \to ({\mathcal S}_4)_{\Sigma_4}.$ Each of the three $\overline J$ terms is an element of $N_*(B\Sigma_4)$.
We then have three versions of $\overline{TR} ( \partial \ \overline{J} ) = \partial (\overline{TR} \ \overline{J}) \in ({\mathcal S}_4)_{\Sigma_4}$.
Given a cocycle $\alpha$, we evaluate each of these three boundaries on the symmetric 4-tensor $\alpha^{\otimes 4}$, using the ${\mathcal S}$ algebra structure of $N^*(X)$ as described in \S4.3.2 and \S4.3.3.
We thus have
$$(\partial\ \overline{TR} \ \overline{J}) (\alpha^{\otimes 4}) = d\ (\overline{TR} \ \overline{J} (\alpha^{\otimes 4})) \in N^*(X).$$
This last expression, for each of the three $\overline{J}$'s, is what we abbreviated as $d (\overline{J} (\alpha))$ in the original statement of Theorem~\ref{Adem2}.
All we have done here is clarify precisely the operad mechanism alluded to in \S0.1.4 by which chains in $N_*(B\Sigma_4)$ act on cocycles $\alpha$.
\paragraph{4.4.5} We have implemented Table Reduction as part of a computer program, along with all the other ingredients needed to make explicit our coboundary formulae for Adem relations.
Note that as explained in \S4.3.3 and discussed in \S3.5.4, for each of the cochains
\begin{equation*}
J = J_\Psi(\widetilde x_q \times \widetilde x_p), \ J_\Psi(\widetilde x_p \times \widetilde x_q), \ J_{(23)}(\widetilde x_q \times \widetilde x_p) \in N_*(E\Sigma_4),
\end{equation*}
it is really the lifts $TR\ J \in {\mathcal S}_4$ of the $\overline{TR}\ \overline{J} $ that we evaluate directly on $\alpha^{\otimes 4}$.
For fixed $(q,p)$, the computer output for the $TR\ J \in {\mathcal S}_4$ is much smaller than the computer output for the $J \in N_*(E\Sigma_4)$.
For a cocycle $\alpha \in H^{-n}(X)$ with fixed small $n$, there is significant further reduction since many surjection generators in ${\mathcal S}_4$ contribute 0 when evaluated on $\alpha^{\otimes 4}$.
Nonetheless, our coboundary formulae for Adem relations quickly get very lengthy, even for cocycles and relations of low degree.\\
This completes our discussion of the main Theorems \ref{Adem1} and \ref{Adem2}.
\section{Combinatorial Appendix}
The first two subsections of the final section of the paper provide proofs of the combinatorial facts about counting partitions that we used to compute the $\smallsmile_n$ operations in $N_*(B\Sigma_2)$ and in $N_*(E\Sigma_2)$.
Then in Subsection 5.3 we provide the details that exploit commutativity of Steenrod operations with suspension to deduce the standard form of Adem relations from the form given in Theorem~\ref{Adem2}.
We find it interesting that this argument proves somewhat more, namely that some of the coboundary terms in our unstable cochain level Adem relations also commute with cochain suspension.
It seems like a good question whether cochain versions of unstable Adem relations can be found that completely commute with cochain suspension.
\subsection{Counting Ordered Partitions}
\paragraph{5.1.1} We begin with the following well-known statement about counting ordered partitions.
The goal is to prove Combinatorial Facts 1 and 2 from \S2.3.2 that were used to evaluate $\smallsmile_n$ products in $N^*(B\Sigma_2)$ and $N^*(E\Sigma_2)$.\\
\begin{lem} \label{l:5.1}
(i): The number of non-negative ordered partitions of an integer $N$ into $M$ summands equals the coefficient of $x^N$ in $(1 + x)^{N+M-1}$, hence is given by the binomial coefficient $\binom {N+M-1}{N}$.
The number of such partitions is also (more obviously) given by the coefficient of $x^N$ in the expansion of $(1 +x + x^2 + \cdots )^M$.\\
(ii): The number of positive ordered partitions of $N$ into $M$ summands is $\binom{N - 1} {N - M} = \binom{N-1}{M-1}$.
\end{lem}
\begin{proof}
(i): We give a correspondence between individual terms of degree $N$ in the expansion of $(1+x)^{N+M-1}$ and non-negative ordered partitions of $N$ into $M$ summands.
The individual terms in the expansion can be viewed as sequences of 0's and 1's of length $N+M-1$, based on whether in each factor $(1+x) $ one selects the $1 = x^0$ or the $x = x^1$.
Such a sequence consists of blocks of 0's and 1's.
Remove one 0 from each block of 0's that lies between two blocks of 1's.
Then the associated ordered partition of $N$ will consist of all the remaining 0's and positive integers corresponding to the number of 1's in each original block of 1's.\\
For the second statement in part (i), in the expansion of $(1 +x + x^2 + \cdots )^M$, to get a term $x^N$ one must choose some power $x^{n_k}$ in the $k^{th}$ factor so that $n_1 + n_2 + \cdots + n_M = N$.\\
(ii): We pass from non-negative partitions to positive partitions by adding 1 to each summand.
Thus, the number of positive ordered partitions of $N$ into $M$ summands is the same as the number of non-negative partitions of $N - M$ into $M$ summands, which is the binomial coefficient $\binom{N-1} {N-M} = \binom{N-1}{M-1}$.
\end{proof}
\paragraph{5.1.2} Now let's consider ordered partitions of $N$ into $M$, $M+1$, or $M+2$ summands, with $M$ specific summands even.
\begin{lem} \label{l:5.2}
(i): The number of non-negative partitions of $N$ into $M$ even summands is the coefficient of $X^N$ in the expansion $(1+x^2)^{N+M-1}$, which is also the coefficient of $x^N$ in $(1 + x^2 + x^4 + \cdots )^M$. \vspace{3pt}
This is the same mod 2 as the coefficient of $x^N$ in $(1 + x + x^2 + \cdots)^{2M}$, which is $\binom{N + 2M -1}{N} = \binom{N+2M-1}{2M-1}$, by Lemma~\ref{l:5.1}(i).\\
(ii): The number mod 2 of positive partitions of $N$ into $M$ even summands is $\binom{N - 1} {N - 2M} = \binom{N - 1}{2M - 1}.$\\
(iii): The number mod 2 of positive partitions of $N$ into $M+1$ summands, all but the first even (or all but the last even) is $\binom{N-1}{N - (2M+1)} = \binom{N-1}{2M}.$ \\
(iv): The number mod 2 of positive partitions of $N$ into $M+2$ summands, all but the first and last even, is $\binom{N-1}{N-(2M+2)} = \binom{N-1}{2M+1}$.
\end{lem}
\begin{proof}
(i) and (ii): The first statement is essentially the same as Lemma~\ref{l:5.1}(i).
We then count mod 2, and use the fact that $(1+ x^2 + x^4 + \cdots )^M \equiv (1 + x + x^2 + ...)^{2M}$ mod 2.
To count positive even partitions, we add 2 to each non-negative term of a partition of $N - 2M$ into $M$ even pieces.\\
(iii): First we count the number mod 2 of non-negative partitions of $N$ into $M+1$ terms where all terms except the first term, or all terms except the last term, are even.
The count is the coefficient of $x^N$ in
$$(1 + x + x^2 + \cdots)(1 +x^2 + x^4 + \cdots)^M \equiv (1 +x + x^2 + \cdots)^{2M+1}.$$
The answer from Lemma~\ref{l:5.1} is $\binom{N+2M}{N}$.
To get positive such partitions, subtract $2M+1$ from $N$.\\
(iv): The mod 2 arithmetic first takes us to the coefficient of $x^N$ in
$$(1+x + x^2 +\ldots)(1+x^2 + x^4 + \cdots)^M(1 + x + x^2 + \cdots) \equiv (1 +x + x^2 + \cdots)^{2M+2}$$
to count non-negative such partitions.
The answer is $\binom{N+2M+1}{N}$.
To get positive such partitions subtract $2M+2$ from $N$, which means subtract 2 for each even piece and 1 for the other two pieces.
\end{proof}
\subsection{Proofs of Theorems \ref{t:binomial repeated} and \ref{t:2.2}}
\paragraph{5.2.1} We now deduce Combinatorial Fact 1 stated in \S2.3.2.
For odd $n = 2m + 1$, we first want to partition $N = j+1$ into $M = m+1$ positive even terms.
The count, explained in Lemma~\ref{l:5.2}(ii) above, is $\binom{j}{2m+1} = \binom{j}{n}$.\\
We have also covered the other part of Combinatorial Fact 1, where $N = i+1$ is partitioned into $m+2$ positive pieces, all but the first and last even.
The answer from Lemma~\ref{l:5.2}(iv) is $\binom{i}{i-(2m+1)} = \binom{i}{n}$.\\
One gets in the same way the claims of Combinatorial Fact 2, when $n = 2m$ is even.
We partition $N = i+1$ into $m+1$ positive pieces, all but the first even.
The count from Lemma~\ref{l:5.2}(iii) is $\binom{i}{i-2m} = \binom{i}{n}$.\\
The combinatorial facts just established imply Theorem~\ref{t:binomial repeated}.
\paragraph{5.2.2} Finally we prove Theorem~\ref{t:2.2}.
In the computation for $E\Sigma_2$, with $|I_1|$ even, we need when $n = 2m$ the number mod 2 of partitions of $N = i+1$ into $m+1$ positive even summands.
By Lemma~\ref{l:5.2}(ii) above, this is $\binom{i}{2m+1} = \binom{i}{n+1}$.
When $n = 2m+1$ and $|I_1|$ is even we need the number mod 2 of partitions of $N = i+1$ into $m+2$ positive summands, all but the last even.
This is also calculated in Lemma~\ref{l:5.2}(iii) as $\binom{i}{2m+2} = \binom{i}{n+1}$.
When $|I_1|$ is arbitrary, the previous counts were $\binom{i}{n}$ in both the $n$ even and $n$ odd cases.
This means the counts with $|I_1|$ odd is the sum mod 2, $\binom{i}{n+1} + \binom{i}{n} = \binom{i+1}{n+1}$, which implies the statements in Theorem~\ref{t:2.2}.
\subsection{Manipulations With the Relations}
\paragraph{5.3.1} We first want to use the relations in Theorem~\ref{Adem2} to prove the Adem relations in their usual form, expressing an inadmissible composition $Sq^aSq^b, a < 2b$, as a sum of admissible compositions,
$$Sq^aSq^b = \sum_i \binom{b-1-i}{a-2i}Sq^{a+b-i}Sq^i.$$
Of course we want cochain level versions of these relations, writing differences of specific cocycles as coboundaries.\\
To keep the notation here consistent with the statement of Theorem~\ref{Adem2}, we will grade cocycles in positive degrees.
In \S2.1.4, we pointed out that on the cocycle level $s(\alpha \smallsmile_i \alpha) = s\alpha \smallsmile_{i+1} s\alpha$, where $s$ is cochain suspension.
This is a strong form at the cocycle level of the commutativity of Squares with suspension.
It then suffices to prove the standard Adem relation on cocycles of very high degree $n$, since, by desuspending, it will also hold on cocycles of lower degree.
In fact, a specific cochain level relation in high degree desuspends to a cochain level relation in lower degrees.
It is unclear how such desuspended relations compare to the unstable relations of Theorem~\ref{Adem2}, although there do exist simply described algorithms for computing such desuspensions in terms of Surj operations or coface operations.\\
First, fix $a$ and $b$.
With $m$ large, we set $n = 2^m-1 + b,\ p = 2^m-1,\ q = 2n -a$.\footnote{We found these values of $n,p,q$ in some lecture notes of J.
Lurie, \cite{13lurie}, although they probably go back to Adem.}
Thus $a = 2n -q,\ b = n-p$.
Suppressing the cocycle $\alpha$ of degree $n$, the cohomology relation in Theorem~\ref{Adem2} is
\begin{equation*}
\sum_\ell \binom{q-\ell}{q-2\ell} Sq^{2n-p-\ell} Sq^{n-q+\ell} \ +\ \sum_\ell \binom{p-\ell}{p-2\ell} Sq^{2n-q-\ell} Sq^{n-p+\ell} \ =\ 0.
\end{equation*}
In the right-half sum, $\ell = 0$ gives $Sq^aSq^b$.
Since $p = 2^m -1$, all its base 2 expansion coefficients are 1's.
Recall that if $A = \sum a_i2^i$ and $B = \sum b_i2^i$, with $a_i, b_i \in \{0, 1\}$, then $\binom{A}{B} = \prod \binom{a_i}{b_i} \pmod{2}$.
Thus, for $\ell> 0$ all the binomial coefficients $\binom{p-\ell}{p-2\ell} \equiv 0 \pmod{2}$, since if $2^i$ is the greatest power of 2 dividing $\ell$ then one will see $\binom{0}{1}$ in the $i^{th}$ factor of the formula for $\binom{p-\ell}{p-2\ell}$.\\
In the left-half sum, set $i = n - q + \ell$.
Then the Square terms are $Sq^{a+b-i}Sq^i$.
The corresponding binomial coefficients are
\begin{align*}
\binom{q-\ell}{q-2\ell} & = \binom{q-\ell} {\ell} = \binom{n-i} {\ell} = \binom{n-i} {q-n+i} \\ & = \binom{n-i}{2n-q-2i} = \binom{n-i}{a-2i} = \binom{2^m +b -1 -i }{a-2i} \\ & \equiv \binom{b-1-i}{a-2i} \pmod2.
\end{align*}
The last congruence holds because $m$ is large, so the $2^m$ just adds an irrelevant 1 in the base 2 expansion.
This proves the usual Adem relation formula.\\
One can also show directly that linear combinations of the relations in Theorem~\ref{Adem2} for any fixed cocycle degree express inadmissible compositions $Sq^aSq^b(\alpha)$ as sums of admissible compositions, plus specific coboundaries.
But that argument is recursive and does not lend itself to determining Adem's general binomial coefficient formulae.
\paragraph{5.3.2} We want to make one more point about the method of \S5.3.1 that finds the exact Adem relations for inadmissible compositions $Sq^aSq^b$ by stabilizing the cocycle dimension and using commutativity of Squares with suspension.
We fixed $a, b$ and chose $n = 2^m-1+b,\ p = 2^m - 1,\ q = 2n - a$.
The integer $m$ can vary.
We then examined the relation of Theorem~\ref{Adem2} for these choices of $(n, p, q)$.
But Theorem~\ref{Adem2} contains more than cohomology information.\footnote{This was the whole point of the paper!} Specifically, Theorem~\ref{Adem2} is a cocycle formula with coboundary terms.\\
Among the coboundary terms are $d(N_{q,p,n}(\alpha))$, affiliated with the right-half Square terms in Theorem~\ref{Adem2}, and $d(N_{p,q,n}(\alpha))$, affiliated with the left-half Square terms.
The affiliations are from Theorem~\ref{Adem1}.
The point of \S5.3.1 was that some binomial arithmetic showed that both halves of the Square terms in Theorem~\ref{Adem2} did not change as $m$ increased.
Specifically, these Square terms gave the usual Adem relations for $Sq^aSq^b$.
Our new point here is that also the expressions for $N_{q,p,n}(\alpha)$ and $N_{p,q,n}(\alpha)$, given in Theorem~\ref{Adem1}, are stable under cochain suspension, as $m$ varies.
First consider
$$N_{q,\, p,\, n}(\alpha) \ =\!\! \sum_{\substack{0< a\le \ell \in \mathbb{Z}[1/2];\ \\ a\equiv \ell\ mod\ \mathbb{Z}}}\binom{p-\ell-a}{p-2\ell} \binom{p- \ell +a}{p-2\ell}Sq^{n -p+\ell +a}(\alpha)\ \smallsmile_{q-p+2\ell+1}\ Sq^{n -p + \ell -a}(\alpha).$$
If $m$ increases by 1, then $n, p$ and $q-p$ increase by $2^m$, and $q$ increases by $2^{m+1}$.
So $n - p$ is unchanged.
Set $M = 2^m$.
Then
$$N_{q+2M,\, p+M,\, n}(s^M\alpha) = s^MN_{q,\, p,\, n}(\alpha).$$
The proof involves binomial arithmetic, similar to what was done in \S5.3.1, and also the cochain suspension formula for $\smallsmile_r$'s mentioned in \S2.1.4 that reads $sx \smallsmile_{r+1} sy = s(x \smallsmile_r y).$ The binomial arithmetic is easy and uses that if $p = 2^m -1$ and $0 < a < \ell \leq p/2$ then $$\binom{2^m+p-\ell -a}{2^m +p-2\ell}\binom{2^m +p-\ell+a}{2^m +p-2\ell} \equiv \binom{p-\ell-a}{p-2\ell} \binom{p-\ell +a}{p-2\ell} \pmod 2,$$ since the $2^m$'s just add irrelevant 1's on the left end of the base two representations of all the other numbers occurring in the binomial coefficients.\\
We also get
$$N_{p+M,\, q+2M,\, n}(s^M\alpha) = s^MN_{p,\, q,\, n}(\alpha).$$
This is trickier.
Note increasing $m$ by 1 decreases both $n-q$ and $p-q$ by $M = 2^m$.
Also, the binomial arithmetic is a bit trickier because in analyzing the summation with binomial coefficients one needs to work with a new variable $j$ with $\ell = M + j$.
This also takes care of the $\smallsmile_{p-q + 2\ell +1}$ product of two Square terms, so that the cochain suspension formula for $\smallsmile_r$'s still works out.\\
It is unclear, to put it mildly, how the other coboundary terms in Theorem~\ref{Adem2} behave under these same $2^m$-fold cochain suspensions.
It seems that it would be quite a nice result if a priori, preferred coboundary formulae for Adem relations could be found that are compatible under cochain suspension.
This would allow constructions of simplicial set three-stage Postnikov tower {\it spectra} that would seem interesting, extending the (easy) constructions of simplicial set, suspension compatible, two-stage Postnikov tower spectra.
The first serious example would be the 2-type of the sphere spectrum, with homotopy groups $\mathbb{Z}, \mathbb{Z}/2, \mathbb{Z}/2$ in degrees 0, 1, 2.
\nocite{24steenrodreduced, 26steenrodepstein}
\bibliographystyle{abbrv}
|
1,116,691,497,730 | arxiv | \section{Introduction}
This paper considers multifidelity estimation of failure probabilities for large-scale applications with expensive-to-evaluate models.
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Failure probabilities are required in, e.g., reliable engineering design and risk analysis. Yet failure probability estimation with expensive-to-evaluate nonlinear models is computationally challenging due to the large number of Monte Carlo samples needed for low-variance estimates.
Efficient failure probability estimation methods aim to reduce the number of samples at which the expensive model is evaluated, e.g., by exploiting variance-reducing sampling strategies, multifidelity/multilevel estimation methods, or sequential sampling approaches.
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Variance reduction can be obtained through importance sampling \cite{owen2013monte}, which allows for order-of-magnitude reductions in the number of samples needed to reliably estimate a small probability. However, importance sampling relies on having a good biasing distribution which in turn requires insight into the system. Surrogate models can provide such insight at much lower computational cost. Multifidelity approaches (see \cite{PWG17MultiSurvey} for a review) that use surrogates for failure probability estimation via sampling have seen great interest recently \cite{dongbin2011,chen_accurate_2013, dongbin2014, Elferson16_MLMCFailure,Ullmann15_MultilevelSubset,Peherstorfer16MFIS,Fagerlund16_MLMCporousflows}, but require that the user selects a good importance sampling density. Multifidelity methods that avoid the selection of a single biasing density and instead use a suite of surrogate models to generate importance sampling densities were proposed in \cite{PKW16MixedMFIS,PKW17MFCE,OwenIS}. Nevertheless, this framework requires all knowledge about the small probability event to be available in the form of biasing densities, and is therefore only applicable to importance sampling estimators.
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Multilevel Monte Carlo~\cite{giles2008multilevel,aslett2017multilevel} methods use a hierarchy of approximations to the high-fidelity model in the sampling scheme. However, those model hierarchies have to satisfy certain error decay criteria, an assumption we do not make here. Subset simulation~\cite{AuBeck_2001SS,papaioannou2015mcmcSS} and line search~\cite{schueller2004critical,angelis2015advanced} can be used directly on the high-fidelity models, and therefore are of a black-box nature.
In this work, in addition to the computationally expensive model, we also have information about the system in form of surrogate models, analytical models, expert elicitation, and reduced models. In other settings where such a variety of information is available, information fusion has been used to combine multi-source probabilistic information into a single estimator, see \cite{clemenWinkler99combiningExperts, Ohagan06ElicitingExpProb, marin10weightingVariance}. Moreover, combining information from multiple models and sources via a weighted multifidelity method can lead to efficient data assimilation strategies \cite{NARAYAN2012}.
Here, we propose a new approach to enable small probability estimation for large-scale, computationally expensive models that draws from prior work in information fusion, importance sampling, and multifidelity modeling.
We use information fusion in combination with multifidelity importance-sampling-based failure probability estimators, where in addition to the variance reduction from importance sampling, we obtain further variance reduction through information fusion.
The proposed multifidelity framework uses the available surrogates to compute multiple unbiased failure probability estimators. We then combine them \textit{optimally} into a new unbiased estimator that has minimal variance amongst all possible linear combinations of those estimators.
The method therefore avoids the selection of the lowest variance biasing density to be used for sampling. Selecting the density that leads to the lowest variance in the failure probability estimator would require additional information, and not even error estimates on the surrogate model would suffice. Thus, we circumvent this step and optimally use all information available to us in form of probability estimators.
This paper is structured as follows: In Section~\ref{sec:lowprob} we illustrate the challenges in small failure probability computation and cover the necessary background material for multifidelity importance sampling used herein. Section~\ref{sec:CompMethods} details our proposed approach of information fusion, importance sampling and multifidelity modeling. We then present in Section~\ref{sec:CDR} a moderately expensive convection-diffusion-reaction test case, where we illustrate the asymptotic behavior of our approach. Section~\ref{sec:resultsjet} discusses a turbulent jet model and demonstrates the computational efficiency of our proposed methods for this computationally expensive model. We close with conclusions in Section~\ref{sec:conclusions}.
\section{Small probability events and importance sampling estimators} \label{sec:lowprob}
We are interested in computing events with small probabilities, e.g., failure events, where the system fails to meet critical constraints. Section~\ref{sec:SmallProb} describes small probability events, Section~\ref{sec:IS} introduces importance sampling and Section~\ref{sec:MFIS} briefly summarizes multifidelity importance sampling.
\subsection{Small probability events} \label{sec:SmallProb}
Let $\Omega$ be a sample space which, together with a sigma algebra and probability measure, defines a probability space.
Define a $d$-dimensional random variable $Z: \Omega \mapsto \mathcal{D} \subseteq \mathbb{R}^{d}$ with probability density $p$, and let $\mathbf{z}$ be a realization of $Z$.
Let $f: \mathcal{D} \subseteq \mathbb{R}^d \mapsto \mathbb{R}^{d'}$ be an expensive-to-evaluate model of high fidelity with corresponding $d'$-dimensional quantity of interest $f(\mathbf{z})\in \mathbb{R}^{d'}$. Let $g: \mathbb{R}^{d'}\mapsto \mathbb{R}$ denote a limit state function that defines failure of the system.
If $g(f(\mathbf{z}))<0$, then $\mathbf{z} \in \mathcal{D}$ is a configuration where the system fails.
This defines a failure set
\begin{equation*}
\mathcal{G}:= \{ \mathbf{z} \in \mathcal{D} \ | \ g(f(\mathbf{z}))<0 \}.
\end{equation*}
Define the indicator function $I_\mathcal{G} : \mathcal{D} \mapsto \{ 0,1 \}$ via
\begin{equation*}
I_\mathcal{G}(\mathbf{z}) = \begin{cases}
1\,,\qquad \mathbf{z} \in \mathcal{G}\, ,\\
0\,,\qquad \text{otherwise}\,.
\end{cases}
\end{equation*}
The standard Monte Carlo estimator of the failure probability
$$
P = \mathbb{E}_p [I_\mathcal{G}[Z]] = \int_{\mathcal{D}} I_\mathcal{G}(\mathbf{z}) p(\mathbf{z}) d\mathbf{z}
$$
uses $n$ realizations $\mathbf{z}_1, \ldots, \mathbf{z}_n$ of the random variable $Z$ and estimates
\begin{equation}
P_n = \frac{1}{n}\sum_{i=1}^n I_\mathcal{G} (\mathbf{z}_i). \label{eq:plainMCest}
\end{equation}
In the special case of small probabilities, standard Monte Carlo may be unfeasible due to the large number of samples needed to obtain good estimators.
Since failure probabilities are generally small, most realizations $\mathbf{z}_i$ will be outside the failure domain $\mathcal{G}$, and conversely, only a small fraction of the $n$ samples lies in the failure region.
The coefficient of variation (also called relative root-mean-squared error) of the estimator $P_n$ is given by
\begin{equation}
e^{\text{CV}} (P_{n}) = \sqrt{\frac{\mathbb{V}[P_n]}{(\mathbb{E}[P_n])^2} } = \sqrt{\frac{P(1-P)}{n P^2} } = \sqrt{\frac{1-P}{n P} } . \label{eq:CVwithP}
\end{equation}
Thus, to obtain estimators with a small coefficient of variation, a large number of samples is necessary. For instance, if the small probability is $P=10^{-4}$ and if we want $e^{\text{CV}}=10^{-1}$ we would need $n=\mathcal{O}(10^{6})$ samples via standard Monte Carlo approaches. This challenge is amplified by the presence of an expensive-to-evaluate model, such as the model of a free plane jet in Section~\ref{sec:resultsjet}.
\subsection{Importance sampling} \label{sec:IS}
Importance sampling achieves variance reduction by using realizations of a random variable $Z': \Omega \mapsto \mathcal{D}$ with probability density $q$. This random variable $Z'$ is chosen such that its probability density function $q$ has higher mass (compared to the nominal density $p$) in the region of the event of interest. For a general introduction to importance sampling, see \cite[Sec.9]{owen2013monte}.
Define the support $\text{supp}(p)=\{ \mathbf{z} \in \mathcal{D} \ | \ p(\mathbf{z})>0 \}$, and let $\text{supp}(p)\subseteq \text{supp}(q)$. Then
\begin{equation}
P = \int_{\mathcal{D}} I_\mathcal{G}(\mathbf{z}) p(\mathbf{z}) d\mathbf{z} = \int_{\mathcal{D}} I_\mathcal{G}(\mathbf{z}) \frac{p(\mathbf{z})}{q(\mathbf{z})} q(\mathbf{z}) d \mathbf{z} \label{eq:Pint}
\end{equation}
is well defined, where $p(\mathbf{z}) / q(\mathbf{z})$ is the \textit{likelihood ratio}---in the context of importance sampling also called \textit{importance weight}.
The importance-sampling estimate of the failure probability $P$ then draws $n$ realizations $\mathbf{z}'_1, \ldots, \mathbf{z}'_n$ of the random variable $Z'$ with density $q$ and evaluates
\begin{equation}
P_n^{\text{IS}} = \frac{1}{n} \sum_{i=1}^n I_\mathcal{G} (\mathbf{z}'_i) \frac{p(\mathbf{z}'_i)}{q(\mathbf{z}'_i)}. \label{eq:IS}
\end{equation}
The variance of the importance sampling estimator is
\begin{equation}
\mathbb{V}[P_n^{\text{IS}}] = \frac{\sigma_q^2}{n}, \label{eq:VarIS}
\end{equation}
where
\begin{equation}
\sigma_q^2 = \int_\mathcal{D} \left ( \frac{I_\mathcal{G} (\mathbf{z}') p(\mathbf{z}') }{q(\mathbf{z}')} - P \right )^2 q(\mathbf{z}')d\mathbf{z}' . \label{eq:ISasympVar}
\end{equation}
If $\text{supp}(p)\subseteq \text{supp}(q)$, and by using \eqref{eq:Pint}, one can show that the importance sampling estimator $P_n^{\text{IS}}$ is an unbiased estimator of the failure probability, i.e.,
\begin{equation*}
\mathbb{E}_q [P_n^{\text{IS}}] = \mathbb{E}_p [I_\mathcal{G}(Z)] = P.
\end{equation*}
The importance sampling estimator $P_n^{\text{IS}}$ has mean $P$ and variance $\sigma_q^2/n$, and by the central limit theorem converges in distribution to the normal random variable $\mathcal{N}(P,\sigma_q^2/n)$.
Constructing a good biasing density that leads to small $\sigma_q^2$ is challenging~\cite{owen2013monte}. We next introduce low-fidelity surrogate models, which are then used to construct biasing densities.
\subsection{Multifidelity Importance Sampling (MFIS)} \label{sec:MFIS}
Recall that by $f:\mathcal{D} \mapsto \mathbb{R}^{d'}$ we denote an expensive-to-evaluate model of high fidelity with corresponding quantity of interest $f(\mathbf{z})\in \mathbb{R}^{d'}$. Let $k$ surrogates
\begin{equation*}
f^{(i)}: \mathcal{D} \mapsto \mathbb{R}^{d'}, \ i=1, \ldots, k
\end{equation*}
of lower fidelities be available, which are cheaper to evaluate than the high-fidelity model $f(\cdot)$. We do not assume any information about the accuracy of the $f^{(i)}(\cdot)$ with respect to the high-fidelity model $f(\cdot)$. Sections~\ref{sec:CDRdiscrete} and \ref{sub:jet_surrogate} detail the specific surrogate models used for the respective applications.
We use the MFIS method (see~\cite{Peherstorfer16MFIS} for details) to obtain $k$ estimators of the failure probability. First, MFIS evaluates the surrogate models $f^{(i)}$ at $m_i$ samples to obtain a surrogate-model specific failure set $\mathcal{G}^{(i)}$. Second, MFIS computes a biasing density $q_i$ by fitting a distribution in form of a Gaussian mixture model to the parameters in the failure set. If no failed samples are found by the surrogate model, i.e., if $\mathcal{G}^{(i)}=\emptyset$, then we set the biasing density to be the nominal density. This leads to $k$ biasing densities $q_1, \ldots, q_k$ from which we get importance sampling estimators
\begin{equation} \label{eq:PniIS}
P_{n_i}^{\text{IS}} =
\frac{1}{n_i} \sum_{j=1}^{n_i} I_\mathcal{G} (\mathbf{z}_{i,j}) \frac{p(\mathbf{z}_{i,j})}{q_i(\mathbf{z}_{i,j})}, \qquad \mathbf{z}_{i,j}\sim q_i, \ j=1,\ldots, n_i,
\end{equation}
for $i=1,\ldots, k $. The variance of the importance sampling estimator is given by \eqref{eq:VarIS} with $n=n_i$ and $\sigma_q=\sigma_{q_i}$, with $\sigma_{q_i}^2$ being the asymptotic variance from \eqref{eq:ISasympVar} with $q=q_i$.
\section{Fusion of multifidelity estimators} \label{sec:CompMethods}
In many practical situations, a range of probability estimators are available, for instance in form of MFIS estimators derived from different biasing densities, in form of analytical models, or estimators derived from expert elicitation~\cite{Ohagan06ElicitingExpProb}.
If one a priori knew which was the lowest variance estimator then a good strategy would be to sample only from that estimator. However, knowing \textit{a priori} which estimator has the lowest variance is a formidable task, and one has to draw samples to assess which estimator has the lowest variance.
In this section, we present our new approach that combines all available estimators in an optimal fashion by solving the following problem.
\begin{problem} \label{probdef}
Given $k$ unbiased estimators, $P_1, \ldots, P_k$ with expected value $P$, i.e. $\mathbb{E}[\Pi] = P, \ i=1, \ldots, k$, find an estimator with expected value $P$ of the form
\begin{equation}
P_{\boldsymbol{\alpha}} = \sum_{i=1}^k \alpha_i \Pi, \label{eq:Palpha}
\end{equation}
such that it attains minimal variance amongst all estimators of the form \eqref{eq:Palpha}. That is, find the optimal weights $\alpha_i \in \mathbb{R}, \ i =1,\ldots,k$ such that
\begin{equation}
\min_{\boldsymbol{\alpha}} \ \mathbb{V}[P_{\boldsymbol{\alpha}}] \quad \textup{s.t.} \quad \mathbb{E}[P_{\boldsymbol{\alpha}}] = P. \label{eq:minprob}
\end{equation}
\end{problem}
\noindent
The fused estimator approach allows to still use information coming from the other (high-variance) estimators, whose samples would have otherwise gone to waste. Moreover, with the proposed method we can estimate small-probabilities for expensive-to-evaluate models by exploiting a variety of surrogates.
We derive expressions for the mean and variance of the fused estimator in Section~\ref{sec:meanVar}. In Section~\ref{sec:OptWeights}, we derive the optimal weights for the fused estimator. Section \ref{sec:Uncorr} then discusses the special case of uncorrelated estimators. Our proposed algorithm is discussed in Section~\ref{sec:fusedIS}, followed by a brief Section~\ref{sec:errComp} that discusses measures of convergence of the estimators.
\subsection{Mean and variance of fused estimator}\label{sec:meanVar}
We start with the observation that if the weights $\alpha_i$ of the fused estimator $P_{\boldsymbol{\alpha}}$ sum to one, then the fused estimator is unbiased:
\begin{equation*}
\sum_{i=1}^k \alpha_i = 1 \quad \Leftrightarrow \quad \mathbb{E}[P_{\boldsymbol{\alpha}}] = \sum_{i=1}^k \alpha_i \mathbb{E}[\Pi] = P \sum_{i=1}^k \alpha_i = P.
\end{equation*}
Let the estimators $P_i$ have corresponding variances $0< \sigma_{i}^2<\infty, \ i=1,\ldots k$. To compute the variance of the fused estimator $P_{\boldsymbol{\alpha}}$ we have to consider covariances between the individual estimators.
Define the Pearson product-moment correlation coefficient as
\begin{equation}
\rho_{i,j} = \frac{\mathbb{C}\text{ov} (\Pi,P_j)}{\sigma_i \sigma_j}, \label{eq:CorrCoef}
\end{equation}
where $\mathbb{C}\text{ov}(\Pi,P_j)=\mathbb{E}[(\Pi- \mathbb{E}[\Pi])(P_j- \mathbb{E}[P_j])] = \mathbb{E}[\Pi P_j] - P^2$.
We also define the symmetric, positive semi-definite \textit{covariance matrix} $\boldsymbol{\Sigma}_{ij} = \mathbb{C}\text{ov}(\Pi, P_j)$ as:
\begin{equation}
\boldsymbol{\Sigma} = \begin{bmatrix}
\sigma_{1}^2 & \sigma_{1}\sigma_{2} \rho_{1,2} &\ldots & \ldots & \sigma_{1}\sigma_{k} \rho_{1,k} \\
\sigma_{2}\sigma_{1} \rho_{2,1} & \sigma_{2}^2 & \sigma_{2}\sigma_{3} \rho_{2,3} & \ldots & \sigma_{2}\sigma_{k} \rho_{2,k} \\
\vdots & \ddots & & & \\
\vdots & & & \sigma_{{k-1}}^2 & \sigma_{{k-1}}\sigma_{k} \rho_{k-1,k} \\
\sigma_{k}\sigma_{1} \rho_{k,1} & \sigma_{k}\sigma_{2} \rho_{k,2} & \ldots & \sigma_{{k}}\sigma_{{k-1}} \rho_{k,k-1} & \sigma_{k}^2 \\
\end{bmatrix}. \label{eq:Mmatrix}
\end{equation}
It is worth noticing that if the estimators $P_1,\ldots, P_k$ are independent, then $\boldsymbol{\Sigma}$ is diagonal. The variance of the fused estimator from \eqref{eq:Palpha} is
\begin{align*}
\mathbb{V}[P_{\boldsymbol{\alpha}}] = \mathbb{V} \left [ \sum_{i=1}^k \alpha_i \Pi \right ]
& = \sum_{i=1}^k \alpha_i^2 \mathbb{V}[\Pi] + 2 \mathbb{C}\text{ov} \left ( \sum_{i=1}^k \alpha_i \Pi, \, \sum_{j=1}^k \alpha_j P_j \right ) \\
& = \sum_{i=1}^k \alpha_i^2 \sigma_i^2 + 2 \sum_{i=1}^k \sum_{j>i}^{k} \alpha_i \alpha_j \sigma_i \sigma_j \rho_{i,j},
\end{align*}
which can be written in vector form as
\begin{equation}
\mathbb{V}[P_{\boldsymbol{\alpha}}] = \boldsymbol{\alpha}^T \boldsymbol{\Sigma} \boldsymbol{\alpha}. \label{eq:varMIScov}
\end{equation}
\noindent In the following section, we provide an explicit formula to find the optimal weights $\boldsymbol{\alpha}$ for the general case of (possibly)-correlated estimators $P_1,\dots,P_k$; while in Section \ref{sec:Uncorr} we discuss the case of independent estimators, such as those constructed with the MFIS method.
\subsection{Optimizing the weights for minimum-variance estimate}\label{sec:OptWeights}
Problem~\eqref{eq:minprob} seeks the optimal $\boldsymbol{\alpha}$ such that the variance in~\eqref{eq:varMIScov} is minimized and $P_{\boldsymbol{\alpha}}$ remains unbiased. In this section, we show that such weights exist, are unique, and present a closed-form solution, provided that the covariance matrix $\boldsymbol{\Sigma}$ is invertible. This is summarized in the following result.
\begin{proposition} \label{prop:weights}
Let $\boldsymbol{P} = [P_1, \ldots, P_k]^T$ be the vector of probability estimators and assume that $\boldsymbol{\Sigma}$ is not singular. Define $\boldsymbol{1}_k = [1, \dots, 1]^T$ as a column-vector of length $k$. The optimization problem \eqref{eq:minprob} has the unique solution
\begin{equation*}
\boldsymbol{\alpha} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1}_k }{ \boldsymbol{1}^T_k \boldsymbol{\Sigma}^{-1} \ \boldsymbol{1}_k }.
\end{equation*}
That is, the minimal variance unbiased estimator $P_{\boldsymbol{\alpha}}$ is such that
\begin{equation*}
P_{\boldsymbol{\alpha}} = \frac{ \boldsymbol{1}^T_k \boldsymbol{\Sigma}^{-1} \ \boldsymbol{P}}{ \boldsymbol{1}^T_k \boldsymbol{\Sigma}^{-1} \ \boldsymbol{1}_k }, \qquad
\mathbb{V}[P_{\boldsymbol{\alpha}}] = \frac{1}{ \boldsymbol{1}^T_k \boldsymbol{\Sigma}^{-1} \ \boldsymbol{1}_k }.
\end{equation*}
\end{proposition}
\begin{proof}
We have seen above that $\sum_{i=1}^k \alpha_i=1$ if and only if $\mathbb{E}[P_{\boldsymbol{\alpha}}] = P$. Define the cost function $J(\boldsymbol{\alpha}) := \mathbb{V}[P_{\boldsymbol{\alpha}}] = \boldsymbol{\alpha}^T \boldsymbol{\Sigma} \boldsymbol{\alpha}$ by using equation~\eqref{eq:varMIScov}. Therefore, the optimization problem~\eqref{eq:minprob} can be written as the quadratic program
\begin{equation}
\min_{\boldsymbol{\alpha}} J(\boldsymbol{\alpha}) = \boldsymbol{\alpha}^T \boldsymbol{\Sigma} \boldsymbol{\alpha}, \quad \text{s.t.} \ \ \boldsymbol{\alpha}^T \boldsymbol{1} =1. \label{eq:optProbAlpha}
\end{equation}
Letting $\mathcal{L}(\boldsymbol{\alpha},\lambda) := \boldsymbol{\alpha}^T \boldsymbol{\Sigma} \boldsymbol{\alpha} + \lambda (\boldsymbol{\alpha}^T \boldsymbol{1} -1) $ denote the Lagrangian cost function associated to \eqref{eq:optProbAlpha}, the optimality conditions are $\nabla_{\boldsymbol{\alpha}} \mathcal{L}(\boldsymbol{\alpha},\lambda) = \boldsymbol{0} $ and $\frac{d \mathcal{L}}{d\lambda}(\boldsymbol{\alpha},\lambda) = 0$.
%
This optimality system is written as
\begin{equation}
\begin{bmatrix} \boldsymbol{\Sigma} & \boldsymbol{1}_k \\ \boldsymbol{1}^T_k & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{\alpha} \\ \lambda \end{bmatrix} = \begin{bmatrix} \boldsymbol{0}_k \\ 1 \end{bmatrix}. \label{eq:QuadProg}
\end{equation}
For invertible $\boldsymbol{\Sigma}$, the unique weights to this quadratic program are then obtained by
\begin{equation}
\boldsymbol{\alpha} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{ \boldsymbol{1}^T_k \boldsymbol{\Sigma}^{-1} \ \boldsymbol{1}_k },
\end{equation}
and the expression for the variance follows by inserting these weights into \eqref{eq:optProbAlpha}. The estimator is obtained by inserting the weights into \eqref{eq:Palpha}.
\end{proof}
The weights can be expressed explicitly in terms of the components of the covariance matrix as
\begin{equation}
\alpha_i = \frac{1}{ \sigma_i^2} \left [\frac{1}{\sum_{l=1}^k \frac{1}{\sigma_l^2}}\left ( 1 + \sum_{l=1}^k \frac{1}{ \sigma_l^2} \sum_{j>l}^{k} \alpha_j \sigma_l \sigma_j \rho_{l,j} \right ) - \sum_{j>i}^{k} \alpha_j \sigma_i \sigma_j \rho_{i,j} \right ]. \label{eq:weightCompNotation}
\end{equation}
Note, that the weights are inversely proportional to the variance of the individual estimators and the weight $\alpha_i$ depends on the covariance between the estimators $\Pi$ and $P_j$. Also, note that if $\Pi$ are correlated some weights may be negative, while for a diagonal $\boldsymbol{\Sigma}$ all weights $\alpha_i$ are \emph{strictly} positive. In the next section, we have a closer look at the uncorrelated case.
\subsection{The special case of uncorrelated estimators} \label{sec:Uncorr}
In the situation where all estimators are uncorrelated, we recover the classical result of the \textit{inverse variance-weighted mean} \cite{meier1953varianceWeighting}.
As a corollary from Proposition \ref{prop:weights} we get the following result.
\begin{corollary}
Consider the setting from Proposition \ref{prop:weights}, and let $\boldsymbol{\Sigma} = \textup{diag}(\sigma_1^2,\ldots, \sigma_k^2)$ be diagonal. Then the unique solution to the optimization problem \eqref{eq:minprob} is given by
\begin{equation}
\alpha_i = \frac{1}{\sigma_i^2 \sum_{i=1}^k \frac{1}{\sigma_i^2}}, \qquad \mathbb{V}[P_{\boldsymbol{\alpha}}] = \frac{1}{\sum_{i=1}^k \frac{1}{\sigma_i^2}} \, . \label{eq:alphaRel}
\end{equation}
\end{corollary}
\noindent A few observations about this special case are in order:
\begin{enumerate}
\item The optimal coefficients $\alpha_i$ of the combined estimator $P_{\boldsymbol{\alpha}}$ are inversely proportional to the asymptotic variance $\sigma_i$ of the corresponding estimator $\Pi$. To reduce the variance via a weighted combination of estimators, smaller weights are assigned to estimators with larger variance.
\item If one variance is small compared to all other ones, say $\sigma_1^2 \ll \sigma_i^2, \ i=2,\ldots,k$, then $\sum_{i=1}^k \frac{1}{\sigma_i^2} \approx \frac{1}{\sigma_1^2}$ so that $\mathbb{V}[P_{\boldsymbol{\alpha}}] \approx \sigma_1^2$. The estimators with large variance cannot reduce the variance of the fused estimator much more.
%
\item If all estimators have equal variance, $\sigma_1^2 = \ldots = \sigma_k^2$, then $\sum_{i=1}^k \frac{1}{\sigma_i^2} = \frac{k}{\sigma_1}$ so that $\mathbb{V}[P_{\boldsymbol{\alpha}}] = \frac{\sigma_1^2}{k} $. Hence, combining the estimators reduces the variance by a factor of $k$.
%
\item Since $0 < \alpha_i<1, \forall i$, it follows from both equations in \eqref{eq:alphaRel} that
\begin{equation}
\mathbb{V}[P_{\boldsymbol{\alpha}}] = \frac{1}{\sum_{i=1}^k \frac{1}{\sigma_i^2}} = {\sigma_i^2} \alpha_i < \sigma_i^2, \forall i \quad \Rightarrow \quad \mathbb{V}[P_{\boldsymbol{\alpha}}] < \min_{i=1,\ldots,k} \sigma_i^2.
\end{equation}
Consequently, we are guaranteed to reduce the variance in $P_{\boldsymbol{\alpha}}$ by combining all estimators in the optimal way described above.
\end{enumerate}
\subsection{Fused multifidelity importance sampling: Algorithm and Analysis} \label{sec:fusedIS}
We now use the general fusion framework to obtain a failure probability estimator. Thus, we solve Problem~\ref{probdef} in the context of importance-sampling-based failure probability estimators so that $\Pi = P_{n_i}^{\text{IS}}$. Our proposed method optimally fuses the $k$ MFIS estimators from \eqref{eq:PniIS}, such that
\begin{equation}
P_{\boldsymbol{\alpha}} = \sum_{i=1}^k \alpha_i P_{n_i}^{\text{IS}}, \label{eq:PalphaIS}
\end{equation}
with the optimal weights chosen as in Proposition~\ref{prop:weights} and $\sum_{i=1}^k n_i = n$. Since estimator $P_{n_i}^{\text{IS}}$ is computed from $n_i$ samples, $P_{\boldsymbol{\alpha}}$ uses $n = \sum_{i=1}^k n_i$ samples.
We now discuss how $P_{\boldsymbol{\alpha}}$ compares to a single importance sampling estimator with $n$ samples. Consider the estimator $P_{j'}^{\text{IS}}$ that uses $n$ samples drawn from a single biasing density $q_{j'}$ for $j' \in \{1, \ldots, k\}$. This estimator would require selection of the lowest biasing density \textit{a priori}, a formidable task. The next results compares $P_{\boldsymbol{\alpha}}$ and $P_{j'}^{\text{IS}}$, and gives a criterion for which the former has lower variance than the latter.
\begin{proposition} \label{prop:compEstimators}
Let $k$ estimators $P_{n_i}^{\text{IS}}$ with $n_1=n_2=\ldots=n_k$ samples be given.
Let $j' \in \{ 1, \ldots, k\}$, and $q_{j'}$ be a biasing density that is used to derive an IS estimator $P_{j'}^{\text{IS}}$ with $n=kn_1$ samples. If
\begin{equation*}
\sigma_{q_j'}^2> \frac{k}{\sum_{i=1}^k \frac{1}{\sigma_i^2}}
\end{equation*}
then the variance of the fused estimator $P_{\boldsymbol{\alpha}}$ in \eqref{eq:PalphaIS} with $n$ samples is smaller than the variance of the estimator with biasing density $q_{j'}$ with $n$ samples, i.e.,
\begin{equation*}
\mathbb{V}[P_{\boldsymbol{\alpha}}] < \mathbb{V}[P_{j'}^{\text{IS}}].
\end{equation*}
\end{proposition}
\begin{proof}
Set $n_i = n/k, \ i=1, \ldots, k$, so that all estimators use the same number of samples. According to equation~\eqref{eq:alphaRel},
\begin{equation*}
\mathbb{V}[P_{\boldsymbol{\alpha}}] = \frac{1}{ \sum_{i=1}^k \frac{n_i}{\sigma_{q_i}^2}} = \frac{k}{n\sum_{i=1}^k \frac{1}{\sigma_{q_i}^2}}
\end{equation*}
as well as $\mathbb{V}[P_{j'}^{\text{IS}}] = \frac{\sigma^2_{q_{j'}}}{n}$, so that
\begin{equation*}
\mathbb{V}[P_{\boldsymbol{\alpha}}] = \frac{k}{n \sum_{i=1}^k \frac{1}{\sigma_{q_i}^2}} < \frac{\sigma^2_{q_{j'}}}{n} = \mathbb{V}[P_{j'}^{\text{IS}}] \qquad \Leftrightarrow \qquad \frac{k}{\sum_{i=1}^k \frac{1}{\sigma_{q_i}^2}} < \sigma^2_{q_{j'}}
\end{equation*}
\end{proof}
The importance sampling estimate~\eqref{eq:PniIS} requires evaluating the high-fidelity model at $n_i$ samples from the biasing density. While not required, we use $n_i = n/k, \ i=1, \ldots,k$ to distribute the computational load evenly. Extension of Proposition~\ref{prop:compEstimators} is straightforward to the case with different number of samples $n_j$ for each estimator $P_j$
The computational procedure is summarized in Algorithm~\ref{alg:MIS}. Here, we denote sampling-based estimates as $\hat{P}_{n_i}^{\text{IS}}$, which are realizations of the estimator $P_{n_i}^{\text{IS}}$.
\begin{algorithm}[H]
\caption{Computing failure probability estimate $\hat{P}_{\boldsymbol{\alpha}}$ via fused importance sampling}
\begin{algorithmic}[1] \label{alg:MIS}
\REQUIRE Nominal distribution $p$, biasing distributions $\{ q_i \}_{i=1}^k$, \# of evaluations $\{ n_i \}_{i=1}^k$, limit state function $g(\cdot)$.
\ENSURE Failure probability estimate $\hat{P}_{\boldsymbol{\alpha}}$ and variance estimate $\mathbb{V}[\hat{P}_{\boldsymbol{\alpha}}]$
%
\FOR[Loop over all surrogates] {$j=1:k$ }
\STATE Draw $\mathbf{z}_{j,1},\ldots, \mathbf{z}_{j,n_j}$ independent realizations from $Z_{j}$ with density $q_j$ and compute
\begin{equation}
\hat{P}_{n_j}^{\text{IS}} = \frac{1}{n_j} \sum_{i=1}^{n_j} I_\mathcal{G} (\mathbf{z}_{j,i}) \frac{p(\mathbf{z}_{j,i})}{q_j(\mathbf{z}_{j,i})} \label{eq:PniIShat}
\end{equation}
\STATE Compute the sample variances
\begin{equation}
\hat{\sigma}_{q_j}^2 = \frac{1}{n_j - 1} \sum_{i=1}^{n_j} \left (I_\mathcal{G}(\mathbf{z}_{j,i})\frac{p(\mathbf{z}_{j,i})}{q_j(\mathbf{z}_{j,i})} - \hat{P}_{n_j}^{\text{IS}} \right )^2 \label{eq:sampleVar}
\end{equation}
\ENDFOR
\STATE Define the vector $\boldsymbol{P} = [\hat{P}_{n_1}^{\text{IS}}, \ldots, \hat{P}_{n_k}^{\text{IS}}]^T$
\STATE Let $\hat{\boldsymbol{\Sigma}} = {\textup diag}(\hat{\sigma}_{q_1}^2/n_1, \ldots, \hat{\sigma}_{q_k}^2/n_k)$
\STATE Compute the fused estimate as in \eqref{eq:optProbAlpha}:
\begin{equation}
\hat{P}_{\boldsymbol{\alpha}} = \frac{ \boldsymbol{1}^T_k \hat{\boldsymbol{\Sigma}}^{-1} \ \boldsymbol{P}}{ \boldsymbol{1}^T_k \hat{\boldsymbol{\Sigma}}^{-1} \ \boldsymbol{1}_k }, \qquad
\mathbb{V}[\hat{P}_{\boldsymbol{\alpha}}] = \frac{1}{ \boldsymbol{1}^T_k \hat{\boldsymbol{\Sigma}}^{-1} \ \boldsymbol{1}_k }
\end{equation}
\end{algorithmic}
\end{algorithm}
\subsection{Error measures and practical computation} \label{sec:errComp}
The failure probability estimate $\hat{P}_{n_i}^{\text{IS}}$ is computed as in \eqref{eq:PniIShat} and the sample variance $\hat{\sigma}_{q_i}^2$ as in \eqref{eq:sampleVar}.
The root-mean-squared-error (RMSE) of the estimate $\hat{P}_{n_i}$ is
\begin{equation}
e^{\text{RMSE}} (\hat{P}_{n_i}) = \sqrt{\frac{\hat{\sigma}_{q_i}^2}{n_i}}, \label{eq:rsme}
\end{equation}
and the relative mean-squared-error, or coefficient of variation is computed as
\begin{equation}
e^{\text{CV}} (\hat{P}_{n_i}) = \sqrt{\frac{\hat{\sigma}_{q_i}^2}{n_i ( \hat{P}_{n_i}^{\text{IS}})^2}}. \label{eq:coefvar}
\end{equation}
\section{Test case: Convection-diffusion-reaction} \label{sec:CDR}
We first consider a PDE model whose solution can be numerically evaluated with moderate computational cost. With this model, we demonstrate the asymptotic behavior of our method because we can afford to sample the high-fidelity model $n = 10^5$ times, which will be too costly for the model in Section~\ref{sec:resultsjet}. The test problem is the convection-diffusion-reaction PDE introduced in Section~\ref{sec:CDRproblem}. Its discretizations and reduced-order models are described in Section~\ref{sec:CDRdiscrete}. Numerical results are presented in Section~\ref{sec:resultsCDR}.
\subsection{Convection-diffusion-reaction PDE model} \label{sec:CDRproblem}
We consider a simplified model of a premixed combustion flame at constant and uniform pressure, and follow the notation and set-up in \cite[Sec.3]{buffoni10MORreactingFlows}. The model includes a one-step reaction of the species
\begin{equation*}
2H_2 + O_2 \ \rightarrow 2 H_2 O
\end{equation*}
in the presence of an additional non-reactive species, nitrogen.
The physical combustor domain is $18mm$ in length ($x$-direction), and $9mm$ in height ($y$-direction), as shown in Figure~\ref{fig:combustor}.
\begin{figure}[H]
\centering
\includegraphics[width=8cm]{cdr_schematic.pdf}
\caption{Set-up of combustor, with details of the boundary conditions in Table~\ref{tbl:bcs}.}
\label{fig:combustor}
\end{figure}
\noindent The velocity field $U$ is set to be constant in the positive $x$ direction, and divergence free. The molecular diffusivity $\kappa$ is modeled as constant, equal and uniform for all species and temperature. The PDE model is given by
\begin{align}
0 & = \kappa \Delta s - U \nabla s + \mathcal{F}(s,\mathbf{z}) \quad \in \tilde{\Omega}
\end{align}
where the state is comprised of the components $s = [T, Y_{H_2}, Y_{O_2}, Y_{H_2O}] $, with the $Y_i$ being the mass fractions of the species (fuel, oxidizer, product), and $T$ denoting the temperature.
Referring to Figure~\ref{fig:combustor}, we have that $\Gamma_D=\Gamma_{D,i} \cup \Gamma_{D,0}$ is the Dirichlet part of the boundary and $\Gamma_N$ combines the top, bottom and right boundary, where Neumann conditions are prescribed. In sum, $\partial \tilde{\Omega} = \Gamma_D \cup \Gamma_N$; the boundary conditions are imposed as given in Table~\ref{tbl:bcs}.
The nonlinear reaction term $\mathcal{F}(s,\mathbf{z}) = [\mathcal{F}_T, \mathcal{F}_{H_2}, \mathcal{F}_{O_2}, \mathcal{F}_{H_2O}](s,\mathbf{z})$ is of Arrhenius type \cite{cuenot96diffusionFlames}, and modeled as
\begin{align}
\mathcal{F}_i(s,\mathbf{z}) & = -\nu_i \left ( \frac{W_i}{\rho} \right ) \left (\frac{\rho Y_F}{W_F} \right )^{\nu_F} \left (\frac{\rho Y_O}{W_O} \right )^{\nu_O} A \exp \left ( - \frac{E}{RT} \right ), \quad i = H_2, \ O_2, \ H_2O\\
\mathcal{F}_T(s,\mathbf{z}) & = Q \ \mathcal{F}_{H_2O}(s,\mathbf{z}).
\end{align}
The parameters of the model are defined in Table~\ref{tbl:param}. The uncertain parameters are the pre-exponential factor $A$ and the activation energy $E$ of the Arrhenius model. The domain for these parameters is denoted as $\mathcal{D}$. In particular, we have that
\begin{equation*}
\mathbf{z} = [A,E] \in \mathcal{D} = [5.5 \times10^{11}, \ 1.5 \times 10^{13}] \times [ 1.5 \times 10^{3}, \ 9.5 \times 10^{3}].
\end{equation*}
\begin{table}
\centering
\caption{Boundary conditions for the combustion model from \cite{buffoni10MORreactingFlows}.}
\begin{tabular}{ c l l}
Boundary & Temperature & Species \\
\hline
$ \Gamma_{D,i} $ & $T=950$K & $Y_{H_2} = 0.0282, \ Y_{O_2} =0.2259, \ Y_{H_2O}=0$ \\
$ \Gamma_{D,0} $ & $T =300$K & $Y_{H_2} = 0, \ Y_{O_2} =0, \ Y_{H_2O}=0$ \\
$ \Gamma_N $ & $\nabla T \cdot \boldsymbol{n} = 0$ & $\nabla Y_i\cdot \boldsymbol{n} = 0 $ \\
\end{tabular}
\label{tbl:bcs}
\end{table}
\begin{table}
\centering
\caption{Parameters for the combustion model from \cite{buffoni10MORreactingFlows}.}
\begin{tabular}{c l l l}
quantity & physical meaning & assumptions & value \\
\hline
$\kappa$ & molecular diffusivity & const., equal, uniform $\forall i$ & $2.0\frac{\text{cm}^2}{\text{s}}$ \\
$U$ & velocity & const. & $50 \frac{\text{cm}}{\text{s}} $ \\
$W_{H_2}$& molecular weight & const. & $2.016 \frac{\text{g}}{\text{mol}}$ \\
$W_{O_2}$& molecular weight & const. & $ 31.9\frac{\text{g}}{\text{mol}}$ \\
$W_{H_2 O}$& molecular weight & const. & $ 18 \frac{\text{g}}{\text{mol}}$ \\
$ \rho $ & density of mixture & const. & $1.39 \times 10^{-3} \frac{\text{g}}{\text{cm}^3}$ \\
$R$ & univ. gas constant & const. & $8.314472 \frac{\text{J}}{\text{mol} \ \text{K}}$\\
$Q$ & heat of reaction & const. & $9800$K \\
$\nu_{H_2}$ & stochiometric coefficient & const. & 2 \\
$\nu_{O_2}$ & stochiometric coefficient & const. & 1 \\
$\nu_{H_2O}$ & stochiometric coefficient & const. & 2 \\
\end{tabular}
\label{tbl:param}
\end{table}
\subsection{Discretization and reduced-order models} \label{sec:CDRdiscrete}
The model is discretized using a finite difference approximation in two spatial dimensions, with 72 nodes in $x$ direction, and 36 nodes in $y$ direction, leading to $10,804$ unknowns in the model.
The nonlinear system is solved with Newton's method. Let $\mathbf{T}(\mathbf{z})$ be the vector with components corresponding to the approximations of the temperature $T(x,y; \mathbf{z})$ at the grid points. The high-fidelity model (HFM) is $f: \mathcal{D} \mapsto \mathbb{R}$ and the quantity of interest is the maximum temperature over all grid points:
\begin{equation*}
f(\mathbf{z}) = \max \ \mathbf{T}(\mathbf{z}).
\end{equation*}
Reduced-order models provide a powerful framework to obtain surrogates for expensive-to-evaluate models. In the case of nonlinear systems, reduced-order models can be obtained via reduced-basis methods \cite{RBbook16}, dynamic mode decomposition \cite{DMDbook16}, proper orthogonal decomposition \cite{berkooz1993proper}, and many others; for a survey, see \cite{MORbook17}.
Here, we compute reduced-order models $f^{(i)}$ for our multifidelity approach via Proper Orthogonal Decomposition and the Discrete Empirical Interpolation Method (DEIM) for an efficient evaluation of the nonlinear term. The training snapshots are generated from solutions to the high-fidelity model on a parameter grid of $50\times50$ equally spaced values $\mathbf{z} \in \mathcal{D}$.
The three surrogate models are built from $2,10,15$ POD basis vectors, and accordingly $2,5,10$ DEIM interpolation points. The corresponding models are denoted as ROM1, ROM2, ROM3, respectively. We denote by $\mathbf{T}_r^{(i)}(\mathbf{z})$ the approximation to the temperature $T(x,y; \mathbf{z})$ via the $i$th ROM.
The surrogate models $f^{(i)}$ are the mappings $f^{(i)}: \mathcal{D} \mapsto \mathbb{R}$ with corresponding quantity of interest denoted as
\begin{equation*}
f^{(i)}(\mathbf{z}) = \max \ \mathbf{T}_r^{(i)}(\mathbf{z}), \quad i=1,\ldots,k.
\end{equation*}
We refer the reader to \cite{buffoni10MORreactingFlows} for more details on the discretization and ROM construction for this convection-diffusion-reaction model.
\subsection{Results for multifidelity fusion of failure probabilities} \label{sec:resultsCDR}
We define a failure of the system when the maximum temperature in the combustor exceeds $2430$K, so that the limit state function is
\begin{equation}
g(f(\mathbf{z})) = 2430 - f(\mathbf{z}),
\end{equation}
and likewise for the reduced-order models $g(f^{(i)}(\mathbf{z})) = 2430 - f^{(i)}(\mathbf{z})$.
To compute the biasing densities, we draw $\hat{m}=20,000$ samples from the uniform distribution on $\mathcal{D}$, compute surrogate-based solutions, and evaluate the limit state function for those solutions. If the limit state function indicates failure of the system for a solution obtained from the $i$th surrogate model, the corresponding parameter is added to $\mathcal{G}^{(i)}$, the failure set computed from the $i$th surrogate model. We compute the biasing densities $q_1,q_2,q_3$ via MFIS (see Section \ref{sec:MFIS}) as Gaussian mixture distributions with a single component. Table~\ref{tbl:timeCDR} shows the computational cost in CPU time of computing the biasing distributions from the various ROMs and the HFM. Computing a biasing density using the high-fidelity model with $\hat{m}=20,000$ samples costs approximately $2.1$ CPU-hours. Constructing the biasing density via the low-fidelity models ROM2 and ROM3 reduces the computational time by a factor of 66 and 58, respectively. Note, that ROM1 is the reduced-order model that is cheapest to execute per model evaluation, but it is also the least accurate. In our case, ROM1 did not produce any samples in the failure region, even after $\hat{m}=10^5$ samples. It is not unexpected that ROM1 is so inaccurate, since only two POD modes are not enough to resolve the important character of this problem. ROM1 is included to demonstrate how the fusion approach can be effective even in the presence of highly inaccurate surrogate models.
\begin{table}[!ht]
\centering
\caption{CPU time to generate the biasing densities, and the number of samples in the failure domain. }
\begin{tabular}{ l | l l l l }
& ROM1 & ROM2 & ROM3 & HFM \\
\hline
\# of samples drawn & $10^5$ & $2\times 10^4$ & $2\times 10^4$ & $2\times 10^4$ \\
\# of samples in failure domain $\mathcal{G}^{(i)}$ & 0 & 13 & 17 & 17 \\
time needed & N.A. & $11.2$[s] & $12.7$[s]& $2.1$[h] \\
\end{tabular}
\label{tbl:timeCDR}
\end{table}
In Figure~\ref{fig:CDR_QoI} we show the quantity of interest, i.e., the maximum temperature. The plots are obtained by generating $m=10^5$ samples from the nominal distribution (left) and the respective biasing distributions (right), and evaluating the HFM at those samples. Figure~\ref{fig:CDR_QoI}, left, shows that the typical range of the quantity of interest is between approximately 1200K and 2440K. However, only the events where the quantity of interest is above 2430K are relevant for the failure probability computation. By using the biasing distributions in Figure~\ref{fig:CDR_QoI}, right, a large portion of the outputs leads to a failure of the system. This indicates that the biasing distributions are successful in generating samples at the failure region of the high-fidelity model.
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.48\textwidth]{CDR_QoInom.pdf}
\hspace{0.3cm}
\includegraphics[width=0.48\textwidth]{CDR_QoI.pdf}
\caption{Quantity of interest $f(\mathbf{z})$ in [K] of HFM ordered by magnitude versus \# of samples $\mathbf{z}$, for $m=10^5$ samples. Left: Samples are from the nominal (uniform) distribution. Right: The parameter samples are drawn from different biasing distributions (biased towards failure above $2430$K). This demonstrates that the biasing distributions are good since the outputs are largely above the failure threshold. Here, ROM1 did not have any parameters in the failure domain, and hence defaulted to being the nominal distribution and is therefore not plotted. }
\label{fig:CDR_QoI}
\end{centering}
\end{figure}
Next, we show results for the fused multifidelity estimator $P_{\boldsymbol{\alpha}}$ with $n$ samples and compare it with importance sampling estimators $\hat{P}_{n_i}^{\text{IS}}$ that only use a single biasing density and also $n$ samples. The fused estimator is obtained via Algorithm~\ref{alg:MIS} with $n_i=\lfloor n/3 \rfloor, i=1,2,3,$ samples by fusing the three surrogate-model-based importance sampling estimators.
For reference purposes, a biasing density is constructed as described above using the HFM with $\hat{m}=20,000$ samples. Based on this density, we compute an importance sampling estimate of the failure probability with $n=10^5$ samples, resulting in $\hat{P}_{10^5}^{\text{IS}} = 8.42 \times 10^{-4}$.
To assess the quality of the fused estimator $P_{\boldsymbol{\alpha}}$, we consider the error measures introduced in Section~\ref{sec:errComp}. In Figure~\ref{fig:CDR_RSME}, left, we show the root mean-squared error of the importance sampling estimators $\hat{P}_{n_i}^{\text{IS}}$ as well as the combined estimator $\hat{P}_{\boldsymbol{\alpha}}$. Figure~\ref{fig:CDR_RSME}, right, shows the coefficient of variation defined in \eqref{eq:coefvar} for the estimators. The fused estimator is competitive in RMSE and coefficient of variation with the estimator using the high-fidelity biasing density, but comes at a much lower computational cost.
Note, that the fused estimator does not use any of the high-fidelity information. We only plotted the high-fidelity estimator for comparison reasons, but the high-fidelity density is not used in our algorithm. Heuristically, we could expect the fused estimator to perform better than the MFIS estimator with high-fidelity-derived biasing density in the following situation. Let the HFM be so expensive that the HF biasing density is built only from a few failure samples, and assume the low-fidelity models are good surrogates, hence able to cheaply explore the failure region. Then the low-fidelity biasing density could be better than the high-fidelity biasing density.
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.495\textwidth]{CDR_RSME.pdf}
\includegraphics[width=0.495\textwidth]{CDR_CoefVar.pdf}
\caption{Left: Root mean-squared error from \eqref{eq:rsme}; Right: Coefficient of variation as defined in \eqref{eq:coefvar} for the convection-diffusion-reaction simulation. }
\label{fig:CDR_RSME}
\end{centering}
\end{figure}
In Table~\ref{tbl:alpha_CDR} we show the weights for the fused estimator $\hat{P}_{\boldsymbol{\alpha}}$. The fused estimator assigns only a small weight $\alpha_1$ to the estimator $\hat{P}_{n_1}^{\text{IS}}$ which uses biasing density $q_1$. This was expected, as the estimator has large variance due to the fact that biasing density $q_1$ is actually the nominal density, see Table~\ref{tbl:timeCDR} as the ROM1 evaluation did not yield any samples in the failure domain.
\begin{table}[H]
\centering
\caption{Weights of the fused estimator $\hat{P}_{\boldsymbol{\alpha}}$ with $n$ samples.}
\begin{tabular}{l | c c c c c c }
& $n=10^2$ & $n=10^3$ & $n=10^4$ & $n=2\times 10^4$ & $n=4\times 10^4$ & $n=10^5$ \\
\hline
$\alpha_1$ & 0 & 0 & 0.005 & 0.001 & 0.002 & 0.005 \\
$\alpha_2$ & 0.587 & 0.471 & 0.331 & 0.294 & 0.415 & 0.742 \\
$\alpha_3$ & 0.413 & 0.529 & 0.664 & 0.705 & 0.583 & 0.253 \\
\end{tabular}
\label{tbl:alpha_CDR}
\end{table}
\subsection{Comparison to subset simulation methods}
To demonstrate the efficiency of our proposed multifidelity method compared to state-of-the-art existing methods in failure probability estimation, we compare our results to subset simulation~\cite{AuBeck_2001SS}, a widely used method for reliability analysis and failure probability estimation. The method defines intermediate failure events
\begin{equation*}
\mathcal{G}_j:= \{ \mathbf{z} \in \mathcal{D} \ | \ g(f(\mathbf{z}))<b_j \}, \ j=1, \ldots, L,
\end{equation*}
for a sequence of threshold levels $b_1 > b_2 > \ldots >b_L=0$ and $L$ being the final level. This ensures that the intermediate failure events are nested as $\mathcal{G}_1 \supset \mathcal{G}_2 \supset\ldots \supset \mathcal{G}_L = \mathcal{G}$. The failure probability can then be expressed as
$$
P = P(I_\mathcal{G}) = \mathbb{E}(I_{\mathcal{G}_1}) \prod_{j=2}^{L} P(I_{\mathcal{G}_j} \vert I_{\mathcal{G}_{j-1}}).
$$
Thus, this method requires sampling from the conditional events $\mathcal{G}_j \vert \mathcal{G}_{j-1}$, and the efficiency of this sampling is pivotal to the success of subset simulation. Markov Chain Monte Carlo (MCMC) methods provide efficient solutions to this problem~\cite{papaioannou2015mcmcSS}. Note, that the $b_j$ cannot be determined in advance, but are found adaptively by specifying an intermediate failure probability $p_0 = P(\mathcal{G}_j \vert \mathcal{G}_{j-1})$. A typical choice is $p_0 = 0.1$ which yields efficient subset simulation results, see~\cite{AuBeck_2001SS}.
Here, we compare our fused importance sampling approach for failure probability estimation to a direct application of subset simulation to the full model. We follow the recent MCMC implementation for subset simulation of \cite{papaioannou2015mcmcSS}. Table~\ref{tbl:SuS} lists the computational results that include the number of levels $L$ that subset simulation needed to arrive at the failure probability estimate, the samples at each level (user defined), the failure probability estimate, and the overall number of samples needed (not known beforehand). All results were averaged over ten independent runs. We also give an approximate coefficient of variation, although we caution that this is \textit{not} the same coefficient of variation defined in \eqref{eq:CVwithP}, since at the intermediate levels, subset simulation produces correlated samples. Thus, we used an approximated coefficient of variation as suggested in ~\cite[Eq.~(19)]{zuev2012bayesianSS}. For a thorough discussion of the coefficient of variation estimation within subset simulation we refer to~\cite[Sec.5.3]{betz2017bayesian}. We observe from Table~\ref{tbl:SuS} that the coefficient of variation monotonically decreases as more samples are added.
%
To compare our proposed multifidelity fusion method with subset simulation, we first note that the estimate from subset simulation $\hat{P}_f$ is biased for finite $N$ (see~\cite[Sec.6.3]{AuBeck_2001SS}), whereas our fused estimator $\hat{P}_\alpha$ is unbiased. Moreover, the numerical results in Table~\ref{tbl:SuS} show that the estimated coefficients of variation are about one order of magnitude larger than the coefficients of variation we reported in Figure~\ref{fig:CDR_RSME}, right. From a computational cost perspective, the estimator with 20,000 samples in subset simulation produces an approximated coefficient of variation of $1.18\times 10^{-1}$ whereas our fused estimator $\hat{P}_{\boldsymbol{\alpha}}$ produces a coefficient of variation of $1.34 \times 10^{-2}$ for the same number of high-fidelity model evaluations. Thus, the fused estimator outperforms subset simulation in this particular example.
%
In sum, our method can successfully take advantage of cheaper low-fidelity methods to get accurate estimators, while the subset simulation method works directly with the full model and therefore does not have access to cheaper surrogate model information.
\begin{table}[H]
\centering
\caption{Results for subset simulation to compute failure probabilities for the convection-diffusion-reaction problem.}
\begin{tabular}{c c c c c }
samples & samples each level & No of levels $L$ & failure Prob. & estimated C.o.V. \\
\hline
2000 & 500 & 4 & $1.06\times 10^{-3}$ & $3.24\times 10^{-1}$ \\
4000 & 800 & 5 & $8.14\times 10^{-4}$ & $2.69\times 10^{-1}$ \\
4000 & 1000 & 4 & $8.80\times 10^{-4}$ & $2.29\times 10^{-1}$ \\
6000 & 1500 & 4 & $9.30\times 10^{-4}$ & $1.92\times 10^{-1}$ \\
10000 & 2000 & 5 & $7.70\times 10^{-4}$ & $1.68\times 10^{-1}$ \\
20000 & 4000 & 5 & $8.22\times 10^{-4}$ & $1.18\times 10^{-1}$ \\
\end{tabular}
\label{tbl:SuS}
\end{table}
\section{Failure probability estimation related to a free plane jet} \label{sec:resultsjet}
We apply the proposed fusion of estimators to quantify the influence of uncertain parameters on the amount of turbulent mixing produced by a free plane jet.
This is a challenging problem, since it involves an expensive-to-evaluate model for which the naive computation of low probabilities requires thousands of hours of computation. We reduce this number significantly with our multifidelity importance sampling framework via fusion of estimators.
The remainder of this section is organized as follows. Section \ref{sec:jetintro} introduces the free plane jet, followed by details of the model and its governing equations in Section~\ref{sub:jet_model}. The uncertain parameters and quantity of interest are defined in Section~\ref{sub:jet_uncertain}. The low-fidelity surrogate models used in this investigation are discussed in Section~\ref{sub:jet_surrogate}. Finally, the results for multifidelity fusion of small probability estimators are presented in Section~\ref{sub:jet_results}.
\subsection{Large-scale application: Free plane jet} \label{sec:jetintro}
Free turbulent jets are prototypical flows believed to represent the dynamics in many engineering applications, such as combustion and propulsion. As such, free jet flows are the subject of several experimental \cite{Gutmark76_planeJet, Gutmark78_impJet, Krothapalli81_mixing}
and numerical investigations \cite{Zhou99_RANS, Ribault99, Stanley02_mixing, Klein03_DNSplanejet, klein15} and constitute an important benchmark for turbulent flows.
Our expensive-to-evaluate model of a free plane jet is based on the two-dimensional incompressible Reynolds-averaged Navier-Stokes (RANS) equations, complemented by the $k-\epsilon$ turbulence model.
Although a RANS model does not resolve all relevant turbulent features of the flow, it represents a challenging large-scale application for the computation of small probabilities. We use this model to investigate the influence of five uncertain parameters on the amount of turbulent mixing produced by the jet. We quantify turbulent mixing using a relatively simple metric: the integral jet width. One of the uncertain parameters is the Reynolds number at the inlet of the jet, which is assumed to vary from 5,000 to 15,000. The other four uncertain parameters correspond to coefficients of the $k-\epsilon$ turbulence model and its boundary condition, as detailed in Section~\ref{sub:jet_uncertain}. Figure~\ref{fig:jet_description} shows a flow field typical of the cases considered here.
\begin{figure}[!ht]
\begin{center}
\subfigure[Contours of turbulent kinetic energy.]{\includegraphics[width=0.49\textwidth]{jet_k}}
\subfigure[Streamlines colored by the intensity of the velocity.]{\includegraphics[width=0.50\textwidth]{jet_stream}}
\end{center}
\caption{Flow field of a two-dimensional plane jet at Reynolds number 10,000, computed with standard coefficients of $k-\epsilon$ turbulence model.}
\label{fig:jet_description}
\end{figure}
\subsection{Modeling and governing equations} \label{sub:jet_model}
We consider a free plane jet in conditions similar to the ones reported in \cite{Klein03_DNSplanejet, klein15}.
Namely, the flow exits a rectangular nozzle into quiescent surroundings with a prescribed top-hat velocity profile and turbulence intensity.
The nozzle has width $D$, and is infinite along the span-wise direction.
The main difference between the free plane jet we considered here and the one described in \cite{Klein03_DNSplanejet, klein15} is the Reynolds number at the exit nozzle.
Here the Reynolds number varies between 5,000 and 15,000.
Our simulation model computes the flow in a rectangular domain $\Omega$ located at a distance $5D$ downstream from the exit of the jet nozzle, as illustrated in Figure~\ref{fig:jet_setup}.
By doing so, modeling the conditions at the exit plane of the jet nozzle is avoided.
Instead, direct numerical simulation data are used to define inlet conditions at the surface $\Gamma_{\text{in}}$.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=3.5in]{setup_jet}
\end{center}
\caption{Illustration of the free plane jet setup.
The diameter of the nozzle is denoted by $D$. The simulation domain $\Omega$ is composed of a $30D \times 10D$ box situated at a distance $5D$ downstream to the nozzle exit.}
\label{fig:jet_setup}
\end{figure}
The dynamics are modeled with the incompressible Reynolds-averaged Navier-Stokes equations, complemented by the $k-\epsilon$ turbulence model \cite{Launder74_ke}:
\begin{align}
(\boldsymbol{v} \cdot \nabla)\boldsymbol{v}
+ \dfrac{1}{\rho}\nabla p
- \nabla \cdot ((\nu + \nu_t)\bar{\bar{S}}(\boldsymbol{v}))
&= 0,
\label{eq:jet_mom}\\
\nabla \cdot \boldsymbol{v} &= 0,
\label{eq:jet_mass}\\
\boldsymbol{v} \cdot \nabla k
- 2\nu_t(\bar{\bar{S}}(\boldsymbol{v}):\bar{\bar{S}}(\boldsymbol{v}))
+ \epsilon
- \nabla \cdot \left(\left(\nu + \dfrac{\nu_t}{\sigma_k}\right)\nabla k \right)
&= 0,
\label{eq:jet_k}\\
\boldsymbol{v} \cdot \nabla \epsilon
- 2C_{1\epsilon}\dfrac{\epsilon\nu_t}{k}(\bar{\bar{S}}(\boldsymbol{v}):\bar{\bar{S}}(\boldsymbol{v}))
+ C_{2\epsilon}\dfrac{\epsilon^2}{k}
- \nabla \cdot \left(\left(\nu + \dfrac{\nu_t}{\sigma_{\epsilon}}\right) \nabla \epsilon \right)
&= 0,
\label{eq:jet_e}
\end{align}
where $\boldsymbol{v} = [v_x, v_y]$ denotes the velocity vector, $p$ denotes pressure, $\rho$ is the density, $\nu$ is the kinematic viscosity, and $\bar{\bar{S}}$ is the strain rate tensor given by
\begin{equation*}
\bar{\bar{S}}(\boldsymbol{v}) = \dfrac{1}{2}(\nabla \boldsymbol{v} + (\nabla \boldsymbol{v})^T).
\end{equation*}
In the $k-\epsilon$ turbulence model, $k$ denotes the turbulent kinetic energy, $\epsilon$ denotes the turbulent dissipation, and $\nu_t$ denotes the turbulent kinematic viscosity, defined as
\begin{equation}
\nu_t = C_{\mu}\dfrac{k^2}{\epsilon} \label{eq:nu_t}.
\end{equation}
The coefficients\footnote{We use $\sigma_k$ and $\sigma_\epsilon$ here as model coefficients, which is typical notation in fluids community. These are only used in this section, and throughout the paper $\sigma$'s are variances.} $C_{\mu}$, $C_{1\epsilon}$, $C_{2\epsilon}$, $\sigma_k$, $\sigma_{\epsilon}$ in \eqref{eq:jet_k}--\eqref{eq:nu_t} are either considered as uncertain parameters, or are functions of uncertain parameters, as detailed in Section~\ref{sub:jet_uncertain}.
At the inlet surface $\Gamma_{\text{in}}$ Dirichlet boundary conditions are imposed.
Data obtained by the direct numerical simulation described in \cite{klein15} (Reynolds number 10,000) are used to determine reference inlet profiles for velocity, $\boldsymbol{v_{\text{ref}}}$, and for turbulent kinetic energy, $k_{\text{ref}}$.
Inlet conditions are allowed to vary by defining a velocity intensity ($U$) scale, which is applied to the reference profiles.
Turbulent dissipation at the inlet is estimated by assuming a mixing length model.
Thus, the boundary conditions at the inlet surface are given by
\begin{align*}
\boldsymbol{v}|_{\Gamma_{\text{in}}} &= U\boldsymbol{v_{\text{ref}}}, &
k|_{\Gamma_{\text{in}}} &= U^2k_{\text{ref}}, &
\epsilon|_{\Gamma_{\text{in}}} = C_{\mu}\dfrac{k^{3/2}}{\ell_m},
\end{align*}
where $\ell_m$ denotes the mixing length parameter.
At the symmetry axis surface, $\Gamma_{\text{sym}}$, no-flux boundary conditions are imposed through a combination of Dirichlet and Neumann conditions of the form
\begin{align*}
v_y|_{\Gamma_{\text{sym}}} &= 0, &
\left.
\dfrac{\partial v_x}{\partial n}
\right|_{\Gamma_{\text{sym}}} &= 0, &
\left.
\dfrac{\partial k}{\partial n}
\right|_{\Gamma_{\text{sym}}} &= 0, &
\left.
\dfrac{\partial \epsilon}{\partial n}
\right|_{\Gamma_{\text{sym}}} &= 0.
\end{align*}
Finally, at the surface $\Gamma_{\text{ff}}$ ``far-field'' conditions that allow the entrainment of air around the jet are imposed through weak Dirichlet conditions, as detailed in \cite{Villa17}.
The complete model includes additional features that make it more amenable to numerical discretization.
The most delicate issue in the solution of the RANS model is the possible loss of positivity of the turbulence variables.
To avoid this issue, we introduce an appropriately mollified (and thus smoothly differentiable) max function to ensure positivity of $k$ and $\varepsilon$.
In addition, if inflow is detected at any point on the far-field boundary, the boundary condition is switched from Neumann to Dirichlet by means of a suitably mollified indicator of the inflow region.
Finally, we stabilize the discrete equations using a strongly consistent stabilization technique (Galerkin Least Squares, GLS, stabilization) to address the convection-dominated nature of the RANS equations.
The complete formulation is shown in \cite{Villa17}.
The model equations described above are solved numerically using a finite element discretization.
The discretization is implemented in FEniCS~\cite{AlnaesBlechta2015a} by specifying the weak form of the residual, including the GLS stabilization and mollified versions of the positivity constraints on $k$ and $\epsilon$ and the switching boundary condition on the outflow boundary.
To solve the nonlinear system of equations that arise from the finite element discretization, we employ a damped Newton method.
The bilinear form of the state Jacobian operator is computed using FEniCS's symbolic differentiation capabilities.
Finally, we use pseudo-time continuation to guarantee global convergence of the Newton method to a physically stable solution (if such solution exists)~\cite{Kelley98}.
The finite element solver is detailed in \cite{Villa17}.
\subsection{Uncertain parameters and quantity of interest} \label{sub:jet_uncertain}
In this investigation five uncertain parameters are considered: velocity intensity at inlet\footnote{Since we keep other physical parameters constant, by varying the velocity intensity we effectively change the Reynolds number.} ($U$), mixing length at inlet ($\ell_m$), and the $k-\epsilon$ turbulence model coefficients $C_{\mu}$, $C_{2\epsilon}$, and $\sigma_k$:
\begin{equation*}
\mathbf{z} = [U, \ell_m, C_{\mu}, C_{2\epsilon}, \sigma_k].
\end{equation*}
The parameter domain is $\mathbf{z} \in \mathcal{D} = [0.5, 1.5 ] \times [ 0.05, 0.15 ] \times [0.01, 0.15 ] \times [ 1.1, 2.5] \times [ 0.5, 2.5 ] $, and the nominal distribution of parameters is uniform in $\mathcal{D}$.
The other two coefficients of the $k-\epsilon$ turbulence model, $C_{1\epsilon}$ and $\sigma_{\epsilon}$, are also uncertain but do not vary independently.
According to Dunn et al.~\cite{dunn:2011}, empirical evidence suggests that $C_{1\epsilon}$ is related to $C_{2\epsilon}$ by
\begin{equation*}
C_{1\epsilon} = \dfrac{C_{2\epsilon} - 0.8}{1.8}.
\end{equation*}
In addition, as noted in \cite{dunn:2011, guillas:2014}, the log-law implies that $\sigma_{\epsilon}$ must follow from
\begin{equation*}
\sigma_{\epsilon} = \dfrac{\kappa^2}{\sqrt{C_{\mu}}(C_{2\epsilon} - C_{1\epsilon})},
\end{equation*}
where $\kappa=0.41$ is the von K\'arman constant.
\begin{figure}[h!]
\begin{center}
\subfigure[{$\mathbf{z} =[0.53, 0.07, 0.13, 1.20, 0.91]$}, $w=1.13$.]{\includegraphics[width=0.45\textwidth, trim={1.5in 0.5in 1.5in 0.5in}, clip]{k_05499}}
\subfigure[{$\mathbf{z} =[0.86, 0.13, 0.08, 1.21, 1.08]$}, $w=2.38$.]
{\includegraphics[width=0.45\textwidth, trim={1.5in 0.5in 1.5in 0.5in}, clip]{k_07119}}\\
\subfigure[{$\mathbf{z} =[0.96, 0.10, 0.08, 1.15, 1.59]$}, $w=2.78$.]{\includegraphics[width=0.45\textwidth, trim={1.5in 0.5in 1.5in 0.5in}, clip]{k_05296}}
\subfigure[{$\mathbf{z} =[1.11, 0.14, 0.12, 2.49, 0.69]$}, $w=4.13$.]{\includegraphics[width=0.45\textwidth, trim={1.5in 0.5in 1.5in 0.5in}, clip]{k_19598}}
\includegraphics[width=0.70\textwidth]{jet_legend}
\end{center}
\caption{Samples of the flow solution computed at different points of the input parameter space $\mathcal{D}$.
The plots show contours of turbulent kinetic energy and velocity streamlines. The white bars denote the integral jet width $w$ associated with each case.}
\label{fig:jet_samples}
\end{figure}
The quantity of interest is the integral jet width measured at $x = 27.5D$:
\begin{equation} \label{eq:w}
w(\boldsymbol{v}; \mathbf{z}) = \dfrac{1}{v_{x_0}D}\int_0^{10D} v_x(x=27.5D,y; \mathbf{z})\,dy,
\end{equation}
where $v_{x_0} = v_x(x=27.5D,y=0; \mathbf{z})$.
Figure~\ref{fig:jet_samples} illustrates a typical solution behavior for this turbulent jet by plotting contours of the turbulent kinetic energy for selected samples in $\mathcal{D}$.
\subsection{Simplified-physics surrogate models} \label{sub:jet_surrogate}
We consider four surrogate models to represent the dynamics of the free plane jet flow.
The models are based on two distinct computational grids (fine and coarse), and on two representations of turbulence effects.
The fine computational grid contains 10,000 elements and 5,151 nodes, while the coarse grid contains 2,500 elements and 1,326 nodes.
Furthermore, the models are based either on the complete $k-\epsilon$ turbulence model described in the previous section, or on a prescribed turbulent viscosity field.
In the latter case, the turbulent viscosity field is estimated by a linear interpolation based on 243 conditions that span the input parameter space $\mathcal{D}$ on a uniform grid (3 points along each of the 5 dimensions). At each of these 243 conditions, the turbulent viscosity field is computed with the $k-\epsilon$ turbulence model and the fine computational grid.
The following four low-fidelity models are increasingly complex in terms of either modeled physics or grid resolution:
\begin{itemize}
\item \textbf{LFM1--CI}: Coarse, interpolated; combines the interpolated turbulence viscosity field with the coarse computational grid (3,978 degrees of freedom); average computational time $25$s;
\item \textbf{LFM2--FI}: Fine, interpolated; combines the interpolated turbulence viscosity field with the fine computational grid (15,453 degrees of freedom); average computational time $72$s;
\item \textbf{LFM3--CKE}: Coarse $k-\epsilon$; combines the $k-\epsilon$ turbulence model with the coarse computational grid (6,630 degrees of freedom); average computational time $109$s;
\item \textbf{HFM}: High-fidelity model; combines the $k-\epsilon$ turbulence model with the fine computational grid (25,755 degrees of freedom); average computational time $590$s.
\end{itemize}
Note that the models based on an interpolated turbulent viscosity field run four to eight times faster than the corresponding models based on the $k-\epsilon$ turbulence model.
This speedup results from eliminating \eqref{eq:jet_k}--\eqref{eq:jet_e} from the governing equations, which leads to a reduction in the total number of degrees of freedom (elimination of variables $k$ and $\epsilon$) and simplifications in the numerical discretization.
Let $\boldsymbol{v}_i$, $i = $ HFM, LFM1, LFM2, LFM3, denote the velocity field computed with the models above.
The high-fidelity model is the mapping from the inputs to the quantity of interest (jet width from \eqref{eq:w}) for a velocity field computed with the most complex representation of the flow dynamics, $\boldsymbol{v}_{\text{HFM}}$:
\begin{equation*}
f: \mathcal{D} \mapsto \mathbb{R}, \qquad f(\mathbf{z}) = w(\boldsymbol{v}_{\text{HFM}}; \mathbf{z}).
\end{equation*}
The surrogate models are defined in a similar fashion as
\begin{equation*}
f^{(i)}: \mathcal{D} \mapsto \mathbb{R}, \qquad f^{(i)}(\mathbf{z}) = w(\boldsymbol{v}_i; \mathbf{z}), \quad i = \text{LFM1, LFM2, LFM3}.
\end{equation*}
\subsection{Results for multifidelity fusion of small probability estimators}
\label{sub:jet_results}
We define a design failure when the jet width is below the value 0.98.
Hence, the limit state function is given by
\begin{equation}
g(f(\mathbf{z})) = f(\mathbf{z}) - 0.98.
\end{equation}
We compute the biasing distributions $q_{i}$ for $ i=\text{LFM1, LFM2, LFM3}$ from the three low-fidelity surrogate models via MFIS (see Section~\ref{sec:MFIS}). For each surrogate, we draw $\hat{m}=20,000$ parameter samples from the uniform distribution on $\mathcal{D}$ and evaluate the limit state function applied to the resulting quantity of interest. If the limit state function indicates failure of the system for a solution obtained from the $i$th surrogate model, the corresponding parameter is added to $\mathcal{G}^{(i)}$, the failure set computed from the $i$th surrogate model. We then fit a multivariate Gaussian to the samples in $\mathcal{G}^{(i)}$, resulting in the biasing densities $q_\text{LFM1}, q_\text{LFM2}, q_\text{LFM3}$.
Evaluation of the limit state function with the threshold value of 0.98 resulted in few samples in the failure region, so we increased it to 1.12 to obtain more samples to compute the biasing density from. For the three surrogate models and the high-fidelity model, the $\hat{m}=20,000$ evaluations yield 21, 21, 30 and 76 samples, respectively, where the QoI falls below that increased threshold. This strategy yields an efficient biasing density as we see below.
%
As reference, we repeat the same process with the high-fidelity model, resulting in the biasing distribution $q_{\text{HFM}}$.
First, we investigate the quality of the biasing distributions. For reference, Figure~\ref{fig:jet_QoI}, left, shows the result of $10^3$ uniform sample evaluations with the four computational models. Note that hardly any samples are below the failure threshold. In contrast, the quantity of interest computed from samples of the four biasing distributions is shown in Figure~\ref{fig:jet_QoI}, right. The biasing distributions give between 10\%-50\% of the 1000 drawn samples in the failure domain.
%
Note, that the y-axis scaling of both figures is different, which also shows that the biased samples result in a tighter range of QoI values than the unbiased samples. Thus, the biasing distributions are indeed biased towards the failure region, and therefore the multifidelity strategy provides a viable way of saving computational time to inform a biasing distribution.
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.49\textwidth]{jet_5d_tmvn_QoInom.pdf}
\includegraphics[width=0.49\textwidth]{jet_5d_tmvn_QoI.pdf}
\caption{Quantity of interest, the width of the jet at $x=27.5D$, for $n=10^3$ samples. Left: The input parameters are drawn from the nominal distribution. Right: The input parameters are drawn from different biasing distributions. Note the large portion of samples falling below the failure threshold, and the different scaling of the $y$-axis.}
\label{fig:jet_QoI}
\end{center}
\end{figure}
The reference failure probability is computed via importance sampling with ${n=10^4}$ samples drawn from the HFM biasing distribution and is ${\hat{P}_{7,500,q_{\text{HFM}}}^{\text{IS}} = 7.25\times 10^{-4}}$.
%
We compute the estimators $P_{n_i}^{\text{IS}}, i=1,2,3$ with $n_i$ samples using the biasing densities $q_\text{LFM1}, q_\text{LFM2}, q_\text{LFM3}$ derived from the three surrogate models. We obtain the fused multifidelity estimator $P_{\boldsymbol{\alpha}}$ as described above in Algorithm~\ref{alg:MIS} with $n_i=\lfloor n/3 \rfloor, i=1,2,3,$ samples by fusing the three surrogate-model-based importance sampling estimators. The fused estimator thus uses a total of $n$ samples. We compare these estimators with an estimator that uses $n$ samples from the HFM biasing density $q_{\text{HFM}}$. The estimators and the error measures are averaged over three independent runs.
%
The coefficient of variation \eqref{eq:coefvar} is shown in Figure~\ref{fig:jet_RSME}, left. The biasing density derived from the high-fidelity model yields the best estimator among all the models, as expected. The fused estimator yields a better coefficient of variation than LFM2 and LFM3, shows almost identical convergence as the estimator using $q_\text{LFM1}$.
%
Table~\ref{tbl:alpha} shows the three weights for the fused estimator $\hat{P}_{\boldsymbol{\alpha}}$ as given in Proposition~\ref{prop:weights}, according to which, the estimates with the lowest variance get assigned the largest weights.
\begin{table}[!ht]
\centering
\caption{Weights of the fused estimator $\hat{P}_{\boldsymbol{\alpha}}$ with $n$ samples.}
\begin{tabular}{l | c c c c c}
& $n=300$ & $n=600$& $n=900$& $n=1200$ \\
\hline
$\alpha_1$ & 0.500 & 0.502 & 0.610 & 0.900\\
$\alpha_2$ & 0.448 & 0.044 & 0.055 & 0.057\\
$\alpha_3$ & 0.052 & 0.453 & 0.335 & 0.043\\
\end{tabular}
\label{tbl:alpha}
\end{table}
The CPU-hours to compute the biasing densities via this approach are shown in Figure~\ref{fig:jet_RSME}, right. Since MC methods are embarrassingly parallel, any practical implementation can take advantage of this. Our numerical experiments were parallelized on a computing cluster with 55 nodes. Each node is a quad-core Intel Xenon E5-1620 with 3.6 GHz and 10MB Cache. The nodes have either 32GB or 64GB RAM.
%
To put the CPU-hours savings achieved by using the high-fidelity model versus the lower-fidelity models to construct biasing densities into perspective, we see that using LFM1 reduces the computational cost by 96\%, LFM2 by 88\% and LFM3 by 81.5\%. If we are using a fused estimator of all three models, we still save more than 65\% computational effort compared to using the HFM, see Figure~\ref{fig:jet_RSME}, right.
%
This significant time difference can have important implications for engineering practice, as it translates into faster evaluation time and savings in CPU-hours .
\begin{figure}[H]
\begin{centering}
\includegraphics[width=0.49\textwidth]{jet_5d_tmvn_CoefVar.pdf}
\includegraphics[width=0.49\textwidth]{jet_5d_tmvn_jet_5d_runtime.pdf}
\caption{Free plane jet application with five uncertain parameters. Left: Coefficient of variation for the different estimators. Right: CPU-hours for construction of the biasing densities used in the estimators in the left plot. }
\label{fig:jet_RSME}
\end{centering}
\end{figure}
\section{Conclusions} \label{sec:conclusions}
We enabled the estimation of small probabilities for expensive-to-evaluate models via a new approach drawing from importance sampling, multifidelity modeling and information fusion. The effectiveness of our proposed approach is demonstrated on a convection-diffusion-reaction PDE, where asymptotic numerical results could be obtained. The strength of the proposed framework is then shown on the target application of the turbulent jet, a challenging problem for small-probability computation due to its high computational cost.
The proposed framework was illustrated for the special case of importance-sampling based estimators, but applies to a much broader class of estimators, as long as the estimators are unbiased. An investigation of correlated estimators and the effect of correlation for variance reduction would be an interesting future direction.
By fusing different estimators, we avoid the difficult biasing density selection problem. We also showed that this strategy always outperforms sampling from the worst biasing density. The numerical results suggest that the fused estimator is often comparable to an estimator that samples from the best biasing density only.
\section*{Acknowledgements}
The authors thank Prof. M. Klein for sharing the DNS data in \cite{klein15} with us.
This work was supported by the Defense Advanced Research Projects Agency [EQUiPS program, award W911NF-15-2-0121, Program Manager F. Fahroo]; the Air Force [Center of Excellence on Multi-Fidelity Modeling of Rocket Combustor Dynamics, award FA9550-17-1-0195]; and the US Department of Energy, Office of Advanced Scientific Computing Research (ASCR) [Applied Mathematics Program, awards DE-FG02-08ER2585 and DE-SC0009297, as part of the DiaMonD Multifaceted Mathematics Integrated Capability Center].
\bibliographystyle{abbrv}
|
1,116,691,497,731 | arxiv | \section{Introduction}
\label{sec1}
The relaxation of closed quantum systems towards equilibrium is a fundamental problem in many-body physics. It is particularly challenging to fully understand this macroscopic process on the basis of microscopic properties \cite{Srednicki1994,Rigol2008,Dziarmaga2010,Polkovnikov2011}. Here, ultracold atomic quantum gases provide an exceptional experimental platform due to the nearly perfect isolation from their environment and the excellent control on a microscopic level. In particular, the possibility to prepare well-defined states far from equilibrium, as well as widely tunable Hamiltonians, has recently attracted a lot of attention, e.g. prethermalization \cite{Berges2004,Gring2012,Langen2013}, relaxation in strongly interacting lattice systems \cite{Kinoshita2006,Hofferberth2007,Cheneau2012,Trotzky2012,Lux2013} and the interplay between thermal and condensate fraction of multicomponent bosons have been studied \cite{Erhard2004}.
Spinor quantum gases are of particular interest since the spin offers an additional degree of freedom, giving rise to complex dynamics involving different relaxation processes on different time scales. Ultracold bosonic quantum gases have been intensively studied and exhibit a rich variety of effects such as texture formation and spin dynamics in spinor Bose-Einstein condensates \cite{Schmaljohann2004,Griesmaier2005,Sadler2006,Klempt2009,Kronjaeger2010}, which can be well described theoretically using a multi-component Gross-Pitaevskii-Equation \cite{MurPetit2006,Santos2006,StamperKurn2012}. Recently, collective spin dynamics was also observed in a thermal Bose gas \cite{Pechkis2013}. Fermions, in contrast, are governed by Pauli blocking and reveal a different behavior. So far, most experiments studied spin 1/2 fermions e.g. the BEC-BCS crossover \cite{Regal2004,Zwierlein2005}, thermodynamic and transport properties \cite{Sommer2011,Koschorrek2013}, collective excitations \cite{Du2008,Natu2009} and magnetic ordering \cite{Jo2009,Conduit2011,Pekker2011,Zhang2011}. Spin-related phenomena in multi-component Fermi gases ($F>1/2$) have been investigated only recently, in the context of spin-mixing dynamics on individual sites of an optical lattice \cite{Krauser2012}, or collective coherent excitations in a trapped system \cite{Heinze2013,Dong2013,Krauser2014}. The latter has been proven to be well described within a Boltzmann equation. The question, how a large-spin fermionic many-body system reaches an equilibrium state via relaxation involving spin and spatial degree of freedom, has not been addressed.
\begin{figure*}[t]
\centering
\includegraphics[width=17.2cm]{Fig1_Relaxation.pdf}
\caption{Schematic description of the relaxation process in a large-spin Fermi gas involving spin and spatial degrees of freedom. (a) The ten spin states of \textsuperscript{40}K. (b) A typical spin-changing collision in the center-of-mass frame and another collision forbidden by the Pauli exclusion principle. (c) Top: Initially all atoms are prepared in a binary spin mixture $m=\pm1/2$. Spin-changing collisions distribute atoms among all other spin states until an approximately balanced population is reached. The Fermi energies for each two-component subsystem is lower than the initial Fermi energy. Bottom: Time evolution of the spatial density for each spin component.}
\label{fig1}
\end{figure*}
In this paper, we study the relaxation dynamics of a trapped fermionic quantum gas of $^{40}$K atoms with large spin of $F=9/2$. Starting from an initial mixture with only a few spin states occupied, we observe a rich relaxation dynamics leading to a redistribution of the atoms among all available spin states. We study the intermediate regime between the collisionless and the hydrodynamic limit.
In the collisionless limit, interactions are weak and can be taken into account on a mean-field level, while the hydrodynamic limit is characterized by stronger interactions which ensure local equilibrium. The dynamics in this intermediate regime is governed by different processes on very different time scales. We identify these processes by deriving a Boltzmann equation from the microscopic Hamiltonian of the large-spin system. This approach describes the time evolution of the system on the level of single particles in contact with the bath of the many-body system \cite{Gallis1990,Hornberger2003}. This corresponds to the intuitive expectation that the system acts as a bath for its own subsystems.
We present a detailed comparison between numerical simulations and experimental data and find good agreement. Our analysis includes the dependence of the relaxation on density as well as on magnetic field. Whereas a higher density enhances the spin relaxation, we find a suppression of spin-changing processes at large magnetic fields due to the quadratic Zeeman shift. The latter effect can be used to control the loss of particles from the subsystem defined by the initially occupied spin components into the initially empty spin states.
Generally, we observe that the relaxation within a subset of spin states, driven by incoherent spin-conserving collisions, happens on a much faster time scale than the redistribution among the spin components due to spin-changing incoherent collisions. The reason is that the spin-changing collisions are driven by the relatively small part of the interactions that breaks the $SU(10)$ symmetry between the spin states. Thus, we encounter a situation similar to prethermalization \cite{Berges2004}, where first a prethermal state is reached, approximately conserving the initial occupations of the single spin states, before the redistribution among all spin states due to slight symmetry breaking sets in. This separation of time scales also allows us to monitor the increase of (effective) temperature within the subsystem of the initially populated spin states, as it is caused by dissipation into empty spin states.
\section{Relaxation processes in a large-spin system}
\label{sec2}
We perform measurements in a quantum degenerate gas of fermionic \textsuperscript{40}K, which has total spin $F=9/2$ in its hyperfine ground state, yielding ten spin states $m=-9/2\ldots+9/2$, as depicted in FIG.~\ref{fig1}(a). We prepare an atomic sample with two spin states occupied (see Appendix \ref{app:A1} for details), confined in a spin-independent dipole trap. Due to the broken $SU(N)$ symmetry in \textsuperscript{40}K resulting from spin-dependent scattering lengths, spin-changing collisions can occur. A microscopic collision process is depicted in FIG.~\ref{fig1}(b): Two particles collide and exchange both spin $m$ and momentum $k$: $(|m_1,k_1\rangle+|m_2,k_2\rangle \rightarrow |m'_1,k'_1\rangle+|m'_2,k'_2\rangle)$. The total spin $S$, the total magnetization $M=m_1+m_2$ as well as the total momentum $k_1+k_2$ have to be conserved in this process. As a particular fermionic feature, the Pauli exclusion principle has to be obeyed, i.e. $m_1 \neq m_2$ and $m'_1 \neq m'_2$. The interplay between the differential quadratic Zeeman energy $\propto m_1^2+m_2^2-{m'}^2_1-{m'}^2_2$ and interaction energy determines whether spin-changing collisions are likely or suppressed. In the presence of spin-changing collisions, the atoms will in general relax into a steady state with population in all ten spin states. Hence, preparation of an initial non-equilibrium state with only a few spin states populated will lead to complex dynamics, in which more and more spin states are gradually occupied (see FIG.~\ref{fig1}(c)). It is a compelling question how the system relaxes towards a steady state.
In FIG.~\ref{fig2} we show exemplarily an experimentally obtained time evolution of the spin occupations in our system. Here, the initial spin configuration is a superposition of all ten spin states created by rotating a mixture of the states $m=\pm1/2$ using rf-pulses \cite{Krauser2014}.
We can clearly identify three different processes occurring on three different time scales:
(i) We observe coherent spin-changing oscillations with a periodicity on the order of hundred ms. (ii) These oscillations are damped with a rate on the order of several hundred ms. (iii) Beyond this, we observe a slow redistribution among the ten spin states on a much longer time scale on the order of tens of seconds.
In the following, we derive a Boltzmann equation, which reproduces the experimentally observed effects and enables us to distinguish, which scattering processes are responsible for each effect. We show that the coherent oscillation (i) are a mean-field effect driven by forward scattering, where $\lbrace k_1,k_2\rbrace=\lbrace k'_1,k'_2\rbrace$. Their damping (ii) is dominated by spin-conserving non-forward collisions $\lbrace k_1,k_2\rbrace\neq\lbrace k'_1,k'_2\rbrace$ and $\lbrace m_1,m_2\rbrace=\lbrace m'_1,m'_2\rbrace$, which lead to a momentum redistribution within the Fermi sea without changing the spin configuration. The long-term redistribution (iii) is governed by non-forward spin-changing collisions, which change the momentum distribution within the Fermi sea as well as the spin configuration $\lbrace k_1,k_2\rbrace\neq\lbrace k'_1,k'_2\rbrace$ and $\lbrace m_1,m_2\rbrace\neq\lbrace m'_1,m'_2\rbrace$.
\begin{figure}[t]
\centering
\includegraphics[width=8.6cm]{Fig2_Relaxation.pdf}
\caption{(a) Measurement of damped spin oscillations and subsequent relaxation towards equilibrium (b), observed in a 3D fermionic quantum gas with large spin. Depicted is the time evolution of the relative populations of all spin-components $\pm m$, starting from an initial superposition of all ten spin states. For the exact experimental configuration, see Appendix \ref{app:A1}. Solid lines are guides-to-the-eye. Note the three time scales of (i) the spin oscillations, (ii) their damping and (iii) the subsequent relaxation of the total system. The redistribution among all spin states occurs on a time scale of $10\,\text{s}$. The magnetic field is $B=0.17\,\text{G}$, particle number $N=4.9\times10^5$ and temperature $T/T_\text{F}=0.22$.}
\label{fig2}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=8.6cm]{Fig3_Relaxation.pdf}
\caption{Numerical simulation of coherent oscillations, damping and relaxation in the 1D case. The initial spin configuration is the same as in FIG.~\ref{fig2}. Axial trapping frequency is $\omega_x=2\pi\times 84\,\text{Hz}$ and radial frequencies are $\omega_{y,z}=2\pi\times 47\,\text{kHz}$, particle number $N=100$ per tube at temperature $T/T_\text{F}=0.2$ and magnetic field $B=1.5\,\text{G}$. As in FIG.~\ref{fig2}, we observe three time scales related to oscillations, damping and relaxation.}
\label{fig3}
\end{figure}
The abovementioned Boltzmann equation includes all these collision processes and captures the non-equilibrium dynamics in a general fashion, applicable to trapped weakly-interacting gases with arbitrary spin. In this approach, the single-particle dynamics is treated as an open system in contact with the environment represented by all the other particles. An approach for deriving a Boltzmann equation was applied successfully to the description of spin dynamics in liquid Hydrogen and Helium \cite{Corruccini1972,Johnson1984,Bashkin1981,Levy1984,Lhuillier1982I,Lhuillier1982II,OwersBradley1997} and later in spin 1/2 Fermi gases \cite{Fuchs2003, Piechon2009}. In this paper we generalize this approach to one- and three-dimensional systems with large spin, accounting for the quadratic Zeeman effect (QZE).
In general, a kinetic equation or Boltzmann equation is used to describe the time evolution of the single-particle density matrix $\hat\rho$. It has the form
\begin{equation}
\frac{d}{dt}\hat\rho-\frac{1}{i\hbar}\left[\hat\rho,\hat H_0\right]=I_\text{coll}\left[\hat\rho\right].
\label{eq:1}
\end{equation}
Here, $\hat H_0$ denotes the single-particle Hamiltonian
\begin{equation}
\hat H_0=\frac{\hat p^2}{2M}+\frac12 M\omega^2\hat x^2+Q\hat S_z^2,
\label{eq:h0}
\end{equation}
which contains the kinetic energy, the harmonic trapping potential and the quadratic Zeeman splitting $Q$ induced by a homogeneous magnetic field. The term $I_\text{coll}\left[\hat\rho\right]$ on the right hand side of (\ref{eq:1}) is called the collision term and is derived from two-particle contact interaction. Due to total spin conservation, collisions are best described in the basis of total spin $|S,M\rangle=\sum_{m_1,m_2} |m_1,m_2\rangle\langle m_1 m_2|SM\rangle$ with the short notation for Clebsch-Gordan coefficients $\langle m_1 m_2|SM\rangle\equiv\langle F,m_1;F,m_2|S,M\rangle$, which we use throughout this paper. In general, scattering in each channel of total spin $S$ depends on a different s-wave scattering length $a_S$. Due to antisymmetrization of the total wave function, s-wave scattering with odd $S$ is forbidden. Thus for ${}^{40}\mathrm{K}$ there are five different scattering lengths present for $S=0,2,4,6,8$ \cite{Krauser2012}. In each collision channel defined by $|S,M\rangle$, particles interact with a contact interaction of strength $g_S^\text{3D}=\frac{4\pi\hbar^2}{M}a_S$, which is used in all 3D calculations. We also consider a 1D system, in which the motion in two transversal directions is frozen out completely by a tight trapping potential (characterized by radial frequencies $\omega_{x,y}$) such that the effective 1D contact interaction parameter is given by $g_S^\text{1D}=2\hbar\sqrt{\omega_y\omega_z}a_S$ (see appendix~ \ref{app:A1} for details). The notation $g_S\equiv g^\text{1D}_S$ for this quantity is used throughout this paper. In \textsuperscript{40}K, the $a_S$ range from 120 to 170 Bohr radii.
We obtain an explicit expression for the collision term in (\ref{eq:1}) using the method originally developed by Lhuillier and Lalo\"e \cite{Lhuillier1982I,Lhuillier1982II,Fuchs2003} for transport properties in Helium. In this approach, collisions are treated as single ``atomic beam'' experiments, where the colliding particles are assumed to be uncorrelated before and after a collision, reminiscent of Boltzmann's original molecular chaos hypothesis, but the scattering process itself is treated on a full quantum level. This approximation is valid for dilute gases where the mean time between collisions is long and the particle number is large. In this regime, binary collisions can be described by the $T$-matrix, which connects the two-body density matrices before and after a collision. Subsequently, the description is reduced to a single-particle level by tracing out the second particle, similar to tracing out a thermal bath in studies of collisional decoherence \cite{Gallis1990,Hornberger2003}.
We calculate the kinetic equation (\ref{eq:1}) in its phase-space representation, where the single-particle density matrix $\rho_{mn}(x,x')$ is expressed by the Wigner function
\begin{equation}
W_{mn}(x,p)=\int\frac{dy}{2\pi\hbar}e^{\frac{ipy}{\hbar}}\rho_{mn}(x+\tfrac{y}{2},x-\tfrac{y}{2}).
\end{equation}
Note that we performed the transformation only with respect to the spatial degrees of freedom. With respect to spin, denoted by the indices, it retains the form of a single-particle density matrix.
The derivation is carried out in detail in section \ref{sec4} and involves a semiclassical gradient expansion of the Wigner function in position and momentum space leading to an equation in matrix form given by
\begin{widetext}
\begin{equation}
\frac{d}{dt} W(x,p)+\partial_0 W(x,p)+\frac{i}{\hbar}\left[Q S_z^2+V^\text{mf}(x), W(x,p)\right]-\frac12\left\lbrace\partial_x V^\text{mf}(x),\partial_p W(x,p)\right\rbrace=I_\text{coll}(x,p)
\label{eq:4}
\end{equation}
where the collision integral reads
\begin{align}
I_{mn}^\text{coll}&(x,p)=-\frac{M}{\hbar^2}\sum_{abl}\left\lbrace\int_{q^2>\epsilon_1}\!\!\!dq\frac{\tilde U_{malb}}{\sqrt{q^2+\Delta_{mlab}}}W_{an}(x,p)W_{bl}(x,p-q)+\int_{q^2>\epsilon_2}\!\!\!dq\frac{\tilde U_{nalb}}{\sqrt{q^2+\Delta_{nlab}}}W_{ma}(x,p)W_{lb}(x,p-q)\right\rbrace\nonumber\\*
&+\frac{M}{\hbar^2}\sum_{abcdl}\int_{q^2>\epsilon_3} dq\frac{U_{malb} U_{ncld}}{\sqrt{q^2+\Delta_{mnlabcd}}} W_{ac}(x,p-\tfrac12 (q-\sqrt{q^2+\Delta_{mnlabcd}}))W_{bd}(x,p-\tfrac12(q+\sqrt{q^2+\Delta_{mnlabcd}})).
\label{eq:5}
\end{align}
\end{widetext}
Here, we define the coupling constants
\begin{align}
U_{acbd}=\sum_{S,M}g_S\left\langle ab|SM\right\rangle\left\langle SM|cd\right\rangle\\*
\tilde U_{acbd}=\sum_{S,M}g^2_S\left\langle ab|SM\right\rangle\left\langle SM|cd\right\rangle
\end{align}
and denote energy shifts induced by the quadratic Zeeman splitting as $\Delta_{abcd}=4MQ(a^2+b^2-c^2-d^2)$ and $\Delta_{mnlabcd}=2MQ(m^2+n^2+2l^2-a^2-b^2-c^2-d^2)$. The infrared cutoffs are given by $\epsilon_1=\frac{MU_{malb}}{\hbar}-\Delta_{mlab}$, $\epsilon_2=\frac{MU_{nalb}}{\hbar}-\Delta_{nlab}$ and $\epsilon_3=\frac{MU_{nalb}}{2\hbar}(U_{malb}+U_{ncld})-\Delta_{mnlabcd}$.
Equation (\ref{eq:4}) contains several terms, each describing a different dynamical process. The free particle motion in the trap is described by $\partial_0=\frac pM\partial_x-M\omega_x^2x\partial_p$. The leading interaction term appears in the commutator $[\cdot,\cdot]$. The commutator drives coherent spin dynamics through the interplay of the quadratic Zeeman effect and a spin-dependent mean-field potential resulting from forward scattering:
\begin{equation}
\label{eq:8}
V_{mn}^\text{mf}(x)=2\sum_{ab}U_{mnab}N_{ab}(x),
\end{equation}
a function of the density $N(x)=\int dp W(x,p)$. In our large-spin system, described by several scattering lengths $a_S$, it is helpful to decompose the mean-field potential (\ref{eq:8}) into two contributions. The first contribution is symmetric with respect to all $N$ spin states and proportional to the mean scattering length. It conserves the occupations of the different spin components. The second term contains that part of the interactions that breaks the $SU(N)$ symmetry between the spin states and describes spin-changing processes. It depends on differences of scattering lengths only and is, thus, typically much smaller than the symmetric term.
The commutator in Eq.~(\ref{eq:4}) vanishes unless the Wigner function possesses off-diagonal elements indicating spin coherence. Moreover, the symmetric spin-conserving mean-field interactions can only contribute if the Wigner function describes an inhomogeneous spin state.
The mean-field potential also appears in the anticommutator $\lbrace\cdot,\cdot\rbrace$. This term results from the subleading order of the semiclassical gradient expansion (where also the spin-independent trapping potential appears) and it is generally smaller than the commutator. It describes spin-dependent forces, that modify the kinetics in the trap. The collision integral (\ref{eq:5}) describes effects beyond mean-field that result from non-forward scattering and generates a dynamics that appears incoherent on the level of a single-particle description. It is quadratic in the scattering lengths. Again we have to distinguish between $SU(N)$ symmetric spin-conserving collision processes on the one hand and spin-changing collisions on the other. The latter processes are described by those terms for which the quadratic Zeeman shifts $\Delta$ are non-zero and they are much smaller than the former, since they depend on the relatively small differences between the scattering lengths only.
\begin{figure}[t]
\centering
\includegraphics[width=8.6cm]{Fig4_Relaxation.pdf}
\caption{Comparison of spin relaxation for 1D and 3D. The initial spin configuration is a mixture of $m=\pm1/2$. (a) Experimental data in a 1D geometry (circles) compared to numerical results (lines) from the 1D Boltzmann equation (\ref{eq:4}) and (dots) from a 1D version of the single-mode approximation (\ref{eq:sma}). Axial trapping frequency is $\omega_x=2\pi\times 84\,\text{Hz}$ and radial frequencies are $\omega_{y,z}=2\pi\times 47\,\text{kHz}$, particle number $N=100$ per tube at temperature $T/T_\text{F}=0.2$ and magnetic field $B=0.12\,\text{G}$. Inset: The system approaches a steady state for longer times. (b) Experimental data (circles) in a 3D configuration compared to calculations (lines) in single-mode approximation (\ref{eq:sma}), $\vec\omega=2\pi\times(33,33,137)\,\text{Hz}$, $N=1.3\times10^5$ and $T/T_\text{F}=0.15$ at $B=0.34\,\text{G}$.}
\label{fig4}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=8.6cm]{Fig5_Relaxation.pdf}
\caption{Density dependence of the spin relaxation rate in 3D, with an initial mixture of atoms in $m=\pm 1/2$. The spin-changing rate is obtained by fitting the solution of coupled rate equations to experimental (points) and theoretical (lines) data. Theoretical values are obtained using the single-mode approximation using Eq.~(\ref{eq:sma}). The magnetic field is $B=0.11\,\text{G}$. We experimentally tune the density by changing the particle number, keeping the temperature constant at $T/T_\text{F}=0.26$.}
\label{fig5}
\end{figure}
The 1D equation (\ref{eq:4}) allows for a numerical treatment with standard methods. An exemplary result is depicted in FIG.~\ref{fig3}, and shows the relaxation dynamics starting from the same initial superposition as in FIG.~\ref{fig2}, but in a 1D setup. The comparison of the two figures allows us to assign each of the three different processes visible to one of the terms in the Boltzmann equation:
(i) The coherent oscillations are driven by the commutator in (\ref{eq:4}), which is linear in differences of scattering lengths and describes forward collisions. (ii) The damping of coherent phenomena arises from the spin-conserving part of the collision integral (\ref{eq:5}), which is quadratic in the scattering lengths. We have checked that spatial dephasing is not responsible as it is suppressed by the dynamically induced long-range nature of mean-field interactions induced by the rapid particle motion in the trap \cite{Ebling2011,Krauser2014}. (iii) The long-term relaxation originates from the spin-changing non-forward collisions in the collision term, quadratic in differences of scattering lengths. Spin-conserving forward scattering does not play a role in the dynamics, it only has a noticeable effect, if spatial symmetry is broken by a magnetic field gradient, as in studies of spin-waves \cite{Fuchs2003,Natu2009,Heinze2013}, which is not the case in our setup. The anticommutator in Eq.~(\ref{eq:4}) leads to a mean-field driven correction to the trapping potential which is however negligible in the experiments considered here.
The collision integral (\ref{eq:5}) enables us to determine whether our system is in the collisionless, hydrodynamic or an intermediate regime. The average collision time in the 3D setup $\tau_\text{3D}\sim (4\pi a^2 n_p v_T)^{-1}$ \cite{Piechon2009}, with the relevant scattering length $a$, peak density $n_p$ and thermal velocity $v_T=\sqrt{k_BT/M}$, ranges from $\sim 30\,\text{ms}$ to $\sim 150\,\text{ms}$ for spin-conserving collisions and $\sim 3\,\text{s}$ to $\sim15\,\text{s}$ in the spin-changing case. Compared to the average trapping frequency of
$\bar\omega=(\omega_x\omega_y\omega_z)^{1/3}\approx2\pi\times 58\,\text{Hz}$, we obtain values for $\bar\omega\tau_\text{3D}$ between $11$ and $55$ for the spin-conserving collisions and between $1100$ and $5500$ in the spin-changing case. The lowest and highest values of $\bar\omega\tau_\text{3D}$ are reached for the lowest and highest densities shown in FIG.~\ref{fig5} respectively. This means we may approach the hydrodynamic regime, where the collision rate is larger than $\bar\omega$ and local equilibrium can be established. On the other hand our system becomes almost collisionless regarding the spin-changing collisions. Generally we are in an intermediate regime. In the 1D case, collision times $\tau_\text{1D}\sim (n_p\omega_y\omega_z a^2/v_T)^{-1}$ are on the order of $0.35\,\text{ms}$ and $35\,\text{ms}$ respectively, meaning that $\omega\tau_\text{1D}\sim 0.2$ and $\omega\tau_\text{1D}\sim 20$, concerning spin-conserving and spin-changing collisions respectively. Hence with respect to the former, the system would be hydrodynamic. However, it is still in an intermediate regime regarding the redistribution of particles among the spin states driven by spin-changing collisions.
\section{Dissipative redistribution of spin occupations}
\label{sec3}
In the following, we focus on the long-term spin relaxation shown in FIG.~\ref{fig2}(b) and FIG.~\ref{fig3}, while recent experiments have studied spin oscillations and their damping \cite{Krauser2014}. In order to restrict the dynamics to this process, we initially prepare a spin mixture consisting of only the spin states $m=\pm1/2$ without coherences. In this case, the coherent oscillations driven by the commutator in Eq.~(\ref{eq:4}) are absent and the Wigner function $W_{mn}$ remains diagonal at all times. In the following we investigate theoretically and experimentally this spin relaxation dynamics in 3D as well as 1D systems. For a direct comparison between theory and experiment, we realize a 1D system employing a deep 2D optical lattice, which confines the atoms into tight elongated tubes \cite{Fertig2005,Kinoshita2006} as described in Appendix \ref{app:A1}. As shown in FIG.~\ref{fig4}(a), the system gradually occupies all spin states and evolves towards a state of almost equal spin populations on a time scale of milliseconds. As a key result, we can well reproduce the experimentally observed dynamics using the full 1D Boltzmann equation without free parameters.
For harmonically trapped 3D systems, where all trap frequencies are about equal, we derive the full 3D version of equation (\ref{eq:4}) as well [see Eq.~(\ref{eq:28}) in section \ref{sec4}]. However, numerical simulations of this equation are too demanding numerically. Nevertheless, the trap-induced motion of the particles is considerably faster than mean-field or relaxation dynamics, which averages the spatial dependence of the interaction via dynamically created long-range interactions. The Wigner function then approximately separates into a product $W_{mn}(\bm x,\bm p,t)\approx M_{mn}(t)\cdot f_0(\bm x,\bm p)$ \cite{Ebling2011,Pechkis2013,Krauser2014}. The spatial part is assumed to be time independent and given by the initial equilibrium distribution $f_0(\bm x,\bm p)=(\exp(\frac{1}{k_BT}[\frac{\bm p^2}{2M}+\frac12 M (\omega_x^2 x^2+\omega_y^2 y^2+\omega_z^2 z^2)-\mu])+1)^{-1}$. We substitute this expression into the 3D kinetic equation (\ref{eq:28}) with the appropriate collision term. Hence, for negligible magnetic fields we find an equation for the matrix $M_{mn}(t)$ given by
\begin{equation}
\frac{d}{dt}M_{mn}=-\lambda\sum_{abcd} T_{mn}^{abcd} M_{ac}M_{bd},
\label{eq:sma}
\end{equation}
where
\[T_{mn}^{abcd}=\frac{M}{4\pi\hbar^4}\left(\tilde U'_{mabd}\delta_{nc}+\tilde U'_{ncbd}\delta_{ma}-\sum_l U'_{malb} U'_{ncld}\right)\]
and
\begin{equation}
\lambda=\frac1N\int d\bm r\int d\bm p\int d\bm q |\bm q|f_0(\bm r,\bm p)f_0(\bm r,\bm p-\bm q).
\label{eq:lambda}
\end{equation}
The 3D coupling constants are given by
\begin{equation}
U'_{acbd}=\frac{4\pi\hbar^2}{M}\sum_{S,M}a_S\left\langle ab|SM\right\rangle\left\langle SM|cd\right\rangle
\end{equation}
and
\begin{equation}
\tilde U'_{acbd}=\frac{16\pi^2\hbar^4}{M^2}\sum_{S,M}a^2_S\left\langle ab|SM\right\rangle\left\langle SM|cd\right\rangle.
\end{equation}
Note that in the 1D case the single-mode approximation has a similar form given by $T_{mn}^{abcd}=\frac{M}{\hbar^2}\left(\tilde U_{mabd}\delta_{nc}+\tilde U_{ncbd}\delta_{ma}-\sum_l U_{malb} U_{ncld}\right)$.
For both single-mode approximations, the quadratic Zeeman shift has been neglected in the above equations, which are thus valid for small magnetic fields only (See Appendix \ref{app:G} for full equations). In FIG.~\ref{fig4} we compare results from a single-mode approximation with experiments in a harmonically trapped Fermi gas yielding a surprisingly good agreement without free parameters. Note the qualitatively comparable behavior on different time scales of milliseconds for 1D and seconds for 3D. On the contrary the damping of the coherent spin oscillations visible in FIG.~\ref{fig2}(a) and FIG.~\ref{fig3} is not described by this approach. This results from the assumption for the single-mode approximation that it completely neglects the multi-mode character of the fermionic many-body system and thus cannot account for spatial redistribution via lateral scattering events.
The high degree of control over all crucial parameters allows for a detailed investigation of the spin redistribution. To obtain further insight into the relaxation mechanisms, we measure the relaxation dynamics (as exemplarily shown in FIG.~\ref{fig4}) for different densities while keeping $T/T_\text{F}$ constant. With higher density the collision rate increases and the relaxation process accelerates, as shown in FIG.~\ref{fig5}. The measured rates correspond to the redistribution of the initially populated components $m=\pm 1/2$ into $m=\pm 3/2,\pm 5/2$ and are well reproduced using the single-mode approximation (\ref{eq:sma}). The rate of spin-changing collisions increases with increasing density in accordance with the density dependence of the integral $\lambda$ (\ref{eq:lambda}).
\begin{figure}[t]
\includegraphics[width=8.6cm]{Fig6_Relaxation.pdf}
\caption{Dependence of spin relaxation on magnetic field. (a) Experimental data, obtained from a 3D experiment (circles) and theoretical results from a single-mode approach (lines). Spin populations are measured after $2\,\text{s}$. (b) Spin populations after $2\, \text{ms}$, as obtained from full 1D simulations. The inset sketches how the interplay of differential QZE and Fermi energy determines the probabilities for lateral spin-changing collisions.}
\label{fig6}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=8.6cm]{Fig7_Relaxation.pdf}
\caption{Temperature increase due to spin redistribution. (a) Time evolution of the temperature difference between a closed (high magnetic field at $B=7.6\,\text{G}$) and a maximally open system (low magnetic field at $B=0.12\,\text{G}$). The shaded area serves as a guide-to-the-eye. The particle number is $N=3.9\times 10^5$ at an initial temperature $T=0.24\,T_\text{F}=65\,\text{nK}$. (b) Results from a simulation of the 1D equation (\ref{eq:4}) at magnetic fields $B=0.1\,\text{G}$ and $B=8\,\text{G}$. The method used to obtain the temperature is discussed in Appendix \ref{app:H}.}
\label{fig7}
\end{figure}
As a second important parameter of the system, we investigate the influence of the magnetic field on the relaxation process. As the Zeeman energy of an atom pair changes during a spin-changing collision, a strong magnetic field suppresses this process by increasing the energy difference between the initial and final spin configuration. In FIG.~\ref{fig6}(a), we depict the experimentally obtained populations of the spin components after $2\,\text{s}$ as a function of the magnetic field strength (see Appendix \ref{app:A2} for details) and compare them to single-mode [FIG.~\ref{fig6}(a)] and 1D calculations [FIG.~\ref{fig6}(b)] after $2\,\text{ms}$. In both cases, the general behavior is very similar and shows a suppression of spin-changing collisions for large magnetic fields. Spin configurations with high values of $|m|$ are energetically significantly separated from the initially populated $m =\pm{1/2}$ and are only occupied at very low field strengths. By changing the magnetic field we can thus tune the magnitude of spin-changing collisions relative to the unaffected spin-conserving collisions up to a complete suppression. This gives us the possibility to view the $m=\pm1/2$ subsystem as a dissipative two-component Fermi gas with a tunable loss mechanism.
We have further investigated the time evolution of the temperature of this subsystem exposed to losses induced by these spin- and momentum changing collisions. We compare two experiments: On the one hand, we perform an experiment at a high magnetic field ($B=7.6\,\text{G}$), where spin-changing collisions are suppressed. On the other hand, we perform a second experiment at a low magnetic field ($B=0.12\,\text{G}$) with strong spin relaxation.
In both cases the system is in the hydrodynamic limit with respect to external degrees of freedom due to the comparatively large spin-conserving interactions. Hence we make the assumption, that at each time the subsystems are close to an intermediate equilibrium state with a well-defined temperature. On the other hand, spin-changing collisions are at least two orders of magnitude weaker and very slowly change the particle number in each subsystem. This situation is reminiscent of prethermalization, first a "prethermal" state is reached under the assumption of conserved quantities, which on a much longer time scale are actually not fully conserved due to a "slightly broken symmetry", leading eventually to full thermalization \cite{Berges2004}. Here, the role of the nearly conserved quantities is played by the occupation numbers of the ten spin states, which change only on a very long time scale. We measure the time evolution of the temperature of the initially populated $m=\pm1/2$ components and compare the temperatures for both cases described above. For large magnetic fields we observe a small heating rate, which we mainly attribute to inelastic photon scattering. However, at low magnetic fields, the heating rate is significantly increased. In FIG.~\ref{fig7}(a), we plot the temperature difference to extract the heating contributions solely generated by spin-changing collisions. This additional increase in temperature is due to hole creation in the Fermi sea \cite{Timmermans2001} by scattering into the unoccupied spin states. We initially prepare a very cold two-component Fermi sea, with only few unoccupied trap levels below the Fermi energy. Losses through spin-changing collisions ``perforate'' this Fermi sea with holes, such that the experimentally obtained temperature increases.
Numerical simulations using the Boltzmann equation (\ref{eq:4}) confirm the experimentally observed heating induced by redistribution [see FIG.~\ref{fig7}(b)].
\section{Microscopic derivation of a large-spin Boltzmann equation}
\label{sec4}
In this section, we describe the derivation of the 1D Boltzmann equation (\ref{eq:4}) in more detail. The reader not immediately interested in the details may skip this section and proceed directly to the conclusions. We follow previous work on the theoretical description of spin-polarized systems of $\text{H}$ or $\text{He}$ \cite{Lhuillier1982I}, called the Lhuillier-Lalo\"e transport equation. We extend it to describe a 1D system with large spin, several scattering channels and a quadratic Zeeman effect. We consider this approach to be suitable for our purpose for a couple of reasons. The entire equation is derived from a microscopic collisional approach, so the collision term is not based on phenomenological assumptions. We avoid the use of a relaxation time approximation, widely used to describe damping in bosonic and fermionic systems \cite{Kavoulakis1998,Massignan2005,Natu2010}, where the collision term is approximated by $I_\text{coll}\left[W\right]=\frac{W_\text{eq}- W}{\tau}$, with a relaxation time $\tau$. The reason is that for a multicomponent system determining the equilibrium state $W_\text{eq}$ is very challenging. Also due to the interplay of many different scattering lengths we expect not one but many different relaxation times for each spin component. Our approach allows to better understand the relaxation process itself, rather than merely its effect on other processes. Furthermore, from a technical point of view, our approach remains quadratic in the Wigner function so it can be numerically simulated using the same standard techniques as the collisionless case \cite{Piechon2009,Natu2009,Ebling2011}.
The idea behind the approach of Lhuillier-Lalo\"e is to interpret the collision integral as the change rate of the state of a single particle $\hat\rho\rightarrow\hat\rho'$ due to binary collisions
\begin{equation}
I_\mathrm{coll}=\frac{\hat\rho'-\hat\rho}{\Delta t}.
\end{equation}
Here $\Delta t$ is the elapsed time interval, which is short compared to any relevant macroscopic dynamics of the system, but nevertheless is longer than the duration of a single collision, which is thus considered to be effectively instantaneous. This quantity will drop out and not appear in the final kinetic equation. With this in mind, we treat collisions in the asymptotic limit, where they are described by the Heisenberg $S$-matrix. It relates the two-body density matrix of both scattering particles before a collision $\hat\rho(1,2)$ with the one after a collision $\hat\rho(1,2)'$. Here $(1,2)$ label the quantum numbers of particles 1 and 2 in first quantization. We obtain
\begin{equation}
\hat\rho(1,2)'=\hat{\mathcal{S}}\hat\rho(1,2)\hat{\mathcal{S}}^\dagger.
\end{equation}
In order to arrive at a single-particle description we trace out particle 2 later. We next assume that particles involved in a collision are uncorrelated, $\hat\rho(1,2)=\hat\rho(1)\otimes\hat\rho(2)$, both before and after the collision, an assumption justified for a system with a large number of particles. This assumption in fact corresponds to Boltzmann's original molecular chaos hypothesis (Stosszahlansatz). For the desired single-particle density matrices before and after a collision we obtain
\begin{equation}
\hat\rho(1)=\frac12\mathrm{Tr}_2\lbrace(\openone-\mathcal P^\mathrm{ex})\hat\rho(1)\otimes\hat\rho(2)(\openone-\mathcal P^\mathrm{ex})\rbrace
\end{equation}
\begin{equation}
\hat\rho'(1)=\frac12\mathrm{Tr}_2\lbrace(\openone-\mathcal P^\mathrm{ex})\hat{\mathcal{S}}\hat\rho(1)\otimes\hat\rho(2)\hat{\mathcal{S}}^\dagger(\openone-\mathcal P^\mathrm{ex})\rbrace
\end{equation}
where we account for the indistinguishability of particles with the operator $\mathcal P^\mathrm{ex}$ exchanging the quantum numbers of particles 1 and 2. Due to fermionic statistics it comes with a minus sign.
This ansatz yields the following expression for the collision integral
\begin{align}
\hat I_\mathrm{coll}\approx\frac{1}{\Delta t}\mathrm{Tr}_2\left\lbrace\frac{\openone-\mathcal P^\mathrm{ex}}{\sqrt2}\left[\hat{\mathcal{S}}\hat\rho(1)\otimes\hat\rho(2)\hat{\mathcal{S}}^\dagger\right.\right.\nonumber\\*
\left.\left.-\hat\rho(1)\otimes\hat\rho(2)\right]\frac{\openone-\mathcal P^\mathrm{ex}}{\sqrt2}\right\rbrace.
\label{eq:6}
\end{align}
The $S$-matrix is related to the $T$-matrix via $\hat{\mathcal{S}}=\openone-2\pi i \hat{\mathcal{T}}$ such that Eq.~(\ref{eq:6}) becomes
\begin{align}
\hat I_\mathrm{coll}\approx\frac{2\pi}{\Delta t}\mathrm{Tr}_2&\!\left\lbrace\frac{\openone\!-\!\mathcal P^\mathrm{ex}}{\sqrt2}\left[i \hat{\mathcal{T}}\hat\rho(1)\otimes\hat\rho(2)-i\hat\rho(1)\otimes\hat\rho(2)\hat{\mathcal{T}}^\dagger\right.\right.\nonumber\\*
&\left.\left.+2\pi\hat{\mathcal{T}}\hat\rho(1)\otimes\hat\rho(2)\hat{\mathcal{T}}^\dagger\right]\frac{\openone\!-\!\mathcal P^\mathrm{ex}}{\sqrt2}\right\rbrace.
\label{eq:9}
\end{align}
This expression then has to be evaluated in the phase-space representation. Before performing the trace operation, we compute the two-body Wigner transform of the expression in braces in Eq.~(\ref{eq:9}). This is a very lengthy exercise we demonstrate in detail in Appendix \ref{app:B}, as well as the subsequent trace, shown in Appendix \ref{app:C}. In the course of these calculations, we require the elements of the $S$-matrix. In the center-of-mass system they are given by
\begin{align}
&\langle 1:k,a;2:-k, b|\hat{\mathcal{T}}|1:k',c;2:-k',d\rangle=\nonumber\\*
&-2\pi i\delta\left(\epsilon_k-\epsilon_{k'}+Q_{abcd}\right)\nonumber\\*
&\times\langle1:a,2:b|\hat T(k,k')|1:c,2:d\rangle,
\label{eq:7}
\end{align}
where $k,k'$ denote the incoming and outgoing wave-vectors of the particles, and $m,n;m',n'$ incoming and outgoing spins respectively. The delta-function assures energy conservation, $\epsilon_k=\frac{\hbar^2 k^2}{2\mu}$ denotes kinetic energy with reduced mass $\mu=\frac{M}{2}$ and $Q_{abcd}\equiv Q(a^2+b^2-c^2-d^2)$ the shift in the quadratic Zeeman energy induced by a spin-changing collision. The on-shell $T$-matrix $\hat T(k,k')$ depends formally on the relative wave-vectors $k,k'$, but for our case of s-wave scattering, the dependence is only on the modulus. As they are related by energy conservation $|k'|=\sqrt{k^2+Q_{abcd}}$, effectively the dependence is only on $k$ or $k'$.
The QZE-shift vanishes for spin-conserving collisions, hence it is absent in the spin 1/2 case and has the effect that scattering processes with a large $Q_{abcd}$ are suppressed, because the $T$-matrix decays $\sim 1/|k'|$ for large $|k'|$. For ${}^{40}\mathrm K$, the splitting is given by the quadratic part of the Breit-Rabi-Formula \cite{BreitRabi}, $Q=-\frac{2\mu_\text{B}^2(g_J-g_I)^2B^2}{9^3a_\text{hfs}}$, with the Bohr magneton $\mu_\text{B}$, nuclear and electronic g-factors $g_I$, $g_J$ and hyperfine structure coefficient $a_\text{hfs}$ \cite{Arimondo1977}.
To account for the spin-dependent interactions we separate the $T$-matrix into channels of total spin $S$ and magnetization $M$ and obtain elements
\begin{align}
\langle1:a,2:b|&\hat T(k,k')|1:c,2:d\rangle\equiv T_{abcd}(k,k')\nonumber\\*
&=\sum_{SM}T_S(k,k')\left\langle ab|SM\right\rangle\left\langle SM|cd\right\rangle.
\end{align}
For a 1D system, the expression for a $T$-matrix in the channel with a coupling constant $g_S$ is
\begin{equation}
T_S(k,k')=\frac{1}{2\pi}\frac{ik'\frac{2\hbar^2}{M}}{1-ik'\frac{2\hbar^2}{Mg_S} }.
\end{equation}
In each scattering channel we perform a low energy expansion in the coupling constant $g_S\rightarrow0$ up to second order, to maintain the unitarity of the $S$-matrix. Since for the 1D case, the expansion in powers of $g_S$ is accompanied by factors of $(k')^{-1}$, we artificially create a singularity in the imaginary part of the $T$-matrix in this procedure. We remedy this problem by choosing a cutoff $|k'|<\frac{Mg_S}{2\hbar^2}$, which is the distance between zero and the maximum of the imaginary part of the $T$-matrix (see Appendix \ref{app:D} for more details). This step is unnecessary in the 3D case discussed by Lhuillier and Lalo\"e \cite{Lhuillier1982I,Lhuillier1982II,Fuchs2003}. The result is then given by
\begin{align}
T_S(k,k')&\approx\frac{g_S}{2\pi}-\begin{cases} 0 & \mbox{if } |k'|\mbox{ $<\frac{Mg_S}{2\hbar^2}$} \\ \frac{iMg_S^2}{4\pi\hbar^2 k'}+\ldots & \mbox{if } |k'|\mbox{ $\geq \frac{Mg_S}{2\hbar^2}$ } \end{cases}
\end{align}
or respectively
\begin{align}
\label{eq:23}
T_{abcd}(k,k')\approx\frac{U_{acbd}}{2\pi}-\begin{cases} 0 & \mbox{if } |k'|\mbox{ $<\frac{MU_{acbd}}{2\hbar^2}$} \\ \frac{i M\tilde U_{acbd}}{4\pi\hbar^2 k'}+\ldots & \mbox{if } |k'|\mbox{ $\geq\frac{MU_{acbd}}{2\hbar^2}$} \end{cases}.
\end{align}
The leading terms linear in $g_S$ correspond to forward scattering, the quadratic terms to backward (lateral in higher dimensions) scattering processes. We do not explicitly denote the spin-dependent cutoff in equations (\ref{eq:5}),(\ref{eq:26}) and (\ref{eq:27}), where it is used in the integrals over $q$.
The expansion of the $T$-matrix is performed in addition to a semiclassical gradient expansion of the Wigner function (see Appendix \ref{app:E} for details) to first order in the linear terms and to zero order in the quadratic expressions. During this procedure we encounter squares of delta functions whose interpretation is described in Appendix \ref{app:F}. Finally we obtain a collisional integral, consisting of three parts
\begin{equation}
I^\mathrm{coll}_{mn}(x,p)=I^\text{mf}_{mn}(x,p)+I^T_{mn}(x,p)+I^{T^2}_{mn}(x,p),
\label{eq:12}
\end{equation}
where $I^\text{mf}$ is linear in $a_S$ and contains the forward scattering part of collisions leading to phase-shifts,
\begin{align}
&I^\text{mf}_{mn}(x,p)\!=\!-\frac{i}{\hbar}\sum_l\!\!\left[V^\text{mf}_{nl}(x)W_{ml}(x,p)\!-\!W_{lm}(x,p)V^\text{mf}_{ln}(x)\right]\nonumber\\*
&\!+\!\frac12\sum_l\!\!\left\lbrace\partial_x V^\text{mf}_{nl}(x)\partial_pW_{ml}(x,p)\!+\!\partial_p W_{lm}(x,p)\partial_x V^\text{mf}_{ln}(x)\right\rbrace.
\end{align}
It coincides with the interaction term obtained from the treatment of the same problem on a simpler mean-field level \cite{Natu2009,Ebling2011}. Due to its effect as a non-linear modification of the trap and magnetic field, in Eq.~(\ref{eq:4}) we have separated this term from the collisional integral and added it to the free motion of the particles in the external fields.
The quadratic terms, which form Eq.~(\ref{eq:5}) contain backward scattering, including momentum exchange between particles. They appear as dissipation on the single-particle level and are given by
\begin{widetext}
\begin{align}
\label{eq:26}
I_{mn}^{T}(x,p)&=-\frac{M}{\hbar^2}\sum_{abl}\left[\int_{q^2>\epsilon_1}\!\!\! dq\frac{\tilde U_{malb}}{\sqrt{q^2+\Delta_{mlab}}}W_{an}(x,p)W_{bl}(x,p-q)+\int_{q^2>\epsilon_2}\!\!\! dq\frac{\tilde U_{nalb}}{\sqrt{q^2+\Delta_{nlab}}}W_{ma}(x,p)W_{lb}(x,p-q)\right]\\*
\label{eq:27}
I_{mn}^{T^2}(x,p)&=\frac{M}{\hbar^2}\sum_{abcdl}\int_{q^2>\epsilon_3}\!\!\! dq\frac{U_{malb} U_{ncld}}{\sqrt{q^2+\Delta_{mnlabcd}}} W_{ac}(x,p-\tfrac12(q-\sqrt{q^2+\Delta_{mnlabcd}}))W_{bd}(x,p-\tfrac12(q+\sqrt{q^2+\Delta_{mnlabcd}})).
\end{align}
The integration domain cutoffs around $q=0$, coming from Eq.~(\ref{eq:23}) are given by $\epsilon_1=\frac{MU_{malb}}{\hbar}-\Delta_{mlab}$, $\epsilon_2=\frac{MU_{nalb}}{\hbar}-\Delta_{nlab}$ and $\epsilon_3=\frac{MU_{nalb}}{2\hbar}(U_{malb}+U_{ncld})-\Delta_{mnlabcd}$.
The corresponding result for a full 3D calculation (see \cite{Fuchs2003} for the case of spin 1/2) reads
\begin{align}
\label{eq:28}
&\frac{d}{dt}W_{mn}(\bm r,\bm p)+\left[\frac{\bm p}{M}\cdot\nabla_r-M\left(\omega_x^2x,\omega_y^2y,\omega_z^2z\right)\cdot\nabla_p+\frac{iQ}{\hbar}(n^2\!-m^2)\right]W_{mn}(\bm r,\bm p)\nonumber\\*
&+\frac{i}{\hbar}\sum_l\left[V^\text{mf}_{nl}(\bm r)W_{ml}(\bm r,\bm p)-W_{lm}(\bm r,\bm p)V^\text{mf}_{ln}(\bm r)\right]\nonumber\\*
&-\frac12\sum_l\left\lbrace\nabla_r V^\text{mf}_{nl}(\bm r)\cdot\nabla_pW_{ml}(\bm r,\bm p)+\nabla_p W_{lm}(\bm r,\bm p)\cdot\nabla_r V^\text{mf}_{ln}(\bm r)\right\rbrace=I_{mn}^\text{coll}(\bm r,\bm p).
\end{align}
The mean-field potential is given by $V_{mn}^\text{mf}(\bm r)=2\int d\bm p\sum_{ab}U'_{mnab}W_{ab}(\bm r,\bm p)$
and the collision term reads
\begin{align}
\label{eq:29}
I^\text{coll}_{mn}(\bm r,\bm p)&=-\frac{M}{4\pi\hbar^4}\int d\bm q\left\lbrace\sum_{abc}\left(\sqrt{q^2+\Delta_{mcab}}\tilde U'_{macb} W_{an}(\bm r,\bm p)W_{bc}(\bm r,\bm p-\bm q)\right.\right.\nonumber\\*
&\hspace{6cm}\left.\left.+\sqrt{q^2+\Delta_{abnc}}\tilde U'_{anbc}W_{ma}(\bm r,\bm p)W_{cb}(\bm r,\bm p-\bm q)\right)\right.\nonumber\\*
&-\left.\frac{1}{2\pi}\int d\Omega\sum_{abcdl}\sqrt{q^2+\Delta_{mnlabcd}}U'_{malb}U'_{ncld}W_{ac}(\bm r,\bm p-\tfrac12(\bm{q}-\bm p')W_{bd}(\bm r,\bm p-\tfrac12(\bm q+\bm p')\right\rbrace
\end{align}
where $\bm p'=\bm e_\Omega\sqrt{q^2+\Delta_{mnlabcd}}$ and $\bm e_\Omega$ denotes the unit vector corresponding to solid angle $d\Omega$.
\end{widetext}
A physical interpretation of this expression can be obtained by looking at the origin of the individual terms. The upper two lines of Eq.~(\ref{eq:29}) originate from the second order of the expansion of the $T$-matrix (\ref{eq:23}), describing the intensity shift in the forward scattered wave \cite{Lhuillier1982I,Fuchs2003}, while the first order only describes a phase shift and appears in the mean-field terms in (\ref{eq:28}). The bottom line of (\ref{eq:29}) contains all lateral scattering processes, hence the explicit angular dependence. In our formalism, the coupling constants $U$, $\tilde U$ include all particle indistinguishability and exchange contributions, which are discussed in greater detail in \cite{Lhuillier1982I}.
\section{Conclusion}
\label{sec5}
We have presented a novel approach to study relaxation dynamics in a closed quantum system, exploiting the unique properties of a large-spin Fermi sea. For this system, we have derived a multicomponent kinetic equation without phenomenological assumptions nor prior knowledge of the equilibrium state. As a key result, we find that this approach is well suited for the quantitative description of weakly interacting fermionic many-body systems with large spin. Both, the comparison of numerical simulations with full spatial resolution to a 1D experiment as well as the comparison of a simplified single-mode approximation to a 3D experiment yield a very good agreement without free parameters. We identify different collisional processes on different time scales and identify spin relaxation as the slowest dynamical process of the system. A variation of the density and the geometry of the system changes the respective spin relaxation rates by several orders of magnitude, ranging from a few milliseconds to several seconds. By tuning the magnetic field, we can precisely control the coupling strengths of individual collision channels, allowing to tune the character of a subsystem of two spin components within the large-spin Fermi sea continuously from an open to a closed system. The spin relaxation manifests itself in a perforation of the Fermi sea accompanied with a temperature increase.
Our results broaden the understanding of many-body relaxation dynamics. In particular, the fermionic character of the system underlines its model character for various systems in nature. The possibility to monitor different spin components individually allows to employ the large-spin Fermi sea for novel studies of decoherence and relaxation processes in quantum many-body systems. Furthermore, spin relaxation dynamics might play an important role for proposed fermionic large-spin phenomena, e.g. quantum-chromodynamic-like color superfluidity or large-spin texture formation.
\begin{acknowledgments}
We acknowledge fruitful discussions with Jannes Heinze and Ludwig Mathey.
This work has been funded by Spanish Ministerio de Ciencia e Innovaci\'on (FIS 2008-00784, AAII-Hubbard, FPI-fellowship), ERC grants QUAGATUA, OSYRIS, EU IP SIQS and
Deutsche Forschungsgemeinschaft (DFG) grant FOR 801 and DFG excellence cluster The Hamburg Centre for Ultrafast Imaging - Structure, Dynamics, and Control of Matter on the Atomic Scale.
\end{acknowledgments}
|
1,116,691,497,732 | arxiv | \section{Introduction}
Facial expressions have been regarded as one of the most common non-verbal cues for conveying human emotions~\cite{surveyAsli, wang2020suppressing}. Therefore, automatic facial expression recognition (FER) from images has received considerable attention in recent years~\cite{sepas2021capsfield, sepas2020facial, roy2021spatiotemporal}. Naturally, deep learning methods have recently become the dominant approach for FER, starting with early work using a shallow CNN and a support vector machine (SVM) in~\cite{tang2013deep}. More sophisticated deep learning solutions have recently been proposed. In \cite{chen2019facial}, a novel attention method which causes the method to focus on moving facial muscles was proposed. In ~\cite{wang2020region} another attention-based method was proposed to rank the importance of facial regions. To tackle data imbalance, which is a common issue in FER, a novel loss function was proposed in~\cite{jiang2020accurate}. In~\cite{kollias2020deep}, a novel data augmentation approach for FER was proposed. By altering the standard skip connections in a ResNet model with a differentiable function, the Bounded Residual Gradient Network (BReG-Net) was proposed in~\cite{hasani2019bounded}. By adding trainable parameters to BReG-Net, the same authors later proposed BReG-NeXt~\cite{hasani2020breg}, which offers state-of-the-art results on AffectNet~\cite{Affectnet} and FER2013~\cite{FER2013} datasets. More recently, self-supervised and contrastive approaches have also begun to gain popularity in this area, but have in many cases used either videos~\cite{roy2021spatiotemporal} or multi-view~\cite{shuvendu2021} data as opposed to the work presented in this study which focuses on a single image.
In order to make use of the existing \textit{relationships} between facial regions, exploiting graph topologies in neural networks for FER has gained popularity~\cite{GCN1,G1,ICLR}. In~\cite{GRNN}, Gabor features were combined and used to feed a gated recurrent unit. In~\cite{G1}, the relationship between facial landmarks were provided by using their psychological semantic relationships. The resulting topology was then processed by a graph network. Despite clear effectiveness of such graph-based methods, \textbf{there is no unique consensus on the topology of the underlying graphs} that should be used for FER. Current graph-based FER methods either construct complete graphs~\cite{GRNN, GCN2,G2} or use pre-defined topologies~\cite{G1,G3,GCN1}, both of which do not capture the optimal connections. The topologies of these graphs can have considerable effects on the performance of networks, including sequential learners. Figure~\ref{fig:banner} shows a face image along with 4 random trees that were built on the extracted facial landmarks. Traversing these trees results in 4 different sequence-based embeddings that can be fed to 4 independent LSTM networks for a 7-class FER task. The corresponding recognition rates are shown below each image in the figure where we see a considerable range given the different topologies.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig2-6thedition.pdf}
\end{center}
\caption{A face image along with 4 random trees. Corresponding recognition rates are written below each image.}
\label{fig:banner}
\end{figure}
In this paper, we propose an end-to-end deep FER architecture to learn an optimal graph of the face in the form of a tree topology whose traversal creates a sequence, which is used to form the input embedding for a sequence learner. To this end, we consider a weighted complete graph, from which we generate a minimum-cost spanning tree and choose the optimal edges. Our end-to-end model
consists of three blocks. In the first block, we perform the tree topology learning. Our optimized tree is then used in a two-stream framework, structure and texture. In the structure stream, we traverse the optimized tree to form a sequence, whose resultant embedding is then fed to an LSTM. This is followed by a soft attention mechanism. In the texture stream, we encode the local patches around the extracted facial landmarks, prior to learning a separate LSTM network. The final embedding is fused with that of the structure stream. To the best of our knowledge, this paper is the first that optimizes the order of relevant facial regions prior to recurrently learning their dependencies for FER. Our experiments on two large-scale public FER datasets, AffectNet~\cite{Affectnet} and FER2013~\cite{FER2013}, set new state-of-the-art results.
In summary, our main contributions are as follows. (\textbf{1}) We propose a novel end-to-end architecture to utilize optimal face tree topologies. (\textbf{2}) Our experiments show that the two-stream solution has a positive impact on the results. Moreover, we observe that given our tree topology optimization, a different face tree is learned for each dataset. Lastly, our solution sets new \textbf{state-of-the-art} values on large-scale AffectNet~\cite{Affectnet} and FER2013~\cite{FER2013} datasets.
\section{Proposed Method}
\subsection{Problem and Solution Overview}
Graphs can model facial components~\cite{G1, angadi2019face} in which vertices denote facial landmarks while edges correspond to the relationship between these landmarks~\cite{jiang2017emotion}. As evident from Figure~\ref{fig:banner}, the topology of such graphs can affect the performance of subsequent models that use it. Nevertheless, no prior work has been proposed to optimally learn the topology of facial graphs. To address this gap, we propose a method to learn the optimum tree capable of characterizing these relationships. Next, in order to learn critical information from the face, we consider two key streams, \textit{structure} and \textit{texture}. The former learns the structure while the latter learns the texture of key patches in the face, both of which consider the optimized tree. Our method traverses the optimized tree to form a sequence, which is then learned recurrently using a sequential learner, in this case an LSTM. Figure~\ref{fig:pipeline} illustrates the architecture of our proposed pipeline.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\columnwidth]{fig-pipeline4.pdf}
\end{center}
\caption{Our proposed pipeline.}
\label{fig:pipeline}
\end{figure}
\subsection{Tree Topology Learning}
We intend to first learn an optimal face tree topology to be used in subsequent components. To this end, we first detect the facial landmarks from the input images using a deep regression architecture~\cite{lv2017deep}. Next, we construct a weighted fully connected graph, $k_n$, from the landmarks. We then find the optimum minimum-cost spanning tree considering the weights of $k_n$, whose traversal generates a sequence by which we form an embedding to feed to a sequential learner. We use the Prim's algorithm~\cite{jiang2009research} to solve the problem of finding a minimum-cost spanning tree. The built tree is then traversed using Preorder depth-first order~\cite{kozen1992depth}, with the starting point being the most centered landmark, i.e., nose tip. During the traversal, when a leaf node is reached, the traversal backtracks to go back to the starting point.
Figure~\ref{fig:TreeTraversal} depicts the tree traversal process.
Our model then updates the learnable weights associated with $k_n$. To this end, we adopt a metaheuristic optimization algorithm as used in~\cite{li2011cooperatively}. For the optimizer's objective function we use the cost function as follows:
\begin{multline}\label{Eq:loss}
J(w)=\frac{1}{3m}(\sum\limits_{i=1}^m L^{foc}_1(\hat{y}^{(i)}, y^i) + \sum\limits_{i=1}^m L^{foc}_2(\hat{y}^{(i)}, y^i ) +\\ \sum\limits_{i=1}^m L^{foc}_3(\hat{y}^{(i)}, y^i)),
\end{multline}
where $m$ is the size of the training set, $L^{foc}_1(x)$ is the focal loss function selected to maximize performance for the fusion step, and $L^{foc}_2(x)$ and $L^{foc}_3(x)$ are respectively the focal loss functions for structure and texture streams. We use focal losses since the data distribution in most FER datasets is highly skewed~\cite{lin2017focal}. Here, $L^{foc}$ is defined as:
\begin{equation}\label{Eq:focal}
L^{foc}(p_b) = -\alpha_b(1-p_b)^\gamma log(p_b),
\end{equation}
where $\gamma$ is the focusing parameter and $p_b$ is the binomial distribution, in which $\hat{y}$ is the probability of $y$ being 1 and $1-\hat{y}$ is the probability of $y$ being 0. We set $\gamma=2$ and $\alpha_b = 0.25$ as recommended in~\cite{lin2017focal}.
At each epoch, the metaheuristic algorithm generates a set of weights associated with $k_n$, from which we generate the minimum-cost spanning tree, whose traversal results in a sequence. Next, this sequence is frozen and the subsequent structure and texture streams are fully trained. The reason for converting the obtained graph to a tree is that the acyclic nature of trees is better suited for traversal, hence sequence generation.
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{tree_new_v4.pdf}
\end{center}
\caption{Details of the face tree topology learning. Traversing the tree results in A-B-C-B-A-D-E-F-E-D-A-G-H-I-H-J-H-G-A, which is then fed to an LSTM network.}
\label{fig:TreeTraversal}
\end{figure}
\subsection{Structure Stream}
The coordinates of the facial landmarks which were extracted in the tree topology learning step, along with the obtained sequence, are the inputs to the structure stream (see Figure~\ref{fig:pipeline}). By using this sequence, we generate an embedding to be fed to an LSTM network with peephole connections~\cite{gers2000recurrent}. Following the LSTM network, we add an attention step~\cite{rocktaschel2015reasoning} to better focus on important parts. To this end, the hidden state vectors $h_i$ of the LSTM cells are multiplied by a set of coefficients $\alpha_i$ to determine the level of attention:
\begin{equation}\label{Eq:attentionWeights}
\alpha_i = \frac{e^{u_i}}{\sum_{k=1}^n e^{u_k}},
\end{equation}
where, $n$ is the number of LSTM units and $u_i$ is calculated as:
\begin{equation}\label{Eq:attentionUi}
u_i = tanh(W_hh_i+b_h).
\end{equation}
In Eq.~\ref{Eq:attentionUi}, $W_h$ and $b_h$ are the trainable weights and biases. The attentive output is calculated $H = \sum_i\alpha_i h_i$, where $H$ is the final embedding of the structure stream.
\subsection{Texture Stream}
Similar to the structure stream, the inputs to the texture stream are the facial landmarks and the sequence generated in the face tree topology learning step. However, unlike the structure stream, we focus on the texture of the facial images. To this end, we first form $n\times n$ patches centered around each facial landmark. We then crop these patches from the images and feed them to a an encoder (ResNet50) pre-trained on the VGG-Face2 dataset~\cite{cao2018vggface2}. Through empirical experimentation, we find the patch size of 17$\times$17 pixels to yield the best results. We stack these embeddings in accordance to the sequence obtained from the tree topology learning step (see Figure~\ref{fig:TreeTraversal}). The resulting embedding is then passed onto an LSTM network followed by a soft attention mechanism.
\subsection{Fusion}
In order to fuse the outputs of the structure and texture streams, we adopt the two-stream fusion strategy used in~\cite{gu2018hybrid, zhang2020rfnet}. This fusion strategy has yielded state-of-the-art results in a number of areas~\cite{gu2018hybrid}. Here, we use two dense layers as encoders for each of the input streams, and generate stream-specific features. Then, soft attention is adopted to learn the weight $\eta$ by using
$\eta=softmax(tanh(W_f[T^*,S^*]+b_f))$,
where $W_f$ and $b_f$ are trainable weights and biases, respectively. $T^*$ and $S^*$ are the stream-specific features for texture and structure. Next, a dense layer is utilized to learn the optimal association across the embedding-specific features.
\section{Experiments}
\noindent \textbf{Implementation.} The entire architecture is implemented using TensorFlow~\cite{abadi2016tensorflow} and is trained using a pair of Nvidia RTX 2080 Ti GPUs. For optimizing the weights of the model, we adopt ADAM optimizer~\cite{kingma2014adam} with the learning rate, first-momentum decay, and second-momentum decay of 0.001, 0.9, and 0.99, respectively.
\noindent \textbf{Datasets.} We test our model on two large in-the-wild datasets: (\textit{\textbf{i}}) \textit{\textbf{AffectNet}}, comprising more than 1 million images with 8 different expression categories, and (\textit{\textbf{ii}}) \textit{\textbf{FER2013}}, which includes 33000 images and has 6 expression categories.
\subsection{Performance}
The recognition rates (RR) obtained by
our model and other state-of-the-art benchmarking methods are presented in Table~\ref{table:results}. For AffectNet dataset,
our model shows an improvement of 1.52\% and 3.28\% over the BreG-NeXt50 and BreG-NeXt32, the current state-of-the-art FER methods.
For FER2013 dataset,
our proposed pipeline outperforms the current state-of-the-art FER method, i.e., BreG-Next50, showing a performance improvement of 1.13\%.
For a head-to-head comparison between
our proposed model and BreG-NeXt50 (best performing benchmark), we present F1 scores for each expression by using a one-vs-all scheme~\cite{galar2011overview} in Figure~\ref{fig:f1score}. Concerning AffectNet,
our model exhibits higher F1 scores than BreG-NeXt50 in all the 8 expression classes except happy and neutral. AffectNet is skewed toward happy and neutral expressions. Despite this imbalance,
our pipeline shows better results in the minority expression classes which are more difficult to recognize. With regards to FER2013,
our pipeline achieves better F1 scores in 4 out of the 7 expressions. The exceptions are surprise, happy, and neutral expressions, in which BreG-NeXt50 performs slightly better.
Figure~\ref{fig:ConfMat} depicts the confusion matrices obtained by our method. For AffectNet, the largest confusions occur for contempt-happy (where contempt is the true class and happy is the predicted class), sad-surprise, and neutral-happy cases. Confusing contempt with happy is common in FER systems, since it is very hard to distinguish between these two even for humans~\cite{hasani2020breg}. In terms of FER2013, the largest confusion occurs in the disgust-happy case. The reason is that disgust and happy are the minority and majority classes respectively. Although we use focal loss to tackle imbalanced data,
some of the minority expressions were still classified as the majority classes. Other than disgust,
our pipeline shows good performance in recognizing true classes in FER2013 dataset.
\subsection{Tree Topology Learning}
The topology evolution of the face trees during can provide insight into the learning process. Figure~\ref{fig:treeEvol} illustrates the convergence of the minimum-cost spanning trees obtained by our method, for AffetNet and FER2013 datasets in iterations 1, 10, 30, and 40. In the first iteration,
a random tree is generated by assigning random weights to the edges of the $k_{50}$. We observe that as training is being performed, face trees tend to resemble the structure of human faces more and more. As an obvious example, face trees formed in iteration 40 for both AffectNet and FER2013 dataset closely resemble human faces around the jaw line, mouth, nose, and eyes, while still being customized for each dataset. Furthermore, in iteration 40, large variations between the two face trees of AffectNet and FER2013 occurs near the mouth and eye regions. This is not surprising, since these regions are highly informative in FER, and therefore these regions are customized for each dataset.
\begin{table}[t]
\centering
\caption{Evaluation metrics on AffectNet and FER2013.}
\label{table:results}
\begin{tabular}{l|l|l|l}
\hline
\textbf{Dataset} & \textbf{Authors} & \textbf{Method} & \textbf{RR} \\
\hline
\multirow{8}{*}{AffectNet} & Zeng et al.~\cite{zeng2018facialincon} & CNN & 57.31 \\
& Hewitt et al.~\cite{hewitt2018cnn} & CNN & 58 \\
& Hua et al.~\cite{hua2019hero} & Ensemble & 62.11 \\
& Chen et al.~\cite{chen2019facial} & Facial mask & 61.50 \\
& Kollias et al.~\cite{kollias2020deep} & Augmentation & 60 \\
& Hasani et al.~\cite{hasani2020breg} & BreG-NeXt32 & 66.74 \\
& Hasani et al.~\cite{hasani2020breg} & BreG-NeXt50 & 68.50 \\
& \textbf{Ours} & Tree learning & \textbf{70.02} \\
\hline
\multirow{6}{*}{FER2013} & Tang et al.~\cite{tang2013deep} & CNN & 69.3 \\
& Mollahosseini et al.~\cite{mollahosseini2016going} & CNN & 66.4 \\
& Vielzeuf et al.~\cite{vielzeuf2017temporal} & CNN & 71.2 \\
& Hasani et al.~\cite{hasani2020breg} & BreG-NeXt32 & 69.11 \\
& Hasani et al.~\cite{hasani2020breg} & BreG-NeXt50 & 71.53 \\
& \textbf{Ours} & Tree learning & \textbf{72.66} \\
\hline
\end{tabular}
\end{table}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.12]{fig6-v8.pdf}
\end{center}
\caption{Comparison of the obtained F1 scores.}
\label{fig:f1score}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.83\linewidth]{fig7-FG.jpg}
\end{center}
\caption{Confusion matrices obtained by our model.}
\label{fig:ConfMat}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.88\columnwidth]{fig3-fg.pdf}
\end{center}
\caption{Evolution of face trees during the training phase.}
\label{fig:treeEvol}
\end{figure}
\begin{table}[t!]
\begin{center}
\caption{Ablation experiments.}
\label{table:abblation}
\footnotesize
\begin{tabular}{c|c|c|c|c}
\hline
\textbf{Removed} & \multicolumn{2}{|c|}{\textbf{RR}} & \multicolumn{2}{|c}{\textbf{Drop in RR}} \\
\cline{2-5}
& \textbf{AffectNet} & \textbf{FER2013} & \textbf{AffectNet} & \textbf{FER2013}\\
\hline
\hline
Tree topology & 66.51\% & 67.11\% & 5.0\% & 7.60\%\\
Structure stream & 64.90\% & 66.02\% & 7.30\% & 9.14\%\\
Texture stream & 62.14\% & 63.17\% & 11.25\% & 13.06\%\\
Fusion strategy & 66.87\% & 68.24\% & 4.50\% & 6.08\%\\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Ablation Study}
To illustrate the contribution of each component of
the proposed pipeline, we perform ablation experiments by systematically removing individual components of the model. Table~\ref{table:abblation} compares the recognition rates obtained by the following models: (\textit{a})
the model without tree topology learning where we adopt a random tree to form the input embeddings; (\textit{b})
the model without the structure stream, in which the coordinates of facial landmarks along with the learned tree topology are utilized to form the embeddings; (\textit{c})
the model without structure stream where the output embedding of ResNet50 along with the learned tree topology is used to form the embeddings; and (\textit{d})
the model with a simple concatenation of the output embeddings from the structure and texture streams, instead of using the 2-stream fusion strategy.
The results presented in Table~\ref{table:abblation} reveal that: (1) tree topology learning is an effective component in
our proposed pipeline whose removal results in significant performance drops of 5.0\% and 7.60\%, respectively for AffectNet and FER2013 datasets; (2) the contribution of the texture stream is higher than that of the structure stream due to the exploitation of the pre-trained ResNet50 in the texture stream; (3) relative contribution of the employed fusion strategy is lower than that of the tree topology learning. This demonstrates the importance of the tree topology learning in the pipeline.
\section{Conclusion}
In this paper we propose a novel end-to-end method called deep face topology network for facial expression recognition. Our model assigns weights to the edges of a complete graph representing all the possible connections between extracted facial landmarks and integrates them into an optimization process. These weights along with the complete graph are used to generate a minimum cost spanning tree (MST), whose traversal generates an optimal sequence. This sequence is used to form an embedding to feed an LSTM network. Our solution has been integrated in an end-to-end two-stream pipeline. The first stream uses the spatial information of facial landmarks, while the other uses the texture information of patches around the landmarks. A fusion strategy has finally been adopted to combine the two. Experiments have been performed with two large-scale AffectNet and FER2013 datasets, where the results show the superiority of our method in comparison to prior work. The main limitation of this work is the optimization speed, which we will address in future works by using policy gradient methods and applying faster algorithms for finding the MST.
{\small
\bibliographystyle{ieee}
|
1,116,691,497,733 | arxiv | \section{Introduction}
For decades after electronic computers were invented, they exponentially increased in power and decreased in size, and the looming advent of the quantum limit was seen as a threat to progress. But in 1982 Feynman published a paper \cite{feynman1} in which he argued that quantum computing could be as much an opportunity as an obstacle. In 1994 Shor published the first quantum algorithm \cite{shor}, a systematic method to factorize integers. On a classical computer factorization is approximately exponential, that is, the time it takes is roughly exponential in the size of the integer $N$. With Shor's algorithm, running on a quantum computer, the time necessary is polynomial in the logarithm of $N$, which is much faster for very large integers. This is an important application, because most encryption relies on the fact that while multiplication is easy and rapid, the inverse problem of factorization is not. Were a macroscopic quantum computer to be built, this would no longer be true. Shortly thereafter Lov Grover \cite{grover} found a quantum algorithm for data base searches. Here the performance improvement is from $O(N)$ to $O(\sqrt(N))$, a quadratic speedup. Clearly at least for some problems, quantum computing can provide enormous advantages. Quantum computers are at least as fast as classical computers in general \cite{bernstein}, and widely believed to be much faster, perhaps exponentially faster, if only we can find the correct algorithms. Very recently a quantum advantage has been proved in a limited context \cite{bravyi}. Other researchers have found quantum advantage to be ``elusive''\cite{Ronnow}. A number of researchers have proved the existence of quantum codes in a variety of contexts \cite{existence, existence2}. But existence is not construction, much less optimization. There is a need for direct comparison, and benchmarking.
Quantum Neural Networks (QNN) are an alternative approach to quantum algorithmic computing. They were first proposed independently by several researchers \cite{orig1, orig2, orig3} decades ago. More recently a number of groups have developed the idea much further. Schutzhold \cite{schutzhold} and Trugenberger \cite{trugenberger} have investigated QNNs for pattern recognition. Schuld et al. \cite{schuld}
have looked at the related problem of linear regression. Some reviews have recently appeared \cite{schuldrev, arunaschalam, Biamonte}. Combining quantum computing and machine learning can benefit both fields: Quantum hardware can accelerate learning \cite{Aimeur} over that for classical networks, while learning techniques can obviate the perennial problem of design of algorithms that take advantage of the quantum nature of reality \cite{2002, 2008, chen, Hentschel}, and take advantage of bootstrapping for scaleability \cite{multi qubit, wiebe}. Some experimental small-scale demonstrations have been made, on photonic systems \cite{Cai}, and on IBM's publically available machines \cite{upcoming}.
Here, we explore the power and the complexity of a QNN, in both classical and quantum computations. We apply standard real- and complex-valued classical neural networks to two well-understood classical tasks of computation: a single-layer perceptron, and iris identification. The complex-valued network, it will be seen, is considerably more powerful than a comparable real-valued network, at both classical tasks, while a fully quantum mechanical network does even better.
More importantly, a QNN can efficiently do fully {\it quantum} calculations. Of course, any classical computer can simulate a quantum system, and in that sense ``do'' a quantum calculation, but with a physical quantum computer we do not need to set up a simulation: the quantum evolution of the system itself does the calculation. As noted above, we are only just beginning to get beyond simulations of QNNs. It is important to map out what kinds of calculations we will want to implement. But what are some truly quantum calculations?
One important quantity is the ``entanglement'' of a quantum system. Entanglement is essentially quantum correlation, a correlation stronger than anything possible classically. The name derives from the fact that if two subsystems are entangled, knowledge of the one subsystem gives knowledge of the other. It is well known that the power of quantum computation relies heavily on the degree of entanglement \cite{neilsen}. Previously, we have shown \cite{2008} that a QNN of this type can successfully calculate a general experimental entanglement witness, applicable to mixed as well as pure states, which can be bootstrapped to multiqubit systems \cite{multi qubit}, and which is robust to noise and to decoherence \cite{2 qubit noise}. We therefore compare also the performance of the classical neural nets on this problem. Our results show that a QNN is considerably better able to compute the entanglement witness than a classical neural network, with much smaller errors and much faster training, and generalized successfully from a much smaller training set.
The organization of the paper is as follows. In Section II, we define each of the three types of neural network that we are comparing: Classical real valued, classical complex, and fully quantum mechanical. Both kinds of classical net are universal approximators. In the fully quantum case presented here, we do not yet have theorems that show precisely how the complexity and power compare; hence this present study. In Section III we compare the performance of each net on classical benchmarking problems. In Section IV we compare their performance on a purely quantum problem: characterization of the entanglement of a quantum system. In Section V we present our conclusions.
\section{Neural Network Approaches }
\subsection{Classical real-valued neural networks}
Standard algorithmic computers are ``Turing complete'', that is, can be programmed to simulate any computer algorithm. Classical, Real-Valued Neural Networks (RVNN) are also ``universal approximators.'' This means essentially that an RVNN can approximate any continuous function that maps inputs to outputs. We will present (though not prove, since the proofs are both known and not our own) the theorems that explain explicitly what this means, in order to be able to compare with a QNN. Proofs can be found in a number of sources \cite{cybenko}. \\
A real-valued neural network with a continuous sigmoidal activation function $\sigma: \mathbf{R}^n \to \mathbf{R} $ can be represented as
\begin{equation}
G(x) = \sum_{j=1}^{N_{l}} \alpha_{j} \sigma \big( {\bf w}_{j}^{T} {\bf x} + \theta_{j} \big)
\end{equation}
\noindent where the sum is over the neurons in the hidden layer, $ {\bf x}, {\bf w}_{j} \in \mathbf{R}^{n} $ are the inputs and weights, respectively; $\alpha , \theta \in \mathbf{R}$ are the scaling factors and biases of the activation function, and where $ {\bf w}_{j}^{T} $ represents the transpose of ${\bf w}_{j}$. \\
It should be noted that $\sigma$ is discriminatory. This means for a measure $\mu \in M(I_{n}) $
\begin{equation}
\int_{I_{n}} \sigma \big( {\bf w}^{T} {\bf x} + \theta \big) d\mu(x) = 0
\end{equation}
\noindent for all ${\bf x} \in \mathbf{R}^{n}$ and $\theta \in \mathbf{R}$ implies that $\mu =0 $. Here $M(I_{n})$ is the space of finite, signed, regular Borel measures on the unit interval, $I_{n}$. This requirement is to ensure that the activation cannot send the affine space ${\bf w}^{T}{\bf x} + \theta$ to the set of measure zero. \\
\textbf{Theorem 1:} Let $\sigma$ be any continuous sigmoidal function. Then finite sums of the form
\begin{equation}
G(x) = \sum_{j=1}^{N_{l}} \alpha_{j} \sigma \big( {\bf y}_{j}^{T} {\bf x} + \theta_{j} \big)
\end{equation}
are dense in the continuous functions on the unit interval, $C(I_{n})$ \\
Theorem 1 tells us that as long as the mapping between the inputs and targets can be described by a continuous function, $G: \mathbf{R}^{n} \to \mathbf{R} $, then for a real-valued neural network with just one hidden layer and any continuous sigmoidal function as the activation function there exist weights $w$, bias $\theta$ and scalar $\alpha$ to approximate this continuous map to any desired accuracy. When the activation is a bounded, measurable sigmoidal function then \cite{cybenko} the finite sum $G(x) = \sum_{j=1}^{N_{l}} \alpha_{j} \sigma \big( {\bf y}_{j}^{T} {\bf x} + \theta_{j} \big)$ is dense in $L^{1}(I_{n})$. \\
An (in house) implementation of a real-valued artificial neural network was used as a baseline. The software NeuralWorks Professional II \cite{NeuralWorks} was also used to verify that the results obtained from commercially available software were comparable to our own implementation.
\subsection{Complex-valued neural networks}
We now extend from $ \mathbf{R}^{n}$ to $\mathbf{C}^{n}$. Clearly it ought to be true that if a calculation can be done in $\mathbf{R}^n$ it can be done in $\mathbf{C}^n$, since the real valued neural network is a special case of the Complex-Valued Neural Net (CVNN). And it is true that the complex-valued net is a universal approximator also, though we need to be a little careful.
\textbf{Theorem 2:} Let $\sigma: \mathbf{C}^n \to \mathbf{C} $ be any complex continuous discriminatory function. Then the finite sums of the product of the form
\begin{equation}
F(z) = \sum_{k=1}^m \beta_k \prod_{l=1}^{s_k} \sigma \big( {\bf W}_{kl}^T {\bf z} + \theta_k \big)
\end{equation}
are dense in $C(I_n)$. Here $\beta_k, \theta_k \in \mathbf{C}$, $ W_{kl}, {\bf z} \in \mathbf{C}^n $ , $m$ is the number of neurons in the hidden layer, and $s_k$ is an arbitrary number depending on how many product terms one wants to do. The product is needed to ensure that the set $ \tau = \big\{ F: \mathbf{C}^n \to \mathbf{C} \big\} $ is an algebra since it will be closed under multiplication \cite{cnn}. This is the difference between Theorem 1 and Theorem 2. Similarly to the real-valued networks, if the activation function, $\sigma: \mathbf{C}^n \to \mathbf{C} $ is any bounded measurable discriminatory function then the finite sum $F(z)=\sum_{k=1}^m \beta_k \prod_{l=1}^{s_k} \sigma \big( {\bf W}_{kl}^T {\bf z} + \theta_k \big) $ is dense in the Lebesgue integrable functions on the unit interval, $L^1(I_n)$. \\
In the complex valued case, the activation function may have essential singularities. But if $\sigma: \mathbf{C}^n \to \mathbf{C} $ is any complex function having an isolated and essential singularity then the finite sum above will be dense in compact subsets of analytic deleted neighborhood of the singularity. \\
This shows that there exists a representation of the CVNN that can solve essentially any problem the RVNN can do. Indeed, we expect that the CVNN will be more powerful or at least more efficient. The archetypal example is the XOR. Since it is a nonlinear problem, a real-valued net needs a hidden layer to solve, but the complex-valued net can do this in a single layer \cite{complex network text}. \\
We are currently working on a new QNN architecture, to which we will be able to apply this theorem directly, that is, a QNN that will be a universal approximator \cite{nams frat}. Here, we use instead simpler versions for the initial benchmarking work, both of the QNN and of the CVNN, which have already been explored in the literature. The implementation of the CVNN used here is largely based on the work of Aizenberg \cite{complex network text}. The major difference from the RVNN is that the neuron inputs, outputs, weights, and biases are numbers in the complex plane. In the RVNN signals are only scaled by their respective weights, while the CVNN allows the scaling and rotation of the signals. This leads to higher functionality in more compact architectures. Similar to a conventional RVNN, the input signals to any given neuron are summed by their respective weights:
\begin{equation}
z = \sum_n {\bf w}_n \cdot {\bf x}_n
\end{equation}
\noindent where ${\bf w}$ and ${\bf x}$ are the complex-valued weights and signals. An activation function is then applied to the sum $z$ at the neuron. The complex activation function $P$ takes the inputs and maps them to a unit circle on the complex plane which constitutes the signal from the neuron to the consequent layer.
\begin{equation}
P(z) = e^{i \cdot arg(z)}
\end{equation}
\noindent The network's scalar inputs and outputs/targets are condensed to the range [0,1] and mapped as points on the complex unit circle using the following mapping function:
\begin{equation}
M(r) = e^{i \cdot \pi r}
\end{equation}
Unlike the RVNN, with this architecture there is no need to take the derivative of the activation function or to apply gradient descent in order for this network to learn. The resulting error from each training pair can be represented as the vector from the output to the target in the complex plane. In other words, given the target $t$ and the output $z$ of an output neuron, the error can be represented as the vector $e$
\begin{equation}
e = t - z = \sum_n \Delta {\bf w}_n \cdot {\bf x}_n
\end{equation}
\noindent given that the resulting error is a contribution of all of the signals to the output neuron. Here $\Delta w_n$ is the contribution of each individual weight to the error. Dividing the error equally among the $N$ incoming weights we arrive at the following required weight adjustments for each training pair per epoch.
\begin{equation}
\Delta {\bf w}_n = \frac{e}{N} \cdot {\bf x}_n^{-1}
\end{equation}
For each subsequent layer the error is backpropagated by dividing the error across the weights relative to the values of the signals they carry. Thus the error gets diluted as it propagates backwards, yet this is applicably manageable given that the higher functionality of the CVNN favors shallower networks for conventional problems.
\subsection{Quantum Neural Network}
Because the quantum benchmark calculation chosen here is that of an entanglement witness, we consider the model for a QNN we first proposed in 2002 for this purpose \cite{2002, 2008}. The system is prepared in an initial state, allowed to evolve for a fixed time, then a (predetermined) measure is applied at the final time. The adjustable parameters necessary are the (assumed time-dependent) couplings and tunneling amplitudes of the qubits; the discriminatory function is provided by the measurement, which we may take to be any nontrivial projection convenient without loss of generality. For the two-qubit system, we choose as output the square of the qubit-qubit correlation function at the final time, which vaires between zero (uncorrelated) and one (fully correlated.) Thus we are considering the map
\begin{equation}
f: C^n \rightarrow [0,1]
\end{equation}
This structure lends itself immediately to the perceptron problem, as well as the entanglement problem.
The network input is the initial state of the system, as given by the density matrix $\rho(0)$. The signal is then propagated through time using Schrodinger's equation
\begin{equation}\label{Schr gen}
\frac{d\rho}{dt}= \frac{1}{i \hbar}[H,\rho]
\end{equation}
where $H$ is the Hamiltonian. Note that we work with the density matrix representation because it is computationally easier than using the state vector (or wave vector) representation. The density matrix form is more general, allowing mixed as well as pure states, but here we will in fact use only pure states. In terms of the state vector of the system $|\psi\rangle$, the density matrix is given by $\rho = |\psi\rangle \langle \psi|$.
For a two-qubit system, we take the Hamiltonian to be:
\begin{equation}
H= \sum_{\alpha=1 }^2 K_{\alpha}X_{\alpha} + \epsilon_{\alpha}Z_{\alpha} + \sum_{\alpha \neq \beta =1 }^2 \zeta_{\alpha\beta}Z_{\alpha} Z_{\beta}
\end{equation}
where $\{X,Z\}$ are the Pauli operators corresponding to each qubit, $\{K\}$ are the amplitudes for each qubit to tunnel between states, $\{\epsilon\}$ are the biases, and $\{\zeta\}$ are the qubit-qubit couplings. Generalization to a fully connected $N$ qubit system is obvious. We choose the usual charge basis, in which each qubit's state is $\pm$1, corresponding to the Dirac notation $|0\rangle$ and $|1\rangle$, respectively.
By introducing the Liouville operator, $L= \frac{1}{\hbar} \big[\cdots, H\big]$, Eq. (\ref{Schr gen}) can be rewritten \cite{peres} as
\begin{equation}\label{schro mod}
\frac{\partial \rho}{\partial t} = -iL\rho
\end{equation}
which has the general solution of
\begin{equation}\label{schro sol}
\rho(t) = e^{-iLt}\rho(0)
\end{equation}
We propagate our initial state, $\rho(0)$, forward in time to the state $\rho(t_f)$ where we will perform a measurement on $\rho(t_f)$, which mathematically is an operator followed by a trace. That measurement, or some convenient mathematical function of that measurement, is the output of the net. That is, the forward operation of the net can be pictured as:\\
\begin{equation}\label{rhoevolmap}
\rho(0) \xrightarrow{ \frac{d\rho}{dt}= \frac{1}{i \hbar}[H,\rho] } e^{-iLt}\rho(0) = \rho(t_f) \rightarrow \mathrm{Trace} (\rho(t_f)M)=\langle M \rangle
\end{equation}
\noindent where $M$ represents the particular measurement chosen. To train the net, the parameters $K$, $\epsilon$ and $\zeta$ are adjusted based on the error, using a quantum analogue \cite{2008} of Werbos's well-known back-propagation technique \cite{werbos}.
There is, clearly, no direct correspondence between these equations for the QNN with those of either the real- or complex-valued neural network, above. The nonlinearity necessary to do learning comes entirely from the (necessarily non-linear) measurement step. We do not know exactly how the complexity of this model compares; work on these theorems is on-going \cite{namthesis}.
However, as a first step, we can investigate the question experimentally, in what follows.
\section{Benchmarking on classical problems}
We first compare the performance of our neural networks on two classical problems: logic gates and pattern classification.
\subsection{Logic Gates}
Perhaps the most basic of neural network architectures is the single-layer perceptron. See Fig. \ref{fig:perceptron}. The perceptron is a single output neuron connected to the input neurons by their respective weighted synapses. The real-valued perceptron is a linear classifier and therefore can approximate linearly separable logic functions such as AND and OR gates; however, because it essentially creates a single linear separator, it necessarily fails to approximate non-linear gates, like the XOR or XNOR.
In contrast, a complex perceptron can calculate the XOR or XNOR, since it can create a nonlinear separation - or, equivalently, uses interference.
\begin{figure}
\includegraphics[width=0.4\textwidth]{perceptron.png}
\caption{Single-layer perceptron architecture.}
\label{fig:perceptron}
\end{figure}
The single layer perceptron was implemented on all four networks. The networks were trained on the 4 possible input pairs, \{(00), (01), (10), and (11)\}, for the outputs of the AND, NAND, OR, NOR, XOR and XNOR gates until the root mean square error (RMS) reached 1\%. See Fig. \ref{fig:rms} for an example. The learning rate of each network was increased to optimal, i.e, the training speed was maximized subject to the constraint that convergence was still achieved. This was done to draw a more equitable comparison on learning speed between the RVNN and CVNN, which have different learning schemes.
To implement logic gates on our QNN model, we must specify the encoding of inputs as a function of the prepared state of the quantum system, and outputs as a function of some measurement on the final state of the quantum system. We take the inputs as being the basis states of the two-qubit system in the so-called ``charge'' basis, that is, the states $|00 \rangle$, $|01 \rangle$, $|10 \rangle$, $|11 \rangle$. We choose the output measure as being the square of the qubit-qubit correlation function at the final time, i.e, $\langle Z_{A}(t_{f})Z_{B}(t_{f})\rangle^{2}$ . An output of one would mean that the two qubits are perfectly correlated to each other at the final time; an output of zero, that they are perfectly uncorrelated. The target values for these outputs will, of course, be different for each logic gate considered. Thus, for example, for the XNOR gate we would want an inital state of $|00\rangle$ or an inital state of $|11\rangle$ to train to a final correlated state but initial states of $|01\rangle$ or of $|10\rangle$ to train to a final uncorrelated state. Because of the inherent continuity of the QNN any initial state close to one of the input states will necessarily produce an final state close to the corresponding output state, and the computation is robust to both noise and decoherence \cite{2 qubit noise}.
Table \ref{table:gates_epochs} shows the number of training epochs required by each ``optimal'' network to reach an RMS error of 1\%, where an epoch is defined as one pass through the 4 training pairs. Note that the nonlinear gates (XOR and XNOR) cannot be done by either of the real classical nets (RVNN and NeuralWorks) with only a single layer. The QNN reached $\leq$1\% error in a single epoch.
\begin{figure}[htb]
\centering
\begin{subfigure}[b]{.49\linewidth}
\centering
\includegraphics[width=.99\textwidth]{AND_rms.png}
\caption{AND}
\end{subfigure}%
\begin{subfigure}[b]{.49\linewidth}
\centering
\includegraphics[width=.99\textwidth]{NAND_rms.png}
\caption{NAND}
\end{subfigure}\\%
\begin{subfigure}[b]{.49\linewidth}
\centering
\includegraphics[width=.99\textwidth]{OR_rms.png}
\caption{OR}
\end{subfigure}%
\begin{subfigure}[b]{.49\linewidth}
\centering
\includegraphics[width=.99\textwidth]{NOR_rms.png}
\caption{NOR}
\end{subfigure}\\%
\begin{subfigure}[b]{.49\linewidth}
\centering
\includegraphics[width=.99\textwidth]{XOR_rms.png}
\caption{XOR}
\end{subfigure}%
\begin{subfigure}[b]{.49\linewidth}
\centering
\includegraphics[width=.99\textwidth]{XNOR_rms.png}
\caption{XNOR}
\end{subfigure}%
\caption{ Example of training RMS as a function of epoch for various logic gates using the (single layer) CVNN perceptron. The RVNN and NeuralWorks networks trained similarly. }\label{fig:rms}
\end{figure}
\begin{table}[h!]
\centering
\begin{tabular}{ |l|l|l|l|l| }
\hline
\multicolumn{5}{|c|}{No. of Epochs to RMS error $\leq$1\%} \\
\hline
Gate & N.Works & RVNN & CVNN & QNN\\
\hline
AND & 11000 & 11146 & 222 & 1 \\
\hline
NAND & 11000 & 11145 & 127 & 1 \\
\hline
OR & 6000 & 5672 & 114 & 1 \\
\hline
NOR & 6000 & 5671 & 172 & 1 \\
\hline
XOR & n/a & n/a & 110 & 1\\
\hline
XNOR & n/a & n/a & 78 & 1\\
\hline
\end{tabular}
\caption{Number of epochs needed for gate training to reach RMS error $\leq$1\% for a (single layer) perceptron using RVNN, CVNN and NeuralWorks (to nearest 1000 epochs) implementations. }
\label{table:gates_epochs}
\end{table}
The real-valued network would not train to an RMS error below 50\% for the XOR and XNOR gates, given a number of epochs up to the order of 10\textsuperscript{6}. In addition, for the linearly separable gates the RVNN required 30-50 times more learning iterations than the CVNN to reach an RMS error of 1\%, making the CVNN training runs computationally much more efficient than RVNN. Note that the single-layer complex-valued perceptron can learn to solve the non-linearly separable XOR and XNOR gates, and do so with an efficiency at least comparable to that for the linearly separable gates as shown in Table \ref{table:gates_epochs}, as mentioned above.
``Quantum-inspired'' networks \cite{bhattacharyya} generally are some version of a CVNN, and, therefore, do have increased power over a RVNN due to their ability to use interference: that is, their performance would be expected to be comparable to our CVNN. However a fully quantum network can do even better. Results of the training of the QNN for the linear and non-linear logic gates are shown in the last column of Table \ref{table:gates_epochs}. Note that the QNN requires only a single epoch to learn any of the linear or the nonlinear gates, and it should be noted that the error continued to decrease below that level. This is an experimental realization of our 2015 theoretical result deriving weights for a quantum perceptron \cite{perceptron}.
\subsection{Iris Classification}
The other archetypal machine learning task we consider is that of pattern classification, for which one of the most common benchmarking problems is the Iris flower classification dataset \cite{irisdataset}. The multivariate dataset contains four features, as inputs: sepal length, sepal width, petal length, and petal width. The corresponding taxonomy of Iris flower (setosa, virginica or versicolor) is represented as a ``one-hot'' vector \cite{harris}. The set contains a total of 150 pairs, 50 of each species, as identified by Fisher in 1936 \cite{fisher}. The difficulty in Iris classification is well demonstrated by the scatterplot shown in Fig. \ref{fig:iris_scatterplot}: two of the the three species cluster together and require a highly non-linear separability function.
The RVNN and CVNN implementations were trained on the dataset to compare their performance for a non-linear multivariate classification problem. The dataset was divided into two randomly selected sets of 75 pairs containing an equal number of each species for training and testing. The networks were first trained on the entirety of the training set. The training pairs were then reduced to 30 and 12 pairs while keeping the testing set at 75 pairs. Both the RVNN and CVNN had a single hidden layer, that is, an architecture (4, $N_h$, 3), where the number of neurons in the hidden layer, $N_h$, was increased from three on up. Unlike the RVNN, the CVNN's testing performance improved substantially as $N_h$ was increased, up to about 100; this is the (optimized) architecture for the results reported in Table \ref{table:iris_results}. ``Testing'' accuracy in Table \ref{table:iris_results} means classification in the correct category if the appropriate output is above 0.5; below, it is counted as incorrect.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{iris_dataset}
\caption{Iris dataset cross-input classification scatterplot \cite{irisdataset}. }
\label{fig:iris_scatterplot}
\end{figure}
For implementation on the QNN, we used the feature data as the coefficients of the input state $|\psi(0)\rangle$ in the Schmidt decomposition. That is, the initial quantum state vector for each input was given by
\begin{equation}
\label{eq:purestate}
|\psi(0)\rangle = a |00\rangle + b e^{i\theta_1)} |01\rangle + c e^{i \theta_2}|10\rangle + d e^{i \theta_3} |11\rangle
\end{equation}
We set all the phase offsets $\{ \theta_{i} \}$ equal to zero (i.e., only real inputs here), and take each amplitude to correspond to a features (i.e., $a,b,c,d $ correspond to sepal length, sepal width, petal length, and petal width.) For the output, we again choose the final time correlation function, taken to be the polynomial $\frac{0.01(a^{2}+b^{2})+c^{2} + 1.8d^{2}}{a^2 + b^2 +c^2 + d^2 }$. This polynomial sorts out the three categories of flowers almost completely. It is possible to choose a more complicated function or higher degree polynomial to create more separation but we chose the simplest polynomial for the ease of training.
\begin{table*}[h]
\centering
\begin{tabular}{l|l|l|l|l|l|l||l|l|l|}
\cline{2-7}
\multicolumn{1}{c|}{} & \multicolumn{3}{c|}{Training RMS (\%)} & \multicolumn{3}{c|}{Testing RMS (\%)} & \multicolumn{3}{c|}{Testing Accuracy (\%)} \\ \hline
\multicolumn{1}{|c|}{Training Pairs} & \multicolumn{1}{c|}{RVNN} & \multicolumn{1}{c|}{CVNN} & \multicolumn{1}{c|}{QNN} & \multicolumn{1}{c|}{RVNN} & \multicolumn{1}{c|}{CVNN} & \multicolumn{1}{c|}{QNN} & \multicolumn{1}{c|}{RVNN} & \multicolumn{1}{c|}{CVNN} & \multicolumn{1}{c|}{QNN} \\ \hline
\multicolumn{1}{|l|}{75}&3.45 &2.06 &0.96 &3.71 &7.78 & 2.31 &100 &100 & 97.3 \\ \hline
\multicolumn{1}{|l|}{30}&0.51 &0.41 &1.1 &4.97 &9.47 & 9.78 &93.3 &96.0 & 97.5 \\ \hline
\multicolumn{1}{|l|}{12}&0.69 &0.09 &0.62 &11.9 &16.4 & 7.48&85.3 &94.7 & 85.5 \\ \hline
\end{tabular}
\caption{Iris dataset training and testing average percentage RMS and identification accuracy for different training set sizes using similar RVNN and CVNN networks, and compared with the QNN. The RVNN was run for 50,000 epochs; the CVNN, for 1000 epochs; and the QNN, for 100 epochs. See text for details of the architectures used.}
\label{table:iris_results}
\end{table*}
Results are shown in Table \ref{table:iris_results}. On this classification problem the advantage of the CVNN over the RVNN is not nearly as pronounced. The RMS training errors for the CVNN are consistently below those of the RVNN, but the testing errors are not. In terms of the classification the CVNN does do better, but not as significantly as with the perceptron. The reason the classification improves while the RMS error does not is that the CVNN tends to a flatter output: thresholding gives the ``correct'' answer, but the separation between classification of ``yes'' (i.e., a value near 1) and ``no'' (i.e. a value near zero) is smaller with the CVNN.
Performance in training, testing, and classification by the QNN is also comparable, to both nets. The major difference among the three is in the amount of training necessary to achieve the results. The RVNN required 50,000 epochs, while the CVNN required only 1000, and the QNN only 100. The ratios of iterations necessary is remarkably similar to the results for the perceptron, shown in Table \ref{table:gates_epochs}. In this sense we can say that the QNN is significantly more efficient at training, in classification as well as in computation.
\section{Quantum computation problem}
We now consider our networks' performance on a purely quantum calculation: an entanglement witness.
\subsection{Benchmarking with entanglement}
An outstanding problem in quantum information is quantifying one of the key elements responsible for the powerful capabilities of quantum computational systems: entanglement. Entanglement is the feature whereby two or more subsystems' quantum states cannot be described independently of each other, that is, that the state of a system AB cannot be correctly represented as the (state of A)$\otimes$(state of B), where $\otimes$ indicates the tensor product. Non-multiplicativity, or entanglement, of states, is fundamental to quantum mechanics. Indeed, all of quantum mechanics can be derived from this characteristic \cite{feynman}; in that sense, entanglement is equivalent to non-commutativity of operators, or to the Heisenberg Uncertainty Principle.
Even though quantifying entanglement has been a prominent subject of research, there is no accepted closed-form representation that is generalizable to quantum states of arbitrary size or properties. Because a quantum state is unentangled if it is separable into its component states, entanglement measures are used to determine the ``closeness'' of a quantum state to the subspace of separable product states. Most of the developed measures involve a subset of possible compositions or require extremizations which are difficult to implement \cite{optimizations}, especially for large systems. One well-known measure is the entanglement of formation which quantifies the resources utilized to construct a specific quantum state. For a two-qubit system we have an explicit formula for the entanglement of formation \cite{wootters}, but generalizations to larger systems are difficult.
The lack of a general analytic solution combined with a base of knowledge on specific sets of states suggests an opportunity for a machine learning approach. Given that the entanglement is an inherent property of the quantum mechanical system, that system in a sense ``knows'' what its entanglement is. Thus, if we are clever, we can devise a measurement or measurements that will extract this information, and, it is hoped, somewhat more efficiently than complete determination of the quantum state (``quantum tomography'') would require, as the number of measurements goes like $2^{2n}$, where $n$ is the number of qubits. It is a theorem \cite{2008} that a single measurement cannot determine the entanglement completely even for a two qubit system; however, one measurement can serve as a ``witness'': that is, a measurement giving a lower bound to the correct entanglement.
Using a training set of only four pure states, we successfully showed \cite{2008} that this model of a QNN can ``learn'' a very general entanglement witness, which tracks very well with the entanglement of formation \cite{wootters}, and which not only works on mixed as well as pure states but can be bootstrapped to larger systems \cite{multi qubit} and, most importantly, is robust to noise and to decoherence \cite{2 qubit noise,robustness}.
Any computation done by a quantum system can be simulated; any function can be approximated. But how complex is this calculation? Theorems 1 and 2 tell us that the weights exist, but how many do we need, and are they easy to find? For the two-qubit system, this is fairly easy to characterize. Note that for the time-propagation part of the computation, each step is linear. Thus, if we have a two qubit system Eq. \ref{rhoevolmap} can be written out as follows. Let $M$ be the 4x4 matrix $M=\{m_{ij}\}$, let the time propagator $T = e^{-iLt}$ be $T=\{T_{ij}\}$, and let $\rho(0) = \{\rho_{ij}\}$.
Then $\rho(t_f) = T\rho(0) =$
\begin{equation*}
\begin{bmatrix}
\sum_{j=1}^4 T_{1j}\rho_{j1} & \sum_{j=1}^4 T_{1j}\rho_{j2} & \sum_{j=1}^4 T_{1j}\rho_{j3} & \sum_{j=1}^4 T_{1j}\rho_{j4} \\
\sum_{j=1}^4 T_{2j}\rho_{j1} & \sum_{j=1}^4 T_{2j}\rho_{j2} & \sum_{j=1}^4 T_{2j}\rho_{j3} & \sum_{j=1}^4 T_{2j}\rho_{j4} \\
\sum_{j=1}^4 T_{3j}\rho_{j1} & \sum_{j=1}^4 T_{3j}\rho_{j2} & \sum_{j=1}^4 T_{1j}\rho_{j3} & \sum_{j=1}^4 T_{3j}\rho_{j4} \\
\sum_{j=1}^4 T_{4j}\rho_{j1} & \sum_{j=1}^4 T_{4j}\rho_{j2} & \sum_{j=1}^4 T_{4j}\rho_{j3} & \sum_{j=1}^4 T_{4j}\rho_{j4} \\
\end{bmatrix} \cdot
\end{equation*}
Therefore,
\begin{equation}\label{quadratic}
\big( Tr[M\rho(t_f)] \big)^2 = \bigg( \sum_{l=1}^4\bigg[ \sum_{k=1}^4\bigg( m_{lk} \sum_{j=1}^4 T_{kj}\rho_{jl} \bigg) \bigg] \bigg)^2
\end{equation}
Eq. \ref{quadratic} forms a second order polynomial in terms of the input elements. It is easy to show \cite{robustness} that the entanglement of formation for a two qubit pure state can be rewritten as a quadratic also:
\begin{equation}
E(|\psi\rangle) = 4a^2d^2 + 4b^2c^2 - 8abcd\cos(\theta_3 - \theta_2 - \theta_1)
\end{equation}
\noindent where the (real) numbers $\{ a,b,c,d, \theta_1, \theta_2, \theta_3 \}$ are defined in Eq. \ref{eq:purestate}. So it should not be altogether surprising that the QNN can learn an entanglement indicator for the two-qubit pure system. But clearly this function, according to Theorems 1 and 2, ought to be able to be approximated by both the RVNN and the CVNN, especially for the relatively simple case where all the phase offsets, \{$\theta_{i}$\}, are zero. As we will see, however, even this turns out to be no easy task for either classical net to learn.
\subsection{Results}
In our previous work \cite{2008} we saw that the QNN was able to learn entanglement for a two-qubit system from a training set of only four pure states. It should also be noted that the QNN had also successfully generalized to mixed states, as well. Here we train, and test, only on pure states; nevertheless, neither the RVNN nor the CVNN was able to accomplish this.
Training for each of the four nets on the entanglement witness is shown in Table \ref{table:entangletrain}. Each network was given the entire $4 \times 4$ density matrix of the state as input. The classical nets again were given a hidden layer, and a single output. The architectures used for the NeuralWorks, RVNN, and CVNN were 16 (input layer), 8 (hidden layer), 1 (output layer). Numbers of epochs were 5 to 10 thousand for the real valued nets; 1000 for the CVNN, and only 20 for the QNN. Again we note the increase in efficiency of training in going from RVNN to CVNN, and in going from CVNN to QNN. For the minimal training set of four, all three classical nets trained below $1\%$ error.
However the testing error, shown in Table \ref{table:entangletest} was quite bad for all the classical nets. Increasing the size of the training set did enable marginally better fitting by all the classical networks, but testing error remained an order of magnitude larger than with the QNN. Increasing $N_h$ did not help. Increasing the size of the training set affected training and testing very little for the QNN, which had already successfully generalized with a set of only four. Neither the NeuralWorks nor the RVNN could learn pure state entanglement well, even with a training set as large as 100. And while the (``quantum-inspired'') CVNN could use interference, it did not generalize efficiently or well, either. This is not surprising when one considers that it is precisely access to the entangled part of the Hilbert space that the CVNN lacks; that is, the state of CVNN cannot itself be entangled. In that sense it is inherently classical.
\def1.5{1.5}
\begin{table}[h]
\centering
\begin{tabular}{l|l|l|l|l|}
\cline{2-5}
\multicolumn{1}{c|}{} & \multicolumn{4}{c|}{Training RMS error (\%)} \\ \hline
\multicolumn{1}{|c|}{Training Pairs} & \multicolumn{1}{c|}{N.Works} & \multicolumn{1}{c|}{RVNN} & \multicolumn{1}{c|}{CVNN} & \multicolumn{1}{c|}{QNN} \\ \hline
\multicolumn{1}{|l|}{100} & 5.66 & 3.74 & 0.97 & 0.04 \\ \hline
\multicolumn{1}{|l|}{50} & 5.96 & 5.89 & 0.53 & 0.09 \\ \hline
\multicolumn{1}{|l|}{20} & 6.49 & 4.97 & 0.04 & 0.2 \\ \hline
\multicolumn{1}{|l|}{4} & 0.00 & 0.93 & 0.01 & 0.2 \\ \hline
\end{tabular}
\caption{Training on entanglement of pure states with zero offsets, for the NeuralWorks, RVNN, CVNN, and QNN.}
\label{table:entangletrain}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}{l|l|l|l|l|}
\cline{2-5}
\multicolumn{1}{c|}{} & \multicolumn{4}{c|}{Testing RMS error(\%)} \\ \hline
\multicolumn{1}{|c|}{Training Pairs} & \multicolumn{1}{c|}{N.Works} & \multicolumn{1}{c|}{ANN} & \multicolumn{1}{c|}{CVNN} & \multicolumn{1}{c|}{QNN} \\ \hline
\multicolumn{1}{|l|}{100} & 7.56 & 5.39 & 3.61 & 0.2 \\ \hline
\multicolumn{1}{|l|}{50} & 7.91 & 10.7 & 6.00 & 0.3 \\ \hline
\multicolumn{1}{|l|}{20} & 13.6 & 15.5 & 9.48 & 0.4 \\ \hline
\multicolumn{1}{|l|}{4} & 48.2 & 51.9 & 55.0 & 0.4 \\ \hline
\end{tabular}
\caption{Testing on entanglement of pure states. Each network was tested on the same set of 25 randomly chosen pure states with zero offset. }
\label{table:entangletest}
\end{table}
\section{Conclusions and Future Work}
The problem of quantifying speedup for quantum algorithms continues to be a difficult one, solved only on a case-by-case basis \cite{shor,grover,bernstein}. The situation with quantum machine learning is similar \cite{Biamonte,Aimeur}. We need both general theorems \cite{namthesis} and benchmarking for specific architectures and learning methods.
The marriage of quantum computing and machine learning can be enormously beneficial to both subfields: with machine learning techniques we can find ``algorithms'' \cite{2008}, do automatic scale-up \cite{multi qubit}, and increase robustness significantly \cite{2 qubit noise,robustness}; while with quantum systems and their access to the enormously greater Hilbert space, machine learning can be both more efficient and more powerful \cite{2008}. In this paper we provide evidence of both. We have shown, here, that a fully quantum neural network can do standard classical benchmarking tasks more efficiently than either a real-valued or a complex-valued classical net. Moreover, a fully QNN can do {\it quantum} calculations efficiently, that is, without explicit simulation. In any physical implementation there will also be additional, natural, nonlinearities we can take advantage of, and, possibly, control and use for machine learning.
There is currently much work being done using ``quantum-inspired'' machine learning \cite{bhattacharyya}. Certainly it is of great value to tease out exactly whence the efficiency, and the complexity, of quantum computing - and specifically quantum neural computing - arise. One way of doing this is precisely to do ``quantum-inspired'' calculations: to allow, e.g., the state of the system to be complex-valued in some sense. Clearly without allowing full superposition this will not be fully quantum mechanical, because it will not include entangled states. There are many more states that are at least partially entangled than there are classical states (in the sense of being product states of subsystems.) But fully quantum mechanical calculations can quickly become dauntingly enormous, so the question is, how much can we get from how little?
There are, of course, many more kinds of machine learning tasks that we did not explore here. Moreover, ours is by no means the only quantum machine learning architecture \cite{Biamonte, chen} or approach \cite{Amin} possible. Chen et al. \cite{chen} also found a quantm advantage for their (very different) quantum neural network model, but, as pointed out in the Introduction, this is not a universal finding \cite{Ronnow}. It will be of great interest to see what kinds of problems can benefit significantly from a fully quantum mechanical approach. It is not at all obvious what these are, or how well simulations will translate to physical implementation. There is much still to do.
\section*{Acknowledgment}
The authors would like to thank William Ingle for helpful discussions.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
|
1,116,691,497,734 | arxiv | \section{Introduction}
The Potts model was introduced in 1952~\cite{Pot52} and has since become one of the most studied models of statistical mechanics.
Each vertex of a graph is assigned one of $q$ spins (colours), with spins of adjacent vertices interacting with a strength depending on the temperature $T>0$ of the system. The classical Ising model is obtained when $q=2$. As in the Ising model, the Potts model becomes increasingly ordered as the temperature decreases, and a phase transition occurs on lattices $\mathbb{Z}^{d}$ with $d \geq 2$. The type of the transition depends on $q$ and $d$, which renders the Potts model an archetype of statistical mechanics model with both continuous and discontinuous phase transitions.
The phase transition and its type can be described by the structure of the set of {\em Gibbs measures}.
These are infinite-volume measures which satisfy the equation of Dobrushin~\cite{Dob72} and Lanford--Ruelle~\cite{LanRue69} (DLR):
for every finite domain, conditionally on the configuration on its exterior, the measure inside the domain is given by the Potts specification with boundary conditions provided by the outside configuration.
The set of Gibbs measures is convex and is thus determined by its extremal points.
This work is restricted to the planar case, which we discuss henceforth.
In terms of Gibbs measures, the phase diagram when $q\geq 2$ may be described as follows:
for $T>T_{c}(q)$ the Gibbs measure is unique and exhibits exponential decay of correlations ({\em free measure});
when $T<T_{c}(q)$ there are exist $q$ distinct ordered Gibbs measures obtained as thermodynamic limits under constant-spin ({\em monochromatic}) boundary conditions~\cite{CoqDumIofVel14}.
The value $T_c(q)$ for which the phase transition occurs was determined in~\cite{BefDum12}.
At $T=T_{c}(q)$, when $2 \leq q \leq 4$ the Gibbs measure is unique~\cite{DumSidTas17} ({\em continuous phase transition});
when $q>4$ the phase transition is discontinuous ~\cite{DumGagHar16,RaySpi20}, and there exist at least $q+1$ distinct Gibbs measures:
one free measure obtained as a limit as $T\searrow T_{c}(q)$ and $q$ monochromatic measures obtained as limits as $T\nearrow T_{c}(q)$.
The goal of the present paper is to prove that in this last case, the above-mentioned Gibbs measures are the only extremal ones.
For the Ising model, the set of Gibbs measures was determined in the celebrated works of Aizenman~\cite{Aiz80} and Higuchi~\cite{Hig81}:
every such measure was shown to be a linear combination of the plus and minus states; the two being equal for $T \geq T_c(2)$.
The original proofs relied on the Griffiths' inequality and other properties specific to the Ising model.
The later proof of Georgii--Higuchi~\cite{GeoHig00} is more robust, but still does not apply to the Potts model,
as it relies on positive association and symmetry of phases.
In the present work, these obstacles are circumvented by considering the graphical representation of the Potts model
known as Fortuin--Kasteleyn (FK) percolation or random-cluster model.
In this model, configurations are assignments of open and closed states to each edge,
and their probability depends on the numbers of open edges and of connected components.
The model has two parameters: an edge-intensity $p \in [0,1]$ akin to the temperature of the Potts model and a cluster-weight $q \geq 1$.
When $q \geq 2$ is an integer, the Edwards--Sokal coupling~\cite{EdwSok88} relates FK-percolation to the $q$-state Potts model.
While FK-percolation satisfies positive association and exhibits a natural symmetry between the primal and dual models,
the proofs previously developed for the Ising model do not apply as the interactions are not nearest-neighbour.
We prove nevertheless that all Gibbs measures for FK-percolation are linear combinations of the wired and the free measures.
Our arguments rely on the self-duality of the square grid, but may be extended to the triangular and hexagonal lattices via the star--triangle transformation.
The typical example of an extremal Gibbs measure which is neither free nor monochromatic is one where several regions with monochromatic behaviour coexist.
The interfaces between these regions need then to have bounded variations; such interfaces are called {\em localised}.
It was conjectured by Dobrushin \cite{Dob68} that interfaces in two-dimensional models are never localised.
When the surface tension is positive, the Ornstein--Zernike theory~\cite{CamIofVel03, Iof98, IofVel10}
may be used to prove Gaussian behaviour of interfaces. This strategy was successfully implemented for the Potts model with $T<T_{c}(q)$~\cite{CamIofVel08,CoqDumIofVel14, CoqVel12}, and showed that the only extremal Gibbs measures are the monochromatic ones.
Unfortunately this approach is still beyond reach for the Potts and FK-percolation models at the point of a discontinuous phase transition.
While it is known in these situations that the correlation length is finite, its relation to the surface tension is not sufficiently well understood to allow the development of the Ornstein--Zernike theory.
It should nevertheless be mentioned that for $q$ very large, the Pirogov-Sinai theory applies to the critical FK-percolation and Potts models~\cite{LaaMesMir91} and may be used~\cite{DobShl85} to characterise the structure of Gibbs measures.
It is worth mentioning that in higher dimensions, interfaces may be localised: at low temperatures the Ising model was shown to exhibit a so-called Dobrushin state~\cite{Dob72} (see also~\cite{Bei75} for a simpler proof).
In closing, let us mention that the approach developed here for FK-percolation also applies to the loop $O(n)$ model on the hexagonal lattice~\cite{DomMukNie81},
in the regime of positive association.
The configurations here are collections of non-intersecting simple cycles (loops) and their probabilities depend on two parameters: the loop-weight $n$ and the edge-weight $x$. When $n\geq 1$ and $nx^{2}\leq 1$, the spin representation of the model was shown to be positively associated~\cite{DumGlaPel21},
and the system exhibits either exponential tails for the lengths of loops ({\em small loops}), or scale-invariant behaviour ({\em macroscopic loops}).
These regimes are reminiscent of the discontinuous and continuous phase transitions of FK-percolation, respectively.
In the former case, our approach shows that every Gibbs measure for the spin representation is a linear combination of the plus and the minus states;
in the latter case, there exists a unique Gibbs measure for the spin representation.
Since the loops are invariant under global sign flip of the spins, the Gibbs measure for the loop $O(n)$ model is unique for all $n\geq 1$ and $x\leq 1/\sqrt n$.
\subsection{Results for the Potts model}
We work on the square lattice, with vertex set $\mathbb{Z}^2$ and edge-set $E(\mathbb{Z}^{2})$ connecting vertices at euclidian distance $1$.
For a finite subgraph $D = (V,E)$ of $\mathbb{Z}^2$ (also referred to as {\em domain} henceforth),
write $\partial D = \{v \in V \,: \, d_D(v) < 4\}$ and $D^{c} = \{v\in \mathbb{Z}^{2}\setminus V\}$.
Let $q\geq 1$ be integer, $T>0$ and $\tau \in \{0,1,\dots,q\}^{\partial D}$.
The $q$-state Potts model on $D$ at temperature $T$ with boundary conditions $\tau$ is the probability measure $\mu_{D,T,q}^\tau$
supported on the set of $\sigma\in \{1,2,\dots,q\}^{V}$ such that $\sigma_{u} = \tau_{u}$ for all $u\in \partial D$ with $\tau_{u}\neq 0$,
and defined by
\begin{equation}\label{eq:def-potts}
\mu_{D,T,q}^\tau[\sigma]=\frac{1}{Z^\tau(D,T,q)} \cdot \exp \big(-\tfrac{1}{T}\cdot \sum_{u\sim v} \mathbbm{1}_{\{\sigma_{u} \neq \sigma_{v}\}}\big),
\end{equation}
where the sum is taken over pairs of adjacent vertices $u,v \in V$ and
the normalising constant $Z^\tau(D,T,q)$ is called the {\em partition function}.
If $\tau$ takes a constant non-zero value on $\partial D$, the boundary conditions are called {\em monochromatic}.
For $i=1,\dots,q$, we use the notation $\mu_{D,T,q}^i$ for the measure $\mu_{D,T,q}^\tau $ with $\tau \equiv i$.
If $\tau$ is constant zero on $\partial D$, the boundary conditions are called {\em free}.
We denote the corresponding measure by $\mu_{D,T,q}^{\rm free}$.
Next we consider infinite-volume measures, that is probability measures on $\{1,\dots,q\}^{\mathbb{Z}^{2}}$.
\begin{definition}
A probability measure $\mu$ on $\{1,\dots,q\}^{\mathbb{Z}^{2}}$ is called {\em Gibbs} for the Potts model with parameters $T >0$ and $q \geq2$ if,
for any finite $D\subset \mathbb{Z}^{2}$ and for every $\tau \in \{1,\dots,q\}^{\partial D}$
for which the conditioning is not degenerate
\begin{equation}\label{eq:DLR-Potts}\tag{DLR-Potts}
\mu[. \, | \, \sigma_{|\partial D} = \tau \text{ and } \mathcal{F}(D^{c})] =\mu_{D,T,q}^\tau,
\end{equation}
where $\mathcal{F}(D^{c})$ is the sigma-algebra of events depending only on the spins in $D^{c}$
and the above refers to the restriction of the left-hand side to $D$.
\end{definition}
A probability measure $\mu$ on $\{1,\dots,q\}^{\mathbb{Z}^{2}}$ is called {\em tail trivial} if for any event $A$ that does not depend on any finite number of spins, either $\mu(A) = 0$ or $\mu(A) = 1$.
The set of Gibbs measures for the Potts model with fixed $T$ and $q$ form a convex subset of probability measures on $\{1,\dots,q\}^{\mathbb{Z}^{2}}$.
The extremal points of this convex set are exactly the tail trivial Gibbs measures~\cite[Thm. 7.7]{Geo11}.
The monochromatic and free infinite-volume measures are defined as thermodynamic limits of the finite-volume measures under the corresponding boundary conditions.
Indeed, let $D_{n}$ be an increasing sequence of finite subgraphs of $\mathbb{Z}^{2}$ that eventually cover all vertices and edges of $\mathbb{Z}^{2}$; we say that $(D_{n})_{n\geq 1}$ increases to $\mathbb{Z}^2$.
It is standard (see~\cite[Prop 6.9]{GeoHagMae01}) that, as $n$ tends to infinity, the measures defined by monochromatic or by free boundary conditions have weak limits that do not depend on $D_{n}$ . We will use the following notation for them:
\[
\mu_{T,q}^i := \lim_{n\to \infty}\mu_{D_{n},T,q}^i \quad \text{and} \quad \mu_{T,q}^{\rm free} := \lim_{n\to \infty}\mu_{D_{n},T,q}^{\rm free}
\]
As thermodynamic limits, $\mu_{T,q}^i$ and $\mu_{T,q}^{\rm free}$ satisfy~\eqref{eq:DLR-Potts} and hence are Gibbs measures.
More generally, it is an immediate consequence of the Spatial Markov property~\eqref{eq:SMP} of the Potts model
that the weak limit of any converging sequence of measures $\mu_{D_{n},T,q}^{\tau_n}$ is Gibbs.
It is standard that the monochromatic measures $\mu_{T,q}^i$ ($i = 1,\dots,q$) are always extremal (see Section~\ref{sec:ES}), though they may all be equal.
The free measure is not always extremal, e.g. in the case $T < T_c$ it is the uniform convex combination of the distinct monochromatic ones (see Remark~\ref{rem:potts-known} below).
The next theorem is the main result of this work and it shows that when $q\geq 2$, every Gibbs measure for the Potts model is a linear combination of the (possibly coinciding) $q+1$ measures defined above: $q$ wired and one free.
\begin{thm}\label{thm:mainPotts}
Let $\mu$ be a Gibbs measure for the Potts model on $\mathbb{Z}^2$ with $q\geq 2$ colours at temperature $T>0$.
Then there exist $\lambda_0,\dots, \lambda_q \geq 0$ summing up to $1$ such that
\begin{align*}
\mu = \lambda_0 \mu_{T,q}^{\rm free} + \lambda_1 \mu_{T,q}^1 +\dots + \lambda_q \mu_{T,q}^q.
\end{align*}
\end{thm}
\begin{rem}\label{rem:potts-known}
The result above is only new for the case $T = T_c(q)$ and $q$ such that the phase transition is discontinuous.
Indeed, for $T > T_{c}(q)$ as well as for $T = T_{c}(q)$ for a continuous phase transition, the Gibbs measure is unique.
When $T < T_{c}(q)$, the extremal Gibbs measures were identified in~\cite{CoqDumIofVel14}:
these are the $q$ monochromatic measures $(\mu_{T}^i)_{i = 1,\dots, q}$.
In particular $\mu_{T,q}^{\rm free} = \sum_{i = 1}^q \tfrac1q \mu_{T,q}^i$.
As discussed below, at the point of a discontinuous phase transition, there exist exactly $q + 1$ extremal measures,
namely $\mu_{T,q}^{\rm free}$ and the $q$ monochromatic measures $\mu_{T,q}^1,\dots, \mu_{T,q}^q$.
\end{rem}
As mentioned in the introduction, for $q = 2$ (which corresponds to the Ising model),
Theorem~\ref{thm:mainPotts} was proven in~\cite{Aiz80,Hig81} and again in ~\cite{GeoHig00}; previously \cite{GalMir72,MesMir75} treated the case of translation invariant measures for the Ising model.
Unfortunately, these arguments do not generalise to higher values of $q$, which is why we will prove Theorem~\ref{thm:mainPotts} using the FK representation.
We should however mention that the works mentioned above served as inspiration for our proof of Theorem~\ref{thm:mainFK} below.
Finally, Theorem~\ref{thm:mainPotts} adapts to the Potts model on the triangular and hexagonal lattices with only minor modifications. See Section~\ref{sec:Hex} for details.
\subsection{Results for FK-percolation}
\label{sec:intro-fk}
Let $D = (V,E)$ be a subgraph of $\mathbb{Z}^2$.
A percolation configuration $\omega$ on $D$ is an element of $\{0,1\}^{E}$.
An edge $e$ is said to be {\em open} (in $\omega$) if $\omega_e=1$, otherwise it is {\em closed}.
When speaking of connections and connected components in $\omega$, we view it as a spanning subgraph of $D$ with edge-set $\{e\in E:\omega_e=1\}$.
For sets of vertices $A$ and $B$, we say that $A$ is connected to $B$ if there exists a path of edges of $\omega$ that connects a vertex of $A$ to a vertex of $B$. This event is denoted by $A \xleftrightarrow{} B$ or $A \xleftrightarrow{\omega} B$.
We also speak of connections in a set of edges~$C$ if all the edges of the path are contained in~$C$; we denote this by $\xleftrightarrow{C}$.
A boundary condition $\xi$ on $ D$ is a partition of $\partial D$.
Vertices of $\partial D$ that are in the same component of $\xi$ are said to be wired together.
When all vertices are wired together, the conditions are called {\em wired} and are denoted by $1$.
When no two vertices are wired together, the conditions are called {\em free} and are denoted by $0$.
When $D$ is finite, the FK-percolation measure on $D$ with edge-intensity $p \in (0,1)$,
cluster-weight $q\geq 1$ and boundary conditions $\xi$ is the probability measure
supported on percolation configurations on $D$ given by
\begin{equation}\label{eq:RCM_def1}
\phi_{D,p,q}^\xi[\omega]=\frac{1}{Z^\xi(D,p,q)} (\tfrac{p}{1-p})^{|\omega|}q^{k(\omega^{\xi})},
\end{equation}
where $|\omega|= \# \{e\in E \colon \omega_{e} = 1\}$ is the number of open edges in $\omega$,
$\omega^{\xi}$ is the graph obtained from $\omega$ by identifying boundary vertices that are wired together in $\xi$,
and $k(\omega^{\xi})$ is the number of connected components in $\omega^{\xi}$;
the normalising constant $Z^\xi(D,p,q)$ is called the {\em partition function}.
We now consider percolation configurations defined on the entire square grid, that is $\omega \in \{0,1\}^{E(\mathbb{Z}^{2})}$.
For a finite domain $D$, such $\omega$ induces boundary conditions $\xi$ on $\partial D$ defined as follows.
We say that two vertices $u,v\in\partial D$ are wired together in $\xi$ if and only if at least one of the following occurs: either $u\xleftrightarrow{\omega\setminus E} v$, or $u\xleftrightarrow{\omega\setminus E} \infty$ and $v\xleftrightarrow{\omega\setminus E} \infty$.
We use the notation $u \xleftrightarrow{\omega\setminus E} \infty$ for the event that $u$ belongs to an infinite cluster in $\omega\setminus E$.
\begin{definition}
A measure $\phi$ on $\{0,1\}^{E(\mathbb{Z}^{2})}$ is called {\em Gibbs} for FK-percolation with parameters $p \in (0,1)$ and $q \geq 1$ if,
for any finite subgraph $D = (V,E)$ of $\mathbb{Z}^{2}$ and for every boundary condition $\xi$ on $D$
for which the conditioning is not degenerate
\begin{equation}\label{eq:DLR-FK}\tag{DLR-FK}
\phi[. \, | \, \omega \text{ induces $\xi$ and } \mathcal{F}(E^{c})] =\phi_{D,p,q}^\xi,
\end{equation}
where $\mathcal{F}(E^{c})$ is the sigma-algebra of events depending only on the edges in $E(\mathbb{Z}^2) \setminus E$
and the above refers to the restriction of the left-hand side to $E$.
\end{definition}
\begin{rem}\label{rem:DLR}
In our definition of FK-percolation, if $\omega$ exhibits several infinite clusters, these are considered as a single one -- that is, they are wired at infinity.
This is an arbitrary choice, but a posteriori bares no importance.
Indeed, with our convention, we will prove that for any Gibbs measure there exists a.s. at most one infinite cluster.
A more general definition would be to weight infinite clusters by some $q' \in [1,q]$.
With any such convention, Lemma~\ref{lem:Kcross} shows that the number of infinite clusters is uniformly bounded.
Thus such a change of convention affects the measure by a bounded Radon-Nikodym derivative.
Hence, the uniqueness of the infinite cluster remains a.s. true, and the change of convention has no importance.
We mention nevertheless that with our convention, the dual of a Gibbs measure is not a priori Gibbs.
\end{rem}
Let $(D_{n})_{n\geq 1}$ be a sequence of domains increasing to $\mathbb{Z}^{2}$.
The monotonicity properties of FK-percolation imply in a standard way (see Section~\ref{sec:fk-background})
that the sequences of measures $\phi_{D_n,p,q}^0$ and $\phi_{D_n,p,q}^1$ converge weakly;
their respective limits $\phi^0_{p,q}$ and $\phi^1_{p,q}$ are called the free and wired infinite-volume measures.
They are the minimal and maximal, respectively, Gibbs measures with parameters $(p,q)$ for the stochastic ordering, and in particular they are tail trivial.
Indeed, for any FK-percolation Gibbs measure $\phi$ with parameters $p\in (0,1)$ and $q \geq 1$,
\begin{align}\label{eq:0phi1}
\phi_{p,q}^0 \leq_{\rm st}\phi \leq_{\rm st}\phi_{p,q}^1.
\end{align}
The next theorem states that $\phi^0_{p,q}$ and $\phi^1_{p,q}$ are the only extremal Gibbs measures for FK-percolation.
\begin{thm}\label{thm:mainFK}
Let $\phi$ be a Gibbs measure for FK-percolation on $\mathbb{Z}^2$ with edge-intensity $p \in (0,1)$ and cluster-weight $q \geq 1$.
Then there exists $\lambda \in [0,1]$ such that
\begin{align*}
\phi = \lambda \phi_{p,q}^0 + (1-\lambda) \phi_{p,q}^1.
\end{align*}
\end{thm}
\begin{rem}
Except when $p = p_c(q) = \frac{\sqrt q}{1 + \sqrt q}$ and $q > 4$, it is known that $\phi_{p,q}^0 = \phi_{p,q}^1$,
and it follows directly from \eqref{eq:0phi1} that the infinite-volume measure is unique
\end{rem}
This result should be compared to its counterpart for the Ising model~\cite{Aiz80, Hig81,GeoHig00}.
The main reason why these proofs do not extend directly is the appearance of the number of clusters in the definition of the measure, which induces infinite-range dependencies. More precisely, the spatial Markov property in FK-percolation is significantly weaker than that of the Ising model: the distribution on a finite subgraph $D$ is not determined by the set of open/closed edges on the boundary of $D$ --- instead the boundary condition is determined by the configuration on the entire complement of $D$. Nevertheless, similarly to the Ising model, FK-percolation on $\mathbb{Z}^{2}$ is positively associated and self-dual at $p=p_{c}(q)$~\cite{BefDum12}. The latter property replaces the symmetry between $+$ and $-$ spins that was used in the previous works.
Our approach follows the ideas of~\cite{GeoHig00}; we sketch it below so as to stress the difference with the Ising model.
First note that the result may be obtained quite easily for translation invariant measures (see Section~\ref{sec:translation_inv}).
Let us proceed by contradiction and assume the existence of an extremal Gibbs measure $\phi$, which is different from $\phi^0$ and $\phi^1$.
Then $\phi$ is tail trivial and must exhibit at least one infinite primal cluster and one infinite dual cluster. The proof of~\cite{GeoHig00} extends to show that in each half-plane there exists at most one infinite primal cluster and one infinite dual one. This implies the existence of one or two bi-infinite interfaces separating primal and dual clusters in the plane. These interfaces must have bounded fluctuations and thus visit a vicinity of $0$. The goal is then to show that the measure is invariant under translations, thus achieving a contradiction.
Denote by $\phi'$ a shift of $\phi$ by an arbitrary integer vector.
As in~\cite{GeoHig00}, we consider a duplicate system, which consists of two independent samples: $\omega$ from $\phi$ and $\omega'$ from $\phi'$. The core of the argument is the construction of arbitrarily large domains $D$ which are measurable in terms of the configurations outside of them and such that the boundary conditions on $D$ induced by $\omega'$ dominate those induced by $\omega$.
It then follows that $\phi'$ stochastically dominates $\phi$. The opposite bound is obtained by symmetry, and thus $\phi = \phi'$,
which is to say that $\phi$ is invariant under translations, as required.
In~\cite[Lemma 5.5]{GeoHig00}, the construction of domains $D$ as above is quite direct,
since in the Ising model it suffices to ensure that $\omega' \geq \omega$ on $\partial D$.
For FK-percolation however, that $\omega' \geq \omega$ on the boundary of $D$ does not imply that the boundary conditions on $D$ induced by $\omega'$ dominates those induced by $\omega$.
Our construction is significantly more involved and requires a sophisticated exploration argument that allows to compare boundary conditions. The exploration relies on a new result for percolation in the half-plane (Proposition~\ref{prop:hp_perco}), which may be of independent interest.
Due to the presence of long-range interactions in FK-percolation, it is not {\em a priori} clear that every thermodynamic limit is a Gibbs measure.
However, this statement can be derived from Theorem~\ref{thm:mainFK} and Lemma~\ref{lem:Kcross} below.
\begin{cor}\label{cor:every-limit-is-gibbs}
Let $p\in (0,1)$, $q\geq 1$, $(D_{n})_{n \geq1}$ be a sequence of domains increasing to $\mathbb{Z}^{2}$
and $(\xi_{n})_{n \geq1}$ be a sequence of boundary conditions on $(D_{n})_{n \geq1}$.
Assume that the sequence of FK-percolation measures $\phi_{D_{n},p,q}^{\xi_{n}}$ has a weak limit.
Then the limiting measure is a Gibbs measure for FK-percolation with parameters $(p,q)$.
Moreover, it is a linear combination of the free and the wired measures.
\end{cor}
Finally, let us mention that the results of this section adapt to FK-percolation on the triangular and hexagonal lattices as described in Section~\ref{sec:Hex}.
\subsection{Loop $O(n)$ model}
Let ${\rm Hex}$ denote the hexagonal lattice of Figure~\ref{fig:tri}.
A loop configuration on ${\rm Hex}$ is any even subgraph of ${\rm Hex}$.
Then, each loop configuration consists of disjoint cycles (loops) and bi-infinite paths.
Let $D$ be a finite subgraph of the hexagonal lattice and let $\xi$ be a loop configuration on ${\rm Hex}$.
For $n,x>0$, the loop $O(n)$ model on $D$ with boundary conditions $\xi$ and edge-weight $x>0$ is
the probability measure supported on loop configurations that coincide with $\xi$ on $D^{c}$ and defined by:
\[
{\sf Loop}_{D,n,x}^{\xi}(\omega) = \frac{1}{Z^{\xi}(D,n,x)}\cdot n^{\# \text{loops}}x^{\# \text{edges}},
\]
where $\# \text{edges}$ stands for the number of edges in $\omega \cap D$
and ${\# \text{loops}}$ for the number of finite loops of $\omega$ that intersect $D$.
Note that the number of bi-infinite paths intersecting $D$ is the same for all $\omega$ in the support ${\sf Loop}_{D,n,x}^{\xi}$,
and we omit it from the count for convenience.
Gibbs measures for the loop $O(n)$ model are defined via the DLR condition in a straightforward fashion.
As for FK-percolation, it is not immediate that thermodynamic limits are Gibbs.
Moreover, it is \emph{a priori} not clear that any Gibbs measures exist.
The following result settles these questions in the regime of positive association for the spin representation of the model
(see Section~\ref{sec:loop-background} for details on the spin representation).
\begin{thm}\label{thm:loop}
Let $n\geq 1$ and $0< x \leq \tfrac{1}{\sqrt{n}}$.
Then, there exists a unique Gibbs measure for the loop $O(n)$ model with edge-weight $x$.
This measure is translation invariant and tail trivial.
Moreover, it is the weak limit of ${\sf Loop}_{D_{k},n,x}^{\xi_{k}}$,
for any sequence of domain $D_{k}$ increasing to ${\rm Hex}$ and any sequence $\xi_{k}$ of loop configurations.
\end{thm}
When $n=1, x\in [0,1]$, the loop $O(n)$ model is in correspondence with the ferromagnetic Ising model on the triangular lattice and the statement is known~\cite{Aiz80,Hig81}.
The statement was recently proven also in the following two ranges of parameters: when $n\in [1,2]$ and $x=x_{c}(n) := \tfrac{1}{\sqrt{2+\sqrt{2-n}}}$~\cite{DumGlaPel21}; when $n\in [1,1+\varepsilon)$ and $x\in (\tfrac{1}{\sqrt{n}}-\varepsilon,\tfrac{1}{\sqrt{n}}]$. In fact a stronger statement was shown there: strong Russo--Seymour--Welsh (RSW) estimates.
Such estimates are expected to hold for all $n\in [0,2]$ and $x \geq x_{c}(n)$.
The authors of the present paper proved~\cite{GlaMan18} the uniqueness of the Gibbs measure at $n=2$ and $x=1$, a result which is not covered by Theorem~\ref{thm:loop}.
However, it is still unproved in this case that the weak limit of any converging sequence of finite volume measures is Gibbs.
When $n$ is large, the behaviour is different and is better understood through the Pirogov--Sinai theory and similar techniques.
More precisely, for some $c,C >0$, if $n > C$ and $nx^6 <c$, the work of Dobrushin--Shlosman~\cite{DobShl85} implies the existence of a unique Gibbs measure, whereas when both $n$ and $nx^6$ are larger than $C$,~\cite{DumPelSamSpi17} proves the existence of at least three distinct extremal Gibbs measures --- these approximate three different ground states and are translations of each other.
Moreover, it is likely that~\cite[Thm~1.8]{DumPelSamSpi17} may be combined with the technique of~\cite{DobShl85}
to prove that these are the only extremal Gibbs measures for the loop $O(n)$ model for parameters as above.
Further background on the loop $O(n)$ model is given in Section~\ref{sec:loop-background}, while the proof of Theorem~\ref{thm:loop} is discussed in Section~\ref{sec:loop_proof}.
This is achieved by adapting the proof of Theorem~\ref{thm:mainFK} to the spin representation of the loop $O(n)$, then transferring the result to the actual loop model.
\paragraph{Organisation of the paper.}
In Section~\ref{sec:background} we give a quick overview of the properties of the three models of interest: FK-percolation, the Potts and the loop $O(n)$ models.
The following two sections deal with the proof of the main result for FK-percolation: Section~\ref{sec:FK_preparation} sets up the context for the duplication argument,
which is then performed in Section~\ref{sec:FK_duplication}.
Corollary~\ref{cor:every-limit-is-gibbs} is proved in Section~\ref{sec:every-limit-is-gibbs}.
Section~\ref{sec:Potts_proof} contains the proof of the main result for the Potts model, namely Theorem~\ref{thm:mainPotts}.
Finally, Section~\ref{sec:extensions} contains two extensions of Theorem~\ref{thm:mainFK}: the same result for the triangular and hexagonal lattices and the proof of Theorem~\ref{thm:loop}.
\paragraph{Acknowledgements}
The authors would like to thank Yinon Spinka for sharing his ideas regarding the exploration argument
and Sebastien Ott for helpful discussions and background.
A significant part of this work was conducted in Fribourg, and the first author is grateful to the University of Fribourg for its hospitality and to the Swiss National Science Foundation for financing these visits.
The second author is supported by the Swiss National Science Foundation and is a member of the NCCR SwissMAP.
\section{Background}\label{sec:background}
For the rest of the paper we will work with the lattice $\mathbb{L} = (V(\mathbb{L}),E(\mathbb{L}))$ whose vertices are
$\{(x,y) \in \mathbb{Z}^2: x+y \text{ even}\}$ and which contains edges between vertices at euclidian distance $\sqrt 2$.
Write $\mathbb{H}$ for the half-plane $\mathbb{R} \times [0,+\infty)$ and $\mathbb{H}^-$ for its vertical reflection $\mathbb{R} \times (-\infty,0]$.
For $n \geq 1$, let the box of size $n$ be defined by $\Lambda_n := [-n,n]^2$.
We will routinely identify these regions to the graphs spanned by the edges of $\mathbb{L}$ contained in them.
Everywhere in this work we assume $q \geq 1$ and $p \in (0,1)$.
\subsection{FK-percolation}\label{sec:fk-background}
We will use standard properties of FK-percolation.
They can be found in~\cite[Chapters 3 and 4]{Gri06}, and we only recall them briefly below.
Fix a finite subgraph~$D = (V,E)$ of~$\mathbb{L}$.
\paragraph{Monotonic properties.}
The partial order on $\{0,1\}^{E(\mathbb{L})}$ is defined by setting that $\omega\geq \omega'$ if $\omega_{e}\geq \omega'_{e}$, for all $e\in E(\mathbb{Z}^{2})$.
An event~$A$ is called {\em increasing} if for any~$\omega\ge\omega'$, $\omega'\in A$ implies that~$\omega\in A$.
For two boundary conditions $\xi$ and $\xi'$ on $D$ we write $\xi\ge\xi'$ if any two boundary vertices wired together in $\xi'$ are also wired together in $\xi$.
Fix~$q\ge1$,~$p \in (0,1)$, and boundary conditions~$\xi\ge\xi'$. Then, for every increasing events~$A$ and~$B$,
\begin{align*}
\phi_{D,p,q}^\xi[A\cap B]&\ge \phi_{D,p,q}^\xi[A]\, \phi_{D,p,q}^{\xi}[B], \tag{PosAssoc}\label{eq:FKG} \\
\phi_{D,p,q}^{\xi}[A]&\ge \phi_{D,p,q}^{\xi'}[A]. \tag{CBC}\label{eq:CBC}
\end{align*}
The inequalities above will respectively be referred to as the {\em positive association} and the {\em comparison between boundary conditions}.
The second inequality may also be stated as $\phi_{D,p,q}^{\xi}$ dominates stochastically $\phi_{D,p,q}^{\xi'}$, which we write
$\phi_{D,p,q}^{\xi}\ge_{\rm st} \phi_{D,p,q}^{\xi'}$.
We direct the reader to~\cite[Ch.~4]{Gri10} for details on stochastic domination.
\paragraph{Spatial Markov property.}
Let $D'$ be a subgraph of $D$ and let $\zeta$ be a percolation configuration on $\mathbb{Z}^2$.
As discussed above (Section \ref{sec:intro-fk}), $\zeta$ defines boundary conditions on $D$ and $D'$~--- denote them by $\xi$ and $\xi'$, respectively. Then,
\begin{equation}\tag{SMP}\label{eq:SMP}
\phi_{D,p,q}^{\xi}[\, \cdot \, \,|\,\omega_e=\zeta_e,\forall e\notin D'] = \phi_{D',p,q}^{\xi'}[\,\cdot\,].
\end{equation}
\paragraph{Finite energy property.}
There exists $\varepsilon = \varepsilon (p,q) > 0$ such that the following holds.
For any subgraph $D = (V,E)$ of $\mathbb{Z}^{2}$, any boundary conditions $\xi$, any $E' \subset E$, and any percolation configuration $\omega_0$,
\[
\phi^{\xi}_{D,p,q}[\omega = \omega_0 \text{ on $E'$}] \geq \varepsilon^{|E'|}.
\]
\paragraph{Infinite-volume limits.}
Due to~\eqref{eq:CBC} and~\eqref{eq:SMP}, for any growing sequence of subgraphs $D_{n}$ increasing to $\mathbb{Z}^{2}$,
the measures $\phi_{D_{n},p,q}^0$ form a stochastically increasing sequence.
Hence their weak limit exists and is independent of the sequence $D_{n}$.
The limit is called the {\em free infinite-volume measure for FK-percolation} and is denoted by $\phi_{p,q}^0$.
Similarly, the sequence $\phi_{D_{n},p,q}^1$ is stochastically decreasing, its weak limit is called the {\em wired infinite-volume measure for FK-percolation} and is denoted by $\phi_{p,q}^1$.
The measures $\phi_{p,q}^0$ and $\phi_{p,q}^1$ are invariant under translations and ergodic, and $\phi^0_{p,q} \leq_{\rm st}\phi^1_{p,q}$.
Depending on the values of $p$ and $q$, these two measures may be equal or not.
\paragraph{Dual model.}
Define the dual lattice of $\mathbb{L}$ as $\mathbb{L}^* = \mathbb{L} + (1,0)$, that is its translate to the right by one unit.
Notice that each vertex of $\mathbb{L}^*$ corresponds to a face of $\mathbb{L}$
and each edge $e$ of $\mathbb{L}$ is crossed by a unique edge $e^*$ of $\mathbb{L}^*$.
We will call the lattice $\mathbb{L}$ and its edges {\em primal}, while those of $\mathbb{L}^*$ are called {\em dual}.
For a subgraph $D = (V,E)$ of $\mathbb{L}$, the dual graph~$D^*=(V^*,E^*)$ of~$D$ is the subgraph of $\mathbb{L}^*$ spanned by the edges dual to those of~$D$.
Given a subgraph configuration~$\omega$, construct its dual configuration~$\omega^*$ on~$D^*$ by
declaring any edge of the dual graph to be open (resp.\ closed) if the
corresponding edge of the primal lattice is closed (resp.\ open) for $\omega$.
Connections in $\omega^*$ are written $\xleftrightarrow{*}$ or $\xleftrightarrow{\omega^*}$.
When $\omega$ is sampled according to $\phi_{D,p,q}^\xi$ for some boundary conditions $\xi$,
its dual configuration $\omega^*$ follows an FK-percolation distribution
with the same parameter~$q$, dual parameter~$p^*$ satisfying~
\begin{align}
\frac{p^*p}{(1-p^*)(1-p)}=q, \label{eq:pp*}
\end{align}
and dual boundary conditions $\xi^*$.
We refer to~\cite{Gri06} for a full description of dual boundary conditions
and simply mention that if $D$ is simply connected and if $\xi$ is induced by a configuration $\omega$ on $\mathbb{L} \setminus D$
with no infinite cluster, then $\xi^*$ are the boundary conditions induced by $\omega^*$.
In particular, the dual of free boundary conditions are the wired ones.
As a consequence of this last fact, if $\omega$ is sampled according to the infinite-volume measure $\phi^0_{p,q}$, then $\omega^*$ has law
$\phi^1_{p^*,q}$.
For each fixed $q$, the unique value of $p$ for which $p = p^*$ is called the {\em self-dual} point.
It may be derived from~\eqref{eq:pp*} and was shown to be the critical point in~\cite{BefDum12b}, so that
\begin{align*}
p_{c}(q) = p_{\rm sd}(q) = \frac{\sqrt q}{1 + \sqrt q}.
\end{align*}
When $p\neq p_{c}(q)$, by~\cite{BefDum12b}, the large-scale behaviour of the model is trivial and $\phi_{p,q}^{0} = \phi_{p,q}^{1}$.
\paragraph{Dichotomy at the self-dual point.}
It was proved in~\cite{DumSidTas17} (see also~\cite{DumTas19} for a more general proof) that when $\phi_{p_{c}(q),q}^{0} \neq \phi_{p_{c}(q),q}^{1}$, the lengths of interfaces between clusters have exponential tails: for some $c>0$
\begin{align}\label{eq:exp_decay}
\phi_{p,q}^0[0\leftrightarrow \partial \Lambda_n] \leq e^{-cn} \quad \text{ and }\quad
\phi_{p,q}^1[(1,0)\xleftrightarrow{*}\partial \Lambda_n] \leq e^{-cn}
\qquad \text{ for all $n\geq 1$}.
\end{align}
Moreover, it was proved in ~\cite{DumSidTas17} that $\phi_{p_{c}(q),q}^{0}=\phi_{p_{c}(q),q}^{1}$ when $1 \leq q \leq 4$. Conversely, when $q>4$ it was proved in~\cite{DumGagHar16} (see also~\cite{RaySpi20} for a shorter proof) that $\phi_{p_{c}(q),q}^{0} \neq \phi_{p_{c}(q),q}^{1}$.
\paragraph{Structure of Gibbs measures.}
It is well-known (see eg.~\cite[Theorem 7.26]{Geo11}) that every Gibbs measure is a linear combination of extremal Gibbs measures.
Thus, in order to describe the set of Gibbs measures of the model, it is enough to characterise its extremal measures,
which are exactly those that are tail trivial~\cite[Thm. 7.7]{Geo11}.
It will be crucial for our arguments that tail trivial Gibbs measures are positively associated (as opposed to other Gibbs measures, which may lack this property).
This fact is well-known but we include the proof for completeness.
\begin{lem}\label{lem:extremal-fkg}
Let $\phi$ be a tail-trivial measure for FK-percolation with parameters $p\in (0,1)$ and $q\geq 1$.
Then $\phi$ is positively associated, that is for any two increasing events $A,B\in \{0,1\}^{E(\mathbb{L})}$, one has~\eqref{eq:FKG}.
\end{lem}
\begin{proof}
For $\omega$ sampled from $\phi$, denote by $\xi_{n}$ the boundary conditions induced by $\omega$ on $\Lambda_{n}$.
Fix two increasing events $A,B$.
Then, by \eqref{eq:DLR-FK} and \eqref{eq:FKG}, $\phi$-a.s., for any $n>0$,
\[
\phi [A\cap B \, | \, \xi_{n}] \geq \phi [A \, | \, \xi_{n}]\, \phi [B \, | \, \xi_{n}].
\]
When regarded as functions of $n$, each probability above is a bounded backward martingale, and therefore converges.
The limit is given by the probability of the corresponding event under $\phi [. \, | \, \mathcal{F}_\infty]$, where $\mathcal{F}_\infty$ is the tail sigma-algebra.
By tail triviality, all three limits are a.s. constant, and equal to the unconditional probabilities of the corresponding events.
\end{proof}
\paragraph{A bound on the number of large clusters.}
The following result is a direct consequence of~\eqref{eq:exp_decay}, and will come in handy at different times.
\begin{lem}\label{lem:Kcross}
Let $p\in (0,1)$ and $q\geq 1$ be such that $\phi_{p,q}^{0} \neq \phi_{p,q}^{1}$.
There exists $K \geq 1$ and $c > 0$ such that,
for any $n$ and any boundary conditions $\xi$ on $\Lambda_{2n}$,
\begin{align}\label{eq:maxK}
\phi_{\Lambda_{2n},p,q}^\xi[\exists K \text{ disjoint clusters intersecting both $\Lambda_n$ and $\partial \Lambda_{2n}$}] \leq e^{-cn}.
\end{align}
\end{lem}
\begin{proof}
We start by proving a version of the statement for $\xi = 0$ and when all edges between $\Lambda_{n-1}$ and $\partial \Lambda_n$ are conditioned to be closed. Write $\partial \Lambda_n \equiv 0$ for the latter event.
By~\eqref{eq:exp_decay}, for some $c'>0$,
\[
\phi^{0}_{\Lambda_{2n}, p,q}[\exists \text{ a crossing from $\Lambda_n$ to $\partial\Lambda_{2n}$} \, | \, \partial \Lambda_n \equiv 0] \leq e^{-c'n}.
\]
Order the points on $\partial\Lambda_{n}$ in an arbitrary order and explore the primal clusters containing them, one after the other.
Conditionally on $\partial \Lambda_n \equiv 0$ and on the first $k$ explored clusters,
the measure in the unexplored part of $\Lambda_{2n}\setminus \Lambda_{n}$ has free boundary conditions.
Thus, applying~\eqref{eq:CBC} and the above bound, we find that for any $k\geq 0$,
\begin{align*}
&\phi^{0}_{\Lambda_{2n}, p,q}[\exists \text{ $k+1$ disjoint clusters intersecting both $\Lambda_n$ and $\partial \Lambda_{2n}$}\, | \, \partial \Lambda_n \equiv 0] \\
&\qquad \leq e^{-c'n}\,\phi^{0}_{\Lambda_{2n}, p,q}[\exists \text{ $k$ disjoint clusters intersecting both $\Lambda_n$ and $\partial \Lambda_{2n}$}\, | \, \partial \Lambda_n \equiv 0].
\end{align*}
Apply this bound repeatedly to conclude that, for any $K\geq 1$,
\[
\phi^{0}_{\Lambda_{2n}, p,q}[\exists \text{ $K$ disjoint clusters intersecting both $\Lambda_n$ and $\partial \Lambda_{2n}$}\, | \, \partial \Lambda_n \equiv 0] \leq e^{-Kc'n}.
\]
We turn our attention to the actual proof of~\eqref{eq:maxK}.
By the finite energy property, conditionally on the existence of
$K$ disjoint clusters crossing from $\Lambda_n$ to $\partial \Lambda_{2n}$, the event $\partial \Lambda_n \equiv 0$ occurs with a probability of at least $\varepsilon^n$, for some constant $\varepsilon>0$
independent of $n$ or $K$. Moreover, changing the boundary conditions from free to $\xi$ affects the measure on $\Lambda_{2n}$ by a Radon-Nikodym derivative bounded between $C^{-n}$ and $C^{n}$, for some constant $C$ independent of $n$ or $K$.
Thus
\[
\phi^{\xi}_{\Lambda_{2n}, p,q}[\exists \text{ $K$ disjoint clusters intersecting both $\Lambda_n$ and $\partial \Lambda_{2n}$}] \leq (C/\varepsilon)^{n}e^{-Kc'n}.
\]
The statement then follows by choosing $K$ sufficiently large.
\end{proof}
\begin{rem}
The proof of Theorem~\ref{thm:mainFK} presented below aims to be generic and to apply to a variety of models with the properties above;
as illustrated by the loop $O(n)$ model.
We will not attempt to describe here the minimal assumption needed for our proof,
but mention that positive association, some form of self-duality and of the spatial Markov property are necessary.
The exponential decay of~\eqref{eq:exp_decay} is also used extensively,
but this is not limiting as the dichotomy theorem of~\cite{DumTas19} applies to models with the properties above
and yields~\eqref{eq:exp_decay} whenever the Gibbs measure is not unique.
\end{rem}
\subsection{Potts model: background and Edwards-Sokal coupling}\label{sec:ES}
The Potts model and FK-percolation are related via the Edwards--Sokal coupling~\cite{EdwSok88}.
We start by stating the coupling and then explain its implications.
Fix an integer $q \geq 2$. For $T > 0$, let $p(T) = 1 - e^{-1/T}$.
\paragraph{Finite volume.}
For a finite domain $D = (V,E)$,
we will define a probability measure $\Phi$ on $\{0,1\}^E \times \{1,\dots, q\}^V$ whose marginals are
the FK-percolation with parameters $(p(T),q)$ and the Potts model with parameters $(T,q)$, respectively.
The measure under boundary conditions $\tau : \partial D \to \{0,1,\dots, q\}$ is defined by
\begin{align}\label{eq:PhiD}
\Phi_D^\tau [(\omega,\sigma)] = \tfrac{1}{Z_D^{\tau}}\big(\tfrac{p}{1-p}\big)^{|\omega|}
\mathbbm{1}_{\{\sigma(u) = \tau(u): \, \forall u\in \partial D\text{ with }\tau(u) \neq 0\}}
\mathbbm{1}_{\{\sigma(u) = \sigma(v): \, \forall uv \in \omega\}},
\end{align}
for all $(\omega,\sigma) \in \{0,1\}^E \times \{1,\dots, q\}^V$, where $Z_D^{\tau}$ is a normalising constant.
Call a pair of configuration $(\omega,\sigma)$ {\em compatible} if any open edge of $\omega$ connects vertices of equal spin in $\sigma$,
which is to say that the second indicator above is equal to $1$.
Call $\sigma$ compatible with $\tau$ if the first indicator above is equal to $1$.
Then $\Phi_D^\xi$ is supported on the set of compatible pairs of configurations with $\sigma$ compatible with $\tau$.
We start by describing why the marginal on $\sigma$ of $\Phi_D^\tau$ is a Potts measure.
Fix a spin configuration $\sigma \in \{1,\dots,q\}^{V}$ coherent with the boundary conditions $\tau$.
Summing $ \Phi_D^\tau [(\omega,\sigma)] $ over all $\omega$ compatible with $\sigma$,
we observe that every pair of adjacent $u,v\in D$ contributes $1$ if $\sigma_{u}\neq \sigma_{v}$ (since $\omega_{uv} = 0$)
and $1+ \tfrac{p}{1-p}$ if $\sigma(u) =\sigma(v)$ (since $\omega_{uv}$ may take values $0$ or $1$).
Due to the choice of $p(T)$, $1+ \tfrac{p(T)}{1-p(T)} = e^{1/T}$.
Hence
\begin{align*}
\sum_\omega \Phi_D^\tau [(\omega,\sigma)] = \tfrac{1}{Z_D^{\tau}}\exp \big(\tfrac{1}{T}\cdot \sum_{u\sim v} \mathbbm{1}_{\{\sigma_{u} = \sigma_{v}\}}\big),
\end{align*}
which coincides with the definition of the $q$-state Potts model given by~\eqref{eq:def-potts}.
As a consequence of the above, we also deduce that the law of $\omega$ conditionally on $\sigma$ is obtained by opening and closing edges independently, with edges $uv$ with $\sigma(u) =\sigma(v)$ having probability $p(T)$ to be open, while all other edges have probability $0$ to be open. \smallskip
We now discuss the marginal on $\omega$.
Fix an edge configuration $\omega\in \{0,1\}^{E}$.
If there exist vertices $u,v \in \partial D$ which are connected in $\omega$ and such that $\tau(u)$ and $\tau(v)$ are distinct and both non-zero, then there exists no $\sigma$ which is coherent with both $\omega$ and $\tau$, and $\Phi_D^\tau[\omega] = 0$.
Assume next that for any two $u,v \in\partial D$ which are connected by a path in $\omega$,
either $\tau_{u}=\tau_{v}$, or $\tau_{u}= 0$, or $\tau_{v}= 0$.
The spin configurations $\sigma$ compatible with $\omega$
are obtained by attributing the same spin to all vertices in each connected component of $\omega$.
Moreover, to ensure that $\sigma$ is coherent with $\tau$, any connected component of $\omega$ that contains a vertex $v \in \partial D$ with $\tau(v) \neq 0$ is necessarily attributed spin $\tau(v)$.
Summing over all spin configurations $\sigma$ compatible with $\omega$ we find
\begin{align*}
\sum_\sigma \Phi_D^\tau [(\omega,\sigma)] = \tfrac{1}{Z_D^{\tau}}\big(\tfrac{p}{1-p}\big)^{|\omega|}q^{k^{\tau}(\omega)},
\end{align*}
where $k^{\tau}(\omega)$ is the number of connected components of $\omega$ that contain no vertex $v \in \partial D$ with $\tau(v) \neq 0$.
In particular, if $\tau$ takes only values $0$ and at most one other value, say $1$, then
$\omega$ has the law of FK-percolation on $D$ with wired boundary conditions on $\{u \in \partial D: \sigma(u) = 1\}$
and free on the rest of $\partial D$.
When $\tau$ takes more than one non-zero value,
$\omega$ has the law of FK-percolation on $D$ with wired boundary conditions on each of $\{u \in \partial D: \tau(u) = i\}$ for $i = 1,\dots,n$,
conditioned that each of these parts of $\partial D$ are disconnected from each other.
Finally, from the above it is immediate that, conditionally on $\omega$, $\sigma$ is uniform among configurations coherent with both $\omega$ and $\tau$.
\paragraph{Infinite volume.}
The Edwards--Sokal coupling in infinite volume may be constructed similarly to that in finite volume.
Let $\mu$ be a Gibbs measure for the Potts model on $\mathbb{L}$.
Define a measure $\Phi$ on pairs of edge and spin configurations $(\omega,\sigma)\in \{0,1\}^{E(\mathbb{L})} \times \{1,\dots, q\}^{V(\mathbb{L})}$
by sampling $\sigma$ according to $\mu$, then sampling $\omega$ by opening each edge $uv$ with probability $p(T)$ if $\sigma(u) =\sigma(v)$ and $0$ otherwise, independently.
Defined in this way, the measure $\Phi$ satisfies a version of the DLR condition:
for any finite domain $D = (V,E)$, conditionally on $\omega$ on $E(\mathbb{Z}^2) \setminus E$ and on $\sigma$ on $V(\mathbb{Z}^2) \setminus V$,
the restriction of $\Phi$ to $D$ is $\Phi_D^\tau$,
where $\tau(u) = 0$ for vertices $u \in \partial D$ which are disconnected in $\omega \cap E^c$ from $V^c$
and $\tau(u) = i$ for all vertices $u \in \partial D$ that are connected in $\omega \cap E^c$ to a vertex $v \in V^c$ with $\sigma(v) = i$.
Note that for the conditioning to be non-degenerate, all vertices $v \in V^c$ connected in $\omega \cap E^c$ to some $u \in \partial D$ need to have same spin.
Observe also that $\tau$ depends only on $\omega$ for the edges of $E^c$ incident to $\partial D$ and
on $\sigma$ for the vertices of~$V^c$ neighbouring vertices of $\partial D$.
\paragraph{Sampling Potts from FK.}
The marginal on $\omega$ of a measure $\Phi$ constructed as above is not {\em a priori} a Gibbs measure for FK-percolation.
However, this is indeed the case when $\Phi$ was constructed from one of the monochromatic Potts measures or from the free one.
It is indeed easy to see that $\mu_{T,q}^{i}$ leads to the wired FK-percolation measure $\phi^{1}_{p(T),q}$ for any $i \in \{1,\dots, q\}$.
Moreover, the procedure can be reversed: $\mu_{T,q}^{i}$ may be sampled from $\phi^{1}_{p(T),q}$ by colouring each finite cluster uniformly in one of the $q$ colours, independently from other clusters, and colouring the unique infinite cluster (if such exists) in colour $i$.
In this way, the tail triviality of $\mu_{T,q}^{i}$ may be deduced from that of $\phi^{1}_{p(T),q}$.
In addition, $\mu_{T,q}^{\mathrm{free}}$ can be sampled from $\phi^{0}_{p(T),q}$ by colouring every (finite or infinite) cluster uniformly and independently of all other clusters.
Thus, $\mu_{T,q}^{\mathrm{free}}$ is tail trivial whenever $\phi^{0}_{p(T),q}$ exhibits no infinite cluster,
and it equals $\tfrac{1}{q}(\mu_{T,q}^{1} + \dots + \mu_{T,q}^{q})$ when $\phi^{0}_{p(T),q}$ exhibits a unique infinite cluster.
\subsection{Loop $O(n)$ model}\label{sec:loop-background}
It is straight-forward that the loop-measure ${\sf Loop}_{D,n,x}$ itself satisfies neither positive association nor the finite energy property due to hard-core constraints: no vertex can have degree one or three.
Fortunately, the spin representation of the loop $O(n)$ model that was introduced in~\cite{DumGlaPel21} (not to be confused with the spin $O(n)$ model) does satisfy these properties when $n\geq 1$ and $x\leq \tfrac{1}{\sqrt{n}}$.
\paragraph{Spin representation.}
Let $F({\rm Hex})$ denote the set of faces of the hexagonal lattice.
For a loop configuration $\omega$ on ${\rm Hex}$, call a spin configuration $\sigma \in \{-1,1\}^{F({\rm Hex})}$ coherent with $\omega$ if the spins at any two neighbouring faces differ if and only if the edge common to the two faces belongs to $\omega$.
This determines $\sigma$ up to a global spin flip and allows to define the push-forward of loop measures to spin configurations.
Under boundary conditions $\tau \in \{-1,1\}^{F({\rm Hex})}$, it is supported on spin configurations $\sigma \in \{-1,1\}^{F({\rm Hex})}$ that agree with $\tau$ on faces outside of $D$ and may be written as:
\begin{equation}\label{eq:def-spin-rep}
{\sf Spin}_{D,n,x}^{\tau}(\sigma) = \frac{1}{Z^{\tau}_{\sf spin}(D,n,x)}\cdot n^{k_{\pm}(\sigma)}x^{\#\{u \sim v \colon \sigma_{u}\neq \sigma_{v}\}},
\end{equation}
where $k_{\pm}(\sigma)$ is the total number of finite spin-clusters in $\sigma$ (both of plus and minus spins) that intersect $D$
and the exponent for $x$ is the number of pairs of adjacent faces with different spins at least one of which is in $D$.
The number of finite loops intervening in ${\sf Loop}_{D,n,x}^{\xi}(\omega)$ appears here as $k_{\pm}(\sigma)$, since each such loop is the external boundary of a finite spin-cluster intersecting $D$; see~\cite{DumGlaPel21} for details on this correspondence.
As in the definition of ${\sf Loop}_{D,n,x}^{\xi}$, $k_{\pm}(\sigma)$ may be modified to include the infinite spin-clusters that intersect $D$ with no effect on the definition of ${\sf Spin}_{D,n,x}^{\tau}$.
We note that, written in this way, the loop $O(n)$ model may be viewed as a mixture of the Ising model and FK-percolation.
In particular, the {\em spatial Markov} and {\em finite energy} properties hold.
Notice that the boundary conditions induced by $\tau$ depend only on the spins of $\tau$ on the faces neighbouring $D$
and on connectivities induced on the boundary faces when $\tau$ outside of $D$ is viewed as a face percolation configuration.
\paragraph{Monotonic properties.}
As for FK-percolation, define the partial order on $\sigma \in \{-1,1\}^{F({\rm Hex})}$ pointwise and say that an event $A \subset \{-1,1\}^{F({\rm Hex})}$ is increasing if $\mathbbm{1}_{A}$ is an increasing function with respect to this order.
It was shown in~\cite{DumGlaPel21} that when $n\geq 1$ and $0<x\leq \tfrac{1}{\sqrt{n}}$, the measure ${\sf Spin}_{D,n,x}^{\tau}$ is positively associated and the measures are stochastically ordered with respect to boundary conditions. That is, for any increasing events $A,B$ and any boundary conditions $\tau \geq \tau'$,
\begin{align}\label{eq:On_mono}
{\sf Spin}_{D,n,x}^{\tau}[A\cap B] \geq {\sf Spin}_{D,n,x}^{\tau}[A] \,{\sf Spin}_{D,n,x}^{\tau}[B] \quad \text{and} \quad
{\sf Spin}_{D,n,x}^{\tau}[A] \geq {\sf Spin}_{D,n,x}^{\tau'}[A].
\end{align}
\paragraph{Infinite-volume measures.}
The notion of Gibbs measures for both the loop model and for the spin representation may be defined using the DLR property.
For the spin model, a choice needs to be made in the way infinite spin-clusters are counted.
We make an arbitrary choice, similarly to that for FK-percolation: all infinite plus spin-clusters are considered wired at infinity, while the infinite minus spin-clusters are counted separately. This choice will ultimately turn out to be irrelevant.
As in FK-percolation, the monotonicity properties imply the existence of the infinite-volume limits with plus and minus boundary conditions
and that these are extremal Gibbs measures. We denote them by ${\sf Spin}_{n,x}^{+}$ and ${\sf Spin}_{n,x}^{-}$, respectively.
If these two measures coincide, then the Gibbs measure for the spin representation of the loop $O(n)$ model (and hence for the model itself) is unique.
\paragraph{Dichotomy.}
Using the monotonicity properties~\eqref{eq:On_mono}, a dichotomy result for the loop~$O(n)$ model similar to that of FK-percolation~\cite{DumSidTas17} was proved in ~\cite{DumGlaPel21}:
for any $n\geq 1$ and $x\leq \tfrac{1}{\sqrt{n}}$, the model either satisfies RSW estimates and exhibits {\em macroscopic loops} or
it exhibits {\em exponential decay} of loop lengths.
The dichotomy result is expected to extend to all $n \geq 0$ and $x > 0$.
Moreover, the macroscopic loops regime is conjectured~\cite{Nie82,BloNie89} to hold for $n\in [0,2]$ and $x\geq x_{c}(n):= \tfrac{1}{\sqrt{2+\sqrt{2-n}}}$, and the model is then expected to exhibit a conformally invariant scaling limit~\cite{KagNie04,Smi06}.
For all other parameters, the model is expected to be in the exponential decay regime.
At the moment, the phase diagram is understood only in several regions of parameters.
For recent results on the two types of behaviour, we direct the reader to the recent works~\cite{CraGlaHarPel20,DumGlaPel21, GlaMan18, GlaMan18b}.
\section{FK-percolation: preparation}\label{sec:FK_preparation}
\subsection{Case of translation invariant measures}\label{sec:translation_inv}
We call a measure $\phi$ on $\{0,1\}^{E(\mathbb{L})}$ translation invariant if it is invariant under the translations by two non-collinear vectors.
\begin{prop}\label{prop:translation_inv}
Let $\phi$ be a Gibbs measure for FK-percolation.
Assume that it is translation invariant and tail trivial.
Then
\begin{align*}
\phi[0\leftrightarrow \infty] = 0 \quad \text{or} \quad \phi[(1,0)\xleftrightarrow{*} \infty] = 0.
\end{align*}
As a consequence, $\phi = \phi_{p,q}^0$ or $\phi = \phi_{p,q}^1$.
\end{prop}
\begin{proof}
It is standard (see eg.~\cite[Theorem 14.15]{Geo11}) that translation invariance and extremality imply ergodicity.
For every integer $N\geq 0$, the event that there exist exactly $N$ infinite clusters is translation invariant and hence has probability $0$ or $1$.
Due to the finite energy property, any two infinite clusters may be merged into a single one with positive probability.
It follows that the number of infinite clusters of $\phi$ is either $0$, $1$ or infinity.
The case of infinitely many infinite clusters may be excluded by the classical argument of Burton and Keane~\cite{BurKea89},
or more directly by Lemma~\ref{lem:Kcross} when $\phi_{p,q}^{0} \neq \phi_{p,q}^{1}$ and by the explicit unique infinite-volume measure when $\phi_{p,q}^{0}=\phi_{p,q}^{1}$.
As the same argument applies to the dual model, we conclude that $\phi$ contains at most one infinite primal cluster and one infinite dual cluster.
Finally,~\cite[Thm.~1.5]{DumRaoTas19} excludes the coexistence of unique infinite primal and dual clusters.
\end{proof}
\subsection{No percolation in half-plane with free boundary conditions}
Denote by $\tau_{1/0}$ the percolation configuration on $\mathbb{L}$ in which all edges of the upper half-plane are open and all other edges are closed.
For a finite domain $D$, the Dobrushin boundary conditions are those induced by $\tau_{1/0}$,
and the measure with these boundary conditions is denoted by $\phi_{D,p,q}^{1/0}$.
The measures on the domains $[-N,N]\times [0,N]$ with Dobrushin boundary conditions form a decreasing sequence,
and therefore converge weakly to a measure on configurations on $\mathbb{H}$, which we denote by $\phi_{\mathbb{H}, p,q}^{1/0}$.
The goal of this section is the following statement.
\begin{prop}\label{prop:no_hp_perco}
Let $q\geq 1$. Then, $\phi_{\mathbb{H}, p_{c}(q), q}^{1/0}[0\leftrightarrow \infty] =0$.
\end{prop}
\begin{proof}
Below we omit $q$ and $p_{c}(q)$ from the notation for brevity.
By construction $\phi_{\mathbb{H}}^{1/0}$ is invariant under horizontal translations and is tail trivial.
Let $\phi$ be the increasing limit of $\phi_{\mathbb{H}}^{1/0}\circ \sigma_{N}$, where $\sigma_{N}$ is the vertical translation by $N$
($\phi_{\mathbb{H}}^{1/0}\circ \sigma_{N}$ is the measure on the half-plane $\mathbb{R} \times [-N,\infty)$, with free boundary conditions on the boundary of said half-plane).
Then $\phi$ is an infinite-volume Gibbs measure which is also invariant under vertical translations, and therefore invariant under all translations.
Now, due to Proposition~\ref{prop:translation_inv}, $\phi$ contains either no primal infinite cluster or no dual infinite cluster.
Let $\rho$ be the automorphism of $\{0,1\}^{E(\mathbb{L})}$, where $\rho(\omega)$ is the configuration obtained
by shifting $\omega^*$ by $(-1,0)$ and reflecting it with respect to the horizontal axis.
Then $\phi\circ \rho$ is also a Gibbs measure and we claim that
\begin{equation}\label{eq:no-hp-perc-reflection}
\phi \leq_{\rm st}\phi\circ \rho.
\end{equation}
Indeed, by the finite energy property, the weak limit
\[
\phi_{\mathbb{R}\times [-M, N]}^{1/0} := \lim_{K\to \infty} \phi_{[-K,K]\times [-M, N]}^{1/0}
\]
exists. Moreover, the measures $\phi_{\mathbb{R}\times [-M, N]}^{1/0}$ are decreasing in $N$ and increasing in $M$.
Finally, due to the monotonicity properties of $\phi_{[-K,K]\times [0, N]}^{1/0}$,
\[
\phi =\lim_{M\to\infty}\, \lim_{N\to\infty}\, \phi_{\mathbb{R}\times [-M, N]}^{1/0} \quad \text{and} \quad
\phi\circ \rho = \lim_{N\to\infty}\, \lim_{M\to\infty}\, \phi_{\mathbb{R}\times [-M, N]}^{1/0},
\]
and~\eqref{eq:no-hp-perc-reflection} follows readily.
Now, by the choice of $\rho$, we have $\phi [0\leftrightarrow \infty] = (\phi\circ\rho) [(1,0) \xleftrightarrow{*} \infty]$.
Furthermore, by~\eqref{eq:no-hp-perc-reflection}, $\phi [0\leftrightarrow \infty] \leq (\phi\circ\rho) [0\leftrightarrow \infty]$.
Finally, Proposition~\ref{prop:translation_inv} applied to $\phi\circ \rho$
proves that at least one of the events $\{(1,0) \xleftrightarrow{*} \infty\}$ and $\{0\leftrightarrow\infty\}$ has probability $0$.
Thus,
\begin{align*}
\phi_{\mathbb{H}}^{1/0}[0\leftrightarrow \infty] \leq \phi [0\leftrightarrow \infty] =0.
\end{align*}
\end{proof}
\begin{cor}\label{cor:infinite_touch}
For $q \geq 1$ and $p = p_c(q)$, let $\phi$ be a tail trivial Gibbs measure for FK-percolation.
Let $\omega$ be a random percolation configuration sampled from $\phi$.
Then there exists at most one infinite primal cluster in the restriction of $\omega$ to $\mathbb{H}$.
If the infinite cluster exists, then it a.s. contains infinitely many points of $\mathbb{R} \times \{0\}$.
The same applies to dual clusters and to all other half-planes.
\end{cor}
\begin{proof}
Write $\mathcal{E}$ for the event that in $\mathbb{H}$, the origin is connected to infinity but not to $(\mathbb{R}\setminus \{0\}) \times \{0\}$.
We first prove that $\phi (\mathcal{E})=0$.
Define the random domain $\mathcal{D} \subset \mathbb{H}$ by removing from $\mathbb{H}$
all primal clusters that intersect $(\mathbb{R}\setminus\{0\}) \times \{0\}$.
In addition, for $n>0$ and a realisation $D$ of $\mathcal{D}$, define $D_{n}:= D\cap \Lambda_{n}$.
Denote by $\xi_{n}$ the boundary condition on $D_{n}$ in which $\partial D_n \cap \partial \Lambda_n$ is wired and the rest of $\partial D_n$ is free.
Due to the construction of $\mathcal{D}$, when $\mathcal{D} = D$, the boundary conditions induced on $D_n$ by the configuration outside of $D_n$ dominate $\xi_n$.
Notice that if $\mathcal{E}$ occurs, $0$ is contained in $D_n$ and is connected to $\partial \Lambda_n$ inside $D_n$.
Thus
\[
\phi [\mathcal{E} \, | \, \mathcal{D} = D]
\leq \phi_{D_{n}, p_{c}(q),q}^{\xi_{n}} [0 \leftrightarrow \partial \Lambda_{n}]
\leq \phi_{\mathbb{H}\cap \Lambda_{n}, p_{c}(q),q}^{1/0} [0 \leftrightarrow \partial \Lambda_{n}].
\]
Letting $n$ tend to infinity, averaging over $D$ and applying Proposition~\ref{prop:no_hp_perco}, we obtain
\[
\phi[\mathcal{E}] \leq \phi_{\mathbb{H}, p_{c}(q),q}^{1/0}[0 \leftrightarrow \infty] = 0,
\]
as desired.
By the finite energy property, we deduce more generally that a.s. there exists no infinite cluster in $\mathbb{H}$ which intersects $\mathbb{R} \times \{0\}$ finitely many times.
In particular, there exists a.s. at most one dual infinite cluster in $\mathbb{H}$ intersecting $\mathbb{R} \times \{0\}$,
since any two such clusters would be separated by a primal infinite cluster which intersects $\mathbb{R} \times \{0\}$ finitely many times.
Moreover, the finite energy property states that if there exist with positive probability two infinite dual clusters in $\mathbb{H}$,
they also have positive probability to both intersect $\mathbb{R} \times \{0\}$. As a consequence, we conclude that $\mathbb{H}$ contains a.s. at most one infinite dual cluster.
The same argument may be applied to the primal model, and we conclude that $\mathbb{H}$ contains at most one infinite primal cluster and that,
if such a cluster exists, it necessarily intersects $\mathbb{R} \times \{0\}$ infinitely many times.
\end{proof}
\subsection{Half-plane percolation in symmetric domains}
\newcommand{{\rm Slit}}{{\rm Slit}}
For even integers $M,N\geq 0$, denote by~${\rm Slit}(M,N)$ the graph obtained by gluing $\mathbb{H}$ and $\mathbb{H}^-$ along the segment $[-M,N] \times \{0\}$.
More precisely, ${\rm Slit}(M,N)$ is obtained from $\mathbb{L}$ by splitting every vertex in $ \big((-\infty,-M) \cup (N,\infty)\big)\times \{0\}$ into two vertices: one is incident only to the edges in $\mathbb{H}$ and the other is incident to the edges in $\mathbb{H}^{-}$.
In particular, the edges of ${\rm Slit}(M,N)$ are in bijection with those of $\mathbb{L}$.
The boundary of ${\rm Slit}(M,N)$ is formed of $\partial \mathbb{H}\setminus ([-M,N] \times \{0\})$ and $\partial \mathbb{H}^-\setminus ([-M,N] \times \{0\})$. For simplicity, we refer to these as $\partial {\rm Slit}(M,N)^+$ and $\partial {\rm Slit}(M,N)^-$, respectively.
When no ambiguity is possible, we omit~$M$ and~$N$ from the notation.
Write $\phi_{{\rm Slit},p,q}^{1/0}$ for the measure on ${\rm Slit}$
with wired boundary conditions on~$\partial {\rm Slit}^+$ and free ones on $\partial {\rm Slit}^-$.
Such a measure may be constructed as a limit of measures in $\Lambda_{R} \cap {\rm Slit}$,
with the boundary conditions on $\partial \Lambda_R$ becoming irrelevant as $R \to \infty$.
See Lemma~\ref{lem:perc-hp-inf-domain} below for a proof of this fact and Figure~\ref{fig:slit} for an illustration.
There is a slight abuse of notation here, since ${\rm Slit}$ and $\Lambda_R \cap {\rm Slit}$ are not formally subgraphs of $\mathbb{L}$.
We will allow it as the structure of these graphs is sufficiently clear and the theory of FK-percolation on general graphs is standard.
We emphasise that $\phi_{{\rm Slit},p,q}^{1/0}$ is not symmetric with respect to primal/dual edges:
the wired conditions are one step closer on both endpoints of $[-M,N]\times \{0\}$ than the free ones.
The main results of this section are the proposition below and its corollary.
\begin{prop}\label{prop:hp_perco}
Let $p\in (0,1)$ and $q\geq 1$. If $\phi_{p,q}^{0} \neq \phi_{p,q}^{1}$,
then there exists a constant $c = c(p,q) > 0$ such that for any $M,N\geq 0$
\begin{align*}
\phi_{{\rm Slit},p,q}^{1/0}[0 \xleftrightarrow{\mathbb{H}} \partial {\rm Slit}^+ ] > c.
\end{align*}
\end{prop}
\begin{rem}
That $\phi_{{\rm Slit},p,q}^{1/0}[0 \xleftrightarrow{} \partial {\rm Slit}^+]$ is uniformly positive
is a simple consequence of duality and of the hypothesis of the proposition;
this appeared already in~\cite[Lem~2.3]{GeoHig00}.
The innovation of the above is that the percolation occurs in the half-plane.
\end{rem}
Before giving the proof of Propositon~\ref{prop:hp_perco}, let us state the following consequence.
\begin{cor}\label{cor:hp_perco}
Let $\phi$ be a Gibbs measure for FK-percolation with $p\in (0,1)$ and $q\geq 1$ that satisfy $\phi_{p,q}^{0} \neq \phi_{p,q}^{1}$.
Then,
\begin{align*}
\phi[0\xleftrightarrow{\mathbb{H}} \infty] > 0\quad \text{ or }\quad \phi[(1,0)\xleftrightarrow{\omega^* \cap \mathbb{H}} \infty] > 0.
\end{align*}
\end{cor}
\begin{figure}
\begin{center}
\includegraphics[width = 0.8\textwidth]{cgood4.pdf}
\caption{The graph ${\rm Slit}$ with $\partial {\rm Slit}^+$ marked by solid black lines (wired) and $\partial {\rm Slit}^-$ marked by dashed lines (free).
For a configuration $\omega$ (in red), modifying the blue edges as indicated produces $\tilde \omega$.
Notice that in $\tilde \omega$ the dual cluster of $(N-1,0)$ has diameter at least $N$ and does not intersect $\partial {\rm Slit}^-$.}
\label{fig:slit}
\end{center}
\end{figure}
Fix for the rest of this section $p\in (0,1), q\geq 1$ such that $\phi_{p,q}^{0} \neq \phi_{p,q}^{1}$ and omit them from the notation.
\begin{proof}[Corollary~\ref{cor:hp_perco}]
Assume that $\phi[(1,0)\xleftrightarrow{\omega^* \cap\mathbb{H}} \infty] =0$.
By the finite energy property, there exists a.s. no infinite dual cluster in $\mathbb{H}$.
Fix some $n>0$ and consider $K>0$ such that,
\[
\phi [\Lambda_{n}\xleftrightarrow{\omega^* \cap\mathbb{H}} \Lambda_{K}^{c}] <1/2.
\]
This means that, with probability at least $1/2$, there exists an open path in $\Lambda_K$ that separates $\Lambda_{n}$ from infinity.
Let $\Gamma$ be the outermost such path and write $-M$ and $N$ for the first coordinates of the endpoints of $\Gamma$ on $\partial \mathbb{H}$.
If no such path exists, write $\Gamma = \emptyset$.
By~\eqref{eq:FKG} and~\eqref{eq:CBC}, we have
\begin{align*}
\phi[0 \xleftrightarrow{\mathbb{H}} \partial \Lambda_n \, | \, \Gamma = \gamma]
\geq \phi^{1/0}_{{\rm Slit}(M,N)}[0 \xleftrightarrow{\mathbb{H}} \partial \Lambda_n] \geq c,
\end{align*}
for any realisation of $\gamma \neq \emptyset$ of $\Gamma$,
where $c >0$ is given by Proposition~\ref{prop:hp_perco} and is independent of $n$.
Averaging over $\gamma$ and using that $n$ is arbitrary, we find $\phi[0 \xleftrightarrow{\mathbb{H}} \infty] \geq c /2 > 0$.
\end{proof}
The rest of the section is dedicated to proving Proposition~\ref{prop:hp_perco}.
For $M,N\geq 0$ and $c > 0$, call a point~$z\in [-M,N]\times \{0\}$ $c$-{\em good} if
\[
\phi_{{\rm Slit}(M,N)}^{{1/0}} [ z \xleftrightarrow{\mathbb{H}} \partial {\rm Slit}(M,N)^+] \geq c.
\]
Our goal is then to show that, for some $c>0$ and every $M,N$, every point is $c$-good.
The main difficulty stems from the fact that the measures $\phi_{{\rm Slit}(M,N)}^{{1/0}}$ for different $M,N$ are not stochastically ordered.
The proof is split into several steps: first we use the finite energy and local surgeries to show that there is a positive proportion of good points (Lemma~\ref{lem:perc-hp-some-good});
then we increase this proportion by changing $c$ (Lemma~\ref{lem:perc-hp-between-good}).
Thus the proportion of $c$-good points may be rendered arbitrarily close to $1$ by choosing $c$ small enough (Lemma~\ref{lem:perc-hp-many-good}).
Finally we use a counting argument to derive from the above that, for some $c>0$, every point is good.
We start with showing that $\phi_{{\rm Slit}}^{{1/0}}$ is well-defined.
\begin{lem}[Dobrushin boundary conditions on ${\rm Slit}$]\label{lem:perc-hp-inf-domain}
Fix $M,N\geq 0$ and write ${\rm Slit} = {\rm Slit}(M,N)$.
Let $(\xi_{k})_{k \geq 1}$ be a sequence of boundary conditions on~$\Lambda_k \cap {\rm Slit}$
which are wired on~$\partial\mathbb{H}$ and free on~$\partial\mathbb{H}^-$.
Then the weak limit of $(\phi_{\Lambda_k \cap {\rm Slit}}^{\xi_k})_{k\geq 1}$ exists and does not depend on the choice of~$(\xi_{k})_{k \geq 1}$.
\end{lem}
\begin{proof}
For $k \geq \max(M,N)$, define the following four boundary conditions on $\Lambda_k \cap {\rm Slit}$
denoted by $\zeta_k^{0,0}$, $\zeta_k^{1,0}$, $\zeta_k^{0,1}$ and $\zeta_k^{1,1}$.
All of them are wired on $\partial\mathbb{H}$ and free on $\partial\mathbb{H}^{-}$. In addition
$\zeta_k^{0,0}$ and $\zeta_k^{0,1}$ are free on $\partial \Lambda_k \cap \mathbb{H}$, while $\zeta_k^{1,0}$ and $\zeta_k^{1,1}$ are wired on this part of the boundary;
$\zeta_k^{0,0}$ and $\zeta_k^{1,0}$ are free on $\partial \Lambda_k \cap \mathbb{H}^-$ and $\zeta_k^{0,1}$ and $\zeta_k^{1,1}$ are wired on $\partial \Lambda_k \cap \mathbb{H}^-$. Then
\begin{align*}
\phi_{\Lambda_k \cap {\rm Slit}}^{\zeta_k^{0,0}} \leq_{\rm st} \phi_{\Lambda_k \cap {\rm Slit}}^{\xi_k} \leq_{\rm st} \phi_{\Lambda_k \cap {\rm Slit}}^{\zeta_k^{1,1}}.
\end{align*}
Thus, the statement follows by proving that
\begin{align}\label{eq:slit1111}
\lim_{k \to \infty} \phi_{\Lambda_k \cap {\rm Slit}}^{\zeta_k^{0,0}} = \lim_{k \to \infty} \phi_{\Lambda_k \cap {\rm Slit}}^{\zeta_k^{1,1}}.
\end{align}
Notice that the two weak limits exist, with the first being increasing, while the second is decreasing.
Fix $\varepsilon > 0$, $r \geq \max(M,N)$ and an increasing event $A$ depending only on the edges in $\Lambda_r \cap {\rm Slit}$.
Then, due to~\eqref{eq:SMP} and~\eqref{eq:CBC},
\begin{align*}
\phi_{\Lambda_k \cap {\rm Slit}}^{\zeta_k^{1,1}}[A] \leq \phi_{\Lambda_k \cap {\rm Slit}}^{\zeta_k^{0,1}}[A \,|\, \Lambda_r \nxlra{\omega^* \cap \mathbb{H}} \partial \Lambda_k].
\end{align*}
Moreover, due to the finite energy property,
\begin{align*}
\phi_{\Lambda_k \cap {\rm Slit}}^{\zeta_k^{0,1}}[\Lambda_r \xleftrightarrow{\omega^* \cap \mathbb{H}} \partial \Lambda_k]
\leq c_0 \phi_{\Lambda_{k}\cap \mathbb{H}}^{0/1}[(1,0) \xleftrightarrow{\omega^* \cap \mathbb{H}} \partial \Lambda_k]
\end{align*}
for some $c_0 = c_0(p,q,r,M,N)$. By Proposition~\ref{prop:no_hp_perco}, the above may be rendered smaller than $\varepsilon$ by taking $k$ large enough.
Thus, we find
\begin{align*}
\phi_{\Lambda_k \cap {\rm Slit}}^{\zeta_k^{1,1}}[A] \leq \phi_{\Lambda_k \cap {\rm Slit}}^{\zeta_k^{0,1}}[A] + \varepsilon
\end{align*}
for $k$ large enough.
The same reasoning applies in $\mathbb{H}^-$, and we find
\begin{align*}
\phi_{\Lambda_k \cap {\rm Slit}}^{\zeta_k^{0,1}}[A] \leq \phi_{\Lambda_k \cap {\rm Slit}}^{\zeta_k^{0,0}}[A] + \varepsilon
\end{align*}
for $k$ large enough.
Combining the last two displays and keeping in mind that $\varepsilon > 0$ is arbitrary, we conclude that
\begin{align*}
\lim_{k \to \infty} \phi_{\Lambda_k \cap {\rm Slit}}^{\zeta_k^{0,0}}[A] = \lim_{k \to \infty} \phi_{\Lambda_k \cap {\rm Slit}}^{\zeta_k^{1,1}}[A]
\end{align*}
for any increasing event $A$ depending on finitely many edges. Finally,~\eqref{eq:slit1111} may be deduced from the above by the monotone class theorem.
\end{proof}
\begin{lem}[Positive proportion of good points]\label{lem:perc-hp-some-good}
There exist~$\alpha,c_0>0$ such that, for any~$M,N >0$, the number of~$c_0$-good points to the left (resp. right) of the origin in ${\rm Slit}(M,N)$
is greater or equal than~$\alpha M$ (resp. $\alpha N$).
\end{lem}
\begin{proof}
Fix small constants~$\alpha, c_0 >0$; we will see below how small they need to be.
Fix $M>0$.
It suffices to prove the property only for the points to the right of the origin and for $N$ sufficiently large.
We will proceed by contradiction and suppose that for some large $N$ the number of $c_0$-good points to the right of the origin is smaller than $\alpha N$.
Recall the automorphism $\rho$ of $\{0,1\}^{E(\mathbb{L})}$
obtained by shifting the dual configuration by $(-1,0)$ and reflecting it with respect to the horizontal axis $\mathbb{R}\times \{0\}$.
Then, $\phi_{{\rm Slit}}^{{1/0}} \circ \rho$ is dominated by $\phi_{{\rm Slit}}^{{1/0}}$ \footnote{The domination, rather than equality, comes from the asymmetry in the boundary of ${\rm Slit}$.}.
Thus, if a point $(u,0)$ is not~$c_0$-good,
then the probability that $(u+1,0)$ is linked to $\partial {\rm Slit}^-$ in $\omega^* \cap \mathbb{H}^-$ is also smaller than~$c_0$.
As a consequence,
\begin{align*}
\phi_{{\rm Slit}}^{{1/0}} \big[\#\{u\in [0,N] \cap 2\mathbb{Z} \colon (u,0) \xleftrightarrow{\mathbb{H}} \partial {\rm Slit}^+ \text{ or } (u+1,0) \xleftrightarrow{\omega^* \cap\mathbb{H}^-} \partial {\rm Slit}^- \} \big]
\leq \alpha N + c_0N.
\end{align*}
Define the following event:
\[
A:= \big\{\#\{u\in [0,N] \cap 2\mathbb{Z} \colon (u,0) \xleftrightarrow{\mathbb{H}} \partial {\rm Slit}^+ \text{ or } (u+1,0) \xleftrightarrow{\omega^* \cap\mathbb{H}^-} \partial {\rm Slit}^- \} \leq 2(\alpha + c_0)N\big\}.
\]
Applying the Markov inequality, we find that
\begin{align}\label{eq:AAaaA}
\phi_{{\rm Slit}}^{{1/0}} [A] \geq \tfrac12.
\end{align}
Since $\phi_{{\rm Slit}}^{{1/0}}$-a.s. $\mathbb{H}$ contains no dual infinite cluster (see Proposition~\ref{prop:no_hp_perco} or the proof of Lemma~\ref{lem:perc-hp-inf-domain}), that $(u,0)$ is connected to infinity in $\mathbb{H}$ implies a.s. that it is also connected to $\partial {\rm Slit}^+$ in $\mathbb{H}$. The same applies in $\mathbb{H}^-$ for dual connections.
For~$\omega \in A$, let $\tilde \omega$ be the configuration
obtained from $\omega$ by
opening the two edges below each dual vertex of $[0,N-1] \times \{0\}$ that is connected to $\partial {\rm Slit}^-$ in $\omega^* \cap \mathbb{H}^-$
and closing the two edges above each primal vertex of $[0,N-1] \times \{0\}$ that is connected to $\partial {\rm Slit}^+$ in $\omega\cap \mathbb{H}$.
See Figure~\ref{fig:slit} for an illustration.
For any configuration $\tilde \omega$ obtained as above, the number of vertices appearing in~\eqref{eq:AAaaA} is zero.
Thus, the two following events occur for $\tilde\omega$
\begin{align*}
(-\infty,0) \times \{0\} \xleftrightarrow{\tilde\omega^* \cap \mathbb{H}} (N - 1,0) \quad \text{and} \quad (-\infty,0]\times \{0\} \xleftrightarrow{\tilde\omega \cap \mathbb{H}^{-}} (N,0).
\end{align*}
Write $\tilde A$ for the intersection of the events above.
Next we will lower bound the probability of $\tilde A$ in terms of that of $A$.
Observe that $\omega$ and $\tilde\omega$ differ on at most $4(\alpha + c_0)N$ edges.
Thus, by the finite energy property, there exists a constant $c_1 = c_1 (p,q) > 0$ independent of $c_0,\alpha$ or $N$ such that
\begin{align*}
\phi[\tilde\omega] \geq \exp[-c_1 (\alpha + c_0)N]\,\phi[\omega] \qquad \forall \omega \in A.
\end{align*}
The function $\omega \mapsto \tilde\omega$ is not one-to-one, but the number of pre-images of any given $\tilde\omega$ is at most
\begin{align*}
S_N(\alpha + c_0) = \sum_{k = 0}^{4(\alpha + c_0)N} \binom{N}{k}.
\end{align*}
Combining the two equations above, we find
\begin{align}\label{eq:Alb}
\phi[\tilde A] \geq \frac{e^{-c_1(\alpha + c_0)N}}{S_N(\alpha + c_0)} \phi[A]
\geq \frac{e^{-c_1(\alpha + c_0)N}}{2S_N(\alpha + c_0)}.
\end{align}
Split now $\tilde A$ into the event $\tilde A^0$, for which $(N-1,0)$ is connected to $\partial {\rm Slit}^-$ in $\tilde \omega^*$,
and its complement $\tilde A^1$.
When $\tilde A^1$ occurs, the dual cluster of $(N-1,0)$ has diameter at least $N$ and is surrounded by open edges.
A standard argument involving conditioning on the dual cluster of $\partial {\rm Slit}^-$ and using~\eqref{eq:SMP} and~\eqref{eq:CBC} yileds
\begin{align}\label{eq:Aub1}
\phi_{{\rm Slit}}^{{1/0}} [\tilde A^{1}]
\leq \phi^{1} [(0,1) \xleftrightarrow{*} \partial \Lambda_N ]
\leq e^{-c N},
\end{align}
with $c >0$ given by~\eqref{eq:exp_decay}.
Assume now that $\tilde\omega \in \tilde A^0$ and close the two edges below $(N,0)$.
In the new configuration, both points $(N-1,-1)$ and $(N+1,-1)$ are disconnected from $\partial {\rm Slit}^+$
and the cluster of one of them has diameter at least~$N$.
Using the finite energy property and the same argument as in \eqref{eq:Aub1}, we conclude that
\begin{align}\label{eq:Aub0}
\phi_{{\rm Slit}}^{{1/0}} [\tilde A^{0}]
\leq c_2 \,\phi^{0} [0 \leftrightarrow{} \partial \Lambda_N ]
\leq c_2 \, e^{-c N},
\end{align}
with $c >0$ given by~\eqref{eq:exp_decay} and $c_2 = c_2(p,q)$ a constant independent of $N$.
Combining~\eqref{eq:Aub1} and~\eqref{eq:Aub0} using the union bound and introducing the result in~\eqref{eq:Alb},
we conclude that
\begin{align*}
S_N(\alpha + c_0) \geq \tfrac{1}{2(1+c_2)}\exp[ cN - c_1(\alpha + c_0)N].
\end{align*}
This is contradictory for $\alpha + c_0$ small enough and $N$ large enough.
\end{proof}
\begin{lem}[Increase proportion of good points]\label{lem:perc-hp-between-good}
There exist~$\alpha, \beta >0$ such that, for any~$c>0$ and any even $-M \leq u \leq v \leq N$ such that $(u,0)$ and $(v,0)$ are both $c$-good,
there exist at least~$\alpha (v-u \vee 0)$ points $(w,0)$ with $u\vee 0 \leq w \leq v$ that are~$\beta c^2$-good.
\end{lem}
\begin{proof}
Fix $c >0$ and $M$, $N$, $u$ and $v$ as above.
By the FKG inequality
\[
\phi_{{\rm Slit}}^{{1/0}} \big[(u,0) \xleftrightarrow{\mathbb{H}} \partial {\rm Slit}^+ \text{ and }(v,0) \xleftrightarrow{\mathbb{H}} \partial {\rm Slit}^+\big]\geq c^2.
\]
When the event above occurs, the measure in the domain bounded by the left-most crossing from~$u$ to $\partial {\rm Slit}^+$ in $\mathbb{H}$
and the right-most crossing from~$v$ to $\partial {\rm Slit}^+$ in $\mathbb{H}$
dominates~$\phi_{\mathrm{Slit}(u,v)}^{{1/0}}$.
Therefore, for each $u\vee 0 \leq w \leq v$ even,
\begin{align}
&\phi_{{\rm Slit}}^{{1/0}} \big[(w,0) \xleftrightarrow{\mathbb{H}} \partial {\rm Slit}^+\big]\nonumber\\
&\qquad \geq\phi_{{\rm Slit}(-u,v)}^{{1/0}} \big[(w,0) \xleftrightarrow{\mathbb{H}} \partial {\rm Slit}(-u,v)^+\big]
\,\phi_{{\rm Slit}}^{{1/0}} \big[(u,0) \xleftrightarrow{\mathbb{H}} \partial {\rm Slit}^+ \text{ and }(v,0) \xleftrightarrow{\mathbb{H}} \partial {\rm Slit}^+\big]\nonumber\\
&\qquad \geq c^2 \phi_{{\rm Slit}(-u,v)}^{{1/0}} \big[(w,0) \xleftrightarrow{\mathbb{H}} \partial {\rm Slit}(-u,v)^+\big].\label{eq:SlitSlit}
\end{align}
Applying Lemma~\ref{lem:perc-hp-some-good} to ${\rm Slit}(-u,v)$ yields the result.
Formally, when $u \geq 0$, we apply Lemma~\ref{lem:perc-hp-some-good} to ${\rm Slit}(0,v-u)$.
\end{proof}
\begin{lem}[Large proportion of good points]\label{lem:perc-hp-many-good}
For any~$\varepsilon >0$, there exists~$c(\varepsilon) >0$ such that, for any~$M,N \geq 0$ even,
the number of~$c$-good points to the right (resp. left) of the origin in ${\rm Slit}(M,N)$
is greater or equal than~$(1-\varepsilon) N/2$ (resp. $(1-\varepsilon) M/2$).
\end{lem}
Notice that in total there are $N/2$ primal points to the right of $0$.
Thus, the above states that the proportion of $c$-good points may be rendered as close to $1$ as desired by choosing $c$ small enough.
\begin{proof}
Take~$\alpha,\beta,c_0>0$ such that both Lemmas~\ref{lem:perc-hp-some-good} and~\ref{lem:perc-hp-between-good} hold.
Fix~$M,N$. We estimate only the number of good points to the right of the origin. The number of good points to the left of the origin can be then estimated in the same way.
By Lemma~\ref{lem:perc-hp-some-good} there are at least~$\alpha N$ points right of $0$ that are~$c_0$-good.
By Lemma~\ref{lem:perc-hp-between-good}, between any two $c_0$-good points $(u,0)$ and $(v,0)$ right of $0$,
there exist at least~$\alpha (|u-v|-1)$ points that are~$\beta c_0^2$-good.
The same holds between $0$ and the left-most $c_0$-good point right of $0$.
Applying this for all pairs of consecutive~$c_0$-good points,
we deduce the existence of at most $(1-\alpha)^2 N$ points that are not~$\beta c_0^2$-good.
Iterating the procedure for $c_{k} = \beta c_{k-1}^2$, we find that there exist at most~$(1-\alpha)^kN$ points that are not~$c_k$-good.
The lemma follows by taking~$k$ large enough that~$(1-\alpha)^k<\varepsilon$.
\end{proof}
\begin{proof}[Proposition~\ref{prop:hp_perco}]
Fix~$\varepsilon = 1/16$. Without loss of generality, assume that $M\geq N$.
Take $c= c(\varepsilon)>0$ such that the conclusion of Lemma~\ref{lem:perc-hp-many-good} holds.
Denote by~$G_{-}$ and~$G_{+}$ the sets of first coordinates of $c$-good points in $[-M, -M+N/2]\times \{0\}$
and in~$[N/2,N]\times \{0\}$, respectively, for ${\rm Slit}(M,N)$.
Then,
\[
|G_{-}|,|G_{+}| \geq \tfrac{1-\varepsilon}4 N, \text{ whence }|G_{-}\times G_{+}| \geq \tfrac1{16} N^2(1- \varepsilon)^2
\geq \tfrac1{32}N^2.
\]
For any~$(u,v) \in G_{-}\times G_{+}$, we have~$v-u \in [M,M+N] \cap 2\mathbb{Z}$.
Thus, there exists at least one even integer~$L\in [M,M+N]$ such that
there are at least $N/16$ pairs~$(u,v) \in G_{-}\times G_{+}$ which satisfy~$v - u = L$.
Fix one such~$L$.
Then the above implies existence of at least $N/16$ even integers $v\in [N/2,N]$ such that $(v-L,v) \in G_{-}\times G_{+}$.
Consider now the set $G_L$ of first coordinates of $c$-good points to the right of $0$ in ${\rm Slit}(L - N,N)$.
By Lemma~\ref{lem:perc-hp-many-good}, $|G_L| \geq (1 - \varepsilon)N$.
Thus, there exists at least one $v \in [N/2,N]$ with $N-v \in G_L$ and $(v - L,v) \in G_{-}\times G_{+}$.
Indeed, the sum of the cardinalities of these two sets is strictly larger than $N/2$.
Fix $v$ as above and write $u = v-L$ and ${\rm Slit} = {\rm Slit}(M,N)$. Then
\begin{align*
\phi_{{\rm Slit}}^{{1/0}} \big[0 \xleftrightarrow{\mathbb{H}} \partial{\rm Slit}^+\big]
&\geq c^2 \,\phi_{{\rm Slit}}^{{1/0}}\big[0 \xleftrightarrow{\mathbb{H}} \partial{\rm Slit}^+ \big|
\,(u,0) \xleftrightarrow{\mathbb{H}} \partial{\rm Slit}^+,\, (v,0) \xleftrightarrow{\mathbb{H}} \partial{\rm Slit}^+\big] \\
&\geq c^2 \,\phi_{{\rm Slit}(-u,v)}^{{1/0}} \big[0 \xleftrightarrow{\mathbb{H}} \partial{\rm Slit}(-u,v)^+\big] \\
&= c^2 \,\phi_{{\rm Slit}(L - N,N)}^{{1/0}} \big[(N-v,0) \xleftrightarrow{\mathbb{H}} \partial{\rm Slit}(L - N,N)^+\big]
\geq c^3.
\end{align*}
The first inequality uses~\eqref{eq:FKG} and the fact that $(u,0)$ and $(v,0)$ are $c$-good for ${\rm Slit}(M,N)$;
the second one uses the same exploration argument as the one leading to~\eqref{eq:SlitSlit};
the equality is obtained after shift by $N-v$;
the third inequality uses that $(N-v,0)$ is $c$-good for ${\rm Slit}(L-N,N)$.
This concludes the proof.
\end{proof}
\section{FK-percolation: duplication argument}\label{sec:FK_duplication}
We are finally ready to prove Theorem~\ref{thm:mainFK}. To do so, we will prove that all extremal Gibbs measures are translation invariant.
\begin{prop}\label{prop:Gibbs=trans_inv}
Let $\phi$ be an extremal Gibbs measure for FK-percolation. Then $\phi$ is translation invariant.
\end{prop}
The above, combined with Proposition~\ref{prop:translation_inv}, immediately implies Theorem~\ref{thm:mainFK}.
\begin{proof}[Theorem~\ref{thm:mainFK}]
Let $\phi$ be an extremal Gibbs measure for FK-percolation. By Proposition~\ref{prop:Gibbs=trans_inv}, $\phi$ is translation invariant
and due to Proposition~\ref{prop:translation_inv}, $\phi$ is equal to $\phi^0$ or $\phi^1$.
Since every Gibbs measure is a linear combination of extremal Gibbs measures, this yields the result.
\end{proof}
The rest of the section is dedicated to proving Proposition~\ref{prop:Gibbs=trans_inv}.
The statement is obvious when $\phi_{p,q}^0=\phi_{p,q}^1$, and we assume henceforth that $p$ and $q$ are such that $\phi_{p,q}^0\neq \phi_{p,q}^1$.
Write $\phi'$ for a translate of $\phi$.
To prove Proposition~\ref{prop:Gibbs=trans_inv}, we will show that $\phi = \phi'$.
By symmetry it suffices to show that $\phi \leq_{\rm st} \phi'$, which we do next.
Let $\mathbb{P}$ be the probability measure governing two independent samples $\omega$ and $\omega'$ with laws $\phi$ and $\phi'$, respectively.
The goal will be to create arbitrarily large random domains $\mathcal{D}$ measurable in terms of $\omega$ and $\omega'$ on $\mathcal{D}^c$
such that the boundary conditions induced by $\omega'$ on $\partial \mathcal{D}$ dominate those induced by $\omega$.
The main difference between our argument and that of~\cite{GeoHig00} appears below, in the way that $\mathcal{D}$ is explored.
Indeed, in~\cite{GeoHig00} (following the previous argument of ~\cite{Aiz80}),
it is shown that $\mathbb{P}$-a.s. there exist infinitely many circuits around $0$ on which $\omega \leq \omega'$.
Then one may take $\mathcal{D}$ to be the domain delimited by the outer-most such circuit contained in a large box $\Lambda_N$.
For the Ising model, this suffices to conclude that the marginal for $\omega$ in $\mathcal{D}$ is dominated by that for $\omega'$.
For FK-percolation, boundary conditions $\xi$ and $\xi'$ induced by $\omega$ and $\omega'$, respectively, are not determined solely by the configurations on $\partial \mathcal{D}$,
but also by the connections between points of $\partial \mathcal{D}$ outside of $\mathcal{D}$.
Thus, with the choice of $\mathcal{D}$ as above, we may not conclude that $\xi \leq \xi'$.
To solve this issue, we will explore $\mathcal{D}$ so that its boundary is formed of one, two or four arcs,
each of which is either open in $\omega'$ or closed in $\omega$.
Unfortunately, the exploration procedure for such a domain is considerably more complicated.
\subsection{Shielding arcs}
As proved in Corollary~\ref{cor:hp_perco}, each half-plane contains at least one infinite cluster, primal or dual.
Corollary~\ref{cor:infinite_touch} implies that such a cluster is unique (if exists).
In light of the above, and of the tail triviality of $\phi$, each half-plane (below we will only refer to $\mathbb{H}$ and $\mathbb{H}^-$)
is in one of the settings below:
\begin{itemize}
\item[(a)] contains a unique infinite primal cluster and no infinite dual cluster, $\phi$-a.s.;
\item[(b)] contains a unique infinite dual cluster and no infinite primal cluster, $\phi$-a.s.;
\item[(c-d)] contains both a unique infinite primal cluster and a unique infinite dual cluster, $\phi$-a.s..
\end{itemize}
In the third case, recall from Corollary~\ref{cor:infinite_touch} that each infinite primal cluster as well as each infinite dual cluster
intersects the horizontal axis infinitely many times. Thus, this case may further be split into two sub-cases:
(c) the primal cluster intersects $(-\infty,0]\times \{0\}$ infinitely many times, while the infinite dual cluster touches $[0,\infty)\times \{0\}$ infinitely many times, or (d) vice-versa.
We then say that we have coexistence in $\mathbb{H}$ with the primal cluster to the left and the dual one to the right, or vice-versa.
Our goal now is to create arcs $\mathcal{S}$ that separate $0$ from $\infty$ in $\mathbb{H}$,
which are explorable from above, and for which the boundary conditions induced by $\omega'$ dominate those induced by $\omega$.
These paths will be called {\em shielding arcs}; a definition is given below and depends on the setting of $\phi$ in $\mathbb{H}$.
Consider a pair of configurations $(\omega, \omega')$.
For $n \geq0$ and $x <0$ and $y > 0$ even,
we say that a path $\varsigma$ in $\mathbb{H}$ is a shielding arc above $\Lambda_n$ between $x$ and $y$
if it does not intersect $\Lambda_n$ and one of the following holds:
\begin{itemize}
\item[(a)] $\varsigma$ runs from $(x,0)$ to $(y,0)$ and is formed entirely of primal edges contained in~$\omega'$;
\item[(b)] $\varsigma$ runs from $(x+1,0)$ to $(y+1,0)$ and is formed entirely of dual edges contained in~$\omega^*$;
\item[(c)] $\varsigma$ runs from $(x,0)$ to $(y+1,0)$ and is
formed of an arc $\varsigma^{(1)}$ of primal edges all of which are contained in $\omega'$,
followed by a single edge of $\mathbb{Z}^2$, followed by an arc $\varsigma^{(2)}$ of dual edges, all contained in $\omega^*$.
Moreover, $\varsigma^{(1)}$ is connected to infinity in $\omega' \cap \mathbb{H}$ by a path that does not intersect~$\varsigma^{(2)}$;
\item[(d)] $\varsigma$ runs from $(x+1,0)$ to $(y,0)$ and is
formed of a sequence of edges of $\omega^*$, followed by a single edge of $\mathbb{Z}^2$, followed by edges of $\omega'$.
Moreover, the primal edges of $\varsigma$ are connected to $\infty$ in $\omega' \cap \mathbb{H}$ by a path that does not intersect the dual edges of~$\varsigma$.
\end{itemize}
In all cases, write ${\rm Above}(\varsigma)$
for the subgraph spanned by the edges of $\mathbb{H}$ separated from $[x,y] \times\{0\}$ by $\varsigma$ as well as those on $\varsigma$.
\begin{prop}\label{prop:shield}
There exists $c > 0$ such that the following holds for all $n\geq 1$.
For any $x \leq 0$ and $y \geq 0$ even, both sufficiently large,
there exists a random variable $\mathcal{S} = \mathcal{S}(\omega,\omega')$
that either takes value $\emptyset$ or is a shielding arc above $\Lambda_n$ between $x$ and $y$
and
\begin{itemize}
\item[(i)] if $\mathcal{S} \neq \emptyset$, then
$\mathcal{S}$ is measurable in terms of the edges in ${\rm Above}(\mathcal{S})$.
In addition, $\mathcal{S} = \emptyset$ is measurable in terms of the edges of $\mathbb{H} \setminus \Lambda_n$;
\item[(ii)] conditionally on $\omega'$, the event $\mathcal{S} \neq \emptyset$ is decreasing in $\omega$;
\item[(iii)]
if we call $\omega'$ {\em promising} when $\mathbb{P}[\mathcal{S} \neq \emptyset \,|\,\omega'] \geq c$,
then the event $\{\omega' \text{ is promising}\}$ is increasing in $\omega'$ and
\[
\mathbb{P}[\omega' \text{ is promising}] \geq c.
\]
\end{itemize}
\end{prop}
The last point implies that $\mathbb{P}[\mathcal{S}\neq\emptyset] \geq c^2$.
In addition to this bound, we would like to say that $\{\mathcal{S}\neq\emptyset\}$ is increasing in $\omega'$ and decreasing in $\omega$, but this is unfortunately not the case,
hence the more complicated expressions in (ii) and (iii).
We should mention that the ingredients already present in~\cite{GeoHig00} suffice to prove the existence of arbitrarily large shielding arcs.
The crucial difference between the above and what is available with the arguments of~\cite{GeoHig00} is that shielding arcs exist with uniformly positive probability {\em between any $x$ and $y$}. This property, which relies on Proposition~\ref{prop:hp_perco}, will allow us to glue shielding arcs in $\mathbb{H}$ and $\mathbb{H}^-$ with no interference (see Corollary~\ref{cor:shield}).
The rest of the section is dedicated to proving Proposition~\ref{prop:shield}.
When there is no coexistence in $\mathbb{H}$ the result is relatively easy; we will mostly focus on the situation when there is coexistence in $\mathbb{H}$. We start with two preparatory results.
When there is coexistence in $\mathbb{H}$,
the primal and the dual infinite clusters are separated by a unique semi-infinite interface starting at $\partial \mathbb{H}$.
Denote it by $\Gamma$ and parametrise starting from $\partial\mathbb{H}$.
Formally $\Gamma$ is an edge-simple path on the medial lattice $\mathbb{Z}^2 + (1/2,1/2)$ of $\mathbb{L}$ (see Fig.~\ref{fig:arcs}),
but the formal details in the definition of $\Gamma$ are of little importance and we omit them here.
Note that $\Gamma$ intersects $\partial \mathbb{H}$ only at its starting point $\Gamma_0$,
which is a point of the form $(n +1/2,0)$, with $n$ even when the infinite primal cluster is on the left and odd when it is on the right.
\begin{lem}\label{lem:hp_perco_pt}
Suppose that $\phi$ is such that there is coexistence in $\mathbb{H}$, with the primal cluster to the left.
For $c>0$ from Proposition~\ref{prop:hp_perco} and any even integer~$x$ and odd integer~$y$
\begin{align}\label{eq:hp_perco_pt}
\phi\big[(x,0)\xleftrightarrow{\mathbb{H}} \infty \,\big|\, \Gamma\big] \geq c \qquad \text{ and }\qquad
\phi\big[(y,0) \xleftrightarrow{\omega^* \cap \mathbb{H}} \infty\,\big|\, \Gamma\big] \geq c,
\end{align}
whenever $\Gamma_0$ is to the right of $(x,0)$ and to the left of $(y,0)$, respectively.
\end{lem}
\begin{figure}
\begin{center}
\includegraphics[width = 0.6\textwidth]{arcs.pdf}
\caption{The exploration path $\Gamma$ staring on the horizontal axis separates the infinite primal and dual clusters of $\mathbb{H}$.
For fixed $\Lambda_N$, the outermost open path connecting $(-\infty,0] \times \{0\}$ to the left side of $\Gamma$ is denoted by $\Sigma$ and may be explored from the outside; the region ${\rm Below}(\Sigma)$ is marked in grey. Conditionally on $\Gamma$ and $\Sigma$, $(x,0)$ is connected to the wired arc above it with uniform positive probability. }
\label{fig:arcs}
\end{center}
\end{figure}
\begin{proof}
We will focus on the first inequality and fix an even integer $x$.
Fix $\varepsilon > 0$.
Let $N = N (\Gamma) \geq 1$ be such that
\begin{align}\label{eq:yxz}
\phi\big[(-\infty, x] \times \{0\} \xleftrightarrow{\mathbb{H} \cap \Lambda_N} \Gamma \,\big| \, \Gamma\big] \geq 1 - \varepsilon,
\end{align}
whenever $\Gamma_0$ is to the right of $x$; otherwise set $N(\Gamma) = 0$.
Write $\Sigma$ for the exterior-most open primal path in $\mathbb{H} \cap \Lambda_N$ that connects $(-\infty, x] \times \{0\}$ to $\Gamma$ if such a path exists;
otherwise set $\Sigma = \emptyset$.
Denote by ${\rm Above}(\Sigma)$ the subgraph spanned by the edges of $\mathbb{H}$ separated from $(x,0)$ by $\Sigma \cup \Gamma$ as well as those on $\Sigma$ and those explored by $\Gamma$.
Let ${\rm Below}(\Sigma)$ be the subgraph spanned by all the other edges. See Figure~\ref{fig:arcs} for an illustration.
Then, both $\Gamma$ and $\Sigma$ are measurable in terms of the edges in ${\rm Above}(\Sigma)$.
It follows that, conditionally on $\Gamma$ and $\Sigma$,
the measure on ${\rm Below}(\Sigma)$ dominates the measure with wired boundary conditions on $\Sigma \cup \Gamma$ and free elsewhere.
Using the above and Proposition~\ref{prop:hp_perco}, we find that
\begin{align*}
\phi\big[(x,0)\xleftrightarrow{\mathbb{H}} \Sigma \cup \Gamma \,\big|\, \Sigma, \Gamma \big] \geq c,
\end{align*}
whenever $\Sigma \neq \emptyset$ and $\Gamma_0$ is to the right of $(x,0)$.
Notice that when the event above occurs, then $(x,0)$ is connected to infinity in $\mathbb{H}$, as it is connected to the primal side of the infinite path $\Gamma$, possibly via $\Sigma$.
Now, summing over all possible realisations of $\Sigma$ different from $\emptyset$ and using~\eqref{eq:yxz} we find
\begin{align*}
\phi\big[(x,0)\xleftrightarrow{\mathbb{H}} \infty \,\big|\, \Gamma \big] \geq c (1- \varepsilon).
\end{align*}
Since $\varepsilon > 0$ is arbitrary, we obtain the desired conclusion.
\end{proof}
\begin{rem}\label{rem:hp_perco_primal}
The same proof shows that if $\mathbb{H}$ contains no dual infinite cluster,
then for any even $x \in \mathbb{Z}$
\begin{align}\label{eq:hp_perco_primal}
\phi[(x,0)\xleftrightarrow{\mathbb{H}} \infty] > c.
\end{align}
A similar statement holds for dual connections when $\mathbb{H}$ contains no primal infinite cluster.
\end{rem}
\begin{lem}\label{lem:hp_perco_pt2}
Suppose that $\phi$ is such that there is coexistence in $\mathbb{H}$ with the primal infinite cluster to the left.
There exists $c' > 0$ such that,
for any $n \geq 1$ and sufficiently large $x \leq 0$ even and $y \geq 0$ odd,
there exist $h \geq n$ and $m\in \mathbb{Z}$ such that
\begin{align}
\phi\big[(x,0)\xleftrightarrow{(\mathbb{R}\times [0,h]) \cap \Lambda_n^c} [m, \infty) \times \{h\}\big] &> c' \qquad \text{and}\nonumber\\
\phi\big[(y,0)\xleftrightarrow{\omega^* \cap (\mathbb{R}\times [0,h]) \cap \Lambda_n^c} (-\infty,m] \times \{h\}\big] &> c'. \label{eq:hp_perco_pt2}
\end{align}
\end{lem}
\begin{proof}
Let $\varepsilon = c/12 > 0$, where $c$ is the constant from Proposition~\ref{prop:hp_perco} (and Lemma~\ref{lem:hp_perco_pt}).
It suffices to prove the statement for $n$ large enough.
Fix $n \geq 1$ large enough that
\begin{equation}\label{eq:contour-through-lambda-n}
\phi\big[\Lambda_{n} \xleftrightarrow{\mathbb{H}} \infty, \, \Lambda_{n} \xleftrightarrow{\omega^* \cap \mathbb{H}} \infty\big] > 1-\varepsilon.
\end{equation}
To start, we claim that
\begin{align}
\lim_{\substack{x \to - \infty\\ x \text{ even}}}\phi\big[(x,0)\xleftrightarrow{\mathbb{H}} \infty \text{ but }(x,0)\nxlra{\mathbb{H} \cap \Lambda_n^c } \infty\big] &= 0\quad \text{ and} \nonumber\\
\lim_{\substack{y \to \infty\\ y \text{ odd}}}\phi\big[(y,0)\xleftrightarrow{\omega^*\cap\mathbb{H}} \infty \text{ but }(y,0)\nxlra{\omega^*\cap \mathbb{H} \cap \Lambda_n^c } \infty\big] &= 0.\label{eq:no_avoiding_La}
\end{align}
The proof of these two equalities is identical, and we focus on the first one.
For $(x,0)$ to be connected to infinity in $\mathbb{H}$ but not in $\mathbb{H} \setminus \Lambda_n$
there needs to exist a dual path in $\mathbb{H}$ from $\Lambda_n$ to $(-\infty, x] \times \{0\}$.
The existence of infinitely many such paths is incompatible with the existence of a primal cluster intersecting $(-\infty, x] \times \{0\}$ infinitely many times, and therefore has probability $0$.
Thus we deduce that only finitely many such dual paths exist a.s., and the Borel--Cantelli lemma implies~\eqref{eq:no_avoiding_La}.
Fix now $x$ and $y$ as in the statement, large enough so that
\begin{equation*}
\phi[(x,0)\xleftrightarrow{\mathbb{H}} \infty \text{ but }(x,0)\nxlra{\mathbb{H} \cap \Lambda_n^c } \infty] < \varepsilon \text{ and }
\phi[(y,0)\xleftrightarrow{\omega^*\cap\mathbb{H}} \infty \text{ but }(y,0)\nxlra{\omega^*\cap \mathbb{H} \cap \Lambda_n^c } \infty] < \varepsilon.
\end{equation*}
For $h > n$ let $\mathcal{H}_x(h)$ be the event that there exists $u \in \partial \Lambda_n$
such that both $u$ and $(x,0)$ are connected to $\mathbb{R} \times \{h\}$ in $(\mathbb{R} \times [0,h]) \setminus \Lambda_n$, but not to each other.
Define $\mathcal{H}_y(h)$ similarly for $y$ and involving dual connections.
Due to the uniqueness of the primal and dual infinite clusters in $\mathbb{H}$, we may pick $h = h(n,x,y)$ large enough such that
\begin{align*}
\phi[\mathcal{H}_x(h)] \leq \varepsilon \qquad \text{ and }\qquad \phi[\mathcal{H}_y(h)] \leq \varepsilon.
\end{align*}
Recall the definition of the infinite interface $\Gamma$ separating the primal and dual infinite clusters in $\mathbb{H}$.
Define $T_{h}$ as the first time when $\Gamma$ intersects $\mathbb{R}\times \{h\}$ and
let $M_{h}$ be the even integer closest to the first coordinate of $\Gamma_{T_{h}}$.
Define $m = m(n,h,x,y) \in \mathbb{Z}$ as the median value for $M_{h}$, so that
\begin{equation}\label{eq:median-point-at-height-h}
\phi(M_{h} \geq m) \geq \tfrac{1}{2} \quad \text{and} \quad \phi(M_{h} \leq m) \geq \tfrac{1}{2}.
\end{equation}
Now that the quantities $h$ and $m$ are fixed, let us prove~\eqref{eq:hp_perco_pt2}.
We will focus on the bound for $x$.
By Lemma~\ref{lem:hp_perco_pt},
\begin{equation*}
\phi\big[(x,0)\xleftrightarrow{\mathbb{H}} \infty \,\big|\, \Gamma_0 \in \Lambda_n \text{ and }M_{h} \geq m \big] \geq c.
\end{equation*}
Now, due to~\eqref{eq:contour-through-lambda-n} and the choice $m$, the event in the conditioning has probability at least $1/2 - \varepsilon$,
and we find
\begin{equation*}
\phi\big[(x,0)\xleftrightarrow{\mathbb{H}} \infty ,\, \Gamma_0 \in \Lambda_n \text{ and }M_{h} \geq m \big] \geq c(1/2 - \varepsilon).
\end{equation*}
Finally, we have
\begin{align*}
&\phi\big[(x,0)\xleftrightarrow{(\mathbb{R}\times [0,h]) \cap \Lambda_n^c} [m, \infty) \times \{h\}\big] \\
&\quad\geq \phi\big[(x,0)\xleftrightarrow{(\mathbb{R}\times [0,h]) \cap \Lambda_n^c } \Gamma \text{ and }M_{h} \geq m \big] \\
&\quad\geq \phi\big[(x,0)\xleftrightarrow{\mathbb{H}} \infty, \, \Gamma_0 \in \Lambda_n \text{ and }M_{h} \geq m \big]
-\phi\big[(x,0)\xleftrightarrow{\mathbb{H}} \infty \text{ but } (x,0)\nxlra{\mathbb{H} \cap \Lambda_n^c} \infty]
-\phi\big[\mathcal{H}_x(h)] \\
&\quad\geq c/2 - 3\varepsilon.
\end{align*}
Due to our choice of $\varepsilon$, we conclude that the above has at least probability $c' = c/4 > 0$.
The same argument applies to $y$ instead of $x$, {\em with the same value of $m$}.
Notice that when $M_h \leq m$, the first dual vertex on $\mathbb{R} \times \{h\}$ visited by $\Gamma$ may be $(m+1,h)$.
In order to yield a $\omega^{*}$-connection from $(y,0)$ to $(-\infty,m]\times \{h\}$, we may modify the configuration in the neighbourhood of $(m,h)$ by the finite energy property.
\end{proof}
\begin{proof}[Proposition~\ref{prop:shield}]
Fix $n$. For simplicity, let us assume that $\phi'$ is the translate of $\phi$ by $(2,0)$; other translates may be treated in a similar way.
We split the proof into several cases.
\medskip
\noindent (a) If $\mathbb{H}$ contains no dual infinite cluster $\phi$-a.s., then the same is valid for $\phi'$.
Due to Remark~\ref{rem:hp_perco_primal},
\begin{align*}
\liminf_{x \to -\infty }\phi'[(x,0)\xleftrightarrow{\mathbb{H} \setminus \Lambda_n} \infty] \geq c \quad \text{ and }\quad
\liminf_{y \to \infty } \phi'[(y,0)\xleftrightarrow{\mathbb{H} \setminus \Lambda_n} \infty] \geq c.
\end{align*}
Indeed, the absence of an infinite dual cluster in $\mathbb{H}$ ensures that
connections to infinity for far enough points $(x,0),(y,0) \in 2\mathbb{Z} \times \{0\}$ may be chosen to avoid $\Lambda_n$
(see also the proof of~\eqref{eq:no_avoiding_La} for a similar argument).
Moreover, due again to the absence of a dual infinite cluster in $\mathbb{H}$ and to~\eqref{eq:FKG},
\begin{align*}
\phi'[(x,0)\xleftrightarrow{\mathbb{H} \setminus \Lambda_n}(y,0)]
\geq \phi'[(x,0)\xleftrightarrow{\mathbb{H} \setminus \Lambda_n}\infty]\, \phi'[(y,0)\xleftrightarrow{\mathbb{H} \setminus \Lambda_n}\infty]
\geq c^2/2,
\end{align*}
for all $x < 0$ and $y > 0$ even and large enough.
Fix now $R = R(n,x,y)$ such that the above connection between $(x,0)$ and $(y,0)$ occurs in $\Lambda_R$ with probability at least $c^2/4$.
Then define $\mathcal{S}$ as the exterior-most path contained in $\omega'$,
with endpoints $(x,0)$ and $(y,0)$ and which is contained in $\mathbb{H} \cap \Lambda_R \setminus \Lambda_n$.
If no such connection exists, set $\mathcal{S} = \emptyset$.
It is immediate that $\mathcal{S}$ is a shielding arc and that it is measurable in terms of the edges in ${\rm Above}(\mathcal{S})$.
Moreover, the existence of $\mathcal{S}$ is increasing in $\omega'$, is independent of $\omega$
and occurs with probability at least $c^2/4$.
\medskip
\noindent
(b) If $\mathbb{H}$ contains no primal infinite cluster $\phi$-a.s., apply the same construction as above to $\omega^*$ instead of $\omega'$.
\medskip
\noindent (c) If there is coexistence in $\mathbb{H}$ with the primal cluster to the left, proceed as follows.
Let $x < 0$ and $y+1 > 0$ be even and odd, respectively, and large enough that Lemma~\ref{lem:hp_perco_pt2} applies to them;
let $h,m$ be the integers given by Lemma~\ref{lem:hp_perco_pt2}.
\begin{figure}
\begin{center}
\includegraphics[width = 0.6\textwidth]{shield2.pdf}
\caption{The blue lines refer to $\omega$ and the red ones to $\omega'$; the solid lines are open primal paths, while the dashed ones are dual.
In the first step of the exploration, the red interface is discovered, starting at $(x,0)$ and up to the first time it hits $[m,\infty) \times \{h\}$.
The rest of the red interface is explored in the second step and the blue interface in the third step.
For the last step to be successful it suffices that $(y+1,0)$ is connected to $(-\infty,m] \times \{h\}$ in $\omega^* \cap (\mathbb{R} \times[0,h]) \cap \Lambda_n^c$.
The path $\mathcal{S}$ is then the upper boundary of the grey region. }
\label{fig:shield}
\end{center}
\end{figure}
We will now describe an exploration algorithm that, if successful, will create $\mathcal{S}$. If the algorithm fails, we set $\mathcal{S} = \emptyset$.
See Figure~\ref{fig:shield} for an illustration.
\smallskip
\noindent {\bf Step 1:}
Explore the left-most connection in $\omega'$ between $(x,0)$ and $[m,+\infty) \times \{h\}$ which is contained in $(\mathbb{R} \times[0,h]) \setminus \Lambda_n$.
If no such connection is found, the algorithm fails.
\smallskip
\noindent {\bf Step 2:}
From the tip of the previous exploration, continue exploring the left-most infinite path in $\omega' \cap (\mathbb{H} \setminus \Lambda_n)$.
If no such path is found, the algorithm fails.
\smallskip
\noindent {\bf Step 3:}
Explore the right-most path in $\omega^*$ starting at $(y+1,0)$, ending at a previously explored edge and contained in $(\mathbb{R} \times[0,h]) \setminus \Lambda_n$.
If no such connection is found, the algorithm fails.
\medskip
When the algorithm is successful, let $D$ be the connected component of $[x,y+1] \times \{0\}$ in the set of unexplored edges of $\mathbb{H}$
and let $\mathcal{S}$ be the curve forming the boundary of $D$ other than $[x,y+1] \times \{0\}$.
Then $\mathcal{S}$ is a shielding arc.
Indeed, after Step 2, the connected component of $(y+1,0)$ in the unexplored edges of $\mathbb{H}$ is bounded entirely by explored edges contained in $\omega'$.
Thus, the $\omega^*$-open path explored in Step 3 forms $\mathcal{S}^{(2)}$, while $\mathcal{S}^{(1)} := \mathcal{S} \setminus \mathcal{S}^{(2)}$ is formed of edges of $\omega'$.
Moreover, the construction of $\mathcal{S}$ shows that $\mathcal{S}$ is measurable in terms of the edges in ${\rm Above}(\mathcal{S})$.
Now let us argue that points $(ii)$ and $(iii)$ are satisfied.
Note that $\omega'$ is promising if and only if both steps 1 and 2 are successful (otherwise $\mathcal{S} = \emptyset$ for any $\omega^*$).
That Step 1 and Step 2 are successful are both increasing events depending only on $\omega'$.
Lemmas~\ref{lem:hp_perco_pt} and~\ref{lem:hp_perco_pt2} provide uniformly positive lower bounds for the probabilities for each of these events.
Applying positive association, we conclude that $\omega'$ is promising with uniformly positive probability.
Finally, as an intersection of increasing events, $\{\omega' \text{ promising}\}$ is also increasing.
Fix a promising $\omega'$.
It is immediate that $\{\mathcal{S} \neq \emptyset\}$ is decreasing in $\omega$ (for fixed $\omega'$).
Moreover, for Step 3 to be successful it suffices (but is not generally necessary) that $(y+1,0)$
be connected to $(-\infty,m] \times \{h\}$ by a $\omega^*$-open path contained in $(\mathbb{R} \times[0,h]) \setminus \Lambda_n$.
This event occurs with uniformly positive probability due to Lemma~\ref{lem:hp_perco_pt2}.
Thus, $\mathcal{S} \neq \emptyset$ has uniformly positive probability conditionally on any promising $\omega'$.
\medskip
\noindent (d) Apply the horizontal reflection of the construction of point (c).
\end{proof}
\subsection{Proof of Proposition~\ref{prop:Gibbs=trans_inv}}
To start, let us state a simple consequence of Proposition~\ref{prop:shield}.
Notice that the equivalent of Proposition~\ref{prop:shield} also applies in $\mathbb{H}^-$.
\begin{cor}\label{cor:shield}
There exists $c > 0$ such that the following holds for all $n\geq 1$.
For any $x \leq 0$ and $y \geq 0$, both sufficiently large,
if we write $\mathcal{S}$ and $\mathcal{S}^-$ for the shielding arcs provided by Proposition~\ref{prop:shield} applied in $\mathbb{H}$ and $\mathbb{H}^-$,
respectively, then
\begin{align*}
\mathbb{P}[\mathcal{S} \neq \emptyset,\, \mathcal{S}^- \neq \emptyset ] \geq c.
\end{align*}
\end{cor}
\begin{proof}
Let $c>0$ be the constant given by Proposition~\ref{prop:shield}.
Fix $n$ and $x < 0$ and $y > 0$ even and large enough for Proposition~\ref{prop:shield} to apply.
Since the events $\{\omega' \text{ is promising for $\mathbb{H}$}\}$ and $\{\omega' \text{ is promising for $\mathbb{H}^-$}\}$
are both increasing in $\omega'$, we conclude by positive association that
\begin{align*}
\mathbb{P}[\omega' \text{ promising for both $\mathbb{H}$ and $\mathbb{H}^-$}] \geq c^2.
\end{align*}
Now, for any such $\omega'$, using the positive association of $\phi$ and property (ii) of Proposition~\ref{prop:shield},
\begin{align*}
\mathbb{P}[\mathcal{S} \neq \emptyset \text{ and }\mathcal{S}^- \neq \emptyset \,|\, \omega'] \geq
\mathbb{P}[\mathcal{S} \neq \emptyset \,|\, \omega']\,\mathbb{P}[\mathcal{S}^- \neq \emptyset \,|\, \omega'] \geq c^2.
\end{align*}
The conclusion follows by combining the two inequalities above and by adjusting $c$.
\end{proof}
\begin{proof}[Proposition~\ref{prop:Gibbs=trans_inv}]
We will show that
\begin{align*}
\phi'[A] \geq \phi[A]
\end{align*}
for any increasing event $A$ that depends on a finite set of edges.
Then the opposite inequality holds by symmetry, and we deduce that $\phi$ and $\phi'$ are equal for all increasing cylinder events.
The monotone class theorem allows then to deduce that $\phi = \phi'$.
For the rest of the proof we focus on showing the inequality above
Fix such an event $A$ and let $n_{0}$ be such that $A$ depends only on the edges in $\Lambda_{n_{0}}$.
Define an increasing sequence of scales $(n_k)_{k \geq 0}$ as follows.
Assuming $n_0,\dots n_{k-1}$ are defined,
let $x_k<0<y_k$ be sufficiently large for Corollary~\ref{cor:shield} to apply with $n = n_{k-1}$.
Let $\mathcal{S}_{k}$ and $\mathcal{S}_{k}^{-}$ be the shielding arcs produced by Corollary~\ref{cor:shield} with $x=x_k$, $y=y_k$, $n= n_{k-1}$.
Then, we may choose $n_{k} > n_{k-1}$ such that
\begin{align}\label{eq:GG'}
\mathbb{P}[\mathcal{S}_{k} \neq \emptyset,\, \mathcal{S}_{k}^- \neq \emptyset \text{ and } \mathcal{S}_{k},\mathcal{S}_{k}^- \subset \Lambda_{n_k} ] \geq c/2,
\end{align}
where $c > 0$ is the constant of Corollary~\ref{cor:shield}.
We will say that $n_k$ is a {\em successful scale} if the event above holds.
As a product of two tail trivial measures, $\mathbb{P}$ is also tail trivial by Fubini's theorem.
Combined with~\eqref{eq:GG'}, the tail triviality of $\mathbb{P}$ implies that there exists a.s. infinitely many successful scales.
For $\varepsilon > 0$, let $K = K(n_0,\varepsilon)$ be such that
\begin{align*}
\mathbb{P}[\exists j \in [1, K] \text{ such that $n_j$ is successful}] \ge 1-\varepsilon.
\end{align*}
Let $J \in [1,K]$ be maximal such that $n_J$ is successful.
Write $\mathcal{S}$ and $\mathcal{S}^-$ for the paths generated by Corollary~\ref{cor:shield} applied to $x_J,y_J$ and $n = n_{J-1}$.
Then $\mathcal{S} \cup \mathcal{S}^-$ delimits a finite domain $\mathcal{D} \subset \Lambda_{n_J}$ which contains $\Lambda_{n_0}$.
When no scale $n_j$ with $1 \leq j\leq K$ is successful, set $\mathcal{S} = \mathcal{S}^{-} = \mathcal{D} = \emptyset$.
We now show that $\mathcal{S}$ and $\mathcal{S}^-$ are measurable in terms of the edges outside of $\mathcal{D}$.
Indeed, for any fixed $j$, $J = j$ is the intersection of the event that $n_j$ is successful and of
the event that no scale $n_k$ with $j < k \leq K$ is successful.
The latter depends only on the edges outside of $\Lambda_{n_j}$, and therefore outside of $\mathcal{D}$.
The former depends on the shielding arcs $\mathcal{S}_{j}$ and $\mathcal{S}_{j}^-$, which were shown in Proposition~\ref{prop:shield} to depend only on the edges outside of $\mathcal{D}$.
Let $\xi$ and $\xi'$ be the boundary conditions induced on $\partial \mathcal{D}$ by $\omega \cap \mathcal{D}^c$ and $\omega'\cap \mathcal{D}^c$, respectively.
Next we claim that $\xi \leq \xi'$.
The proof of this fact depends on the situations in $\mathbb{H}$ and $\mathbb{H}^-$.
It is always the case that $\partial \mathcal{D}$ may be split into at most four arcs $\mathcal{S}^{(1)},\dots, \mathcal{S}^{(4)}$
of alternating type: $\mathcal{S}^{(1)}$ and $\mathcal{S}^{(3)}$ are formed of primal edges which are in $\omega'$,
while $\mathcal{S}^{(2)}$ and $\mathcal{S}^{(4)}$ are formed of dual edges which are in $\omega^*$;
these arcs are joined by single edges of $\mathbb{Z}^2$.
When the boundary of $\mathcal{D}$ is formed of only one or two arcs, we set $\mathcal{S}^{(3)} = \mathcal{S}^{(4)} = \emptyset$
and potentially $\mathcal{S}^{(1)} = \emptyset$ or $\mathcal{S}^{(2)} = \emptyset$.
If both $\mathbb{H}$ and $\mathbb{H}^-$ contain no infinite primal cluster, then $\partial \mathcal{D}$ is a circuit in $\omega^*$,
and $\xi$ is the free boundary condition. Since the free boundary condition is minimal, $\xi \leq \xi'$.
Similarly, when both $\mathbb{H}$ and $\mathbb{H}^-$ contain no infinite dual cluster, then $\partial \mathcal{D}$ is a circuit of primal edges of $\omega'$, and again $\xi \leq \xi'$.
The former case is realised when $\phi = \phi^{0}$ and the latter one when $\phi = \phi^{1}$.
In retrospect, all other cases are excluded.
The boundary of $\cal\mathcal{D}$ is formed of two arcs in most other cases:
when $\mathbb{H}$ and $\mathbb{H}^-$ each contain a single infinite cluster but of different types,
when one of $\mathbb{H}$ and $\mathbb{H}^-$ contains a single infinite cluster and the other has coexistence,
or when the both $\mathbb{H}$ and $\mathbb{H}^-$ have coexistence, with the primal cluster on the same side.
In all of these cases $\xi'$ is wired on $\mathcal{S}^{(1)}$ and is arbitrary on the rest of $\partial \mathcal{D}$,
while $\xi$ is free on $\mathcal{S}^{(2)}$ and arbitrary on $\mathcal{S}^{(1)}$. It follows directly that $\xi' \geq \xi$.
Finally, for $\partial \mathcal{D}$ to have four arcs, both $\mathbb{H}$ and $\mathbb{H}^-$ need to have coexistence,
with the primal cluster on opposite sides.
Notice then that our construction of shielding arcs guarantees that $\mathcal{S}^{(1)}$ and $\mathcal{S}^{(3)}$
are both connected to infinity in $\omega' \setminus \mathcal{D}$.
Due to our convention for accounting for infinite clusters in~\eqref{eq:DLR-FK}, the arcs $\mathcal{S}^{(1)}$ and $\mathcal{S}^{(3)}$ are wired together in $\xi'$.
As such, $\xi'$ dominates any boundary condition that is free on $\mathcal{S}^{(2)}$ and $\mathcal{S}^{(4)}$.
Thus we conclude that $\xi \leq \xi'$ also in this case.
Now, due to~\eqref{eq:DLR-FK} applied to $\phi$ and $\phi'$ and to the fact that $\mathcal{D}$ is explored from the outside,
we find
\begin{align*}
\phi'[A]-\phi[A]
\geq \mathbb{E}\big[(\phi_{\mathcal{D}}^{\xi'}[A] - \phi_{\mathcal{D}}^{\xi}[A])\mathbbm{1}_{\{\mathcal{D} \neq \emptyset\}} \big] -\mathbb{P}[\mathcal{D} = \emptyset]
\geq -\varepsilon.
\end{align*}
Since $\varepsilon$ is arbitrary, we conclude that $\phi'[A] \geq\phi[A]$, which was our goal.
\end{proof}
\section{Thermodynamical limits for FK-percolation: proof of Corollary~\ref{cor:every-limit-is-gibbs}}
\label{sec:every-limit-is-gibbs}
In this section we prove Corollary~\ref{cor:every-limit-is-gibbs}.
The proof is immediate when $\phi_{p,q}^{0} = \phi_{p,q}^{1}$, since for any finite subgraph $D$ of $\mathbb{L}$ and any boundary conditions $\xi$
\begin{align*}
\phi^0_{D,p,q} \leq_{\rm st}\phi^\xi_{D,p,q} \leq_{\rm st}\phi^1_{D,p,q},
\end{align*}
and the free and wired measures on $D$ have the same limit as $D$ increases to $\mathbb{L}$.
Henceforth we focus on $p\in (0,1)$ and $q\geq 1$ for which $\phi_{p,q}^0\neq \phi_{p,q}^1$; the proof uses the following lemmas.
For $\xi$ a boundary condition on some finite subgraph $D = (V,E)$ of $\mathbb{L}$,
a wired component of $\xi$ is an element of the partition $\xi$ that contains at least two vertices.
We start by proving Corollary~\ref{cor:every-limit-is-gibbs} for sequences of boundary conditions with at most one wired component.
\begin{lem}\label{lem:every-limit-is-gibbs1}
Let $p\in (0,1)$ and $q\geq 1$ be such that $\phi_{p,q}^0\neq \phi_{p,q}^1$.
Let $(D_{n})_{n\geq 1}$ be a sequence of domains increasing to $\mathbb{L}$
and $(\xi_{n})_{n\geq 1}$ be a sequence of boundary conditions containing at most one wired component.
Assume that the sequence of FK-percolation measures $\phi_{D_{k},p,q}^{\xi_{k}}$ has a weak limit $\phi$.
Then $\phi$ is a Gibbs measure for FK-percolation with parameters $p,q$, and
\begin{align}\label{eq:noinf_primal_dual}
\phi[0\xleftrightarrow{} \infty \text{ and }(1,0)\xleftrightarrow{*} \infty] = 0.
\end{align}
\end{lem}
\begin{proof}
Fix $p\in (0,1)$ and $q\geq 1$ for which $\phi_{p,q}^0\neq \phi_{p,q}^1$ and omit them from notation for the rest of the proof.
Let $(D_{n}, \xi_{n})_{n\geq 1}$ and $\phi$ be as in the statement.
We will prove that $\phi$ is a Gibbs measure for FK-percolation;
\eqref{eq:noinf_primal_dual} then follows from Theorem~\ref{thm:mainFK}.
Fix a finite subgraph $G$ of $\mathbb{L}$.
To start, we show that there exists $C_0 = C_0(G) > 0$ such that for any $R$ and $n$ such that $G \subset \Lambda_R \subset D_n$,
\begin{align}\label{eq:bcbc}
\phi_{D_n}^{\xi_n}\big[ \exists x,y \in \partial G \text{ s.t. }
x\nxlra{\omega^{\xi_n} \setminus G} y \text{ but } x\xleftrightarrow{ G^c} \partial \Lambda_R \text{ and } y\xleftrightarrow{ G^c} \partial \Lambda_R \big]
\leq C_0 e^{-c R},
\end{align}
where $c >0$ is given by~\eqref{eq:exp_decay}.
\begin{figure}
\begin{center}
\includegraphics[width = 0.6\textwidth]{one_wired_comp.pdf}
\caption{A domain $D_n$ with boundary conditions containing a single wired component; the wired component is formed of three arcs which are wired together.
If $x$ and $y$ are both connected to $\partial \Lambda_R$, but not to each other, then at least one of them is connected to $\partial \Lambda_R$,
but not to the wired component of $\partial D_n$; in the figure, $y$ is such.
After exploring the cluster of the wired component of $\partial D_n$ (in red), $y$ is (almost) surrounded by free boundary conditions. }
\label{fig:one_wired_comp}
\end{center}
\end{figure}
Fix $R$ and $n$ as above and write $\mathcal{C}$ for the connected component of the wired component of $\xi_n$ in $\omega \setminus G$.
Then~\eqref{eq:SMP} implies that, for any realisation ${\rm C}$ of $\mathcal{C}$,
the restriction of $\phi_{D_n}^{\xi_n}[.|\mathcal{C} = {\rm C}]$ to $D_n \setminus {\rm C}$ is $\phi_{D_n \setminus {\rm C}}^{\zeta}$,
where $\zeta$ is free at all vertices of $\partial (D_n \setminus {\rm C})$ not contained in $G$ and wired on $\partial (D_n \setminus {\rm C})\cap G$. See Figure~\ref{fig:one_wired_comp} for an illustration.
Now, for the event in~\eqref{eq:bcbc} to occur, it is necessary that at least one vertex of $\partial G$
is connected to $\partial \Lambda_{R}$ in $D_n \setminus {\rm C}$.
Indeed, for $x$ and $y$ to be disconnected in $\omega^{\xi_n} \setminus G$, at least one of them needs to not be contained in ${\rm C}$.
Thus,
\begin{align*}
\phi_{D_n}^{\xi_n}\big[ \exists x,y \in \partial G \text{ s.t. } &
x\nxlra{\omega^{\xi_n} \setminus G} y \text{ but } x\xleftrightarrow{ G^c} \partial \Lambda_R \text{ and } y\xleftrightarrow{ G^c} \partial \Lambda_R \big]\\
&\leq \sum_{{\rm C}} \phi_{D_n \setminus {\rm C}}^{\zeta}[G\leftrightarrow \partial \Lambda_{R}] \,\phi_{D_n}^{\xi_n}[\mathcal{C} = {\rm C}]\\
&\leq \sum_{{\rm C}} { C}_1 \phi_{D_n \setminus {\rm C}}^{0}[G\leftrightarrow \partial \Lambda_{R}] \,\phi_{D_n}^{\xi_n}[\mathcal{C} = {\rm C}]\\
&\leq \sum_{{\rm C}} { C}_1 \phi^{0}[G\leftrightarrow \partial \Lambda_{R}] \,\phi_{D_n}^{\xi_n}[\mathcal{C} = {\rm C}]\\
&\leq { C}_1 \, |\partial G| \, \exp\big[-c (R -{\rm rad}(G))\big].
\end{align*}
where the sums are over all possible realisations ${\rm C}$ of $\mathcal{C}$.
The second inequality follows by modifying the boundary conditions on $\partial (D_n \setminus {\rm C}) \cap G$,
which affect the probability of the event by a constant $C_1$ depending only on $|\partial G|$.
The third inequality uses~\eqref{eq:CBC}.
The last inequality is simply~\eqref{eq:exp_decay};
${\rm rad}(G)$ stands for the minimal $r$ such that $G \subset \Lambda_r$.
This concludes the proof of~\eqref{eq:bcbc}. \medskip
We turn to the proof of~\eqref{eq:DLR-FK} for $\phi$.
Fix a boundary condition $\chi$ on $G$ and an event $B \in \mathcal{F}(G^c)$.
Assume $\phi[\{\omega \text{ induces $\chi$ on $G$}\}\cap B] >0$.
Our goal is to prove that
\begin{align}\label{eq:zsup0}
\phi[A \cap \{\omega \text{ induces $\chi$ on $\partial G$}\} \cap B] =
\phi_{G}^\chi[A] \phi[\{\omega \text{ induces $\chi$ on $\partial G$}\} \cap B],
\end{align}
for any event $A \in \mathcal{F}(G)$.
By the monotone class theorem, it suffices to prove the above for events $B$ that depend on finitely many edges.
We assume this henceforth.
Fix some $\varepsilon > 0$ and let $R \geq 1$ be such that
$G \subset \Lambda_R$, $B \in \mathcal{F}(\Lambda_R\setminus G)$ and for any $n$ such that $\Lambda_{R} \subset D_n$,
\begin{align}\label{eq:zsup}
\phi_{D_n}^{\xi_n}\big[ \exists x,y \in \partial G \text{ s.t. } &
x\nxlra{\omega^{\xi_n} \setminus G} y \text{ but } x\xleftrightarrow{ G^c} \partial \Lambda_R \text{ and } y\xleftrightarrow{ G^c} \partial \Lambda_R \big] < \varepsilon.
\end{align}
The existence of such an $R$ follows from our choice of $B$ and~\eqref{eq:bcbc}.
Notice that, due to the convergence, the above also holds for $\phi$.
Fix now $n$ large enough that $\Lambda_{R} \subset D_n$ and
\begin{align}\label{eq:zsup2}
\sup_{C \in \mathcal{F}(\Lambda_{R})}\big|\phi_{D_n}^{\xi_n}[C] - \phi[C]\big| < \varepsilon.
\end{align}
The existence of such an $n$ is a consequence of the notion of weak convergence for the product topology.
For $\omega$ a configuration on $\mathbb{L}$ or $D_n$,
write $\zeta(\omega)$ for the boundary conditions on $\partial G$ induced by $\omega$
and $\zeta_R(\omega)$ for those induced by $\omega \cap \Lambda_R$, where the boundary of $\Lambda_R$ is considered wired.
That is, $x,y \in \partial G$ are wired in $\zeta_R(\omega)$ if they are connected to each other in $\omega \cap \Lambda_R \cap G^c$,
or if they are both connected to $\partial \Lambda_R$ in $\omega \cap G^c$.
Then, for any event~$A \in \mathcal{F}(G)$,
\begin{align}
\phi[A \cap \{\zeta(\omega) = \chi\} \cap B]
&= \phi[A \cap \{\zeta_R(\omega) = \chi\} \cap B] + \delta_1, \nonumber\\
&= \phi_{D_n}^{\xi_n}[A \cap \{\zeta_R(\omega) = \chi\} \cap B] + \delta_1 + \delta_2 \nonumber\\
&= \phi_{D_n}^{\xi_n}[A \cap \{\zeta(\omega^{\xi_n})= \chi \} \cap B] + \delta_1 + \delta_2 +\delta_3 \nonumber\\
&= \phi_{G}^{\chi}[A]\, \phi_{D_n}^{\xi_n}[\{\zeta(\omega^{\xi_n})= \chi\} \cap B] + \delta_1 + \delta_2 +\delta_3.
\label{eq:zsup5}
\end{align}
where $\delta_1$, $\delta_2$ and $\delta_3$ are quantities defined by the successive differences of the probabilities above
and the last equality is the application of~\eqref{eq:SMP} for $\phi_{D_n}^{\xi_n}$.
We bound $\delta_1$, $\delta_2$ and $\delta_3$ as follows.
For $\zeta_R(\omega)$ and $\zeta(\omega^{\xi_n})$ to differ, the event in~\eqref{eq:zsup} needs to occur.
Thus $|\delta_3|\leq \varepsilon$. Since any two infinite clusters are considered to be wired together, the same argument can be applied to $\phi$ to show that $|\delta_1 | \leq \varepsilon$.
Lastly, by~\eqref{eq:zsup2}, $|\delta_2 | \leq \varepsilon$.
Applying ~\eqref{eq:zsup5} to the full event, we find
\begin{align*}
\big| \phi[\{\zeta(\omega) = \chi\} \cap B] -
\phi_{D_n}^{\xi_n}[\{\zeta(\omega^{\xi_n})= \chi\} \cap B] \big| \leq 3\varepsilon.
\end{align*}
Finally, combining the above with~\eqref{eq:zsup5}, yields
\begin{align*}
\big| \phi[A \cap \{\zeta(\omega) = \chi\} \cap B]
- \phi_{G}^{\chi}[A]\, \phi[\{\zeta(\omega) = \chi\} \cap B]\big| \leq 6\varepsilon.
\end{align*}
Since $\varepsilon$ is arbitrary, the above implies~\eqref{eq:zsup0}, and the proof is complete.
\end{proof}
\begin{lem}\label{lem:every-limit-is-gibbs2}
Fix $p\in (0,1)$ and $q\geq 1$ for which $\phi_{p,q}^0\neq \phi_{p,q}^1$.
For any $K \geq 1$ and $\varepsilon >0$, there exists $N = N(K,\varepsilon)\geq 1$ such that,
for any domain $D$ containing $\Lambda_N$ and any boundary conditions $\xi$ on $D$ with at most $K$ wired components,
\[
\phi_{D,p,q}^\xi[0\xleftrightarrow{} \partial D \text{ and }(1,0)\xleftrightarrow{*} \partial D ] < \varepsilon.
\]
\end{lem}
\begin{proof}
Fix $p$ and $q$ and omit them from the notation.
Let us first treat the case $K =1$. We proceed by contradiction, that is, we suppose the existence of $\varepsilon > 0$
and of a sequence of finite graphs $D_n$ increasing to $\mathbb{L}$ and boundary conditions $\xi_n$, each with at most one wired component,
and such that
\begin{align}\label{eq:absasumpt}
\phi_{D_n}^{\xi_n}[0\xleftrightarrow{} \partial D_n \text{ and }(1,0)\xleftrightarrow{*} \partial D_n ] \geq \varepsilon \qquad \text{ for all $n\geq 1$}.
\end{align}
The space $\{0,1\}^{E(\mathbb{L})}$ is compact in the product topology,
and therefore the sequence of measures $\phi_{D_n}^{\xi_n}$ has subsequential weak limits.
If $\phi$ is such a limit, Lemma~\ref{lem:every-limit-is-gibbs1} shows that
$\phi[0\xleftrightarrow{} \infty \text{ and }(1,0)\xleftrightarrow{*} \infty ] =0$, which contradicts~\eqref{eq:absasumpt}.
This concludes the proof for~$K = 1$.
\medskip
Let now $K$ be arbitrary. For a finite graph $D$ and boundary conditions $\xi$,
write $\tilde\xi$ for the boundary conditions obtained by wiring together all wired components of $\xi$.
If $\xi$ has at most $K$ wired components, then
\begin{align*}
\phi_{D}^{\xi}[0\xleftrightarrow{} \partial D \text{ and }(1,0)\xleftrightarrow{*} \partial D ]
\leq q^K \phi_{D}^{\tilde\xi}[0\xleftrightarrow{} \partial D \text{ and }(1,0)\xleftrightarrow{*} \partial D ].
\end{align*}
Note that $\tilde \xi$ has at most one wired component, and we may apply the case $K=1$ to the probability on the right-hand side.
In conclusion, one may choose $N$ large enough to render the right-hand side of the above arbitrarily small,
which concludes the proof of the lemma.
\end{proof}
\begin{lem}\label{lem:every-limit-is-gibbs3}
Fix $p\in (0,1)$ and $q\geq 1$ for which $\phi_{p,q}^0\neq \phi_{p,q}^1$.
For any $\varepsilon >0$, there exists $N = N(\varepsilon)\geq 1$ such that,
for any domain $D$ containing $\Lambda_N$ and any boundary conditions $\xi$ on $D$,
\begin{align*}
\phi_{D,p,q}^\xi[0\xleftrightarrow{} \partial D \text{ and }(1,0)\xleftrightarrow{*} \partial D ] < \varepsilon.
\end{align*}
\end{lem}
\begin{proof}
Fix $p$ and $q$ and omit them from the notation.
Let $K,c$ be the constants given by Lemma~\ref{lem:Kcross} so that, for any $n>0$ and any boundary conditions $\xi$ on~$\Lambda_{2n}$,
\begin{align}\label{eq:KC1}
\phi^{\xi}_{\Lambda_{2n}}[\exists K \text{ disjoint clusters intersecting both $\Lambda_n$ and $\partial \Lambda_{2n}$}] \leq e^{-cn}.
\end{align}
By Lemma~\ref{lem:every-limit-is-gibbs2}, we may pick $n$ sufficiently large that,
for any domain $D$ containing $\Lambda_{n}$ and any boundary conditions $\xi$ on $D$ with at most $K$ wired components,
\begin{align}\label{eq:KC2}
\phi^{\xi}_{D}[0\leftrightarrow \partial D \text{ and } (1,0)\xleftrightarrow{*} \partial D] < \varepsilon/2.
\end{align}
Moreover, we may assume that $e^{-cn}<\varepsilon/2$.
Fix $N=2n$ and let $D$ be a domain containing $\Lambda_{N}$ and $\xi$ be an arbitrary boundary condition on $\partial D$.
Let $\mathcal{C}$ be the union of the connected components in $D \setminus \Lambda_n$ that intersect~$\partial D$.
For any realisation $C$ of $\mathcal{C}$, the restriction of $\phi_D^\xi[.|\, \mathcal{C} = C]$ to $D\setminus C$ is
the measure $\phi_{D\setminus C}^\zeta$, where $\zeta$ is a boundary condition with as many wired components as there are clusters that intersect both $\partial D$ and $\Lambda_n$.
By the choice of $K$,
\begin{align*}
&\phi_{D}^\xi[0\xleftrightarrow{} \partial D \text{ and }(1,0)\xleftrightarrow{*} \partial D ] \\
&\qquad\leq \sum_{C} \phi_{D\setminus C}^\zeta[0\xleftrightarrow{} \partial (D\setminus C) \text{ and }(1,0)\xleftrightarrow{*} \partial (D\setminus C) ] \, \phi_{D}^\xi[\mathcal{C} = C]\\
&\qquad\leq \varepsilon/2 +
\phi^{\xi}_{D}[\zeta \text{ contains more than $K$ wired components}]
\leq \varepsilon.
\end{align*}
The first inequality is due to~\eqref{eq:KC2}, while the second to~\eqref{eq:KC1}.
\end{proof}
We are finally ready to proceed to the main goal of this section, namely the proof of Corollary~\ref{cor:every-limit-is-gibbs}.
\begin{proof}[Corollary~\ref{cor:every-limit-is-gibbs}]
As already mentioned, the proof is immediate when $\phi_{p,q}^{0}=\phi_{p,q}^{1}$.
Fix $p\in (0,1)$ and $q\geq 1$ for which $\phi_{p,q}^0\neq \phi_{p,q}^1$ and let $\phi_{D_n,p,q}^{\xi_n}$ be a sequence of measures as in the statement, converging to some infinite-volume measure $\phi$.
The proof is essentially the same as that of Lemma~\ref{lem:every-limit-is-gibbs1},
the only difference is that~\eqref{eq:bcbc} is unavailable when $\xi_n$ has more than a single wired component.
However, in light of Lemma~\ref{lem:every-limit-is-gibbs3}, for any $r$ such that $G \subset \Lambda_r$ and $n$ sufficiently large for $D_n$ to contain $\Lambda_{2r}$,
\begin{align*}
\phi_{D_n}^{\xi_n}\big[\zeta(\omega^{\xi_n}) \neq\zeta_{2r}(\omega) \big]
&\leq
\sup_{\xi}\phi_{\Lambda_{2r}}^{\xi}\big[G \xleftrightarrow{} \partial \Lambda_{2r} \text{ and } G \xleftrightarrow{*} \partial \Lambda_{2r}\big] \\
&\leq
c_0(G)\,\sup_{\xi}\phi_{\Lambda_{2r}}^{\xi}\big[0 \xleftrightarrow{} \partial \Lambda_{2r} \text{ and } (1,0)\xleftrightarrow{*} \partial \Lambda_{2r}\big]
\xrightarrow[r \to \infty]{} 0,
\end{align*}
where the supremum is over all boundary conditions $\xi$ on $\partial \Lambda_{2r}$,
the first inequality is due to~\eqref{eq:SMP} and the second to the finite energy property; $c_0(G)$ is a constant depending on~$G$.
The rest of the proof is identical to that of Lemma~\ref{lem:every-limit-is-gibbs1}.
\end{proof}
\section{The Potts model}\label{sec:Potts_proof}
\subsection{Preparations}
\begin{lem}\label{lem:Phi2}
Let $\mu$ be a tail trivial Gibbs measure for the Potts model on $\mathbb{L}$ with parameters $T>0$ and $q\geq 2$
that is different from $\mu^{\mathrm{free}}$, $\mu^{1}$, \dots, $\mu^{q}$.
Then $\Phi$ is also tail trivial and $\omega$ contains at least two infinite clusters of different colours.
\end{lem}
\begin{proof}
Fix $T$ and $q$ and omit them from the notation.
The tail triviality of $\Phi$ follows readily from that of $\mu$ by Fubini's theorem because,
given the spin configuration, the edges are open or closed independently (with probability that depends on the spins).
We turn to the existence of infinite clusters and proceed by contradiction.
Assume that, for $(\omega,\sigma)$ sampled from $\Phi$,
all infinite clusters of $\omega$ have a.s. the same colour in $\sigma$.
By the tail triviality of $\Phi$, this colour is constant -- assume it to be~$1$.
Write $\phi$ for the $\omega$-marginal of $\Phi$.
Fix $n>0$ and consider $N > n$.
Let us sample a pair of spin and edge configurations $(\sigma, \omega)$ from $\Phi$ in several steps.
First we sample both configurations on $\Lambda_{N}^{c}$.
By the Gibbs property, the conditional distribution of $\omega$ inside $\Lambda_N$
is that of FK-percolation in $\Lambda_N$ with wired boundary conditions,
conditioned on that no two points of $\partial \Lambda_N$ that have different colour in $\sigma$ are connected (see Section~\ref{sec:ES}).
Sample now $\omega$ in $\Lambda_N \setminus \Lambda_n$ according to this distribution.
Let $\mathcal{E}(n,N)$ be the event that no point of $\partial \Lambda_N$ of colour different from $1$ is connected to $\Lambda_n$.
Notice that $\mathcal{E}(n,N)$ depends on $\omega$ in $\Lambda_n^c$ and on $\sigma$ in $\Lambda_N^c$.
Moreover, if $\mathcal{E}(n,N)$ occurs, any completion of $\omega$ in $\Lambda_n$ ensures that
no two points of $\partial \Lambda_N$ that have different colour in $\sigma$ are connected.
As in the previous section, write $\zeta(\omega)$ for the boundary condition on $\partial \Lambda_n$
induced by $\omega$, so that two vertices of $\partial \Lambda_n$ are wired if and only if they are connected in $\omega \cap \Lambda_n^c$ or both belong to infinite clusters.
Then, by the Gibbs property for FK-percolation, the law of $\omega$ in $\Lambda_n$
conditionally on $\sigma$ on $\Lambda_N^c$ and $\omega$ on $\Lambda_n^c$ such that $\mathcal{E}(n,N)$ occurs is~$\phi_{\Lambda_{n}}^{\zeta(\omega)}$.
Finally, by our assumption, $\Phi[\mathcal{E}(n,N)] \xrightarrow[N\to \infty]{}1$.
We conclude that there exists a sequence of random boundary conditions $\zeta_n$ for FK-percolation on $\Lambda_n$ such that
$$ \phi = \lim_{n \to \infty }\phi_{\Lambda_n}^{\zeta_n}.$$
In light of Corollary~\ref{cor:every-limit-is-gibbs}, $\phi$ is a linear combination of $\phi^{0}$ and $\phi^{1}$.
Moreover, by tail triviality, it is equal to one of these two measures.
It then follows that $\mu$ is either $\mu^{\rm free}$ or $\mu^{1}$, which contradicts our assumption.
\end{proof}
\subsection{Proof of Theorem~\ref{thm:mainPotts}}
Let $\mu$ be an extremal Gibbs measure for the Potts model with parameters $T > 0$ and $q \ge 2$.
As already discussed, the only case of interest is when $T = T_{c}(q)$ and $q$ is such that $\phi_{p_{c}(q),q}^0\neq \phi_{p_{c}(q),q}^1$.
We remind the reader that this occurs if and only if $q >4$.
For the whole proof $q$, $T$ and $p= p(T)$ are fixed such that $\phi_{p,q}^0\neq \phi_{p,q}^1$; they will be omitted from the notation.
Let $\Phi$ be the Edwards-Sokal measure associated to $\mu$.
For $i \in \{1,\dots, q\}$, write $\omega^i$ for the edges of colour $i$ in $(\omega,\sigma)$, that is the edges $uv$ of $\omega$ such that $\sigma(u) = \sigma(v) = i$.
\begin{rem}
It may be tempting to try to apply Theorem~\ref{thm:mainFK} to the $\omega$-marginal of~$\Phi$.
Notice however that this measure {\em a priori} might not be Gibbs for FK-percolation:
its restriction to finite domains is an FK-percolation measure {\em conditioned} that boundary vertices of different colours are disconnected.
We will rather consider the configurations $\omega^i$ and attempt to prove
that at most one of these contains an infinite cluster. Lemma~\ref{lem:Phi2} allows us then to conclude.
The configurations $\omega^i$ do not have FK-percolation distributions, but will be shown to be dominated by certain
FK-percolation mesures, which due to Theorem~\ref{thm:mainFK} are of the form $\lambda_i \phi^1 + (1- \lambda_i)\phi^0$.
Then, using duality, we will prove that for any $i \neq j$, $\lambda_i + \lambda_j < 2$,
so that at most one of $\omega^i$ and $\omega^j$ contains a.s. an infinite cluster.
\end{rem}
Fix $\delta > 0$ a small constant to be chosen below; it will be allowed to depend on $q$, but not on the values $n$ and $N$ appearing hereafter.
Fix $n$ such that, for any $N \geq n$,
\begin{align}
& \phi^1[\Lambda_N \leftrightarrow{} \partial \Lambda_{2N}] > \tfrac12 \text{ and } \label{eq:choicen1}\\
& \Phi[\exists K \text{ disjoint clusters intersecting $\partial \Lambda_{2N}$ and $\Lambda_N$}] < \tfrac12, \label{eq:choicen2}
\end{align}
where $K$ is given by Lemma~\ref{lem:Kcross} \footnote{Lemma~\ref{lem:Kcross} does not apply exactly to $\Phi$, but the proof may easily be adapted.}.
It follows from Corollary~\ref{cor:every-limit-is-gibbs} that we can fix $N = N(n) \geq 2n$ such that, for any boundary condition $\xi$ on $\partial \Lambda_N$,
\begin{align}\label{eq:choiceN}
\inf_{\lambda \in [0,1]} \big\|(\phi_{\Lambda_N}^\xi)_{|\Lambda_{2n}} - ( \lambda \phi^1 + (1- \lambda)\phi^0)_{|\Lambda_{2n}} \big\|_{\rm TV} \leq \delta,
\end{align}
where the above refers to the distance in total variation between the restrictions of $\phi_{\Lambda_N}^\xi$ and $\lambda \phi^1 + (1- \lambda)\phi^0$ to $\Lambda_{2n}$.
Define the domain $\mathcal{D}$ explored from the outside in $\Phi$ as follows.
Explore all interfaces between open and closed edges inside $\Lambda_{2N} \setminus \Lambda_N$ starting on $\partial \Lambda_{2N}$
and ending either on $\partial \Lambda_{2N}$ or on $\partial \Lambda_{N}$; see Figure~\ref{fig:potts_flowers} for an illustration.
Let $\mathcal{D}$ be the connected component of $0$ in the set of unexplored edges.
We define $\tau : \partial \mathcal{D} \to \{0,1,\dots, q\}$ as the boundary conditions induced by the configuration outside of $\mathcal{D}$.
More precisely, for each vertex $v \in \partial \mathcal{D}$ which is on the free side of an exploration, set $\tau_{v} = 0$;
if $v$ is on the wired side of an exploration, set $\tau_{v}$ for the colour of the primal open edges adjacent to $v$ in the exploration.
Then, since $\mathcal{D}$ is explored from the outside, for any realisation $D$ of $\mathcal{D}$,
\begin{align*}
\Phi[.\, |\, \mathcal{D}=D, (\omega, \sigma) \text{ on } D^{c}] _{|D}= \Phi_{D}^{\tau}.
\end{align*}
We emphasise that the event $\mathcal{D} = D$ and the boundary conditions $\tau = \tau (\omega,\sigma)$ are determined by $(\omega,\sigma)$ on $D^{c}$.
Assume that $\mathcal{D}= D$ and fix a realisation of $(\omega,\sigma) \text{ on } D^c$ and thus the boundary conditions $\tau$.
Notice that the boundary of $D$ is naturally split into a sequence of primal and dual arcs, with each of the primal arcs being monochromatic.
For each colour $i$, write $D^{i}$ for the domain obtained from $D$ by deleting all edges which are adjacent to vertices $u \in \partial D$ with $\tau_{u}\notin \{0,i\}$.
We now define boundary conditions $\xi^{i}$ and $\zeta^{i}$ for FK-percolation on $D^{i}$.
Notice that $\partial D^i$ shares vertices with $\partial D$, namely those on the primal arcs of $\partial D^i$ of colour $i$, as well those on the free arcs of $\partial D^i$.
In $\xi^{i}$, each primal arc of $\partial D$ of colour $i$ is wired, while the rest of the boundary is free.
In $\zeta^i$, all primal arcs of $\partial D$ of colour $i$ are wired together and the rest of the boundary is free.
Thus $\xi^i$ and $\zeta^i$ differ only in the wiring between the different arcs of $\partial D$ of colour $i$.
For simplicity, we will write $\phi_{D}^{\zeta^i}$ for the measure $\phi_{D^i}^{\zeta^i}$,
where the edges in $D \setminus D^{i}$ are closed a.s.. The same applies to~$\xi^i$.
Below we state a series of results for the fixed realisation $D$ of $\mathcal{D}$, with the fixed configurations $(\omega,\sigma)$ outside,
and with the boundary conditions $\tau$, $\xi^i$ and $\zeta^i$ that these induce.
\begin{figure}
\begin{center}
\includegraphics[width = 0.32\textwidth, page = 1]{potts_flower.pdf}
\includegraphics[width = 0.32\textwidth, page = 4]{potts_flower.pdf}
\includegraphics[width = 0.32\textwidth, page = 3]{potts_flower.pdf}
\caption{{\em Left:} the exploration of the interfaces originating on $\partial \Lambda_{2N}$ reveals the edges in the grey region;
$\mathcal{D}$ is formed of the edges entirely contained in the white region.
{\em Middle:}~The boundary conditions
$\zeta^i$ and $\xi^i$ induced on $\mathcal{D}^{i}$ where $i$ corresponds to the blue colour.
The difference between them is that in $\zeta^{i}$ the two blue arcs are wired together, while in $\xi^i$ they are not.
{\em Right:} The boundary conditions $\xi^j$ induced on $\mathcal{D}^{j}$ where $j$ corresponds to the red colour.
When considering the dual measure, the bold dotted contours correspond to wired boundary conditions.
Thus the dual measure dominates that in the middle picture (after shift by~$(1,0)$). }
\label{fig:potts_flowers}
\end{center}
\end{figure}
\begin{lem}\label{lem:p1}
If $(\omega,\sigma)$ is sampled according to $\Phi_{D}^{\tau}$,
then the law of $\omega^{i}$ is dominated by~$\phi_{D}^{\zeta^{i}}$.
\end{lem}
\begin{proof}
Let $(\omega,\sigma)$ be sampled according to $\Phi_{D}^{\tau}$ and
write $\mathcal{C}$ for the union of all clusters of $\omega$ of colours $j\neq i$ intersecting $\partial D$.
Then, for any realisation $C$ of $\mathcal{C}$, the spatial Markov property for $\Phi$ ensures that the first marginal of
$\Phi_D^\tau(.|\mathcal{C} = C)$ follows the FK-percolation measure $\phi_{D \setminus C}^\chi$,
where $\chi$ are the boundary conditions that are wired on the vertices of $\partial (D \setminus C) \cap \partial D$ of colour $i$ in $\tau$, and free elsewhere.
This corresponds to the measure $\phi_{D}^{\zeta^i}$ conditioned on the event that all edges of, or adjacent to $C$ are closed.
As the latter is a decreasing event,~\eqref{eq:FKG} implies that the law of $\omega \setminus \mathcal{C}$ is stochastically dominated by $\phi_{D}^{\zeta^i}$.
Finally, by construction $\omega^i$ is contained in $\omega \setminus \mathcal{C}$, which concludes the proof.
\end{proof}
Write $\mathcal{K}$ for the number of clusters of $\omega \cap D^c$ that intersect both $\partial \Lambda_{2N}$ and $\partial \Lambda_{N}$.
\begin{lem}\label{lem:p2}
Let $K\in \mathbb{N}$ and assume that $(\omega,\sigma)$ outside of $D$ are such that $\mathcal{K} = K$.
Then the Radon-Nikodym derivative of $\phi_{D}^{\xi^{i}}$ with respect to $\phi_{D}^{\zeta^{i}}$ is bounded by $q^K$.
In particular, for any event $A$ depending only on the edges in $D$,
\begin{align*}
\phi_{D}^{\zeta^{i}}[A] \leq q^K \phi_{D}^{\xi^{i}}[A].
\end{align*}
\end{lem}
\begin{proof}
As already stated, the only difference between $\xi^i$ and $\zeta^i$ is that the primal arcs of $\partial D$ of colour $i$ are wired together in $\zeta^{i}$ but not in $\xi^{i}$.
There exist at most $K$ such arcs, as they are in bijection with the primal clusters in $\Lambda_{2N} \setminus \Lambda_{N}$ of colour $i$ which cross from $\partial \Lambda_{2N}$ to $\partial\Lambda_{N}$.
Thus the weight of a configuration under $\phi^{\zeta^{i}}_{D}$ differs from that under $\phi^{\xi^{i}}_{D}$ by a multiplicative factor between $1$ and $q^{K}$.
This also holds for the corresponding partition functions, and the statement follows.
\end{proof}
\begin{lem}\label{lem:p3}
For $i \neq j$, if $\omega^{i}$ is sampled according to $\phi_{D}^{\xi^{i}}$,
then the law of its dual configuration $(\omega^{i})^*$ translated by $(1,0)$
dominates $\phi_{D}^{\xi^{j}}$.
\end{lem}
In the above, we refer to the restrictions of the two measures to the intersection of $(D^{i})^* + (1,0)$ and $D^j$.
We will actually only be interested in the effect of these two measures on the edges immediately adjacent to $0$,
which are indeed contained in this intersection.
\begin{proof}
Fix $i \neq j$.
Write $E_i$ for the set of edges on the primal arcs of $\partial D$ of colour $i$
(these edges are not part of $D$, only their endpoints are).
Write $F_i$ for the set containing the edges whose duals form the dual arcs of $\partial D$,
as well as all edges that are adjacent to the primal arcs of $\partial D$ of colours different from $i$.
Then $\omega^{i}$ has the law of $\phi^{0}$ restricted to $D$ and conditioned on that all edges of $E_i$ are open and all those of $F_i$ are closed and belong to the same dual cluster in $D^{c}$.
The law of $\omega^{j}$ can be described in a similar way using the corresponding events $E_{j}$ and $F_{j}$:
\[
\omega_{j} \sim \phi^{0}[\cdot \,|\, \text{$\omega \equiv 1$ on $E_{j}$, $\omega \equiv 0$ on $F_{j}$, all edges of $F_{j}^*$ are connected in $\omega^* \cap D^c$}].
\]
Define the sets of edges $E_{i}'$ and $F_{i}'$ that are obtained from $E_{i}$ and $F_{i}$, respectively,
by considering the dual edges and translating these by $(1,0)$.
The above, along with the duality between $\phi^0$ and $\phi^1$
imply that $(\omega^{i})^*$ translated by $(1,0)$ has the law of $\phi^{1}$ conditioned on that all edges of $E_{i}'$ are closed and those of $F_{i}'$ are open and belong to the same cluster in $D^{c}$:
\[
\omega_{i}^{*}+(1,0) \sim \phi^{1}[\cdot \,|\, \text{$\omega \equiv 0$ on $E_{i}'$, $\omega \equiv 1$ on $F_{i}'$, all edges of $F_{i}'$ are connected in $D^c$}].
\]
In order to prove that $\omega_{i}^{*}$ translated by $(1,0)$ dominates $\omega_{j}$1 it suffices to study the events on which the measures above are conditioned.
First observe that each edge of $E_i'$ may be written $e^* + (1,0)$ with $e \in E_i$. This means that all edges in $E_i'$ are incident to a vertex of $\partial D$ of colour $i$, whence $E_i' \subset F_j$.
Applying the reverse transformation, we get similarly that $E_i \subset F_j'$.
Finally, being in the same dual cluster is a decreasing condition and being in the same primal cluster is an increasing condition. The statement follows by the FKG inequality.
\end{proof}
For $i\in \{1,\dots, q\}$, write $\lambda_i = \lambda_i(D,\tau)$ for the value $\lambda \in [0,1]$ that minimises
\begin{align}\label{eq:lambdasc}
\big\|(\phi_{\mathcal{D}}^{\xi^{i}})_{|\Lambda_{2n}} - ( \lambda \phi^1 + (1- \lambda)\phi^0)_{|\Lambda_{2n}} \big\|_{\rm TV}.
\end{align}
\begin{cor}\label{cor:lambdas}
Let $e$ be the edge between $(0,0)$ and $(1,1)$. Then, for any $i,j \in \{1,\dots, q\}$ distinct,
\begin{align}\label{eq:lambdas}
\lambda_i + \lambda_j \leq 1 + \frac{2\delta}{\phi^1[e \text{ open}] -\phi^0[e \text{ open}]}.
\end{align}
\end{cor}
\begin{proof}
Due to the choice of $N$ specified in \eqref{eq:choiceN}, the infimum of~\eqref{eq:lambdasc} is smaller than $\delta$.
If $f$ denotes the edge between $(0,0)$ and $(-1,1)$,
Lemma~\ref{lem:p3} implies that
\begin{align*}
1
& = \phi_{\mathcal{D}}^{\xi^{1}}[e \text{ open}] + \phi_{\mathcal{D}}^{\xi^{1}}[e \text{ closed}] \\
&\geq \phi_{\mathcal{D}}^{\xi^{1}}[e \text{ open}] + \phi_{\mathcal{D}}^{\xi^{2}}[f \text{ open}]
\geq (\lambda_1 + \lambda_2) \phi^1[e \text{ open}] + (2- \lambda_1 - \lambda_2)\phi^0[e \text{ open}] - 2\delta,
\end{align*}
where we used that $\phi^0[e \text{ open}] = \phi^0[f \text{ open}]$.
Moreover, since $\phi^0$ is dual to $\phi^1$, $\phi^1[e \text{ open}] = 1 -\phi^0[e \text{ open}] >1/2$, and basic algebra yields~\eqref{eq:lambdas}.
\end{proof}
Let us now conclude the proof of the theorem.
\begin{proof}[Theorem~\ref{thm:mainPotts}]
By the choice~\eqref{eq:choicen2} of $n$, for any $i\in \{1,\dots,q\}$,
\begin{align*}
\Phi[\Lambda_n \nxlra{~\omega^i\,\,\,~} \Lambda_{2n}^c]
&\geq \Phi\big[\phi_{\mathcal{D}}^{\zeta^{i}}[\Lambda_n \nxlra{~\omega^i\,\,\,~} \Lambda_{2n}^c] \big]\\
&\geq \tfrac12 \Phi\big[\phi_{\mathcal{D}}^{\zeta^{i}}[\Lambda_n \nxlra{~\omega^i\,\,\,~} \Lambda_{2n}^c] \,\big|\, \mathcal{K} \leq K \big]\\
&\geq \tfrac12 q^{-K}
\Phi\big[ \phi_{\mathcal{D}}^{\xi^{i}}[\Lambda_n \nxlra{~\omega^i\,\,\,~} \Lambda_{2n}^c] \,\big|\, \mathcal{K} \leq K \big] \\
&\geq \tfrac12 q^{-K}
\Phi\big[ \lambda_i \phi^1[\Lambda_n \nxlra{~\omega^i\,\,\,~} \Lambda_{2n}^c] + (1- \lambda_i)\phi^0[\Lambda_n \nxlra{} \Lambda_{2n}^c] - \delta \,\big|\, \mathcal{K} \leq K \big] \\
&\geq \tfrac12 q^{-K}\Big( \Phi[1 - \lambda_i\,|\,\mathcal{K} \leq K ] \phi^0[\Lambda_n \nxlra{} \Lambda_{2n}^c] - \delta\Big),
\end{align*}
where $\Phi$ governs the choice of $\mathcal{D}$ and $\tau$,
and hence of $\zeta^{i}$, $\xi^i$ and ultimately $\lambda_i=\lambda_i(\mathcal{D},\tau)$.
In the above, the first line uses Lemma \ref{lem:p1},
the second uses~\eqref{eq:choicen2}, the third uses Lemma \ref{lem:p2}, and the fourth uses the definition of $\lambda_{i}$ and the choice \eqref{eq:choiceN} of $N$.
We proceed by contradiction and assume that $\mu$ is different from $\mu^{\mathrm{free}}$ and $\mu^i$ with $i \in \{1,\dots, q\}$.
Then, due to Lemma~\ref{lem:Phi2}, $\Phi$ contains at least two infinite clusters of different colours.
Permute the colours so that there exist infinite clusters of colours $1$ and $2$.
Then, due to Corollary~\ref{cor:lambdas},
\begin{align*}
\Phi[\lambda_1\,|\,\mathcal{K} \leq K] +\Phi[\lambda_2\,|\,\mathcal{K} \leq K] \leq 1 + \frac{2\delta}{\phi^1[e \text{ open}] -\phi^0[e \text{ open}]} \leq 3/2,
\end{align*}
provided that $\delta$ is small enough.
Thus, at least one of $\Phi[\lambda_1|\mathcal{K} \leq K]$ and $\Phi[\lambda_2 |\mathcal{K} \leq K]$ is smaller than $3/4$.
Assume with no loss of generality that it is the former.
Recall that, since there is no percolation under $\phi^0$, $\phi^0[\Lambda_n \nxlra{} \Lambda_{2n}^c] \to 1$ as $n\to1$.
Thus, assuming $\delta < 1/4$, we find that $\limsup_{n \to \infty}\Phi[\Lambda_n \xleftrightarrow{\omega^1} \Lambda_{2n}^c] < 1$.
This contradicts our assumption that there exists percolation in $\omega^1$ $\Phi$-a.s..
\end{proof}
\section{Extensions}\label{sec:extensions}
In this section we describe how the proof of Theorem~\ref{thm:mainFK} may be extended to two other settings, namely FK-percolation on the triangular and hexagonal lattices and the loop $O(n)$ model. Since the argument requires only minor modifications, we only provide sketches.
\subsection{FK-percolation and Potts models on the triangular and hexagonal lattices}\label{sec:Hex}
Consider the triangular and hexagonal lattices of Figure~\ref{fig:tri} defined in the plane $\mathbb{C}$ as follows.
The triangular lattice, denoted ${\rm Tri}$, has vertex set $\{u + v e^{i\pi/3}: u,v \in \mathbb{Z}\}$ and edges between vertices at euclidian distance $1$.
The hexagonal lattice ${\rm Hex}$ is the dual of ${\rm Tri}$: it has a vertex in the centre of each face of ${\rm Tri}$ and edges between vertices corresponding to neighbouring faces.
\begin{figure}
\begin{center}
\raisebox{-0.5\height}{\includegraphics[width = 0.4\textwidth]{TH.pdf}}\qquad
\raisebox{-0.5\height}{\includegraphics[width = 0.5\textwidth]{STT.pdf}}
\caption{
{\em Left:} The triangular lattice ${\rm Tri}$ and its dual hexagonal lattice ${\rm Hex}$.
{\em Right:} The star-triangle transformation $T$ transforms the triangle $\bf T$ into the star $\bf S$ by adding a central vertex $O$
and replacing the edges between $A$, $B$ and $C$ by those between $O$ and $A$, $B$ and $C$.
}
\label{fig:stt}\label{fig:tri}
\end{center}
\end{figure}
The FK-percolation models on ${\rm Tri}$ and ${\rm Hex}$ are defined similarly to that on $\mathbb{L}$.
In particular, one may define infinite-volume measures $\phi^{i}_{{\rm Tri}, p,q}$ and $\phi^{i}_{{\rm Hex}, p,q}$ via thermodynamical limits
for $i \in \{0,1\}$, $q \geq 1$ and $p \in (0,1)$.
Moreover, the duality relation described in Section~\ref{sec:fk-background} adapts to this setting and relates the models on ${\rm Tri}$ and ${\rm Hex}$,
with the edge-weights related by~\eqref{eq:pp*}.
In spite of the primal and dual models acting on different lattices, a notion of self-duality may be defined in this context using the star-triangle transformation.
\begin{lem}\label{lem:stt}
Consider the triangle and star graphs ${\bf T}$ and ${\bf S}$ of Figure~\ref{fig:stt} with $\partial {\bf T} = \partial {\bf S} = \{A,B,C\}$.
For any $q \geq 1$, any boundary conditions $\xi$, for $p$ such that
\begin{align}\label{eq:p_c_tri}
\Big(\tfrac{p}{1-p}\Big)^3 + 3\Big(\tfrac{p}{1-p}\Big)^2 = q,
\end{align}
there exists a coupling $\mathbb{P}$ of $\omega' \sim \phi_{{\bf T},p,q}^\xi$ and
$\omega \sim \phi_{{\bf S},p^*,q}^\xi$ such that
\begin{align*}
U \xleftrightarrow{\omega} V \Leftrightarrow U \xleftrightarrow{\omega'} V \quad \forall U,V \in \{A,B,C\},\, \mathbb{P}\text{-a.s.}.
\end{align*}
\end{lem}
The coupling above may be described as a random map $T$ associating to a configuration $\omega$ the configuration $\omega'$.
When $\omega$ contains at least one open edge, there exists a unique configuration $\omega'$ that yields the same connections between $A$, $B$ and $C$ as $\omega$, and we set $T(\omega) = \omega'$.
When $\omega$ contains no open edges, $T(\omega)$ is chosen at random between the configurations on ${\bf S}$ with zero or one open edges:
the configuration with zero open edges is chosen with probability $(\tfrac{p}{1-p})^3/q$, while each configuration with exactly one open edge is chosen with probability $(\tfrac{p}{1-p})^2/q$.
Finally, the randomness in $T$ is independent of that in $\omega$.
The proof of Lemma~\ref{lem:stt}, as well as of the explicit coupling described above appeared in several papers, for instance in~\cite[Prop. 2.2]{GriMan13}.
Using the star-triangle transformation, it was proved in~\cite{BefDum12} that~\eqref{eq:p_c_tri} determines the critical point $p_c({\rm Tri}, q)$ of FK-percolation on the triangular lattice,
and that the critical point for the hexagonal lattice is given by its dual $p_c({\rm Hex}, q)=p_c({\rm Tri}, q)^*$.
Moreover, the dichotomy of~\cite{DumSidTas17} adapts to this setting. Finally, when $p\in (0,1)$ and $q\geq 1$, it was shown in~\cite{DumLiMan18} that
\begin{align*}
\phi_{{\rm Tri},p,q}^0\neq \phi_{{\rm Tri}, p,q}^1 &\Longleftrightarrow q > 4 \text{ and } p = p_c({\rm Tri}, q)\qquad \text{ and }\\
\phi_{{\rm Hex},p,q}^0\neq \phi_{{\rm Hex},p,q}^1 &\Longleftrightarrow q > 4 \text{ and } p = p_c({\rm Hex}, q).
\end{align*}
The proof of~\cite{BefDum12} is based on the observations that, for $p = p_c({\rm Tri}, q)$,
the star-triangle transformation $T$ may be applied to each upward pointing triangle in ${\rm Tri}$
and the resulting graph is ${\rm Hex}$ shifted up by $1/\sqrt3$.
Moreover, if the configuration on ${\rm Tri}$ is chosen according to $\phi^{i}_{{\rm Tri}, p,q}$ for some $i \in \{0,1\}$,
then the resulting one is the shift of $\phi^{i}_{{\rm Hex}, p^*,q}$ by $(0,-1/\sqrt3)$
As stated in the introduction, Theorem~\ref{thm:mainFK} extends to the triangular and hexagonal lattices.
\begin{thm}\label{thm:mainFK_tri}
Let $\phi$ be a Gibbs measure for FK-percolation on ${\rm Tri}$ or ${\rm Hex}$ with edge-intensity $p \in (0,1)$ and cluster-weight $q \geq 1$.
Then there exists $\lambda \in [0,1]$ such that
\begin{align*}
\phi = \lambda \phi_{{\rm Tri},p,q}^0 + (1-\lambda) \phi_{{\rm Tri},p,q}^1 \quad \text{ or } \quad \phi = \lambda \phi_{{\rm Hex},p,q}^0 + (1-\lambda) \phi_{{\rm Hex},p,q}^1,
\end{align*}
depending on the setting.
\end{thm}
The proof is exactly similar to that of Theorem~\ref{thm:mainFK}, with only one notable difference. We may focus on ${\rm Tri}$, as the result for ${\rm Hex}$ is obtained by duality. Only the setting $p = p_c({\rm Tri})$ requires attention; for all other values of $p$ the Gibbs measure is unique.
In Proposition~\ref{prop:no_hp_perco} and Lemma~\ref{lem:perc-hp-some-good}, we used an automorphism $\rho$ of the set of configurations
which was mapping the primal model to an (approximate) vertical symmetry of the dual model.
In the present case, $\rho$ should be replaced with the composition of the application of $T$ to each upward pointing triangle,
the symmetry with respect to the horizontal axis $\mathbb{R} \times \{0\}$, and the shift by $(0,1/\sqrt 3)$.
Using this transformations, the proofs of both Proposition~\ref{prop:no_hp_perco} and Lemma~\ref{lem:perc-hp-some-good} adapt readily. The rest of the proof is identical.
In the same way as in Corollary~\ref{cor:every-limit-is-gibbs}, it may be deduced from Theorem~\ref{thm:mainFK_tri}
that any thermodynamical limit of FK-percolation measures on the triangular and hexagonal lattices is Gibbs.
Finally, the proof of Theorem~\ref{thm:mainPotts} may also be adapted to the triangular and hexagonal lattices, with only minor modifications. In addition to using Theorem~\ref{thm:mainFK_tri} instead of Theorem~\ref{thm:mainFK}, one only needs to adapt the proof of Lemma~\ref{lem:p3},
where auto-duality plays a particular role. This may easily be done using duality combined with the star-triangle transformation, as descried above.
\subsection{Proof for loop $O(n)$ model}\label{sec:loop_proof}
Fix for the whole section $n\geq 1$ and $0 < x \leq {1}/{\sqrt n}$.
We start by considering the structure of Gibbs measures for the spin system (see the discussion of Section~\ref{sec:loop-background} for details).
As for FK-percolation, using monotonicity, one may define the Gibbs measures ${\sf Spin}_{n,x}^{+}$ and ${\sf Spin}_{n,x}^{-}$ for the spin system
as limits of finite-volume measures. By construction, these are translation invariant and tail trivial.
Moreover, due to~\eqref{eq:On_mono}, any Gibbs measure for the spin system is contained between these two for the stochastic ordering.
The relevant result for the spins system is the following
\begin{prop}\label{prop:GibbsSpin}
Any Gibbs measure for the spin system with parameters $(n,x)$
is a linear combination of ${\sf Spin}_{n,x}^{+}$ and ${\sf Spin}_{n,x}^{-}$.
\end{prop}
\begin{proof}
We only sketch this proof as it is very similar to that of Theorem~\ref{thm:mainFK}.
If the parameters are such that the model is in the macroscopic loops regime, then ${\sf Spin}_{n,x}^{+} = {\sf Spin}_{n,x}^{-}$,
and this is the only Gibbs measure for the spin system.
If the model is in the exponential decay regime, then ${\sf Spin}_{n,x}^{+} \neq {\sf Spin}_{n,x}^{-}$,
but the proofs of Sections~\ref{sec:FK_preparation} and~\ref{sec:FK_duplication} apply with only minor changes.
Indeed, the only significant alteration is in the transformation $\rho$ used in the proofs of Proposition~\ref{prop:no_hp_perco} and Lemma~\ref{lem:perc-hp-some-good}:
it should be replaced here with a vertical symmetry composed with a sign flip.
\end{proof}
\begin{rem}
The proof for the spin system may appear closer to those for the Ising model from~\cite{Aiz80},~\cite{Hig81} or~\cite{GeoHig00}.
Note however that the boundary conditions in the DLR equation for the spin system are not determined entirely by the spins on the boundary of the domain,
and therefore the more complex argument of the present paper is necessary.
\end{rem}
Next we turn to the Gibbs measures for the loop model. The key is the following lemma.
\begin{lem}\label{lem:LtoS}
Let ${\sf Loop}$ be a Gibbs measure for the loop model with parameters $(n,x)$.
Consider the measure ${\sf Spin}$ on $ \{-1,1\}^{F({\rm Hex})}$ obtained by sampling a configuration $\omega$ from ${\sf Loop}$,
then sampling uniformly one of the two spin configurations coherent with $\omega$.
Then ${\sf Spin}$ is a Gibbs measure for the spin system with parameters $(n,x)$.
\end{lem}
\begin{proof}
Consider a domain $D$.
Recall that a boundary condition $\tau$ for the spin system on $D$
contains an assignment of spins for the faces neighbouring $D$ and connectivity information between these faces in the spin percolation.
A boundary condition for the loop model on $D$ is a pairing of certain vertices of $\partial D$, acting as the endpoints of the various segments of loops in $D^c$.
It is immediate from the correspondence between loops and spins that
there is a two-to-one function mapping spin boundary conditions $\tau$ on loop boundary conditions $\xi(\tau)$.
Precisely, for any coherent loop and spin configuration $\omega,\sigma$ on the full lattice, we have
\begin{align*}
\omega \text{ induces $\xi(\tau)$ on $D$}\, \Leftrightarrow \, \sigma \text{ induces $\tau$ or $-\tau$ on $D$}.
\end{align*}
Fix a spin boundary condition $\tau$ and write $\xi =\xi(\tau)$ for the corresponding loop boundary condition.
Let $\omega$ be a random configuration chosen according to ${\sf Loop}$ and $\sigma$ be a spin configuration chosen uniformly between the two configurations coherent with $\omega$; $\sigma$ has law ${\sf Spin}$.
Conditioning on that $\sigma$ induces $\tau$ is the same as conditioning that $\omega$ induces $\xi$ and that the choice of the spins is such that $\sigma$ corresponds to $\tau$ on the faces neighbouring $D$.
Since the latter choice is independent of $\omega$,
we conclude that the law of $\omega$ in $D$ conditioned on $\{\sigma \text{ induces } \tau\}$
is the same as that conditioned on $\{\omega \text{ induces } \xi\}$,
and is therefore ${\sf Loop}_{D,n,x}^{\xi}$ due to the Gibbs property for ${\sf Loop}$.
Now, due to the properties of the correspondence between loop and spin configurations \cite[Prop. 3]{DumGlaPel21},
we conclude that
$${\sf Spin}[.\,|\, \sigma \text{ induces } \tau] = {\sf Spin}_{D,n,x}^{\tau}.$$
The same holds when conditioning additionally on any event depending only on the spins on $D^c$, which proves the DLR property for ${\sf Spin}$.
\end{proof}
\begin{proof}[Theorem~\ref{thm:loop}]
We start with the existence.
A Gibbs measure for the loop model may be obtained by considering the paths separating $+$ from $-$ spins in a configuration chosen according to ${\sf Spin}_{n,x}^{+}$.
Since ${\sf Spin}_{n,x}^{+}$ is translation invariant and tail trivial, the same holds for the associated loop measure.
We turn to the uniqueness. Let ${\sf Loop}$ be a Gibbs measure for the loop model, and construct ${\sf Spin}$ as in Lemma~\ref{lem:LtoS}.
Then, due to Proposition~\ref{prop:GibbsSpin}, ${\sf Spin} = \lambda {\sf Spin}_{n,x}^{+} + (1-\lambda) {\sf Spin}_{n,x}^{-}$ for some $\lambda \in [0,1]$.
Observe now that ${\sf Spin}_{n,x}^{-}$ may be obtained from ${\sf Spin}_{n,x}^{+}$ by a global sign flip.
As such, the laws of the interfaces between $+$ and $-$ clusters in spin configurations sampled from
${\sf Spin}_{n,x}^{+}$ and ${\sf Spin}_{n,x}^{-}$, respectively, are the same.
In conclusion, ${\sf Loop}$ does not depend on $\lambda$, and is therefore unique.
\end{proof}
\bibliographystyle{abbrv}
|
1,116,691,497,735 | arxiv | \section{Introduction}
Clean graphene has shown to be an excellent platform to investigate various electron optical experiments, ranging from Fabry-P\'{e}rot (FP) resonances \cite{Campos12b,Grushina13,Rickhaus13,Varlet14,Shalom16,Calado15}, snake states \cite{Rickhaus15,Taychatanapat14}, electron guiding \cite{Rickhaus15_2}, magnetic focusing \cite{Taychatanapat13,Calado14}, Veselago lensing or angle-dependent transmission studies of negative refraction \cite{Lee15,Chen16}. The \textit{pn}-junctions are basic building blocks of many of these experiments, which for example are used to confine electrons into FP cavities, where the \textit{pn}-junctions play the role of semireflective mirrors. The study of FP resonances have already revealed themselves as a powerful tool to investigate various aspects of graphene. Examples are the $\pi$-shift of the FP resonances at low magnetic field originating from Klein tunnelling \cite{Klein29} in single layer graphene \cite{Shytov08,Young09,Varlet16} or the FP resonances in ultraclean, suspended devices that proved ballistic transport over several microns \cite{Rickhaus13, Grushina13}.
Encapsulation of graphene in hexagonal boron nitride (hBN) \cite{Dean10} can not only allow for quasi-ballistic transport (the mean free path is comparable or larger than the device geometry), but can also be used for the creation of a superlattice in graphene due to the periodic potential modulation of the Moir\'{e} pattern. Such a pattern stems from a small lattice mismtach of graphene and hBN.
The modified bandstructure of graphene, which includes the emergence of satellite Dirac peaks (satellite DPs), is experimentally observable only for small misalignment angles ($\theta$) between hBN and graphene \cite{Woods14,Yankowitz12}. The formation of a Moir\'{e} superlattice has been demonstrated in various spectroscopy \cite{Yankowitz12,Yu14,Shi14}
and transport \cite{Hunt13,Ponomarenko13,Dean13,Gorbachev14,L_Wang15,Wang15,Greenaway15,Kumar16,Lee16} experiments. Besides experiments, considerable effort has been invested to calculate the band-structure of graphene under the influence of such a Moir\'{e} superlattice \cite{Wallbank13,Moon14,Wallbank15}.
Here we demonstrate confinement using band engineering based on locally gated Moir\'{e} superlattices and the appearance of FP resonances defined by the main and satellite DPs playing the role of reflective barriers. Although several aspects of FP cavities have been investigated, such as the effect of the \textit{pn}-junction smoothness on the visibility of the FP resonances \cite{Rickhaus13}, the electronic tunability of the cavity size has not been studied. First, we will discuss samples with a large misalignment angle between the hBN and graphene. We find that the cavity size is strongly dependent on carrier density in and outside the cavity. Then, we turn to aligned samples with a superlattice structure and take advantage of the large tunability of the cavity size to obtain additional information on the modulated Moir\'{e} band-structure which is yet not fully understood. Our findings are consistent with the electron-hole symmetry breaking of the Moir\'{e} superlattice \cite{Wallbank13}.
\begin{figure*}[htbp!]
\centering
\includegraphics[width=\textwidth]{Cavity_length.pdf}
\caption{\textbf{Fabry-P\'{e}rot resonances measured in a two-terminal \textit{pnp}-configuration. a,} False-color SEM image with the measurement configuration sketched. The contacts are shown in yellow, the top-gate in red and the graphene between the two hBN-layers is shown in cyan. Scale-bar equals \SI{500}{nm}. \textbf{b,} Numerical derivative of the conductance as a function of global back-gate ($V_{\text{BG}}$) and local top-gate ($V_{\text{TG}}$). Here it is replotted as a function of charge-carrier density in the outer ($n_{\text{out}}$) and inner cavity ($n_{\text{in}}$). The three most important FP resonances present in the system are indicated. \textbf{c,} Visibilty $\Delta G/\left<2G\right>$ of the FP resonances indicated in (b) in comparison. \textbf{d,} Extracted cavity length of the central cavity as a function of $n_{\text{in}}$, as indicated with the cyan arrow in (b), for different values of $n_{\text{out}}$. \textbf{e,} The same data as shown in (d), but now plotted as a function of $n_{\text{in}}/n_{\text{out}}$ for experiment (red) and theory (blue, for transport simulation see Supporting Information). The black, dashed line corresponds to the width of the top-gate measured with the SEM. The blue, dashed lines are a guide to the eye for the theoretical data and the theoretical data shifted by \SI{30}{nm} respectively.}
\label{fig:Regular_PNP}
\end{figure*}
The device geometry of our two-terminal \textit{pnp}-junction is shown in Fig.~\ref{fig:Regular_PNP}a: a global back-gate and a local top-gate is used to separately control the charge carrier density in the inner ($n_{\text{in}}$) and outer ($n_{\text{out}}$) regions thus forming \textit{pp'p}- or \textit{pnp}-junctions. In order to reduce the cross-talk between the gates we used a thin (\SI{\sim 30}{nm}) hBN layer. Therefore $n_{\text{out}}$ can be calculated according to $n_{\text{out}}=V_{\text{BG}} \cdot (1/C_{\text{SiO}_{2}}+1/C_{\text{hBN,b}})^{-1}$, where $C_{\text{SiO}_{2}}$ and $C_{\text{hBN,b}}$ are the geometrical capacitances of the SiO$_{2}$ and bottom hBN respectively, and $V_{\text{BG}}$ and $V_{\text{TG}}$ the applied gate-voltages. In analogy, $n_{\text{in}}$ is given by $n_{\text{in}}= V_{\text{BG}} \cdot (1/C_{\text{SiO}_{2}}+1/C_{\text{hBN,b}})^{-1}+V_{\text{TG}} \cdot C_{\text{hBN,t}}$ where $C_{\text{hBN,t}}$ is the top hBN capacitance. The Cr/Au contacts dope the graphene in its proximity \textit{n}-type, independent of $V_{\text{BG}}$. The conductance is measured in a two-terminal configuration as a function of $V_{\text{BG}}$ and $V_{\text{TG}}$ (which can be directly converted to $n_{\text{in}}$ and $n_{\text{out}}$), as shown in Fig.~\ref{fig:Regular_PNP}b. The overall device length is roughly \SI{1}{\micro m}, with a top-gate width of \SI{230}{nm} being centered in the middle of the device. In the measured device, a high mobility of the electrons and holes ($\mu_{\text{e,h}}$) was extracted from field effect measurements, yielding $\mu_{\text{e}}$\SI{\sim 150000}{cm^2 V^{-1} s^{-1}} and $\mu_{\text{h}}$\SI{\sim 50000}{cm^2 V^{-1} s^{-1}} respectively.
All structures were fabricated following the procedure given by Wang \textit{et al.} \cite{Wang13} with some additional improvements. A detailed description of the fabrication is given in the Supporting Information. The measurements were done using low frequency lock-in technique in a variable temperature insert (VTI) with a base-temperature around $T=1.6\,$K.
The derivative of the differential conductance with respect to $n_{\text{in}}$ and $n_{\text{out}}$ are shown in Fig.~\ref{fig:Regular_PNP}b for $n_{\text{out}}<0$. The visible fringes are FP resonances appearing in the \textit{pp'p} regime (bottom part, $n_{\text{in}}<0$) and \textit{pnp} regime (upper part, $n_{\text{in}}>0$). Specific cuts within an extended (red) or limited (blue, green) gate-range are shown in Fig.~\ref{fig:Regular_PNP}c. While the FP resonances in the inner (outer) cavities are tuned predominantly by $n_{\text{in}}$ ($n_{\text{out}}$) as indicated with the blue (green) arrow in Fig.~\ref{fig:Regular_PNP}b, the FP resonances between the contacts depends on both densities as indicated by the red arrow.
The three types of FP resonances are very different in their visibility $\Delta G/(2 \left<G\right>)$. Here $\Delta G$ is the difference between the conductance at constructive and destructive interference, and $\left<G\right>$ denotes the mean conduction in between oscillation maximum and minimum. The FP resonances between the contacts yield the lowest visibility ($\sim1\%$), those in the outer cavities yield an intermediate visibility ($\sim4\%$) and FP resonances in the inner cavity yield the highest visibility ($\sim11\%$).
In ballistic graphene, the FP visibilities depend on the transmission/reflection properties of the confining boundaries, namely the \textit{pn}-interface. The transmission/reflection probabilities of a \textit{pn}-interface
strongly depends on the angle of the incoming charge carrier \cite{Cheianov06}.
In the case of a ``sharp" \textit{pn}-junctions ($d\ll\lambda_{F}$, where $d$ denotes the distance over which the density changes and $\lambda_{F}$ is the Fermi wavelength), transmission of charge carriers is possible up to large angles measured with respect to the \textit{pn}-junction normal. In contrast, for a very ``smooth" \textit{pn}-junction ($d\gg\lambda_{F}$), only charge carriers at low angles are transmitted (in our device this is around \SI{\sim 20}{\degree}, although depends strongly on the gate voltages). Since our conductance measurement is not angle resolved, the measured signal averages over all possible angles and leads to smearing of the FP resonances. It turns out that for a smooth \textit{pn}-junction, which transmits a narrower range of angles, the highest FP visibilities can be observed \cite{Rickhaus13}.
In our system we discriminate between two types of \textit{pn}-junctions: i) the \textit{pn}-junction created in proximity of the contacts for $n_{\text{out}}<0$ and ii) the \textit{pn}-junction created using the global back-gate and local top-gate. We note that (ii) is much smoother compared to (i) (see Supporting Information). Following the above given argument one sees that the FP resonance visibility is highest/lowest when the cavity is defined by two softer/sharper \textit{pn}-junctions.
We will now extract the effective cavity length, $L$ from the FP oscillations in the central cavity, which deviates in most cases from the physical width of the top-gate. Assuming a FP resonator with a hard-wall potential (fixed width of the cavity), the cavity width can be extracted from the FP oscillations.
Constructive interference forms if the path-difference between directly transmitted and twice reflected waves is equal to $2 L=j\lambda_{\text{F}}$, where $\lambda_{\text{F}}$ is the Fermi wavelength and $j$ is an integer.
The $j$-th FP resonance can be rewritten as $L\sqrt{n_{\text{j}}}=j\sqrt{\pi}$ using $\lambda_{\text{F}}=2\pi/ k_{\text{F}}=2 \sqrt{\pi/n}$ which is valid for single layer graphene. For two neighboring peaks, for example $j$-th peak at density $n_\text{j}$ and $(j+1)$-th peak at density $n_{\text{j+1}}$:
\begin{equation}\label{cavity-length}
L=\frac{\sqrt{\pi}}{\sqrt{n_{\text{j+1}}}-\sqrt{n_{\text{j}}}} .
\end{equation}
We note here that in Equation~\ref{cavity-length} $L$ is independent of $n_{\text{out}}$ which is an oversimplification of the problem.
An alternative way to define the cavity length is to measure the distance between the two zero-density points of the right and left \textit{pn}-junctions. However, since the position of the \textit{pn}-interface is experimentally not directly accessible we will use Equation~\ref{cavity-length} to deduce the cavity size. More discussion on this is given in the Supporting Information.
The cavity length was extracted by taking various linecuts comparable to the one indicated with the blue arrow in Fig.~\ref{cavity-length}b and then using Equation~\ref{cavity-length}. We observe an increase of $L$ with increasing $n_{\text{in}}$ (for fixed $n_{\text{out}}$), and a decrease of $L$ with increasing $n_{\text{out}}$ (for fixed $n_{\text{in}}$) as shown in Fig.~\ref{fig:Regular_PNP}d. Consequently $L$ does depend on $n_{\text{out}}$, as expected. Surprisingly, by plotting $L$ as a function of $n_{\text{in}}/n_{\text{out}}$, all data-points lie on one universal curve which is shown in Fig.~\ref{cavity-length}e, independent of the exact position within Fig.~\ref{fig:Regular_PNP}b from which they have been extracted from. Within the applied gate-range, $L$ varies substantially, by up to \SI{200}{nm}, which corresponds to a shift of around \SI{100}{nm} per \textit{pn}-junction.
The evolution of $L$ as a function of $n_{\text{in}}/n_{\text{out}}$ extracted from the experiment was compared with the one extracted from a transport simulation (based on the method described in Ref. \cite{Rickhaus13}) using Equation~\ref{cavity-length}. The latter reveals good qualitative agreement with the experiment as shown in Fig.~\ref{cavity-length}e. The most significant difference between experiment and theory are: i) an off-set of $\Delta L \sim 30\;$nm from the theory to the experiment and ii) a disagreement between the trends of the two curves when approaching very low charge carrier densities in the outer cavity, corresponding to large values of $n_{\text{in}}/n_{\text{out}}$ (the same is true when depleting the inner cavity, not shown here). In the experiment the cavity length $L$ saturates while it increases continuously in the theory. We believe that the reason for this is that the measured sample bears a residual doping ($n^{*}$) which is not present in theory. The residual doping causes that for values below $n^{*}$ the electrostatic gates are unable to further deplete the graphene, thus for $V_{\text{BG}} \rightarrow 0$ the effective value of $n_{\text{in}}/n_{\text{out}}$ (tuning $L$) remains fixed. The off-set of the two curves (i) might origin from a too narrow top-gate in theory if the top-gate in the experiment was measured with an error of \SI{\sim 30}{nm}.
\begin{figure*}[htbp!]
\centering
\includegraphics[width=\textwidth]{Cavity_length_Moiree.pdf}
\caption{\textbf{Fabry-P\'{e}rot resonancesin in the presence of a Moir\'{e} superlattice. a,} Numerical derivative of the conductance as a function of $n_{\text{in}}$ and $n_{\text{out}}$. The red, cyan and green lines indicate the regular FP resonances described in Fig.~\ref{fig:Regular_PNP}. Additional FP resonances emerge if $n_{\text{in}}$ or $n_{\text{out}}$ is tuned above or below a satellite Dirac peak. These FP resonances are indicated with orange and purple arrows. The position of the main and satellite DPs are indicated with black and purple/orange triangles respectively. \textbf{b,} DOS of graphene as a function of energy in the presence of a Moir\'{e} superlattice with hBN for three different parameter sets used in the calculation performed by Wallbank \textit{et al.} Figure adapted from Ref. \cite{Wallbank13}. Black, orange and purple triangles indicate main and satellite Dirac-peaks respectively. The blue, dashed line indicates the DOS for unperturbed graphene. \textbf{c,d,} High-resolution measurements ($d/dn_{\text{in}}G\;$(e$^2$/h)) of the regions where the inner cavity is tuned beyond the satellite DP and the additional FP resonances are present. \textbf{e,} Position of the individual FP resonance peaks (lower panel) and their relative spacing $\Delta n_{\text{in}}$ (upper panel) is plotted as a function of $n_{\text{in}}$. The black, dashed line indicates the values expected from theory. If the semitransparent boundaries are defined via the satellite DP the spacing is further increased. However, no precise trend of $\Delta n_{\text{in}}$ (increasing or decreasing) as a function of $n_{\text{in}}$ can be seen from the few data-points extracted.}
\label{fig:Moiree_PNP}
\end{figure*}
Now we turn to graphene which is aligned with a small twist angle with respect to one of the hBN layers. The resulting band-reconstruction includes additional satellite DPs which are indicated with the purple and orange triangles in Fig.~\ref{fig:Moiree_PNP}a where the derivative of the conductance is plotted. The main DP is indicated with the black triangle. At low doping the semitransparent boundaries in the bipolar region are defined by the main DP (comparable to Fig.~\ref{fig:Regular_PNP}b) and FP resonances within inner and outer cavity are observed, marked by the green and cyan arrows, respectively.
Whereas in Fig.~\ref{fig:Regular_PNP}b,c the contacts are the only boundaries leading to FP resonances in the unipolar regime (red arrow), additional semitransparent boundaries, formed by the satellite DPs, emerge at high doping in the presence of a superlattice. As a result, novel FP oscillation are visible at high doping as indicated with the purple and orange arrows in Fig.~\ref{fig:Moiree_PNP}a, which are absent in Fig.~\ref{fig:Regular_PNP}b (no Moir\'{e} superlattice). This new set of FP oscillation resembles again the pattern known from the bipolar regime and the transition from FP resonances across the whole sample to FP resonances within the inner and outer cavity is a direct consequence of the satellite DPs forming these additional semitransparent boundaries. There are also charge carriers that bounce between the contacts and the satellite DP boundary, leading to weak resonances as a function of $n_{\text{out}}$ (see Supporting Information).
In the following, we compare the visibility between FP resonances across the main DP and the satellite DP within the inner cavity for $n_{\text{in}}>0$ and $n_{\text{in}}<0$ separately ($\mu_{e}$\SI{\sim 100000}{cm^2 V^{-1} s^{-1}}, $\mu_{h}$\SI{\sim 50000}{cm^2 V^{-1} s^{-1}}.
Charge carriers for $n_{\text{in}}<0$ (hole side) bouncing between boundaries formed by the satellite DPs ($\Delta G\sim0.9\%$) show a visibility that is 40$\%$ lower than the visibility of the main DP oscillation ($\Delta G \sim1.5\%$), as shown in Fig.~\ref{fig:Moiree_PNP}a. In contrast, for $n_{\text{in}}>0$ (electron side), the visibility is reduced by 85$\%$ ($\Delta G\sim0.4\%$ at the satellite DP compared to $\Delta G\sim3\%$ at the main DP).
To interpret these observations, we consider a family of possible Moir\'{e} minibands for graphene on hBN substrate, which was calculated by Wallbank \textit{et al.} \cite{Wallbank13} using a general symmetry-based approach. Because the relatively large number of model-dependent parameters of the symmetry breaking potential
can strongly influence the obtained Moir\'{e} perturbation, the focus was on the generic features for different sets of parameters used. In Fig.~\ref{fig:Moiree_PNP}b, the DOS for three different sets of parameters are plotted. Details on the parameters can be found in Ref. \cite{Wallbank13}.
For the case of $n_{\text{in}}<0$, when crossing the satellite DP on the hole side, the DOS between inner and outer cavity decreases to zero (independent of the parameters used for the calculations of the DOS as given in Ref. \cite{Wallbank13}) for all $k_{x,y}$, as indicated with the purple triangle in Fig.~\ref{fig:Moiree_PNP}b. This result is experimentally supported by capacitance spectroscopy of hBN-graphene-hBN heterostructures in the presence of a Moir\'{e} superlattice \cite{Yu14}. The vanishing DOS leads to similar reflection/transmission coefficients as for the main DP, resulting in a comparable visibility of the two FP resonances. A representative cut at fixed $n_{\text{out}}<0$ is shown in Fig.~\ref{fig:Moiree_PNP}d.
For the electron side with $n_{\text{in}}>0$, the significantly reduced visibility is in qualitative agreement with the band-structures shown in Fig.~\ref{fig:Moiree_PNP}b, since the DOS at the satellite DP (indicated with an orange triangle) is reduced (depending on the parameters used in the calculation), but never vanishes. A representative cut at fixed $n_{\text{out}}>0$ is shown in Fig.~\ref{fig:Moiree_PNP}c. A direct implication of the finite DOS is that only some charge-carriers have a nonzero reflection coefficient, thus contributing to the FP resonances, while the remaining ones account for a background current. It is worth noting, that even though the visibility of the FP resonances across the electron side satellite DP ($n_{\text{in,out}}>0$, Fig.~\ref{fig:Moiree_PNP}c) are significantly reduced, they seem to be more regular over a wider gate-range compared to the FP resonances across the hole satellite DP ($n_{\text{in,out}}<0$, Fig.~\ref{fig:Moiree_PNP}d). However, this could be as well just sample-specific since the mobility in the hole side was significantly lower compared to the electron side.
We now compare the evolution of the position of the individual FP resonance peaks ($|n_{\text{out}}|\sim \text{const.}$) and their relative spacing $\Delta n_{\text{in}}$ as a function of $n_{\text{in}}$, which are plotted in the lower and upper panel of Fig.~\ref{fig:Moiree_PNP}e. The latter analysis is performed instead of the cavity length analysis since Equation~\ref{cavity-length} does not hold any more if the Fermi-energy is tuned beyond the satellite DP. This is because in Equation~\ref{cavity-length} a circular Fermi-surface and electron like dispersion is assumed, which does not hold in the presence of a strong band-modulation. In contrast, mapping the FP resonance peak position as a function of charge-carrier density is free of any assumptions and therefore independent of the band-structure. If the semitransparent interfaces are defined via the main DP, the evolution of $\Delta n_{\text{in}}$ between neighboring peaks as a function of $n_{\text{in}}$ is in good agreement with the values extracted from theory (transport simulation in the absence of a Moir\'{e} superlattice), which is indicated with the black, dashed line in Fig.~\ref{fig:Moiree_PNP}e. If the semitransparent interfaces are defined via the satellite DPs, the density-spacing between the FP resonance peaks is further increased on both, the electron and hole side.
Although a precise extraction of the cavity size is not possible with Equation~\ref{cavity-length}, an estimate can still be given. We have found a cavity size of $L$\SIrange{\sim 250}{310}{nm} on the electron side if the density is measured from the main DP, whereas setting the density to zero at the satellite DP gives an unphysical cavity size ($L$\SIrange{\sim 80}{200}{nm}). On the electron side the density of states seems almost unaltered, as can be seen in Fig.~\ref{fig:Moiree_PNP}b, however the band-structure is substantially modified, which can be seen in Ref.~\cite{Wallbank13}. The band-structure consists of two nonisotropic bands above the satellite DPs which makes the situation rather complex, with different visibilities and angle dependent transmissions for the two bands.
On the hole side the cavity size analysis yields cavity sizes of $L$\SIrange{\sim 280}{360}{nm} if the density is measured from the main DP. At the satellite DP the density of states decreases to zero, meaning that a real Dirac point is formed. Therefore one might expect, that the density for the FP oscillations should be measured from the satellite DP. However by setting the charge-carrier density to zero at the hole satellite DP, the analysis gives unphysical results. We note, that close to the hole satellite DP Equation~\ref{cavity-length} should be valid if counting the charge-carrier density from the satellite DP, since the band-structure is isotropic. Besides the most pronounced FP resonances indicated with the purple arrow in Fig.~\ref{fig:Moiree_PNP}d, a second set of resonances with a much shorter period seem to appear in the very vicinity of the hole satellite DP (see inset of Fig.~\ref{fig:Moiree_PNP}d). For these resonances the extracted cavity length leads only to reasonable values ($L$\SI{\sim 300}{nm}) when counting the charge carrier density starting from the satellite DP. Unfortunately the residual doping ($n_{\mathrm{res}}$\SIrange{\sim 0.4e15}{1e15}{m^{-2}}) prevents us from resolving more of these features in the very vicinity of the satellite DP. A possible explanation for the different behavior of the two sets of FP oscillation ($n_{\text{out}}<0$) might be the following:
The small resonances are only observed up to densities of $\Delta n$\SI{<2e15}{m^{-2}} (where $\Delta n$ is measured from the satellite DP), where the Moir\'{e} miniband remains close to Dirac like (linear dispersion relation). The Fermi energy corresponding to theses small oscillations is consequently exclusively tuned in the linear part of the Moir\'{e} miniband (see Supporting Information).
From the band-structures we would expect that the small oscillations should be visible up to higher doping values, namely values which are larger by a factor of up to $\sim 10$ depending on the model. This estimate is based on the energy-spacing between the satellite DP and the first van Hove singularity at higher energies, extracted from theory (see Supporting Information). Note that the extracted energy-spacing corresponds to an upper bound within which the dispersion relation might be considered as linear. The Fermi energy corresponding to the stronger resonances indicated with the purple arrow, does partially reside outside of the linear region of the Dirac cones where the band-structure becomes more complex, including singularities and band-overlaps.
We note that in the work of Lee \textit{et al.} \cite{Lee16} they suggest that model 3, which is shown in Fig.~\ref{fig:Moiree_PNP}b, is the one describing the experimental situation best. Furthermore, for model 1, where multiple satellite DPs appear near every main DP, Equation~\ref{cavity-length} would also have to be multiplied by a factor of $\sqrt{g_{\text{SDP}}}$, where $g_{\text{SDP}}$ is the additional degeneracy of the satellite Dirac cones. Although by including additional degeneracies, and counting the density from the satellite DP, the strong oscillations could give reasonable cavity sizes, however, for the small oscillations, the additional degeneracies would result in too large cavity sizes. Moreover, most experimental evidence points to a single Dirac cone at the satellite DP.
We also note here, that the visibility of these resonances are quite low, which could also introduce some experimental error in the analysis.
In conclusion, we first analysed a \textit{pnp}-junction in the absence of a Moir\'{e} superlattice where the varying visibility of the different types of FP resonances could be linked to sharp and smooth \textit{pn}-junctions in the system. Furthermore, the change of the effective cavity length depending on $n_{\text{in}}$ and $n_{\text{out}}$ could be mapped via the FP resonances.
In a second sample, the presence of a Moir\'{e} superlattice gave rise to satellite DPs and the modified band-structure resulted in confinement of electronic trajectories and in the appearance of FP resonances. Although the oscillations are formed in the Moir\'{e} minibands, they can mostly be described as if they would originate from the non-modified band-structures. Further studies will be needed to explain these findings.
Our results show that confinement of electrons can be obtained using miniband engineering. Future studies can investigate the angle dependent transmission properties of such an interface. Moreover combining the studies of transverse magnetic focusing \cite{Lee16} and bias spectroscopy studies of FP resonances or detailed studies of the magnetic field dependence of FP resonances can reveal further details on the band reconstruction.\\
\textbf{ACKNOWLEDGMENTS}\\
This work was funded by the Swiss National Science Foundation, the Swiss Nanoscience Institute, the Swiss NCCR QSIT, the ERC Advanced Investigator Grant QUEST, and the EU flagship project graphene. M.-H.L. and K.R. acknowledge financial support by the Deutsche Forschungsgemeinschaft (SFB 689). The authors thank John Wallbank, David Goldhaber-Gordon, Simon Zihlmann and L\'{a}szlo Oroszl\'{a}ny for fruitful discussions.\\
\bibliographystyle{PRL}
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1,116,691,497,736 | arxiv | \section{\uppercase{Introduction}}
Most comets are observed to show activity when they reach heliocentric distances $r_\mathrm{H} \lesssim 5$--6 AU, where the most abundant cometary volatile, water ice, begins to sublimate as a result of increasing insolation. However, a few comets have been observed to exhibit activity at greater heliocentric distances. This distant activity cannot be explained by sublimation of water ice, but other mechanisms including crystallization of amorphous ice (Prialnik \& Bar-Nun 1992) and sublimation of supervolatile species (A'Hearn et al.~2012) may be responsible. The reason why only a few distant comets have been observed to be active is twofold. Firstly, when comets are far away from the Sun, they are intrinsically less active because of their lower temperatures. Secondly, distant comets tend to be extremely faint, introducing an observational bias.
Fortunately, thanks to ever-advancing technology and better sky coverage by ongoing sky surveys, recent years have witnessed an increasing number of discoveries of distant comets (e.g., C/2006 S3 (LONEOS) discovered at $r_{\mathrm{H}} = 14.3$ AU, C/2010 U3 (Boattini) at $r_{\mathrm{H}} = 18.4$ AU, both pre-perihelion), making a better understanding of activity in distant comets possible.
Comet C/2017 K2 (hereafter ``K2") was detected by Pan-STARRS at Haleakala, Hawai`i on UT 2017 May 21 (Wainscoat et al.~2017), when it was $r_\mathrm{H} \approx 16$ AU from the Sun. The current orbital solution by the JPL Horizons ephemeris service identifies it as a long-period comet, with perihelion distance $q = 1.811$ AU, eccentricity $e = 1.00034$, inclination $i = 87\degr.6$, and a perihelion passage on UT 2022 Dec 21.\footnote{The elements are at epoch TT 2017 Jun 10.0, retrieved on 2017 Oct 04.}
In earlier work (Jewitt et al.~2017), we used the Hubble Space Telescope (\textit{HST}) to set a limit to the size of the nucleus ($R_{\mathrm{N}} \lesssim 9$ km) and established that the coma of K2 consists of large ($\gtrsim 0.1$ mm) sized dust grains released over a period of years. We identified a prediscovery detection from 2013 at heliocentric distance $r_{\mathrm{H}} = 23.7$ AU. Meech et al.~(2017) argued instead that the coma grains are small ($\sim$1 $\mu$m) and, using a sublimation model, inferred a nucleus radius $14 \le R_{\mathrm{N}} \le 80$ km. Both papers conclude that the activity is likely driven by the sublimation of a supervolatile ice (CO, CO$_2$, N$_2$, or O$_2$ according to Jewitt et al.~(2017) and CO according to Meech et al.~(2017)). Other, non-equilibrium processes are also possible.
In this paper, we present archival, serendipitous prediscovery observations of K2, and we explore the orbit of K2 using Monte Carlo simulations.
\section{\uppercase{Observations}}
We used the Solar System Object Image Search (SSOIS; Gwyn et al.~2012) of the Canadian Astronomy Data Centre (CADC), to find K2 in archival data. Ten $U$-band images were found, taken at the 3.6-m Canada-France-Hawaii Telescope (CFHT) on UT 2013 May 10, 12 and 13, using the MegaCam prime focus imager. The offsets from ephemerides by JPL Horizons and the Minor Planet Center (MPC) at the time of our prediscovery were enormous, $\sim$$+3\arcmin$ in RA and $-11\arcmin$ in declination for the former, and $\sim$$+1\arcmin$ in RA and $-3\arcmin$ in declination for the latter, but the rate of angular motion was fully consistent with both sources. Detailed descriptions of the observations and the image of the comet are given in Jewitt et al.~(2017). Judging from its non-stellar appearance ($\mathrm{FWHM} = 1\arcsec.5 \pm 0\arcsec.1$, compared to $\mathrm{FWHM} \approx 0\arcsec.9$ for nearby background stars), the comet was active in 2013, although it is slightly trailed in the data (0\arcsec.98 in length) because of the long exposure and its non-sidereal motion.
Unfortunately none of the available star catalogs provide $U$-band magnitude data for stars in the field of view (FOV) of the CFHT data. Therefore, we calibrated $U$-band magnitudes of field stars then adjacent to the comet using the Keck-I 10-m telescope. Observations on UT 2017 September 20 were taken using a $u'$-broadband filter (central wavelength 3404 \AA, full width at half maximum transmission 3750 \AA) under photometric conditions with the Low Resolution Imaging Spectrometer (LRIS; Oke et al.~1995). The image scale was 0\arcsec.135 pixel$^{-1}$. We calibrated field stars in the 2013 CFHT field using observations of photometric standard stars PG1648+536E and PG1633+099A from the catalog by Landolt (1992).
Using image search software documented in Clark (2014), we also identified K2 in archival images from the Catalina Sky Survey (CSS) taken as early as 2015 November (see Table \ref{tab_geo} for details, and Figure \ref{fig_K2_CSS} for images), although with barely detectable motion with respect to the background sources, because of low angular resolution. The detections were consolidated by checking red plates from the Second Digitized Sky Survey (DSS2-red) having a similar limiting magnitude but a much better resolution. The CSS images from 2015 and 2016 were taken through a 0.7-m f/1.8 Schmidt telescope equipped with an unfiltered 4K $\times$ 4K CCD having an image scale of 2\arcsec.50 pixel$^{-1}$ and a FOV of 8.1 deg$^{2}$. Those from 2017 were obtained through the same telescope but equipped with an unfiltered 11K $\times$ 11K CCD having a scale of 1\arcsec.50 pixel$^{-1}$ and a FOV of 19.4 deg$^{2}$. All the images use an individual exposure time of $t_\mathrm{exp} = 30$ s, except that the set from 2015 has $t_\mathrm{exp} = 45$ s. The point-source $V$-band limiting magnitude of the images is $\sim$20.
We also searched for the comet in images taken between UT 2014 May 12 and 15 by the Palomar Transient Factory (PTF) 1.2-m diameter telescope (Law et al.~2009). The images, with an exposure time of 60 s and a FOV of $0\degr.6 \times 1\degr.1$, have a point-source $V$-band limiting magnitude of $\sim$20.5 (Waszczak et al.~2017), and a scale of 1\arcsec.01 pixel$^{-1}$. We had no success in detecting the comet in the individual images. Neither could we see anything above the noise level of the background in a stacked image coadded from the consecutive four-days of data with registration on the calculated motion of the comet. More recent PTF archival data from 2015 and 2016 cover the region of the comet, however, they are still proprietary, and we have no access to them. Given what we found with the CSS archival data, the comet should be detectable therein.
Prediscovery observations by the Pan-STARRS survey were reported as early as UT 2014 Mar 20 (Meech et al.~2017). We made no attempts to identify K2 in archival images prior to the CFHT observations in 2013, because the SSOIS inquiry shows no serendipitous observations from sufficiently large telescopes or with sufficiently long exposure times.
\section{\uppercase{Results}}
\label{sec_rslt}
We performed aperture photometry on all the available images. Using an aperture 2\arcsec.3 in radius, we obtained the apparent magnitude of K2 in a stacked image, coadded from all the CFHT individual frames from UT 2013 May 10, 12 and 13 with registration on the apparent motion of the comet, as $m_U = 22.83 \pm 0.08$. This value benefits from the Keck calibration data described above and supercedes the coarse estimate ($m_U = 23.7 \pm 0.3$) by Jewitt et al.~(2017).
We also performed aperture photometry on the CSS images. Since they were obtained without a photometric-standard filter, we had to first determine the zero-points of the images ($ZP$) by introducing $k_\mathrm{c}$, which is the color term satisfying the following equation
\begin{equation}
m_{\ast,V} - ZP_\lambda + 2.5 \log \left( \frac{\mathcal{F}_{\ast}}{t_\mathrm{exp}} \right) - k_\mathrm{c} \left( m_{\ast,B} - m_{\ast,V} \right) = 0
\label{eq_zp},
\end{equation}
\noindent where $m_{\ast, \lambda}$ is the star magnitude in some bandpass, and $\mathcal{F}_{\ast}$ is the signal in ADU of the star measured within photometric apertures of 15\arcsec.0 and 12\arcsec.0 in radius, respectively for the images taken by the old and new CCDs, which are approximately twice the FWHM of the field stars. The sky background was computed in annuli having inner and outer radii of $\sim$3$\times$ and $5 \times$ FWHM for the data, respectively. We utilised the AAVSO Photometric All-Sky Survey Data Release 9 (APASS-DR9; Henden et al.~2016) and reduced the zero-points of all the CSS images in the $V$ band from the least-squares fit (see Figure \ref{fig_CSS_ZP} as an example). The color of the comet ($m_U - m_V = 1.11 \pm 0.03$, $m_B - m_V = 0.74 \pm 0.02$, $m_V - m_R = 0.45 \pm 0.02$, from our Keck observation), is assumed to be unchanged. Then the measured flux of comet K2, which was obtained by applying the same photometric aperture and sky annulus that we used for stars, is converted to the apparent $V$-band magnitude. The errors stem mainly from the uncertainty in the determination of the zero-points ($\sim$0.2 mag), as well as the low signal-to-noise ratio of the comet.
Our results are summarised in Table \ref{tab_mag}. The temporal evolution of the apparent magnitude of K2 is shown in Figure \ref{fig_lc_2017K2}(a), in which we have included photometry from Meech et al.~(2017).\footnote{Magnitude data by Meech et al.~(2017) from the prediscovery Pan-STARRS observations were converted from Sloan-$r'$ to $V$-band magnitude using a transformation equation from Jordi et al.~(2006).} Note that different sizes of photometric apertures have been employed (see Table \ref{tab_mag}). Meech et al.~(2017) scaled their measurements by means of curves of growth to a set that would have been obtained using an aperture of 5\arcsec.0 in radius, which is much smaller than aperture sizes we applied to the CSS images. As the comet is a diffuse source, $\sim$9\arcsec~in radius in June 2017 (Jewitt et al.~2017), we expect that their measurements systematically underestimate the brightness (as is evident in Figure \ref{fig_lc_2017K2}). The closer the comet, the more significant is the difference. For example, for a steady-state coma with sufficient signal-to-noise ratio, the aperture correction in the prediscovery Pan-STARRS observations from $\mathit{\Delta} = 22.2$ to 16.3 AU results in a difference of $\sim$0.3 mag, with the more recent photometry being too faint. Indeed, this phenomenon can be readily seen in Figure \ref{fig_lc_2017K2}(a). However, our attempts to correct the Pan-STARRS photometry to an aperture of fixed linear (as opposed to angular) radius based on the surface brightness profile obtained in Jewitt et al.~(2017) failed to give satisfactory results. We simply decided not to perform aperture corrections.
In order to investigate the intrinsic brightness of the comet, the effect of the varying viewing geometry ought to be eliminated, so we compute the absolute magnitude from
\begin{equation}
m_V \left(1,1,0 \right) = m_V \left( r_\mathrm{H}, \mathit{\Delta}, \alpha \right) - 5 \log \left(r_\mathrm{H} \mathit{\Delta} \right) + 2.5 \log \phi \left( \alpha \right)
\label{eq_H},
\end{equation}
\noindent where $\alpha$ is the phase angle, and $\phi (\alpha)$ is the phase function of the coma. For the latter, we use the empirical phase function of dust (Marcus 2007; Schleicher \& Bair 2011; \url{http://asteroid.lowell.edu/comet/dustphase.html}), and normalised at $\alpha=0$\degr. The result is shown in Figure \ref{fig_lc_2017K2}(b). We then estimate the effective cross-section, $C_\mathrm{e}$, using
\begin{equation}
C_\mathrm{e} = \frac{\pi r_{\oplus}^2}{p_{V}} 10^{-0.4 \left[m_{V} \left(1,1,0 \right) - m_{\odot,V}\right]}
\label{eq_Xsect},
\end{equation}
\noindent where $r_\oplus \approx 1.5 \times 10^{8}$ km is the mean Sun-Earth distance, $p_{V}$ is the $V$-band geometric albedo, assumed to be $p_V = 0.04 \pm 0.02$ (see Lamy et al.~2004), and $m_{\odot,V} = -26.74$ is the $V$-band apparent magnitude of the Sun. We plot the temporal variation of the effective cross-section of K2 in Figure \ref{fig_xs_2017K2}, and summarise our photometry in Table \ref{tab_mag}.
\section{\uppercase{Discussion}}
\subsection{Dust Dynamics}
\label{sec_dd}
We attempted to constrain properties of dust grains of comet K2 in the high-resolution post-discovery \textit{HST} observation by Jewitt et al.~(2017) using our Monte Carlo model, in which dust grains are released from the nucleus with a range of non-zero initial velocities, and are subsequently subject to the solar radiation pressure force and the gravitational force from the Sun, whose ratio is denoted as $\beta$. The grain size is related to $\beta$ by $\beta \propto \left(\rho_{\mathrm{d}} \mathfrak{a} \right)^{-1}$, where $\mathfrak{a}$ is dust grain size, and $\rho_{\mathrm{d}}$ is bulk density. Together with the release time of the dust particles from the observed epoch $\tau$, their trajectories can be uniquely determined. We assumed that the dust particles follow a simple power-law size distribution, i.e., $\mathrm{d} n \left(\mathfrak{a} \right) \propto \mathfrak{a}^{-3.5} \mathrm{d}\mathfrak{a}$, where $\mathrm{d} n $ is the number of dust grains having radii between $\mathfrak{a}$ and $\mathfrak{a} + \mathrm{d} \mathfrak{a}$, and that their ejection speeds are described by an empirical relationship $v_{\mathrm{ej}} \propto \mathfrak{a}^{-0.5}$. Similar Monte Carlo models have been widely applied elsewhere (e.g., Fulle 1989; Ishiguro 2008; Moreno 2009; Ye \& Hui 2014; Hui et al.~2017).
We tested our simulation with different combinations of minimum, $\mathfrak{a}_{\min}$, and maximum, $\mathfrak{a}_{\max}$, particle radius, earliest dust release time from the observed epoch, $\tau_{0}$, and $v_{\mathrm{ej}}$. A successful model should be able to match the observed morphology of the coma. The model is insensitive to the size of the largest dust grains, $\mathfrak{a}_{\max}$, because such particles are rare and carry a negligible fraction of the total scattering cross-section for the power-law index assumed. In Figure (\ref{fig_mdl_17K2}) we show two models computed using $\mathfrak{a}_{\max} = 2$ mm, $v_{\mathrm{ej}}$ = 1.9 m s$^{-1}$ and $\tau_{0} = 1500$ days, consistent with the CFHT observation that K2 has been active at least since 2013 May. The remaining dust parameters can then be obtained without much ambiguity. By inspection, we determine that large dust grain (radii of $0.5 \lesssim \mathfrak{a} \lesssim 2$ mm) models closely simulate the morphology of comet K2 in the \textit{HST} data (c.f.~the middle- and right-hand panels of Figure \ref{fig_mdl_17K2}). On the other hand, models including smaller particles (radii of $0.01 \lesssim \mathfrak{a} \lesssim 2$ mm) show a clear radiation-pressure-swept tail which is not present in the data (c.f.~left and right-hand panels of the Figure). The Monte Carlo models thus support the inference made by Jewitt et al.~(2017) to the effect that the coma is dominated by submillimeter particles, but contradicts the one by Meech et al.~(2017), who assumed 2 $\mu$m-sized dust particles in their sublimation model. We found that this conclusion cannot be mitigated even if more recent ejection times, $\tau_{0}$, and higher ejection speeds are adopted. To conclude, the observed dust particles of comet K2 must be large, at least submillimeter sized, to avoid the formation of an observable, radiation pressure swept tail.
\subsection{Mass Loss}
\label{sec_ml}
There is a large scatter in the absolute magnitudes (see Figure \ref{fig_lc_2017K2}(b)) in part due to the aperture issue discussed in Section (\ref{sec_rslt}). Nevertheless, the comet appears to brighten in data from 2017 compared to 2013. The brightening corresponds to a maximum possible increase in the scattering cross-section $\Delta C_\mathrm{e} \approx \left(4.5 \pm 2.1 \right) \times 10^{4}$ km$^{2}$. The mean mass-loss rate of the comet, denoted as $\overline{\dot{M}}$, can be estimated from
\begin{equation}
\overline{\dot{M}} = \frac{4\rho_{\mathrm{d}} \Delta C_\mathrm{e} \bar{\mathfrak{a}}}{3\Delta t}
\label{eq_mloss},
\end{equation}
\noindent where $\rho_{\mathrm{d}}$ and $\bar{\mathfrak{a}}$ are the bulk density and the mean radius of the dust grains, respectively. With a nominal $\rho_{\mathrm{d}} = 0.5$ g cm$^{-3}$, $\bar{\mathfrak{a}} \approx 1$ mm, and $\Delta t \approx 1.2 \times 10^{8}$ s, Equation (\ref{eq_mloss}) yields $\overline{\dot{M}} \approx \left(2.4 \pm 1.1 \right) \times 10^{2} $ kg s$^{-1}$. The uncertainty only incorporates the error from photometry together with the error in albedo. Our estimate for the mass-loss rate is larger than $\overline{\dot{M}} \sim 60$ kg s$^{-1}$ by Jewitt et al.~(2017), mainly because we adopted a larger mean grain size based on our Monte Carlo simulations. Given the fact that the maximum dust dimension cannot be confidently constrained and that there are many other unknowns, such a difference is not significant.
A lower limit to the size of the nucleus of comet K2 can be estimated by assuming that the activity is supported by equilibrium sublimation of exposed ices. We solve the energy equilibrium equation between insolation, thermal emission and sublimation,
\begin{equation}
\left(1 - A_{\mathrm{B}} \right) S_{\odot}\left( \frac{r_{\oplus}}{r_{\mathrm{H}}} \right)^{2} \cos \zeta = \epsilon \sigma T^{4} + L\left( T \right) f_{\mathrm{s}} \left( T \right)
\label{eq_sub},
\end{equation}
\noindent in which $A_{\mathrm{B}}$ is the Bond albedo, $S_{\odot} = 1361$ W m$^{-2}$ is the solar constant, $\cos \zeta$ is the effective projection factor for the surface ($\cos \zeta = 1$ for a subsolar scenario, and $\cos \zeta = 1/4$ corresponding to an isothermal nucleus), $\epsilon$ is the emissivity, $\sigma = 5.67 \times 10^{-8}$ W m$^{-2}$ K$^{-4}$ is the Stefan-Boltzmann constant, $T$ is the surface temperature in K, and $L$ (in J kg$^{-1}$) and $f_{\mathrm{s}}$ (in kg m$^{-2}$ s$^{-1}$) are the latent heat of the sublimating material, and the mass flux of the sublimated ice, respectively, both as functions of temperature. The heat conduction towards the nucleus interior is ignored. For simplicity, we assign $\epsilon = 0.9$ and $A_{\mathrm{B}} = 0.01$ (e.g., Buratti et al.~2004) and analyse the sublimation of CO (whose volatility is representative of other potential supervolatiles like N$_2$ and O$_2$) and CO$_{2}$ at the time-averaged heliocentric distance $r_{\mathrm{H}} = 17.1$ AU during the period 2013--2017.
We adopted empirical thermodynamic parameters of CO and CO$_2$ respectively listed in Prialnik et al.~(2004) and Cowan \& A'Hearn (1979), solved Equation (\ref{eq_sub}) and obtained $4.9 \times 10^{-6} \le f_{\mathrm{s}} \le 2.0 \times 10^{-5}$ kg m$^{-2}$ s$^{-1}$ for CO, and $8.6 \times 10^{-11} \le f_{\mathrm{s}} \le 2.2 \times 10^{-6}$ kg m$^{-2}$ s$^{-1}$ for CO$_{2}$, where the lower limits correspond to isothermal sublimation and the upper ones are from subsolar sublimation. In order to supply the mass-loss rate inferred from photometry, the minimum surface area, $A_{\mathrm{s}} = \overline{\dot{M}} / f_{\mathrm{s}}$, has to be in the range $12 \lesssim A_{\mathrm{s}} \lesssim 48$ km$^2$ for sublimation of CO, and $1.1 \times 10^{2} \lesssim A_{\mathrm{s}} \lesssim 2.8 \times 10^{6}$ km$^2$ for CO$_{2}$. These are equivalent to equal-area circles of radii $R_{\mathrm{N}} \sim \sqrt{A_{\mathrm{s}} / \pi} \gtrsim 2$ km and $R_{\mathrm{N}} \gtrsim 6$ km, respectively. Given that the upper limit to the nucleus radius from the HST measurement is $R_{\mathrm{N}} <$ 9 km, we see that sublimation of CO (and N$_2$, O$_2$) is easily capable of supplying the coma even if only a small fraction of the nucleus surface is active, while CO$_2$ must be sublimating from near the subsolar point, if it is present.
We then proceed to estimate the critical grain size, $\mathfrak{a}_{\mathrm{c}}$, of dust particles, which can be lifted off from the surface by the gas-drag force $F_{\mathrm{D}} = C_{\mathrm{D}} \pi \mathfrak{a}^2 \mu \mathfrak{m}_{\mathrm{H}} N v_{\mathrm{th}}^{2}$, where $C_{\mathrm{D}}$ is the dimensionless drag coefficient, $\mu$ is the molecular weight ($\mu = 28$ for CO, and $\mu = 44$ for CO$_2$), $\mathfrak{m}_{\mathrm{H}} = 1.67 \times 10^{-27}$ kg is the mass of the hydrogen atom, $N$ is the number density of the molecule, and $v_{\mathrm{th}}$ is the thermal speed of the gas. By equating the gas-drag force and the gravitational force at the surface and ignoring spinning of the body, with simple algebra we derive the critical grain dimension as
\begin{equation}
\mathfrak{a}_{\mathrm{c}} = \frac{9C_{\mathrm{D}} \overline{\dot{M}}}{32 \pi^2 G \rho \rho_{\mathrm{d}} R^3} \sqrt{\frac{2 k_{\mathrm{B}} T}{\pi \mu \mathfrak{m}_{\mathrm{H}}}}
\label{eq_ac},
\end{equation}
\noindent where $G = 6.67 \times 10^{-11}$ m$^{3}$ kg$^{-1}$ s$^{-2}$ is the gravitational constant, $\rho$ is the density of the nucleus, and $k_{\mathrm{B}} = 1.38 \times 10^{-23}$ J K$^{-1}$ is the Boltzmann constant. Assuming a unity gas-to-dust production ratio, along with $C_{\mathrm{D}} = 1$ and $\rho = \rho_{\mathrm{d}}$, Equation (\ref{eq_ac}) yields $\mathfrak{a}_{\mathrm{c}} \lesssim 4$ mm for sublimation of CO, and $\mathfrak{a}_{\mathrm{c}} \lesssim 0.2$ mm for CO$_2$, which is in line with Jewitt et al.~(2017). Note that our dust model suggests the size of the dust grains $\mathfrak{a} \gtrsim 0.5$ mm. We thus prefer CO (and materials of similar volatility) sublimation as the cause of the activity, but since there are many approximations in our model (e.g.~the neglect of rotation and the neglect of contact forces at the nucleus surface, see Gundlach et al.~2015), we feel that it would be premature to rule out CO$_2$ as the activity driver.
\subsection{Orbital Evolution}
\label{sec_orb}
We next examine the dynamical evolution of comet K2 in an attempt to understand its recent history. We downloaded the astrometric measurements of the comet from the MPC, which include our own astrometry from the prediscovery archival images. The measurements were debiased following Farnocchia et al.~(2015). The code \textit{EXORB9}, a part of the \textit{SOLEX12} package developed by A. Vitagliano, was exploited for orbit determination. Weights on each set of observations were adjusted to approximately accommodate ad hoc astrometric residuals whenever they were found aggressive. Twenty-four observations with residuals greater than 1\arcsec.5 either in RA or declination were discarded, leaving 336 observations (93\% of the total number) to be fitted by orbit determination. An optimised solution was thereby obtained, having a weighted rms of 0\arcsec.472. Our derived orbital elements are generally similar to those in the solution by JPL Horizons (Table \ref{tab_orb}), despite different choices of the weighting scheme and the threshold for filtering bad-residuals astrometry (D. Farnocchia, private communication). We then generated 10$^4$ clones of the nominal orbit according to the associated covariance matrix of the orbital elements, and performed backward N-body integration in \textit{MERCURY6} (Chambers 1999) using the 15th-order RADAU integrator (Everhart 1985) for the past 1 kyr, and the hybrid symplectic algorithm for the past 1 Myr to investigate the dynamical evolution of K2.\footnote{Switching to the hybrid integration scheme is a measure to reduce the computation time, at the cost of losing some accuracy. Since we have no interest in examining orbits of individual clones, but are looking at the total statistics, the result is not influenced.} Gravitational perturbations from the eight major planets and Pluto, post-Newtonian corrections (Arminjon 2002), and the influence of the galactic tide, which is a major perturber of the Oort cloud (e.g., Heisler \& Tremaine 1986; Fouchard et al.~2005), were included in the simulation. The possible distant giant planet claimed by Trujillo \& Sheppard (2015) and Batygin \& Brown (2016) was not considered, since the evidence for this body remains equivocal (Shankman et al.~2017). Neither have we included stellar perturbations, although the frequency of encounters with stellar systems passing within 1 pc of the Sun is estimated to be as many as $11.7 \pm 1.3$ Myr$^{-1}$ (Garc\'{i}a-S\'{a}nchez et al.~2001), yet 73\% of the encounters are with M dwarfs having low masses ($\lesssim 0.4 M_{\odot}$, where $M_{\odot}$ is the solar mass).
We did not incorporate possible non-gravitational acceleration of K2 in the orbital solution. To test the impact of this neglect, we employed \textit{EXORB9} to repeat the aforementioned procedures to solve for non-gravitational parameters $A_{j}$ ($j=1,2,3$) as defined in Marsden et al.~(1973) but obeying an empirical momentum-transfer law from sublimation of CO and CO$_2$ in a hemispherical scenario, following the method in Hui \& Jewitt (2017). No detection of non-gravitational acceleration was made above the formal uncertainty levels (well below $1\sigma$). Solving for the non-gravitational parameters barely helps reduce the rms of the fit (to 0\arcsec.471), justifying our omission of the non-gravitational effect.
The orbital evolution of K2 in terms of the reciprocal of the semimajor axis ($a^{-1}$), perihelion distance $q$, eccentricity $e$ and inclination $i$ in the past 1 kyr is shown in Figure \ref{fig_oscele_1ky_17K2}. Note that the orbital elements are still referred to the heliocentric reference system. We can see that the ranges of $q$ and $i$ sway increasingly with time in the past 1 kyr, whereas $a^{-1}$ gradually approaches $\sim$10$^{-5}$ AU$^{-1}$ and $e$ tends to creep $< 1$. The examined orbital elements exhibit zigzagging oscillations with a dominant period of $\sim$11.9 yr, close enough to the orbital period of Jupiter to indicate non-negligible gravitational perturbations from the gas giant.
Now we move on to results from the backward integration for the past 1 Myr. Starting from now, we change the reference origin to the barycenter of the solar system. We obtain median values $a_{\mathrm{med}}^{-1} = \left(3.61 \pm 1.71 \right) \times 10^{-5}$ AU$^{-1}$ and $e_{\mathrm{med}} < 1$ from the clones (See Figure \ref{fig_baryorb_1Myr_17K2}, the assigned uncertainty is the standard deviation). Only 173 ($\sim$1.7\%) of the total clones have originally hyperbolic orbits and so we conclude that the comet is very unlikely to be of interstellar origin. Instead, K2 is probably from the Oort spike, which consists of a mix of dynamically new and old comets (Kr{\'o}likowska \& Dybczy{\'n}ski 2010; Fouchard et al.~2013). We cannot determine whether the comet is dynamically new or old from our backward integration, because the integration time (1 Myr) is shorter than the orbital period of the comet, $P = a^{3/2} \gtrsim 2$ Myr. Only from the region with $a^{-1} < 2.5 \times 10^{-6}$ AU$^{-1}$ in the Oort spike, are dynamically old comets completely absent (Kr{\'o}likowska \& Dybczy{\'n}ski 2017). As a result, whether K2 penetrated into the planetary region during the previous perihelion passage must be regarded as unsettled. Analysis of the forward integration of the orbit of K2 is not performed, because we have concern that intensified sublimation activity as the comet approaches the Sun will intensify non-gravitational effects.
Although detected as early as 2013, K2 managed to repeatedly escape detection by the major sky surveys. Why was comet K2 not discovered much earlier? The two important reasons, we suspect, are the high inclination and the low rate of its angular motion. The majority of sky surveys are optimised for making discoveries of small bodies that move at much higher speeds, such as near-Earth and main-belt asteroids. Furthermore, angualr resolution has been generally sacrificed for wider-FOV coverage, making discovery of slow-moving objects even more difficult. Although surveys like the Outer Solar System Origin Survey\footnote{\url{http://www.ossos-survey.org/about.html}} are dedicated to transneptunian objects, and should have had capability to detect objects moving as slowly as K2, they mainly search along the ecliptic plane.
\section{\uppercase{Summary}}
Key conclusions of our study about comet C/2017 K2 (PANSTARRS) are summarised as follows.
\begin{enumerate}
\item The comet was recorded serendipitously by the CFHT and the CSS on many occasions since 2013. At $r_{\mathrm{H}} = 23.7$ AU, K2 is the most distant comet ever observed on the way to perihelion.
\item The combined archival photometry suggests that the activity of the comet has been slowly increasing since 2013, as it approaches the Sun.
\item By means of our Monte Carlo simulation of the dust motion, we confirm that dust properties estimated by Jewitt et al.~(2017) during the \textit{HST} observation are valid, i.e., predominant dust grains of the comet are $\gtrsim 0.5$ mm in radius, with ejection speeds of $\sim$1--3 m s$^{-1}$, and have been released in a continuous manner since 2013 May.
\item By assuming a cometary albedo $p_V = 0.04 \pm 0.02$, the mass-loss rate of comet K2 during the period of 2013--2017 was estimated to be $\overline{\dot{M}} \approx \left(2.4 \pm 1.1 \right) \times 10^{2}$ kg s$^{-1}$, which requires a minimum active surface area of $\sim$12 km$^{2}$ if the activity is driven by sublimation of CO, and $\sim$110 km$^{2}$ for CO$_2$. The nucleus must be at least of kilometer-size to sustain the observed activity by sublimation.
\item Monte Carlo simulations of the pre-entry orbit of the comet give original (1 Myr ago) reciprocal semimajor axis $a_{\mathrm{med}}^{-1} = \left(3.61 \pm 1.71 \right) \times 10^{-5}$ AU$^{-1}$ (referred to the barycenter of the solar system). We find that some 98\% of orbital clones originate from within the Oort spike.
\end{enumerate}
\acknowledgements
{
We thank Eric Christensen, Davide Farnocchia, Aldo Vitagliano, and Quan-Zhi Ye for their generous help, and the anonymous referee for a speedy review. This research is in part based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. The facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency, and the AAVSO Photometric All-Sky Survey (APASS), funded by the Robert Martin Ayers Sciences Fund, were used. Some of our data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The products of the Digitized Sky Surveys, which were produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166, were employed in this research. The NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the NASA, has been made use of. This work is funded by a grant from NASA to D.J.
}
\clearpage
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1,116,691,497,737 | arxiv | \section{Introduction} \label{sec:intro}
The field of Galactic astronomy has entered a transformative era. Large-scale surveys, such as APOGEE, GAIA and GALAH, are providing millions of high-quality spectroscopic and astrometric measurements of stars across the Milky Way \citep{Rave2006, Apogee,Galah,Lamost,Gaia}. Future large-scale surveys, which will release even more high-quality data, are on the horizon \citep{4most,SDSS5,Weave}.
In this landscape of high-volume high-quality stellar astronomy, fully extracting the scientifically relevant information from stellar spectra remains a difficult problem. Classically, this has been done by comparing observations to synthetic spectra generated from theoretical models \citep[e.g.][]{aspcap}. However, the precision with which stellar labels can be derived under such an approach is ultimately limited by the faithfulness with which synthetic spectra reproduce observations. Because of computational constraints and gaps in knowledge, synthetic spectra do not perfectly match observations, something sometimes referred to as the ``synthetic gap" \citep{cyclestar}. Computational models used to generate synthetic spectra use incomplete stellar line lists and usually must make simplifying assumptions, such as, that stellar atmospheres are one-dimensional, in hydrostatic equilibrium, and in local thermodynamic equilibrium. In addition, even beyond these issues, observations are affected by further systematics such as telluric lines introduced by the earth's atmosphere \cite[e.g.][]{SDSSdataproducts} and telescope imperfections/aberrations.
Ultimately this synthetic gap limits our ability to extract information from stellar spectra. In \cite{CramerRaoTing,HowManyElements} it was shown that stellar spectra contain more chemical information than is captured in bulk metallicity and $\alpha$-enhancement alone. The precision of derived individual stellar abundances from large surveys, however, may be limited by an inability to fully extract information from spectra, given approximate models, rather than by the signal to noise of observations.
\mkn{made some edits here, OK?}
This is problematic because a lot of interesting science requires measuring chemical similarity between stars with a precision beyond that currently delivered by large stellar surveys modelling pipelines. In particular, high precision chemical measurements are needed for strong chemical tagging \citep{TaggingFreeman}. This is an ambitious Galactic archaeology endeavour aiming to identify stellar siblings - stars born from the same molecular cloud - using chemical information derived from spectroscopy long after clusters gravitationally dissipate. In practice, whether such chemical tagging is theoretically possible at scale is still an open question, but may be answered with large-scale surveys like GALAH \citep{Galah}. For this form of chemical tagging to be successful, stellar siblings must share a near-identical chemical composition with sufficient variability in chemical compositions between clusters. Even if strong chemical tagging, reveals itself to be impossible at large scale, precise chemical similarity measurements would still be useful in reconstructing the broad nature of our galaxy's evolution \citep[e.g.][]{Coronado2020, Kamdar2020}.
These issues motivate the development of methods capable of extracting information from stellar spectra. and overcoming the synthetic gap between observations and theoretical models. Several data-driven methods have been developed for this purpose. Methods such as those proposed \cite{Ness2015,Casey2016,Leung2018,Ting2019,cyclestar} allow for improving precision of stellar labels through leveraging data-driven interpolators between stellar spectra and labels, reducing the impact of noise and systematics on derived parameters. However, as such approaches still rely on synthetic spectra they do not fully alleviate issues with systematic errors from mismatching theoretical spectra. Recently, methods for finding chemically similar stars directly from stellar spectra without reliance on synthetic spectra have been developed \cite{Bovy2016,pca_dimensionality,Cheng2020TestingTC}. This category of method works by removing the effect of non-chemical parameters on stellar spectra, thus isolating the chemical information within the spectra. Such approach are not without drawbacks. Although they remove the dependency on synthetic models, they still require a comprehensive and precise determination of all non-chemical factors of variation, additionally they must make simplifying assumptions regarding the cross-dependencies between chemical and non chemical factors of variations which may have an impact on accuracy.
In this paper, we present a new approach for identifying chemically similar stars from spectroscopic data which we name Relevant Scaled Component Analysis" (RSCA) because of it's similarities with Relevant Component Analysis \citep{relevantComponentAnalysis}. Our approach is rooted in the machine learning subfield of metric learning. Instead of estimating individual chemical abundances, our approach projects spectra directly into a lower dimensional subspace in which distances between spectra are made to encode a useful notion of chemical similarity between stars. Crucially, as our approach for transforming stellar spectra does not rely at any stage on synthetic spectra or quantities derived from these, its performance is not hindered by inaccuracies in stellar modelling allowing for superior performance.
A novelty of our work is that instead of using synthetic spectra to learn this notion of chemical similarity, we make use of spectra from know open-clusters with open-cluster membership information. Open clusters are groups of stars born together that remain gravitationally bound after birth and up to present days. They are relatively rare as most stellar clusters dissipate rapidly after birth \citep{openclusterreview}. However, they are extremely useful tools in modern Galactic astronomy. In particular, open clusters, which can be identified using astrometry, display near-identical chemical abundances although small scatter may exist at the 0.01 to 0.02 dex level \cite[e.g.][]{Bovy2016,AtomicDiffusionLiu, Ness2018}. \mkn{this statement is not true there is literature that reports these are homogeneous only to the 0.01-0.02 level - i.e. Bovy2017?, Ness2018, Liu2018, de Souto 2017?, Da Silva, 20XX. Theoretically, they are expected to potentially show some small intrinsic scatter, as a function of initial mass and mixing i.e. Krumholtz and Bland-Hawthorn 20XX. This is not a problem for this paper, but we should correct the prior sentence and acknowledge that they represent common birth sites and with that the chemical variability that is typical at fixed birth place and time - whatever that is}\ddm{Kindoff curious about how can we be confident that these levels of homogeneity are inconsistent with instrumental systematics and synthetic spectra misspecification.}. \mkn{I agree it might be some systematic but I think leave as is as we have referenced appropriately and flows well - note a scatter is also predicted under theory i.e Bland-Hawthorn and Krumholz 2010.} Open clusters have found many uses in modern astronomy. For example to obtain high-precision measurements of the radial abundance gradients in the Milky Way \cite{abundancegradientold,abundancegradientnew} or to benchmark and calibrate stellar survey abundance measurements \cite{aspcap}. Here, we use open clusters as a golden standard for learning a notion of chemical similarity. In our approach, we take the viewpoint that if open clusters are indeed chemically homogeneous, then a successful metric for encoding chemical similarity will be one in which open-cluster stellar siblings are highly clustered.
Our algorithm, RSCA, has several properties that make it suitable for the task at hand, of measuring chemical similarity between stars:
\begin{itemize}
\item It is fully \textit{data-driven}. Chemical similarity is measured without any reliance or dependency on theoretical models. This offers a measure of chemical similarity that is independent from the systematics introduced in traditional stellar modelling \citep[e.g.][]{Jofre2017}, and offers a means of validating existing discoveries.
\item It is \textit{computationally efficient}. Processing spectra from a large stellar survey \mkn{how large? 10$^5$, 10$^6$ stars?} can be done in minutes \mkn{is this because it is all linear? worth mentioning}. The most computationally intensive step of the approach is a Principal Component Analysis decomposition.
\item It is \textit{interpretable}. In its current formulation, measuring chemical similarity using our method amounts to evaluating Euclidean distances between stars projected on a hyperplane of the stellar spectra space.
\item It is \textit{precise}. We find the method, using spectra, to be more effective at identifying stellar siblings from open-clusters than is possible using stellar abundance measurements. We believe this to be in large part because our method bypasses the synthetic gap introduced by spectral modelling. Furthermore, our experiments suggest that the performance could be further improved, for example, with a larger dataset of open-cluster stars \mkn{or taking into account the error on the flux which we do not currently do?}.
\end{itemize}
The paper is organized as follows. We briefly introduce Principal Component Analysis in Section \ref{sec:PCA}, which is a core component of our algorithm. In Section \ref{sec:metric}, we outline the conceptual ideas behind our approach for measuring chemical similarity. In Section \ref{sec:algo}, we dive deeper, and present our algorithm, RSCA. This is implemented using open-clusters observed by the APOGEE survey in Sections \ref{sec:experiments}, and evaluated in light of the field distribution of stars. Its trade-offs and implications are discussed in Section \ref{sec:discussion}.
\mkn{edits throughout intro - minor changes, pls check}
\section{Principal Component Analysis}\label{sec:PCA}
The principal components of a dataset $X$, of shape $N_{D} \times N_{F}$ containing $N_{D}$ data points and $N_{F}$ features, are an ordered orthogonal basis of the feature space with special properties. In the principal component basis, basis vectors are ordered by the amount of variance they capture. They have the property that for any j, the hyperplane spanned by the first j-axes of the basis is the j-dimensional hyperplane, which maximally captures the data variance.
The principal component basis corresponds to the unit-norm eigenvectors of the covariance matrix of $X$ ordered by eigenvalue magnitude. This can be obtained through diagonalization of the covariance matrix. The principal component basis can also be formulated as the maximum likelihood solution of a probabilistic latent model which is known as Probabilistic Principal Component Analysis (PPCA) (see \cite{bishop}). This probabilistic formulation is useful in that it enables one to obtain the principal components for a dataset containing missing values by marginalizing over these.
Principal Component Analysis (PCA) is an algorithm for dimensionality reduction that consists of performing a change-of-basis to the principal component basis and discarding all but the \mkn{in this paragraph don't you mean j not k - you use 'j' above? a bit repetitive here perhaps as this paragraph intersects with the prior paragraphs a fair bit?}\ddm{I though it would be better to swap letters because j refers to any PCA dimension while k refers specifically to the number of PCA components (which is a hyperparameter)} \mkn{can we discuss, it is still the j-largest principle components, right? You can use k as the hyperparam symbol, but I think for consistency should be j-largest (?) } k-largest principal components. In PCA, the number of principal components used k is a hyperparameter controlling the trade-off between the amount of information preserved in the dataset $X$ after compression and the degree of compression.
As we will make use of further in this paper, the principal components also allow for generating a sphering transformation. This is a linear transformation of the dataset to a new representation, in which the covariance matrix of the dataset $X$ is the identity matrix. Sphering using PCA is carried out by performing a change-of-basis to a modified principal component basis in which the principal components are divided by the square root of their associated eigenvalues.
\section{Chemical similarity as metric learning}\label{sec:metric}
Our goal is to identify chemically similar stars from their spectra, for stars that span a range of stellar parameters (i.e. effective temperatures and surface gravities). We therefore build our algorithm to identify stars originating from common open clusters, using open cluster spectra.
For our method, we assume that open clusters are close to chemically homogeneous because of their common birth origin \citep{Ness2018} but are not special in any other way (at least in terms of their spectra). That is to say, we assume that the only information within spectra useful for recognizing open-clusters are the chemical features of the spectra, and so that a model which identifies open-clusters from the spectra will need to extract the chemical information within spectra
We frame the task of building such a model recognizing open-clusters as a metric learning task. That is to say, we build a data-driven model converting stellar spectra into a representation in which Euclidean distances convey the uncalibrated probability of stars originating from a shared open-cluster. To accomplish this, the training objective of our data-driven algorithm can be understood as transforming stellar spectra into a representation in which the distances between intracluster stars is minimized and the distance between intercluster stars is maximized.
Distances in the representation resulting from such an optimizing procedure will organically quantify the chemical similarity of stars. Non-chemical factors of variation, such as for example stellar temperatures and instrumental systematics, will not contribute to the representation as their presence would make distances between stellar siblings larger. Instead, such a representation will only contain those factors of variation of spectra that are discriminative of open-clusters, ie the chemical factors of variation. Crucially, chemical factors of variation will contribute to distances in the representation in proportion to how precisely they can be estimated from stellar spectra. Stronger chemical features like metallicity, which can be estimated with higher precision, will be more strongly amplified than weaker chemical features.
The utility of this data-driven approach is that it is independent of imperfect model atmosphere approximations and other issues affecting synthetic spectra. This provides a high fidelity technique to turn to specific applications within Galactic archaeology, such as chemical tagging of stars that are most chemically similar \citep{TaggingFreeman}.
\mkn{reworked paragraph below:}
Our approach only uses the spectra of stars. These spectra contain chemical information - as found in spectral lines - which is predictive of the age and birth abundances of a star. The assumption underpinning our work is that this chemical information will be the only information within stellar spectra useful for distinguishing open-clusters and so will be the only information captured by our model. If this assumption is true, then the representation induced by the model will measure a form of chemical similarity between stellar spectra.
However, since open-clusters, in addition to sharing a common age and near-identical birth abundances, are also gravitationally bound, they can be identified from their spatial proximity if such information is available in the spectra. As such spatial information does not robustly transfer towards identifying dissolved clusters, it mustn't be captured by our model. In this work, we apply our algorithm to pseudo-continuum normalized spectra with diffuse interstellar bands masked, which we assume not to contain any information about spatial location so as for our representation after training to only contain chemical information. That pseudo-continuum normalized spectra do not encode any spatial information is plausible since after continuum normalization, the spectrum should not contain any information about stellar distance. With the impact of reddening removed and diffuse interstellar bands masked, a spectrum should also not contain any information about the stellar extinction and interstellar medium along the line of sight of the star. We examine the validity of these assumptions in latter sections. Ultimately, it is worth emphasizing that our method only exploits features proportionally to their discriminativeness at recognizing open-clusters. Therefore, we can expect our model to not heavily rely on non-robust features (provided that these are significantly less informative than robust features). This also relies on our open cluster training data being representative of the parameter space that should be marginalized out; i.e. our model does not learn to associate inter cluster stars via \mbox{$\log g$}\ and \mbox{$T_{\rm eff}$}, which could happen if the evolutionary state of observed cluster stars was similar within clusters and different between clusters.
\mkn{do you also take in the error on each flux bin? Or not? let's state this explicitly, as spectra comes in such a range of SNR we have to address this. So this isn't clarified how this feeds into your algorithm, but are you using this ? When you get your Euclidean distance to determine similarity are you not using the error on the flux or is this already accounted for in the prior steps? could you clarify this is probably the biggest thing I am missing here. If you don't use this, okay, but I think we need a discussion of how this works over a range of SNR and a test that SNR is not one of the strongest inter-cluster association features - any euclidean distance will scale directly with SNR if not included?}\ddm{Wouldn't it be the case that open-cluster stars don't share a common SNR but a common observing time. I've done some checks and I don't think that our model is exploiting any SNR dependence.}
\begin{figure*}
\includegraphics[width=\linewidth ,page=7,trim={2cm 0cm 1cm 0cm},clip]{figures/diagram.pdf}
\caption{Schematic depiction of our algorithm. The algorithm proceeds by first encoding stellar spectra into a lower dimensional representation made two-dimensional for illustrative purposes. In this representation, stellar siblings - which are represented by same-coloured dots - are not initially identifiable by their Euclidean distance in the basis (represented by black arrows). The objective of the metric-learning algorithm (dashed blue) is to find a new basis in which distances are informative about which stars are stellar siblings. This objective is realized through three linear steps: a sphering transformation on the dataset, a reparametrization to a suitable basis, and a scaling of the basis vectors.}
\label{fig:diagram_algorithm}
\end{figure*}
\section{Algorithm}\label{sec:algo}
\subsection{Overview}
For the remainder of this paper, we denote $X_{\rm clust}$ as the matrix representation of a dataset containing spectra of known open-cluster stars and analogously define $X_{pop}$ as the matrix representation of a larger dataset of stellar spectra with unknown cluster membership. These matrices will be respectively of shapes $N_{\rm d_{clust}} \times N_{\rm b}$ and $N_{\rm d_{pop}} \times N_{\rm b}$, where $N_{\rm d_{clust}}$ is the number of open-cluster stars, $N_{\rm d_{pop}}$ the number of of stars in the large dataset and $N_b$ the number of spectral bins. We assume access to only a limited number of open-cluster stars such that $N_{\rm d_{clust}} < < N_{\rm d_{pop}}$. We also assume that the spectra are pseudo-continuum normalized spectra, with diffuse interstellar bands masked in a process following that described in Appendix \ref{appendix:masking}. Pseudo-continuum spectra are normalized rest-frame spectra in which the effects of interstellar reddenning and atmospheric absorption are removed, in a process described in \cite{Apogee}.
Before explaining our metric-learning algorithm in detail, we briefly provide upfront a high-level overview of it's inner workings. At its core, our algorithm RSCA works by individually scaling one-by-one the basis vectors of a general-purpose representation of stellar spectra. Basis vectors are scaled proportionally to their usefulness at identifying open clusters which is evaluated from a comparison between the variance amongst stellar siblings and amongst field stars along the basis vector. Not all basis representations are appropriate for this scaling. Since the scaling is done by measuring the variance along basis vectors it is only appropriate when these variances fully capture the variability amongst stellar spectra and amongst stellar siblings. So as to ensure that the variances do indeed capture the full variability, before the scaling procedure, we apply a preliminary set of transformation designed to transform to a basis in which the covariance amongst stellar siblings and the covariance amongst field stars are diagonal (matrices). As there are then no off-diagonal terms in the covariance matrices this ensures that the variances along basis vectors fully capture the covariance information amongst stellar siblings and amongst field stars.
RSCA accomplishes the transformation from stellar spectra to a metric-learning representation through a set of linear transformations applied to the spectra. Since the composition of linear transformations is itself a linear transformation, RSCA can also be parametrized as a linear transformation. We provide in Figure \ref{fig:diagram_algorithm} a graphical depiction of the linear transformations involved in the algorithm. Pseudo-code for the algorithm is also provided in Appendix \ref{appendix:pseudocode} and the full source code of our project which contains a python implementation for our algorithm is also publicly available online.
We now walk step-by-step through the successive linear-transformations involved in the RSCA algorithm.
\mkn{do you want to give this a name at all? i.e. rather than calling this `the algorithm' everywhere?}
\subsection{Encoding Spectra}
In the first step of our approach, denoted as \textbf``1) Compress Spectra" in Figure \ref{fig:diagram_algorithm}, Principal Component Analysis, applied to $X_{pop}$, is used to convert stellar spectra into a lower dimensional representation. This dimensionality reduction step serves to make the algorithm more data-efficient which is crucial given the small number of open-clusters within our dataset.
As some stellar bins are flagged as untrustworthy, we use Probabilistic Principal Component Analysis (PPCA), a variant of the PCA algorithm which can accommodate missing values. After finding principal components of $X_{pop}$, we compress the datasets by discarding all but the $D$ largest principal components, where D is a hyperparameter requiring tuning. Datasets $X_{pop}$ and $X_{clust}$ are encoded into $Z_{pop}$ and $Z_{clust}$ of shape $N_{\rm d_{pop}} \times N_{\rm D}$ and $N_{\rm d_{clust}} \times N_{\rm D}$.
\subsection{Metric Learning}
\subsubsection{Assumptions}
The metric-learning part of the algorithm consists of a set of three linear transformations to the PCA-compressed spectra. The success of the metric-learning algorithm relies on a series of assumptions about the PCA-compressed spectra being satisfied. Performance should be robust to within small departures from these assumptions, but will still ultimately be tied to how well these assumptions are respected.
First, we assume that the datapoints (ie stars) in $Z_{pop}$ are well approximated as being drawn from a multivariate Gaussian distribution. That is to say that if we define $\mu_{pop}$ and $\Sigma_{pop}$ as the mean and covariance of $Z_{pop}$, then the stars within $Z_{pop}$ can be assumed as being samples drawn from $z_{pop} \sim N(\mu_{pop},\Sigma_{pop})$.
Next we make the assumption that individual clusters are themselves approximately Gaussian in the PCA-compressed space. That is to say we posit that the members of open-clusters are well approximated as being samples drawn from a distribution $z_{clust} \sim N(\mu_{clust},\Sigma_{clust})$. Crucially, we make the assumption that all open-clusters share the same covariance matrix $\Sigma_{clust}$ and only differ in their mean $\mu_{clust}$.
Relating these assumptions back to Figure \ref{fig:diagram_algorithm}. Our set of assumption come down to assuming that in the panel succeeding ``1) Encode Spectra" every colour's markers, where each marker colour represents stellar siblings, follow a multivariate Gaussian with a common covariance matrix and that that the markers for field stars (not represented) are themselves also drawn from a multivariate Gaussian.
It is worth emphasizing that the most important of these assumptions is the assumption that the stars within clusters are distributed following a shared covariance matrix (ie clusters have the same shape irrespective of their location in the representation) is by far the most important. It is this assumption which allows a linear transformation - i.e. a transformation that acts the same across the whole representation- to be an effective approach for measuring chemical similarity. As an example of the role of this translation invariance assumption, in the "physical" representation of stellar spectra - formed by concatenating physical and chemical stellar parameters. This assumption of cluster translation invariance can be interpreted as assuming that the scatter amongst stars in physical and chemical parameters should be the same for all clusters irrespective of the clusters parameters which is a sensible assumption.
\ddm{Old version commented out}
\subsubsection{Sphering \& Reparmetrization}
The sphering and reparametrization steps of our algorithm are a set of two successive linear transformations serving to transform to a vector representation of stellar spectra in which $\Sigma_{pop}$ and $\Sigma_{clust}$, the covariance matrices associated to field stars and to clusters, are diagonal matrices. They proceed as follows:
The sphering step applied after the PCA compression and denoted in Figure \ref{fig:diagram_algorithm} as \textbf{``2) Sphere"}, consists in linearly transforming the vector representation of spectra such that the dataset $Z_{pop}$ has a covariance matrix of identity after transformation. This linear transformation takes the form of a sphering transformation applied to $Z_{pop}$, where in our experiments we use the PCA-sphering scheme.
In the reparametrization step of our algorithm, applied after the sphering transform, the covariance matrix $\Sigma_{clust}$ is diagonalized. As an intermediary step in doing so, a dataset approximately distributed according according to $N(0,\Sigma_{clust})$ is created from the open-cluster dataset by subtracting to every star's vector representation the mean representation of all star's belonging to the same cluster $\widehat{\mu}_{clust}$, such that each cluster becomes zero-centered. It is worth noting that as $\widehat{\mu}_{clust}$ is estimated from a limited number of samples it will not exactly match with the true $\mu_{clust}$ and so the resultant dataset will only approximately be distributed according to $N(0,\Sigma_{clust})$.
As PCA basis vectors correspond to (unit-norm) eigenvectors of covariance matrices, the PCA basis obtained by applying PCA to the zero-centered open-cluster dataset parametrizes a transformation to a representation in which $\Sigma_{clust}$ is (approximately) diagonal. A change-of-basis to this PCA basis thus parametrizes the desired diagonalization of the cluster covariance matrix. This change-of-basis is denoted in Figure \ref{fig:diagram_algorithm} as \textbf{``3) Reparametrize"}. Because the basis vectors of the PCA basis have by construction unit norm, the covariance matrix $\Sigma_{pop}$ will still be the identity matrix after this change-of-basis and hence both $\Sigma_{clust}$ and $\Sigma_{pop}$ will be diagonal matrices as desired.
\subsection{Scaling factor}\label{sec:scaling}
In the final step of the metric learning algorithm, denoted by \textbf{``4) Rescale"} in Figure \ref{fig:diagram_algorithm}, basis vectors are scaled proportionally to their dimension's usefulness at recognizing open clusters. This is done by applying separately to each dimension of the representation a scaling factor. Here, the independent scaling of dimensions is justified by $\Sigma_{clust}$ and $\Sigma_{pop}$ being diagonal covariance matrices.
We design our scaling factor such that, after transformation, distances between pairs of stars quantify the ratio between the probability that pairs of stars originate from the same (open) cluster and the probability that they do not. To put it another way, we seek to scale dimensions such that pairs of stars that are more likely to originate from the same cluster as compared to originating from different clusters have a smaller separation (ie Euclidean distance) in the representation than less likely pairs.
Under our set of assumptions, random stellar siblings (intra-cluster stars) are distributed as $z_{clust} \sim N(\mu_{clust_i},\sigma_{clust_i})$ where $\mu_{clust_i}$ and $\sigma_{clust_i}$ are the mean and standard deviation along dimension $i$ (at this stage in the algorithm). Accordingly, using the standard formula for the sum of normally distributed variables, the one-dimensional distance along $i$ between pairs of random stellar siblings $z_{clust_{1}}$ and $z_{clust_{2}}$ follows a half-normal distribution $d_{clust_i} \sim |z_{clust_1}-z_{clust_2}|=|N(0,2 \sigma_{\rm clust_i})|$. Likewise, the distance $d_{pop_{i}}$ between pairs of random stars follows a similar half-normal distribution $d_{pop_i} \sim |N(0,2 \sigma_{\rm pop_i})|$ where $\sigma_{\rm pop_i}$ is the standard deviation amongst field stars along dimension i.
For a pair of stars observed a distance $d_{i}$ away from each other along a dimension $i$, the ratio between the probability of the pair originating from the same cluster and the probability of the pair not originating from the same cluster is:
\begin{equation}
r_{i}(d_i)=\frac
{
p(d_{clust_i}=d_{i})
}
{
p(d_{pop_i}=d_{i})
}=
|
\frac
{
N(0,2 \sigma_{\rm clust_i})
}
{
N(0,2 \sigma_{\rm pop_i})
}|
\end{equation}
which evaluates to (as distances $d_{i}$ are by design greater than 0)
\begin{equation}
r_{i}(d_i) = A_i e^{\frac{-d_{i}^{2}}{2\sigma_{r_{i}}^{2}}}
\end{equation}
where
\begin{equation}
A_i=\frac{\sigma_{pop_i}}{\sigma_{clust_i}}
\end{equation} and
\begin{equation}
\sigma_{r_{i}}= \frac{\sigma_{\rm clust_{i}}\sigma_{\rm pop_{i}}}{\sqrt{\sigma_{\rm pop_{i}}^2-\sigma_{\rm clust_{i}}^2}}
\end{equation}
As dimensions are assumed to be independent, the probability ratio accounting for all dimensions is the product of the probability ratio of the separate dimensions:
\begin{equation}
r =
\prod_{i=0}^{D}
\frac
{
N(0,2 \sigma_{\rm clust_i})
}
{
N(0,2 \sigma_{\rm pop_i})
}
=
Ce^{
-\frac{1}{2}
\sum_{i=0}^{D}
(\frac{d_i}{\sigma_{r_i}})^2
}
\end{equation}
where $C =\prod_{i=0}^{D}A_{i}$.
From this expression, it can be seen that multiplying dimensions by a scaling factor of $\frac{1}{\sigma_{r_i}}$ leads to a representation in which the probability ratio $r$ is a function of Euclidean distance and where pairs of stars with smaller Euclidean distance separation have a higher probability of originating from the same open-cluster as compared to their probability of originating from different cluster than pairs with larger Euclidean separation.
It is thus clear that dividing dimensions by a scaling factor of $\frac{1}{\sigma_{r_i}}$ induces a representation where distances directly encode the probability of stars originating from the same cluster, as desired for our metric-learning approach. However, using this expression as a scaling factor requires evaluating the $\sigma_{clust_i}$'s and $\sigma_{pop_i}$'s along all dimensions. Because the representation has been sphered, the population's standard deviation is unity along all directions ($\sigma_{pop_i}=1$). We estimate the intra-cluster standard deviations using a pooled variance estimator:
\begin{equation}
\sigma_{\rm clust_i}^{2}
=\frac{
\sum_{j=1}^{k}\left(n_{j}-1\right) \sigma_{ij}^{2}
}{
\sum_{j=1}^{k}\left(n_{j}-1\right)
}
\end{equation}
where $\sigma_{ij}^{2}$ refers to the sample variance along dimension i for the sample of stars belonging to an open-cluster j containing $n_{j}$ stars in $X_{clust}$. To make the algorithm more robust to the presence of any outliers in the dataset, such as misclassified stellar siblings, we use the median absolute deviation (MAD) as an estimator for the sample standard deviation $\sigma_{ij}$. That is
\begin{equation}
\mathrm{MAD}=\operatorname{median}\left(\left|X_{i}-\tilde{X}\right|\right)
\end{equation}
where $X_{i}$ and $\tilde{X}$ are respectively the data values and median along a dimension. \ddm{I've added the bit about MAD at the end here. I've been using this instead of std for ages but didn't bother to write it down until now (it's particularly helpful for the models on stellar abundances more so than on spectra if my memory is correct).}
To better understand the effect of our scaling factor on the representation it is applied to, it is instructive to look into the impact it has on the dimensions of a representation. When stars belonging to the same cluster have, along a dimension, a similar standard deviation to the full population of stars (ie $\sigma_{clust} \approx \sigma_{pop}$), the dimension carries no information for recognizing cluster member stars and the scaling factor accordingly fully suppresses it $\sigma_{r} \rightarrow \infty$. On the other hand, for dimensions where the population standard deviation $\sigma_{pop}$ is significantly larger than the cluster standard deviation $\sigma_{clust}$, the population standard deviation is no longer relevant and $\sigma_{r} \approx \sigma_{clust}$. That is to say the scaling devolves into measuring distances relative to the number of standard deviations away from the cluster standard deviation.
\section{Experiments}\label{sec:experiments}
We validate our approach for encoding chemical similarity by testing its performance on real data obtained by the APOGEE survey data release 16 \citep{ApogeeDR16}. The APOGEE survey \citep{Apogee} is an infrared, high resolution, high signal-to-noise spectroscopic survey. The APOGEE survey uses a 300-fiber spectrograph \citep{APOGEEspectrographs} installed at the Sloan Digital Sky Survey telescope located at Apache Point Observatory \citep{ApachePointObservatory}.
\subsection{Dataset Preparation}
For our experiments we use spectra from the public APOGEE data release DR16 \citep{ApogeeDR16} to create $X_{clust}$ and $X_{pop}$, our datasets of field and open-cluster stars. Our field dataset $X_{clust}$ contains spectra for 151,145 red-giant like stars matching a set of quality cuts on the 16th APOGEE data release described below. Our open-cluster dataset contains spectra for 185 stars distributed across 22 open clusters, obtained after further quality cuts using the OCCAM value-added catalogue \cite{Donor_2020}, a catalogue containing information about candidate open-cluster observed by APOGEE. We also create baseline datasets $Y$ and $Y_{clust}$ containing stellar abundances for the stars in $X$ and $X_{clust}$. We include abundances for 21 species in $Y$ and $Y_{clust}$: C, CI, N, O, Na, Mg, Al, Si, S, K, Ca, Ti, TiII, V, Cr, Mn, Fe, Co, Ni, Cu, Ce. These abundances are derived from the X\_H entry in the allStar FITS file.
To create the dataset of field stars $X_{pop}$, we make the following dataset cuts. With the intention of only preserving red-giant stars, we discard all but those stars for which $4000<\mbox{$T_{\rm eff}$}<5000$ and $1.5<\mbox{$\log g$}<3.0$, where we use the $\mbox{$T_{\rm eff}$}$ and $\mbox{$\log g$}$ derived by the ASPCAP pipeline. In addition, we further exclude any stars for which some stellar abundances of interest were not successfully estimated by the ASPCAP pipeline by removing any star containing abundances set to -9999.99 for any of our 21 species of interest \ddm{and all spectra for which the STAR\_BAD flag is set in ASPCAPFLAG}. The pseudo-continuum spectra of those remaining stars, as found in the AspcapStar FITS file, were used to create the matrix $X_{pop}$ in which each column contains the spectra of one star.
To create the dataset of open-cluster member stars $X_{clust}$, we cross-match our filtered dataset with the OCCAM value-added catalogue \cite{Donor_2020} so as to identify all candidate open-clusters observed by APOGEE. We only keep those spectra of stars with open-cluster membership probability CG\_PROB$>$0.8 \citep{gaudinclusters}. After this cross-match, we further filtered the dataset by removing those clusters containing only a single member stars as these are not useful for us. Additionally, we further discard one star with Apogee ID "2M19203303+3755558" found to have a highly anomalous metallicity. After this procedure, 185 OCCAM stars remain, distributed across 22 clusters. We do not cut any stars based on their signal-to-noise ratio. The stars in $X_{pop}$ have a median signal-to-noise ratio of 157.2 and interquantile range of 102.0-272.4 while those in $X_{clust}$ have a median signal-to-noise ratio of 191.4 and interquantile range of 117.7-322.6. \mkn{snr distribution for clusters and field?}\ddm{Not describing SNR with standard deviation because it's not appropriate given the non-symmetry of distribution values (lots of high SNR outliers contribute to leading to high std).}
Because of cosmic rays, bad telluric line removal or instrumental issues, the measurements for some bins of stellar spectra are untrustworthy. We censor such bad bins to prevent them from impacting our low-dimensional representation. Censored bins are treated as missing values in the PPCA compression. In this work, we have censored any spectral bin for which the error (as found in the AspcapStar FITS file error array) exceeds a threshold value of 0.05. Additionally, we censor for all stars in the dataset, those wavelength bins located near strong interstellar absorption features. More detail about the model-free procedure for censoring interstellar features can be found in Appendix \ref{appendix:masking}.
\mkn{Can we use the earlier defined nomenclature pop and clust to link to the distance? Is your distance metric calculated entirely consistently for both open cluster and test population in the following section? I propose you call this $d$ and use this everywhere, your Euclidean distance, including in figures}
\subsection{Measuring Chemical Similarity}\label{sec:measuringchemicalsimilarity}
Evaluating how good a representation is at measuring the chemical similarity of stars requires a goodness of fit indicator for assessing the validity of its predictions. We use the "doppelganger rate" as our indicator. This is defined as the fraction of random pairs of stars appearing as similar or more similar than stellar siblings according to the representation, where similarity is measured in terms of distance $d$ in the studied representation. It is worth noting that this approach towards estimating doppelganger rates is related but different from the probabilistic approach presented in \cite{Ness2018}.
We estimate doppelganger rates on a per-cluster basis. For each cluster in $X_{clust}$, the doppelganger is calculated as the fraction of pairs composed of one cluster member and one random star whose distance in the studied representation, $d_{intra-family}$ is $\leq$ than the median distance amongst all cluster pairs $d_{inter-family}$ . That is, $d_{inter-family}$ are pairs composed of two confirmed cluster members within $X_{clust}$ and $d_{inter-family}$ is a pair with one random field star selected from $X_{pop}$ and one cluster member from the studied cluster in $X_{clust}$. When calculating $d_{inter-family}$, we only consider pairs of stars with similar extinction and radial velocity, that is to say with $\Delta \rm AK\_TARG<0.05$ and $\Delta \rm VHELIO\_AVG<5$. By only comparing stars at similar extinction and similar velocity, we ensure that any model being investigated cannot reduce it's doppelganger rate through exploiting extinction or radial velocity information in the spectra.
So as to facilitate comparisons between different representations, we aggregate the per-cluster doppelganger rates into a "global" doppelganger rate which gives an overall measurement of a representations effectiveness at identifying open-clusters. The global doppelganger rate associated to a representation is obtained by averaging the per-cluster doppelganger rates through a weighted average in which clusters are weighted by their size in $X_{clust}$.
\mkn{I got a little lost here in the paragraph below. I think you saying you do this for open clusters individually (and I know this) but I think a bit hard to figure out exactly what is happening for a first time reader. I am happy to try to re-word a little but reluctant to as not 100 percent sure I follow. I tried to add in some nomenclature to try to clarify please correct if not right/we could use this throughout? I think the trick in clarifying this will be to link to the algorithm description/nomenclature which you could use throughout the paper to help clarify when you are talking about what. i.e. you introduce all the terems in the algorithm but don't propagate them to this section? }
\mkn{are you sure this is not 100 percent? Because a random star would do not better than a cluster pair, so you would get as many of each? }\ddm{If we do no better than random, then both distributions are identical and so the fraction below the median is 50 percent.}
\mkn{Because metallicity will be the biggest discriminator, how does your doppelganger rate change if you only compare to field stars with the same metallicity of the clusters. In that, the test across all metallicity done in this way will always give you a low doppelganger rate as your MDF is so broad, so there is a low probability you are comparing to stars of the same metallicity, where as your sample of clusters are all at the same metallicity, so the algorithm could work somewhat poorly and still give you a small doppelganger rate? I think the test you have is all okay but worth doing again only allowing intra-pair comparisons at the same [Fe/H] to demonstrate that it leverages (coarse/broad) chemical information.}
There is an added subtlety to assessing a representation through its global doppelganger rate. There are very few open-cluster stars in the dataset. Therefore, RSCA as a data-driven procedure applied to open-clusters, is susceptible to overfitting to the open-cluster dataset. To prevent overfitting from affecting results, we carry out a form of cross-validation in which clusters are excluded from the dataset used for the derivation of their own doppelganger rate. In this scheme, calculating the global doppelganger rate of a representation requires repeated application of our algorithm, each time on a different subset with one cluster removed, as many times as there are open clusters.
We caution that our cross-validation approach has some implications on the derived doppelganger rates. Because, every cluster's doppelganger rate is evaluated on a slightly different data subset, the quoted distances and doppelganger rates are not comparable from cluster to cluster. In particular, smaller clusters, whose doppelganger rates are derived from a larger number of open-cluster stars, are expected to have marginally lower doppelganger rates.
\subsection{PCA Dimensionality}
\begin{figure}
\includegraphics[width=\columnwidth]{figures/pca/pca_dimensionality.pdf}
\caption{Global doppelganger rates as a function of the number of PCA components used to encode spectra. Performance with cross-validation is shown in blue while performance without cross-validation is shown in green.}
\label{fig:global_doppelganger}
\end{figure}
The number of principal components used in the encoding stage of our metric learning approach is an important hyperparameter requiring tuning. In Figure \ref{fig:global_doppelganger}, we plot the doppelganger rate against the number of principal components, both with and without using the cross-validation procedure described in section \ref{sec:measuringchemicalsimilarity}. Results without cross-validation display significant overfitting and are mostly shown in an effort to highlight the importance of the cross-validation procedure.
This Figure illustrates how, unsurprisingly, the algorithms performance is strongly dependent on the PCA dimensionality. With doppelganger rates consistently decreasing with increasing PCA dimensionality up until a dimensionality of size 30 after which doppelganger rates start increasing because of overfitting. Since on our studied dataset, the algorithm reaches its peak performance for 30 PCA components, all further quoted results and figures will accordingly use the first 30 PCA components.
That our algorithm's performance still improves up to a dimensionality of 30 is interesting. This demonstrates that a hyperplane of at least size 30 is required for capturing the intrinsic variations of APOGEE stellar spectra, a number noticeably larger than the 10 dimensional hyperplane found in \cite{pca_dimensionality}, although methodological differences between studies may partially explain such differences. For example, the PCA fit in \cite{pca_dimensionality} was applied on spectra displaying limited instrumental systematics and with non-chemical imprints on the spectra preliminary removed. It should also be noted that this does not mean that the chemical space is 30 dimensional. Some PCA dimensions may capture instrumental systematics or non-chemical factors of variation. Also, since chemical species leave non-linear imprints, because of the linearity of PCA each non-linear chemical dimension may require multiple PCA components to be fully captured.
\subsection{RSCA interpretability}
We now study which spectral features are leveraged by the RSCA algorithm when recognizing open-clusters. As in the RSCA basis, dimensions are scaled proportionally to their perceived usefulness at measuring chemical similarity, the factors of variation judged most important by RSCA will correspond to the most strongly scaled dimensions of the representation. Figure \ref{fig:interpretation} plots the relationship between the three most strongly scaled dimensions and metallicity for a representation obtained by running the RSCA algorithm with a PCA dimensionality of 30.
\begin{figure*}
\includegraphics[width=0.24\textwidth]{figures/interpretation/feature1.pdf}
\includegraphics[width=0.24\textwidth]{figures/interpretation/feature2.pdf}
\includegraphics[width=0.24\textwidth]{figures/interpretation/feature3.pdf}
\includegraphics[width=0.24\textwidth]{figures/interpretation/vrad.pdf}
\caption{\mkn{not sure how this looks on black and white printer? could use light grayscale for the points to make easier to see if obscured when not in colour. Also note in caption how many stars in this figure = of interest to know/remind a reader how much data is being plotted/used. }Three features judged most important by metric learning approach plotted against [Fe/H] for the 151,145 stars in $X_{field}$ and the fourth most-important feature plotted against VHELIO\_AVG (radial velocity ASPCAP label). Location of the 185 stars in $X_{clust}$ (the open-cluster dataset used to train the metric learning model) are shown by orange markers.}
\label{fig:interpretation}
\end{figure*}
As seen from the leftmost panel, there is a close relationship between "Feature \#1" - the most strongly scaled direction in our metric-learning transformation - and the ASPCAP [Fe/H] label. The relationship is close to a one-to-one mapping which illustrates how this feature traces the metallicity content of the stellar spectra. Because "Feature \#1" (as a direction in a hyperplane of stellar spectra) is a linear function of the stellar spectra and metallicity is a non-linear feature, some degree of scatter in the relationship is expected.
The relationship between "Feature \#2" and [Fe/H] (second panel) exhibits the same bimodality as observed when plotting alpha enhancements [$\alpha$/Fe] against [Fe/H] \citep{Leung2018}. This provides strong evidence that "Feature \#2" captures $\alpha$-element enhancements. It is particularly noteworthy that we are able to recover the $\alpha$-element bimodality when the open-clusters in our dataset (orange markers) are located only in one of the bimodality's. This demonstrates the metric model's excellent capacity to extrapolate to abundance values outside the range of value's covered in the open-cluster training dataset and provides evidence that the model may still be effective for stars atypical of those in the open-cluster dataset.
The relationship between "Feature \#3" and metallicity (third panel) is not as easily interpreted. Given the non-linear nature of metallicity, it is possible that it encodes residual metallicity variability not captured by "Feature \#1" but it is also possible that it contains some further independent chemical dimension.
This figure illustrates a nice property of RSCA. Because dimensions of the RSCA representation correspond to eigenvectors of the covariance matrix $\Sigma_{clust}$, the RSCA algorithm, at least to first order, assigns the distinct factors of variations within spectra to separate dimensions of the representation. That is to say that the dimensions of the RSCA representation capture distinct factors of variation, such as the metallicity or the alpha-element abundance, rather than a combination of factors of variation. Additionally, the most important factors of variation for recognzing open-clusters occupy the dimensions with the largest scaling factors. This property makes RSCA particularly versatile. For example RSCA can be used to separate out thin and thick disk stars or to select low metallicity stars. Additionally, it is likely that because of this property RSCA could be used to search for hidden chemical factors of variation within stellar spectra, although this has not been attempted in this paper.
We found that some of the dimensions of the RSCA representation showed trends with radial velocity. An example of a dimension showing a trend with radial velocity is shown in the last panel of Figure \ref{fig:interpretation} and an investigation into the detailed causes of the radial velocity trends is presented in appendix \ref{appendix:radial}. The existence of such trends indicates that even after our pseudo-continuum normalization procedure RSCA is still capable, at least weakly, to exploit radial velocity information in the spectra to recognize stellar siblings. Because only a subset of the dimensions show such trends, a representation tracing only chemistry can be obtained by only keeping those dimensions which show no trends with radial velocity. In this work, we propose to only keep the first 3 dimensions of the representation. While this choice may appear particularly stringent, as we will show in coming sections, these 3 dimensions contain the bulk of the discriminative power of the representation.
\subsection{Comparison to abundances}\label{sec:results}
In this section we compare the effectiveness at measuring chemical similarity of our data-driven approach with that achievable from stellar abundances. To do so, we compare the doppelganger rates that are achieved by RSCA to those obtained from stellar labels, the results of such a comparison are shown in Figure \ref{fig:ablation}. We remind that the doppelganger rates are evaluated for pairs of stars at the same extinction and radial velocity. This guarantees that the doppelganger rate cannot be artificially reduced through our model exploiting information relating to radial velocity or extinction. Per-cluster doppelganger rates are also provided in Appendix \ref{appendix:local}.
From this Figure, we see that all steps of the RSCA are crucial for obtaining low doppelganger rates when working directly with spectra - as seen from how low doppelganger rates are only obtained with full application of our metric-learning approach (red). On the other hand, when working with stellar abundances, the RSCA approach appears to bring only limited benefits - as seen from the only slight difference in doppelganger rates between measuring distances in the raw abundance space (blue) and in the transformed space (red) that is obtained by application of our full-metric learning approach minus the PCA compression. This result is not surprising and reflects how most steps of the RSCA approach are designed with the aim of generating a representation comparable to stellar labels, that is to say one where all factors of variation other than chemical factors of variation are removed. The RSCA approach does however still bring some benefits when working with abundances as seen from the lower doppelganger rates.
This Figure also shows that excluding chemical species improves the doppelganger rate, as seen from how the doppelganger rate is lower when using a subset of species (right) than when using the full set of abundances (center). This can appear counter-intuitive as it implies that more data leads to worsened performance but in this specific case, where the uncertainties on abundances are not accounted for, it can be justified by the low intrinsic dimensionality of chemical space. Since many species contain essentially the same information, adding species with higher uncertainty into the representation adds noise into the representation without contributing any additional information beneficial for recognizing open-clusters. We expect that such an effect would disappear when accounting for uncertainties on stellar labels, but it is still a good illustration of the brittleness of abundance-based chemical tagging. \ddm{Fixed this paragraph to clarify the role of not incorporating uncertainties.}
When generating these results, we manually investigated many different combinations of chemical species and show in "abundance subset" the combination which was found to result in the lowest doppelganger rate which is the combination of stellar labels Fe,Mg,Ni,Si,Al,C,N. The doppelganger rate from this combination of species - of 0.023 -, despite being the smallest doppelganger rate achieved from stellar labels, is noticeably higher than the doppelganger rate obtained from stellar spectra of 0.020. That our method is able to produce better doppelganger rates from spectra than from stellar labels highlights the existence of information within stellar spectra not being adequately captured by stellar labels. While stellar labels are derived from synthetic spectra which only approximately replicate observations, our fully data-driven model makes direct use of the spectra translating into lower doppelganger rates.
\begin{figure}
\includegraphics[width=\columnwidth]{figures/ablation/ablation.pdf}
\caption{Global doppelganger rates estimated for varying metric-learning approaches and representations. On the x-axis, "spectra" refers to doppelganger rates obtained from spectra $X$ after dimensionality reduction with PPCA to a 30-dimensional space, "all abundances" to doppelganger rates obtained from a representation formed from the full set of APOGEE abundances in $Y$, "abundance subset" to doppelganger rates obtained using a representation formed only from the abundances for the following species: Fe,Mg,Ni,Si,Al,C,N. Global doppelganger rates "on raw" (blue) are obtained by measuring distances in the raw representation without any transformation of the representation, "on scaled" (green) are obtained by applying the scaling transform on the raw representation without preliminary application of the sphering and reparametrization transform, "on transformed" are obtained by applying all steps of the proposed metric learning approach. As the implementation of the PPCA algorithm used in this paper yielded stochastic PCA components, doppelganger rates from spectra correspond to the mean across 10 runs with error bars corresponding to the standard deviation amongst runs.}
\label{fig:ablation}
\end{figure}
\subsection{Dimensionality of chemical space}
That the PCA representation is thirty dimensional does not mean that all thirty dimensions carry information useful for recognizing open-clusters. To get a grasp of the dimensionality of the chemical space probed by RSCA, we calculated the doppelganger rates for RSCA representations in which all but the dimensions with the largest scaling factor are discarded. We calculate doppelganger rates multiple times, each with a different number of dimensions preserved. The results of this investigation are shown in Table \ref{tab:dimensionality}. From this Table, we see that the dimensionality of spectra appears to be, at least to first degree, extremely low-dimensional. The top two dimensions of the RSCA model (as shown in Figure \ref{fig:interpretation}) are capable of matching the performance obtained from stellar labels whilst the top four dimensions exceed the performance from using the full representation. The four-dimensional representation is even more effective at recognizing chemically identical stars than the full RSCA representation which itself was more effective than stelar labels. It is not a new result that the dimensionality of chemical space probed by APOGEE is low-dimensional. Recent research suggest that, at the precisions captured by APOGEE labels, chemical abundances live in a very low-dimensional space. For example, it was found in \cite{Ness2019,HowManyElements} that [Fe/H] and stellar age could predict all other elemental abundances to within or close to measurement precision. However while previous analysis have depended on abundances to show this, here we can do this directly from spectra. As our methodology directly picks up on factors of variation and, if not controlled for, is capable of picking up on weak factors of variation such as diffuse interstellar bands we can be confident that any remaining chemical factors of variation are either i) highly non-linear for our model to not be capable of picking up on them, ii) very weak spectral features, or iii) not particularly discriminative of open-clusters as would be the case for chemical variations arising from internal stellar processes. \ddm{for example induced by accretion of planetary materials or internal stellar processes.}.
\begin{table}
\centering
\begin{tabular}{p{1.5cm} p{3.0cm}}
\hline
N & Doppelganger rate \\
\hline
1 & $0.0962 \pm 0.0212$ \\
2 & $0.0219 \pm 0.0025$ \\
3 & $0.0198 \pm 0.0028$ \\
4 & $0.0182 \pm 0.0021$ \\
5 & $0.0180 \pm 0.0021$ \\
6 & $0.0188 \pm 0.0017$ \\
7 & $0.0184 \pm 0.0017$ \\
30 & $0.0199 \pm 0.0015$ \\
\end{tabular}
\caption{Global doppelganger rate obtained by the RSCA model applied to stellar spectra in which all but the N most strongly scaled dimensions are discarded. As the implementation of the PPCA algorithm used in this paper yielded stochastic PCA components, doppelganger rates from spectra correspond to the mean across 10 runs with error bars corresponding to the standard deviation amongst runs.}
\label{tab:dimensionality}
\end{table}
\subsection{Impact of Dataset Size}
\begin{figure}
\includegraphics[]{figures/datasize/datasize.pdf}
\caption{Expected global doppelganger rates when training a metric-learning model on only a subset of all open-clusters in $X_{clust}$ with a number of clusters given by the x-axis. Results for different PCA dimensionalities used for compressing stellar spectra are represented by different colored lines. Clusters used in the expectated doppelganger rate calculations were chosen randomly from $X_{clust}$, and quoted results are for the average of 50 repeated trials.}
\label{fig:datasize}
\end{figure}
Our method learns to measure chemical similarity directly from open-clusters without reliance on external information. Because of this, it's performance will be tightly linked to the quality and quantity of data available. Figure \ref{fig:datasize} attempts to estimate our method's dependency on the size of the open-cluster dataset. In this figure, for varying PCA dimensionalities, we plot the expected doppelganger rate for an open cluster dataset containing a given quantity of open-clusters whose number is given by the x-axis. We estimate the expected doppelganger rate for a given number of open-clusters by estimating and averaging the doppelganger rates for all data subsets containing that number of clusters. From this figure, we see that the larger PCA dimensionalities still benefit from the addition of open-clusters. This is suggestive that performance would likely further improve with access to additional open-clusters. Such larger datasets may also enable the usage of more complex non-linear metric-learning approaches which may yield further improvements not captured in this figure.
\section{Discussion}\label{sec:discussion}
\mkn{discussion of not leveraging error on pixels, checked for SNR dependence? How does fidelity of this approach scale with SNR? }
We have presented a novel approach for identifying chemically similar stars from spectroscopy based on training a metric-learning model on the spectra of open-cluster stars. This approach has several appealing properties. It is end-to-end data-driven in the sense that it does not rely on synthetic models nor labels derived from synthetic models. This method only makes use of open-cluster members, which can themselves be identified with minimal reliance on theoretically-derived quantities \citep{openclusteridentification1,openclusteridentification2,openclusteridentification3}. This makes the method insensitive to the domain gap of synthetic spectra. Additionally, where traditional spectral fitting approaches require instrumental systematics to be fully suppressed, lest they further exacerbate the domain gap, our fully data-driven approach, at least in theory, automatically learns on its own to ignore most variations due to instrumental systematics.
We expect that the approach that we have developed will perform particularly well in a number of regimes. For example, low resolution spectra, where blended features lead to compounding model inaccuracies. Additionally, for M-type stars, where molecular features complicate the retrieval process. In general, our method will likely be an efficient and effective approach where theoretical models are inaccurate or the observed spectra itself is plagued by complex systematics. \ddm{Any citations and is this correct.}
Although our data-driven algorithm shows excellent performance, one may wonder whether there is still room for further improvements, particularly since strong chemical tagging, if ever possible, will require dramatic improvements in our chemical similarity measurements \citep{HowManyElements}. There are reasons to be hopeful here.
Firstly, our data-driven model comes with clear limitations which are not inherent to the approach, but rather imposed by our modelling choices. There are many algorithmic choices for building a metric learning model optimized at distinguishing open-clusters \citep[e.g.][]{LargeMarginNearestNeighborClassification,Neighbourhood,relevantComponentAnalysis,pml1Book} and ours is only one of many choices. In particular, the ability of RSCA to extract the chemical content of stellar spectra is constrained by its linear nature. Although this linearity is convenient for avoiding over-fitting, enabling better out-of-distribution performance and cross-validation, it also artificially limits the precision with which "pure" features containing only chemical signal can be learned. One can be hopeful that a suitably regularized non-linear metric learning models, such as for example twin neural network \citep{siamese,pml1Book}, could surpass our model. However, building such a model from our limited number of open-clusters would present its own unique challenges. \mkn{what is the evidence that the model is not flexible enough? }\ddm{That the "real" features of spectra are non-linear.}
Secondly, by being entirely data-driven, the performance of this approach is inexorably linked to the quality and quantity of data available. This makes it poised to benefit from new open-cluster discoveries and/or deliberately targeted observations.\ddm{Improvements may also be possible by leveraging other source of chemically similar stars such as wide binaries.}
Our method also comes with caveats. Data-driven methods do not extrapolate well outside of their training dataset. As such, performance may be lower for clusters that are atypical compared to those in the open-cluster reference set. Since open clusters are typically younger stars \citep{openclusterreview}, this means performance may be decreased on older cluster stars.However, given the existence of a tight relationships between RSCA dimensions and chemical parameters for all stars in $X_{field}$, such an effect is likely to be small. Additionally, our model makes no use of the error information in spectra, which is valuable information that could likely be squeezed-out for even better performance.
Another downside of our approach is its coarse-grained nature. While stellar labels provide a fine-grained view into chemical similarity with a breakdown into chemical composition of individual species, our approach only provides a coarse-grained measurement of chemical similarity. This limits the types of scientific problems the approach can be used to answer. However, it may very well be possible to extend the method from measuring overall chemical similarity to measuring individual elemental abundances. A way in which this could possibly be done is through applying it to windows centered on the locations of stellar lines instead of to the full spectrum. Also, it is not always clear what exact information is captured by the representation, and in particular there is always a risk, despite all of our checks, that the model is acting on non-chemical information within the spectra.
\section{Conclusion}
Large-scale Galactic surveys, the likes of APOGEE, LAMOST and GALAH, have collected hundreds of thousands of high-quality stellar spectra across the galaxy. These surveys are vastly broadening our understanding of the Milky Way. How best to analyse these spectra however still remains an open-question. One limitation is that traditional spectral fitting methods currently do not make full use of the information in stellar spectra. This is largely because our stellar models are approximations.
In this paper we developed a fully data-driven, linear metric learning algorithm operating on spectra for extracting the chemical information within stellar families. Through experiments on APOGEE, we demonstrated that our metric learning model identified stars within open clusters more precisely compared to using stellar labels suggesting an improved ability to discriminate between chemically identical stars. \ddm{We further found that our model's capacity to distinguish open-clusters could largely be attributed to a two-dimensional subspace of our final representation which was found to approximately coincide with metallicity and alpha-elemental abundances. The low-dimensionality at which our model's capacity at recognizing open-clusters plateaued supports the idea that the dimensionality of chemical space probed by APOGEE is, for Galactic archaeology purposes, very low-dimensional. However, we do find hints of further dimensions potentially containing chemical information.}
There are several reasons why our metric-learning approach could be favoured over using stellar labels. It can be applied to spectra of stars for which we do not yet have very good synthetic spectra and so would otherwise not be able to analyze well. It is completely independent of our theoretical knowledge of stellar atmospheres and so could be used to validate existing astronomical results in a way which is independent of any biases which may exist in our synthetic spectra. Finally and perhaps most importantly, whereas the traditional derivation of stellar labels is fundamentally limited by our inability to generate faithful synthetic spectra, our metric learning approach does not suffer from such a limitation. This means that by improving the quality of the training dataset and the metric-learning approach used, performance may be further improved.
\begin{acknowledgements}
DDM is supported by the STFC UCL Centre for Doctoral Training in Data Intensive Science (grant number ST/P006736/1).
Funding for the Sloan Digital Sky
Survey IV has been provided by the
Alfred P. Sloan Foundation, the U.S.
Department of Energy Office of
Science, and the Participating
Institutions.
SDSS-IV acknowledges support and
resources from the Center for High
Performance Computing at the
University of Utah. The SDSS
website is www.sdss.org.
SDSS-IV is managed by the
Astrophysical Research Consortium
for the Participating Institutions
of the SDSS Collaboration including
the Brazilian Participation Group,
the Carnegie Institution for Science,
Carnegie Mellon University, Center for
Astrophysics | Harvard \&
Smithsonian, the Chilean Participation
Group, the French Participation Group,
Instituto de Astrof\'isica de
Canarias, The Johns Hopkins
University, Kavli Institute for the
Physics and Mathematics of the
Universe (IPMU) / University of
Tokyo, the Korean Participation Group,
Lawrence Berkeley National Laboratory,
Leibniz Institut f\"ur Astrophysik
Potsdam (AIP), Max-Planck-Institut
f\"ur Astronomie (MPIA Heidelberg),
Max-Planck-Institut f\"ur
Astrophysik (MPA Garching),
Max-Planck-Institut f\"ur
Extraterrestrische Physik (MPE),
National Astronomical Observatories of
China, New Mexico State University,
New York University, University of
Notre Dame, Observat\'ario
Nacional / MCTI, The Ohio State
University, Pennsylvania State
University, Shanghai
Astronomical Observatory, United
Kingdom Participation Group,
Universidad Nacional Aut\'onoma
de M\'exico, University of Arizona,
University of Colorado Boulder,
University of Oxford, University of
Portsmouth, University of Utah,
University of Virginia, University
of Washington, University of
Wisconsin, Vanderbilt University,
and Yale University.
\end{acknowledgements}
\section{Introduction} \label{sec:intro}
The field of Galactic astronomy has entered a transformative era. Large-scale surveys, such as APOGEE, GAIA and GALAH, are providing millions of high-quality spectroscopic and astrometric measurements of stars across the Milky Way \citep{Rave2006, Apogee,Galah,Lamost,Gaia}. Future large-scale surveys, which will release even more high-quality data, are on the horizon \citep{4most,SDSS5,Weave}.
In this landscape of high-volume high-quality stellar astronomy, fully extracting the scientifically relevant information from stellar spectra remains a difficult problem. Classically, this has been done by comparing observations to synthetic spectra generated from theoretical models \citep[e.g.][]{aspcap}. However, the precision with which stellar labels can be derived under such an approach is ultimately limited by the faithfulness with which synthetic spectra reproduce observations. Because of computational constraints and gaps in knowledge, synthetic spectra do not perfectly match observations, something sometimes referred to as the ``synthetic gap" \citep{cyclestar}. Computational models used to generate synthetic spectra use incomplete stellar line lists and usually must make simplifying assumptions. This includes for example that stellar atmospheres are one-dimensional, in hydrostatic equilibrium, and in local thermodynamic equilibrium. In addition, even beyond these issues, observations are affected by further systematics such as telluric lines introduced by the earth's atmosphere \cite[e.g.][]{SDSSdataproducts} and telescope imperfections/aberrations.
Ultimately this synthetic gap limits our ability to extract information from stellar spectra. In \cite{CramerRaoTing,HowManyElements} it was shown that stellar spectra contain more chemical information than is captured in bulk metallicity and $\alpha$-enhancement alone. The precision of derived individual stellar abundances from large surveys, however, may be limited by an inability to fully extract information given approximate models, rather than by the signal to noise of observations.
This is problematic because a lot of interesting science requires measuring chemical similarity between stars with a precision beyond that currently delivered by large stellar surveys modelling pipelines. In particular, high precision chemical measurements are needed for strong chemical tagging \citep{TaggingFreeman}. This is an ambitious Galactic archaeology endeavour aiming to identify stellar siblings - stars born from the same molecular cloud - using chemical information derived from spectroscopy long after clusters gravitationally dissipate. In practice, whether such chemical tagging is theoretically possible at scale is still an open question, but may be answered with large-scale surveys like GALAH \citep{Galah, Buder2021}. For this form of chemical tagging to be successful, stellar siblings must share a near-identical chemical composition with sufficient variability in chemical compositions between clusters. Even if strong chemical tagging, reveals itself to be impossible at large scale, precise chemical similarity measurements would still be useful in reconstructing the broad nature of our galaxy's evolution \citep[e.g.][]{Coronado2020, Kamdar2020}.
These issues motivate the development of methods capable of extracting information from stellar spectra. and overcoming the synthetic gap between observations and theoretical models. Several data-driven methods have been developed for this purpose. Methods such as those proposed \cite{Ness2015,Casey2016,Leung2018,Ting2019,cyclestar,Das_2019} allow for improving precision of stellar labels through leveraging data-driven interpolators between stellar spectra and labels, reducing the impact of noise and systematics on derived parameters. However, as such approaches still rely on synthetic spectra they do not fully alleviate issues with systematic errors from mismatching theoretical spectra. Recently, methods for finding chemically similar stars directly from stellar spectra without reliance on synthetic spectra have been developed \cite{Bovy2016,pca_dimensionality,Cheng2020TestingTC,de_Mijolla_2021}. This category of method works by removing the effect of non-chemical parameters on stellar spectra, thus isolating the chemical information within the spectra. Such approach are not without drawbacks. Although they remove the dependency on synthetic models, they still require a comprehensive and precise determination of all non-chemical factors of variation. Additionally, they must make simplifying assumptions regarding the cross-dependencies between chemical and non-chemical factors of variations which may have an impact on accuracy.
In this paper, we present a new approach for identifying chemically similar stars from spectroscopic data which we name ``Relevant Scaled Component Analysis" (RSCA) because of its similarities with Relevant Component Analysis \citep{relevantComponentAnalysis}. Our approach is grounded in the machine learning subfield of metric learning. Instead of estimating individual chemical abundances, we project spectra directly into a lower dimensional subspace in which distances between spectra are made to encode a useful notion of chemical similarity between stars. Crucially, as our approach for transforming stellar spectra does not rely at any stage on synthetic spectra or quantities derived from these, its performance is not hindered by inaccuracies in stellar modelling.
A novelty of our work is that instead of using synthetic spectra to learn this notion of chemical similarity, we make use of spectra from know open clusters with open cluster membership information. Open clusters are groups of stars born together that remain gravitationally bound after birth and up to the present day. They are relatively rare, as most stellar clusters dissipate rapidly after birth \citep{openclusterreview}. However, they are extremely useful tools in modern Galactic astronomy. In particular, open clusters, which can be identified using astrometry, display near-identical chemical abundances although small scatter may exist at the 0.01 to 0.02 dex level and up to $<$0.05 dex level for some elements \cite[e.g.][]{Bovy2016,AtomicDiffusionLiu, Ness2018,Cheng2020TestingTC}. Open clusters have found many uses in modern astronomy. For example to obtain high-precision measurements of the radial abundance gradients in the Milky Way \citep{abundancegradientold,abundancegradientnew} or to benchmark and calibrate stellar survey abundance measurements \citep{aspcap}. Here, we use open clusters as a golden standard for learning a notion of chemical similarity. In our approach, we take the viewpoint that if open clusters are indeed chemically homogeneous, then a successful metric for encoding chemical similarity will be one in which open cluster stellar siblings are highly clustered.
Our algorithm, RSCA, has several properties that make it suitable for the task at hand, of measuring chemical similarity between stars:
\begin{itemize}
\item It is fully \textit{data-driven}. Chemical similarity is measured without any reliance or dependency on theoretical models. This offers a measure of chemical similarity that is independent from the systematics introduced in traditional stellar modelling \citep[e.g.][]{Jofre2017}, and offers a means of validating existing discoveries.
\item It is \textit{computationally efficient}. As the method is linear, processing spectra from the full APOGEE stellar survey can be done in minutes. The most computationally intensive step of the approach is a Principal Component Analysis decomposition.
\item It is \textit{interpretable}. In its current formulation, measuring chemical similarity using our method amounts to evaluating Euclidean distances between stars projected on a hyperplane of the stellar spectra space.
\item It is \textit{precise}. We find the method, using spectra, to be more effective at identifying stellar siblings from open clusters than is possible using stellar abundance measurements. We believe this to be in large part because our method bypasses the synthetic gap introduced by spectral modelling. Furthermore, our experiments suggest that the performance could be further improved, for example, with a larger dataset of open cluster stars or by taking into account the error on the flux which we do not currently do.
\end{itemize}
The paper is organized as follows. In Section \ref{sec:metric}, we outline the conceptual ideas behind our approach for measuring chemical similarity. We then briefly introduce Principal Component Analysis in Section \ref{sec:PCA}, which is a core component of our algorithm. In Section \ref{sec:algo}, we dive deeper, and present our algorithm, RSCA. This is implemented using open clusters observed by the APOGEE survey in Sections \ref{sec:experiments}, and evaluated in light of the field distribution of stars. Its trade-offs and implications are discussed in Section \ref{sec:discussion}.
\section{Concepts and Assumptions}
\subsection{Chemical similarity as metric learning}\label{sec:metric}
The characteristics within a stellar spectrum are caused by the interplay of many factors of variation. These include chemical and physical parameters of the star and the instrumental systematics associated with the telescope, as well as interstellar dust along the line of sight. Measuring chemical similarities requires disentangling the imprint left on the spectra by chemical factors of variation from that left by the other non-chemical factors of variation. Our goal is to identify chemically similar stars from their spectra, for stars that span a range of physical stellar parameters (i.e. effective temperatures and surface gravities). We approach this task from a data-driven perspective, and build an algorithm for identifying stars that are as chemically similar as birth siblings, using open cluster spectra.
For our method, we assume that open clusters are close to chemically homogeneous because of their common birth origin \citep{Ness2018} but are not special in any other way (at least in terms of their spectra). That is to say, we assume that the only information within spectra useful for recognizing open clusters are the chemical features of the spectra, and so that a model which identifies open clusters from the spectra will need to do so by extracting the chemical information within spectra
We frame the task of building such a model recognizing open clusters as a metric learning task. That is to say, we build a data-driven model converting stellar spectra into a representation in which Euclidean distances convey the uncalibrated probability of stars originating from a shared open cluster. To accomplish this, the training objective of our data-driven algorithm can be understood as transforming stellar spectra into a representation in which the distance between intra-cluster stars is minimized and the distance between inter-cluster stars is maximized.
Distances in the representation resulting from such an optimizing procedure will organically quantify the chemical similarity of stars. Non-chemical factors of variation, such as stellar temperatures and instrumental systematics, will not contribute to the representation as their presence would make distances between stellar siblings larger. Instead, such a representation will only contain those factors of variation of spectra that are discriminative of open clusters, ie the chemical factors of variation. Crucially, chemical factors of variation will contribute to distances in the representation in proportion to how precisely they can be estimated from stellar spectra. Stronger chemical features will be more strongly amplified than weaker chemical features.
The utility of this data-driven approach is that it is independent of imperfect model atmosphere approximations and other issues affecting synthetic spectra. This provides a high fidelity technique to turn to specific applications within Galactic archaeology, such as chemical tagging of stars that are most chemically similar \citep{TaggingFreeman}.
The assumption underpinning our work is that this chemical information will be the only information within stellar spectra useful for distinguishing open clusters and so will be the only information captured by our model. If this assumption is true, then the representation induced by the model will measure a form of chemical similarity between stellar spectra.
However, since open clusters, in addition to sharing a common age and near-identical birth abundances, are also gravitationally bound, they can be identified from their spatial proximity if such information is available in the spectra. As such spatial information does not robustly transfer towards identifying dissolved clusters, it mustn't be captured by our model. In this work, we apply our algorithm to pseudo-continuum normalized spectra with diffuse interstellar bands masked, which we assume not to contain any information about spatial location so as for our representation after training to only contain chemical information. Assuming pseudo-continuum normalized spectra do not encode any spatial information is plausible, since after continuum normalization, the spectrum should not contain significant information about stellar distance. With the impact of reddening removed and diffuse interstellar bands masked, a spectrum should also not contain any information about the stellar extinction and interstellar medium along the line of sight of the star. We examine the validity of these assumptions in latter sections. Ultimately, it is worth emphasizing that our method only exploits features proportionally to their discriminativeness at recognizing open clusters. Therefore, we can expect our model to not heavily rely on non-robust features (provided that these are significantly less informative than robust features). This also relies on our open cluster training data being representative of the parameter space that should be marginalized out; i.e. our model does not learn to associate inter cluster stars via \mbox{$\log g$}\ and \mbox{$T_{\rm eff}$}, which could happen if the evolutionary state of observed cluster stars was similar within clusters and different between clusters.
\begin{figure*}
\includegraphics[width=\linewidth ,page=7,trim={2cm 0cm 1cm 0cm},clip]{figures/diagram.pdf}
\caption{Schematic depiction of RSCA. The algorithm proceeds by first encoding stellar spectra into a lower dimensional representation made two-dimensional for illustrative purposes. In this representation, stellar siblings - which are represented by same-coloured dots - are not initially identifiable by their Euclidean distance in the basis (represented by black arrows). The objective of the metric-learning algorithm (dashed blue) is to find a new basis in which distances are informative about which stars are stellar siblings. This objective is realized through three linear steps: a sphering transformation on the dataset, a reparametrization to a suitable basis, and a scaling of the basis vectors.}
\label{fig:diagram_algorithm}
\end{figure*}
\subsection{Principal Component Analysis}\label{sec:PCA}
Our metric-learning algorithm, RSCA, first uses Principle Component Analysis (PCA) to transform the data into a (lower-dimensional) basis that represents the primary variability of the ensemble of spectra we work with.
The principal components of a dataset $X$, of shape $N_{D} \times N_{F}$ containing $N_{D}$ data points and $N_{F}$ features, are an ordered orthogonal basis of the feature space with special properties. In the principal component basis, basis vectors are ordered by the amount of variance they capture. They have the property that for any k, the hyperplane spanned by the first k-axes of the basis is the k-dimensional hyperplane, which maximally captures the data variance. In PCA, the number of principal components used, k, is a hyperparameter controlling the trade-off between the amount of information preserved in the dataset $X$ after compression and the degree of compression.
The principal component basis corresponds to the unit-norm eigenvectors of the covariance matrix of $X$ ordered by eigenvalue magnitude. This can be obtained through diagonalization of the covariance matrix. The principal component basis can also be formulated as the maximum likelihood solution of a probabilistic latent model which is known as Probabilistic Principal Component Analysis (PPCA) (see \cite{bishop}). This probabilistic formulation is useful in that it enables one to obtain the principal components for a dataset containing missing values by marginalizing over these.
As we will make use of further in this paper, the principal components also allow for generating a sphering transformation. This is a linear transformation of the dataset to a new representation, in which the covariance matrix of the dataset $X$ is the identity matrix. Sphering using PCA is carried out by performing a change-of-basis to a modified principal component basis in which the principal components are divided by the square root of their associated eigenvalues.
\section{Relevant Scaled Component Analysis Algorithm}\label{sec:algo}
The input to RSCA is individual stellar spectra, some of which belong to open clusters, and some of which are a reference field sample. The output of RSCA is, for each spectra, a $N_{k}$ vector, in which dimensions are scaled such that distances between $N_{k}$ vectors of pairs of stars encode chemical similarity between those stars.
We step through this in detail below.
\subsection{Overview}
Let us define $X_{\rm clust}$ as the matrix representation of a dataset containing the spectra of known open cluster stars. Analogously, let us define $X_{pop}$ as the matrix representation of a larger dataset of stellar spectra in the field (with unknown cluster membership). These matrices are respectively of shapes $N_{\rm d_{clust}} \times N_{\rm b}$ and $N_{\rm d_{pop}} \times N_{\rm b}$, where $N_{\rm d_{clust}}$ is the number of open cluster stars, $N_{\rm d_{pop}}$ the number of of stars in the large dataset and $N_b$ the number of spectral bins. For our purposes, we assume access to only a limited number of open cluster stars such that $N_{\rm d_{clust}} < < N_{\rm d_{pop}}$. We also assume that the spectra in these matrices are pseudo-continuum normalized spectra, with diffuse interstellar bands masked in a process following that described in Appendix \ref{appendix:masking}. Pseudo-continuum spectra are normalized rest-frame spectra in which the effects of interstellar reddening and atmospheric absorption are removed, in a process described in \cite{Apogee}.
RSCA takes as inputs $X_{clust}$ and $X_{pop}$. Through a series of linear-transformation to these matrices, RSCA maps these matrices into new matrices of shape $N_{\rm d_{clust}} \times N_{\rm K}$ and $N_{\rm d_{pop}} \times N_{\rm K}$ whose entries are the stellar spectra transformed to a metric-learning representation of dimensionality $K$. Euclidean distances in this new metric-learning representation can then be used to measure chemical similarity between spectra. As all the steps of RSCA are linear transformations, the mapping converting from spectra to the metric learning representation can be parameterized by an $N_{\rm K} \times N_{\rm bins}$ matrix and used to convert unseen spectra (or for visualization purposes).
We provide in Figure \ref{fig:diagram_algorithm} a graphical depiction of the linear transformations involved in the RSCA algorithm. RSCA works by first projecting the spectra onto a set of basis vectors with PCA (Step 1). For visualization purpose, this basis is made two-dimensional although it would normally be higher dimensional. After this PCA compression (Step 1), stellar siblings are represented as same-coloured dots whose xy coordinates correspond to coordinates in the PCA basis. Once in the PCA basis a series of linear transformation are applied to the data. For improved clarity, we keep the data fixed throughout our algorithm visualization and represent linear transformations as change-of-basis (black arrows). Step 2 and 3 of the algorithm, find a new basis which more aptly captures spectral variability amongst stellar siblings and Step 4 of the algorithm rescales basis vectors of this basis based on a comparison between their spectral flux variance amongst stellar siblings and amongst field stars. The outcome of the RSCA algorithm is a new representation of the spectra in which dimensions that are unhelpful for discriminating stellar siblings are minimised in amplitude (through a stretching-out of basis vectors). Conversely, dimensions that are helpful in recognizing stars within the same open clusters are made larger (through a squeezing of basis vectors). The $K$ vector of principle components for each star can be collapsed into a measure of chemical similarity through a Euclidean distance measure between stars of their scaled representation, output from RSCA, where $d$ = $\sqrt{(n_{K}- n'_{K})^2}$ for any pair of stars $n,n'$.
We now walk step-by-step through the successive linear-transformations involved in the RSCA algorithm. For following along, the pseudo-code for RSCA is provided in Appendix \ref{appendix:pseudocode} and the full source code of our project which contains a Python implementation is made available online \footnote{https://github.com/drd13/RSCA}.
\subsection{Step 1: Compress the spectra with PCA to reduce the risk of overfitting}
In the first step of our approach, denoted as \textbf``1) Compress Spectra", in Figure \ref{fig:diagram_algorithm}, we apply PCA to $X_{pop}$ to convert the population of stellar spectra into a lower dimensional representation. This dimensionality reduction step serves to make the algorithm more data-efficient which is crucial given the risk of overfitting from the small number of open clusters within our dataset.
As some stellar bins are flagged as untrustworthy, we use Probabilistic PCA (PPCA), a variant of the PCA algorithm which can accommodate missing values. After finding principal components of $X_{pop}$, we compress the data by discarding all but the $K$ largest principal components, where $K$ is a hyperparameter requiring tuning. Then, datasets $X_{pop}$ and $X_{clust}$ are each transformed using the $K$ basis vectors, which we call $Z_{pop}$ and $Z_{clust}$; the representation of the spectra in the PCA basis of $X_{pop}$. These haves shapes of $N_{\rm d_{pop}} \times N_{\rm K}$ and $N_{\rm d_{clust}} \times N_{\rm K}$.
\subsection{Metric Learning: Sphering, Reparameterization and Rescaling}
Step (1) in our procedure of our PCA compression is a pre-processing step. The steps that follow fall into the realm of a general-purpose metric-learning algorithm. These rely on assumptions about the PCA-compressed spectra being satisfied. Performance should be robust to within small departures from these assumptions, but will still ultimately be tied to how well these assumptions are respected. We lay out our assumptions for steps (2)-(4) below.
\subsubsection{Assumptions}
First, we assume that the data (i.e. spectra) in $Z_{pop}$ are well approximated as being drawn from a multivariate Gaussian distribution. That is to say that if we define $\mu_{pop}$ and $\Sigma_{pop}$ as the mean and covariance of $Z_{pop}$, then the stars within $Z_{pop}$ can be assumed as being samples drawn from $z_{pop} \sim N(\mu_{pop},\Sigma_{pop})$.
Next, we make the assumption that individual clusters are themselves approximately Gaussian in the PCA-compressed space. That is to say we posit that the members of open clusters are well approximated as being samples drawn from a distribution $z_{clust} \sim N(\mu_{clust},\Sigma_{clust})$. Crucially, we make the assumption that all open clusters share the same covariance matrix $\Sigma_{clust}$ and only differ in their mean $\mu_{clust}$. This is perhaps our strongest and most important assumption. That is, that the stars within different clusters are distributed following a shared covariance matrix (i.e. clusters have the same shape irrespective of their location in the representation). It is this assumption which allows a linear transformation i.e. a transformation that acts the same across the whole representation, to be an effective approach for measuring chemical similarity. This assumption of what is effectively cluster translation invariance, can be interpreted as assuming that the scatter amongst stellar spectra in physical and chemical parameters should be the same for all clusters irrespective of the clusters parameters which is a sensible assumption. Connecting these assumptions back to Figure \ref{fig:interpretation}, each step 2-4 requires that stars in any population, within clusters and within the field, follow a multivariate Gaussian with an invariant covariance matrix for each individual cluster.
\subsubsection{Step 2: Sphering to transform the population covariance matrix into the identity matrix}
Together, the sphering and reparametrization steps of RSCA serve to transform $\Sigma_{pop}$ and $\Sigma_{clust}$ to a vector representation of stellar spectra in which $\Sigma_{pop}$ and $\Sigma_{clust}$, the covariance matrices of the field stars and clusters, respectively, are diagonal matrices. As there are then no off-diagonal terms in the covariance matrices, this ensures that the variances along basis vectors fully capture the covariance information amongst stellar siblings and amongst field stars.
In Step \textbf{``2) Sphere"} of our algorithm, in Figure \ref{fig:diagram_algorithm}, we linearly transform the vector representation of spectra such that the dataset $Z_{pop}$ has a covariance matrix of identity after transformation. This linear transformation takes the form of a sphering transformation applied to $Z_{pop}$, where in our experiments we use the PCA-sphering scheme. This has the utility of fully capturing the variability amongst stellar spectra and amongst stellar siblings.
\subsection{Step 3: Reparameterization to diagonalize the cluster covariance matrix}
In steps 1 and 2 of RSCA we do operations on the full field population and full cluster population, respectively. In steps 3 and 4, we transform and scale to recognise stellar sibling likeness compared to the field, so consider stellar variability of the stars within individual clusters individually.
In the next step of our algorithm after the sphering transform, \textbf{``3) Reparametrize"} in Figure \ref{fig:diagram_algorithm} represents a change of basis.
In this step, the covariance matrix $\Sigma_{clust}$, is diagonalized. Since we do not have direct access to $\Sigma_{clust}$, as an intermediary step in doing so, a dataset approximately distributed according according to $N(0,\Sigma_{clust})$ is created from the open cluster dataset. This is done by subtracting each star's vector representation from the mean representation of all stars belonging to the same cluster $\widehat{\mu}_{clust}$, such that each cluster becomes zero-centered. It is worth noting that as $\widehat{\mu}_{clust}$ is estimated from a limited number of samples, it will not exactly match with the true $\mu_{clust}$ and so the resultant population will only approximately be distributed according to $N(0,\Sigma_{clust})$.
As PCA basis vectors correspond to (unit-norm) eigenvectors of covariance matrices, the PCA basis obtained by applying PCA to the zero-centered open cluster dataset parametrizes a transformation to a representation in which $\Sigma_{clust}$ is (approximately) diagonal. A change-of-basis to this PCA basis thus parametrizes the desired diagonalization of the cluster covariance matrix. Because the basis vectors of the PCA basis have by construction unit-norm, the covariance matrix $\Sigma_{pop}$ will still be the identity matrix after this change-of-basis and hence both $\Sigma_{clust}$ and $\Sigma_{pop}$ will be diagonal matrices as desired.
\subsection{Step 4: Scaling to maximise discriminative power in identifying chemically similar stars}\label{sec:scaling}
In the final step of the metric learning algorithm, denoted by \textbf{``4) Rescale"} in Figure \ref{fig:diagram_algorithm}, basis vectors are scaled proportionally to their dimension's usefulness at recognizing open clusters. This is done by applying separately to each dimension of the representation a scaling factor. Here, the independent scaling of dimensions is justified by $\Sigma_{clust}$ and $\Sigma_{pop}$ being diagonal covariance matrices.
We use the separate individual clusters within $X_{clust}$ and field populations $X_{pop}$, to measure variance in each dimension to determine our scaling factor. We design our scaling factor such that, after transformation, distances between pairs of random stars quantify the ratio between the probability that pairs of stars originate from the same (open) cluster and the probability that they do not. To put it another way, we seek to scale dimensions such that pairs of stars that are more likely to originate from the same cluster as compared to originating from different clusters have a smaller separation (i.e. Euclidean distance) in the representation than less likely pairs.
Under our set of assumptions, along a dimension $i$ amongst the K dimensions, random stellar siblings (intra-cluster stars) are distributed as $z_{clust} \sim N(\mu_{clust_i},\sigma_{clust_i})$ where $\mu_{clust_i}$ and $\sigma_{clust_i}$ are the mean and standard deviation along dimension $i$ (at this stage in the algorithm). Accordingly, using the standard formula for the sum of normally distributed variables, the one-dimensional distance along $i$ between pairs of random stellar siblings $z_{clust_{1i}}$ and $z_{clust_{2i}}$ follows a half-normal distribution $d_{clust_i} \sim |z_{clust_{1i}}-z_{clust_{2i}}|=|N(0,2 \sigma_{\rm clust_i})|$. Likewise, the distance $d_{pop_{i}}$ between pairs of random field stars follows a similar half-normal distribution $d_{pop_i} \sim |N(0,2 \sigma_{\rm pop_i})|$ where $\sigma_{\rm pop_i}$ is the standard deviation amongst field stars along dimension $i$.
For a pair of stars observed a distance $d_{i}$ away from each other along a dimension $i$, the ratio between the probability of the pair originating from the same cluster (intra-cluster) and the probability of the pair not originating from the same cluster (inter-cluster) is:
\begin{equation}
r_{i}(d_i)=\frac
{
p(d_{clust_i}=d_{i})
}
{
p(d_{pop_i}=d_{i})
}=
|
\frac
{
N(0,2 \sigma_{\rm clust_i})
}
{
N(0,2 \sigma_{\rm pop_i})
}|
\end{equation}
which evaluates to (as distances $d_{i}$ are by design greater than 0)
\begin{equation}
r_{i}(d_i) = A_i e^{\frac{-d_{i}^{2}}{2\sigma_{r_{i}}^{2}}}
\end{equation}
where
\begin{equation}
A_i=\frac{\sigma_{pop_i}}{\sigma_{clust_i}}
\end{equation} and
\begin{equation}
\sigma_{r_{i}}= \frac{\sigma_{\rm clust_{i}}\sigma_{\rm pop_{i}}}{\sqrt{\sigma_{\rm pop_{i}}^2-\sigma_{\rm clust_{i}}^2}}
\end{equation}
As dimensions are assumed to be independent, the probability ratio accounting for all dimensions is the product of the probability ratio of the separate dimensions:
\begin{equation}
r =
\prod_{i=0}^{K}
\frac
{
N(0,2 \sigma_{\rm clust_i})
}
{
N(0,2 \sigma_{\rm pop_i})
}
=
Ce^{
-\frac{1}{2}
\sum_{i=0}^{D}
(\frac{d_i}{\sigma_{r_i}})^2
}
\end{equation}
where $C =\prod_{i=0}^{K}A_{i}$.
From this expression, it can be seen that multiplying dimensions by a scaling factor of $\frac{1}{\sigma_{r_i}}$ leads to a representation in which the probability ratio $r$ is a function of Euclidean distance and where pairs of stars with smaller Euclidean distance separation have a higher probability of originating from the same open cluster as compared to their probability of originating from different cluster than pairs with larger Euclidean separation.
It is clear that dividing dimensions by a scaling factor of $\frac{1}{\sigma_{r_i}}$ induces a representation where distances $d$ which are measured between the scaled reconstructed representation. directly encode the probability of stars originating from the same cluster, as desired for our metric-learning approach. However, using this expression as a scaling factor requires evaluating the $\sigma_{clust_i}$'s and $\sigma_{pop_i}$'s along all dimensions. Because the representation has been sphered, the population's standard deviation is unity along all directions ($\sigma_{pop_i}=1$). We estimate the intra-cluster standard deviations using a pooled variance estimator:
\begin{equation}
\sigma_{\rm clust_i}^{2}
=\frac{
\sum_{j=1}^{k}\left(n_{j}-1\right) \sigma_{ji}^{2}
}{
\sum_{j=1}^{k}\left(n_{j}-1\right)
}
\end{equation}
where $\sigma_{ji}^{2}$ refers to the sample variance along dimension i for the sample of stars belonging to an open cluster j containing $n_{j}$ stars in $X_{clust}$. To make the algorithm more robust to the presence of any outliers in the dataset, such as misclassified stellar siblings, we use the median absolute deviation (MAD) as an estimator for the sample standard deviation $\sigma_{ji}$. That is
\begin{equation}
\mathrm{MAD}=\operatorname{median}\left(\left|X_{i}-\tilde{X}\right|\right)
\end{equation}
where $X_{i}$ and $\tilde{X}$ are respectively the data values and median along a dimension.
To better understand the effect of our scaling factor on the representation it is applied to, it is instructive to look into the impact it has on the distances between stars along dimensions of a representation. When stars belonging to the same cluster have, along a dimension, a similar standard deviation to the full population of stars (i.e. $\sigma_{clust} \approx \sigma_{pop}$), the dimension carries no information for recognizing cluster member stars and the scaling factor accordingly fully suppresses it $\sigma_{r} \rightarrow \infty$. On the other hand, for dimensions where the population's standard deviation $\sigma_{pop}$ is significantly larger than the cluster standard deviation $\sigma_{clust}$, the population's standard deviation is no longer relevant and $\sigma_{r} \approx \sigma_{clust}$. That is to say the scaling devolves into measuring distances relative to the number of standard deviations away from the cluster standard deviation.
\section{Experiments on APOGEE Data}\label{sec:experiments}
We validate our approach for encoding chemical similarity by testing its performance on real data obtained by the APOGEE survey data release 16 \citep{ApogeeDR16}. The APOGEE survey \citep{Apogee} is an infrared, high resolution, high signal-to-noise spectroscopic survey. The APOGEE survey uses a 300-fiber spectrograph \citep{APOGEEspectrographs} installed at the Sloan Digital Sky Survey telescope located at Apache Point Observatory \citep{ApachePointObservatory}.
\subsection{Dataset Preparation}
For our experiments we use spectra from the public APOGEE data release DR16 \citep{ApogeeDR16} to create $X_{clust}$ and $X_{pop}$, our datasets of field and open cluster stars. Our field dataset $X_{pop}$ contains spectra for 151,145 red-giant like stars matching a set of quality cuts on the 16th APOGEE data release described below. Our open cluster dataset $X_{clust}$ contains spectra for 185 stars distributed across 22 open clusters, obtained after further quality cuts using the OCCAM value-added catalogue \cite{Donor_2020}, a catalogue containing information about candidate open cluster observed by APOGEE. We also create baseline datasets $Y$ and $Y_{clust}$ containing stellar abundances for the stars in $X$ and $X_{clust}$. We include abundances for 21 species in $Y$ and $Y_{clust}$: C, CI, N, O, Na, Mg, Al, Si, S, K, Ca, Ti, TiII, V, Cr, Mn, Fe, Co, Ni, Cu, Ce. These abundances are derived from the X\_H entry in the allStar FITS file.
To create the dataset of field stars $X_{pop}$, we make the following dataset cuts. With the intention of only preserving red-giant stars, we discard all but those stars for which $4000<\mbox{$T_{\rm eff}$}<5000$~K and $1.5<\mbox{$\log g$}<3.0$~dex, where we use the $\mbox{$T_{\rm eff}$}$ and $\mbox{$\log g$}$ derived by the ASPCAP pipeline. In addition, we further exclude any stars for which some stellar abundances of interest were not successfully estimated by the ASPCAP pipeline by removing any star containing abundances set to -9999.99 for any of our 21 species of interest. We also exclude all spectra for which the STAR\_BAD flag is set in ASPCAPFLAG. The pseudo-continuum spectra of those remaining stars, as found in the AspcapStar FITS file, were used to create the matrix $X_{pop}$ in which each column contains the spectra of one star.
To create the dataset of open cluster member stars $X_{clust}$, we cross-match our filtered dataset with the OCCAM value-added catalogue \cite{Donor_2020} so as to identify all candidate open clusters observed by APOGEE. We only keep those spectra of stars with open cluster membership probability CG\_PROB$>$0.8 \citep{gaudinclusters}. After this cross-match, we further filtered the dataset by removing those clusters containing only a single member stars as these are not useful for us. Additionally, we further discard one star with Apogee ID ``2M19203303+3755558" found to have a highly anomalous metallicity. After this procedure, 185 OCCAM stars remain, distributed across 22 clusters. We do not cut any stars based on their signal-to-noise ratio. The stars in $X_{pop}$ have a median signal-to-noise ratio of 157.2 and interquantile range of 102.0-272.4 while those in $X_{clust}$ have a median signal-to-noise ratio of 191.4 and interquantile range of 117.7-322.6.
Because of cosmic rays, bad telluric line removal or instrumental issues, the measurements for some bins of stellar spectra are untrustworthy. We censor such bad bins to prevent them from impacting our low-dimensional representation. Censored bins are treated as missing values in the PPCA compression. In this work, we have censored any spectral bin for which the error (as found in the AspcapStar FITS file error array) exceeds a threshold value of 0.05. Additionally, we censor for all stars in the dataset, those wavelength bins located near strong interstellar absorption features. More detail about the model-free procedure for censoring interstellar features can be found in Appendix \ref{appendix:masking}.
\subsection{Measuring Chemical Similarity}\label{sec:measuringchemicalsimilarity}
Evaluating how good a representation is at measuring the chemical similarity of stars requires a goodness of fit indicator for assessing the validity of its predictions. We use the "doppelganger rate" as our indicator. This is defined as the fraction of random pairs of stars appearing as similar or more similar than stellar siblings according to the representation, where similarity is measured in terms of distance $d$ in the studied representation. It is worth noting that this procedure for estimating doppelganger rates is related but different from the probabilistic approach presented in \cite{Ness2018}.
We estimate doppelganger rates on a per-cluster basis by measuring distances between pairs of stars in the RSCA output representation. For each cluster in $X_{clust}$, the doppelganger is calculated as the fraction of pairs composed of one cluster member and one random star whose distance in the studied representation, $d_{intra-family}$ is $\leq$ than the median distance amongst all cluster pairs $d_{inter-family}$ . That is, $d_{intra-family}$ are pairs composed of two confirmed cluster members within $X_{clust}$ and $d_{inter-family}$ is a pair with one random field star selected from $X_{pop}$ and one cluster member from the studied cluster in $X_{clust}$. When calculating $d_{inter-family}$, we only consider pairs of stars with similar extinction and radial velocity, that is to say with $\Delta \rm AK\_TARG<0.05$ and $\Delta \rm VHELIO\_AVG<5$. By only comparing stars at similar extinction and similar velocity, we ensure that any model being investigated cannot reduce it's doppelganger rate through exploiting extinction or radial velocity information in the spectra.
So as to facilitate comparisons between different representations, we aggregate the per-cluster doppelganger rates into a ``global" doppelganger rate which gives an overall measurement of a representations effectiveness at identifying open clusters. The global doppelganger rate is obtained by averaging the per-cluster doppelganger rates through a weighted average in which clusters are weighted by their size in $X_{clust}$.
There is an added subtlety to assessing a representation through its global doppelganger rate. There are very few open cluster stars in the dataset. Therefore, RSCA as a data-driven procedure applied to open clusters, is susceptible to overfitting to the open cluster dataset. To prevent overfitting from affecting results, we carry out a form of cross-validation in which clusters are excluded from the dataset used for the derivation of their own doppelganger rate. In this scheme, calculating the global doppelganger rate of an RSCA representation requires repeated application of our algorithm, each time on a different subset with one cluster removed, as many times as there are open clusters.
We caution that our cross-validation approach has some implications on the derived doppelganger rates. Because, every cluster's doppelganger rate is evaluated on a slightly different data subset, the quoted distances and doppelganger rates are not comparable from cluster to cluster.
\subsection{PCA Dimensionality}
\begin{figure}
\includegraphics[width=\columnwidth]{figures/pca/pca_dimensionality.pdf}
\caption{Global doppelganger rates as a function of the number of PCA components used to encode spectra. Performance with cross-validation is shown in blue while performance without cross-validation is shown in green.}
\label{fig:global_doppelganger}
\end{figure}
The number of principal components used in the compression, or encoding stage of RSCA (Step 1) is an important hyperparameter requiring tuning. In Figure \ref{fig:global_doppelganger}, we plot the doppelganger rate against the number of principal components, both with and without using the cross-validation procedure described in section \ref{sec:measuringchemicalsimilarity}. Results without cross-validation display significant overfitting and are mostly shown in an effort to highlight the importance of the cross-validation procedure.
This Figure illustrates how, unsurprisingly, RSCA's performance is strongly dependent on the PCA dimensionality. Doppelganger rates decrease with increasing PCA dimensionality, up until a dimensionality of size 30. At K $>$ 30, doppelganger rates start increasing because of overfitting. Since on our studied dataset, RSCA reaches its peak performance for 30 PCA components, all further quoted results and figures will accordingly use the first 30 PCA components.
That RSCA's performance still improves up to a dimensionality of 30 is interesting. This demonstrates that a hyperplane of at least size 30 is required for capturing the intrinsic variations of APOGEE stellar spectra, a number noticeably larger than the 10 dimensional hyperplane found in \cite{pca_dimensionality}. Methodological differences between studies may partially explain such differences. For example, the PCA fit in \cite{pca_dimensionality} was applied on spectra displaying limited instrumental systematics and with non-chemical imprints on the spectra preliminary removed. It should also be noted that this does not mean that the chemical space is 30 dimensional. Some PCA dimensions may capture instrumental systematics or non-chemical factors of variation, such as residual sky absorption and emission and interstellar dust imprints. Also, since chemical species leave non-linear imprints, because of the linearity of PCA, each non-linear chemical dimension may require multiple PCA components to be fully captured.
\subsection{RSCA interpretability}
We now study which spectral features are leveraged by the RSCA algorithm when recognizing open clusters. In the RSCA rescaled basis (Step 4), the dimensions are scaled proportionally to their perceived usefulness at measuring chemical similarity. Therefore, factors of variation judged most important by RSCA will correspond to the most strongly scaled dimensions of the representation (ie with the largest $\frac{1}{\sigma_{r_i}}$). Figure \ref{fig:interpretation} shows the relationship between the three features with scaled dimensions with the largest amplitudes and metallicity, [Fe/H] and [$\alpha/Fe$] for a representation obtained by running the RSCA algorithm with a PCA dimensionality of 30.
\begin{figure*}
\includegraphics[width=0.24\textwidth]{figures/interpretation/feature1.pdf}
\includegraphics[width=0.24\textwidth]{figures/interpretation/feature2.pdf}
\includegraphics[width=0.24\textwidth]{figures/interpretation/feature3.pdf}
\includegraphics[width=0.24\textwidth]{figures/interpretation/vrad.pdf}
\caption{Three features judged most important by metric learning approach plotted against [Fe/H] and [$\alpha$/Fe] for the 151,145 stars in $X_{pop}$ and the fourth most-important feature plotted against VHELIO\_AVG (radial velocity ASPCAP label). Location of the 185 stars in $X_{clust}$ (the open cluster dataset used to train the metric learning model) are shown by orange markers.}
\label{fig:interpretation}
\end{figure*}
As seen from the leftmost panel, there is a close relationship between "Feature \#1" - the RSCA dimension with the largest associated scaling factor and the ASPCAP [Fe/H] label. The relationship is close to a one-to-one mapping, which illustrates how this feature traces the metallicity content of the stellar spectra. Because "Feature \#1" (as a direction in a hyperplane of stellar spectra) is a linear function of the stellar spectra and metallicity is a non-linear feature, some degree of scatter in the relationship is expected.
The relationship between "Feature \#2" and [Fe/H] (second panel) exhibits the same bimodality as observed when plotting alpha enhancements [$\alpha$/Fe] against [Fe/H] \citep{Leung2018}. This indicates that "Feature \#2" captures $\alpha$-element enhancements. It is particularly noteworthy that we are able to recover the $\alpha$-element bimodality when the open clusters in our dataset (orange markers) are located only in the low-$\alpha$ sequence. This demonstrates the metric model's capacity to extrapolate to abundance values outside the range of values covered in the open cluster training dataset. This provides evidence that the model may still be effective for stars atypical of those in the open cluster dataset.
The relationship between "Feature \#3" and metallicity (third panel) is not as easily interpreted. Given the non-linear nature of metallicity, it is possible that it encodes residual metallicity variability not captured by "Feature \#1" but it is also possible that it contains some further independent chemical dimension.
This figure illustrates a nice property of RSCA. Because dimensions of the RSCA representation correspond to eigenvectors of the covariance matrix $\Sigma_{clust}$, the RSCA algorithm, at least to first order, assigns the distinct factors of variation within spectra to separate dimensions of the representation. That is, to say that the dimensions of the RSCA representation capture distinct factors of variation, such as the metallicity or the alpha-element abundance, rather than a combination of factors of variation. Additionally, the most important factors of variation for recognzing open clusters occupy the dimensions with the largest scaling factors. This property makes RSCA particularly versatile. For example RSCA can be used to separate out high and low $\alpha$-abundance stars in the disk, or to select low-metallicity stars. Additionally, it is likely that because of this property RSCA could be used to search for hidden chemical factors of variation within stellar spectra, although this has not been attempted in this paper.
We found that some of the dimensions of the RSCA representation showed trends with radial velocity. An example of a dimension showing a trend with radial velocity is shown in the last panel of Figure \ref{fig:interpretation} and an investigation into the detailed causes of the radial velocity trends is presented in Appendix \ref{appendix:radial}. The existence of such trends indicates that even after our pseudo-continuum normalization procedure, RSCA is still capable, at least weakly, to exploit radial velocity information in the spectra to recognize stellar siblings. Because only a subset of the dimensions show such trends, a representation tracing only chemistry can be obtained by only keeping those dimensions which show no trends with radial velocity. In this work, we propose to only keep the first three dimensions of the representation. While this choice may appear particularly stringent, as we will show in coming sections, these three dimensions contain the bulk of the discriminative power of the representation (see Table \ref{tab:dimensionality}).
\subsection{Comparison of using RSCA versus measured abundances in calculating chemical likeness}\label{sec:results}
In this section, we compare the effectiveness of measuring chemical similarity using data-driven approach, with that achievable from using measured stellar abundances. To do so, we compare the doppelganger rates that are obtained by RSCA to those from using stellar abundance labels. The results of such a comparison are shown in Figure \ref{fig:ablation}. For this Figure, doppelganger rates are measured from abundances consistently with the RSCA approach (see Figure caption for more detail), such that any differences in performance can be attributed to underlying differences in the information content of the representation. We remind the reader that the doppelganger rates are evaluated for pairs of stars at the same extinction and radial velocity. This guarantees that the doppelganger rate cannot be artificially reduced through our model exploiting information relating to radial velocity or extinction. Per-cluster doppelganger rates are also provided in Appendix \ref{appendix:local}.
From this Figure, we see that executing all steps of the RSCA is crucial for obtaining low doppelganger rates when working directly with spectra. This is seen from how the low doppelganger rates are only obtained with full application of our metric-learning approach (red). On the other hand, when working with stellar abundances, the RSCA approach appears to bring only limited benefits - as seen from the only slight difference in doppelganger rates between measuring distances in the raw abundance space (blue) and in the transformed space (red) that is obtained by application of our full-metric learning approach minus the PCA compression. This result is not surprising and reflects how most steps of the RSCA approach are designed with the aim of generating a representation comparable to stellar labels. That is, to say one where all factors of variation other than chemical factors of variation are removed. The RSCA approach does however still bring some benefits when working with abundances as seen from the lower doppelganger rates.
This Figure also shows that excluding chemical species improves the doppelganger rate. This is seen from how the doppelganger rate is lower when using a carefully chosen subset of species (right) than when using the full set of abundances (center). This can appear counter-intuitive as it implies that more data leads to worsened performance but in this specific case, where the uncertainties on abundances are not accounted for, it can be justified by the low intrinsic dimensionality of chemical space. Since many species contain essentially the same information, adding species with higher uncertainty into the representation adds noise into the representation. It does this without contributing any additional information beneficial for recognizing open clusters. We expect that such an effect would disappear when accounting for uncertainties on stellar labels, but it is still a good illustration of the brittleness of abundance-based chemical tagging.
The combination of species shown in red is the set of species which were found, after manual investigation, to yield the lowest doppelganger rates. This is the combination of stellar individual element abundance labels Fe,Mg,Ni,Si,Al,C,N (with respect to Fe with the exception of Fe which is with respect to H). The doppelganger rate from this combination of species, of 0.023, despite being the smallest doppelganger rate achieved from stellar labels, is higher than the doppelganger rate obtained from stellar spectra of 0.020 (2 percent). That our method is able to produce better doppelganger rates from spectra than from stellar labels highlights the existence of information within stellar spectra not being adequately captured by stellar labels. While stellar labels are derived from synthetic spectra which only approximately replicate observations, our fully data-driven model makes direct use of the spectra translating into lower doppelganger rates.
\begin{figure}
\includegraphics[width=\columnwidth]{figures/ablation/ablation.pdf}
\caption{Global doppelganger rates estimated for varying metric-learning approaches and representations. On the x-axis, ``spectra'' refers to doppelganger rates obtained from spectra $X$ after dimensionality reduction with PPCA to a 30-dimensional space, ``all abundances" to doppelganger rates obtained from a representation formed from the full set of APOGEE abundances in $Y$, ``abundance subset" to doppelganger rates obtained using a representation formed only from the abundances for the following species: Fe,Mg,Ni,Si,Al,C,N. Global doppelganger rates "on raw" (blue) are obtained by measuring distances in the raw representation without any transformation of the representation, ``on scaled'' (green) are obtained by applying the scaling transform on the raw representation without preliminary application of the sphering and reparametrization transform (Steps 1 and 4 for spectra and only Step 4 for abundances which do not need dimensionality reduction), "on transformed" are obtained by applying all steps of the proposed metric learning approach (Steps 1,2,3 and 4 for spectra and Steps 2,3,4 for abundances). As the implementation of the PPCA algorithm used in this paper yielded stochastic PCA components, doppelganger rates from spectra correspond to the mean across 10 runs with error bars corresponding to the standard deviation amongst runs.}
\label{fig:ablation}
\end{figure}
\begin{figure}
\includegraphics[]{figures/datasize/datasize.pdf}
\caption{Expected global doppelganger rates when training a metric-learning model on only a subset of all open clusters in $X_{clust}$ with a number of clusters given by the x-axis. Results for different PCA dimensionalities used for compressing stellar spectra are represented by different colored lines. Clusters used in the expectated doppelganger rate calculations were chosen randomly from $X_{clust}$, and quoted results are for the average of 50 repeated trials.}
\label{fig:datasize}
\end{figure}
\subsection{Dimensionality of chemical space}
That the PCA representation is 30 dimensional does not mean that all 30 dimensions carry information useful for recognizing open clusters. To get a grasp of the dimensionality of the chemical space captured by RSCA, we calculated the doppelganger rates for RSCA representations in which only the dimensions with the largest scaling factors are kept (ie other dimensions are excluded from the doppelganger rate distance calculations). We calculate doppelganger rates multiple times, each with a different number of dimensions preserved. The results of this investigation are shown in Table \ref{tab:dimensionality}. From this Table, we see that the dimensionality of spectra appears to be, at least to first degree, extremely low. The top two dimensions of the RSCA model (as shown in Figure \ref{fig:interpretation}) are capable of matching the performance obtained from stellar labels whilst the top four dimensions exceed the performance from using the full representation. The four-dimensional representation is even more effective at recognizing chemically identical stars than the full RSCA representation which itself was more effective than stellar labels. It is not a new result that the dimensionality of chemical space probed by APOGEE is low. Recent research suggest that, at the precisions captured by APOGEE labels, chemical abundances live in a very low-dimensional space, for stars of the disk. For example, it was found in \cite{Ness2019,HowManyElements} that [Fe/H] and stellar age could predict all other elemental abundances to within or close to measurement precision. However, while previous analysis have depended on abundances to show this, here we can do this directly from spectra. As our methodology directly picks up on factors of variation and, if not controlled for, is capable of picking up on weak factors of variation such as diffuse interstellar bands, we can be confident that any remaining chemical factors of variation are either i) highly non-linear for our model to not be capable of picking up on them, ii) very weak spectral features, or iii) not particularly discriminative of open clusters as would be the case for chemical variations arising from internal stellar processes for example induced by accretion of planetary materials or internal stellar processes.
\begin{table}
\centering
\begin{tabular}{p{1.5cm} p{3.0cm}}
\hline
N & Doppelganger rate \\
\hline
1 & $0.0962 \pm 0.0212$ \\
2 & $0.0219 \pm 0.0025$ \\
3 & $0.0198 \pm 0.0028$ \\
4 & $0.0182 \pm 0.0021$ \\
5 & $0.0180 \pm 0.0021$ \\
6 & $0.0188 \pm 0.0017$ \\
7 & $0.0184 \pm 0.0017$ \\
30 & $0.0199 \pm 0.0015$ \\
\end{tabular}
\caption{Global doppelganger rate obtained by the RSCA model applied to stellar spectra in which all but the N most strongly scaled dimensions of a 30 dimensional RSCA representation are discarded. As the implementation of the PPCA algorithm used in this paper yielded stochastic PCA components, doppelganger rates from spectra correspond to the mean across 10 runs with error bars corresponding to the standard deviation amongst runs.}
\label{tab:dimensionality}
\end{table}
\subsection{Impact of Dataset Size}
Our method learns to measure chemical similarity directly from open clusters without reliance on external information. Because of this, its performance will be tightly linked to the quality and quantity of data available. Figure \ref{fig:datasize} attempts to estimate our method's dependency on the size of the open cluster dataset. In this figure, for varying PCA dimensionalities, we plot the expected doppelganger rate for an open cluster dataset containing a given quantity of open clusters, whose number is given by the x-axis. We estimate the expected doppelganger rate for a given number of open clusters by estimating and averaging the doppelganger rates for all data subsets containing that number of clusters. From this figure, we see that the larger PCA dimensionalities still benefit from the addition of open clusters. This is suggestive that performance would likely further improve with access to additional open clusters. Such larger datasets may also enable the usage of more complex non-linear metric-learning approaches which may yield further improvements not captured in this figure.
\section{Discussion}\label{sec:discussion}
We have presented a novel approach for identifying chemically similar stars from spectroscopy based on training a metric-learning model on the spectra of open cluster stars. This approach has several appealing properties. It is end-to-end data-driven in the sense that it does not rely on synthetic models nor labels derived from synthetic models. This method only makes use of open cluster members, which can themselves be identified with minimal reliance on theoretically-derived quantities \citep{openclusteridentification1,openclusteridentification2,openclusteridentification3}. This makes the method insensitive to the domain gap of synthetic spectra. Additionally, where traditional spectral fitting approaches require instrumental systematics to be fully suppressed, lest they further exacerbate the domain gap, our fully data-driven approach, at least in theory, automatically learns on its own to ignore most variations due to instrumental systematics.
We expect that the approach that we have developed will perform particularly well in a number of regimes. For example, low resolution spectra, where blended features lead to compounding model inaccuracies. Additionally, for M-type stars, where molecular features complicate the retrieval process. In general, our method will likely be an efficient and effective approach where theoretical models are inaccurate or the observed spectra itself is plagued by complex systematics. This is for example the case for M dwarfs \citep{Birky_2020,Behmard_2019}.
Although our data-driven algorithm shows excellent performance, one may wonder whether there is still room for further improvements, particularly since strong chemical tagging, if ever possible, will require improvements in our chemical similarity measurements \citep{HowManyElements}. There are reasons to be hopeful here.
Firstly, our data-driven model comes with clear limitations which are not inherent to the approach, but rather imposed by our modelling choices. There are many algorithmic choices for building a metric learning model optimized at distinguishing open clusters \citep[e.g.][]{LargeMarginNearestNeighborClassification,Neighbourhood,relevantComponentAnalysis,pml1Book} and ours is only one of many choices. In particular, the ability of RSCA to extract the chemical content of stellar spectra is constrained by its linear nature. Although this linearity is convenient for avoiding over-fitting, enabling better out-of-distribution performance and cross-validation, it also artificially limits the precision with which ``pure" features containing only chemical signal can be learned. One can be hopeful that a suitably regularized non-linear metric learning models, such as for example twin neural network \citep{siamese,pml1Book}, could surpass our model. However, building such a model from our limited number of open clusters would present its own unique challenges.
Secondly, by being entirely data-driven, the performance of this approach is inexorably linked to the quality and quantity of data available. This makes it poised to benefit from new open cluster discoveries and/or deliberately targeted observations. Improvements may also be possible by leveraging other source of chemically similar stars such as wide binaries.
Our method also comes with caveats. Data-driven methods do not extrapolate well outside of their training dataset. As such, performance may be lower for clusters that are atypical compared to those in the open cluster reference set. Since open clusters are typically younger stars \citep{openclusterreview}, this means performance may be decreased on older cluster stars. However, given the tight relationships between RSCA dimensions and chemical parameters for stars in $X_{pop}$, such an effect is likely to be small. Additionally, our model makes no use of the error information in spectra, which is valuable information that could likely be squeezed-out for even better performance.
Another downside of our approach is its coarse-grained nature. While stellar labels provide a fine-grained view into chemical similarity with a breakdown into chemical composition of individual species, our approach only provides a coarse-grained measurement of chemical similarity. This limits the types of scientific problems the approach can be used to answer. However, it may very well be possible to extend the method from measuring overall chemical similarity to measuring individual elemental abundances. A way in which this could possibly be done is through applying it to windows centered on the locations of stellar lines instead of to the full spectrum. Also, it is not always clear what exact information is captured by the representation, and in particular there is always a risk, despite all of our checks, that the model is acting on non-chemical information within the spectra.
\section{Conclusion}
Large-scale Galactic surveys, the likes of APOGEE, LAMOST and GALAH, have collected hundreds of thousands of high-quality stellar spectra across the galaxy. These surveys are vastly broadening our understanding of the Milky Way. How best to analyse these spectra however still remains an open-question. One limitation is that traditional spectral fitting methods currently do not make full use of the information in stellar spectra. This is largely because our stellar models are approximations.
In this paper we developed a fully data-driven, linear metric learning algorithm operating on spectra for extracting the chemical information within stellar families. Through experiments on APOGEE, we demonstrated that our metric learning model identifies stars within open clusters more precisely compared to using stellar labels which indicates an improved ability to discriminate between chemically identical stars. We further found that our model's capacity to distinguish open clusters could largely be attributed to a two-dimensional subspace of our final representation which was found to approximately coincide with metallicity and $\alpha$-elemental abundances. That our model's capacity at recognizing open clusters plateaus at N$\sim$ 4 supports the idea that the dimensionality of chemical space probed by APOGEE is, for Galactic archaeology purposes, low, in the disk. However, we do find hints of further dimensions potentially containing chemical information.
There are several reasons why our metric-learning approach could be favoured over using stellar labels. It can be applied to spectra of stars for which we do not yet have very good synthetic spectra and so would otherwise not be able to analyze well. It is completely independent of our theoretical knowledge of stellar atmospheres and so could be used to validate existing astronomical results in a way which is independent of any biases which may exist in our synthetic spectra. Finally and perhaps most importantly, whereas the traditional derivation of stellar labels is fundamentally limited by our inability to generate faithful synthetic spectra, our metric learning approach does not suffer from such a limitation. This means that by improving the quality of the training dataset and the metric-learning approach used, performance may be further improved.
\begin{acknowledgements}
DDM is supported by the STFC UCL Centre for Doctoral Training in Data Intensive Science (grant number ST/P006736/1). DDM thanks Serena Viti for helpful discussions.
Melissa K Ness is supported in part by a Sloan Foundation Fellowship.
Funding for the Sloan Digital Sky
Survey IV has been provided by the
Alfred P. Sloan Foundation, the U.S.
Department of Energy Office of
Science, and the Participating
Institutions.
SDSS-IV acknowledges support and
resources from the Center for High
Performance Computing at the
University of Utah. The SDSS
website is www.sdss.org.
SDSS-IV is managed by the
Astrophysical Research Consortium
for the Participating Institutions
of the SDSS Collaboration including
the Brazilian Participation Group,
the Carnegie Institution for Science,
Carnegie Mellon University, Center for
Astrophysics | Harvard \&
Smithsonian, the Chilean Participation
Group, the French Participation Group,
Instituto de Astrof\'isica de
Canarias, The Johns Hopkins
University, Kavli Institute for the
Physics and Mathematics of the
Universe (IPMU) / University of
Tokyo, the Korean Participation Group,
Lawrence Berkeley National Laboratory,
Leibniz Institut f\"ur Astrophysik
Potsdam (AIP), Max-Planck-Institut
f\"ur Astronomie (MPIA Heidelberg),
Max-Planck-Institut f\"ur
Astrophysik (MPA Garching),
Max-Planck-Institut f\"ur
Extraterrestrische Physik (MPE),
National Astronomical Observatories of
China, New Mexico State University,
New York University, University of
Notre Dame, Observat\'ario
Nacional / MCTI, The Ohio State
University, Pennsylvania State
University, Shanghai
Astronomical Observatory, United
Kingdom Participation Group,
Universidad Nacional Aut\'onoma
de M\'exico, University of Arizona,
University of Colorado Boulder,
University of Oxford, University of
Portsmouth, University of Utah,
University of Virginia, University
of Washington, University of
Wisconsin, Vanderbilt University,
and Yale University.
\end{acknowledgements}
|
1,116,691,497,738 | arxiv | \section{Introduction}
\label{sec:intro}
Planets with strong magnetic fields may generate radio and/or X-ray emission when interacting with energetic charged particles.
It is well known that Jupiter emits radio waves from its auroral region due to the cyclotron-maser instability \citep[e.g.][]{wu1979,zarka1998,treumann2006}.
Exoplanets could also generate radio emission through similar mechanisms, depending on their intrinsic magnetic fields and the properties of surrounding plasmas, e.g. stellar wind particles and particles from Io-like moons.
This process could provide an avenue to discover planets that are otherwise extremely difficult to find --- those orbiting highly evolved stars.
Unlike traditional methods of discovering exoplanets, in which the planetary signal is a tiny perturbation on light from the star, low-frequency planetary radio emission might be an arena where planets are vastly brighter than their stars.
This paper is an exploration of the surprisingly diverse range of physical processes that lead to this emission and the prospects for detecting radio emission from planets around giant stars with current and near-future low-frequency radio observatories.
Observations of radio emission from solar system planets imply an empirical relation: the auroral radio power is proportional to the input solar wind energy going into each planetary magnetosphere. This is commonly referred to as the ``radiometric Bode's law'' \citep{desch+kaiser1984,zarka2001}.
Extrapolating this scaling to exoplanets, radio emission from exoplanetary systems has been examined.
\citet{farrell1999}, \citet{zarka2001}, and \citet{lazio2004} estimated that a few of the known exoplanetary systems may have radio flux of $\sim$1~mJy level, due to the small orbital distance of the planets.
\citet{stevens2005} gave improved estimates of the stellar mass-loss rate based on X-ray flux and re-evaluated the radio flux of known exoplanetary systems.
\citet{griesmeier2005} took account of the high stellar activity in the early stage of the planetary system and proposed that the young system would be a good candidate in which to search for radio emission.
\citet{griesmeier2007a, griesmeier2007b} discussed the effects of the detailed properties of stellar wind in the proximity of the stars. They considered not only the kinetic energy of the stellar wind but also the magnetic energy of the stellar wind and coronal mass ejection.
Note that in these papers, the scaling relations for the planetary magnetic field differ from paper to paper.
In \citet{reiners2010}, the authors adopted a new scaling relation of planetary magnetic fields based on \citet{christensen_et_al2009}.
\citet{jardine2008} modeled the reconnection between the magnetic field of a close-in planet and that of the host star to obtain estimates that are not based on the ``radiometric Bode's law.''
Although observational searches for these radio signatures are underway, no clear detection has been claimed \citep{bastian2000,george2007,smith2009,stroe2012,hallinan2013,murphy2015}, while there are some promising initial results \citep{lecavelier_et_al2013,sirothia2014}.
When stars less than $\sim$8~$M_\sun$ evolve off the main sequence, they evolve through the red-giant branch (RGB) and the asymptotic-giant branch (AGB) phases where their radii and luminosities increase by orders of magnitude.
Jovian planets in orbit around such stars can migrate inward or outward due to the interplay between tidal torques and the mass-loss process on the post-main sequence \citep{nordhaus_et_al2010, kunitomo2011, mustill+villaver2012, spiegel2012, nordhaus+spiegel2013}.
During this time, such planets can be transiently heated to hot-Jupiter temperatures ($\gtrsim$1000~K) at distances out to tens of AU, depending on the star's mass; such planets are termed ``Red-Giant Hot Jupiters'' (RGHJs), though the term can refer to planets orbiting either RGB and AGB stars \citep{spiegel+madhusudhan2012}.
Such planets are also subject to interactions with a massive (but slow) stellar wind, as the mass-loss rate of evolved stars is significant, ranging from $\sim$10$^{-8}$~$M_\odot$~yr$^{-1}$ to $\sim$10$^{-5}$~$M_\odot$~yr$^{-1}$ with the highest values for AGB stars \citep[e.g.,][]{reimers1975, schild1989, vassiliadis1993, schoier2001, vanloon2005}.
On the assumption that the radio emission is correlated with the stellar wind, planetary companions around evolved stars could also generate bright radio emission.
Based on this speculation, \citet{ignace2010} examined radio emission from known substellar-mass companions around cool evolved stars.
They found that the low ionization fraction of the stellar wind of evolved stars suppresses their radio emission and leads to weak radio emission.
In addition, they considered a scenario in which post-bow shock heating could produce ionized hydrogen atoms. This indicates a configuration of radio-wave generation very different from that of the solar system, and the lower-limit estimate, at least, is well below the detection limit.
In this paper, we consider a further plausible scenario, where the accretion of the massive stellar wind onto the planet would emit UV and X-ray photons that ionize the stellar wind in the vicinity of the planet.
The ionized stellar wind particles would then interact with the planetary magnetic field in the same way as the solar system planets do.
Thus, by extrapolating the ``radiometric Bode's law,'' we provide more optimistic estimates compared to the previous result.
In addition, we introduce two major advances beyond the work of \citet{ignace2010} that are related to the observability of radio emission from RGHJs.
First, we estimate the plasma frequency cut-off of the stellar wind, which turns out to be one of the major obstacles to detecting radio emission from RGHJs.
Second, we discuss the planetary parameter space to search for radio emission, employing scaling laws for the planetary magnetic field, noting that the survey of exoplanets around highly evolved stars is not complete.
In Section \ref{s:assumptions}, we present the framework to obtain the frequency and the flux of planetary auroral radio emission, and we describe our models for the stellar wind and planetary magnetosphere.
Section \ref{s:result} presents the estimates of the spectral radio flux of RGHJs and compares the predictions with what might be expected from canonical hot Jupiters as well as those from Jupiter-twins.
Section \ref{s:observability} gives the prospects for the signal detection with the current/future instruments.
Estimates for the known late-type (M-type) evolved stars are also included.
Finally, Section \ref{s:conc} concludes the paper with a brief summary.
\section{Model}
\label{s:assumptions}
In this section, we describe our framework to estimate radio emission from RGHJs.
First, we introduce our scheme to compute the frequency and intensity of planetary radio emission in Sections \ref{ss:model_frequency} and \ref{ss:model_intensity}, respectively.
Then, two ingredients for the framework --- the strength of the planetary magnetic field and the properties of the stellar wind --- will be presented in Sections \ref{ss:magneticfield} and \ref{ss:stellarwind}, respectively.
In Section \ref{ss:ionization}, we consider the ionization around the planets, which is a crucial factor to determine the efficiency of the interaction between the planetary magnetosphere and the stellar wind.
\subsection{Frequency of Radio Emission}
\label{ss:model_frequency}
As ionized electrons flow along planetary magnetic field lines, auroral radio waves are emitted at the local cyclotron frequency.
The upper limit is around the cyclotron frequency of the planetary surface magnetic field, $\nu_{\rm cyc,\,max}$:
\begin{equation}
\nu_{\rm cyc,\,max} = \frac{eB}{2\pi m_e c} \approx 28 {\rm~MHz} \left( \frac{B}{10 \rm~G} \right) \label{eq:fcyc}
\end{equation}
where $B$ is the strength of the magnetic field at the planetary surface, $e$ and $m_e$ are the charge and mass of electrons, respectively, and $c$ is the speed of light.
Radio emission from an exoplanet is observable from the ground (on Earth) only when its frequency is greater than both the plasma frequency of Earth's ionosphere $\nu_{\rm plasma}^\oplus$ and the maximum plasma frequency along the line of sight $\nu_{\rm plasma}^{\rm los}$:
\begin{equation}
\nu_{\rm cyc,\,max} > \nu_{\rm plasma}^\oplus \;\;\; \mbox{and} \;\;\; \nu_{\rm cyc,\,max} > \nu_{\rm plasma}^{\rm los}
\end{equation}
The plasma frequency may be expressed as
\begin{eqnarray}
\nu_{\rm plasma} & = & \sqrt{\frac{n_e e^2}{\pi m_e}} \\
& = & 8979 {\rm~Hz} \times \left( \frac{n_e}{\rm cm^{-3}} \right)^{1/2} \, .
\label{eq:fplasma}
\end{eqnarray}
In the Earth's ionosphere, the electron number density is less than $10^6$~cm$^{-3}$, which implies that $\nu_{\rm plasma}^\oplus \lsim 10$~MHz.
Along the line of sight, the maximum plasma frequency $\nu_{\rm plasma}^{\rm los}$ is typically governed by the density of stellar wind particles around the planet, which will be specified in Section \ref{ss:stellarwind} below.
Another factor that affects the radio emission is the plasma frequency at the site of radio-wave generation. This is because the cyclotron-maser instability as a mechanism to generate intense radio emission is efficient when the {\it in situ} plasma frequency is small in comparison to the local cyclotron frequency \citep[][]{treumann2006}.
Future work will be required to estimate the local plasma density and examine this condition.
In this paper, we assume that this condition is satisfied at least at some region along the poloidal magnetic field lines.
\subsection{Flux of Radio Emission}
\label{ss:model_intensity}
The auroral radio spectral flux of exoplanets observed at the Earth, $F_{\nu}$, can be expressed by:
\begin{equation}
F_{\nu} = \frac{P_{\rm radio}}{\Omega l^2 \Delta \nu}
\label{eq:Fnu}
\end{equation}
where $P_{\rm radio}$ is the energy that is deposited as auroral radio emission of the considered frequency range, $\Omega$ is the solid angle of the emission, $l$ is the distance between the target and the Earth, and $\Delta \nu$ is the frequency bandwidth.
We estimate the radio emission of exoplanets, $P_{\rm radio}$, simply by scaling the Jovian auroral radio emission, $P_{\rm radio,\,J}$, with the input energy from stellar wind, in the same manner as \citet{griesmeier2005,griesmeier2007a,griesmeier2007b}.
The scaling is based on the empirical/apparently good correlation between the radio emission intensity of solar system planets and the input kinetic energy, $P_{\rm inp,\,k}$, or the magnetic energy of the solar wind, $P_{\rm inp,\,m}$, \citep[the ``radiometric Bode's law'';][]{desch+kaiser1984,zarka2001}, i.e.,
\begin{eqnarray}
P_{\rm radio} &\propto & P_{\rm inp} \\
P_{\rm inp,\,k} &=& m_p n v^3 \cdot \pi r_{\rm mag} ^2 \label{eq:Pinp_kin}, \\
P_{\rm inp,\,m} &=& (B_{\star\bot }^2/8 \pi ) \, v \cdot \pi r_{\rm mag} ^2 \label{eq:Pinp_mag} \, ,
\end{eqnarray}
where $m_p$ is the proton mass, $n$ is the number density of the stellar wind, $v$ is the relative velocity of the stellar wind particles to the planet, $ B_{\star\bot }$ is the interstellar magnetic field perpendicular to the stellar wind flow, and $r_{\rm mag}$ is the distance from the center of the planet to the magnetic stand-off point, described below.
It is not clear from observations of the solar system planets which of the above two relations (Eqs.~\ref{eq:Pinp_kin} and \ref{eq:Pinp_mag}) most accurately captures the ``true'' relationship between input ingredients and output radio power \citep{zarka2001}.
In this paper, we assume that the radio emission scales with the input kinetic energy (i.e., that $P_{\rm radio} \propto P_{\rm k,\,inp}$, as per Eq.~\ref{eq:Pinp_kin}) and consider the possible effects of a dense stellar wind.
Note that if Eq.~\ref{eq:Pinp_mag} is actually the better predictor of radio power, it is difficult to estimate the emission from RGHJs at this point, because magnetic fields of evolved stars are not well constrained for highly evolved stars; most observations set only upper limits \citep[e.g.,][]{konstantinova2010,konstantinova2013,petit2013,tsvetkova2013,auriere2015}.
In the case of the M-type giant EK boo, a surface magnetic field $\sim$0.1-10~G has been measured.
In this particular case, given the large stellar radius ($R_\star \sim 210 R_{\odot }$), the magnetic moment may be about $10^3$$\times$ larger than that of the Sun and therefore may also increase the planetary radio emission.
However, due to our present ignorance about the strength of magnetic fields of evolved stars, we leave the magnetic model of radio emission for RGHJs for future work.
In reality, the total power of Jovian auroral emission varies greatly over time.
The average is of the order of $1.3\times 10^{10}$~W. During highly active periods, it averages more like $8.2\times 10^{10}~$W, and the emission can reach powers as high as $4.5 \times 10^{11}$~W during peak activity \citep{zarka_et_al2004}.
Here, we employ $P_{\rm radio,\,J}=2.1\times 10^{11}$~W as a canonical value, following \citet{griesmeier2005,griesmeier2007b}.
The magnetic stand-off point is where stellar wind ram pressure and planetary magnetic pressure are in approximate balance:
\begin{equation}
m_p n v ^2 \sim \frac{B^2}{8\pi}\left( \frac{r_{\rm mag}}{R_p} \right)^{-6} \label{eq:stand-off}.
\end{equation}
The outflow ram pressures from heated planets are negligible compared to these pressures, as discussed in Appendix \ref{ap:outflow}.
Therefore,
\begin{equation}
r_{\rm mag} \sim R_p \left( 8\pi m_p n \right) ^{-1/6} v^{-1/3} B^{1/3} \label{eq:stand-off-radius}
\end{equation}
The radius obtained for Jupiter from this equality is about half of the actual magnetospheric radius \citep[][]{griesmeier2005}.
To estimate $r_{\rm mag}$ for RGHJs, we scale this radius according to the parameter dependence of equation~(\ref{eq:stand-off-radius}).
For exoplanets, we assume that the solid angle of the emission ($\Omega$) is the same as that of Jupiter.
In reality, the solid angles of auroral radio emission from Jupiter, Saturn, and Earth are $\sim$1.6, $\sim$6.3, and $\sim$3.5 steradians, respectively \citep{desch+kaiser1984}, which are on the same order and will not significantly affect our order-of-magnitude estimate of radio emission.
The bandwidth, $\Delta \nu$, is assumed to be proportional to the representative frequency of the emission, which is the cyclotron frequency, following \citet{griesmeier2007b}.
\subsection{Assumptions for Planetary Magnetic Field}
\label{ss:magneticfield}
In order to obtain the frequency and the intensity of the radio emission, we need to compute the strength of the magnetic field at the planetary surface, $B$.
We do so simply by scaling the Jovian magnetosphere at the surface $B_J \sim 10$ G according to the planetary mass and age, based on the scaling relation described below.
Several scaling relations for planetary magnetic field strength have been proposed \citep[e.g.][]{busse1976,russel1978,stevenson1979,mizutani1992,sano1993,starchenko2002,christensen2006,christensen_et_al2009}, which are summarized and compared with numerical simulations in \citet{christensen2010}.
We employ the scaling law proposed in \citet{christensen_et_al2009} and used in \citet{reiners2010} to explore the evolution of planetary magnetic fields:
\begin{equation}
B_{\rm dynamo} ^2 \propto f_{\rm ohm}\;\rho_{\rm dynamo}^{1/3} \; (Fq_o)^{2/3} \, , \label{eq:Bscaling}
\end{equation}
where $B_{\rm dynamo}$ is the mean magnetic field in the dynamo region, $f_{\rm ohm}$ is the ratio of ohmic dissipation to the total dissipation, $\rho_{\rm dynamo} $ is the mean density in the dynamo region, $F$ is an efficiency factor of order unity, and $q_o$ is the convected flux at the outer boundary of the dynamo region (see \citealt{christensen_et_al2009} for a comprehensive description).
This scaling law is based on the assumption that the ohmic dissipation energy is a fraction of the available convected energy and was found to be in good agreement with both the numerical experiments (over a wide parameter space) and with known objects from the Earth to stars.
Here, $f_{\rm ohm}$ and $F$ are assumed to be constant for the bodies considered in this paper.
Dipole magnetic field strength at the planetary surface, denoted by $B$, is then scaled by
\begin{equation}
B \propto B_{\rm dynamo} \left( \frac{r_{\rm dynamo}}{R_p} \right)^3 \, . \label{eq:Bscaling2}
\end{equation}
where $r_{\rm dynamo}$ is the radius of the outer boundary of the dynamo region.
The scaling law for $B_{\rm dynamo}$, Equation (\ref{eq:Bscaling}), is reasonable only for rapidly rotating objects.
Unlike canonical hot Jupiters, RGHJs are not tidally locked to their host stars, so they probably have the rapid rotation needed for Eq.~\ref{eq:Bscaling} to be a useful ansatz \citep{spiegel+madhusudhan2012}.
In this paper, we assume that RGHJs indeed are rapidly rotating so that they generate planetary magnetic fields through the same mechanism as Jupiter.
In order to evaluate $\rho _{\rm dynamo}$ and $q_o$, we need a model of the internal planetary structure.
We consider Jupiter-like gaseous planets and assume that the planetary radius is constant at $R_p = R_{p,{\rm J}}$, as numerical calculations show that the radii of gaseous planets over the range of $0.1 M_{p,{\rm J}} < M_p < 10M_{p, {\rm J}}$ (with core mass less than 10\%) converge to $0.8 R_{p,{\rm J}} < R_p < 1.2R_{p, {\rm J}}$ within 1 Gyr \citep{fortney2007, spiegel+burrows2012, spiegel+burrows2013}.
For the density profile, we follow \citet{griesmeier2007b} and assume a polytropic gas sphere with index $n=1$, which results in:
\begin{equation}
\rho [r] = \left( \frac{\pi M_p}{4 R_p^3} \right) \frac{\sin \left[ \pi \frac{r}{R_p} \right]}{\left( \pi \frac{r}{R_p} \right)} \, . \label{eq:rho_r}
\end{equation}
We determine the radius of the outer boundary of the dynamo region, $r_{\rm dynamo}$, by assuming that the hydrogen becomes metallic when $\rho (r)$ exceeds the critical density $\rho_{\rm crit} = 0.7\,\mbox{g/cm}^3$ \citep{2006explanets_guillot, griesmeier2007b}.
The density of the metallic core, $\rho _{\rm dynamo}$ is obtained by averaging the density in the core.
In the case of Jupiter, $r_{\rm dynamo, J} = 0.85 R_{\rm J}$ and $\rho_{\rm dynamo, J} = 1.899~{\rm g/cm}^3$.
The scaling of convected heat flux at the outer boundary, $q_o$, is obtained by dividing the age-dependent net planetary luminosity, $L_p$, by the surface area of the core region, i.e., $4\pi r_{\rm dynamo}^2$.
The time-dependent luminosity is taken from equation (1) of \citet{burrows_et_al2001} \citep[see also][]{marley2007}.
Ignoring the relatively weak dependence on average atmospheric Rosseland mean opacity leads to:
\begin{equation}
L_p \sim 6.3\times10^{23} \; {\rm erg} \; \left( \frac{t}{4.5 \rm~Gyr} \right)^{-1.3} \left( \frac{M_p}{M_{\rm J}} \right)^{2.64} \, .
\label{eq:burrowsLum}
\end{equation}
Therefore, we have
\begin{equation}
q_o \sim q_{o, {\rm J}} \left( \frac{t}{4.5 \rm~Gyr} \right)^{-1.3} \left( \frac{M_p}{M_{\rm J}} \right)^{2.64} \left( \frac{r_{\rm dynamo}}{r_{\rm dynamo,\,J}} \right)^{-2}\, .
\label{eq:burrowsHeatFlux}
\end{equation}
\subsection{Assumptions for Stellar Wind}
\label{ss:stellarwind}
The other key ingredient for radio emission is the stellar wind.
The number density of particles in the stellar wind, $n$, can be expressed as
\begin{equation}
n = \frac{\dot M_\star}{4\pi a^2 m_p v_{\rm sw}} \, ,
\label{eq:n}
\end{equation}
where $\dot M_\star$ is the stellar mass-loss rate, $a$ is the orbital distance from the star, $m_p$ is the proton mass, and $v_{\rm sw}$ is the velocity of the stellar wind.
For the solar wind, $\dot M_\odot \sim 2\times 10^{-14} M_{\odot}$/yr and $v_{\rm sw} \sim 400~{\rm km/s}$ \citep[e.g.,][]{hundhausen1997}.
The mass-loss rate of red giants is typically $\dot M_\star \sim 10^{-8}-10^{-7} M_{\odot}$/yr \citep{reimers1975}, and the rate can be as high as $10^{-5} M_\odot{\rm /yr}$ during the AGB phase \citep{schild1989, vassiliadis1993, schoier2001, vanloon2005}.
Therefore, we have
\begin{equation}
\frac{\dot M_\star}{\dot M_{\odot}} \sim 10^6 - 10^9 \, . \label{eq:scale_Mdot}
\end{equation}
The stellar wind velocity, $v_{\rm sw}$, becomes smaller as the star evolves.
The wind velocity is typically of the order of the escape velocity at a distance of several times the stellar radius \citep{suzuki2007}, i.e., $\sim$$\sqrt{2GM_{\star}/R'_{\star}}$, where $R'_{\star}$ is several times $ R_{\star}$.
For a star with radius $R_{\star }=100~R_{\odot }$, this results in $v_{\rm sw}\sim 30~{\rm km/s}$.
Therefore, a solar-mass red giant with radius $R_{\star}=100R_{\odot}$ produces a stellar wind that is slower by an order of magnitude than that of the Sun, $v_{\odot}$.
Based on equation (\ref{eq:scale_Mdot}), the number density of the stellar wind (equation \ref{eq:n}) is normalized as follows:
\begin{eqnarray}
n &=& 1.8 \times 10^6 \; {\rm cm^{-3}} \times \left( \frac{a}{5 \; {\rm AU}} \right)^{-2} \notag \\
&&\times \left( \frac{\dot M_\star}{10^{-8} M_{\odot }{\rm /yr}} \right) \left( \frac{v_{\rm sw}}{30~\mbox{km/s}} \right)^{-1} \, . \label{eq:n_normalized}
\end{eqnarray}
The velocity term in equation (\ref{eq:Pinp_kin}), which is the relative velocity between the planet and the stellar wind particles falling onto the planet, depends on the stellar wind velocity, the infall velocity (i.e., the acceleration due to planet's gravitational field), and the planetary orbital velocity.
All three of these terms might be of the same order of magnitude:
as described above, $v_{\rm sw}$ is $\sim$30~km/s;
the infall velocity from the planetary gravity is $\sim$10-25~km/s depending on the planetary mass (ranging from $1 M_{\rm J} to 10 M_{\rm J}$);
and the orbital velocity is $10-30~{\rm km}\,{\rm s}^{-1}$ depending on the orbital distance (ranging from $1-5~{\rm AU}$).
In the next section, we simply consider
\begin{equation}
\frac{v}{v_{\odot}} \sim 10^{-1} \, \label{eq:scale_v}
\end{equation}
a fiducial value for the normalization.
\subsection{Ionization of Stellar Wind Particles Around the Planet}
\label{ss:ionization}
As discussed in \citet{ignace2010}, as stars evolve, the ionization fraction of the stellar wind diminishes to the order of $\sim$10$^{-3}$ \citep{drake1987}.
Since only charged particles interact with a planetary magnetic field, this suggests inefficient interaction with the planetary magnetosphere and hence, low input energy for radio emission.
However, for a highly evolved star, the velocity of the stellar wind eventually becomes slower than the escape velocity of the planetary companion and hence, stellar wind particles will accrete onto the planets.
\citet{spiegel+madhusudhan2012} considered Bondi--Hoyle accretion where the accretion radius is
\begin{eqnarray}
R_{\rm acc} &=& \frac{2GM_p}{v_{\rm rel}^2} \\
&\sim & 4 ~R_{\rm J} \left( \frac{M_p}{M_{\rm J}} \right) \left( \frac{v_{\rm rel}}{30 ~{\rm km~s^{-1}}} \right)^{-2} \, .
\end{eqnarray}
The accretion luminosity $L_{\rm acc}$ and temperature $T_{\rm acc}$ are
\begin{eqnarray}
L_{\rm acc} \sim && 10^{25} \; {\rm erg\;s}^{-1} \left( \frac{\dot M_\star}{10^{-8} M_{\odot }/{\rm yr}} \right) \notag \\
&& \times \left( \frac{M_p}{M_{\rm J}} \right)^3 \left( \frac{M_\star}{M_{\odot }} \right)^{-2} \, . \label{eq:Lacc}\\
T_{\rm acc} \sim && 2 \times 10^5 ~\mbox{K} \left( \frac{M_p}{M_{\rm J}} \right) \, . \label{eq:Tacc}
\end{eqnarray}
The accretion onto planets, therefore, leads to emission of UV/X-ray photons whose characteristic energy $\sim$$k_B T_{\rm acc}$ exceeds the ionization energy of hydrogen, $E_{\rm Rydberg} = 13.6$~eV ($\lambda = 91.2\rm~nm$).
Therefore, UV/X-ray radiation from accretion will create a local ionized region around the planet.
Let us consider the ionization profile around the planet.
We suppose a state where the ionization and recombination rates are in equilibrium.
Denoting the ionization fraction by $x$, the equilibrium state at a distance $r$ from the planet may be represented by
\begin{equation}
\frac{\dot N_X}{ 4 \pi r^2 } e^{- \tau } n ( 1-x ) \sigma _{\rm H} [E_{\rm photon}] = (n x )^2 \beta [T_e] \label{eq:equilibrium}
\end{equation}
\begin{equation}
\tau = \int _{R_{\rm J}}^r n(1-x) \sigma_{\rm H} [E_{\rm photon}] dr \, ,
\end{equation}
where $\dot N_X$ is the source rate of the photons that can ionize hydrogen, $E_{\rm photon}$ is the energy per photon, and $\sigma _{\rm H}$ is the cross-section of H atoms for X-ray photons.
Per \citet{verner1996}, $\sigma _{\rm H}$ scales as
\begin{equation}
\sigma _{\rm H} [E_{\rm photon}] \sim 6.3 \times 10^{-18} \;{\rm cm}^2 \cdot \left( \frac{E_{\rm photon}}{E_{\rm Rydberg}} \right)^{-3} \, , \label{eq:sigma_H}
\end{equation}
where $\beta [T_e]$ is the ``class B'' recombination coefficient as a function of the electron temperature, $T_e$.
We adopt the value at $T_e \sim 10^4$~K, $\beta \sim 2.6 \times 10^{-13}\;{\rm cm^3/sec} $, in the following \citep{pequignot1991}; when the electron temperature is varied from $10^3$ to $10^5$~K, $\beta $ varies $1.5 \times 10^{-12}-3.2 \times 10^{-14}\;{\rm cm^3}\,{\rm sec}^{-1}$.
The source rate is obtained by counting the number of photons with energy exceeding $E_{\rm Rydberg}$, which is approximately given by dividing the X-ray accretion luminosity by the characteristic photon energy produced:
\begin{equation}
\dot{N}_X \sim \frac{L_{\rm acc}}{k_{\rm B} T_{\rm acc}} \, ,
\end{equation}
where $k_{\rm B}$ is Boltzmann's constant.
As a result,
\begin{eqnarray}
\dot{N}_X \sim
\left\{
\begin{array}{ll}
4 \times 10^{35} \; {\rm sec^{-1}} & \;\;\;(M_p = M_{\rm J}) \\
4 \times 10^{37} \; {\rm sec^{-1}} & \;\;\;(M_p = 10 M_{\rm J})
\end{array} \, .
\right.
\end{eqnarray}
For accretion onto very massive planets, the characteristic photon energy is so high that the released energetic electrons may also ionize other atoms in the vicinity.
The cross-section\footnote{This is close to the geometric cross-section of the Bohr radius.} of hydrogen for electrons is $\sim$4$\times$10$^{-17}$~cm$^2$ \citep{fite1958}, which implies a mean free path for ionized electrons of $\sim$2$R_{\rm J}$ in the surrounding medium.
As a result, nearly 2/3 of released energetic electrons will ionize a hydrogen atom within $\sim$2$R_{\rm J}$, and more than 99\% will ionize a hydrogen atom within $\sim$10$R_{\rm J}$.
In principle, photons with energy $E_{\rm photon}$ have the potential to ionize $E_{\rm photon}/E_{\rm Rydberg}$ hydrogen atoms.
To account for this, we consider two limiting possibilities:
After an ionizing collision, the energy can (\emph{i}) be split evenly between the two electrons, or it can (\emph{ii}) go entirely into the kinetic energy of one electron and not at all into that of the other (of course, any split in between these extremes is possible, too).
Note that conservation of momentum and energy imply that the proton will not acquire a significant fraction of the energy of the collision.\footnote{Were it otherwise, we would have to take into account what fraction of a rubber ball's kinetic energy is imparted to the kinetic energy of the Earth when bouncing a ball.}
Scenario (\emph{i}) implies a cascade where an electron with energy $E_{\rm in}$ ionizes an atom, producing two electrons (an ionizing electron plus the released electron) with energy $ ( E_{\rm in} - E_{\rm Rydberg} ) /2 $ for each.
For example, in an idealized case with $M_p = 10 M_{\rm J}$, $k_B T_{\rm acc} \sim 172\,{\rm eV} \sim 12 \, E_{\rm Rydberg}$, a photoionization could produce an electron with energy of $(12-1) = 11 E_{\rm Rydberg} $, then a second ionization by that electron would produce two photons with energy of $(11-1)/2 = 5 E_{\rm Rydberg}$, and the third ionization by these two photons would produce four photons with energy of $(5-1)/2 = 2 E_{\rm Rydberg}$, etc.
The cascade can proceed to the fourth order at the maximum.
Under scenario (\emph{ii}), the cascade proceeds with the initial $E_{\rm in}$ electron leading to an electron with kinetic energy $E_{\rm in} - E_{\rm Rydberg}$ and another electron with zero kinetic energy.
Clearly, this cascade can produce a maximum total of $E_{\rm in} / E_{\rm Rydberg}$ free electrons.
In reality, not all released electrons result in further ionization interactions.
If $\xi$ represents the fraction of released electrons that proceed to the next ionization, then the number of ionized atoms $N_i$ released through this cascade (\emph{i}) is
\begin{eqnarray}
\nonumber N_i & = & (1-\xi) + 2\xi (1-\xi) + 4 \xi^2 (1-\xi) + 8 \xi^3 \\
\label{eq:N_i1} & = & 1 + \xi + 2 \xi^2 + 4 \xi ^3 \, .
\end{eqnarray}
Alternatively, cascade (\emph{ii}) leads to
\begin{eqnarray}
\nonumber N_i & = & \frac{1 - \xi^k}{1 - \xi} \\
\label{eq:N_i2} & = & 1 + \xi + \xi^2 + \cdots + \xi^{k - 1} \, ,
\end{eqnarray}
where $k \equiv E_{\rm in} / E_{\rm Rydberg}$ is the maximum number of ionizations for the given initial electron energy.
This limit leads to a value for $N_i$ that is not dramatically different from that of limit (\emph{ii}).
In Appendix \ref{sec:AppendixA}, we estimate the M{\o}ller scattering cross-section and show that cascade (\emph{ii}) --- unequal recoil energies --- is probably more realistic.
Ultimately, $\dot{N}_X $ in equation (\ref{eq:equilibrium}) is replaced by
\begin{equation}
\dot{N}_X \rightarrow N_i \dot{N}_X \, .
\end{equation}
For a $10 M_{\rm J}$ planet, $N_i$ is probably approximately in the range 5---10.
\begin{figure}[htbp]
\plotoneh{ionizationfraction2.pdf}
\caption{Profile of ionization fraction measured from the surface of the planet, due to UV/X-ray from the accretion of stellar wind onto the planet.
Solid lines show the solutions without the correction for the secondary ionization by ionized electrons, and the dashed line shows the solution for a 10$M_{\rm J}$ planet taking the correction into account with efficiency factor $N_i=6$.
The vertical arrows show the Str\"omgren radius estimated simply using equation (\ref{eq:stromgren}).
The dotted vertical lines indicate the location of the magnetic stand-off radii, $r_{\rm mag}$, obtained using equation (\ref{eq:stand-off-radius_RGHJ}) below. }
\label{fig:ionizationfraction}
\end{figure}
The ionization fraction $x$ as a solution of equation (\ref{eq:equilibrium}) is shown in Figure \ref{fig:ionizationfraction}.
When $\tau \sim n \sigma_H r$ is much smaller than unity and thus the $e^{-\tau}$ term can be ignored (as is the case for $M_p = 10 M_{\rm J}$), the solution is simply
\begin{eqnarray}
x[r] &=& \frac{-1 + \sqrt{1+4C[r]}}{2C[r]} \\
C[r] &\equiv & \frac{4 \pi n \beta [T_e] r^2}{\dot{N} \sigma_H[E_{\rm photon}]} \, .
\end{eqnarray}
The solid lines show the ionization fraction corresponding to no additional ionization by electrons (i.e., $\xi=0$, or $N_i = 1$) and the dashed line shows the solution with $N_i=6$ for a $10 M_{\rm J}$ planet.
In the figure, the vertical lines show the magnetic stand-off radius obtained using equation (\ref{eq:stand-off-radius_RGHJ}) below.
While the photon rate is larger for more massive planets, the strong dependence of the cross-section on photon energy (equation \ref{eq:sigma_H}) leads to a decrease in the radius of the ionized region.
Nevertheless, {\it a substantial amount of ionized plasma is expected around the magnetic stand-off radius, despite the initially low ionized fraction of the stellar wind}.
The extent of the ionized region may also be roughly estimated as the Str\"omgren radius \citep{stromgren1939}:
\begin{equation}
r_{\rm stromgren} = \left( \frac{3}{4\pi} \frac{\dot N_X}{n^2 \beta } \right)^{1/3} \, ,
\end{equation}
which gives
\begin{eqnarray}
r_{\rm stromgren} \sim
\left\{
\begin{array}{ll}
67 ~R_J & \;\;\;(M_p = M_{\rm J}) \\
310 ~R_J & \;\;\;(M_p = 10 M_{\rm J})
\end{array}
\right. \label{eq:stromgren}
\end{eqnarray}
for a planet 5~AU from a red giant host.
The Str\"omgren radii for 1-$M_{\rm J} $ and 10-$M_{\rm J}$ planets are also indicated in Figure \ref{fig:ionizationfraction}.
Note that this is a very interesting and different regime of the Str\"omgren sphere from the commonly-considered case (photoionization around O/B-type stars). Around RGHJs, the Str\"omgren sphere does not delineate a sharp edge of ionization, because of the smaller source rate and smaller photon-hydrogen cross-section (in this case parameter ``$a$'' in equation 13 of \citealt{stromgren1939} is not small) on account of X-ray photons interacting more weakly with neutral hydrogen than UV photons near the ionization limit.
The Str\"omgren radii are indicated by vertical arrows in Figure \ref{fig:ionizationfraction}.
Crucially, X-ray and UV emission from accretion onto the planet should ionize a significant fraction of the incoming stellar wind, thereby allowing radio waves to be generated from this interaction.
In reality, the temperature and luminosity based on the Bondi--Hoyle accretion (equations \ref{eq:Lacc} and \ref{eq:Tacc}) describes the situation only approximately.
Precisely, only the neutral portion of the stellar wind can accrete onto the planet without interacting with the planetary magnetic field, and the ionized portion would lose some energy at the bow shock before it accretes.
On the other hand, the ionized plasma interacting with the magnetic field could have a cross-section larger than the Bondi--Hoyle accretion radius.
The detailed electromagnetic structure around RGHJs therefore requires elaborate numerical simulations that are beyond the scope of this paper.
In the following sections, we aim to give order-of-magnitude estimates of radio emission, observability, etc., which ought to be robust with respect to uncertainties in the details of the ionization process.
\section{Estimates}
\label{s:result}
\subsection{Planetary Magnetic Field and Frequency of Radio Emission}
\label{ss:Bplanet}
\begin{figure}[bhtp]
\plotoneh{rho_r_dynamo.pdf}
\plotoneh{qBf_Christensen.pdf}
\caption{Upper panels: radius and average density of the dynamo region as a function of planetary mass.
Lower panels: evolutions of heat flux at the outer boundary, planetary magnetic field, and maximum cyclotron frequency, for planets with varying masses ($M_{\rm J}$: blue, $5M_{\rm J}$: green, and $10M_{\rm J}$: red).}
\label{fig:planetaryB}
\end{figure}
Figure \ref{fig:planetaryB} shows the computed radius ($r_c$) and average density ($\rho_c$) of the dynamo region as a function of planetary mass, as well as the heat flux at the outer boundary of the core ($q_o$), the estimated strength of the planetary magnetic field ($B$), and the corresponding cyclotron frequency ($\nu_{\rm cyc}$) as functions of planetary mass and the age.
Substituting equation (\ref{eq:burrowsHeatFlux}) into equation (\ref{eq:Bscaling2}), and given that $r_{\rm dynamo} $ does not change significantly, the magnetic field is approximately:
\begin{equation}
B \sim B_{\rm J} \left( \frac{M_p }{M_{\rm J}} \right)^{1.04} \left( \frac{t}{4.5~\rm{Gyr}} \right)^{-0.43} \label{eq:scalingB}
\end{equation}
under this model, where $B_{\rm J}$ is the magnetic field strength of Jupiter; in this paper, we roughly consider $B_{\rm J} \sim 10~{\rm G}$. Reasonably, the resultant values agree with \citet{reiners2010}, who adopted the same scaling law for the planetary magnetic field; we show this figure just for completeness.
Note that the cyclotron frequency of Jovian planets typically falls between 10~MHz and 1~GHz.
In this regime, there are a number of current and near-future radio observatories including the Giant Metrewave Radio Telescope (GMRT), Low-Frequency Array (LOFAR), Hydrogen Epoch of Reionization Array (HERA), Square Kilometer Array (SKA), and potential upgrades to the Very Large Array (VLA) (see Section \ref{ss:detectability}).
Since $\nu_{\rm cyc,\,max} > \nu_{\rm plasma}^\oplus $, the radio emission will not be hindered by Earth's ionosphere cut-off.
On the other hand, it may experience opacity due to the plasma of the stellar wind particles around the planet.
The maximum plasma frequency along the line of sight, $\nu_{\rm plasma}^{\rm los}$, corresponds to that in the vicinity of the planet, if the planet is on the near side of its star to the Earth.
Therefore, substituting equation~(\ref{eq:n_normalized}) to $n_e$ in equation~(\ref{eq:fplasma}),
\begin{eqnarray}
\label{eq:fplasmalos} \nu_{\rm plasma}^{\rm los} & \sim &
8979 {\rm~Hz} \times \left( \frac{\dot M_\star}{4 \pi a^2 m_p v_{\rm sw}} \times 1\rm~cm^3 \right)^{1/2} \\
\label{eq:fplasmalos_scaled} &=& 12 {\rm~MHz} \times \left( \frac{a}{5~\mbox{AU}}\right)^{-1} \left(\frac{v_{\rm sw}}{30~\mbox{km/s}} \right)^{-1/2} \notag \\
& & \times \left( \frac{\dot M_\star}{10^{-8} M_{\odot }{\rm /yr}}\right)^{1/2} \label{eq:fplasma_RG} \, .
\end{eqnarray}
We can see emission only from where $\nu_{\rm cyc,\,max} > \nu_{\rm plasma}^{\rm los}$.
The detectable parameter space will be presented in more detail in the next section.
\subsection{Flux of RGHJ Radio Emission in Comparison with Canonical HJs}
\label{ss:brightness}
\begin{figure*}[bp]
\plotonesc{radio_emission_mass_orbit_ver3.pdf}
\caption{Radio flux in unit of Jy from a planetary companion to a red giant with mass loss rate $10^{-8} M_{\odot }/\mbox{yr}$ (top) and that to an AGB star with mass loss rate $10^{-5} M_{\odot }/\mbox{yr}$ (bottom).
The systems are located at 100 pc away.
The doubly-hatched regions show the parameter spaces where the planetary radio emission would not be observable at all because the maximum frequency of the emission (cyclotron frequency at the planetary surface, $\nu_{\rm cyc}$) is below the plasma frequency cut-off, $\nu_{\rm plasma}^{\rm los}$.
The regions hatched with vertical lines show the parameter spaces where the frequencies of bulk radio emission is below $\nu_{\rm plasma}^{\rm los}$, i.e., $\nu_{\rm plasma}^{\rm los} > 0.1 \nu_{\rm cyc}$.}
\label{fig:radio}
\end{figure*}
The magnetic stand-off radius (equation \ref{eq:stand-off-radius}) may be written as follows using the stellar mass-loss rate:
\begin{eqnarray}
\nonumber r_{\rm mag}
&=& r_{\rm mag,\,J} \left( \frac{B}{B_{\rm J}} \right)^{1/3} \left( \frac{a}{5.2\mbox{AU}} \right)^{1/3} \\
&& \times \left( \frac{\dot M_\star}{\dot M_\odot} \right)^{-1/6} \left( \frac{v}{v_{\odot }} \right)^{-1/6} \label{eq:stand-off-radius2}
\end{eqnarray}
The typical value for RGHJs is found by substituting relevant values for stellar wind parameters described in Section \ref{ss:stellarwind}:
\begin{eqnarray}
\nonumber r_{\rm mag} &\sim & 14 \, R_J \left( \frac{B}{B_{\rm J}} \right)^{1/3} \left( \frac{a}{5\mbox{AU}} \right)^{1/3} \\
&& \times \left( \frac{\dot M_\star}{10^{-8} M_{\odot }{\rm /yr}} \right)^{-1/6} \left( \frac{v}{10^{-1}v_{\odot }} \right)^{-1/6}
\label{eq:stand-off-radius_RGHJ}
\end{eqnarray}
where we employ $r_{\rm mag,\,J} = 84~R_{\rm J}$ \citep{joy2002}.
Substituting equation (\ref{eq:stand-off-radius2}) to equation (\ref{eq:Pinp_kin}), the scaling of the radio emission is expanded as follows:
\begin{eqnarray}
P_{\rm k,\,inp} &=& m_{\rm p} nv^3 \cdot \pi r_{\rm mag}^2 \\
&=& P_{\rm k,\,inp,\,J} \left( \frac{B}{B_{\rm J}} \right)^{2/3} \left( \frac{a}{5.2~\mbox{AU}} \right)^{-4/3} \notag \\
&& \times \left( \frac{\dot M_\star}{\dot M_{\odot}} \right)^{2/3} \left( \frac{v}{v_{\odot}} \right)^{5/3} \label{eq:Pkinp}
\end{eqnarray}
We may compare the radio emission power of RGHJs at 5 AU with that of canonical hot Jupiters set at 0.05 AU.
In order to perform this comparison, equation (\ref{eq:Pkinp}) can be re-normalized as follows:
\begin{eqnarray}
P_{\rm k,\,inp}
&\approx & 140 ~P_{\rm k,\,inp,\,J} \left( \frac{B}{B_{\rm J}} \right)^{2/3} \left( \frac{a}{5~\mbox{AU}} \right)^{-4/3} \notag \\
&& \times \left( \frac{\dot M_\star}{10^{-8} M_{\odot}{\rm /yr}} \right)^{2/3} \left( \frac{v}{10^{-1} v_{\odot}} \right)^{5/3} \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \mbox{(for RGB stars' companions)} \notag \\
&\approx & 14000 ~P_{\rm k,\,inp,\,J} \left( \frac{B}{B_{\rm J}} \right)^{2/3} \left( \frac{a}{5~\mbox{AU}} \right)^{-4/3} \notag \\
&& \times \left( \frac{\dot M_\star}{10^{-5} M_{\odot}{\rm /yr}} \right)^{2/3} \left( \frac{v}{10^{-1} v_{\odot}} \right)^{5/3} \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \mbox{(for AGB stars' companions)} \notag \\
&\approx & 490 ~P_{\rm k,\,inp,\,J} \left( \frac{B}{B_{\rm J}} \right)^{2/3} \left( \frac{a}{0.05~\mbox{AU}} \right)^{-4/3} \notag \\
&& \times \left( \frac{\dot M_\star}{\dot M_{\odot}} \right)^{2/3} \left( \frac{v}{v_{\odot}} \right)^{5/3} \label{eq:P_HJ} \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \mbox{(for canonical hot Jupiters)} \notag
\end{eqnarray}
Here, we have normalized the magnetic field strength of canonical hot Jupiters with $B_{\rm J}$, considering the uncertainty of the magnetic fields of tidally locked planets.
Note that some models of planetary magnetic field strength predict that tidally locked planets have weaker magnetic fields due to their slow rotation \citep[e.g.][]{griesmeier2004}.
Although the orbital velocity of canonical hot Jupiters has been ignored in equation (\ref{eq:P_HJ}), the Keplerian velocity at 0.05~AU around a solar-mass star is $\sim$130~km~s$^{-1}$, which results in only a $<$10\% increase of the relative velocity.
Using the expressions above, we find that the radio spectral flux density observed at the Earth is:
\begin{eqnarray}
F_{\nu} &=& \frac{P_{\rm radio}}{\Omega d^2 \nu_{\rm cyc}} \\
&\approx & 5.2\times 10^{-8} \mbox{Jy} \left( \frac{d}{100\mbox{~pc}} \right)^{-2} \notag \\
&&\times \left( \frac{B}{B_{\rm J}} \right)^{-1/3} \left( \frac{a}{5~\mbox{AU}} \right)^{-4/3} \notag \\
&& \times \left( \frac{\dot M_\star}{\dot M_{\odot}} \right)^{2/3} \left( \frac{v}{v_{\odot}} \right)^{5/3} \label{eq:F_nu} \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \mbox{(for a Jupiter-twin)} \notag \\
&\approx & 0.70 \times 10^{-5} \mbox{Jy} \left( \frac{d}{100\mbox{~pc}} \right)^{-2} \notag \\
&&\times \left( \frac{B}{B_{\rm J}} \right)^{-1/3} \left( \frac{a}{5~\mbox{AU}} \right)^{-4/3} \notag \\
&& \times \left( \frac{\dot M_\star}{10^{-8} M_{\odot}/{\rm yr}} \right)^{2/3} \left( \frac{v}{10^{-1} v_{\odot}} \right)^{5/3} \label{eq:F_nu_RGHJ} \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \mbox{(for RGB stars' companions)} \notag \\
&\approx & 0.70 \times 10^{-3} \mbox{Jy} \left( \frac{d}{100\mbox{~pc}} \right)^{-2} \notag \\
&&\times \left( \frac{B}{B_{\rm J}} \right)^{-1/3} \left( \frac{a}{5~\mbox{AU}} \right)^{-4/3} \notag \\
&& \times \left( \frac{\dot M_\star}{10^{-5} M_{\odot}/{\rm yr}} \right)^{2/3} \left( \frac{v}{10^{-1} v_{\odot}} \right)^{5/3} \label{eq:F_nu_AGB} \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \mbox{(for AGB stars' companions)} \notag \\
&\approx & 2.4 \times 10^{-5} \mbox{Jy} \left( \frac{d}{100\mbox{~pc}} \right)^{-2} \notag \\
&&\times \left( \frac{B}{B_{\rm J}} \right)^{-1/3} \left( \frac{a}{0.05~\mbox{AU}} \right)^{-4/3} \notag \\
&& \times \left( \frac{\dot M_\star}{\dot M_{\odot}} \right)^{2/3} \left( \frac{v}{v_{\odot}} \right)^{5/3} \label{eq:F_nu_RGHJs} \\
&& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \mbox{(for canonical hot Jupiters)} \notag
\end{eqnarray}
Thus, RGHJs are expected to be intrinsically as bright as the closest hot Jupiters.
Compared with the equivalent Jupiter-like planets around main sequence stars, the massive stellar wind of late red giants can increase the radio emission from planetary companions by more than two orders of magnitude, which allows us to explore 10 times more distant systems, i.e., 1000 times more volume.
This at least partially compensates for the small population of evolved stars.
Equation (\ref{eq:F_nu_RGHJ}) gives a spectral flux density one to two orders of magnitude smaller than the prediction of equation (5) in \citet{ignace2010} if we assume the same set of fiducial values and full ionization.
This is primarily because their formulation did not incorporate the effect of a compressed planetary magnetosphere due to a massive stellar wind, while the scaling law for the planetary magnetic field strength is also different.
Using this model for planetary magnetic field strength, we may consider the radio spectral intensity and maximum frequency of a given planetary mass and age.
Figure \ref{fig:radio} shows contours of radio spectral flux density from planetary companions at 4.5~Gyr with varying masses and orbital distances.
The target system is located at a distance of 100~pc from the Earth; the flux is scaled by the distance as a quadratic function.
The observable energy flux is limited by the plasma cut-off frequency in the vicinity of the planets, given by equation (\ref{eq:fplasmalos_scaled}).
The doubly hatched regions in Figure \ref{fig:radio} indicate the regions of parameter space where
$\nu_{\rm cyc,\,max} < \nu_{\rm plasma}^{\rm los}$, i.e., the radio emission from the planet cannot be observed from the Earth.
In reality, the peak of the auroral radio emission does not always occur at the maximum frequency.
Instead, it usually exists at lower frequencies.
Here, we also consider one more criterion, $\nu_{\rm cyc,\,max} > 10 \nu_{\rm plasma}^{\rm los}$ as a conservative measure for the observability of the bulk of radio emission.
In Figure \ref{fig:radio}, this conservative region is shown as hatched regions with vertical lines.
This indicates that the plasma cut-off due to a dense stellar wind is a significant obstacle to detecting planetary companions around evolved stars.
For a red-giant system, the most promising targets are systems with massive companions with $M_p \gtrsim 5M_{\rm J}$, although systems with smaller companions at distant orbits are also accessible.
For an AGB-star system, however, only very massive planets at distant orbits are marginally detectable, because the stellar wind is so dense that there is significant radio opacity due to the plasma from the accreting stellar wind.
\section{Observability}
\label{s:observability}
In this section, we discuss the observability of the estimated radio emission.
An obvious potential obstacle is the intrinsic radio emission from host red giant stars themselves; if they are bright relative to the radio flux from the planets, it is significantly more difficult to identify the planetary contribution.
We will see in Section \ref{ss:RGradio} that radio flux from the planets will probably be larger in a certain range of parameter space.
Then, in Section \ref{ss:actualMgiants}, we estimate the radio flux from hypothetical Jupiter-like planets around known M giants.
In Section \ref{ss:detectability}, we examine the sensitivities and limitations of several current and future
radio instruments at relevant frequencies.
Finally, Section \ref{ss:timevariability} notes the polarization and the time variability of the radio emission as keys to discern the signals from the background noise.
\subsection{Intrinsic Radio Emission of Red-Giants Stars}
\label{ss:RGradio}
\begin{figure}[tbp]
\plotoneh{cartoon2_5Mj_100pc_smooth.pdf}
\caption{A cartoon of the radio emission spectra of an RGHJ with 5$M_{\rm J}$ and the host red giants with 100 $R_{\odot }$.
The spectrum of the RGHJ is modeled after Jovian radio spectra, e.g.\ Figure 8 of \citet{zarka_et_al2004} and Figure 2 of \citet{griesmeier2007a}; the contribution from Io-DAM is not shown here.
The spectra of the host red giant are modeled simply by extrapolating observed radio spectra above 1~GHz with a power law.
The shaded region represents the portion of the spectrum below the plasma frequency of the stellar wind.}
\label{fig:cartoon}
\end{figure}
Over the last four decades there have been numerous radio continuum observations of nearby red giants, many undertaken with the aim of understanding the extended atmospheres and mass-loss mechanisms of K- and M-type giants~\citep[e.g.,][]{Newell1982, Knapp1995, Skinner1997, Lim1998, OGorman2013}.
Almost all observations have been carried out at frequencies $\geq$5~GHz, probing thermal Bremsstrahlung emission from the large, partially ionized envelopes surrounding the giant stars~\citep{drake1986}.
The spectral index of this emission is of order unity, with a range of reported values for various sources between 0.8 and 1.6~\citep{OGorman2013}.
Only two published studies describe attempts to detect continuum emission from single (non-binary) red giant stars
at frequencies below 1 GHz ($L$ band); both yielded null results.
First, a program that observed a sample of nine M-type giants at 430 MHz with the Arecibo Telescope failed to detect any of the sources down to flux density 10 mJy \citep{Fix1976}.
Later, with the Molonglo Observatory Synthesis Telescope (MOST), a sample of eight K- and M-type giants were observed at 843 MHz; the upper limits of their flux densities were placed at approximately 1 mJy \citep{Beasley1992}.
Current facilities could achieve far deeper sensitivity on similar targets.
For example, as listed in Table~\ref{tab:sens}, the GMRT at 150 MHz reaches a sensitivity better than 0.5~mJy in under an hour.
However, any re-attempts to detect single red giants at wavelengths near 1 meter remain to be presented in the literature.
The only red giants that have been detected at meter wavelengths are those in interacting binary systems like those of the RS CVn type.
For example, \citet{vandenOord1994} detected plasma maser emission from II Pegasi at 360 MHz and 609 MHz with the WSRT.
Although single dwarf-type main sequence stars are also known to exhibit bright, coherent radio flares~\citep{Bastian1990}, no analog to this non-thermal emission process has been proposed to exist for evolved stars.
Altogether, we lack firm observational constraints on the brightness of M giants near 30-300 MHz frequencies.
However, assuming thermal Bremsstrahlung emission continues to dominate in this wavelength regime, we can extrapolate down from the reported centimeter flux densities, using the specific spectral index.
For example, \citet{OGorman2013} reported $\alpha$~Boo ($R_\star = 25.4 R_{\odot }$, $T=4286$ K, $d=11.26$ pc) at 1~GHz is about 70 $\mu$Jy, and $\alpha$~Tau ($R_\star = 44.2 R_{\odot }$, $T=3910$ K, $d=20.0$ pc) at 3~GHz is about 40 $\mu$Jy.
Using these data, we extrapolate the radio flux from the host star $F_{\star}$ to the lower frequency range with a simple power law as follows:
\begin{eqnarray}
F_{\star}(\nu ) & \sim & (0.4-1.4) \times 10^{-5} ~\mbox{Jy} \times \\
\nonumber && \left( \frac{d}{100 ~\mbox{pc}} \right)^{-2} \!\! \left( \frac{R_{\star }}{100~R_{\odot}} \right)^2 \!\! \left( \frac{\nu}{1~\mbox{GHz}} \right)^{\alpha_* } \, ,
\end{eqnarray}
where the power index $\alpha_* $ is of order unity.
Figure \ref{fig:cartoon} is a cartoon of the radio spectrum of a planet with $5~M_{\rm J}$ and that of the host red giant star with 100~$R_{\odot }$, both of which are placed at a distance of 100 pc.
Note that the thermal contribution from the accretion is negligible in this figure.
The spectral shape is modeled by simply scaling the Jovian radio spectrum (e.g. Figure 8 of \citealt{zarka_et_al2004}).
The continuum lines from the star in the cases of $\alpha_* =0.8$ and $2.0$ are shown as two cases.
As Figure \ref{fig:cartoon} indicates, with a reasonable range of spectral index, the radio emission from the star is smaller by an order of magnitude than the expected radio flux from RGHJs at frequencies below 300~MHz.
For less massive planets (with weaker magnetic fields), the distinction is even clearer, because the peak flux is larger and the peak frequencies are smaller.
Therefore, in the following, we ignore the radio emission from the red giant as a noise source.
\subsection{Radio Flux from Nearby M Giants}
\label{ss:actualMgiants}
We now examine how realistic it is that RGHJ radio emission might be detectable with current and near-future instruments.
In order to do so, we consider if known red giant stars were to host planetary companions and estimate the resulting radio flux density.
We use the list of M-type red giants from Table 4 of \citet{dumm1998}, which includes estimates of stellar masses, radii, and effective temperatures.
Their data do not unambiguously distinguish RGB from early AGB stars, so probably about 40\% of the stars on the list are actually early AGB stars.
For each red giant, the mass-loss rate is estimated using the improved Reimers' equation \citep{reimers1975} given by \citet{schroder2005,schroder2007}:
\begin{equation}
\dot M_\star \sim 8 \times 10^{-14} ~M_\odot{\rm /yr} ~\left( \frac{\tilde L \tilde R}{\tilde M}\right) \left( \frac{T_{\rm eff}}{4000} \right)^{3.5} \left( 1 + \frac{g_{\odot }}{4300 g_{\star}} \right) \, , \label{eq:mass-loss}
\end{equation}
where $\tilde L = L_\star/L_\odot$ is the stellar luminosity, $\tilde R = R_\star/R_\odot$ is the stellar radius, $\tilde M = M_\star/M_\odot$ is the stellar mass, and $g_\star$ is the surface gravity.
The velocity of the stellar wind is assumed to be the escape velocity at $\sim$$4 R_{\star}$:
\begin{equation}
v \sim \sqrt{\frac{2GM_\star}{4R_{\star }}} \, .
\end{equation}
The age of the system is taken to be our crude estimate of the main-sequence lifetime, as a function of stellar mass.
We simply assume:
\begin{equation}
t \sim 10\,{\rm Gyr}\, \left( \frac{M_{\star}}{M_{\odot }} \right)^{-2.5} \, .
\end{equation}
Combining this relation with equations (\ref{eq:fcyc}) and (\ref{eq:scalingB}) results in
\begin{equation}
\nu_{\rm cyc,\,max} \sim 20 \,{\rm MHz}\, \left( \frac{M_{\star}}{M_{\odot }} \right)^{1.075} \left( \frac{M_p}{M_{p,\,{\rm J}}} \right)^{1.04} \, .
\end{equation}
In addition, the distance to the system is obtained based on the parallax data from {\it Hipparcos} data sets\footnote{\url{http://www.rssd.esa.int/index.php?project=HIPPARCOS}}.
Using these data, we estimate the spectral flux of radio emission by specifying planetary mass and orbital distance of a hypothetical planetary companion.
We consider [A] a Jupiter-twin whose maximum cyclotron frequency is $\nu _{\rm cyc,\,max} \sim 20\,(M_{\star }/M_{\odot})$~MHz, and [B] a larger Jovian planet with $M_p \sim 10M_p$ whose maximum cyclotron frequency is $\nu _{\rm cyc,\,max} \sim 200\,(M_{\star }/M_{\odot})$~MHz.
Figure \ref{fig:observability} displays the estimated radio flux from planets [A] and [B] around M-type red giants within 300~pc.
To span a reasonable range of orbital distances, we place the planets at 1 AU and at 5 AU from their host stars.
The shape of the marker indicates whether the radio emission can escape from the system (circles), i.e., $\nu _{\rm cyc,\,max} > \nu_{\rm plasma}^{\rm los}$, or not (X's).
Assuming Jupiter-mass planets at 1 AU (5 AU) which typically have the maximum cyclotron frequency at 10---100~MHz depending on the age, 148 (12) systems have potential to emit radio waves at a flux density level above $\sim$10$~\mu$Jy out to $\sim$100---300~pc.
Those systems with the highest mass-loss rates, which are intrinsically the brightest systems, would probably be unobservable because of the plasma cut-off frequency of the stellar wind.
For more massive planets, with maximum cyclotron frequency $\sim$100---1000~MHz, the radio flux density is lowered due to its dependence on $B^{-1/3}$ (Equation (\ref{eq:F_nu})), but is more likely to reach Earth because it is less extinguished by plasma opacity.
10-$M_{\rm J}$ planets at $\sim$1~AU may be detectable out to $\sim$200~pc.
A more realistic estimate of the expected number of detections may be evaluated by combining these results with the probability for a star to have a planet with a certain mass and orbit.
Observations of main-sequence stars imply that the number of Jovian planets is in the range of several percent to $\sim$10\% of the number of stars \citep[e.g.][]{cumming2008,cassan2012}.
Assuming the same population for RG systems, the expected number of actual detections from RGHJs would be of order unity.
A more detailed examination using the empirical planet mass function by \citet{cumming2008} is given in Appendix \ref{s:expectedvalue}.
\begin{figure}[tbhp]
\plotoneh{radio_M-RG_1Mp.pdf}
\plotoneh{radio_M-RG_10Mp.pdf}
\caption{Radio flux from hypothetical RGHJs of 1$M_{\rm J}$ (upper panel) and 10$M_{\rm J}$ (lower panel) at 1~AU (blue) and 5~AU (black).
X symbols indicate that the radio emission is not observable because cyclotron frequency is less than plasma density around the planet. }
\label{fig:observability}
\end{figure}
\subsection{Detectability with Current and Near-future Low-frequency Radio Instrumentation}
\label{ss:detectability}
\begin{deluxetable*}{cccccc}
\tabletypesize{\footnotesize}
\tablecolumns{6}
\tablecaption{Detectability of Hot Jupiters with Current
and Future Radio Telescopes.\label{tab:sens}}
\tablehead{
Instrument & Band & \mbox{$\sigma_{\mbox{\tiny RMS}}$}\tablenotemark{a} & $\tau$\tablenotemark{b} & Resolution & $\sigma_c$\tablenotemark{c}\\
& (MHz) & ($\mu$Jy bm$^{-1}$) & (hr) & (\arcsec) & ($\mu$Jy bm$^{-1}$)}
\startdata
LOFAR-LBA & 30-80 & 1100 & 230000 & 10 & 57 \\
LOFAR-HBA & 110-200 & 180 & 6500 & 4.1 & 1.1 \\
HERA & 50-250 & 76 & 1200 & 410 & 36000\\
GMRT & 130-190 & 162 & 5400 & 16.5 & 57 \\
SKA-Low (part) & 50-200 & 5.9 & 7.2 & 7.6 & 9.7 \\
SKA-Low (full) & 50-350 & 1.8 & 0.7 & 4.8 & 1.5 \\
VLA-LOBO & 50-350 & 77 & 1200 & 8.6 & 10.4
\enddata
\tablenotetext{a}{RMS noise in a 1-hour observation, computed for imaging of the full band reported in this table (Equation~\ref{eq:rms_sens}).}
\tablenotetext{b}{The integration time in hours required to reach an RMS of 2~$\mu$Jy (i.e.\ 5-$\sigma$ detection of a typical source at 100~pc with 10~$\mu$Jy bm$^{-1}$ average flux).}
\tablenotetext{c}{Source confusion RMS contribution.}
\end{deluxetable*}
We consider in this section several current and near-future radio observatories that operate at frequencies $\nu \lesssim 350~\rm MHz$ and can reach continuum sensitivities less than 1~mJy in a few hours.
The results are summarized in Table~\ref{tab:sens}.
The primary limitation to detectability is a telescope's sensitivity, characterized by the root mean square (RMS) noise fluctuations in the sky and receiver.
The RMS noise \mbox{$\sigma_{\mbox{\tiny RMS}}$}\ in an interferometric observation with integration time $\tau$, bandwidth $\Delta \nu$, effective area $A_{\rm eff}$ ($\approx0.7 \pi (D/2)^2$ for interferometers comprising antennas with dish size $D$), and $N$ antennas is \citep[see e.g.][]{condon+ransom2016}
\begin{equation}
\mbox{$\sigma_{\mbox{\tiny RMS}}$} = \frac{2 k \mbox{$T_{\mbox{\tiny sys}}$}}{A_{\rm eff} \, \sqrt{\Delta \nu \, \tau N (N-1)}}
\left(\frac{\rm 10^{26} Jy}{\rm W m^{-2} Hz^{-1}}\right) \, .
\label{eq:rms_sens}
\end{equation}
Here, \mbox{$T_{\mbox{\tiny sys}}$}\ is the blackbody-temperature equivalent of the total system noise, which is the sum of the instrument or receiver noise temperature \mbox{$T_{\mbox{\tiny rx}}$}\ and the noise contribution from the sky \mbox{$T_{\mbox{\tiny sky}}$}.
We estimate \mbox{$T_{\mbox{\tiny sky}}$}\ using the low-frequency sky noise temperature fit from \cite{Rogers+Bowman2008}:
\begin{equation}
\mbox{$T_{\mbox{\tiny sky}}$} = T_{150} \left( \frac{\nu}{150~\rm MHz}\right)^{-\beta} + \mbox{$T_{\mbox{\tiny CMB}}$}
\label{eq:Tsky}
\end{equation}
where $T_{150} = 283.2~\rm K$ is the \mbox{$T_{\mbox{\tiny sky}}$}\ at 150~MHz, $\beta=2.47$, and $\mbox{$T_{\mbox{\tiny CMB}}$} = 2.73~\rm K$ is the contribution from the cosmic microwave background (CMB).
While this relation was fit to data taken in the 100---200~MHz range, the authors found it to be consistent with published measurements from 10 to 408~MHz.
An additional factor that affects detectability is source confusion, the imaging noise associated with unresolved interloping radio sources in an observation.
Source confusion is characterized by the standard deviation $\sigma_c$ in the surface brightness of an image due to one or more unresolved sources in the beam solid angle (for a review, see \citealt{Condon1974,Condon2012}).
To estimate $\sigma_c$, reported in Table~\ref{tab:sens}, we use the relations provided by \cite{condon+ransom2016}, reproduced below:
\begin{eqnarray}
\sigma_c & \approx & 0.2 \left( \frac{\nu}{1 \rm~GHz} \right)^{-0.7} \left( \frac{\theta}{1\arcmin} \right)^2\!\!\!\!,~(\theta > 10\arcsec) \\
\sigma_c & \approx & 2.2 \left( \frac{\nu}{1 \rm~GHz} \right)^{-0.7} \left( \frac{\theta}{1\arcmin} \right)^{10/3}\!\!\!\!\!\!\!\!\!\!,~(\theta < 10\arcsec).
\end{eqnarray}
As Table~\ref{tab:sens} illustrates, source confusion is likely to exceed the 2-$\mu$Jy detection limit of a typical RGHJ source for most telescopes under consideration.
However, source confusion can be mitigated through differencing observations that cancel out background sources.
Potential axes for such cancellation are polarization and time.
The source confusion limit is dominated by active galactic nuclei, which have low polarization fractions $\lesssim2.5\%$ \citep{Stil2014} and slowly vary with time.
Therefore, both the circularly polarized nature of the cyclotron emission from planets and the time variability (discussed in \S\ref{ss:timevariability}) partially mitigate the source confusion limit, allowing planetary radio emission, like other time-variable sources, to be detected below the imaging confusion limit.
We therefore expect confusion not to be a strong limitation to the detectability of radio emission from RGHJs.
The LOFAR operates as two separate arrays defined by their high and low bands, differing configurations, and antenna designs, which are called respectively the High Band Antennas (HBA) and Low Band Antennas (LBA).
LOFAR-LBA operates in the 10---90~MHz range, and LOFAR-HBA operates in the 100---240~MHz range \citep{vanHaarlem2013}.
We consider it as two separate instruments here.
LOFAR-LBA's most sensitive band is centered at 60~MHz, and LOFAR-HBA's is at 150~MHz.
Given the strong frequency dependence of the effective aperture of a dipole antenna, and that the receiver noise temperature for LOFAR $\mbox{$T_{\mbox{\tiny rx}}$} \sim \mbox{$T_{\mbox{\tiny sky}}$}$, we use the LOFAR Image Noise calculator\footnote{Heald; \url{http://www.astron.nl/~heald/test/sens.php}}, which is based on {\it SKA Memo 113}\footnote{Nijboer, Pandey-Pommier, \& de Bruyn; \url{http://www.skatelescope.org/uploaded/59513\_113\_Memo\_Nijboer.pdf}}, to obtain the sensitivity and imaging parameters in Table~\ref{tab:sens}.
The HERA\footnote{\url{http://reionization.org/}} is a telescope array under construction in South Africa that will ultimately consist of 352 14-m parabolic dishes, operating from 50 to 250~MHz \citep{pober_et_al2014}.
Unlike the other telescope observatories considered here, HERA is a dedicated experiment for making power spectral measurements of cosmic reionization over wide fields of view.
Similar to its progenitor, the Precision Array to Probe the Epoch of Reionization (PAPER; \citealt{parsons_et_al2010}), HERA's dishes are deployed in an extremely compact configuration \citep{parsons_et_al2012} and statically point toward zenith.
While these features make HERA less ideal for targeted observations, HERA's substantial collecting area, long observing campaigns, and wide field of view make it a powerful survey instrument.
The GMRT is an array comprising 30 25-meter antennas, operating at frequencies 130---2000~MHz.
The lowest band (130---190~MHz) exhibits receiver noise $T_{\rm rx}$ comparable to the sky noise.
To be conservative, we use the GMRT online calculator.\footnote{\url{http://gmrt.ncra.tifr.res.in/~astrosupp/obs\_setup/sensitivity.html}}
The low-frequency component of the SKA, recently rebaselined to have 0.4~km$^2$ collecting area in 130,000 elements with baselines extending to 65~km, will operate from 50 to 350~MHz.\footnote{\url{https://www.skatelescope.org/news/worlds-largest-radio-telescope-near-construction/}}
We estimate the noise in the lower half of the band ($\nu<200~\rm MHz$), appropriate for lower mass ($\lesssim 10 M_{\rm J}$) separately from the full band, which could be used to detect $\sim$30-$M_{\rm J}$ objects.
We again adopt the \cite{Rogers+Bowman2008} relation for \mbox{$T_{\mbox{\tiny sys}}$}\ (Equation~(\ref{eq:Tsky})) and assume $\mbox{$T_{\mbox{\tiny rx}}$} \approx 60~\rm K$.
We caution that these assumptions are optimistic, and the time estimates could in reality be a factor of a few worse.
The low-frequency receivers (28---80~MHz; 4-Band) in the VLA have $\mbox{$T_{\mbox{\tiny rx}}$} \sim 260 \rm~K$, similar to the receiver noise temperatures for the Long Wavelength Array (LWA; \citealt{Hicks2012}).
As noted in \citet{Hicks2012}, this noise level is subdominant to \mbox{$T_{\mbox{\tiny sky}}$}\ by at least -6~dB.
Future upgrades to the VLA such as the LOw Band Observatory (LOBO; \citealt{Kassim2015IAU}) are being considered, and could potentially cover the full 30---470~MHz band.
We estimate the sensitivity of LOBO assuming the receiver noise extrapolates linearly from $\mbox{$T_{\mbox{\tiny rx}}$} \approx 60~\rm K$ (internal VLA memo) in P-Band (230---470~MHz) to $\mbox{$T_{\mbox{\tiny rx}}$} \approx 260~\rm K$ in 4-band, and consider just the 50---350~MHz portion of the full 30---470~MHz band.
The noise estimates shown in Table \ref{tab:sens} indicate that, at the frequencies relevant for the detection of radio emission from Jupiter-size planets, current instrument sensitivities are generally too low.
On the other hand, for more massive exoplanets ($M_p \sim 10 M_{\rm J}$) where the bulk of the radio emission would exist above 50~MHz, the low-frequency component of SKA will be able to detect the signal within half a day from planets with $M_p \sim 10~M_{\rm J}$ at $\sim$1~AU within $\sim$200~pc, or those at $\sim$5~AU within $\sim$100~pc.
VLA-LOBO could also work for a few nearby systems.
\subsection{Discriminating the Signal from the Background}
\label{ss:timevariability}
Since the confusion limit is on the same order as the signal (Table \ref{tab:sens}), it is crucial to consider the ways to discriminate the signal from these sources.
There are two key features that makes planetary radio emission distinct: the circular polarization and the time variability \citep{hess2011,zarka2015}.
Planetary auroral radio emission is nearly completely circularly polarized (e.g., \citealt{dessler1983}, and references therein).
In contrast, active galactic nuclei --- the major confusion sources --- have low polarization fractions $\lesssim$2.5\% \citep{Stil2014}.
The other key feature is the significant time variability of the planetary radio emission.
The factors that influence time variability are listed below:
\paragraph{Modulation due to planetary spin rotation ( $\sim$a few hours)}
Radio bursts from Jupiter tend to be correlated with its spin phase \citep[e.g.][]{dessler1983} due to the misalignment between the magnetic axis and the spin axis.
Similarly, high-temporal-resolution observations of an RGHJ system could reveal periodicity at the planet's spin frequency.
\paragraph{Modulation due to the presence of satellites ($\sim$a few days)}
Jupiter's radio bursts are affected by the location of its moon Io, as well \citep{dessler1983}, because Io disturbs the surrounding electromagnetic field.
Likewise, if an RGHJ has a moon that emits plasma, then the radio flux from such a system is likely to be modulated by the orbital motion of the moon \citep[see, e.g.,][]{noyola2014}.
\paragraph{Variability of the stellar wind ($\sim$a few months?)}
If the stellar wind passing the planet is variable, any modulation of the plasma density would also lead to time variability of planetary radio emission.
Such variability might be expected to occur on timescales of several months to half a year, as many M giants have semi-regular periodicities on the order of a few hundred days \citep{Kiss:1999aa}.
\paragraph{Modulation due to orbital motion of the planet ($\sim$a few years)}
If the planetary orbit is close to the edge-on configuration, the radio emission is likely to disappear or significantly dim at certain orbital phases, due to the increased plasma cut-off frequency along the path and/or secondary eclipse \citep{lecavelier_et_al2013}.
The plasma cut-off frequency increases at some phases because, when the planet is on the far side of the star from Earth, the path of the radio emission toward Earth comes through the vicinity of the star, where the plasma density may be high enough to be partially or completely opaque to the emission, depending on the ionization fraction of the stellar wind.
\section{Summary and Discussion}
\label{s:conc}
In this paper, we estimate the radio brightness of distant ``hot Jupiters'' around evolved stars (RGHJs).
Unlike \citet{ignace2010}, we consider that UV/X-ray photons from accretion onto the planets partially ionize the stellar wind in the vicinity of planets, which would otherwise be mostly neutral for a cool star's wind.
This process yields the free electrons that are crucial to producing the radio emission.
Based on such a picture, the dense stellar wind of an RGB or AGB star would interact with the planetary magnetic field and would add power in the form of kinetic energy into the magnetosphere of an RGHJ.
We find that the intrinsic brightness of radio emission from RGHJs could be comparable to or greater than that of canonical hot Jupiters in close-in orbits around main-sequence stars and $>$100 times brighter than distant Jupiter-twins around main-sequence stars.
This implies that they can be searched for 10 times further away or in a volume 1000 times as large.
This can compensate for the rareness of the evolved stars at least partly.
Thus, RGHJs will serve as reasonable targets in future searches for exoplanetary radio emission.
A major obstacle to observing this radio emission is the plasma cut-off frequency of the (ionized) stellar wind.
Due to the great density of the stellar wind, the cut-off frequency is as high as $\sim$12~MHz for typical red giants and $\sim$400~MHz for typical AGB stars, making planetary-mass companions to AGB stars difficult or impossible to see via their auroral radio emission.
The most promising targets are massive planetary companions ($M_p \gtrsim 5 M_{\rm J}$) to red giant stars, with magnetic fields stronger than that of Jupiter (and, therefore, higher cyclotron frequencies).
Similarly, if the plasma frequency in the region where the bulk of the radio emission is generated is large relative to the cyclotron frequency, this could pose a serious obstacle to observability.
The radio flux from a system with a Jovian exoplanet at a distance of 100~pc would typically be on the order of $\sim$10~$\mu$Jy.
Such signals would be detectable with SKA within half a day.
For a few nearby systems, a possible upgrade to the VLA such as LOBO would also provide reasonable integration times and resolving power.
In both cases, it is critical to consider polarization and/or time variability of the planetary radio flux, in order to separate it from confused sources in the same resolution element.
The radio emission from RGHJs may soon prove to be a valuable tool for surveying exoplanets in a region of parameter space around highly evolved stars where traditional exoplanet-discovery methods, such as transit and radial-velocity observations, are less sensitive.
Once this planetary radio emission is detected, it will provide a unique approach for studying RGHJ properties.
As the auroral radio flux is significantly modulated in association with several physical processes described in Section \ref{ss:timevariability}, the time variability is useful to infer the planetary spin rotation period or the presence of satellites.
If spectra are obtained, the upper cut-off frequency can tell us the magnetic field strength at the planetary surface, which, when combined with constraints on the planetary mass via, e.g., radial-velocity observations, tests the proposed scaling law for the planetary dynamo.
Additionally, the lower cut-off frequency of the RGHJ emission could probe the stellar wind properties at the planetary orbit.
And finally, the planetary radio power in such an extreme environment as an RG's massive stellar wind will provide valuable data points to test the empirical ``radiometric Bode's law.''
\vspace{0.5in}
\acknowledgements
{\sc Acknowledgments}
Y.F. is supported from the Grant-in-Aid No. 25887024 by the Japan Society for the Promotion of Science and from an appointment to the NASA Postdoctoral Program at NASA Goddard Institute for Space Studies, administered by Oak Ridge Affiliated Universities.
D.S.S. gratefully acknowledges support from a fellowship from the AMIAS group.
J.N. acknowledges support from NASA grant HST AR-12146.04-A and NSF grant AST-1102738.
We thank David Hogg for encouraging us to pursue calculations of exoplanetary radio emission.
We thank Greg Novak for helpful conversations.
We thank Jake VanderPlas for developing the XKCD-style plotting package for matplotlib, and acknowledge its use.
A.R.P. is grateful for support from NSF grants 1352519 and 1440343.
Y.F. greatly acknowledges insightful and helpful discussions with Tomoki Kimura and Hiroki Harakawa.
The portion of this research for which T.M. is responsible was performed under a National Research Council Research Associateship Award at the Naval Research Laboratory (NRL).
Basic research in radio astronomy at NRL is supported by 6.1 Base funding.
|
1,116,691,497,739 | arxiv |
\subsection{Contraction of Pairs}
In this section we investigate properties of the contraction operation in our goal to extend Theorem \ref{theo:P123} to contractible pairs. More specifically, we characterize representations of pairs $(G,C)$ satisfying $(P2),(P3)$. We show that (i) the contraction operation preserves $\enpt$ edges, (ii) the order of contractions is irrelevant and (iii) the contraction operation preserves $(P2)$, $(P3)$.
\begin{lemma}\label{lem:contraction-pairs}
Let $\rep$ be a representation for the pair $(G,G')$, and let $e=\set{p,q} \in E(G')$. If $\contract{(G,G')}{e}$ is defined then
$\contractppq$ is a representation for the pair $\contract{(G,G')}{e}$.
\end{lemma}
\begin{proof}
By Lemma \ref{lem:contraction-union} $\contractppq$ is an $\ept$ representation for $\contractge$. It remains to show that it is an $\enpt$ representation for $\contract{G'}{e}$, i.e. that for any two paths $P_{p'},P_{q'} \in \contractppq$, the edge $e'=\set{p',q'}$ is in $E(\contract{G'}{e}) \iff P_{p'} \sim P_{q'}$. Let $P_s = P_p \cup P_q$ and $s$ be the vertex obtained by the contraction. We assume first that $P_s \notin \set {P_{p'},P_{q'}}$. Then $e' \in E(\contract{G'}{e}) \iff e' \in E(G') \iff P_{p'} \sim P_{q'}$ as required. Now we assume without loss of generality that $P_{p'}=P_s$ and we recall that $e=\set{p,q} \in E(G')$ is the contracted edge. We have to show that $e'=\set{s,q'} \in E(\contract{G'}{e}) \iff P_s \sim P_{q'}$. We observe that
\begin{eqnarray}
\set{s,q'} \in E(\contract{G'}{e}) & \iff & \set{p,q'} \in E(G') \vee \set{q,q'} \in E(G') \nonumber\\
& \iff & P_p \sim P_{q'} \vee P_q \sim P_{q'}\label{eqn:one}
\end{eqnarray}
and
\begin{eqnarray}
P_s \sim P_{q'} & \iff & (P_p \cup P_q) \cap P_{q'} \neq \emptyset \wedge \split(P_p \cup P_q, P_{q'})=\emptyset \nonumber\\
& \iff & (P_p \cap P_{q'} \neq \emptyset \vee P_q \cap P_{q'} \neq \emptyset) \wedge \split(P_p, P_{q'})=\emptyset \wedge \split(P_q, P_{q'})=\emptyset.\label{eqn:two}
\end{eqnarray}
Clearly, (\ref{eqn:two}) implies (\ref{eqn:one}). To conclude the proof, assume that ($\ref{eqn:one}$) holds. Then $P_p \cap P_{q'} \neq \emptyset \vee P_q \cap P_{q'} \neq \emptyset$. Now assume, by way of contradiction, that (\ref{eqn:two}) does not hold. Then $\split(P_p,P_{q'}) \neq \emptyset \vee \split(P_q,P_{q'}) \neq \emptyset$ implying $P_p \nsim P_{q'} \vee P_q \nsim P_{q'}$. Combining with (\ref{eqn:one}) this implies that exactly one of $P_p \sim P_{q'}$ and $P_q \sim P_{q'}$ holds. Therefore without loss of generality $P_p \sim P_{q'}, P_q \nsim P_{q'}$. Then $e'=\set{p,q'} \in E(G')$ and $e \triangle e' \in E(G)$, therefore $\contract{(G,G')}{e}$ is undefined, thus constituting a contradiction to the assumption of the lemma.
\end{proof}
Let $\bar{E}=\set{e_1, e_2, \ldots, e_k} \subseteq E(G')$. For every $k>1$ we define $\contract{(G,G')}{e_1,\ldots,e_k} \defined \contract{\contract{(G,G')}{e_1,\ldots,e_{k-1}}}{e_k}$ provided that both contractions on the right hand side are defined, otherwise it is undefined. The following Lemma follows from Lemma \ref{lem:contraction-pairs} and states that the order of contraction of the edges is irrelevant.
\begin{lemma}\label{lem:orderIrrelevantInContraction}
Let $(G,G')$ be a pair, $\bar E= \set{e_1, e_2, \ldots, e_k} \subseteq E(G)$, and $\pi$ a permutation of the integers $\set{1,\ldots, k}$. Then $\contract{(G,G')}{e_1,\ldots,e_k}$ is defined if and only if $\contract{(G,G')}{e_{\pi(1)},\ldots,e_{\pi(k)}}$ is defined. Moreover when they are defined $\contract{(G,G')}{e_1,\ldots,e_k}=\contract{(G,G')}{e_{\pi(1)},\ldots,e_{\pi(k)}}$.
\end{lemma}
\begin{proof}
Assume that $\contract{(G,G')}{e_1,\ldots,e_k}$ is defined. By $k-1$ successive applications of Lemma \ref{lem:contraction-pairs} we conclude that a representation of $\contract{(G,G')}{e_1,\ldots,e_k}$ can be obtained from a representation of $(G,G')$ by applying a sequence of $k-1$ union operations. The result of $k-1$ union operations yield a set of paths. As union is commutative and associative, this result will remain the same, (i.e. a set of paths) if we change the order of the operations. On the other hand as union preserves split vertices, and the result does not contain split vertices, there are no split vertices at any given step of new sequence of union operations. We conclude that $\contract{(G,G')}{e_{\pi(1)},\ldots,e_{\pi(k)}}$ is defined. The other direction holds by symmetry. Whenever both $\contract{(G,G')}{e_1,\ldots,e_k}$ and $\contract{(G,G')}{e_{\pi(1)},\ldots,e_{\pi(k)}}$ are defined we have
$\contract{(G,G')}{e_1,\ldots,e_k}$=$(\contract{G}{e_1,\ldots,e_k},\contract{G'}{e_1,\ldots,e_k})$=
$(\contract{G}{e_\pi(1),\ldots,e_\pi(k)},\contract{G'}{e_\pi(1),\ldots,e_\pi(k)})=\contract{(G,G')}{e_{\pi(1)},\ldots,e_{\pi(k)}}$.
\end{proof}
Based on this result, we denote the contracted pair as $\contract{(G,G')}{\bar{E}}$ and say that $\bar{E}$ is contractible.
If $\repn{2}=\contract{\repn{1}}{P_p, P_q}$ for two paths $P_p, P_q \in \pp_1$ we denote this by $\repn{1} \rightsquigarrow_U \repn{2}$. The relation $\gtrsim_U$ is the reflexive-transitive closure of $\rightsquigarrow_U$, and $\repn{1} \lesssim_U \repn{2}$ is equivalent to $\repn{2} \lesssim_U \repn{1}$.
$(G_1,G'_1) \rightsquigarrow_C (G_2, G'_2)$ if $(G_2, G'_2) = \contract{(G_1, G'_1)}{e}$ for some $e \in E(G'_1)$. The relation $\gtrsim_C$ is the reflexive-transitive closure of $\rightsquigarrow_C$, and $\repn{1} \lesssim_C \repn{2}$ is equivalent to $\repn{2} \lesssim_C \repn{1}$.
By Lemma \ref{lem:contraction-pairs}, $\lesssim_U$ is homomorphic to $\lesssim_C$.
Following the above definitions, a non-contractible pair of graphs is said to be \emph{contraction-minimal}, because it is minimal in the partial order $\lesssim_C$.
We proceed by showing that the contraction operation preserves assumptions $(P2),(P3)$.
\begin{lemma}\label{lem:BBRTopDoesNotComeFromContraction}
Let $\set{p,q,r}$ be a $BBR$ triangle of $\contract{(G,G')}{e}$ with $\set{p,r}$ being the red edge. Then $q \in V(G)$, i.e. $q$ is not the vertex obtained by the contraction.
\end{lemma}
\begin{proof}
Assume, by contradiction that $e=\set{q',q''}$ and $q$ is the vertex obtained by the contraction of $e$. Assume without loss of generality that $\set{p,q'}$ and $\set{q'',r}$ are edges of $G'$. Then both $\set{p,q''}$ and $\set{r,q'}$ are non-edges of $G$, because otherwise $e$ is not contractible. Then $\set{p,q',q'',r}$ is a hole of size $4$ with blues edges, a contradiction.
\end{proof}
\begin{lemma}\label{lem:ContractionPreservesP2P3}
(i) If $(P2)$ holds for $(G,G')$ then $(P2)$ holds for $\contract{(G,G')}{e}$.\\
(ii) If $(P3)$ holds for $(G,G')$ then $(P3)$ holds for $\contract{(G,G')}{e}$.\\
\end{lemma}
\begin{proof}
(i) Assume, by contradiction, that $(G,G')$ does not have an induced sub-pair isomorphic to $\kp$ and without loss of generality $\contract{(G,G')}{e}$ has a sub-pair isomorphic to $\kp$ induced by the vertices $U=\set{p,q,r,s}$ where $p$ and $s$ are the endpoints of the subgraph isomorphic to $P_4$. Let $v$ be the vertex created by the contraction of $e$. If $v \notin U$ then the sub-pair induced by $U$ is also a sub-pair of $(G,G')$, contradicting our assumption. Therefore $v \in U$. By Lemma \ref{lem:BBRTopDoesNotComeFromContraction} we have that $v \notin \set{q,r}$. Therefore, let without loss of generality $v=p$, $e=\set{p',p''}$ and $p''$ is adjacent to $q$ in $G'$. $\set{p',q}$ is a non-edge of $G$, because $e$ is contractible. As $\set{p,s}$ and $\set{p,r}$ are edges of $\contractge$, $\set{p',s}$ or $\set{p'',s}$ is an edge of $G$, and $\set{p',r}$ or $\set{p'',r}$ is an edge of $G$. If $\set{p'',s}$ is an edge of $G$ then $\set{p'',r}$ is a non-edge of $G$ since otherwise $\set{p'',q,r,s}$ induce a sub-pair isomorphic to $\kp$ in $(G,G')$. Therefore $\set{p',r}$ is an edge of $G$. Then $\set{p',p'',q,r}$ induces a hole of size 4 with blue edges, a contradiction. Thus $\set{p',s}$ is an edge of $G$, since $\set{p',q}$ and $\set{p'',s}$ are non-edges of $G$, $\set{p',p'',q,s}$ induce a hole of length 4 with blue edges, a contradiction.
(ii) Assume, by contradiction, that $(G,G')$ has a representation $\rep$ satisfying $(P3)$ and that no representation of $\contract{(G,G')}{e}$ satisfies $(P3)$. Let $v$ be the vertex created by the contraction of $e=\set{v',v''}$. Then by Lemma \ref{lem:contraction-union} $\contract{\rep}{P_{v'},P_{v''}}$ is a representation of $\contract{(G,G')}{e}$ and it contains a red edge-clique $\set{p,q,r}$. If $v \notin \set{p,q,r}$ then $\set{p,q,r}$ is an edge-clique of $\rep$, contradicting our assumption. Assume without loss of generality that $v=p$. Let $e_0$ be an edge of $T$ defining the edge-clique $\set{v,q,r}$. Since $e_0 \in P_{v'} \cup P_{v''}$, without loss of generality $e_0 \in P_{v'}$. Then $\set{v',q,r}$ induces an edge-clique on $e_0$. This is clearly a red edge-clique since $e$ is contractible, a contradiction.
\end{proof}
We now describe how minimal representations of $\contract{(G,G')}{e}$ can be obtained from minimal representations of $(G,G')$.
\begin{lemma}\label{lem:MinimizationAfterUnion}
Let $\rep$ be a minimal representation, $\repprime$ a representation such that $\repprime \lesssim \contractppq$ and $\repprime \approxeq \contractppq$. Let $e$ be an edge of $T$ involved in a minimal sequence of minifying operations $s$ that obtains $\repprime$ from $\contractppq$. There is an operation of $s$ and a path $P$ such that the operation removes $e$ from $P$ ($tr(P,e)$, or $contract(e)$ and $e \in P$) where at least one of the following holds:\\
(i) $e$ is a tail of $P_p \cap P_q$, $P \cap P_p \cap P_q=\set{e}$ and $P \cap (P_p \cup P_q) \supsetneq \set{e}$.\\
(ii) $e$ is incident to an internal vertex $u$ of $P_p \cup P_q$, $e$ is a tail of $P$, $P$ is not in a pie with center $u$.
\end{lemma}
\begin{proof}
Let $G=\eptgp$ and $G'=\enptgp$ and consider an operation $op$ of $s$. Without loss of generality we can assume that $op$ is the first operation of $s$, by Lemma \ref{lem:MinOperationsExchangeable}. Furthermore, by Lemma \ref{lem:MinOperationsMonotone}, the representation obtained by applying $op$ is also equivalent to $\contractppq$. Therefore without loss of generality $op$ is the only operation of $s$. Note that $op$ is defined on $\contractppq$ except when $op$ is $tr(P_p \cup P_q,e)$. In this case $e$ is a tail of $P_p$ or $P_q$ (or both). In the following discussion, whenever we apply $op$ to $\rep$, we mean that we apply $tr(P_p,e)$ or $tr(P_q,e)$ one of which is well defined on $\rep$.
By the minimality of $\rep$, $op$ cannot be applied to $\rep$. More precisely, if $op$ is applied, either an edge of $G$ becomes a non-edge, or a red edge of $(G,G')$ becomes a blue edge. We term such an edge of $(G,G')$ \emph{an affected edge} and the corresponding paths of $\rep$ \emph{affected pair of paths}. Let $\set{r,s}$ be an affected edge of $(G,G')$. If $\set{P_r,P_s} \cap \set{P_p,P_q} = \emptyset$ then $P_r,P_s$ is a pair of affected paths in $\contractppq$, contradicting the fact that $op$ can be applied to $\contractppq$. We conclude that $\set{P_r,P_s} \cap \set{P_p, P_q} \neq \emptyset$. Assume without loss of generality that $P_s=P_p$, i.e. $P_r,P_p$ is an affected pair of paths. We consider two disjoint cases:
Case 1) $\set{r,p}$ becomes a non-edge after applying $op$. Then $P_r \cap P_p = \set{e}$ for some edge $e$ of $T$, and after the removal of $e$ the intersection becomes empty. On the other hand $P_r \cap (P_p \cup P_q) \supsetneq \set{e}$, because otherwise $P_r$ and $P_p \cup P_q$ constitute an affected pair of paths in $\contractppq$. Then $e$ is a tail of $P_p \cap P_q$ (see Figure \ref{fig:MinimizationAfterUnion} a) and $P_r$ is the claimed path $P$ (note that possibly $r=q$ as opposed to the figure). In this case i) holds.
Case 2) $\set{r,p}$ is a red edge, and it becomes a blue edge after applying $op$. Then $P_r \nsim P_p$ (therefore $r \neq q$) and $P_r,P_p$ do not split after applying $op$. Therefore $\split(P_r,P_p)=\set{u}$ for an endpoint $u$ of $e$, $e$ is a tail of exactly one of $P_r, P_p$, and $u$ is an internal vertex of $P_p$ thus of $P_p \cup P_q$. Let $\set{P,P'} = \set{P_r, P_p}$ such that $e$ is a tail of $P$, and $e \notin P'$. If $P$ is not in a pie with center $u$ then ii) holds. Otherwise $P$ has two neighbors $\set{P',P''}$ in this pie. $e \in P''$ because $e \notin P$ and $e$ is an edge incident to $u$, the center of the pie. Recalling that $e$ is a tail of $P$ we conclude that after the removal of $e$ from $P$, its intersection with $P''$ becomes empty. Therefore i) holds.
\end{proof}
\begin{figure}[htbp]
\centering
\include{figure/MinimizationAfterUnion}
\caption{Possible minifying operations on $\contractppq$.}
\label{fig:MinimizationAfterUnion}
\end{figure}
\begin{lemma}\label{lem:EverySplitVertexIsACenterofBrokenPlanarTour}
Every split vertex of a path $P$ of a broken planar tour is a center of a pie containing $P$.
\end{lemma}
\begin{proof}
By construction, every split vertex of a path $P$ of a tour is a center of a pie containing $P$. We will show that the same holds for a broken planar tour. Let $P_1',P_2'$ be two paths of a broken planar tour such that $v \in \split(P_1',P_2')$. These paths are sub-paths of two paths $P_1,P_2$ of a tour and $v \in \split(P_1,P_2)$. Then $v$ is a center of a pie containing $P_1,P_2$ and also other paths. Each one of the other paths has at least one sub-path in the broken planar tour that crosses $v$. These paths, together with $P_1',P_2'$ constitute a pie with center $v$ of the broken planar tour. We conclude that every split vertex of $P$ is a center of a pie, and therefore case (ii) of Lemma \ref{lem:MinimizationAfterUnion} is impossible.
\end{proof}
We notice that by Lemma \ref{lem:EverySplitVertexIsACenterofBrokenPlanarTour} it follows that the case (ii) of Lemma \ref{lem:MinimizationAfterUnion} is impossible.
\subsection{Background}
Given a tree $T$ and a set $\pp$ of non-trivial simple paths in $T$, the Vertex Intersection Graph of Paths in a Tree (\vpt) and the Edge Intersection Graph of Paths in a Tree ($\ept$) of $\pp$ are denoted by $\vptgp$ and $\eptgp$, respectively. Both graphs have $\pp$ as vertex set. $\vptgp$ (resp. $\eptgp$) contains an edge between two vertices if the corresponding two paths intersect in at least one vertex (resp. edge). A graph $G$ is $\vpt$ (resp. $\ept$) if there exist a tree $T$ and a set $\pp$ of non-trivial simple path in $T$ such that $G$ is isomorphic to $\vptgp$ (resp. $\eptgp$). In this case we say that $\rep$ is an $\vpt$ (resp. $\ept$) representation of $G$.
In this work we focus on edge intersections of paths, therefore whenever we are concerned with intersection of paths we omit the word "edge" and simply write that two paths \emph{intersect}. The graph of edge intersecting and non-splitting paths of a tree ($\enpt$) of a given representation $\rep$ as described above, denoted by $\enptgp$, has a vertex $v$ for each path $P_v$ of $\pp$ and two vertices $u,v$ of $\enptgp$ are adjacent if the paths $P_u$ and $P_v$ intersect and do not split (that is, their union is a path). A graph $G$ is an $\enpt$ graph if there is a tree $T$ and a set of paths $\pp$ of $T$ such that $G$ is isomorphic to $\enptgp$. We note that when $T$ is a path $\eptgp=\enptgp$ is an Interval Graph. Therefore the class $\enpt$ includes all Interval Graphs.
$\ept$ and $\vpt$ graphs have applications in communication networks. Assume that we model a communication network as a tree $T$ and the message routes to be delivered in this communication network as paths on $T$. Two paths conflict if they both require to use the same link (vertex). This conflict model is equivalent to an $\ept$ (a $\vpt$) graph. Suppose we try to find a schedule for the messages such that no two messages sharing a link (vertex) are scheduled in the same time interval. Then a vertex coloring of the $\ept$ ($\vpt$) graph corresponds a feasible schedule on this network.
$\ept$ graphs also appear in all-optical telecommunication networks. The so-called Wavelength Division Multiplexing (WDM) technology can multiplex different signals onto a single optical fiber by using different wavelength ranges of the laser beam \cite{CGK92,R93}. WDM is a promising technology enabling us to deal with the massive growth of traffic in telecommunication networks, due to applications such as video-conferencing, cloud computing and distributed computing \cite{DV93}. A stream of signals traveling from its source to its destination in optical form is termed a \emph{lightpath}. A lightpath is realized by signals traveling through a series of fibers, on a certain wavelength. Specifically, Wavelength Assignment problems (WLA) are a family of path coloring problems that aim to assign wavelengths (i.e. colors) to lightpaths, so that no two lightpaths with a common edge receive the same wavelength and certain objective function (depending on the problem) is minimized. \emph{Traffic Grooming} is the term used for combination of several low-capacity requests into one lightpath using Time Division Multiplexing (TDM) technology \cite{GRS98}. In this context a set of paths can be combined into one lightpath, thus receiving the same color, as long as they satisfy the following two conditions:
\begin{itemize}
\item The load condition: at most $g$ lightpaths using the same fiber may receive the same color, where $g$ is an integer termed \emph{the grooming factor}.
\item The union of the paths receiving the same color should constitute a path or a set of disjoint paths.
\end{itemize}
It follows that a feasible solution of the traffic grooming problem is a vertex coloring of the graph $\eptgp \setminus \enptgp$ where each color class induces a sub-graph of $\eptgp$ with clique number at most $g$. Therefore it makes sense to analyze the structure of these graph pairs, i.e. the two graphs $\eptgp$ and $\enptgp$ defined on the same set of vertices.
\subsection{Related Work}
$\ept$ and $\vpt$ graphs have been extensively studied in the literature. Although $\vpt$ graphs can be characterized by a fixed number of forbidden subgraphs \cite{Leveque2009Characterizing}, it is shown that $\ept$ graph recognition is NP-complete \cite{GJ85}. Edge intersection and vertex intersection give rise to identical graph classes in the case of paths in a line and in the case of subtrees of a tree. However, $\vpt$ graphs and $\ept$ graphs are incomparable in general; neither $\vpt$ nor $\ept$ contains the other. Main optimization and decision problems such as recognition \cite{Fanica1978211}, the maximum clique \cite{Fanica2000181}, the minimum vertex coloring \cite{Golumbic:2004:AGT:984029} and the maximum stable set problems \cite{Spinrad1995181} are polynomial-time solvable in $\vpt$
whereas recognition and minimum vertex coloring problems remain NP-complete in $\ept$ graphs \cite{Golumbic1985151}. In contrast, one can solve in polynomial time the maximum clique \cite{GJ85} and the maximum stable set \cite{RobertE1985221} problems in $\ept$ graphs.
After these works on $\ept$ and $\vpt$ graphs in the early 80's, this topic did not get focus until very recently. Current research on intersection graphs is concentrated on the comparison of various intersection graphs of paths in a tree and their relation to chordal and weakly chordal graphs \cite{Golumbic2008Equivalences,GolumbicLS08}. Also, some tolerance model is studied via $k$-edge intersection graphs where two vertices are adjacent if their corresponding paths intersect on at least $k$ edges \cite{Golumbic2008Kedge}. Besides, several recent papers consider the edge intersection graphs of paths on a grid (e.g \cite{BiedlS10}).
In \cite{BESZ13} we defined the family of $\enpt$ graphs and proved basic properties of these graphs. Namely we show that this family includes the families of trees, cycles, cliques, and interval graphs. We then focus on the representation of $\enpt$ holes. The main result of \cite{BESZ13} is an algorithm that, under a certain assumption, finds the representation of an $\enpt$ cycle given together with its corresponding $\ept$ graph. Through this algorithm we characterize all representations of $\enpt$ holes satisfying the assumption.
\subsection{Our Contribution}
In this work our main objective is to characterize minimal $\enpt$ representations of chordless cycles. To achieve this goal we assume that the input is augmented by the underlying $\ept$ graph. For any given pair of graph $(G,C)$ where $C$ is an Hamiltonian cycle of $G$, we provide the minimal representation (if such a representation exists) of $(G,C)$ satisfying some assumption defined in Section \ref{subsec:eptn-OurPreviousResults}, $(P3)$. We also show that the problem is $\nph$ in general, i.e. when the assumption does not necessarily hold. Our approach can be summarized as follows. Given a pair $(G,C)$, we first remove $\kp$ and contract all contactable $\enpt$ edges. In \cite{BESZ13}, we characterized minimal representations of contraction-minimal and $\kp$-free pairs. A contraction operation corresponds to a union of two paths in the representation. We then study the reverse operations of contractions (union in the representation) and characterization of representations of $\kp$s.
In Section \ref{sec:prelim} we give definitions and preliminaries. In Section \ref{sec:uncontraction} we present basic results related to contraction and describe an algorithm for pairs which are $\kp$-free satisfying the assumption $(P3)$. In Section \ref{sec:K4P4} we characterize the representations of $\kp$, we define aggressive contraction operation and using these results we present an algorithm that returns the minimal representation of a given pair $(G,C)$ satisfying $(P3)$. In Section \ref{sec:General} we show that in general, in the case where the assumption does not hold, there does not exist a polynomial-time algorithm that provides the minimal representation of a given pair $(G,C)$, unless $\textsc{P}=\textsc{NP}$.
\subsection{Representations of $\kp$ and Small Cycles}\label{subsec:k4p4}
We denote a set of $4$ vertices inducing a sub-pair isomorphic to $\kp$ as an ordered quadruple where the first vertex is one of the endpoints of the the induced $P_4$, the second vertex is its neighbor and so on. $(p,q,r,s)$ is a $\kp$ of $(G,G')$ whenever $\set{p,q,r,s}$ induces a sub-pair $\kp$ of $(G,G')$ and $p,s$ are the endpoints of the induced sub-path isomorphic to $P_4$. Clearly, $(p,q,r,s)=(s,r,q,p)$.
We start with a Lemma that characterize representations of $\kp$ pairs in general. This lemma will be useful in developing our results. Then we present the unique minimal representation of $(G,C_5)$ pairs containing a $\kp$. Together with Lemma \ref{lem:unique-c5} this completes the characterization of all the $(G,C_5)$ pairs because a $(G,C_5)$ satisfies $(P3)$ vacuously. We continue by proving more properties of minimal representations of induce $\kp$ sub-pairs of pairs $(G,C)$ with at least $6$ vertices. Using these properties we show that a $(G,C_6)$ satisfying $(P3)$ does not contain sub-pairs isomorphic to $\kp$.
\newcommand{\bigcap \pp_K}{\bigcap \pp_K}
\begin{lemma}\label{lem:K4P4Representation}
Let $K=(i,i+1,i+2,i+3)$ be a $\kp$, $\rep$ be a representation of $K$, and $\bigcap \pp_K \defined P_i \cap P_{i+1} \cap P_{i+2} \cap P_{i+3}$. There is a path $core(K)$ of $T$ with endpoints $u,v$ such that:\\
(i)~$\split(P_i,P_{i+2})=\set{u}$, $\split(P_{i+1},P_{i+3})=\set{v}$, $P_{i+1}$ (resp. $P_{i+2}$) does not cross $u$ (resp. $v$).\\
(ii)~$\emptyset \neq \bigcap \pp_K \subseteq \left( P_{i+1} \cap P_{i+2} \right) \subseteq core(K)$. In particular $u \neq v$.\\
(iii)~At least one of $P_i, P_{i+3}$ crosses both endpoints of $core(K)$ and $\emptyset \neq \split(P_i, P_{i+3}) \subseteq \set{u,v}$.\\
(iv)~$P_{i+1} \cup P_{i+2}$ crosses both endpoints of $core(K)$.\\
(v)~The removal of the edges of $P_{i+1} \cup P_{i+2}$ from $T$ disconnects $P_i$ from $P_{i+3}$.
\end{lemma}
\begin{proof}
(i) Assume, by way of contradiction, that $\abs{\split(P_i, P_{i+2})}=2$. Let these two vertices be $w,w'$. As $P_{i+1} \sim P_i$ and $P_{i+1} \sim P_{i+2}$ we conclude that $P_{i+1} \subseteq p_T(w,w')$. $P_{i+3} \nsim P_{i+1}$ therefore $P_{i+3}$ splits from $P_{i+1}$ in at least one vertex $w''$ that is an intermediate vertex of $p_T(w,w')$. Then $P_{i+3}$ splits from $P_{i+2}$ at $w''$ contradicting the fact that $\set{i+2,i+3}$ is an $\enpt$ edge. Therefore $\abs{\split(P_i, P_{i+2})}=1$ and by symmetry, $\abs{\split(P_{i+1},P_{i+3})}=1$. Let $\split(P_i, P_{i+2}) = \set{u}$ and $\split(P_{i+1}, P_{i+3}) = \set{v}$. We define $core(K) = p_T(u,v)$. For the rest of the claim, assume by contradiction that $P_{i+1}$ crosses $u$. Then either $P_{i+1} \nsim P_i$ or $P_{i+1} \nsim P_{i+2}$, contradicting the the fact that $\set{i,i+1}$ and $\set{i+1,i+2}$ are $\enpt$ edges.
At this point we can uniquely define the following edges that will be used in the rest of the proof: $e_i$ (resp. $e_{i+2}$) is the edge of $P_i \setminus P_{i+2}$ (resp. $P_{i+2} \setminus P_i$) incident to $\split(P_i,P_{i+2})$, and $e_{i+1}$ and $e_{i+3}$ are defined similarly. Note that $e_i \neq e_{i+2}$ and $e_{i+1} \neq e_{i+3}$, but the definition does not exclude the possibility that, for instance $e_i = e_{i+1}$.
(ii) A claw-clique of size $4$ contains exactly one $\enpt$ edge, however a path isomorphic to $P_4$ contains three edges. Therefore the representation of $K_4$ is an edge-clique. Let $e$ be an edge defining this edge-clique, i.e. $e \in \bigcap \pp_K$. The removal of $e$ from $T$ disconnects it into two subtrees. In order to prove that $\bigcap \pp_K \subseteq core(K)$ it suffices to show that $u$ and $v$ are in different subtrees. Assume, by way of contradiction that $u,v$ are in the same subtree $T_r$ with root $r$ where $r$ is an endpoint of $e$. Let $r'$ be the least common ancestor of $u,v$ in $T_r$ (possibly $u=v$ in which case $r'=u=v$). All the $4$ paths contain $e$ and cross $r'$ (so that each one crosses at least one of $u,v$), i.e. they ``enter'' $r'$ from the same edge $e'$ (where possibly $r'=r$ and $e'=e$). If $r' \notin \set{u,v}$ then as $P_{i+1}$ crosses $v$ and $P_{i+2}$ crosses $u$, $r' \in \split(P_{i+1}, P_{i+2})$, contradicting $P_{i+1} \sim P_{i+2}$. Therefore we can assume without loss of generality that $r'=u$. Then the edges $e_i$ and $e_{i+2}$ are incident to $r'$. Then $P_{i+1}$ (resp. $P_{i+3}$) contains $e_i$ (resp. $e_{i+2}$) because $P_{i+1} \sim P_i$ (resp. $P_{i+3} \sim P_{i+2}$). Therefore $r' \in \split(P_{i+1}, P_{i+2})$, contradicting $P_{i+1} \sim P_{i+2}$. Therefore $u$ and $v$ are in different subtrees, i.e. $e \in p_T(u,v)=core(K)$. As $e$ can be any edge defining the edge-clique this implies that $\bigcap \pp_K \subseteq core(K)$. It remains to prove that $P_{i+1} \cap P_{i+2} \subseteq core(K)$. For this purpose, it is sufficient to show that both of $P_{i+1}$ and $P_{i+2}$ have one endpoint in $core(K)$. Indeed, assume without loss of generality that $P_{i+1}$ does not have an endpoint in $core(K)$. Then $P_{i+1}$ crosses $u$ and does not include at least one of the edges $e_i, e_{i+2}$. Therefore $P_{i+1} \nsim P_i$ or $P_{i+1} \nsim P_{i+2}$, a contradiction.
Consult Figure \ref{fig:K4P4Representation} for the rest of the proof.
(iii) By the above discussion $u$ (resp. $v$) is an intermediate vertex of $P_i$ and $P_{i+2}$ (resp.$P_{i+1}$ and $P_{i+3}$), and they all intersect in at least one edge $e \in core(K)$. In order to see the first part of the claim assume, by way of contradiction, that both of $P_i$ and $P_{i+3}$ have an endpoint in $core(K)$, in this case $\bigcap \pp_K$ is between these two endpoints. Therefore $P_i \sim P_{i+3}$, a contradiction.
We now proceed to show the rest of the claim: Let $w \in \split(P_i,P_{i+3})$. $e_i \notin P_{i+3}$ because otherwise $P_{i+3} \nsim P_{i+2}$, and by symmetry $e_{i+3} \notin P_i$. Therefore, $w$ is on $core(K)$. On the other hand $w$ is not an intermediate vertex of $core(K)$. Indeed, consider the two sub-paths obtained by removing $e$ from $core(K)$. If $w$ is an intermediate vertex of $core(K)$, then at least one of $P_{i+3} \nsim P_{i+2}$, $P_i \nsim P_{i+1}$ holds, depending on the sub-path $w$ belongs. We conclude $w \in \set{u,v}$. Together with $P_i \nsim P_{i+3}$, this implies the claim. Note that $\split(P_i, P_{i+3})=\set{u,v}$ if and only if both of $P_i$ and $P_{i+3}$ cross both endpoints $u,v$ of $core(K)$.
(iv) As $\set{i+1,i+2}$ is an $\enpt$ edge, $Q \defined P_{i+1} \cup P_{i+2}$ is a path. Moreover, $e_{i+1} \in P_{i+1}$ and $e_{i+2} \in P_{i+2}$, therefore $\set{e_{i+1},e_{i+2}} \subseteq Q$, implying the claim.
(v) It suffices to show that $core(K)$ separates $P_i$ and $P_{i+3}$. Suppose that after the removal of $core(K)$ the two paths still intersect. This is possible only if $e_{i+3} \in P_i$ or $e_i \in P_{i+3}$. Assume without loss of generality that $e_{i+3} \in P_i$. Then $P_i \nsim P_{i+1}$, a contradiction.
\end{proof}
\begin{figure}[htbp]
\centering
\include{figure/K4P4Representation}
\caption{Representations of $\kp$ pairs where $\split(P_i,P_i+3) = \set{u,v}$ and $\split(P_i,P_i+3) \subsetneq \set{u,v}$, respectively}\label{fig:K4P4Representation}
\end{figure}
Pairs $(G,C_5)$ with induced $\kp$ pairs are different than bigger cycles in a few respects. Therefore we analyse this case separately. We recall that a pair $(G,C_5)$ satisfies $(P3)$ vacuously, and that in Section \ref{subsec:c5c6} we found the unique minimal representation of a pair $(G,C_5)$ that satisfies $(P2)$. We now investigate the representation of a pair $(G,C_5)$ that does not satisfy $(P2)$.
\begin{theorem}\label{thm:C5WithK4P4}
If $(G,C_5)$ does not satisfy $(P2)$ then (i) $G$ is isomorphic to the graph depicted in Figure \ref{fig:C5WithK4P4}, and (ii) $(G,C_5)$ has a unique minimal representation also depicted in Figure \ref{fig:C5WithK4P4}.
\end{theorem}
\begin{proof}
Assume without loss of generality $K=(0,1,2,3)$ is a $\kp$ of $(G,C_5)$, and let $core(K) = P_T{u,v}$. If $\split(P_0,P_3)=\set{u,v}$ then $P_4 \subseteq core(K)$, implying that $P_4 \sim P_1$ or $P_4 \sim P_2$, i.e. at least one of $\set{1,4}$ or $\set{2,4}$ is an $\enpt$ edge, a contradiction.
Assume without loss of generality $\split(P_0,P_3)=\set{u}$, and that $P_3$ crosses both $u$ and $v$. Then $P_0$ has one endpoint $u'$ in $core(K)$, and $P_0 \cap P_3 = p_T(u,u')$.
As $P_4 \sim P_0$ and $P_4 \sim P_3$, we have $P_4 \sim p_T(u,u')$ and $P_4$ does cross $u$. Therefore $P_4$ intersects $core(K)$. By Lemma \ref{lem:K4P4Representation} (iv) $core(K) \subseteq P_1 \cup P_2$. We conclude that $P_4 \cap (P_1 \cup P_2) \neq \emptyset$, i.e. $P_4 \cap P_1 \neq \emptyset$ or $P_4 \cap P_2 \neq \emptyset$. As $\set{4,1}$ and $\set{4,2}$ are not $\enpt$ edges, we have that $P_4 \nsim P_1$ or $P_4 \nsim P_2$. On the other hand $P_4$ does not cross $u$ and by Lemma \ref{lem:K4P4Representation} (i), $P_2$ does not cross $v$, thus $\split(P_2,P_4)=\emptyset$. Therefore $P_4 \nsim P_1$ and $P_4 \parallel P_2$. Moreover, $\split(P_4,P_1)=\set{v}$, i.e. $P_4$ crosses $v$. Therefore one endpoint $u''$ of $P_4$ is in $p_T(u,u')$, and must be between $u'$ and the endpoint of $P_1$ in $core(K)$.
It is easy to see that the path $\bigcap \pp_K$ can be contracted to one edge $e$ without affecting the relationships between the paths. Similarly, any edge between $u$ and $e$, and any edge between $e$ and $u''$ can be contracted. The path $p_T(u',u'')$ can be contracted to one edge, and the path $p_T(u',v)$ can be contracted to a single vertex $v$. This leads to the representation in Figure \ref{fig:C5WithK4P4}.
\end{proof}
\begin{figure}[htbp]
\centering
\include{figure/C5withK4P4}
\caption{The unique $(G,C_5)$ pair that does not satisfy $(P2)$ and its unique minimal representation}\label{fig:C5WithK4P4}
\end{figure}
We now observe a property of the representations of $(G,C_5)$ in order to demonstrate the first family of non-$\enpt$ graphs.
\begin{lemma}
$G+C_5$ is not an $\enpt$ graph whenever $G$ is not a complete graph.
\end{lemma}
\begin{proof}
A pair $(G',C_5)$ satisfies $(P3)$ vacuously. If $(G',C_5)$ satisfies $(P2)$ then by Lemma \ref{lem:unique-c5}, its unique minimal representation is the one depicted in Figure \ref{fig:eptn-c5}. Otherwise, by Theorem \ref{thm:C5WithK4P4}, its unique minimal representation is the one depicted in Figure \ref{fig:C5WithK4P4}.
Let $i \in V(G)$. $i$ is adjacent to every vertex of $C_5$. We observe that in both cases above a) $P_i$ is a sub-path of $p_T(u,v)$, and b) there is a specific edge $e$ of $p_T(u,v)$ that is also in $P_i$. Therefore, for any two vertices $i,j \in V(G)$ $P_i$ and $P_j$ are intersecting sub-paths of $p_T(u,v)$, thus $P_i \sim P_j$. We conclude that $G$ is a complete graph.
\end{proof}
We now extend Lemma \ref{lem:K4P4Representation}. As opposed to Lemma \ref{lem:K4P4Representation} that investigates the structure of a $\kp$ regardless of any specific context, the next lemma provides us with further properties of minimal representations satisfying $(P3)$ of pairs $(G,C)$.
\begin{lemma}\label{lem:K4P4RepresentationsInBigCycles}
Let $K=(i,i+1,i+2,i+3)$ be a $\kp$ of a pair $(G,C)$ satisfying $(P3)$ with at least $6$ vertices. Let $\rep$ be a minimal representation of $(G,C)$ and let $\pp_K=\set{P_i: i\in K}$.\\
(i) $\bigcap \pp_K=\set{e}$ for some edge $e$ which is used exclusively by the paths of $\pp_K$, i.e. $e \in P_j \Rightarrow j \in K$.\\
(ii) $e$ divides $T$ into two subtrees $T_1, T_2$ such that $T_1$ is a cherry of $<T,\pp_K>$ with center $w_1$. We denote this subtree as $cherry(K)$.\\
(iii) $\split(P_i, P_{i+3})= \set{w_2} \subseteq V(T_2)$.\\
(iv) $N_G(j)=K$ if and only if $\split(P_j,P_i) \cup \split(P_j,P_{i+3})=\set{w_1}$. The unique vertex $j$ satisfying this condition is one of $i+1,i+2$.
\end{lemma}
\begin{proof}
Consult Figure \ref{fig:MinimalK4P4InACycle} for this proof.
(i) Let without loss of generality $i=0$. By Lemma \ref{lem:K4P4Representation}, $\bigcap \pp_K$ is not empty. By contradiction assume that a path $P_j \notin \pp_K$ intersects $\bigcap \pp_K$. Then $K \cup \set{j}$ is an edge-clique of $G$. We claim that this $K_5$ contains at least one red triangle, contradicting $(P3)$. Indeed, as $C$ has at least $6$ vertices, $j$ is adjacent in $C$ to at most one vertex $k \in \set{0,3}$. $K \setminus \set{k}$ contains one red edge. The endpoints of this edge together with $j$ constitute a red edge-clique. Therefore, no path of $\pp \setminus \pp_K$ intersects $\bigcap \pp_K$. Then no intermediate vertex of $\bigcap \pp_K$ is a split vertex. By the minimality of $\rep$, $\bigcap \pp_K$ consists of one edge, say $e$.
(ii) Let $T_1, T_2$ be the subtrees obtained by the removal of $e$ from $T$. As $V(G) \setminus K$ is a connected component of $G$, the union of the paths $\pp \setminus \pp_K$ is a subtree $T'$ of $T$. $T'$ is a subtree of $T_1$ or a subtree of $T_2$, because otherwise there is at least one path of $\pp \setminus \pp_K$ using $e$, contradicting (i). Without loss of generality let $T_2$ be the subtree containing $T'$, and $T_1$ be the subtree that intersects only paths of $\pp_K$. By Lemma \ref{lem:K4P4Representation} (ii), $T_1$ contains exactly one endpoint of $core(K)$. For $i \in \set{1,2}$, let $w_i$ be the endpoint of $core(K)$ that is in $T_i$. $w_1$ is the only split vertex in $T_1$ because it contains only paths of $\pp_K$. As the representation is minimal, there are no edges between $e$ and $w_1$, as otherwise they could be contracted. Any subtree of $T_1$ starting with an edge incident to $w_1$ can be contracted to one path because the subtree does not contain split vertices. Moreover this path can be contracted to one edge, because all the paths entering the subtree intersect in its first edge. There are only two such subtrees, therefore $T_1$ is isomorphic to $P_3$ and $w_1$ is its center.
(iii) Assume that $\abs{\split(P_0, P_3)}=2$. Then by Lemma \ref{lem:K4P4Representation}, $\split(P_0, P_3)=\set{w_1,w_2}$, i.e. $w_1$ is an internal vertex of both $P_0$ and $P_3$. In this case, one can remove from $P_0$ its unique edge in $T_1$ without affecting the relationships between the paths. This contradicts the minimality of $\rep$. Indeed, a) any change in $T_1$ affects relationships between paths of $\pp_K$ only, b) $\bigcap \pp_K$ is not affected, therefore all the paths of $\pp_K$ still intersect, c) $\set{w_1} = \split(P_0, P_3) = \split(P_0,P_2)$ and $\set{w_2} = \split(P_1, P_3)$ hold after the tail removal.
Now assume that $\split(P_0, P_3)=\set{w_1}$. $P_0$ crosses $w_2$ because $\split(P_0,P_2)=\set{w_2}$. Then $P_3$ does not cross $w_2$. As $P_4 \sim P_3$, $P_4,P_2,P_0$ intersect in the last edge of $P_3$, and thus constitute a red edge-clique, contradicting $(P3)$. We conclude that $\split(P_0, P_3)=\set{w_2}$.
(iv) First assume $j \notin \set{i+1,i+2}$. Clearly, $N_G(j) \neq K$. Moreover, we have $\split(P_j,P_i) \cup \split(P_j,P_{i+3}) \neq \set{w_1}$. Indeed, if $j \notin K$ then $w_1$ is not a vertex of $P_j$ and if $j \in \set{i,i+3}$ the condition holds because (iii).
We now assume $j \in \set{i+1,i+2}$. By Lemma \ref{lem:K4P4Representation} (v), the removal of $P_1 \cup P_2$ disconnects $P_0$ from $P_3$. Then the tree $T'$ intersects $P_1 \cup P_2$. Therefore, at least one of $P_1, P_2$ intersects $T'$. By Lemma \ref{lem:K4P4Representation} i) one of $P_1, P_2$ does not cross $w_2$, i.e. does not intersect $T_2$ which in turn includes $T'$, a contradiction. We conclude that exactly one of $P_1, P_2$ intersects $T'$. In other words exactly one of $1,2$ is adjacent to $V(G) \setminus K$. Assume $N_G(i+1)=K$. Then $P_{i+1} \cap T_1 \neq \emptyset$, therefore $\split(P_{i+1},P_i) = \emptyset$, i.e. $\split(P_{i+1},P_{i+3})=\set{w_1}$, concluding the claim. The case $N_G(i+2)=K$ is symmetric.
\end{proof}
\begin{figure}[htbp]
\centering
\include{figure/MinimalK4P4InACycle}
\caption{A minimal representation of a pair $(G,C)$ with an induced $\kp$ with $N_G(i+1)=K$.}\label{fig:MinimalK4P4InACycle}
\end{figure}
We term, as \emph{isolated}, the vertex $j \in \set{i+1,i+2}$ of $K=(i,i+1,i+2,i+3)$ satisfying $N_G(j)=K$ whose existence and uniqueness are guaranteed by Lemma \ref{lem:K4P4RepresentationsInBigCycles} (iv). We recall that $(i,i+1,i+2,i+3)=(i+3,i+2,i+1,i)$, and in view of this result, we introduce an alternative notation: We denote $K$ as $[i,i+1,i+2,i+3]$ if $i+1$ is its isolated vertex, and as $[i+3,i+2,i+1,i]$ otherwise.
\begin{lemma}\label{lem:OtherPathsEnteringTheCore}
Let $K=[i,i+1,i+2,i+3]$ a $\kp$ of a pair $(G,C)$ with at least $6$ vertices, $\rep$ a minimal representation of $(G,C)$ satisfying $(P3)$. If there is a path $P_j \notin \pp_K$ intersecting $core(K)$, then $j=i-1$ and $\abs{core(K)}=2$, otherwise $\abs{core(K)}=1$.
\end{lemma}
\begin{proof}
Let $\bigcap \pp_K = \set{e}$, and assume that $j \notin K$ and $P_j \cap core(K) \neq \emptyset$. Recall that $e \notin P_j$. If $P_j$ splits from $core(K)$ then it splits from each one of $P_i, P_{i+2}, P_{i+3}$. In particular $\set{j,i,i+2}$ constitutes a red edge-clique, thus violating $(P3)$. If $P_j \subseteq core(K)$ then $P_j \sim P_{i+2}$ implying $j \in \set{i+1,i+3} \subset K$, contradicting our assumption. Therefore $P_j$ crosses the endpoint $w_2$ of $core(K)$. Then $P_j$ intersects with each one of $P_i, P_{i+2}, P_{i+3}$ in the last edge of $core(K)$. Therefore a) $P_j \nsim P_{i+2}$ because $j \notin \set{i+1,i+3}$, and b) $P_i \nsim P_{i+2}$. If $P_j \nsim P_i$ then $\set{j,i,i+2}$ constitutes a red edge-clique, violating $(P3)$. Therefore $P_j \sim P_i$, implying $j=i-1$. Note that $P_{i+1} \cap P_{i-1} = \emptyset$ because $i+1$ is isolated. $P_{i-1} \cap core(K)$ consists of a single edge $e'(\neq e)$, because otherwise they can be contracted to a single edge without affecting the relationships between the paths $P_{i-1},P_i,P_{i+2},P_{i+3}$ that are the only paths that intersect the contracted edges. Then $core(K)$ consists of the two edges $e,e'$. If $P_{i-1}$ does not intersect $core(K)$ then $\pp_K$ are the only paths that intersect $core(K)$. Therefore, all the edges of $core(K)$ can be contracted to one edge.
\end{proof}
\begin{lemma}\label{lem:NoAggressiveContractibleC6}
A pair $(G,C)$ with $6$ vertices satisfying $(P3)$ does not contain an induced $\kp$.
\end{lemma}
\begin{proof}
Assume without loss of generality that $[0,1,2,3]$ is a $\kp$ of $(G,C)$. Let $\rep$ be a representation of $(G,C)$ satisfying $(P3)$. For $i \in \set{0,3}$ let $T_i$ be the unique connected component of $T \setminus core(K)$ intersecting $P_i$. By Lemma \ref{lem:OtherPathsEnteringTheCore}, $P_4$ does not cross $w_2$. Therefore $P_4$ is completely in $T_3$. As $P_4 \cap P_5 \neq \emptyset$, $P_5$ intersects $T_3$. If $P_5$ is completely in $T_3$ then $P_5 \parallel P_0$, otherwise $P_5 \nsim P_0$. Both cases contradict the fact that $\set{5,0}$ is an edge of $C$.
\end{proof}
\subsection{Intersection of $\kp$ pairs and Aggressive Contraction }\label{subsec:K4P4IntersectionsAgressiveContraction}
We now focus on pairs with at least $7$ vertices. We start by analyzing the intersection of their $\kp$ sub-pairs.
\begin{lemma}\label{lem:twok4p4s}
Let $(G,C)$ be a pair with at least $7$ vertices satisfying $(P3)$, and $K=[i,i+1,i+2,i+3]$ a $\kp$ of $(G,C)$.\\
(i) There is at most one $\kp$, $K' \neq K$ such that $E(C[K]) \cap E(C[K']) \neq \emptyset$ and if such a $\kp$ exists then $K' = [i+5,i+4,i+3,i+2]$ (and therefore $\set{i+2, i+4}$ is an edge of $G$).\\
(ii) If $\set{i+2 , i+4}$ is an edge of $G$ then $K'=[i+5,i+4,i+3,i+2]$ induces a $\kp$ of $(G,C)$.
\end{lemma}
\begin{proof}
Let without loss of generality $i=0$.
(i) Since $1$ is isolated, $1 \notin K'$. Therefore if $E(C[K]) \cap E(C[K']) \neq \emptyset$ for some $\kp$ $K'$ then $E(C[K]) \cap E(C[K']) = \set{\set{2,i}}$, i.e. $K' = (2,3,4,5)$. As $3$ is adjacent to $1$, $3$ is not isolated in $K'$. Therefore, $K'=[5,4,3,2]$.
(ii) Assume $\set{2,4}$ is an edge of $G$ and that, by way of contradiction, $K'=\set{2,3,4,5}$ is not a $\kp$. Consult Figure \ref{fig:K4P4PairCharacterization} for the following discussion. For $j \in \set{0,3}$ let $T_j$ be the connected component of $T \setminus core(K)$ intersecting $P_j$. As $P_4 \sim P_3$, Lemma \ref{lem:OtherPathsEnteringTheCore} implies that $P_4$ is completely in $T_3$. $P_4 \nsim P_2$, by our assumption. Let $w_3$ be the endpoint of $P_3$ in $T_3$ and $w_4$ be the split vertex of $P_2$ and $P_4$. Then $w_3 \in p_T(w_2,w_4)$ (possibly $w_3=w_4$). $P_5$ does not intersect at least one of $P_2$ and $P_3$, because otherwise $K'$ is a $\kp$. Then it does not intersect $P_3$. The union of the paths $P_6, \ldots P_{n-1}$ constitutes a subtree $T'$ of $T$ that intersects both $P_0$ and $P_5$. Therefore there is at least one path $P_j \in \set{P_6, \ldots P_{n-1}}$ crossing the last edge of $P_3$ (incident to $w_3$). Then $\set{2,4,j}$ is an edge-clique defined by this edge. Moreover, a) $P_2 \nsim P_4$, b) $P_j \nsim P_2$ because $j \notin \set{1,3}$, $P_j \nsim P_4$ because $j \notin \set{3,5}$. Therefore $\set{2,4,j}$ is a red edge-clique, contradicting the assumption that $(P3)$ is satisfied.
\end{proof}
\begin{figure}[htbp]
\centering
\include{figure/K4P4PairCharacterization}
\caption{Proof of Lemma \ref{lem:twok4p4s}}\label{fig:K4P4PairCharacterization}
\end{figure}
By the above lemma $\kp$ sub-pairs may intersect only in pairs. We term two intersecting $\kp$ pairs as \emph{twins}, and a $\kp$ not intersecting with another as a \emph{single} $\kp$.
Given a $\kp$ $K=[i,i+1,i+2,i+3]$ of a pair $(G'',C'')$ satisfying $(P3)$, the \emph{aggressive contraction} operation is the replacement of the vertices $i+2,i+3$ by a single vertex $(i+2).(i+3)$. We denote the resulting pair $(\contract{G''}{e},\contract{C''}{e})$ (where $e=\set{i+2,i+3}$) as $\aggressivegcdpk$. The following lemma characterizes the aggressive contraction operation in the representation domain.
\begin{lemma}\label{lem:aggressiveContraction}
Let $(G'',C'')$ be a pair with at least $7$ vertices, $\repdprime$ be a representation of it satisfying $(P3)$, and
$K=[i,i+1,i+2,i+3]$ be a $\kp$ of $(G'',C'')$. Then:\\
$\aggressivegcdpk$ is a pair satisfying $(P3)$ and a representation $\repprime$ of $(G',C') = \aggressivegcdpk$ satisfying $(P3)$ is obtained from $\repdprime$ by first removing $cherry(K)$ and also $cherry(K')$ if $K$ and $K'$ are twins, and then applying the union operation to $P_{i+2}$ and $P_{i+3}$.
\end{lemma}
\begin{proof}
Let without loss of generality $i=0$. Recall that by Lemma \ref{lem:twok4p4s}, $\set{2,4}$ is an edge of $G''$, if and only if $K$ is a twin. Figure \ref{fig:ACofOneK4P4} illustrates the following two steps in the case that $K$ is a single.
(Step $1$) We remove $cherry(K)$ (and also $cherry(K')$ when $K$ and $K'$ are twins) from $T''$. By Lemma \ref{lem:K4P4RepresentationsInBigCycles} we know that by removing cherries we don't lose any edge intersection, and we loose exactly one split vertex per cherry, namely the center of the cherry. This vertex (or vertices) is $\split(P_1, P_3)$ (and also $\split(P_2, P_4)$ when $K$ is a twin). Thus the edge $\set{1,3}$ (and also $\set{2,4}$ when $K$ is a twin) becomes blue. As no new red edges are introduced, the resulting representation does not contain red edge-cliques, i.e. satisfies $(P3)$.
(Step $2$) We contract the resulting graph on the edge $\set{2,3}$. We claim that this contraction is defined. Indeed assume by contradiction that $\set{2,3}$ participates in a $BBR$ triangle. This $BBR$ triangle is one of $\set{1,2,3}$ and $\set{2,3,4}$. Then one of $\set{1,3}$ and $\set{2,4}$ is a red edge, contradicting the fact that these edges (if exist) becomes blue after step $1$. This contraction corresponds to the union operation on the paths $P_2,P_3$, and by Lemma \ref{lem:ContractionPreservesP2P3} the resulting graph satisfies $(P3)$.
\end{proof}
\subsection{Algorithm}\label{subsec:K4P4Algorithm}
\newcommand{\textsc{FindMinimalRepresentation-P3}}{\textsc{FindMinimalRepresentation-P3}}
\newcommand{\textsc{MakeCherry}}{\textsc{MakeCherry}}
\begin{figure}[htbp]
\centering
\include{figure/aggresiveContraction-oneK4P4}
\caption{Aggressive contraction of a single $\kp$}\label{fig:ACofOneK4P4}
\end{figure}
\begin{figure}[htbp]
\centering
\include{figure/aggresiveContraction-twoK4P4}
\caption{Aggressive contraction of twins}\label{fig:ACOfTwoK4P4s}
\end{figure}
Lemma \ref{lem:aggressiveContraction} implies an algorithm for finding the unique minimal representation of
pairs satisfying $(P3)$. Algorithm $\textsc{FindMinimalRepresentation-P3}$ is a recursive algorithm that processes a single $\kp$ or a twin of $\kp$s at every invocation. The processing is done by applying aggressive contraction to convert the involved $\kp$(s) to $(K_3,P_3)$ (s), solving the problem recursively, and finally transforming the representation of the $(K_3, P_3)$ to a representation of a $\kp$. In the Build Representation phase, Algorithm $\textsc{FindMinimalRepresentation-P3}$ performs the reversal of steps 1 and 2 described in Lemma \ref{lem:aggressiveContraction}, (see Figures \ref{fig:ACofOneK4P4}, \ref{fig:ACOfTwoK4P4s}).
\alglanguage{pseudocode}
\begin{algorithm}
\caption{$\textsc{FindMinimalRepresentation-P3}(G'',C'')$}
\label{alg:algtwo}
\begin{algorithmic}[1]
\Require {$C''=\set{0,1,\ldots,\abs{V(G'')}-1}$ is an Hamiltonian cycle of $G''$ and $\abs{V(G'')} \geq 6$}
\Ensure {A minimal representation $\repbar$ of $(G'',C'')$ satisfying $(P3)$ if any}
\If {$(G'',C'')$ is $\kp$-free}
\State \Return $\textsc{FindMinimalRepresentation-P2-P3}(G'',C'',\wdtgdprime)$
\EndIf
\State
\Statex \textbf{Aggressive Contraction:}
\State Pick a $\kp$, $K=[i,i+1,i+2,i+3]$ of $(G'',C'')$.
\Comment Renumber vertices if necessary.
\State $(G',C') \gets \aggressivegcdpk$.
\State
\Statex \textbf{Recurse:}
\State $\repbarprime \gets \textsc{FindMinimalRepresentation-P3}(G',C')$.
\State
\Statex \textbf{Build Representation:}
\State $\repbar \gets \repbarprime$.
\State Replace $P_{(i+2).(i+3)}$ by two copies $P_{i+2}$ and $P_{i+3}$ of itself.
\If {$i+2$ is adjacent to $i+4$ in $G''$}
\State
\Comment $K'=[i+5,i+4,i+3,i+2]$ is the twin of $K$ in $(G'',C'')$
\State \textsc{MakeCherry}($\repbar, i+4, i+2$).
\Else
\Comment $K$ is a single
\State $w \gets$ the endpoint of $P_{i+2}$ which is not in $core(K)$.
\State \textsc{AdjustEndpoint} $(\repbar, G'', P_{i+2}, w)$.
\EndIf
\State \textsc{MakeCherry}($\repbar, i+1, i+3$).
\State
\Statex \textbf{Validate:}
\If {$\eptg{\ppbar}=G''$ and $\ppbar$ satisfies $(P3)$}
\State \Return $\repbar$
\Else
\State \Return ``NO''
\EndIf
\Function{\textsc{MakeCherry}}{$\repbar,p,q$}
\State Let $v \in V(\bar{T})$ be the common endpoint of $P_p, P_q$.
\State Add two new vertices $v', v''$ and two edges $\set{v,v'}, \set{v,v''}$ to $\bar{T}$.
\State Extend $P_p$ so that the endpoint $v$ is moved to $v'$.
\State Extend $P_q$ so that the endpoint $v$ is moved to $v''$.
\EndFunction
\end{algorithmic}
\end{algorithm}
A \emph{broken tour with cherries} is a representation obtained by adding cherries to a broken tour.
\begin{theorem}
$\prbpthree$ can be solved in polynomial time. YES instances have a unique solution, and whenever $n \geq 6$ this solution is a broken planar tour with cherries.
\end{theorem}
\begin{proof}
As the case $\abs{V(G'')} < 6$ is already solved, we will show that for any given pair $(G'',C'')$ with $\abs{V(G'')} \geq 6$, $\textsc{FindMinimalRepresentation-P3}$ solves $\prbpthree$. If $(G'',C'')$ is a ''NO'' instance, then the instance has no representation satisfying $(P3)$. In this case then the algorithm returns ``NO'' at the validation phase. Therefore we assume that $(G'',C'')$ is a ''YES'' instance, and prove the claim by induction on the number $k$ of induced $\kp$ pairs of $(G'',C'')$.
If $k=0$ then $(G'',C'')$ does not contain any $\kp$ pairs, therefore satisfies $(P2)$. In this case the algorithm invokes $\textsc{FindMinimalRepresentation-P2-P3}$ and the claim follows from Theorem \ref{thm:FindMinimalRepresentation}.
Otherwise $k>0$. We assume that the claim holds for any $k'<k$ and prove that it holds for $k$. In this case, as the pair contains at least one $\kp$, one such pair $K$ is chosen arbitrarily by the algorithm and aggressively contracted. The resulting pair $(G',C')=\aggressivegcdpk$ has the following properties:
\begin{itemize}
\item{Satisfies $(P3)$.} (By Lemma \ref{lem:aggressiveContraction})
\item{The number of $\kp$ pairs is less then $k$.}
\item{$\abs{V(G')} \geq 6$.} This is because $\abs{V(G')}=\abs{V(G'')}-1$ and $\abs{V(G'')}>6$. Indeed, if $\abs{V(G'')}=6$, we have $k=0$ by Lemma \ref{lem:NoAggressiveContractibleC6}.
\end{itemize}
Therefore, $(G',C')$ satisfies the assumptions of the inductive hypothesis. Then, $\repbarprime$ is the unique minimal representation of $\aggressivegcdpk$ satisfying $(P3)$. It remains to show that the representation $\repbarprime$ is obtained from the representation $\repbar$ returned by the algorithm, by applying the steps described in Lemma \ref{lem:aggressiveContraction}.
Let without loss of generality $K=[i,i+1,i+2,i+3]$. By Lemma \ref{lem:twok4p4s}, $K$ has a twin $K'=[i+5,i+4,i+3,i+3]$ if and only if $\set{i+2,i+4}$ is an edge of $G''$. The algorithm checks the existence of this edge and takes two different actions, accordingly.
If $K$ is not a twin then step 2, i.e. the union operation is reversed by breaking apart the path $P_{(i+2).(i+3)}$ into two paths $P_{i+2}$ and $P_{i+3}$. Then step 1 is reversed by invoking procedure $\textsc{MakeCherry}$ (see Figure \ref{fig:ACofOneK4P4}).
If $K$ is a twin, then $cherry(K)$ and $cherry(K')$ are uniquely determined by Lemma \ref{lem:K4P4RepresentationsInBigCycles} (ii) and procedure $\textsc{MakeCherry}$ acts accordingly. This determines all the endpoints of $P_i,P_{i+1},P_{i+2},P_{i+3},P_{i+4},P_{i+5}$ that are different from the representation $\repbarprime$ (see Figure \ref{fig:ACOfTwoK4P4s}).
\end{proof}
\section{Introduction}\label{sec:intro}
\input{introduction}
\section{Preliminaries}\label{sec:prelim}
\input{preliminaries}
\section{$\enpt$ Holes Satifying $(P2)$ and $(P3)$}\label{sec:uncontraction}
\input{EPTNBasics}
\input{uncontraction}
\section{$\enpt$ Holes Satifying $(P3)$}\label{sec:K4P4}
\input{k4p4}
\section{$\enpt$ Holes: the General Case}\label{sec:General}
\input{general}
\section{Conclusions and Future Work}\label{sec:conclusion}
\input{summary}
\small
\bibliographystyle{abbrvwithurl}
\subsection{Definitions}\label{subsec:prelim-definitions}
\runningtitle{Notation:}
Given a graph $G$ and a vertex $v$ of $G$, we denote by $\delta_G(v)$ the set of edges of $G$ incident to $v$, by $N_G(v)$ the set consisting of $v$ and its neighbors in $G$, and by $d_G(v)=\abs{\delta_G(v)}$ the degree of $v$ in $G$. A vertex is called a \emph{leaf} (resp. \emph{intermediate vertex}, \emph{junction}) if $d_G(v)=1$ (resp. $=2$, $\geq 3$). Whenever there is no ambiguity we omit the subscript $G$ and write $d(v)$.
Given a graph $G$, $\bar{V} \subseteq V(G)$ and $\bar{E} \subseteq E(G)$ we denote by $G[\bar{V}]$ and $G[\bar{E}]$ the subgraphs of $G$ induced by $\bar{V}$ and by $\bar{E}$, respectively.
The \emph{union} of two graphs $G, G'$ is the graph $G \cup G' \defined (V(G) \cup V(G'), E(G) \cup E(G'))$. The \emph{join} $G+G'$ of two disjoint graphs $G,G'$ is the graph $G \cup G'$ together with all the edges joining $V(G)$ and $V(G')$, i.e. $G + G' \defined (V(G) \cup V(G'), E(G) \cup E(G') \cup (V(G) \times V(G')))$.
Given a (simple) graph $G$ and $e \in E(G)$, we denote by $\contractge$ the (simple) graph obtained by contracting the edge $e$ of $G$, i.e. by coinciding the two endpoints of $e=\set{p,q}$ to a single vertex $p.q$, and then removing self loops and parallel edges. Let $\bar{E}=\set{e_1, e_2, \ldots e_k} \subseteq E(G)$. We denote by $\contract{G}{e_1,\ldots,e_k}$ the graph obtained from $\contract{G}{e_1,\ldots,e_{k-1}}$ by contracting the (image of the) edge $e_k$. The effect of such a sequence of contractions is equivalent to contracting every connected component of $G[\set{e_1,\ldots,e_k}]$ to a vertex. Therefore the order of contractions is not important, i.e. for any permutation $\pi$ of $\set{1,\ldots,k}$ we have $\contract{G}{e_1,\ldots,e_{k-1}}=\contract{G}{e_{\pi(1)},\ldots,e_{\pi(k-1)}}$. Based on this fact, we denote by $\contract{G}{\bar{E}}$ the graph obtained by contracting the edges of $\bar{E}$ (in any order).
For two vertices $u,v$ of a tree $T$ we denote by $p_T(u,v)$ the unique path between $u$ and $v$ in $T$. A vertex $w$ of a path $P$ that is not an endpoint of $P$ is termed an \emph{internal vertex} of $P$. We also say that $P$ \emph{crosses} $w$. A \emph{cherry} of a tree $T$ is a connected subgraph of $T$ consisting of two leaves of $T$ connected to an internal vertex of $T$.
\runningtitle{Intersections and union of paths:}
Given two paths $P,P'$ in a graph, we write $P \parallel P'$ to denote that $P$ and $P'$ are non-intersecting, i.e. \emph{edge-disjoint}. The \emph{split vertices} of $P$ and $P'$ is the set of junctions in their union $P \cup P'$ and is denoted by $\split(P,P')$. Whenever $P$ and $P'$ intersect and $\split(P, P') = \emptyset$ we say that $P$ and $P'$ are \emph{non-splitting} and denote this by $P \sim P'$. In this case $P \cup P'$ is a path or a cycle. When $P$ and $P'$ intersect and $\split(P, P') \neq \emptyset$ we say that they are \emph{splitting} and denote this by $P \nsim P'$. Clearly, for any two paths $P$ and $P'$ exactly one of the following holds: $P \parallel P'$, $P \sim P'$, $P \nsim P'$.
When the graph $G$ is a tree, the union $P \cup P'$ of two intersecting paths $P,P'$ on $G$ is a tree with at most two junctions, i.e. $\abs{\split(P,P')} \leq 2$ and $P \cup P'$ is a path whenever $P \sim P'$.
\runningtitle{The VPT, EPT and ENPT graphs:}
Let $\pp$ be a set of paths in a tree $T$. The graphs $\vptgp, \eptgp$ and $\enptgp$ are graphs such that $V(\enptgp) = V(\eptgp) = V(\vptgp)=\set{p | P_p \in \pp}$. Given two distinct paths $P_p,P_q \in \pp$, $\set{p,q}$ is an edge of $\enptgp$ if $P_p \sim P_q$, and $\set{p,q}$ is an edge of $\eptgp$ (resp. $\vptgp$) if $P_p$ and $P_q$ have a common edge (resp. vertex) in $T$. See Figure \ref{fig:vpt-ept-eptn} for an example. From these definitions it follows that
\begin{observation}
$E(\enptgp) \subseteq E(\eptgp)\subseteq E(\vptgp)$.
\end{observation}
\begin{figure}[htbp]
\centering
\include{figure/vpt-ept-eptn}
\caption{A host tree $T$, a collection of paths $\pp = \left\{ P_1, P_2, P_3, P_4, P_5 \right\} $ defined on $T$ and the corresponding graphs $\vptgp, \eptgp$ and $\enptgp$.}
\label{fig:vpt-ept-eptn}
\end{figure}
Two graphs $G$ and $G'$ such that $V(G)=V(G')$ and $E(G') \subseteq E(G)$ are termed a \emph{pair (of graphs)} denoted as $(G,G')$. If $\eptgp=G$ (resp. $\enptgp=G$) we say that $\rep$ is an $\ept$ (resp. $\enpt$) representation for $G$. If $\eptgp=G$ and $\enptgp=G'$ we say that $\rep$ is a representation for the pair $(G,G')$. Given a pair $(G,G')$ the sub-pair induced by $\bar{V} \subseteq V(G)$ is the pair $(G[\bar{V}],G'[\bar{V}])$. Clearly, any representation of a pair induces representations for its induced sub-pairs, i.e. the pairs have the hereditary property.
A \emph{cherry} of a representation $\rep$ is a cherry of $T$ with leaves $v,v'$ such that $v$ (resp. $v'$) is an endpoint of exactly one path $P$ (resp. $P'$) of $\pp$.
Throughout this work, in all figures, the edges of the tree $T$ of a representation $\rep$ are drawn as solid edges whereas the paths on the tree are shown by dashed, dotted, etc. edges. Similarly, edges of $\enptgp$ are drawn with solid or blue lines whereas edges in $E(\eptgp) \setminus E(\enptgp)$ are dashed or red. We sometimes refer to them as blue and red edges respectively. For an edge $e=\set{p,q}$ we use $\split(e)$ as a shorthand for $\split(P_p,P_q)$. We note that $e$ is a red edge if and only if $\split(e) \neq \emptyset$.
\runningtitle{Cycles, Chords, Holes, Outerplanar graphs, Weak dual trees:}
Given a graph $G$ and a cycle $C$ of it, a \emph{chord} of $C$ in $G$ is an edge of $E(G) \setminus E(C)$ connecting two vertices of $V(C)$. The \emph{length} of a chord connecting the vertices $i$,$j$ is the length of a shortest path between $i$ and $j$ on $C$. $C$ is a \emph{hole} (chordless cycle) of $G$ if $G$ does not contain any chord of $C$. This is equivalent to saying that the subgraph $G[V(C)]$ of $G$ induced by the vertices of $C$ is a cycle. For this reason a chordless cycle is also called an \emph{induced} cycle.
An \emph{outerplanar} graph is a planar graph that can be embedded in the plane such that all its vertices are on the unbounded face of the embedding. An outerplanar graph is Hamiltonian if and only if it is biconnected; in this case the unbounded face forms the unique Hamiltonian cycle. The \emph{weak dual} graph of a planar graph $G$ is the graph obtained from its dual graph, by removing the vertex corresponding to the unbounded face of $G$. The weak dual graph of an outerplanar graph is a forest, and in particular the weak dual graph of a Hamiltonian outerplanar graph is a tree~\cite{CH67}. When working with outerplanar graphs we use the term \emph{face} to mean a bounded face.
\subsection{$\ept$ Graphs} \label{subsec:prelim-eptn-knownresults}
We now present definitions and results from \cite{GJ85} that we use throughout this work.
A \emph{pie} of a representation $\rep$ of an $\ept$ graph is an induced star $K_{1,k}$ of $T$ with $k$ leaves $v_0, v_1, \ldots, v_{k-1} \in V(T)$, and $k$ paths $P_0, P_1, \ldots P_{k-1} \in \pp$, such that for every $0 \leq i \leq k-1$ both $v_i$ and $v_{(i+1)\mod k}$ are vertices of $P_i$. We term the central vertex of the star as the \emph{center} of the pie (See Figure \ref{fig:ept-cycle}). It is easy to see that the $\ept$ graph of a pie with $k$ leaves is the hole $C_k$ on $k$ vertices. Moreover, this is the only possible $\ept$ representation of $C_k$ when $k \geq 4$.
\begin{figure}
\centering
\include{figure/ept-cycle}
\caption{The representation of $\ept$ cycle; a pie}
\label{fig:ept-cycle}
\end{figure}
\begin{theorem}\cite{GJ85}\label{thm:golumbicpie}
If an $\ept$ graph contains a hole with $k \geq 4$ vertices, then every representation of it contains a pie with $k$ paths.
\end{theorem}
Let $\pp_e \defined \set{p \in \pp|~e \in p}$ be the set of paths in $\pp$ containing the edge $e$. A star $K_{1,3}$ is termed a \emph{claw}. For a claw $K$ of a tree $T$, $\pp[K] \defined \set{p \in \pp|~p \textrm{~uses two edges of~}K}$. It is easy to see that both $\eptg{\pp_e}$ and $\eptg{\pp[K]}$ are cliques. These cliques are termed \emph{edge-clique} and \emph{claw-clique}, respectively. Moreover, these are the only possible representations of cliques.
\begin{theorem}\cite{GJ85}\label{thm:golumbiccliques}
Any maximal clique of an $\ept$ graph with representation $\rep$ corresponds to a subcollection $\pp_e$ of paths for some edge $e$ of $T$, or to a subcollection $\pp[K]$ of paths for some claw $K$ of $T$.
\end{theorem}
Note that a claw-clique is a pie with $3$ leaves.
\subsection{$\enpt$ Graphs and $\ept,\enpt$ Graph Pairs} \label{subsec:eptn-OurPreviousResults}
In this section we present definitions and results from \cite{BESZ13-TR1} that we use throughout this work, introduce new terms, and prove basic results.
\runningtitle{Equivalent and minimal representations:}
We say that the representations $\repn{1}$ and $\repn{2}$ are \emph{equivalent}, and denote by $\repn{1} \approxeq \repn{2}$, if their corresponding $\ept$ and $\enpt$ graphs are isomorphic under the same isomorphism (in other words, if they constitute representations of the same pair of graphs $(G,G')$).
We write $\repn{1} \rightsquigarrow \repn{2}$ if $\repn{2}$ can be obtained from $\repn{1}$ by one of the following two \emph{minifying operations}:
\begin{itemize}
\item{} Contraction of an edge $e$ of $T_1$ (and of all the paths in $\pp_1$ using $e$). We denote this operation as $contract(e)$.
\item{} Removal of an initial edge (\emph{tail}) $e$ of a path $P$ of $\pp_1$. We denote this operation as $tr(P,e)$.
\end{itemize}
The partial order $\gtrsim$ is the reflexive-transitive closure of the relation $\rightsquigarrow$, and $\repn{1} \lesssim \repn{2}$ is equivalent to $\repn{2} \gtrsim \repn{1}$.
$\rep$ is a \emph{minimal} representation, if it is minimal in the partial order $\lesssim$ restricted to its equivalence class $[\rep]_\approxeq$, i.e. over all the representations representing the same pair as $\rep$. Throughout the work we aim at characterizing minimal representations. Whenever a sequence of minifying operations contains two operations $tr(P,e)$ and $contract(e)$ in this order, the first operation can be deleted from the sequence, to obtain a shorter and equivalent sequence. A sequence that can not be shortened in this way and contains the biggest number of $contract(e)$ operations is termed a \emph{minimal} sequence. Whenever $\repn{1} \gtrsim \repn{2}$ we consider only minimal sequences of minifying operations that transform $\repn{1}$ to $\repn{2}$. We observe two properties of such sequences.
\begin{lemma}\label{lem:MinOperationsExchangeable}
Let $\repn{1} \gtrsim \repn{2}$, and $s$ be a minimal sequence of minifying operations transforming $\repn{1}$ to $\repn{2}$. Then every permutation of $s$ transforms $\repn{1}$ to $\repn{2}$ too.
\end{lemma}
\begin{proof}
If $contract(e)$ is an operation of $s$ then there is no other operation in $s$ involving $e$. This is because such an operation is impossible after $contract(e)$, and if it appears before $contract(e)$ it contradicts the minimality of $s$. To conclude the result, we observe that any two successive operations in $s$ are interchangeable. Indeed, for two distinct edges $e,e'$ the operations $contract(e), contract(e')$ (resp. $contract(e),tr(P,e')$) are interchangeable, and for two not necessarily distinct edges $e,e'$ the operations $tr(P,e),tr(P',e')$ are interchangeable.
\end{proof}
\begin{lemma}\label{lem:MinOperationsMonotone}
If $\repn{1} \gtrsim \cdots \gtrsim \repn{n}$ and $\repn{1} \approxeq \repn{n}$ then $\repn{1} \approxeq \cdots \approxeq \repn{n}$.
\end{lemma}
\begin{proof}
Let $G_i=\eptg{\pp_i}$ and $G'_i=\enptg{\pp_i}$. We observe that both minifying operations are monotonic in the sense that they neither introduce neither new intersections, nor new splits. Namely, for $1 \leq i < n$, $E(G_{i+1}) \subseteq E(G_i)$ and $E(G_{i+1}) \setminus E(G'_{i+1}) \subseteq E(G_i) \setminus E(G'_i)$. As $\repn{1} \approxeq \repn{n}$ we have $(G_1,G'_1)=(G_n,G'_n)$, i.e. $E(G_n)=E(G_1)$ and $E(G_n) \setminus E(G'_n) = E(G_1) \setminus E(G'_1)$. Therefore $E(G_1) = \cdots = E(G_n)$ and $E(G_1) \setminus E(G'_1) = \cdots = E(G_n) \setminus E(G'_n)$, concluding $(G_1,G'_1) = \cdots =(G_n,G'_n)$.
\end{proof}
\runningtitle{$\ept$ holes:}
A pie of $\rep$ is an independent set of $\enptg{\pp}$. Therefore combining with Theorem \ref{thm:golumbicpie} we have the following lemma that we use in our proofs:
\begin{lemma}\label{lem:NoBlueHole}
If there is a representation for a pair $(G,G')$ then there is no blue edge (i.e. edge of $G'$) in a hole of length greater than 3 in $G$.
\end{lemma}
\begin{proof}
A hole of size a least $4$ is represented by a pie. For any two paths $P_p,P_q$ of this pie, we have either $P_p \nsim P_q$ or $P_p \parallel P_q$, therefore $\set{p,q}$ is not an $\enpt$ edge.
\end{proof}
Combining Lemma \ref{lem:NoBlueHole} with Theorem \ref{thm:golumbicpie}, we obtain the following characterization of pairs $(C_k, G')$. For a pair $(C_k$, $G'$), exactly one of the following holds:
\begin{itemize}
\item{$k>3$.} In this case $C_k$ is represented by a pie. Therefore $G'$ is an independent set. In other words, $C_k$ consists of red edges. We term such a hole, a red hole.
\item{$k=3$ and $C_k$ consists of red edges.} $G'$ is an independent set. We term such a hole a red triangle.
\item{$k=3$ and $C_k$ contains exactly one $\enpt$ (blue) edge ($G' = P_1 \cup P_2$).} We term such a hole a $BRR$ triangle, and its representation is an edge-clique.
\item{$k=3$ and $C_k$ contains two $\enpt$ (blue) edges ($G' = P_3$).} We term such a hole a $BBR$ triangle, and its representation is an edge-clique.
\item{$k=3$ and $C_k$ consists of blue edges ($G' = C_3$).} We term such a hole a blue triangle.
\end{itemize}
\runningtitle{$\ept$ contraction:}
Let $\rep$ be a representation and $P_p,P_q \in \pp$ such that $P_p \sim P_q$. We denote by $\contractppq$ the representation that is obtained from $\rep$ by replacing the two paths $P_p,P_q$ by the path $P_p \cup P_q$, i.e. $\contractppq \defined \left<T,\pp \setminus \set{P_p,P_q} \cup \set{P_p \cup P_q}\right>$. We term this operation a \emph{union}. Lemma \ref{lem:contraction-union} follows from the below observation.
\begin{observation}\label{obs:splitofunion}
For every $P_p,P_q,P_r \in \pp$ such that $P_p \sim P_q$, $\split(P_p \cup P_q, P_r) = \split(P_p,P_r) \cup \split(P_q,P_r)$.
\end{observation}
\begin{lemma}\label{lem:contraction-union}
\cite{BESZ13-TR1} Let $\rep$ be a representation for the pair $(G,G')$, and let $e=\set{p,q} \in E(G')$. Then
$\contractge$ is an $\ept$ graph. Moreover $\contractge=\eptg{\contractppq}$.
\end{lemma}
\runningtitle{Contraction of pairs:}
The definition of the contraction operation extends to pairs: The contraction of an $\enpt$ edge does not necessarily correspond to the union operation in the $\enpt$ representation. More concretely, let $P_p$,$P_q$ and $P_{q'}$ be such that $P_p \sim P_q$, $P_p \sim P_{q'}$ and $P_q \nsim P_{q'}$. Then $\contract{G'}{\set{p,q}}$ is not isomorphic to $\enptg{\contractppq}$ as $ \set{q',p.q} \notin E(\enptg{\contractppq})$. Let $(G,G')$ be a pair and $e \in E(G')$. If for every edge $e' \in E(G')$ incident to $e$, the edge $e''= e \triangle e'$ (forming a triangle together with $e$ and $e'$) is not an edge of $G$ then $\contract{(G,G')}{e} \defined (\contractge,\contract{G'}{e})$, otherwise $\contract{(G,G')}{e}$ is undefined. Whenever $\contract{(G,G')}{e}$ is defined we say that $(G,G')$ is \emph{contractible} on $e$, or when there is no ambiguity about the pair under consideration we say that $e$ is \emph{contractible}. We say that a pair is contractible if it is contractible on some $\enpt$ edge. Clearly, $(G,G')$ is not contractible if and only if every edge of $G'$ is contained in at least one $BBR$ triangle.
\runningtitle{Weak Dual Trees:}
The definition of weak dual tree is extended from Hamiltonian outerplanar graphs to any Hamiltonian graph as follows. Given a pair $(G,C)$ where $C$ is a Hamiltonian cycle of $G$, a weak dual tree of $(G,C)$ is the weak dual tree $\wdtg$ of an arbitrary Hamiltonian maximal outerplanar subgraph $\opg$ of $G$. $\opg$ can be built by starting from $C$ and adding to it arbitrarily chosen chords from $G$ as long as such chords exists and the resulting graph is planar.
By definition of a dual graph, vertices of $\wdtg$ correspond to faces of $\opg$. By maximality, the faces of $\opg$ correspond to holes of $G$. The degree of a vertex of $\wdtg$ is the number of red edges in the corresponding face of $\opg$. To emphasize the difference, for an outerplanar graph $G$ we will refer to \emph{the} weak dual tree of $G$, whereas for a (not necessarily outerplanar) graph $G$ we will refer to \emph{a} weak dual tree of $G$.
The following lemma describes the effect of contraction on weak dual trees.
\begin{lemma}\label{lem:contraction-wdt}
\cite{BESZ13-TR1} Let $(G,C)$ be a pair satisfying $(P2),(P3)$ and let $\wdtg$ be a weak dual tree of $(G,C)$. (i) There is a bijection between the contractible edges of $(G,C)$ and the intermediate vertices of $\wdtg$. (ii) The tree obtained from $\wdtg$ by smoothing out the intermediate vertex corresponding to a contractible edge $e$ is a possible weak dual tree of $\contractgce$.
\end{lemma}
\runningtitle{Representations of $\enpt$ holes:}
Our goal in this work is to characterize the representations of $\enpt$ holes. More precisely we characterize representations of pairs $(G,C_n)$ where $C_n$ is a Hamiltonian cycle of $G$. For this purpose we define the following problem.
\begin{center} \fbox{\begin{minipage}{11.9cm}
\noindent \prb
{\bf Input:} A pair $(G,C_n)$ where $C_n$ is a Hamiltonian cycle of $G$
{\bf Output:} A minimal representation $\rep$ of $(G,C_n)$ if such a representation exists, ``NO'' otherwise.
\end{minipage}}\\
\end{center}
For $n=3$ the only possible pair is $(C_3,C_3)$ whose unique minimal representation is by 3 identical paths consisting of one edge each.
Let $T$ be a tree with $k$ leaves and $\pi=(\pi_0,\ldots,\pi_{k-1})$ a cyclic permutation of the leaves. The \emph{tour} $(T,\pi)$ is the following set of $2k$ paths: $(T,\pi)$ contains $k$ \emph{long} paths, each of which connecting two consecutive leaves $\pi_i, \pi_{i+1 \mod k}$. $(T,\pi)$ contains $k$ \emph{short} paths, each of which connecting a leaf $\pi_i$ and its unique neighbor in $T$.
A \emph{planar embedding} of a tour is a planar embedding of the underlying tree such that any two paths of the tour do not cross each other. A tour is \emph{planar} if it has a planar embedding. Note that a tour $(T,\pi)$ is planar if and only if $\pi$ corresponds to the order in which the leaves are encountered by some DFS traversal of $T$.
Consider the following three properties
\begin{itemize}
\item{$(P1)$:} $(G,C_n)$ is not contractible.
\item{$(P2)$:} $(G,C_n)$ is $(K_4, P_4)$-free, i.e., it does not contain an induced sub-pair isomorphic to a $(K_4, P_4)$.
\item{$(P3)$:} Every red triangle of $(G,C_n)$ is a claw-clique, i.e. corresponds to a pie of $\rep$.
\end{itemize}
Note that $(P1)$ and $(P2)$ are properties of pairs and $(P3)$ is a property of representations. We say that $(P3)$ holds for a pair $(G,C)$ whenever it has a representation $\rep$ satisfying $(P3)$. It is convenient to define the following problem.
\begin{center} \fbox{\begin{minipage}{11.9cm}
\noindent \prbpthree
{\bf Input:} A pair $(G,C_n)$ where $C_n$ is a Hamiltonian cycle of $G$ and $n \geq 4$.
{\bf Output:} A minimal representation $\rep$ of $(G,C_n)$ that satisfies $(P3)$ if such a representation exists, ``NO '' otherwise.
\end{minipage}}\\
\end{center}
In this work we extend the following results of \cite{BESZ13-TR1}.
\begin{theorem}\label{thm:BuildPlanarTourCorrect}
\cite{BESZ13-TR1} Instances of $\prbpthree$ satisfying properties $(P1),(P2)$ can be solved in polynomial time. YES instances have a unique solution, and whenever $n \geq 5$ this solution is a planar tour.
\end{theorem}
\begin{theorem}\label{theo:P123}
\cite{BESZ13-TR1} If $n>4$ the following statements are equivalent:\\
(i) $(G,C_n)$ satisfies assumptions $(P1-3)$.\\
(ii) $(G,C_n)$ has a unique minimal representation satisfying $(P3)$ which is a planar tour of a weak dual tree of $G$.\\
(iii) $G$ is Hamiltonian outerplanar and every face adjacent to the unbounded face $F$ is a triangle having two edges in common with $F$, (i.e. a BBR triangle).
\end{theorem}
For $n=4$ there are two possible pairs, namely $\kp$ and $(K_4 - e, C_4)$, each of which satisfying $(P1),(P2)$ and having a unique minimal representation. Therefore in this work we assume $n \geq 5$.
The opposite of a sequence of union operations that create one path is termed \emph{breaking apart}. Namely, breaking apart a path $P$ is to replace it with paths $P_1, \ldots ,P_k$ such that $\cup_{i=1}^k P_i=P$, $\forall 1 \leq i < k, P_i \cap P_{i+1} \neq \emptyset$, and $P_i \subseteq P_j$ if and only if $i=j$.
A \emph{broken tour} is a representation obtained from a tour by subdividing edges and breaking apart long paths of a tour. Clearly, if the tour is planar the broken tour is also planar, i.e. has a planar embedding.
\subsection{Small Cycles: the pairs $(G,C_5)$ and $(G,C_6)$}\label{subsec:c5c6}
We now return to the study of the representations of pairs $(G',C')$ satisfying $(P2),(P3)$. Without loss of generality we let $V(G')=V(C')=\set{0,1,\ldots,n-1}$, $n \geq 5$ and note that all arithmetic operations on vertex numbers are done modulo $n$.
In this Section we analyze the special cases of $n \in \set{5,6}$. This cases are special because our technique for the general case is based on contraction of cycles to smaller ones and assumes that the representation of a non-contractible pair is a planar tour (Theorem \ref{theo:P123}). However this theorem does not hold when $n=4$. The following lemma analyzes the case $n=5$. We note that in this case $(P3)$ holds vacuously.
\begin{lemma}\label{lem:unique-c5}
If $(G',C_5)$ satisfies $(P2)$ then (i) $G'$ is the graph depicted in Figure \ref{fig:eptn-c5}, and (ii) $(G',C_5)$ has a unique minimal representation also depicted in Figure \ref{fig:eptn-c5}.
\end{lemma}
\begin{proof}
(i) $G'$ contains at least two non-crossing red edges, because otherwise there is a hole of size $4$ with blue edges. Without loss of generality, let these edges be $\set{1,3}$ and $\set{1,4}$. If one of $\set{2,4}$ or $\set{0,3}$ is a red edge, then we have a $\kp$, contradicting our assumption. If $\set{0,2}$ is a red edge, then we have a hole of size $4$ containing blue edges, contradicting Lemma \ref{lem:NoBlueHole}. Therefore $\set{1,3}$ and $\set{1,4}$ are the only red edges in this pair.
(ii) We contract $\set{3,4}$ of $(G',C_5)$ and obtain the pair $(G,C_4)$ with one red edge $\set{1,3.4}$. This pair has a unique minimal representation $\repprime$ characterized in \cite{BESZ13-TR1}. Any representation of $(G', C_5)$ is obtained by splitting the path $P'_{3.4}$ of $\repprime$ into two overlapping paths and making sure that both of them split from $P_1$. This leads to the minimal representation depicted in Figure \ref{fig:eptn-c5}.
\end{proof}
\begin{figure}[htbp]
\centering
\include{figure/eptn-c5}
\caption{(a) The unique $\enpt$ representation of $C_5$ satisfying $(P2)$ and (b) corresponding pair $(G,C_5)$.}
\label{fig:eptn-c5}
\end{figure}
\begin{lemma}\label{lem:NoContractibleC6}
If $(G',C_6)$ satisfies $(P2)$ and $(P3)$ then it is not contractible.
\end{lemma}
\begin{proof}
Assume, by way of contradiction, that $(G',C_6)$ satisfies $(P2)$ $(P3)$ and the edge $e=\set{0,1}$ is contractible. Therefore, $\set{0,2}$ and $\set{5,1}$ are non-edges of $G'$. $\set{2,5}$ is also a non-edge, because otherwise $\set{0,1,2,5}$ is a hole of size $4$ with blue edges. Then $\set{0,1}$ must be in a $BRR$ triangle. From the two possible options remaining, assume without loss of generality that this triangle is $\set{0,1,4}$. At least one of $\set{2,4}$ and $\set{1,3}$ is an edge of $G'$ because otherwise $\set{1,2,3,4}$ is a hole of size $4$ with blue edges. On the other hand, if both of them are edges then $\set{1,2,3,4}$ is a $\kp$, a contradiction. Therefore exactly one of them is an edge of $G'$. We analyze these cases separately.
\begin{itemize}
\item{$\set{2,4}$ is an edge of $G'$, $\set{1,3}$ is not an edge of $G'$:} In this case $\set{0,3}$ is not an edge, because otherwise $\set{0,1,2,3}$ is a hole of size $4$ with blue edges. Then $\set{3,5}$ is not an edge, because otherwise $\set{0,1,2,3,5}$ is a hole of size $5$ with blue edges. Then $\set{5,0,1,2,3}$ induces a path on $4$ vertices in $G'$. Since none of the paths $P_0,P_1,P_2,P_3,P_5$ split from another, their union is a graph with maximum degree two, i.e. every representation of them is an interval representation where no three paths intersect at one edge. Now $P_4 \sim P_5$ and $P_4 \sim P_3$. Therefore, $P_4$ intersects all of $P_0,P_1$ and $P_2$ and does not split from them. Then $\set{4,0},\set{4,1},\set{4,2}$ are blue edges, a contradiction.
\item{$\set{1,3}$ is an edge of $G'$, $\set{2,4}$ is not an edge of $G'$:} Assume by way of contradiction $\set{0,1}$ is contracted, the contracted pair is the same as the pair in Figure \ref{fig:eptn-c5} (b) where contracted edge $\set{0,1}$ corresponds to vertex $1$ of $(G,C_5)$. We will show that $1$ can not be a vertex obtained by a contraction. Let $\set{1',1''}$ be the contracted edge. For the following discussion consult Figure \ref{fig:eptn-c5} (a). One endpoint of each one of $P_{1'}$, $P_{1''}$ is the same as the endpoints of $P_1$ since $P_1 = P_{1'} \cup P_{1''}$. $P_{1'}$ (resp. $P_{1''}$) can not cross $v$ since otherwise $\set{1',2}$ (resp. $\set{0,1''}$) is a blue chord. $P_{1'} \sim P_{1''}$, therefore there exist some edge $e$ such that $P_{1'} \cap P_{1''} \ni e$ and $e \in p_T(u,v)$. But $e \in p_T(u,v) \subseteq P_3 \cup P_4$ then either $\set{1',3}$ or $\set{1'',4}$ (or both) is a blue chord.
\end{itemize}
\end{proof}
\subsection{The General Case}\label{subsec:P23Representations}
\newcommand{\textsc{FindMinimalRepresentation-P2-P3}}{\textsc{FindMinimalRepresentation-P2-P3}}
\newcommand{\textsc{AdjustEndpoint}}{\textsc{AdjustEndpoint}}
Algorithm \ref{alg:findminimalrepresentation} is a recursive description of $\textsc{FindMinimalRepresentation-P2-P3}$. It follows the paradigm of obtaining a non-contractible pair by successive contractions, and then reversing the corresponding union operations in the representation. The reversal of the union operation, i.e. the breaking apart of a path is done by duplicating the path and then moving one endpoint of each path to an appropriate internal vertex of the original path, and possibly subdividing an edge. The key to the correctness of the algorithm is the following lemma that among others enables us to consider only one minifying operation.
\begin{lemma}\label{lem:MinimizationOfBrokenPlanarTours}
Let $\rep$ be a minimal representation of $(G,C)$, $\repprime$ a broken planar tour representation such that $\repprime \lesssim \contractppq$ and $\repprime \approxeq \contractppq$. Every operation in a minimal sequence of operations that obtains $\repprime$ from $\contractppq$ is a $contract(e)$ operation, where $e$ is a tail of $P_p \cap P_q$.
\end{lemma}
\begin{proof}
Consider an operation in a minimal sequence of minifying operations as in the statement of the lemma. Let $e$ be the edge involved in the operation, and let $P_r$ be a path whose existence is guaranteed by Lemma \ref{lem:MinimizationAfterUnion}. By Lemma \ref{lem:EverySplitVertexIsACenterofBrokenPlanarTour}, case (ii) of Lemma \ref{lem:MinimizationAfterUnion} is impossible. Then case (i) of the lemma holds, i.e. there is a path $P_r$ such that a) the minifying operation removes $e$ from $P_r$, b) $e$ is a tail of $P_p \cap P_q$, c) $P_r \cap P_p \cap P_q = \set{e}$, and d) $P_r \cap (P_p \cup P_q) \supsetneq \set{e}$.
The minifying operation is either $contract(e)$ or $tr(P_r,e)$. We will show that if $tr(P_r,e)$ can be applied, i.e. no affected pair after applying $tr(P_r,e)$, then $contract(e)$ can also be applied. For the following discussion consult Figure \ref{fig:MinimizationAfterUnion} a) where $\split(P_r,P_p) = \emptyset$, i.e. the dotted part of $P_r$ adjacent to $e$ in the figure, is empty.
Without loss of generality we assume that $e$ is a tail of $P_p$. Since $e$ is not a tail of $P_p \cup P_q$, we have $r \neq p.q$. $e$ divides $T$ into two subtrees $T_1, T_2$. As $e$ is a tail of $P_p$, $P_p$ can not intersect both subtrees. We assume without loss of generality that $T_2 \cap P_p = \emptyset$. Let $\bar{\pp}$ denote the set of paths of $\contractppq$, i.e. $\bar{\pp}=\pp \setminus \set{P_p, P_q} \cup \set{P_p \cup P_q}$ and $e'$ be the edge adjacent to $e$ in $P_r \cap (P_p \cup P_q)$. Every path of $P \in \bar{\pp}$ that contains $e$ contains also $e'$, because otherwise $P \cap P_r = \set{e}$ and $(P,P_r)$ would constitute an affected pair of $tr(P_r,e)$. For $k \in \set{1,2}$, let $\pp_k = \set{P \in \bar{\pp}|~ P \cap T_k \neq \emptyset \wedge e \textrm{~is a tail of~} P}$. Note that by definition, $\pp_1 \cap \pp_2 = \emptyset$. As $e' \in T_2 \cap P_r$, we have $P_r \in \pp_2$. We note that for every path $P_s \in \pp_2$, $P_p \sim P_s$, i.e. $\set{p,s}$ is an edge of $C$. As the degree of $p$ is $2$ in $C$ and both of $q$ and $r$ neighbors of $p$ in $C$, we conclude that $\pp_2 = \set{P_r}$. On the other hand, $\pp_1=\emptyset$ because for every path $P_s \in \pp_1$, $\set{s,r}$ is an affected pair of $tr(P_r,e)$ (as $P_s \cap P_r = \set{e}$). Therefore $\pp_1 \cup \pp_2 = \set{P_r}$, i.e. the only path with tail $e$ is $P_r$.
Assume by way of contradiction that there exists an affected pair $\set{s,t}$ of $contract(e)$. As $e' \in P_s \cap P_t$, they intersect after the contraction. Therefore $\set{s,t}$ is a red-edge that becomes blue after the contraction. This can happen only if $e$ is a tail of exactly one of $P_s, P_t$. Therefore, $r \in \set{s,t}$ from the above discussion. But then $\set{s,t}$ constitute an affected pair of $tr(P_r,e)$, contradicting to our initial assumption. We conclude that $contract(e)$ has no affected pairs.
\end{proof}
\alglanguage{pseudocode}
\begin{algorithm}
\caption{$\textsc{FindMinimalRepresentation-P2-P3}(G',C')$}
\label{alg:findminimalrepresentation}
\begin{algorithmic}[1]
\Require {$C'=\set{0,1,\ldots,\abs{V(G')}-1}$ is an Hamiltonian cycle of $G'$ and $\abs{V(G')} > 5$}
\Ensure {A minimal representation $\repbarprime$ of $(G',C')$ satisfying $(P3)$ if any}
\If {$(G',C')$ is contraction-minimal}
\If{$G'$ is outerplanar}
\State \Return $\textsc{BuildPlanarTour}(G',C')$
\Else
\State \Return ``NO''
\EndIf
\EndIf
\State
\Statex \textbf{Contract:}
\State Pick an arbitrary contractible edge $e = \set{i,i+1}$ of $C'$
\State $(G,C) \gets \contract{(G',C')}{e}$
\State Let $j$ be the vertex of $(G,C)$ created by the contraction of the edge $e$
\State
\Statex \textbf{Recurse:}
\State $\repbar \gets \textsc{FindMinimalRepresentation-P2-P3}(G,C)$.\label{lin:recurse}
\State
\Statex \textbf{Uncontract:}
\State $\repbarprime \gets \repbar$
\State Let $u$ and $v$ be the endpoints of $P_j$ such that
\State ~~~~$u$ (resp. $v$) is contained in $P_{i-1}$ (resp. $P_{i+2}$)
\State Replace $P_j \in \bar{\pp}'$ by two copies $P_i$ and $P_{i+1}$ of itself
\State $\textsc{AdjustEndpoint} (\repbarprime, G, i, u)$
\State $\textsc{AdjustEndpoint} (\repbarprime, G, i+1, v)$
\Statex \textbf{Validate:}
\If {$\eptg{\ppbar'}=G'$ and $\repbarprime$ satisfies $(P3)$}
\State \Return $\repbarprime$
\Else
\State \Return ``NO''
\EndIf
\Function{\textsc{AdjustEndpoint}}{$\repbar, G, p, w$}
\Comment $w$ is the endpoint of $P_p$ to be adjusted
\State $e_w$ denotes the tail of $P_p$ incident to $w$
\State $\xx_w$ denotes $\set{P_x: e_w \in P_x \textrm{~and~} \set{p,x} \notin E(G)}$
\State $\yy_w$ denotes $\set{P_y: P_p \cap P_y = \set{e_w} \textrm{~and~} \set{p,y} \in E(G)}$
\While {$\yy_w = \emptyset$ and $\xx_w \neq \emptyset$}
\State $tr(P_p,e_w)$
\EndWhile
\If {$\xx_w \neq \emptyset$} \Comment Also $\yy_w \neq \emptyset$ as the while loop terminated
\State Subdivide $e_w$ into two edges $e_w, e_w'$ \Comment Revert the minifying operation
\For {$P_x \in \xx_w$}
\State $tr(P_w,e_{w'})$
\EndFor
\State $tr(P_p,e_w)$
\EndIf
\EndFunction
\end{algorithmic}
\end{algorithm}
\begin{figure}[htbp]
\centering
\include{figure/AdjustEndpoint2}
\caption{The effect of union and minifying operations, and the reversal of this effect by Procedure $\textsc{AdjustEndpoint}$ (invoked with $p=i$).}\label{fig:AdjustEndpoint}
\end{figure}
\begin{theorem}\label{thm:FindMinimalRepresentation}
Instances of $\prbpthree$ satisfying property $(P2)$ can be solved in polynomial time. YES instances have a unique solution, and whenever $n \geq 6$ this solution is a broken planar tour.
\end{theorem}
\begin{proof}
If $n=5$ the result follows from Lemma \ref{lem:unique-c5}. If $(G',C')$ is a "NO" instance, then $\textsc{FindMinimalRepresentation-P2-P3}$ returns ``NO'' in the validation phase. Therefore we assume that $n \geq 6$, and that $(G',C')$ is a "YES" instance, i.e. it has at least one representation satisfying $(P3)$. We will show that for any pair $(G',C')$ satisfying $(P2)$, and a minimal representation $\repprime$ of $(G',C')$ that satisfies $(P3)$, the representation $\repbarprime$ returned by $\textsc{FindMinimalRepresentation-P2-P3}$ is a broken planar tour and
\[
\repbarprime \cong \repprime \textrm{~and~} \repbarprime \lesssim \repprime.
\]
We will prove by induction on the number $k$ of contractible edges of $(G',C')$. If $k=0$ then $(G',C')$ is not contractible, therefore satisfies $(P1)$. In this case the algorithm invokes $\textsc{BuildPlanarTour}$ and the claim follows from Theorem \ref{thm:BuildPlanarTourCorrect}.
Otherwise $k>0$. We assume that the claim holds for $k-1$ and prove that it holds for $k$. As $(G',C')$ contains at least one contractible edge, one such edge $\set{i,i+1}$ is chosen arbitrarily by the algorithm and contracted. The resulting pair $(G,C)=\contract{(G',C')}{\set{i,i+1}}$ has the following properties:
\begin{itemize}
\item{Satisfies $(P2),(P3)$.} (By Lemma \ref{lem:ContractionPreservesP2P3})
\item{The number of contractible edges is $k-1$.}
\item{$\abs{V(G)} \geq 6$.} This is because $\abs{V(G)}=\abs{V(G')}-1$ and $\abs{V(G')}>6$. Indeed, if $\abs{V(G')}=6$, we have $k=0$ by Lemma \ref{lem:NoContractibleC6}.
\end{itemize}
Therefore, $(G,C)$ satisfies the assumptions of the inductive hypothesis. Let $\repprime$ be a minimal representation of $(G',C')$ satisfying $(P3)$. Then $\contract{\repprime}{P_i,P_{i+1}}$ is a representation of $(G,C)=\contract{(G',C')}{\set{i,i+1}}$. By the inductive hypothesis, $\repbar$ is a broken planar tour that satisfies
\[
\repbar \cong \contract{\repprime}{P_i,P_{i+1}} \textrm{~and~} \repbar \lesssim \contract{\repprime}{P_i,P_{i+1}}.
\]
In other words $\repbar$ is obtained from $\repprime$ by replacing the two paths $P_i, P_{i+1}$ with the path $P_i \cup P_{i+1}$, then applying a (possibly empty) sequence of minifying operations. By Lemma \ref{lem:MinimizationOfBrokenPlanarTours}, these minifying operations are $contract(e)$ for a tail $e$ of $P_i \cap P_{i+1}$. In the Uncontract phase, $\textsc{FindMinimalRepresentation-P2-P3}$ performs a reversal of these transformations. See Figure \ref{fig:AdjustEndpoint} for the following discussion. One endpoint of each one of $P_i$ and $P_{i+1}$ is an endpoint of $P_i \cup P_{i+1}$. Therefore one needs to determine only one endpoint of each one of $P_i$ and $P_{i+1}$. First $P_i \cap P_{i+1}$ is duplicated and the so obtained paths are called $P_i$, $P_{i+1}$.
For $p \in \set{i,i+1}$, let $w$ be the endpoint of $P_p$ to be adjusted. $e_w$ denotes the tail of $P_p$ incident to $w$. We denote by $\xx_w$ the set of paths containing $e_w$ such that vertices of $G'$ corresponding to these paths are not adjacent to $p$. We denote by $\yy_w$ the set of paths intersecting $P_p$ only on $e_w$ and whose corresponding vertices in $G'$ are adjacent to $p$. If $\yy_w$ is empty (that is, every path that intersects $P_p$ also intersects $P_p \setminus \set{e_w}$, $e_w$ can be safely removed from $P_p$ without losing intersections. If $\xx_w$ is non-empty this removal is a necessary operation. The algorithm performs these tail removals as long as they are necessary and safe. If at the end of this loop, $\xx_w$ is empty then we are done. Otherwise $\xx_w$ and $\yy_w$ are non-empty, then $e_w$ can not be safely removed from $P_p$. In this case $\textsc{AdjustEndpoint}$ subdivides $e_w$ (thus reversing the minifying operation $contract(e)$) and removes one tail from $P_p$ and one tail from every path $X \in \xx_w$, so that $P_p$ does not intersect $X$ but still intersects every path $Y \in \yy_w$.
\end{proof}
|
1,116,691,497,740 | arxiv | \section{Introduction}
\bigskip
In \cite{NOT}, Ng defined the notion of an action $\alpha$ of a
small category $G$ on another small category $H$ and show that its
semi-direct product $H\times_\alpha G$ is a small category if either
$G$ is a groupoid or $G^{(0)} = H^{(0)}$ (see \cite[Propositions 8
\& 10]{NOT}). On the other hand, Exel has recently defined in
\cite{EXE} the notion of semigroupoids as well as semigroupoid
$C^*$-algebras. Since a small category is a semigroupoid, Ng asked
whether one can define a kind of ``action'' $\beta$ of $G$ on
$C^*(H)$ such that $C^*(H)\times_\beta G \cong C^*(H\times_\alpha
G)$.
\medskip
The aim of this article is to give an answer to this question. More
precisely, we will define ``quasi-actions'' of groupoids on
$C^*$-algebras and their crossed products. For any action $\alpha$
of a groupoid $G$ on a small category $H$, there exists a
subcategory $H_r\subseteq H$ such that every element $h\in H_r$ is
in the domain of some $\alpha_g$ (such an action is called
``regular'') and that $H_r\times_\alpha G = H\times_\alpha G$.
Moreover, one can define a quasi action $\tilde\alpha$ of $G$ on
$C^*(H_r)$ such that the crossed product $C^*(H_r)\times_{\tilde\alpha}
G$ is $*$-isomorphic to $C^*(H_r\times_\alpha G) =
C^*(H\times_\alpha G)$. On the other hand, according to
\cite[Proposition 2]{NOT}, a groupoid $G$ can be ``represented'' by
a group bundle $\{G_\xi: \xi\in G^{(0)} /R\}$ for an equivalence
relation $R$ on $G^{(0)}$. In this case, one has a decomposition
$H_r = \cup_{\xi\in G^{(0)} /R} H_\xi$ such that $\alpha$ induces a
transitive action $\tilde\alpha_\xi$ of the group $G_\xi$ on $H_\xi$
for each $\xi\in G^{(0)} /R$ and that $C^*(H_r)\times_{\tilde\alpha} G
\cong \bigoplus_{\xi\in G^{(0)} /R} C^*(H_\xi)\times_{\tilde\alpha_\xi}
G_\xi$. Consequently, in order to understand $C^*(H\times_\alpha
G)$, one can study a collection of the crossed products of
transitive groupoid actions on the categories.
\medskip
\emph{Acknowledgements:} The author would like to thank Prof.
Chi-Keung Ng for his invaluable conversation and for pointing out
some problems in an earlier version of this work.
\bigskip
\section{\bf Preliminaries and basic definitions}
\medskip
At first, let's recall the definition of small category, whose
notations will be used through out the paper. One can find the
definition of category in \cite {CAT}.
\begin{defn}
\label{defn1}{\rm (a) A small category $\Lambda$ is a category with
its object space being a set, or equivalently, all its morphisms
form a set. We denote the object space by $\Lambda^{(0)}$, and
composable morphism pairs by $\Lambda^{(2)}$, which is a subset of
$\Lambda\times\Lambda$.\\ \\(b) Let $f, g\in\Lambda$. We shall say
that $f$ divides $g$, or that $g$ is a multiple of $f$, in symbols
$f\mid g$, if there exists $h\in\Lambda$ such that $fh=g$. We say
that $f$ and $g$ intersect if they admit a common multiple, writing
f$\Cap$g; otherwise we will say that $f$ and $g$ are disjoint,
writing $f\perp g$.}
\end{defn}If we identify a small category
with all its morphisms, every small category is a semigroupoid (see
the definition in \cite[2.1]{EXE}). It is also well known that any
groupoid is just a small category with every morphism being
invertible. Moreover, if $H$ is a small category, we set $H^{(0)}$
as its unit space, also let $\textbf s$ and $\textbf t$ be the
source and target map of each morphism being (element) respectively.
Especially, these notations are available also for the groupoid
case.
\medskip
\begin{eg}
\label{eg2}{\rm Let X be a set and R be an equivalence relation on
X. Suppose that $G_{\xi}$ is a group for any $\xi\in(X/R)$. Then (X,
R, $\{G_{\xi}\}_{\xi\in(X/R)}$) is called a group bundle over the
equivalence classes of R. Then, let $$\mathcal {G}=\{(x, g,
y):\xi\in X/R;x, y\in\xi;g\in G_{\xi}\}$$and $\textbf t, \textbf
s:\mathcal {G}\rightarrow\ X$ are defined by $\textbf t(x, g, y)=x$
and $\textbf s(x, g, y)=y$. Moreover, let
$$(x, g, y)(y, h, z)=(x, gh, z)\ and\ (x, g, y)^{-1}=(y, g^{-1}, x)$$
for any $x,y,z\in\xi$ with $xRy$ and $yRz$, and for any $g,h\in
G_\xi$. Through this way, $\mathcal {G}$ is endowed with a groupoid
structure. Indeed, every groupoid is of this type as proved in
\cite[Proposition 2]{NOT}. More precisely, for a set $X$, there is a
one to one correspondence between groupoids with unit space X and
group bundles over the equivalence classes of equivalence relations
on X, where R is defined by $x \sim y$ if $G^x_y\triangleq \textbf
t^{-1}(x)\cap \textbf s^{-1}(y)$ is non-empty for x, y in $G^{(0)}$,
and $G_{\xi}=G_{x}^{x}$ for some fixed element $x\in\xi$. The
original thought lies in \cite[1.1]{REN}.}
\end{eg}
\medskip
\medskip
\begin{eg}
\label{par}\rm Let $Par(B(\mathcal {H}))=\{(\mathcal{M}, S): S\ {\rm
partial\ isometry},\ (\ker S)^{\perp}=\mathcal{M} \}$ for some
Hilbert space $\mathcal{H}$. Define $\textbf s(\mathcal{M}, S)=\mathcal{M}$
and $\textbf t(\mathcal{M}, S)=S(\mathcal{M})$. Then we have
$Par(B(\mathcal {H}))$ is a groupoid. By Gelfand-Naimark theory, any
$C^*-$algebra $A$ admits a groupoid structure on its partial
isometries denoted by $Par(A)$.
\end{eg}
\medskip
Let's recall from \cite[Definition 4]{NOT} the definition of actions
of a groupoid on a small category as well as a remark followed. The
main motivation of this definition comes from the definition of the
Lie groupoid actions on manifolds.
\begin{defn}
\label{eg4}\rm Let $G$ and $H$ be small categories. Suppose that
$\varphi: H^{(0)} \rightarrow G^{(0)}$. We let
$$G\times^\varphi H\ :=\ \{ (g,h)\in G\times H: \textbf s(g) = \varphi(\textbf t(h)) = \varphi (\textbf s(h)) \}.$$
A \emph{left action of $G$ on $H$ with respect to $\varphi$} is a
map $(g,h)\mapsto \alpha_g(h)$ from $G\times^\varphi H$ to $H$ such
that for any $(g',g)\in G^{(2)}$, $(h',h)\in H^{(2)}$ and $u\in
H^{(0)}$ with $(g,h),(g,u), (g,h')\in G\times^\varphi H$, we have:
\begin{enumerate}
\item[\rm (I).] $\alpha_g(\textbf s(h)) = \textbf s(\alpha_g(h))$;
\item[\rm (II).] $\alpha_g(\textbf t(h)) = \textbf t(\alpha_g(h))$;
\item[\rm (III).] $\varphi(\alpha_g(u)) = \textbf t(g)$;
\item[\rm (IV).] $\alpha_{\varphi(\textbf t(h))} (h) = h$;
\item[\rm (V).] $\alpha_{g'}(\alpha_{g}(h)) = \alpha_{g'g}(h)$;
\item[\rm (VI).] $\alpha_g(h' h) = \alpha_g(h')\alpha_g(h)$.
\end{enumerate}
For simplicity, we say that ($\varphi, \alpha$) (or just $\alpha$)
is a left action of G on H. As a convention, We will always assume
that $\varphi$ is surjective, and denote for each $g\in G$,
$$H^g\triangleq\{h\in H: \textbf s(g)=\varphi(\textbf s(h))=\varphi(\textbf
t(h))\}.$$\end{defn}
\medskip
\begin{prop}\rm\cite[Proposition 8]{NOT}
\label{rem6} Suppose that $G$ is a groupoid acting on a small
category $H$ by a left action $(\varphi, \alpha)$ and define the
semi-direct product category $H\times_\alpha G$ by
$$H\times_\alpha G\ \triangleq \ \{(h,g)\in H\times G: \textbf t(g) = \varphi(\textbf s(h)) = \varphi (\textbf t(h))\}.$$
For any $(h,g)\in H\times_\alpha G$, we set
$$\textbf s(h,g)\ \triangleq\ \alpha_{g^{-1}}(\textbf s(h)) \qquad {\rm and} \qquad \textbf t(h,g)\ \triangleq\ \textbf t(h)$$
(here, we identify $u\in H^{(0)}$ with its canonical image $(u,
\varphi(u)) \in H\times_\alpha G$). Moreover, if $(h,g),(h',g')\in
H\times_\alpha G$ satisfying $\textbf s(\alpha_{g^{-1}}(h)) = \textbf t(h')$, we
define
$$(h,g)(h',g')\ \triangleq\ (h\alpha_g(h'), gg').$$
This turns $H\times_\alpha G$ into a small category. If, in
addition, $H$ is a groupoid, then $H\times_\alpha G$ is also a
groupoid with
$$(h,g)^{-1} = (\alpha_{g^{-1}}(h^{-1}), g^{-1}).$$
\end{prop}
\medskip
The semi-direct product $H\times_{\alpha} G$ has a simple criterion
for the disjointness of two elements given by the following
proposition.
\begin{prop}
\label{prop7} Let G be a groupoid, and H be a small category, and
$(\varphi, \alpha)$ is a left action of G on H. For any $(h, g)$,
$(h^{'}, g^{'})\in H\times_\alpha G$, we have $(h, g)\perp$($h^{'},
g^{'}$) iff $h\perp h^{'}$ or $g\perp g^{'}$.
\end{prop}
\begin{proof}
Sufficiency is obvious, and we only prove the necessity, i. e. , if
$(h, g)\perp\ $($h^{'}, g^{'}$) and $g\Cap g^{'}$, $h\perp h^{'}$
must hold. Otherwise, if $h\Cap h^{'}$, by definition, there exists
$k, k^{'}$ and $l, l^{'}$ such that $hk=h^{'}k^{'}$ and
$gl=g^{'}l^{'}$. Since $(h, k)\in H^{(2)}$, then $\textbf
s(h)=\textbf t(k)$, so
$$\textbf s(\alpha_{g^{-1}}(h))=\alpha_{g^{-1}}(\textbf s(h))=\alpha_{g^{-1}}(\textbf t(k))=\textbf t(\alpha_{g^{-1}}(k))$$
and
$$\textbf s(\alpha_{g^{'-1}}(h^{'}))=\alpha_{g^{, -1}}(\textbf s(h^{'}))=\alpha_{g^{'-1}}(\textbf t(k^{'}))=\textbf t(\alpha_{g^{'-1}}(k^{'})). $$
Moreover,
$$(h, g)(\alpha_{g^{-1}}(k), l)=(h\alpha_{g}\alpha_{g^{-1}}(k), gl)=(h
\alpha_{\varphi(\textbf t(k))}(k), gl)=(hk, gl). $$
$$(h^{'}, g^{'})(\alpha_{g^{'-1}}(k^{'}), l^{'})=(h^{'}\alpha_{g^{'}}\alpha_{g^{'-1}}(k^{'}), g^{'}l^{'})=(h^{'}
\alpha_{\varphi(\textbf t(k^{'}))}(k^{'}), g^{'}l^{'})=(h^{'}k^{'},
g^{'}l^{'}). $$ On the other hand, $(hk, gl)=(h^{'}k^{'},
g^{'}l^{'})$ which gives the contradiction that $(h, g)\Cap$($h^{'},
g^{'}$). This completes the proof.
\end{proof}
\medskip
Next, we define $C^*-$algebra for small category as in \cite{EXE}.
\begin{defn}\rm
\label{defn8}\cite[4.1]{EXE} Let $\Lambda$ be a small category and
let B be the unital $C^*$-algebra. A mapping $S:\Lambda\rightarrow B$ will be
called a representation of
$\Lambda$ in B, if for every $f, g\in\Lambda$, \\
\begin{enumerate}
\item[\rm (I).]$S_{f}$ is a partial isometry,
\item[\rm (II).]
\[
S_{f}S_{g}=\left\{\begin{array}{ll}
S_{fg}, &{\rm if}\ (f, g)\in\Lambda^{(2)},\\
0, &{\rm otherwise}.
\end{array}
\right.
\]
\end{enumerate}
Moreover the initial projections $Q_{f}=S_{f}^{*}S_{f}$, and the
final projections $P_{g}=S_{g}S_{g}^{*}$, are required to commute
amongst themselves and to satisfy
\begin{enumerate}
\item[\rm (III).]$P_{f}P_{g}=0, if f\perp g$,
\item[\rm (IV).]$Q_{f}P_{g}=P_{g}$, if $(f, g)\in\Lambda^{(2)}. $
\end{enumerate}
\end{defn}
\medskip
Note that we always have: $Q_{f}P_{g}=0$ if $(f,
g)\not\in\Lambda^{(2)}$. We now recall the definition of
semigroupoid $C^*$-algebra\ from \cite{EXE} (see also \cite{BLA}). The
semigroupoid $C^*$-algebra\ is the $C^*$-algebra\ $C^{*}(\Lambda)$ generated by a
family of partial isometries $\{S_{f}\}_{f\in\Lambda}$ subject to
the relation that the correspondence $f\rightarrow S_{f}$ is a
representation with the universal property that for every
representation T of $\Lambda$ in a unital $C^*$-algebra\ B there exists a
unique *-homomorphism
$$\varphi :C^{*}(\Lambda)\rightarrow B, $$
such that $\varphi(S_{f})=T_{f}$, for every $f\in\Lambda$.
\medskip
\begin{defn}\rm\label{defn10}Let G be a groupoid, and H be a small category, and $(\varphi,
\alpha)$ is a left action of G on H. We call the action {\bf
regular}, if $\varphi(\textbf s(h))=\varphi(\textbf t(h))$ for any
$h\in H$.
\end{defn}
\medskip
\begin{lem}
\label{l1}Let G, H and $(\varphi, \alpha)$ be as above. Define
$H_r\triangleq \{h\in H:\varphi(\textbf s(h))=\varphi(\textbf
t(h))\}$, then $H_r$ is a subcategory of H, and $C^*(H\times_\alpha
G)\cong C^*(H_r\times_\alpha G)$.
\end{lem}
\begin{proof}It follows from definition that $H\times_\alpha G=H_r\times_\alpha
G$.
\end{proof}
\medskip
\section{\bf Quasi $C^*-$ dynamical system and crossed product}
\medskip
Now we introduce the definition of the crossed product of a $C^*$-algebra\ by
a (discrete) groupoid. \begin{defn}\rm\label{quasi}Suppose that $A$
is a $C^*$-algebra\ and $G$ is a discrete groupoid. A \emph{quasi action} of
$G$ on $A$ is a map $\beta$ from $G$ to $\mathcal {N}(A)\triangleq
\{(\varphi, \mathcal{D}(\varphi))\ |\ \varphi:\ A\rightarrow A\ {\rm
is\ a\
*-homomorphism};\ \mathcal{D}(\varphi)\subseteq A\ {\rm is\ a\
closed\
*-subalgebra};\ \varphi|_{\mathcal{D}(\varphi)}:\mathcal{D}(\varphi)\rightarrow \varphi(A)\ {\rm is\ a\ *-isomorphism}\}$, satisfying
\begin{enumerate}
\item[\rm (I).] if $(s, t)\in G^{(2)}$, then
$\beta_t(A)=\mathcal{D}(\beta_s)$, $\mathcal
{D}(\beta_{st})=\mathcal {D}(\beta_t)$, and
$\beta_{st}=\beta_s\beta_t$.
\item[\rm (II).] if $(s, t)\notin G^{(2)}$, then $\beta_s\beta_t=0$.
\end{enumerate}
Also we call $(A, G, \beta)$ a $quasi\ C^{*}-dynamical\ system$.
\end{defn}
\medskip
By definition, one always has
$\mathcal{D}(\beta_g)=\mathcal{D}(\beta_{\textbf s(g)})$, and
$\beta_{e}|_{\mathcal{D}(\beta_e)}=id$, for any $e\in G^{(2)}$. We
now construct a quasi $C^{*}-$dynamical system by the regular action
of a groupoid on some small category.
\medskip
\begin{prop}
\label{eg9} Let $\alpha$ be a regular action of groupoid $G$ on a
small category $H$. Assume the category $C^*$-algebra\ $C^{*}(H)$ is
generated by a family of partial isometries $\{S_{h}\}_{h\in H}$.\\
{\bf (a)} for every $g\in G$, the $C^*-$algebra
$C^*(H^g)=C^*(H^{\textbf s(g)})$ can be identified with the closed
*-subalgebra generated by $\{S_h\}_{h\in
H^g}$.\\
{\bf (b)} there exists a quasi action of $G$ on $C^*(H)$ such that
for any $g\in G$, $\tilde{\alpha}_g|_{C^*(H^{\textbf
s(g)})}:C^*(H^{\textbf s(g)})\rightarrow C^*(H^{\textbf t(g)})$
is the *-isomorphism given by $\tilde{\alpha}_g(S_h)=S_{\alpha_g(h)}$.\\
In this way, we obtain a quasi $C^*-$dynamical system $(C^*(H), G,
\tilde{\alpha})$.
\end{prop}
\begin{proof}{\bf (a)} For every $g\in G^{(0)}$, let $C^*(H^g)$ be generated by
$\{T_{h}\}_{h\in H^g}$. Define $$\mathcal{L}:H^g\rightarrow C^*(H)$$
by $$\mathcal{L}(h)=S_{h}\ \ (h\in H^g).$$ One can verify that
$\mathcal{L}$ induces a representation of $H^g$ in $C^*(H)$. By the
universal property, we get a
*-homomorphism (denoted by $\tilde{\mathcal{L}}$) from $C^*(H^g)$ to
$C^*(H)$ with $\tilde{\mathcal{L}}(T_h)=S_h$. On the other hand, we
define
$$\mathcal{L}':H\rightarrow C^*(H^g)$$by
\[
\mathcal{L}'(h)=\left\{\begin{array}{ll}
T_h, &t\in H^g\\
0, &{\rm otherwise}.
\end{array}
\right.
\]
Since the action is regular, we can verify that $\mathcal{L}'$
induces a representation of $H$ in $C^*(H^g)$. Again by the
universal property, we get a
*-homomorphism(denoted by $\tilde{\mathcal{L}'}$) from $C^*(H)$ to $C^*(H^g)$ with
$\tilde{\mathcal{L}'}(S_h)=T_h$. Obviously, $\tilde{\mathcal{L}'}$ is the
left inverse of $\tilde{\mathcal{L}}$, thus $\tilde{\mathcal{L}}$ is isometric, hence complete the proof of {\bf (a)}.\\
\\
{\bf (b)} For every $g\in G$, define $$\tilde{\alpha}_g:H\rightarrow
C^*(H^{\textbf t(g)})$$ by
\[
\tilde{\alpha}_g(h)=\left\{\begin{array}{ll}
S_{\alpha_g(h)}, &{\rm if}\ h\in H^g\\
0, &{\rm otherwise}.
\end{array}
\right.
\]
For $h_1,h_2\in H^g$, if $(h_1,h_2)\in H^{(2)}$, since
$\textbf s(\alpha_g(h_1))=\alpha_g(\textbf s(h_1))=\alpha_g(\textbf t(h_2))=\textbf t(\alpha_g(h_2))$,
we have $(\alpha_g(h_1), \alpha_g(h_2))\in H^{(2)}$; and if
$(h_1,h_2)\notin H^{(2)}$, we have $(\alpha_g(h_1),
\alpha_g(h_2))\notin H^{(2)}$. Moreover, if $h_1\perp h_2$, we also
have $\alpha_g(h_1)\perp \alpha_g(h_2)$. Otherwise, if
$\alpha_g(h_1)\Cap \alpha_g(h_2)$, there exists $k_1,k_2\in G$ such
that $\alpha_g(h_1)k_1=\alpha_g(h_2)k_2$. Consider the action by
$\alpha_{g^{-1}}$ on both sides, and this gives the contradiction
that $h_1\perp h_2$. Noticing the regularity of $\alpha$ and the
fact that $\alpha_g$ keeps the composability and disjointness of any
two elements in $H^g$ , one can verify that $\tilde{\alpha}_g$ is a
representation of $H$ in $C^*(H^{\textbf t(g)})$. By the universal
property, we get a
*-homomorphism (also denoted by $\tilde{\alpha}_g$) from
$C^*(H)$ to $C^*(H^{\textbf t(g)})$ given by
$\tilde{\alpha}_g(S_h)=S_{\alpha_g(h)}$ for $h\in H^g$. Similarly, we
can construct a *-homomorphism
$\tilde{\alpha}_{g^{-1}}:C^*(H)\rightarrow C^*(H^{\textbf s(g)})$ given
by
\[
\tilde{\alpha}_{g^{-1}}(S_k)=\left\{\begin{array}{ll}
S_{\alpha_{g^{-1}}(k)}, &{\rm if}\ k\in H^{g^{-1}}\\
0, &{\rm otherwise}.
\end{array}
\right.
\]
It is not hard to see that the maps $\tilde{\alpha}_g|_{C^*(H^{\textbf s(g)})}$ and
$\tilde{\alpha}_{g^{-1}}|_{C^*(H^{\textbf t(g)})}$ are the inverses of each
other, so $\tilde{\alpha}_g\in \mathcal{N}(C^*(H))$ holds for every
$g\in G$. One can also verify
$$\tilde{\alpha}:G\rightarrow \mathcal {N}(C^*(H))$$ sending $g\in G$ to $\tilde\alpha_g$ satisfies the conditions in Definition \ref{quasi}, hence we have $\tilde\alpha$
is a quasi action of $G$ on $C^*(H)$. It completes the proof of {\bf
(b)}.
\end{proof}
\medskip
\begin{defn}\rm\label{defn11}We call $(\pi, u)$ is a covariant
representation of a quasi $C^*-$dynamical system $(A,G,\beta)$, if
$\pi$ is a *-representation of $C^*-$algebra $A$ on some Hilbert
space $\mathcal {H}$, and u is a groupoid homomorphism from G to
$Par(B(\mathcal {H}))$ with $u_su_t=0$ for any $(s,t)\notin
G^{(2)}$, satisfying the compatible conditions that
$u(g)\pi(a)u(g)^*=\pi(\beta_g(a))$ and
$u(g)\pi(a)=\pi(\beta_g(a))u(g)$ for any $a\in \mathcal
{D}(\beta_g)$.
\end{defn}
By definition, one always has $u(g)^*=u(g^{-1})$ for any $g\in G$.
\medskip
\begin{prop}\label{cov}
Let $(A, G, \beta)$ be a quasi $C^*-$dynamical system. Assume that
$\pi$ is a *-representation on a Hilbert space $\mathcal {H}$.
Define $(\tilde{\pi}, u)$ for $(A,G,\beta)$ on $l^2(G,\mathcal {H})$
by:$$(\tilde\pi(a)f)(s)=\pi(\beta_{s^{-1}}(a))(f(s))\ \ \ (a\in A),$$
\[
(u_{t}f)(s)=\left\{\begin{array}{ll}
f(t^{-1}s), &{\rm if}\ (t^{-1},s)\in G^{(2)}\\
0, &{\rm otherwise}.
\end{array}
\right.
\]
Then $(\tilde\pi, u)$ is a covariant representation of $(A, G, \beta)$.
Consequently, for any quasi $C^*-$dynamical system, covariant
representations always exist.
\end{prop}
\begin{proof} It is obvious that $\tilde{\pi}$ is a
*-representation of $A$ on $l^2(G,H)$. For each $t\in G$, $\ker
(u_t)=\{f\in l^2(G,H): f(s)=0\ {\rm when}\ (t, s)\in G^{(2)}\}$.
Hence $\ker(u_t)^{\perp}=\{f\in l^2(G,H): f(s)=0\ {\rm when}\ (t,
s)\notin G^{(2)}\}$, and $u_t$ acts on $\ker(u_t)^{\perp}$
isometrically, hence a partial isometry. It is not difficult to
check that $u$ induces a groupoid homomorphism, and by definition we
have $u(s)u(t)=0$ for any $(s,t)\notin G^{(2)}$. To verify
$(\tilde{\pi}, u)$ is covariant representation, for any $a\in \mathcal
{D}(\beta_t)$ if $(t^{-1}, s)\in G^{(2)}$, we have
$$(u(t)\tilde\pi(a)u(t^{-1})f)(s)=(\tilde\pi(a)u(t^{-1})f)(t^{-1}s)=\pi(\beta_{s^{-1}t}(a))(f(s))=(\tilde\pi(\beta_t(a))f)(s);$$
and if $(t^{-1}, s)\notin G^{(2)}$, we also have
$$(u(t)\tilde\pi(a)u(t^{-1})f)(s)=0=\pi(\beta_{s^{-1}}\beta_t(a))(f(s))=(\tilde\pi(\beta_t(a))f)(s).$$
To show that $u(t)\tilde\pi(a)=\tilde\pi(\beta_t(a))u(t)$, if
$(t^{-1},s)\in G^{(2)}$, we have
$$(u(t)\tilde\pi(a)f)(s)=(\tilde\pi(a)f)(t^{-1}s)=(\pi(\beta_{s^{-1}t}(a))(f(t^{-1}s))=(\tilde\pi(\beta_t(a))u(t)f)(s);$$
and if $(t^{-1},s)\notin G^{(2)}$, we also have
$$(u(t)\tilde\pi(a)f)(s)=0=(\tilde\pi(\beta_t(a))u(t)f)(s).$$ It completes the proof.
\end{proof}
\medskip
Let $(A, G, \beta)$ be as defined in Definition \ref{quasi}. (The following notations are mainly derived from
\cite[8.1]{Dav}.) In this case, the Haar system gives the counting
measure. The space of finitely supported $A-valued$ functions is the
algebra $A[G]$ of all finite sums $f=\Sigma_{t\in G}a_{t}t$ with
$a_{t}\in\mathcal {D}(\beta_{t^{-1}})$ for each $t\in G$. Whence if
$g=\Sigma_{u\in G}b_{u}u\in A[G]$, then set
$$fg=\sum_{s\in G}\left(\sum_{(t^{-1}, s)\in G^{(2)}}a_{t}\beta_{t}(b_{t^{-1}s})\right)s,
\eqno(1a)$$and$$f^{*}=\sum_{t\in G}\beta_{t}(a_{t^{-1}}^{*})t.
\eqno(1b)$$Notice that in formula($1a$), $b_{t^{-1}s}\in\mathcal
{D}(\beta_{s^{-1}t})=\mathcal {D}(\beta_{t})$, hence it makes sense.
It is not hard to prove that $a_{t}\beta_{t}(b_{t^{-1}s})\in\mathcal
{D}(\beta_{s^{-1}})$.
Also $(s^{-1}, s)\in G^{(2)}$ and the sum always makes sense.
\medskip
\begin{prop}
\label{induce}Any covariant representation $(\pi,u)$ of quasi
$C^*-$dynamical system $(A, G, \beta)$ yields an $l^1-$contractive
*-representation of $A[G]$ by $$\sigma(f)=\sum_{t\in G}\pi(a_t)u_t.\
\ (f=\sum_{t\in G}a_t t\in A[G])$$
\end{prop}
\begin{proof}Indeed, $$\sigma(f)^*=\sum_{t\in G} u_t^*\pi(a_t)^*=\sum_{t\in G}u_{t^{-1}}\pi(a_t^*)u_tu_{t^{-1}}=\sum_{s\in G}\pi(\beta_s(a^*_{s^{-1}}))u_s=\sigma
(f^*)$$ and notice that if $(t,v)\in G^{(2)}$, then
$\textbf s(t)=\textbf s(v^{-1})$, so we have $b_v\in\mathcal {D}(\beta_t)$, hence
\begin{eqnarray}\sigma(f)\sigma(g)&=&\sum_{t\in G}\sum_{v\in
G}\pi(a_t)u_t\pi(b_v)u_v\nonumber\\
&=&\sum_{t\in G}\sum_{v\in G}\pi(a_t)\pi(\beta_t(b_v))u_{tv}\ \ (t,v)\in G^{(2)}\nonumber\\
&=&\sum_{s\in G}(\sum_{(t^{-1},s)\in
G^{(2)}}\pi(a_t\beta_t(b_{t^{-1}s})))u_s=\sigma(fg). \nonumber
\end{eqnarray}
Since $\pi$ is norm contractive and non-zero partial isometry has
norm 1, it follows that $\sigma$ is $l^1-$contractive. It completes
the proof.
\end{proof}
\medskip
\begin{defn}\rm\label{cross} Let $(A,G,\beta)$ be a quasi $C^*-$dynamical system.
The crossed product $A\times_\beta G$ is the enveloping $C^*$-algebra\ of
$A[G]$. That is, one defines a $C^*$-algebra\ norm by
$$\norm{f}=\sup_{\sigma}\norm{\sigma(f)}$$as $\sigma$ runs over all
*-representations of $A[G]$ which is $l^{1}-$contractive. Then
$A\times_{\beta}G$ is just the $C^*-$completion of $A[G]$ with this
$C^*$-norm. \end{defn}
\medskip
\begin{rem}\rm \label{cros}In Definition \ref{cross}, $\norm{f}=\sup_{\sigma}\norm{\sigma(f)}\neq
0$ if $f\in A[G]$, and $f\neq 0$. To see this, choose $\pi$ to be a
faithful *-representation of $A$ on some Hilbert space
$\mathcal{H}$, and construct a covariant representation of $(A, G,
\beta)$ as in Proposition \ref{cov}. For $g=\sum_{t\in G}a_tt\in
A[G]$ with $a_t\neq 0$, choose $h\in \mathcal{H}$ and $h\notin
\ker(\pi(\beta_{e})(a_t))$ $(e=\textbf s(t^{-1}))$. For $f\in l^2(G,
\mathcal{H})$ defined by
\[
f(s)=\left\{\begin{array}{ll}
h, &{\rm if}\ s=t^{-1}\\
0, &{\rm otherwise},
\end{array}
\right.
\]
one can verify that $\sigma(g)(f)(e)\neq 0$, where $\sigma$ is as
defined in Proposition \ref{induce}. Consequently, $A[G]$ is
naturally embedded into $A\times_\alpha G$. \end{rem}
\medskip
By Proposition \ref{induce}, Definition \ref{cross} and Remark \ref{cros}, we have that the crossed product $A\times_\beta G$ has the property that for any
covariant representation $(\pi, u)$ of a quasi $C^*-$dynamical
system $(A,G,\beta)$, there is a representation of $A\times_\beta G$
into $C^*(\pi(A), u(G))$ obtained by setting
$$\sigma(f)=\sum_{t\in G}\pi(a_t)u_t.\ \ (f=\sum_{t\in G}a_t t\in
A[G])$$
\bigskip
\section{\bf Main applications}
Let us first recall two well-known lemmas in $C^*-$algebra theory.
\begin{lem}\label{lem}(a) Let A be a $C^*-$algebra, and S be a partial
isometry in A, with final projection $SS^*=P_0$ and initial
projection $S^*S=Q_0$. Then for any projections P and Q satisfying
$S=PS=SQ$, we have $P_0\leq P$, and $Q_0\leq Q$. ($P_0$ and $Q_0$
are minimal respect to the above property) Moreover, if there is a
$T\in Par(A)$ with $TS=Q_0$, $ST=P_0$, $TT^*=Q_0$ and $T^*T=P_0$,
then $T=S^*$.\\
\\
(b) Let $\{S_i\}_{i\in I}$ be a family of partial isometries in
$B(\mathcal{H})$ with $\{Q_i\}_{i\in I}$ and $\{P_i\}_{i\in I}$
being the initial and final projections respectively, such that for
any $i,\ j\in I$ with $i\neq j$, we have $Q_i\perp Q_j$ and
$P_i\perp P_j$ hold. Then the sum $\sum_{i\in I}S_i$ converges
strongly, and $\sum_{i\in I}S_i\in Par(B(\mathcal{H}))$.
\end{lem}
\medskip
\begin{thm}
Let H be a small category, G a groupoid, $(\varphi, \alpha)$ is an
action of G on H. Let $H_r=\{h\in H:\varphi(\textbf
s(h))=\varphi(\textbf t(h))\}$. Then $C^{*}(H\times_{\alpha}G)\cong
C^{*}(H_r)\times_{\tilde\alpha}G. $ Here $\cong$ means *-isomorphism,
and $C^*(H_r)\times_{\tilde\alpha} G$ corresponds to the quasi
$C^*-$dynamical system $(C^*(H_r),G,\tilde{\alpha})$ introduced in
Proposition \ref{eg9}.
\end{thm}
\begin{proof}
With the above comments, it is enough to consider the case when the
action is regular. \\
\medskip
Let $\Lambda=H\times _{\alpha}G$. Assume that
$C^{*}(H)$ is generated by a family of partial isometries
$\{S_{h}\}_{h\in H}$, and $C^*(H\times_\alpha G)$ is generated by
$\{S_{(h,g)}\}_{(h,g)\in H\times_{\alpha} G}$. Define
$$T:\Lambda\rightarrow C^*(H)[G]\subseteq C^{*}(H)\times_{\tilde\alpha}G$$ by $$T(h, g)=S_{h}g.$$
It is well defined since $(h, g)\in\Lambda$ implies that $S_h$ lies
in the domain of $\tilde\alpha_{g^{-1}}$. We shall verify the
conditions in Definition \ref{defn8} to show that $T$ is actually a
representation. \\
\begin{enumerate}\item[\rm (I).]Since \begin{eqnarray}(S_{h}g)(S_{h}g)^{*}(S_{h}g)&=&(S_{h}g)(\tilde\alpha_{g^{-1}}(S_{h}^{*})g^{-1})(S_{h}g)\nonumber\\
&=&S_{h}S_{h}^{*}S_{h}g=S_hg\nonumber
\end{eqnarray} we have that $T(h, g)$ is a
partial isometry.
\item[\rm (II).]If $(h_1, g_{1})(h_2, g_{2})\in \Lambda^{(2)}$, then $\textbf s(\alpha_{g^{-1}}(h_1))=\textbf t(h_2)$,
and so
$(S_{h_{1}}g_{1})(S_{h_{2}}g_{2})=S_{h_{1}}S_{\alpha_{g_{1}}(h_{2})}g_{1}g_{2}$.
Suppose that $(h_1, g_{1})(h_2, g_{2})\not\in \Lambda^{(2)}$. If
$(g_{1}, g_{2})\not\in G^{(2)}$, it follows that
$(S_{h_{1}}g_{1})(S_{h_{2}}g_{2})=0$. If $(g_{1}, g_{2})\in
G^{(2)}$, we also have $(S_{h_{1}}g_{1})(S_{h_{2}}g_{2})=0$ since
$(h_1, \alpha_{g_1}(h_2))\not\in H^{(2)}$. In sum, we have
\[T(h_{1}, g_{1})T(h_{2}, g_{2})=\left\{
\begin{array}{ll}
T((h_{1}, g_{1})(h_{2}, g_{2})) & (h_{1}, g_{1})(h_{2}, g_{2})\in\Lambda^{(2)}, \\
0 & otherwise.
\end{array}
\right.
\]
Note that the initial projection for $T(h_1, g_1)$ is $Q(h_{1},
g_{1})=\tilde\alpha_{g_{1}^{-1}}(Q_{h_{1}})g_{1}^{-1}g_{1}$ and the
final projection for $T(h_2, g_2)$ is $P(h_{2},
g_{2})=P_{h_{2}}g_{2}g_{2}^{-1}$. If $(g_{1}, g_{2})\in G^{(2)}$, we
have that
$g_{1}^{-1}g_{1}=g_{1}^{-1}g_{1}g_{2}g_{2}^{-1}=g_{2}g_{2}^{-1}=g_{2}g_{2}^{-1}g_{1}^{-1}g_{1}$,
hence \begin{eqnarray}Q(h_{1}, g_{1})P(h_{2},
g_{2})&=&\tilde\alpha_{g^{-1}}(Q_{h_{1}})P_{h_{2}}g_{1}^{-1}g_{1}g_{2}g_{2}^{-1}\nonumber\\
&=&P_{h_{2}}\tilde\alpha_{g^{-1}}(Q_{h_{1}})g_{2}g_{2}^{-1}g_{1}^{-1}g_{1}=P(h_{2}, g_{2})Q(h_{1}, g_{1}).\nonumber\\
\end{eqnarray} On the other hand, if
$(g_{1}, g_{2})\not\in G^{(2)}$, we also have $Q(h_{1},
g_{1})P(h_{2}, g_{2})=P(h_{2}, g_{2})Q(h_{1}, g_{1})=0. $\
Similarly, we can prove that initial projections $Q(h_{1}, g_{1})$
and final projections $P(h_{2}, g_{2})$ are mutually commutative.
\item[\rm (III).] When $(h_{1}, g_{1})\perp (h_{2}, g_{2})$, we
have proved in Proposition \ref{prop7} that $h_{1}\perp h_{2}$ or
$g_{1}\perp g_{2}$, which implies that $P(h_{1}, g_{1})P(h_{2},
g_{2})=0$.
\item[\rm (IV).]Finally, if $(h_{1}, g_{1})(h_{2},
g_{2})\in\Lambda^{(2)}$, we have $(\alpha_{g_{1}^{-1}}(h_{1}),
h_{2})\in H^{(2)}$ Hence
\begin{eqnarray}Q(h_{1}, g_{1})P(h_{2},
g_{2})&=&\tilde\alpha_{{g}^{-1}_{1}}(Q_{h_{1}})P_{h_{2}}g_{2}g_{2}^{-1}=Q_{\alpha_{{g}^{-1}_{1}}(h_{1})}P_{h_{2}}g_{2}g_{2}^{-1}\nonumber\\
&=&P_{h_{2}}g_{2}g_{2}^{-1}=P(h_{2}, g_{2}) \nonumber\end{eqnarray}
\end{enumerate}So we have shown that T is indeed a representation of
$\Lambda$ in $C^{*}(H)\times_{\tilde\alpha}G$,and by the universal
property, there is a unique
*-homomorphism $$\tilde{T}:C^{*}(H\times_{\alpha}G)\rightarrow
C^{*}(H)\times_{\tilde\alpha}G$$extending T, with the property $\tilde{T}(S_{(h,g)})=S_h g$.\\
\medskip
\medskip
On the other hand, we now construct a covariant representation
$(\pi, u)$ of the quasi $C^*-$dynamical system $(C^*(H), G,
\tilde\alpha)$ in $B(\mathcal {H})$, where $C^*(H\times_\alpha G)$ is
represented faithfully on $\mathcal{H}$. Define $\pi:H\rightarrow
B(\mathcal {H})$, by $$\pi(h)=S_{(h, \varphi(\textbf s(h))}.$$ We can
verify that $\pi$ defines a representation of H in $B(\mathcal
{H})$. Then by the universal property, we get a
*-homomorphism (also denoted by $\pi$) $\pi:C^*(H)\rightarrow
B(\mathcal {H})$, by $$\pi(S_h)=S_{(h, \varphi(\textbf s(h))}.$$ Define
$u:G\rightarrow Par(B(\mathcal {H}))$, by $$u(g)=\sum_{e\in
H^{(0)}\cap H^{g^{-1}}}S_{(e, g)}.\ \ (*)$$ We now verify that
$u(g)$ is well defined. For any $g_0\in G^{(0)}$, we have
$S_{(e,g_0)}$ is a projection for any $e\in H^{g_0}$, because either
$S_{(e,g_0)}$ is 0, or otherwise an idempotent and norm 1 element.
Therefore, $S^*_{(e,g)}=S_{(\alpha_ {g^{-1}}(e), g^{-1})}$ holds. In
fact, consider $Q=S_{(\alpha_ {g^{-1}}(e), g^{-1})}S_{(e, g)}$ which
is obviously a projection, then by checking the minimality concerned
in Lemma \ref{lem} (a), that is, $S_{(e, g)}=S_{(e, g)}Q$, and for
any projection $Q'$ satisfying $S=SQ'$, we have $Q=QQ'$, hence $Q$
is the initial projection for $S_{(e, g)}$. The same procedure works
for verifying the other conditions of Lemma \ref{lem} (a). It
follows that $S^*_{(e,g)}=S_{(\alpha_ {g^{-1}}(e), g^{-1})}$. We can
now verify that the right hand side of formula (*) is strongly
convergent. Since the pairs (e, g) in the above formula are mutually
disjoint, we have $\{Q_{(e,g)}\}$ and $\{P_{(e,g)}\}$ satisfy the
conditions in lemma \ref{lem} (b). (Note that $(e_1,g)$ and
$(e_2,g)$ are disjoint if $e_1\neq e_2$, hence
$S^*_{(e_1,g)}S_{(e_2,g)}=S^*_{(e_1,g)}P_{(e_1,g)}P_{(e_2,g)}S_{(e_2,g)}=0$
by (iii) of definition \ref{defn8}, which is an important result
that we have used silently). So u(g) is well-defined. One can check
that u defines a groupoid homomorphism, and $u(g_1)u(g_2)=0$ if
$(g_1,g_2)\notin G^{(2)}$. To prove that $(\pi, u)$ is a covariant
representation of $(C^*(H), G, \tilde\alpha)$, it is enough to prove
that for $a=S_h$ such that $h\in H^{g}$, we have (the following
convergence corresponds to the strong topology)
\begin{eqnarray}u(g)\pi(S_h)u(g)^*&=&S_{(\textbf t(\alpha_g(h)),g)}S_{(h,g^{-1}g)}S_{(\textbf s(h),
g^{-1})}=S_{(\alpha_g(h),gg^{-1})}\nonumber\\
&=&\pi(S_{\alpha_g(h)})=\pi(\tilde\alpha_g(S_h)),\nonumber\end{eqnarray}
and $u(g)\pi(S_h)=\pi(\tilde\alpha_g(S_h))u(g)$ since
$$\pi(S_h)=S_{(h,g^{-1}g)}=\pi(S_h)u(g)^*u(g).$$ Notice
that $\pi(S_h)u(g)=S_{(h, gg^{-1})}S_{(\textbf s(h),g)}=S_{(h,g)}\in
C^{*}(H\times_{\alpha}G)$ for any $S_hg\in C^*(H)[G]$, and by the
universal property, we have a
*-homomorphism
$$\tilde{T{'}}:C^{*}(H)\times_{\tilde\alpha}G\rightarrow
C^{*}(H\times_{\alpha}G). $$ $\tilde{T{'}}$ satisfies $\tilde{T{'}}(S_h
g)=S_{(h,g)}$ for any $(h,g)\in H\times_\alpha G$. It is not hard to
see that $\tilde{T}$ and $\tilde{T{'}}$ inverse each other, hence complete
the proof of this theorem.
\end{proof}
\medskip
Given a family of $C^*$-algebras\ $\{A_{\lambda}\}_{\lambda\in\Lambda}$, we
denote $\bigoplus_{\lambda}^{c_{0}}A_{\lambda}$ for those $
(a_{\lambda})\in \Pi_{\lambda\in\Lambda}A_{\lambda}$ such that for
each $\varepsilon>0$, there exists a finite subset F of $\Lambda$
for which $\norm{a_{\lambda}}<\varepsilon$ if
$\lambda\in\Lambda\setminus F$. Also
$\bigoplus_{\lambda}^{c_{00}}A_{\lambda}$ denote those
$({a_{\lambda}})$ with finite support.
\medskip
\begin{thm}
Assume that $\alpha$ is a regular action of a groupoid G on a small
category H. Let $G^{(0)}$ be the unit space of $G$, and G is
isomorphic to the groupoid given by the group bundle $(G^{(0)}, R,
\{G_{\xi}\}_{\xi\in(G^{(0)}/R)})$ over the equivalence classes of
the equivalence relations R on $G^{(0)}$, as in Example \ref{eg2}.
Let $H_{\xi}\triangleq \{h\in H:\varphi(s(h))\in G_{\xi}\}$, and
$\alpha_\xi$ be the action of $G_\xi$ on $H_\xi$ inherited from
$\alpha$. Then $C^{*}(H)\times_{\tilde\alpha}G\cong
\bigoplus_{\xi}^{c_{0}}C^{*}(H_{\xi})\times_{\tilde\alpha_\xi}G_{\xi}$.
\end{thm}
\begin{proof}
Assume that $C^*(H_\xi)$ is generated by a family of partial
isometries $T_h$, and $C^*(H)$ is generated by $S_h$. We define
$\pi: H_\xi \rightarrow C^*(H),$ by $$\pi(h)=S_h.$$ Then by the
universal property, we get a *-homomorphism $\tilde\pi: C^*(H_\xi)
\rightarrow C^*(H),$ by $$ \pi(T_h)=S_h.$$ On the other hand, since
the action is regular, we construct a
*-homomorphism from $C^*(H)$ to $C^*(H_\xi)$ which is the left inverse of $\tilde\pi$, as we did in Proposition \ref{eg9}. Thus,
we have that $\tilde\pi$ is isometry, and we reasonably view
$C^*(H_\xi)$ to be a *-subalgebra of $C^*(H)$. Consider the
decomposition operator
$$\mathcal {L}:C^{*}(H)[G]\rightarrow
\bigoplus_{\xi}^{c_{00}}C^{*}(H_{\xi})[G_{\xi}].
$$Firstly, noticing that the algebraic operations on each $C^*(H_{\xi})[G_{\xi}]$ is the restriction of the operations on $C^*(H)[G]$, one can check that $\mathcal {L}$ a *-isomorphism between these two
*-algebras. Secondly, $\mathcal {L}$ is $l^1$ contractive since the $c_{00}$ norm of any element in $\bigoplus_{\xi}^{c_{00}}C^{*}(H_{\xi})[G_{\xi}]$
is always bounded by $l^1$ norm of the corresponding element in $C^*(H) [G]$. By universal property, we get a *-homomorphism $\tilde{\mathcal {L}}:C^{*}(H)\times_{\tilde\alpha}G\rightarrow
\bigoplus_{\xi}^{c_{0}}C^{*}(H_{\xi})\times_{\tilde\alpha_\xi}G_{\xi}$ extending $\mathcal {L}$. Combining with density of the two *-algebras $C^{*}(H) [G]$ and $\bigoplus_{\xi}^{c_{00}}C^{*}
(H_{\xi})[G_{\xi}]$, we have $\tilde{\mathcal {L}}$ is a *-isomorphism, hence complete the proof.
\end{proof}
Each $G_{\xi}$ above is a transitive
subgroupoid of G in the term of \cite[1.1]{REN}, that is the map (r,
d) from $G_{\xi}$ to $G_{\xi}^{(0)}\times G_{\xi}^{(0)}$ is onto;
equivalently the orbit space $G_{\xi}^{(0)}/G_{\xi}$ is single,
whence $G(x)\triangleq t^{-1}(x)\cap s^{-1}(x)$ are isomorphic for
all $x\in\xi$. It is obvious that the each action $\alpha_\xi$ is
regular. As a result, in order to study $C^*(H\times_\alpha G)$, one
can study a collection of the crossed products of regular transitive
groupoid actions on small categories.
\bigskip
|
1,116,691,497,741 | arxiv | \section{Introduction}
\subsection{Setting problem}
In this paper, we consider the initial-boundary problem of semilinear wave equation
\begin{align}\label{Wave-problem}
\begin{cases}
u_{tt} - \Delta u = f(u), \,\,\, & x \in \Omega,\,\, t>0, \\
u(x,0) =u_0(x),\,\,\, u_t(x,0) = u_1(x) &x \in \overline{\Omega}, \\
u(x,t)=0\,\,\, & x \in \partial \Omega, t\geq 0,
\end{cases}
\end{align}
where $\Omega$ is a bounded domain in $\mathbb{R}^n$. A total energy of problem \eqref{Wave-problem} is
\begin{align*}
E(t)&=\frac{1}{2} \int_{\Omega} u^2_t dx +\frac{1}{2} \int_{\Omega} |\nabla u|^2 dx - \int_{\Omega} [F(u) - \sigma] dx,
\end{align*}
where $F(u)=\int_0^uf(s)ds$ and $\sigma$ is some positive constant. Then a law of conservation of energy states that
\begin{equation}
E(t)=E(0):=\frac{1}{2} \int_{\Omega} u^2_1 dx +\frac{1}{2} \int_{\Omega} |\nabla u_0|^2 dx - \int_{\Omega} [F(u_0) - \sigma] dx.
\end{equation}
A simple computation gives
\begin{align*}
\frac{dE(t)}{dt} &=\frac{d}{dt}\left[ \frac{1}{2} \int_{\Omega} u^2_t dx + \frac{1}{2} \int_{\Omega} |\nabla u|^2 dx - \int_{\Omega} [F(u) - \sigma] dx \right]\\
& = \int_{\Omega}u_t u_{tt}dx + \int_{\Omega} \nabla u \cdot \nabla u_t dx - \int_{\Omega} f(u)u_tdx \\
& = \int_{\Omega} [ \Delta u + f(u) ]u_t dx - \int_{\Omega} \Delta u u_t dx - \int_{\Omega} f(u)u_tdx\\
& = 0.
\end{align*}
In this paper we use the following notations:
\begin{align*}
||u||^2 := || u ||^2_{L^2(\Omega)} = \int_{\Omega } |u(x,t)|^2 dx, \,\, \text{ and } \,\,
||u||^{\gamma}:= || u ||^{\gamma}_{L^{\gamma}(\Omega)} =\int_{\Omega } |u(x,t)|^{\gamma} dx.
\end{align*}
\subsection{A new condition on nonlinearity}
Let $f(u)$ satisfy the following assumptions:
\begin{itemize}
\item[(i)] $f \in C^1$, $f(0)=f'(0)=0$.
\item[(ii)] (a) $f$ is monotone and convex for $u>0$, concave for $u<0$; \\
(b) $f$ is convex for $-\infty < u < + \infty$.
\item[(iii)] Let $2<\alpha \leq \gamma <2n/(n-2)$, then
\begin{equation}
|uf(u)|\leq \gamma| F(u)-\sigma|,
\end{equation}
and
\begin{equation}\label{cont_1}
\alpha F(u)\leq uf(u)+\beta u^2 + \alpha \sigma,
\end{equation}
where $F(u) = \int_0^u f(s) ds $, $\sigma>0$, and $\beta < \frac{\lambda_1(\alpha -2)}{2}$ with $\lambda_1$ is the first eigenvalue of Laplacian.
In the case (b), \eqref{cont_1} holds for $u>0$.
\end{itemize}
For our convenience, these assumptions on the nonlinearity $f(u)$ denoted as $(H)$. Also, we have two cases such $(H)-(a)$ and $(H)-(b)$, respectively. For example, $f(u)=|u|^{p-1}u$ is the case $(H)-(a)$ and $f(u)=u^p$ is the case $(H)-(b)$.
From the assumption $(H)$ one leads the lemma that describes the properties of nonlinearity $f(u)$.
\begin{lem}
Let $f(u)$ satisfy $(H)$. There are two cases:\\
$(H)-(a)$ $f$ is monotone and is convex for $u>0$, concave for $u<0$, then
\begin{itemize}
\item[(i)] $f(u)\geq 0$ for $u>0$, and $f(u)\leq 0$ for $u<0$.
\item[(ii)] $uf(u)\geq 0$ and $F(u)\geq 0$ for $-\infty <u<+\infty$.
\item[(iii)] $F(u)$ is increasing on $0<u<+\infty$ and decreasing on $-\infty < u <0$.
\end{itemize}
$(H)-(b)$ $f$ is convex for $-\infty < u < + \infty$, then
\begin{itemize}
\item[(iv)] $f(u)\geq 0$ for $-\infty < u < +\infty$.
\item[(v)] $uf(u)\geq 0$ and $F(u)\geq 0$ for $u>0$; $uf(u)\leq 0$ and $F(u)\leq 0$ for $u<0$.
\item[(vi)] $F(u)$ is increasing on $-\infty < u < +\infty$.
\end{itemize}
\end{lem}
Here we omit the proof.
Note that new condition \eqref{cont_1} on nonlinearity $f(u)$ includes the following cases:
\begin{align}
(2+\epsilon) F(u) &\leq uf(u), \label{eq-0.1}\\
(2+\epsilon) F(u) &\leq uf(u) + \sigma\label{eq-0.2}, \\
(2+\epsilon) F(u) &\leq uf(u) + \beta u^2 + \sigma \label{eq-0.3},
\end{align}
where $0<\beta \leq \epsilon\lambda_1 /2$, $\sigma>0$ and $F(u)=\int_0^u f(s)ds$.
Condition \eqref{eq-0.1} is used by Philippin and Proytcheva \cite{PP-06} to prove the blow-up solutions for the semilinear heat equation. It is a special case of an abstract condition by Levine and Payne \cite{LP2-74}. Later, Bandle and Brunner \cite{BB-98} relaxed this condition with \eqref{eq-0.2}. Recently, a new condition \eqref{eq-0.3} was introduced by Chung and Choi \cite{Chung-Choi} for blow-up solution of the semilinear heat equation. Moreover, blow-up solutions of nonlinear porous medium equations and pseudo-parabolic equations with the new condition are studied in \cite{ST-21, RST-21}.
Note that condition \eqref{eq-0.3} extends \eqref{eq-0.1} and \eqref{eq-0.2}. The difference between \eqref{eq-0.3} and \eqref{eq-0.2} is that \eqref{eq-0.3} depends on the domain due to term $\beta u^2$. Constant $\beta$ depends on the domain through the first eigenvalue $\lambda_1$. If the first eigenvalue $\lambda_1$ is arbitrary small then \eqref{eq-0.3} gets closer to \eqref{eq-0.2}.
Condition \eqref{eq-0.3} is equivalent to
\begin{equation*}
\frac{d}{du}\left( \frac{F(u)}{u^{2+\epsilon}} - \left(\frac{\sigma}{2+\epsilon}\right) \frac{1}{u^{2+\epsilon}} - \frac{\beta}{\epsilon} \frac{1}{u^{\epsilon}} \right) \geq 0, \,\,\, u>0.
\end{equation*}
Then it is easy to see that
\begin{align*}
&\eqref{eq-0.1} \,\,\, \text{ holds if and only if } \,\, F(u)=u^{2+\epsilon}h_1(u),\\
&\eqref{eq-0.2} \,\,\, \text{ holds if and only if } \,\, F(u)=u^{2+\epsilon}h_2(u)+b,\\
&\eqref{eq-0.3} \,\,\, \text{ holds if and only if } \,\, F(u)=u^{2+\epsilon}h_3(u)+au^2+b,
\end{align*}
for some positive constants $\epsilon$, $b$, and $a<\lambda_1/2$, where $h_1$, $h_2$, and $h_3$ are nondecreasing functions on $(0,+\infty)$. Note that the constants $\epsilon$, $b$, and $a$ shall be different in each case.
\begin{lem}\label{lem}
Let $f$ be a function satisfying $\eqref{eq-0.3}$ and $uf(u)\geq \lambda u^2$ for $u>0$ along with $\lambda > \lambda_1$. Then condition $\eqref{eq-0.3}$ implies that there exists $m>0$ such that $h_3(u)>0$ for $u>m$. In this case, it is possible to find $\mu >0$ such that $uf(u)\geq \mu u^{2+\epsilon}$ for $u\geq m$. It means that conditions \eqref{eq-0.2} and \eqref{eq-0.3} are equivalent.
\end{lem}
\begin{proof}[Proof of Lemma \ref{lem}]
Recall the facts $0<a<\lambda_1/2$ and $\lambda > \lambda_1$ that allows to obtain
\begin{equation*}
F(u) \geq \frac{\lambda}{2}u^2 > \frac{\lambda_1}{2}u^2,
\end{equation*}
then
\begin{equation*}
u^{2+\epsilon}h_3(u) = F(u) -au^2 -b \geq \frac{\lambda -\lambda_1}{2} u^2 -b,
\end{equation*}
which goes to $+\infty$ as $u\rightarrow +\infty$. So there is $m>1$ such that $h_3(m)>0$, which implies that
\begin{equation*}
F(u) \geq u^{2+\epsilon}h_3(u), \,\,\, u\geq m.
\end{equation*}
Applying it into condition \eqref{eq-0.3}, we get
\begin{align*}
u^{2+\epsilon} h_3(m) &\leq uf(u) + \beta u^2 + \sigma \\
& \leq (1+ \epsilon/2)uf(u) + \sigma, \,\,\, u\geq m>1.
\end{align*}
That gives
\begin{equation*}
uf(u) \geq \mu u^{2+\epsilon}, \,\, u\geq m>1,
\end{equation*}
for some constant $\mu>0$.
Next we show that \eqref{eq-0.2} and \eqref{eq-0.3} are equivalent. Since $0<\beta \leq \epsilon \lambda_1/2$ and $uf(u)\geq \lambda u^2>\lambda_1 u^2$ for $u>0$, it follows from \eqref{eq-0.3} that
\begin{equation*}
\epsilon_1 F(u) + (2+\epsilon_2)F(u) \leq uf(u) + \frac{\epsilon \lambda_1}{2} u^2 + \sigma,
\end{equation*}
where $\epsilon_1= \frac{\epsilon \lambda_1}{\lambda}>0$ and $\epsilon_2 = \epsilon -\epsilon_1>0$. Then we have
\begin{align*}
uf(u) +\sigma &\geq (2+\epsilon_2)F(u) + \epsilon_1 \int_0^u [f(s) -\lambda s]ds\\
&\geq (2+\epsilon_2)F(u),
\end{align*}
which is \eqref{eq-0.2}. This completes the proof.
\end{proof}
\subsection{Literature overview}
There are huge amount of papers devoted to problem \eqref{Wave-problem}, and it is impossible to cover everyone. We focus on a study of long-term behavior of evolution equations. Early works of this direction are by Keller \cite{Keller-57}, Sattinger \cite{Sattinger-68}, J\"orgens \cite{Jorgens-70}, Tsutsumi \cite{Tsutsumi-1, Tsutsumi-2, Tsutsumi-3}, Levine \cite{Levine73} and others, where they investigated conditions on the initial data and source term $f$ for which solutions of \eqref{Wave-problem} blow up in a finite time. In 1975, Payne-Sattinger \cite{PS-75} introduced a potential wells method to study problem \eqref{Wave-problem}, and its extensions to parabolic and abstract operator equations. The potential wells method gives a better intuitive understanding of the phenomena of instability, because we could get this intuition by considering one-dimensional mechanical analogue of problem \eqref{Wave-problem} as $\ddot{x} = -x + f(x)$. It becomes one of the most powerful tool for studying nonlinear evolution equations, (see more in the book \cite{QS-book}).
In 2002, Vitillaro extended the potential wells theory for the wave equation with an internal nonlinear source and a nonlinear boundary damping as follows
\begin{align*}
\begin{cases}
u_{tt} - \Delta u = |u|^{p-2}, \,\,\, & x \in \Omega,\,\, t>0, \\
u(x,0) =u_0(x),\,\,\, u_t(x,0) = u_1(x) &x \in \overline{\Omega}, \\
u=0\,\,\, & \text{ on } \,\, \Gamma_0, t\geq 0,\\
\frac{\partial u}{\partial \nu } = \alpha(x) |u_t|^{m-2}u_t & \text{ on } \,\, \Gamma_1, t\geq 0,
\end{cases}
\end{align*}
where $\Omega \subset \mathbb{R}^n$ is a regular bounded domain with smooth boundary given as the disjoint union $\partial \Omega= \Gamma_0 \cup \Gamma_1$.
In 2006, Liu-Zhao \cite{LZh-06} generalized and improved the results of Payne-Sattinger \cite{PS-75}, by obtaining the invariant sets and vacuum isolation of solutions, and a threshold result of global existence and nonexistence of solutions with condition $(H)-(a)$ on nonlinearity $f(u)$, (see also \cite{LZh-03} and \cite{LZh-04}). Later, Runzhang Xu \cite{Xu-10} complemented the result of Liu-Zhao \cite{LZh-06} by considering the critical initial energy $E(0)=d$. We refer other works of Runzhang Xu and his collaborators in this direction \cite{XS-13}, \cite{XZh-17}, and \cite{XLN-20}.
In 2020, Chen-Chen proved the unsolved case $(H)-(b)$ which fills important gaps regarding problem \eqref{Wave-problem}. By introducing a family of potential wells, the invariant sets and vacuum isolation of solutions were obtained. Moreover, the global nonexistence of solution and asymptotic behavior of solution were proved for semilinear parabolic equations.
In addition to the references given in detail above, the literature is rich with results on wave equations along with a damped term. Many pioneering papers such as Lions and Strauss \cite{LS-65}, Nakao and Nanbu \cite{NN-75}, as well as works by Glassey \cite{Glassey} and Ball \cite{Ball-77} are worthy of mention. Moreover, we would like to mention \cite{CCM-04, CZT-20, DSS-20, Esuivel-03, GS-06, Ikehata-96, LX-20, MRV-18, NO-93} and references therein.
\subsection{Main goal of the paper} A main aim of this work is to study a global existence and blow-up of solutions to initial-boundary problem \eqref{Wave-problem} with a new condition on a nonlinearity $f(u)$ as follows
\begin{equation*}
\alpha\int_0^u f(s)ds \leq uf(u) + \beta u^2 +\alpha \sigma,
\end{equation*}
for some positive constants $\alpha$, $\beta$, and $\sigma$, where $\beta < \frac{\lambda_1(\alpha -2)}{2}$ with $\lambda_1$ being the first eigenvalue of Laplacian. The new condition was introduced by Chung-Choi \cite{Chung-Choi} for a heat equation. By introducing a family of potential wells, we are able to establish the invariant sets, vacuum isolation of solutions, and a threshold results of global existence and nonexistence of solutions with the new condition on a source term. Moreover, critical initial conditions $E(0)=d$ are discussed for global existence and blow-up solutions.
\subsection{Plan of the paper}
In Section \ref{Sec2}, we introduce the potential wells theory along with its properties. In Section \ref{Sec3}, we use the family of potential wells to obtain invariant sets and vacuum isolating solutions of problem \eqref{Wave-problem}. In Section \ref{Sec4} and \ref{Sec5}, we prove the global existence and nonexistence of solutions for problem \eqref{Wave-problem} with initial conditions $E(0)<d$ and $E(0)=d$. In Section \ref{Sec6}, we provide a brief conclusion of results.
\section{Potential Wells Theory}\label{Sec2}
In this section, we introduce a family of potential wells. First, we would like to illustrate idea of potential wells by giving a simple example such as one-dimensional mechanical analogue of problem \eqref{Wave-problem}
\begin{equation}\label{eq-1D}
\ddot{x} = -x + f(x),
\end{equation}
where $x$ is a real number. This example was given by Payne-Sattinger \cite{PS-75} and it describes a mechanical system with one degree of freedom, while problem \eqref{Wave-problem} may be considered as a system with an infinite number of degree of freedom.
The potential energy of equation \eqref{eq-1D} is defined by
\begin{equation*}
V(x) = \frac{x^2}{2} - F(x),
\end{equation*}
where $F(x) = \int_0^x f(s)ds$. Imagine that potential energy $V(x)$ has the qualitative shape with a local minimum at $x=0$ and a local maximum at $x = x_*$.
The set
\begin{equation*}
W = \{ x: V(x)<d, \,\, x < x_* \}
\end{equation*}
describes a potential well. The total energy of equation \eqref{eq-1D} is
\begin{equation*}
E(t) = \frac{\dot{x}^2}{2} + \frac{x^2}{2} - F(x)= \frac{\dot{x}^2}{2} + V(x).
\end{equation*}
As we know that the total energy of \eqref{eq-1D} is conserved under the motion. Then by the conservation of energy we have
\begin{itemize}
\item If $E(0)<d$ and $x(0)\in W$, then $x(t)\in W$ for all $t>0$.
\item If $E(0)<d $ and $x(0)>x_*$, then $x(t) > x_* $ for all $t>0$.
\end{itemize}
In order to cross into potential well $W$, the total energy would have to be greater than the depth of potential well $d$.
This idea can be extended to the infinite dimensional case \eqref{Wave-problem}. The \textbf{potential energy} associated with \eqref{Wave-problem} is the functional
\begin{align*}
J(u) & = \frac{1}{2} || \nabla u ||^2 - \int_{\Omega} [F(u)-\sigma] dx.
\end{align*}
We define the \textbf{Nehari functional} by
\begin{equation*}
I(u) = || \nabla u ||^2 - \int_{\Omega } uf(u)dx, \,\,\, u \in H_0^1(\Omega),
\end{equation*}
where $I(u)$ can be considered as the extension of point $x_*$ in one dimension analogue.
The depth of potential wells is defined by
\begin{equation*}
d := \inf \{J(u): u \in H_0^1(\Omega)\backslash \{ 0 \}, I(u)=0\}.
\end{equation*}
Also the quantity $d$ can be understood as a mountain pass energy. For instance, there is the valley. If we pour water into the valley containing a local minimum at the origin, as the water level rises in the direction of the potential energy and passes at a local maximum $d$.
Note that in all proofs, we should treat slightly differently the cases $(a)$ and $(b)$ for assumption $(H)$ for nonlinearity $f(u)$.
\begin{lem}\label{lem-1}
Let $f(u)$ satisfy $(H)$, then we have
\begin{itemize}
\item[(a)] $uf'(u) - f(u) \geq 0$ holds in the case $(H)-(a)$ with equality holding only for $u=0$, and in the case $(H)-(b)$ for $u \geq 0$.
\item[(b)] $|F(u)-\sigma|\leq A |u|^{\gamma}$ for some $A>0$ and all $u\in R$.
\item[(c)] $F(u)-\sigma \geq B |u|^{\lambda}$ for $|u|\geq 1$ with $B = \frac{F(\widetilde{u})}{\widetilde{u}^{\lambda}}>0$ and $\lambda = \frac{\alpha}{1+\beta}>2$.
\end{itemize}
\end{lem}
\begin{proof}[Proof of Lemma \ref{lem-1}]
$(a)$ If $f(u)$ satisfies $(H)-(a)$, then the proof is based on the second variation of functional $J$ at a critical point $w$. Let
\begin{equation*}
i(\tau):= J(w+\tau v) = \frac{1}{2} \int_{\Omega } |\nabla (w+ \tau v)|^2 dx - \int_{\Omega }F(w+\tau v)dx,
\end{equation*}
where $v \in C_0^{\infty}(\Omega)$. The second variation of $J$ is
\begin{equation*}
i''(0) = \int_{\Omega } |\nabla v|^2 dx - \int_{\Omega } f'(w)v^2 dx.
\end{equation*}
If $w$ is a local minimum of functional $J$, then $i''(0)\geq 0$. Then $w=0$ gives $i''(0) = \int_{\Omega } |\nabla v|^2dx\geq 0$. On other hand, $w$ is a non-trivial critical point, then $i''(0)\leq 0$. Let $w$ be an admissible test function, then we compute
\begin{align*}
i''(0) &= \int_{\Omega } |\nabla w|^2 dx - \int_{\Omega } f'(w)w^2 dx\\
& = - \int_{\Omega } w(\Delta w + f'(w)w)dx \\
& = -\int_{\Omega } w( f'(w)w - f(w))dx,
\end{align*}
it follows that $ w( f'(w)w - f(w)) \geq 0$.
In the case $(H)-(b)$ $, $ $f$ is convex for $-\infty < u < + \infty$, this gives $f(0)\geq f(u)+f'(u)(0-u)$ that gives $uf'(u)\geq f(u)$.
$(b)$ From $(H)$, the growth condition $|uf(u)|\leq \gamma |F(u)-\sigma|$ implies
\begin{align*}
\frac{|f(u)|}{|F(u)-\sigma|} \leq \frac{\gamma}{|u|} \,\,\, \text{ for } \,\,\, u \neq 0.
\end{align*}
Taking any known fixed $\widetilde{u} \neq 0$, we integrate above inequality from $\widetilde{u}$ to $u$, which gives
\begin{align*}
\int_{\widetilde{u}}^u \frac{|f(s)|}{|F(s)-\sigma|} ds &\leq \int_{\widetilde{u}}^u \frac{\gamma}{|s|}ds,
\end{align*}
and by using $F'(u)=f(u)$, then we have
\begin{align*}
\ln \frac{|F(u)-\sigma|}{|F(\widetilde{u})-\sigma|}\leq \ln \frac{|u|^{\gamma}}{|\widetilde{u}|^{\gamma}},
\end{align*}
that is $|F(u)-\sigma| \leq A |u|^{\gamma}$ where $A = \frac{|F(\widetilde{u})-\sigma|}{|\widetilde{u}|^{\gamma}}$.
$(c)$ From $(H)$ we have $\alpha F(u) \leq uf(u) + \beta u^2 +\alpha \sigma$. For $|u|\geq 1$, we write
\begin{align*}
\frac{\alpha}{|u|} \leq \frac{|f(u)|}{F(u)-\sigma} + \frac{\beta |u|}{F(u)-\sigma} \leq (1+\beta) \frac{|f(u)|}{F(u)-\sigma},
\end{align*}
then we get
\begin{equation}
\frac{|f(u)|}{F(u)-\sigma} \geq \frac{\lambda}{|u|} ,
\end{equation}
where $\lambda= \frac{\alpha}{1+\beta}$. As the previous approach on $(b)$, we get $F(u)-\sigma \geq B |u|^{\lambda} $ where $B = \frac{F(\widetilde{u})}{\widetilde{u}^{\lambda}}$.
\end{proof}
\begin{rem}
Note that from Lemma \ref{lem-1} and condition $(H)$ we have the following properties:
\begin{itemize}
\item $|f(u)|\leq \gamma A|u|^{\gamma-1}$ for all $u \in R$.
\item $uf(u) + \beta u^2 \geq \alpha B|u|^{\lambda}$ for $u\geq 1$.
\end{itemize}
\end{rem}
\begin{lem}\label{lem-2}
Let $f(u)$ satisfy $(H)$. Let $u\in H_0^1(\Omega)$, $||\nabla u ||\neq 0$, and
\begin{align}
\phi(\epsilon) = \frac{1}{\epsilon} \int_{\Omega } u f(\epsilon u) dx.
\end{align}
Then we have the following properties of function $\phi$:
\begin{itemize}
\item $\phi(\epsilon)$ is increasing on $0< \epsilon < \infty$.
\item $\lim_{\epsilon \rightarrow 0}\phi(\epsilon)=0$ and $\lim_{\epsilon \rightarrow +\infty}\phi(\epsilon)=+\infty$.
\end{itemize}
In the case $(H)-(b)$, there is additional assumption $u(x)> 0$.
\end{lem}
\begin{proof}[Proof of Lemma \ref{lem-2}]
Let
\begin{align*}
\frac{d\phi(\epsilon)}{d\epsilon} &= \frac{1}{\epsilon^2} \int_{\Omega } [\epsilon u^2 f'(\epsilon u) - uf(\epsilon u)] dx \\
&= \frac{1}{\epsilon^3} \int_{\Omega } \epsilon u[\epsilon u f'(\epsilon u) - f(\epsilon u)] dx >0.
\end{align*}
By $(a)$ in Lemma \ref{lem-1}, we see that $\phi(\epsilon)$ is increasing on $0<\epsilon < \infty$.
The property $\lim_{\epsilon \rightarrow 0}\phi(\epsilon)=0$ comes from the fact that
\begin{align*}
|\phi (\epsilon)| \leq \frac{1}{\epsilon^2} \int_{\Omega } |\epsilon u f(\epsilon u)|dx \leq \frac{\gamma A}{\epsilon^2} \int_{\Omega } |\epsilon u|^{\gamma}dx = \gamma \epsilon^{\gamma -2} A || u ||^{\gamma}_{\gamma}.
\end{align*}
The property $\lim_{\epsilon \rightarrow +\infty}\phi(\epsilon)=+\infty$ can be shown by using the fact $uf(u)\geq 0$ and $(c)$ in Lemma \ref{lem-1} as follows
\begin{align*}
\phi(\epsilon) &= \frac{1}{\epsilon^2}\int_{\Omega } \epsilon u f(\epsilon u) dx \geq \frac{1}{\epsilon^2}\int_{\Omega_{\epsilon} } \epsilon u f(\epsilon u) dx\\
& \geq \frac{1}{\epsilon^2}\int_{\Omega_{\epsilon} } [\alpha F(\epsilon u) - \beta \epsilon^2 u^2 - \alpha \sigma ]dx \\
& \geq \alpha B \epsilon^{\lambda -2} \int_{\Omega_{\epsilon} } |u|^{\lambda} dx - \beta \int_{\Omega_{\epsilon} } u^2 dx,
\end{align*}
where $\Omega_{\epsilon} = \{ x \in \Omega \,\, | \,\, u(x)\geq \frac{1}{\epsilon} \}$ and
\begin{equation*}
\lim_{\epsilon \rightarrow + \infty} \int_{\Omega_{\epsilon} } |u|^{\lambda} dx = || u ||^{\lambda}_{\lambda} >0.
\end{equation*}
This completes the proof.
\end{proof}
When nonlinearity term $f(u)$ satisfies condition $(H)$, then all non-trivial critical points are a priori unstable equilibria for semilinear wave equation \eqref{Wave-problem}. The next lemma will provide the local minimum of $J$, a critical point $\epsilon^*$ and positiveness of a depth of potential well.
\begin{lem}\label{lem-3} Let $f(u)$ satisfy $(H)$, $u\in H_0^1(\Omega)$, and $||\nabla u ||\neq 0$. Then
\begin{itemize}
\item[(i)] $ \lim_{\epsilon \rightarrow 0}J(\epsilon u)=0$.
\item[(ii)] $ \frac{d}{d\epsilon} J(\epsilon u) = \frac{1}{\epsilon} I(\epsilon u)$.\\
In addition, we assume $u(x)>0$ in the case $(H)-(b)$.
\item[(iii)] $ \lim_{\epsilon \rightarrow + \infty}J(\epsilon u)= -\infty$.
\item[(iv)] There exists a unique $\epsilon^*=\epsilon^*(u)>0$ such that
\begin{equation}
\frac{d}{d\epsilon} J(\epsilon u)|_{\epsilon=\epsilon^*} = 0.
\end{equation}
\item[(v)] $J(\epsilon u)$ is increasing on $0 \leq \epsilon \leq \epsilon^*$, decreasing on $\epsilon^* \leq \epsilon < \infty$ and takes the maximum at $\epsilon = \epsilon^*$.
\item[(vi)] $I(\epsilon u)>0$ for $0<\epsilon < \epsilon^*$, $I(\epsilon u)<0$ for $\epsilon^*<\epsilon < \infty$ and $I(\epsilon^*u)=0$.
\end{itemize}
\end{lem}
\begin{proof}[Proof of Lemma \ref{lem-3}]
The proof of (i) follows from $|F(u)-\sigma|\leq A |u|^{\gamma}$ and
\begin{equation*}
J(\epsilon u) = \frac{\epsilon^2}{2} || \nabla u||^2 - \int_{\Omega } [F(\epsilon u) -\sigma ]dx.
\end{equation*}
The proof of (ii) follows from
\begin{equation*}
\frac{d}{d\epsilon} J(\epsilon u) = \epsilon || \nabla u||^2 - \int_{\Omega } uf(\epsilon u)dx = \frac{1}{\epsilon}I(\epsilon u).
\end{equation*}
First, we consider the case (iii) for $(H)-(a)$. Let
\begin{align*}
J(\epsilon u) &= \frac{\epsilon^2}{2} || \nabla u||^2 - \int_{\Omega } [F(\epsilon u) -\sigma] dx\\
&\leq \frac{\epsilon^2}{2} || \nabla u||^2 - B \epsilon^{\lambda} \int_{\Omega_{\epsilon} } |u|^{\lambda}dx.
\end{align*}
Since $\lambda >2$, $J(\epsilon u)$ goes to $-\infty$ for $\epsilon \rightarrow +\infty$. This guaranties the existence of a critical point $\epsilon^*$. The case $(H)-(b)$ can be proved in the same way with additional assumption $u>0$.
The case (iv), the uniqueness of critical point, can be proved by supposing that there two roots of equation $\frac{dJ(\epsilon u)}{d\epsilon}=0$ as $\epsilon_1<\epsilon_2$. Then we have
\begin{equation*}
\epsilon_1 || \nabla u||^2 - \int_{\Omega } uf(\epsilon_1 u)dx =0,
\end{equation*}
and
\begin{equation*}
\epsilon_2 || \nabla u||^2 - \int_{\Omega } uf(\epsilon_2 u)dx =0.
\end{equation*}
Elimination of $|| \nabla u||^2 $ from above equations gives
\begin{equation*}
\int_{\Omega } u \left[\frac{f(\epsilon_2 u)}{\epsilon_2}- \frac{f(\epsilon_1 u)}{\epsilon_1} \right]dx = 0.
\end{equation*}
This expression can be rewritten by the following substitution $w=\epsilon_1 u$ and $\epsilon = \epsilon_2/\epsilon_1>1$ as
\begin{equation}\label{eq-2.4}
\phi(\epsilon)-\int_{\Omega } wf(w)=0.
\end{equation}
It is easy to see from Lemma \ref{lem-2} that equation \eqref{eq-2.4} does not hold.
The case (v) and (vi) follow from Lemma \ref{lem-2} and
\begin{equation*}
\frac{d}{d\epsilon} J(\epsilon u) = \frac{1}{\epsilon} I(\epsilon u)= \epsilon ( || \nabla u||^2 - \phi(\epsilon)).
\end{equation*}
The proof is completed.
\end{proof}
\begin{rem} In this remark, we would like to show how the notation of $I(u)$ is adopted.
Note that for a unique $\epsilon^*=\epsilon^*(u)>0$ we have
\begin{equation*}
\frac{d}{d\epsilon} J(\epsilon u)|_{\epsilon=\epsilon^*} = 0,
\end{equation*}
and $J(\epsilon^*u)$ can be considered as the highest level attained when leaving the {\em potential well} $W$ along a ray in the direction $u$. Now the depth $d$ of potential well is defined by
\begin{equation*}
d = \inf_{u\neq 0} J(\epsilon^*u).
\end{equation*}
Then $u$ can be normalized so that $\epsilon^*=1$, which gives
\begin{equation*}
I(u)\equiv \frac{d}{d\epsilon^*} J(\epsilon^* u)|_{\epsilon^*=1} = || \nabla u||^2 - \int_{\Omega } uf(u)dx.
\end{equation*}
Then we have
\begin{equation*}
d := \inf \{J(u): u \in H_0^1(\Omega)\backslash \{ 0 \}, I(u)=0\}.
\end{equation*}
\end{rem}
In order to introduce the family of potential wells $\{W_{\delta}\}$ and $\{V_{\delta}\}$, we define
\begin{align*}
I_{\delta}(u) = \delta || \nabla u||^2 - \int_{\Omega } uf(u)dx, \,\,\, \delta >0,
\end{align*}
and
\begin{equation*}
d(\delta) := \inf \{J(u): u \in H_0^1(\Omega)\backslash \{ 0 \}, I_{\delta}(u)=0\}.
\end{equation*}
The following lemmas will describe how space $H_0^1(\Omega)$ is divided into two parts and the properties of inside and outside parts are discussed. For a graphical illustration, we will refer to Figure 1 in Liu-Zhao's paper \cite{LZh-06}.
\begin{lem}\label{lem-4}
Let $f(u)$ satisfy $(H)$. Suppose that $u \in H_0^1(\Omega)$ and
\begin{align*}
r(\delta) = \left( \frac{\delta}{a C^{\gamma}_*} \right)^{\frac{1}{\gamma-2}} \,\,\, \text{and} \,\,\, a = \sup \frac{u f(u)}{|u|^{\gamma}},
\end{align*}
where $C_*$ is the embedding constant form $H_0^1(\Omega)$ into $L^{\gamma}(\Omega)$. Then we have
\begin{itemize}
\item[(i)] If $0<|| \nabla u || < r(\delta)$, then $I_{\delta}(u)>0$. In the case $\delta =1$, if $0< || \nabla u || <r(1)$, then $I(u)>0$.
\item[(ii)] If $I_{\delta}(u)<0$, then $|| \nabla u || >r(\delta)$. In the case $\delta =1$, if $I(u)<0$, then $|| \nabla u || >r(1)$.
\item[(iii)] If $I_{\delta}(u)=0$, then $|| \nabla u || \geq r(\delta)$. In the case $\delta =1$, if $I(u)=0$, then $|| \nabla u || \geq r(1)$.
\end{itemize}
\end{lem}
Note that $(i)$ says that if $||\nabla u||$ is inside a ball of radius $r(\delta)$, then one lies inside part of space $H_0^1(\Omega)$. $(ii)$ says that if one lies outside part of space $H_0^1(\Omega)$ then $||\nabla u||$ is outside a ball of radius $r(\delta)$. $(iii)$ says that one lies on the boundary then $||\nabla u||$ is on surface or outside of a ball with radius $r(\delta)$.
\begin{proof}[Proof of Lemma \ref{lem-4}]
(i) From $0<|| \nabla u ||<r{(\delta)}$ we have
\begin{align*}
\int_{\Omega } u f(u) dx &\leq \int_{\Omega } |u f(u)| dx \leq a \int_{\Omega } |u|^{\gamma} dx = a || u||_{\gamma}^{\gamma}\\
&\leq a C_*^{\gamma} || \nabla u||^{\gamma-2}|| \nabla u||^{2}< \delta || \nabla u||^2,
\end{align*}
which gives $I_{\delta}(u)>0$.
(ii) From $I_{\delta}(u)<0$ we get
\begin{equation*}
\delta || \nabla u ||^2 < \int_{\Omega } u f(u) dx \leq a || u ||^{\gamma}_{\gamma} \leq aC_*^{\gamma} || \nabla u||^{\gamma -2} || \nabla u||^{2},
\end{equation*}
which implies $||\nabla u || >r(\delta)$.
(iii) From $I_{\delta}(u)=0$ we obtain
\begin{equation*}
\delta || \nabla u ||^2 \leq \int_{\Omega } u f(u) dx \leq a || u ||^{\gamma}_{\gamma} \leq aC_*^{\gamma} || \nabla u||^{\gamma -2} || \nabla u||^{2},
\end{equation*}
which implies $||\nabla u || \geq r(\delta)$.
\end{proof}
Next lemma describes the properties of $d(\delta)$ for a family of potential wells.
\begin{lem}\label{lem-5}
Let $f(u)$ satisfy $(H)$. Then
\begin{itemize}
\item[(i)] $d(\delta)>a(\delta)r^2(\delta)$ for $0<\delta< \frac{\alpha}{2} -\frac{\beta}{\lambda_1}$, where $a(\delta)= \frac{1}{2} - \frac{\delta}{\alpha} - \frac{\beta}{\lambda_1\alpha}$.
\item[(ii)] $\lim_{\delta \rightarrow 0}d(\delta)=0$ and there exists a unique $b$, $\frac{\alpha}{2} - \frac{\beta}{\lambda_1} \leq b \leq \frac{\gamma }{2}$ such that $d(b)=0$ and $d(\delta)>0$ for $0<\delta<b$.
\item[(iii)] $d(\delta)$ is strictly increasing on $0\leq \delta \leq 1$, strictly decreasing on $1\leq \delta \leq \infty$ and takes the maximum $d=d(1)$ at $\delta =1$.
\end{itemize}
\end{lem}
\begin{proof}[Proof of Lemma \ref{lem-5}]
We begin the proof by considering case $(H)-(a)$.
(i) If $I_{\delta}(u) =0$ and $|| \nabla u||\neq 0$, then we have $|| \nabla u|| \geq r(\delta)$ from Lemma \ref{lem-4}. By applying $I_{\delta}(u)=0$, the Poincar\'e inequality, and Lemma \ref{lem-4}, we have
\begin{align*}
J(u) &= \frac{1}{2} || \nabla u ||^2 - \int_{\Omega } [F(u)-\sigma] dx\\
&\geq \frac{1}{2} || \nabla u ||^2 - \frac{1}{\alpha} \int_{\Omega }uf(u)dx- \frac{\beta}{\alpha} \int_{\Omega }u^2dx \\
& \geq \left[\frac{1}{2} - \frac{\delta}{\alpha} - \frac{\beta}{\lambda_1\alpha} \right]|| \nabla u ||^2\\
& \geq a(\delta) r^2(\delta),
\end{align*}
where $0<\delta< \frac{\alpha}{2} -\frac{\beta}{\lambda_1}$. For the case $(H)-(b)$, the proof is same with additional condition $u>0$.
$(ii)$ Lemma \ref{lem-2} implies that for $u \in H_0^1(\Omega)$, $||\nabla u||\neq 0$ and any $\delta >0$ we can define a unique $\epsilon(\delta) = \phi^{-1}(\delta || \nabla u||^2)$ such that
\begin{equation}\label{eq-3}
\epsilon^2\phi(\epsilon) = \int_{\Omega } \epsilon u f(\epsilon u) dx=\delta || \nabla (\epsilon u)||^2 .
\end{equation}
Then for $I_{\delta}(\epsilon u)=0$ we get
\begin{equation*}
\lim_{\delta \rightarrow 0} \epsilon (\delta) =0 \,\, \text{ and } \,\, \lim_{\delta \rightarrow +\infty} \epsilon (\delta) = +\infty.
\end{equation*}
From Lemma \ref{lem-3} it follows that
\begin{align*}
\lim_{\delta \rightarrow 0} J(\epsilon u) = \lim_{\epsilon \rightarrow 0}J(\epsilon u)= 0 \,\, &\text{ and } \,\, \lim_{\delta \rightarrow 0} d(\delta) = 0, \\
\lim_{\delta \rightarrow +\infty} J(\epsilon u) = \lim_{\epsilon \rightarrow +\infty}J(\epsilon u)= -\infty \,\, &\text{ and } \,\, \lim_{\delta \rightarrow + \infty} d(\delta) = -\infty.
\end{align*}
Existence of $d(b)=0$ for $b\geq \alpha/2 -\lambda_1 \beta$ and $d(\delta)>0$ for $0<\delta<b$ follows from the part $(i)$ and above expressions. Upper bound of $b\leq \gamma/2$ follows from the facts $|uf(u)|\leq \gamma |F(u)-\sigma|$ and $I_{\delta}(u)=0$, then
\begin{align*}
J(u) &= \frac{1}{2} || \nabla u||^2 - \int_{\Omega } [F(u)-\sigma]dx \\
& \geq \frac{1}{2} || \nabla u||^2 - \gamma^{-1} \int_{\Omega } uf(u)dx\\
& = \left( \frac{1}{2} - \frac{\delta}{\gamma}\right) || \nabla u ||^2 < 0, \,\,\, \text{ for } \,\,\, \delta > \gamma/2.
\end{align*}
In the case $(H)-(b)$, by assumption $u(x)> 0$ we can prove in the same way.
For the case $(iii)$, it is enough to show that $d(\delta')<d(\delta'')$ for any $0<\delta'<\delta''<1$ or $1<\delta''<\delta' <b$. In other word, we need to prove that for some $c(\delta',\delta'')>0$, we have
$$J(v)>J(u)-c(\delta',\delta''),$$
for any $u\in H_0^1(\Omega)$ and $v\in H_0^1(\Omega)$ with $I_{\delta''}(u)=0$, $|| \nabla u|| \neq 0$ and $I_{\delta'}(u)=0$, $|| \nabla v|| \neq 0$, respectively.
First, we define $\epsilon(\delta)$ for $u$ as \eqref{eq-3}, then $I_{\delta}(\epsilon(\delta)u)=0$ and $\epsilon(\delta'')=1$. Let us have $g(\epsilon)=J(\epsilon u)$, then
\begin{align*}
\frac{d}{d\epsilon} g(\epsilon) = \frac{1}{\epsilon} \left( || \nabla (\epsilon u) ||^2 - \int_{\Omega } \epsilon u f(\epsilon u) dx \right) = (1-\delta )\epsilon ||\nabla u ||^2,
\end{align*}
where we use the fact $\delta || \nabla u ||^2 = \int_{\Omega } uf(u)dx$.
Now we choose $v=\epsilon(\delta')u$, then $I_{\delta'}(v)=0$ and $|| \nabla v || \neq 0$. For $0<\delta'<\delta''<1$, we get
\begin{align*}
J(u) - J(v) &= J(\epsilon(\delta'') u) - J(\epsilon(\delta') u) \\
&= g(1) - g(\epsilon(\delta')) \\
&= (1-\delta')\epsilon(\delta')|| \nabla u||^2(1-\epsilon(\delta'))\\
&> (1-\delta'')\epsilon(\delta')r^2(\delta'')(1-\epsilon(\delta')) \equiv c(\delta',\delta'').
\end{align*}
For $1<\delta''<\delta'<b$, we have
\begin{align*}
J(u) - J(v) &= J(\epsilon(\delta'') u) - J(\epsilon(\delta') u)\\
&= g(1) - g(\epsilon(\delta')) \\
&> (\delta'' -1)r^2(\delta'')\epsilon(\delta'') (\epsilon(\delta'')-1).
\end{align*}
This completes the proof.
\end{proof}
\begin{lem}\label{lem-6}
Let $f(u)$ satisfy $(H)$ and $0<\delta<\frac{\alpha}{2}-\frac{\beta}{\lambda_1}$. Then the following properties hold
\begin{itemize}
\item[(i)] Suppose that $J(u)\leq d(\delta)$ and $I_{\delta}(u)>0$, then
\begin{equation*}
0<|| \nabla u||^2 < \frac{d(\delta)}{a(\delta)}.
\end{equation*}
In the case $\delta=1$, if $J(u)\leq d$ and $I(u)>0$, then
\begin{equation*}
0<|| \nabla u||^2 < \frac{d}{a(1)}.
\end{equation*}
\item[(ii)] Suppose that $J(u)\leq d(\delta)$ and $I_{\delta}(u)=0$, then
\begin{equation*}
0<|| \nabla u||^2 < \frac{d(\delta) }{a(\delta)}.
\end{equation*}
In the case $\delta=1$, if $J(u)\leq d$ and $I(u)=0$, then
\begin{equation*}
0<|| \nabla u||^2 < \frac{d}{a(1)}.
\end{equation*}
\item[(iii)] Suppose that $J(u)\leq d(\delta)$ and $|| \nabla u||^2>d(\delta)/a(\delta)$, then $I_{\delta}(u)<0$. \\
In the case $\delta=1$, if $J(u)\leq d$ and $|| \nabla u||^2>{d}/{a(1)}$, then $I(u)<0$.
\end{itemize}
\end{lem}
\begin{proof}[Proof of Lemma \ref{lem-6}] The case $(i)$ follows from
\begin{align*}
J(u) &= \frac{1}{2} || \nabla u ||^2 - \int_{\Omega } [F(u)- \sigma] dx\\
&\geq \frac{1}{2} || \nabla u ||^2 - \frac{1}{\alpha} \int_{\Omega }uf(u)dx- \frac{\beta}{\alpha} \int_{\Omega }u^2dx \\
& > \left[\frac{1}{2} - \frac{\delta}{\alpha} - \frac{\beta}{\lambda_1\alpha} \right]|| \nabla u ||^2 \\
& = a(\delta) || \nabla u ||^2 .
\end{align*}
The proof of cases $(ii)$ and $(iii)$ follows from the similar argument. The proofs hold for the case $(H)-(b)$ with additional assumption $u(x)>0$.
\end{proof}
Then a family of potential wells can be defined for $0<\delta<b$ as follows
\begin{align*}
W_{\delta} &:= \{ u \in H_0^1(\Omega) \,\,| \,\, I_{\delta}(u)>0, J(u)<d(\delta) \} \cup \{0\},\\
\overline{W}_{\delta} & := W_{\delta}\cup \partial W_{\delta}=\{ u \in H_0^1(\Omega) \,\,| \,\, I_{\delta}(u)\geq 0, J(u) \leq d(\delta) \},
\end{align*}
In the case $\delta=1$, we have
\begin{align*}
W & := \{ u \in H_0^1(\Omega) \,\,| \,\, I(u)>0, J(u)<d \} \cup \{0\},\\
\overline{W} & := \{ u \in H_0^1(\Omega) \,\,| \,\, I(u)\geq 0, J(u)\leq d \}.
\end{align*}
In addition, we define
\begin{align*}
V_{\delta}&:= \{ u \in H_0^1(\Omega) \,\,| \,\, I_\delta(u)<0, J(u)<d(\delta) \},\\
V &:=\{ u \in H_0^1(\Omega) \,\,| \,\, I(u)<0, J(u)<d \}.
\end{align*}
From the definition of $W_{\delta}$, $V_{\delta}$ and Lemma \ref{lem-5}, we derive the following properties:
\begin{lem}\label{lem-7}
If $0<\delta'<\delta''\leq 1$ and $1\leq \delta''<\delta' <b $, then $W_{\delta'} \subset W_{\delta''}$ and $V_{\delta'} \subset V_{\delta''}$, respectively.
\end{lem}
\begin{lem}\label{lem-8}
Let $0<J(u)<d$ for some $u \in H_0^1(\Omega)$. Let $\delta_1<\delta_2$ are two roots of equation $J(u)=d(\delta)$. Then the sign of $I_{\delta}(u)$ is unchangeable for $\delta_1<\delta<\delta_2$.
\end{lem}
\begin{proof}[Proof of Lemma \ref{lem-8}]
The proof follows from contradiction argument. $||\nabla u|| \neq 0$ comes from $J(u)>0$. If sign of $I_{\delta}(u)$ is changeable for $\delta_1<\delta <\delta_2$, then there exists $\delta^* \in (\delta_1,\delta_2)$ such that $I_{\delta^*}(u)=0$. Thus by the definition of $d(\delta)$ we have $J(u)\geq d(\delta^*)$ that contradicts
\begin{equation*}
J(u)=d(\delta_1)=d(\delta_2) < d(\delta^*),
\end{equation*}
this completes the proof.
\end{proof}
\section{Invariant sets and vacuum isolating of solutions}\label{Sec3}
In this section, the invariance sets and the vacuum isolating of solutions are discussed for semilinear wave equation \eqref{Wave-problem}.
Recall the total energy of semilinear wave equation \eqref{Wave-problem}
\begin{equation*}
E(t) = \frac{1}{2} || u_t||^2 + \frac{1}{2} || \nabla u ||^2 - \int_{\Omega } [F(u)- \sigma] dx \equiv \frac{1}{2} || u_t||^2 + J(u).
\end{equation*}
\begin{thm}\label{thm-inv-sets}
Let $f(u)$ satisfy $(H)$, $u_0(x)\in H_0^1(\Omega)$, and $u_1(x) \in L^2(\Omega)$. Assume that $0<e<d$, then equation $d(\delta)=e$ has two roots $\delta_1<\delta_2$. Therefore, we formulate the following properties
\begin{itemize}
\item[$(i)$] All solutions of problem \eqref{Wave-problem} with $E(0)=e$ belong to $W_{\delta}$ for $\delta_1 <\delta < \delta_2$, provided $I(u_0)>0$ or $||\nabla u_0|| =0$.
\item[$(ii)$] All solutions of problem \eqref{Wave-problem} with $E(0)=e$ belong to $V_{\delta}$ for $\delta_1 <\delta < \delta_2$, provided $I(u_0)<0$.
\end{itemize}
\end{thm}
\begin{proof}[Proof of Theorem \ref{thm-inv-sets}]
In the case $(i)$, we suppose that $u(x,t)$ is any solution of problem \eqref{Wave-problem} with $E(0)=e$ and $I(u_0)>0$ or $||\nabla u_0|| =0$. If $I(u_0)>0$, then we have
\begin{equation}\label{eq-3.1}
a(\delta) || \nabla u_0||^2 + \frac{1}{\alpha} I_{\delta}(u_0) < J(u_0) \leq d(\delta),
\end{equation}
and
\begin{equation}\label{eq-3.2}
\frac{1}{2} || u_1||^2 + J(u_0) = E(0) = d(\delta_1)=d(\delta_2)< d(\delta), \,\,\, \delta_1 < \delta <\delta_2.
\end{equation}
From \eqref{eq-3.2} and the definition of $d(\delta)$ we get $I_{\delta}(u_0)>0$ and $J(u_0)<d(\delta)$ that is $u_0(x) \in W_{\delta}$ for $0<\delta<b$.
Now we prove $u(t) \in W_{\delta}$ for $\delta_1<\delta<\delta_2$ and $0<t<T$, where $T$ is the maximal existence time of $u(t)$. Arguing by contradiction, there must exist $t_0 \in (0,T)$ such that $u(t_0)\in \partial W_{\delta}$ for some $\delta \in (\delta_1, \delta_2)$, and $I_{\delta}(u(t_0))=0$, $|| \nabla u(t_0)|| \neq 0$ or $J(u(t_0))=d(\delta)$. From
\begin{equation}\label{eq-3.3}
\frac{1}{2} || u_t||^2 + J(u) = E(0) < d(\delta), \,\,\, \delta \in (\delta_1,\delta_2), \,\, t \in (0,T),
\end{equation}
it is easy to see that $J(u(t_0))\neq d(\delta)$. Also, if $I_{\delta}(u(t_0))=0$ and $|| \nabla u(t_0)|| \neq 0$, then by the definition of $d(\delta)$ we have $J(u(t_0))\geq d(\delta)$ that contradicts \eqref{eq-3.3}.
In the case $(ii)$, we assume again that $u(x,t)$ is any solution of problem \eqref{Wave-problem} with $E(0)=e$ and $I(u_0)<0$. As a previous case, $u_0(x)\in V_{\delta}$ can be established by Lemma \ref{lem-8} and \eqref{eq-3.2}.
The proof of $u(t) \in V_{\delta}$ for $\delta \in (\delta_1,\delta_2)$ follows from arguing by contradiction. Let $t_0 \in (0,T)$ be the first time such that $u(t) \in V_{\delta}$ for $t \in [0,t_0)$ and $u(t_0) \in \partial V_{\delta}$, i.e. $I_{\delta}(u(t_0))=0$ or $J(u(t_0))=d(\delta)$ for some $\delta \in (\delta_1,\delta_2)$. From \eqref{eq-3.3} it follows that $J(u(t_0))\neq d(\delta)$. If $I_{\delta}(u(t_0))=0$, then $I_{\delta}(u(t))<0$ for $t \in (0,t_0)$ and Lemma \ref{lem-4} yield $|| \nabla u(t)||>r(\delta)$ and $|| \nabla u(t_0)||\geq r(\delta)$. Hence by the definition of $d(\delta)$ we have $J(u(t_0))\geq d(\delta)$ which contradicts \eqref{eq-3.3}.
\end{proof}
\begin{cor}\label{cor_1}
If the assumption $E(0)=e$ in Theorem \ref{thm-inv-sets} is replaced by $0<E(0)\leq e$, then the conclusion of Theorem \ref{thm-inv-sets} also holds.
\end{cor}
\begin{thm}
Let $f(u)$, $u_i(x)$ for $i=0,1$, $e$ and $(\delta_1,\delta_2)$ be the same as those in Theorem \ref{thm-inv-sets}. Then for any $\delta \in (\delta_1,\delta_2)$ both sets $W_{\delta}$ and $V_{\delta}$ are invariant, thereby both sets
\begin{equation*}
W_{\delta_1\delta_2} = \bigcup _{\delta_1<\delta<\delta_2} W_{\delta}, \,\, \text{ and }\,\,\, V_{\delta_1\delta_2} = \bigcup _{\delta_1<\delta<\delta_2} V_{\delta},
\end{equation*}
are invariant respectively under the flow of \eqref{Wave-problem}, provided by $0<E(0)\leq e$.
\end{thm}
Note that if $0<E(0)\leq e$, then $I(u_0)=0$ and $||\nabla u_0 ||\neq 0$ are impossible, which comes from Theorem \ref{thm-inv-sets}. We see that for the set of all solutions of problem \eqref{Wave-problem} with $0<E(0)\leq e$ there exists a vacuum region
\begin{equation*}
U_e = \{ u \in H_0^1(\Omega) \,\, | \,\, || \nabla u|| \neq 0 \text{ and } I_{\delta}(u)=0, \,\,\, \delta_1 <\delta<\delta_2 \}
\end{equation*}
such that there is not any solution of problem \eqref{Wave-problem} in $U_e$.
The vacuum region $U_e$ become bigger and bigger with decreasing of $e$. As the limit case we get
\begin{equation*}
U_0 = \{ u \in H_0^1(\Omega) \,\, | \,\, || \nabla u|| \neq 0 \text{ and } I_{\delta}(u)=0, \,\,\, 0 <\delta<b \}.
\end{equation*}
In the case $E(0)\leq 0$ we have the following invariant set of solutions for problem \eqref{Wave-problem}.
\begin{lem}\label{lem-9}
All nontrivial solutions of problem \eqref{Wave-problem} with $E(0)=0$ satisfy
\begin{equation*}
|| \nabla u || \geq \left( \frac{1}{2AC_*^{\gamma}} \right)^{\frac{1}{\gamma-2}}.
\end{equation*}
\end{lem}
\begin{proof}[Proof of Lemma \ref{lem-9}]
First, we get $J(u)\leq 0$ for $0\leq t< T$, where $T$ is the maximal existence time of $u(t)$, from the energy equality
\begin{equation*}
\frac{1}{2}|| u_t||^2 + J(u) = E(0) =0.
\end{equation*}
Then
\begin{align*}
\frac{1}{2}|| \nabla u||^2 &\leq \int_{\Omega} [F(u)-\sigma]dx \leq A \int_{\Omega} |u|^{\gamma}dx\\
&=A|| u||^{\gamma}_{\gamma} \leq AC_*^{\gamma} || \nabla u||^{\gamma-2}|| \nabla u||^2.
\end{align*}
This completes the proof.
\end{proof}
\begin{thm}\label{thm-E-neg}
Let $u_0(x)\in H_0^1(\Omega)$ and $u_1(x)\in L^2(\Omega)$. Suppose that $E(0)=0$ or $E(0)<0$ with $||\nabla u_0|| \neq 0$. Then all solutions of problem \eqref{Wave-problem} belong to $V_{\delta}$ for $\delta \in (0,\alpha/2-\beta/\lambda_1 )$.
\end{thm}
\begin{proof}[Proof of Theorem \ref{thm-E-neg}]
By using the conservation law of energy and for $\delta \in (0,\alpha/2- \beta/\lambda_1)$, we have
\begin{equation*}
J(u) \geq a(\delta)||\nabla u||^2 + \frac{1}{\alpha}I_{\delta}(u),
\end{equation*}
we write the following expression
\begin{align}\label{eq-3.4}
E(0) \geq \frac{1}{2}|| u_t||^2 + a(\delta)||\nabla u||^2 + \frac{1}{\alpha}I_{\delta}(u).
\end{align}
If $E(0)<0$, then \eqref{eq-3.4} gives $I_{\delta}(u)<0$ and $J(u)<0<d(\delta)$ for $\delta \in (0,\alpha/2-\beta/\lambda_1 )$. If $E(0)=0$ and $|| \nabla u_0||\neq 0$, then Lemma \ref{lem-9} shows that $|| \nabla u||$ is positive for $t\in [0,T)$. So from \eqref{eq-3.4} we have $I_{\delta}(u)<0$ and $J(u)<0<d(\delta)$ for $\delta \in (0,\alpha/2-\beta/\lambda_1 )$.
\end{proof}
\section{Global existence of solutions for semilinear wave equation}\label{Sec4}
In this section, we prove a global existence of solutions of semilinear wave equation \eqref{Wave-problem} by using the family of potential wells.
\begin{defn}
Let $u \in L^{\infty}(0,T;H_0^1(\Omega))$ and $u_t \in L^{\infty}(0,T;L^2(\Omega))$. Then a function $u(x,t)$ is called a {\em weak solution} of problem \eqref{Wave-problem} on $\Omega \times [0,T)$ if
\begin{equation*}
(u_{t},v) + \int_0^t (\nabla u, \nabla v) d\tau = \int_0^t (f(u),v)d\tau + (u_1,v),
\end{equation*}
for every $v \in H_0^1(\Omega)$, $u(x,0)=u_0(x)\in H_0^1(\Omega)$, and $t \in (0,T)$.
\end{defn}
\subsection{Initial condition $ 0<E(0)<d$ and $I(u_0)>0$}
\begin{thm}\label{thm-global}
Let $f(u)$ satisfy $(H)$. Let $u_0(x) \in H_0^1(\Omega)$, $u_1(x)\in L^2(\Omega)$ and $2\beta < \lambda_1(\alpha -2)$. Suppose that $0<E(0)<d$ and $I(u_0)>0$ or $|| \nabla u_0||=0$. Then problem \eqref{Wave-problem} admits a global weak solution $u(t) \in L^{\infty}(0,\infty; H_0^1(\Omega))$, $u_t(t)\in (0,\infty;L^2(\Omega)) $ and $u(t) \in W$ for $t \in [0,\infty)$.
\end{thm}
\begin{lem}[Lemma 1.3 in \cite{Lions}]\label{lem-Lions}
Let $D$ be a bounded domain in $\mathbb{R}^n \times \mathbb{R}_t$, $g_{\mu}$ and $g$ are functions from $L^q(D)$, $1<q<\infty$, so
\begin{equation*}
|| g_{\mu} ||_{L^q(D)} \leq C, \,\,\,\, g_{\mu} \rightarrow g \,\, \text{ in } D.
\end{equation*}
Then $g_{\mu} \rightarrow g $ weakly star in $L^q(D)$.
\end{lem}
\begin{proof}[Proof of Theorem \ref{thm-global}]
Here we follow the proof from Liu-Zhao \cite{LZh-06}. Let $w_j(x)$ be a system of base functions in $H_0^1(\Omega)$. Then we are able to construct the approximate solutions $u_m(x,t)$ of problem \eqref{Wave-problem} as follows
\begin{equation*}
u_m(x,t) = \sum_{j=1}^m g_{jm}(t)w_j(x) \,\, \text{ for } \,\, m=1,2,\ldots
\end{equation*}
satisfying
\begin{equation}\label{eq-4.1}
(u_{m}'',w_s) + (\nabla u_m, \nabla w_s) = (f(u_m),w_s) \,\, \text{ for } \,\, s=1,2,\ldots,m,
\end{equation}
\begin{equation}\label{eq-4.2}
u_m(x,0) = \sum_{j=1}^m a_{jm}w_j(x) \rightarrow u_0(x) \,\, \text{ in } \,\, H_0^1(\Omega),
\end{equation}
\begin{equation}\label{eq-4.3}
u_{m}'(x,0) = \sum_{j=1}^m b_{jm}w_j(x) \rightarrow u_1(x) \,\, \text{ in } \,\, L^2(\Omega).
\end{equation}
Note that $u_t=u'$. If we multiply \eqref{eq-4.1} by $g'_{sm}(t)$ and sum over $s$, then this gives
\begin{equation*}
(u_{m}'',u_{m}') + (\nabla u_m, \nabla u_{m}') = (f(u_m),u_{m}') \,\, \text{ for } \,\, m=1,2,\ldots.
\end{equation*}
This implies conservation of energy $E_m(t)$, which allows to write for $0\leq t < \infty$
\begin{align}\label{eq-4.4}
E(t)&=\frac{1}{2} || u_{m}'||^2 + \frac{1}{2} || \nabla u_m||^2 - \int_{\Omega }[F(u_m)-\sigma] dx \nonumber\\
&=\frac{1}{2} || u_{m}'||^2 +J(u_m) = E_m(0) < d,
\end{align}
and for $I(u_{m}(x,0))>0$ we have $u_m \in W$ for sufficiently large $m$ and $0\leq t<\infty$.
Now we obtain a priori estimates. Let
\begin{align*}
J(u_m) &= \frac{1}{2} || \nabla u_m||^2 - \int_{\Omega } [F(u_m)- \sigma] dx \\
&\geq \frac{1}{2} || \nabla u_m||^2 - \frac{1}{\alpha} \int_{\Omega } u_m f(u_m) dx - \frac{\beta}{\alpha} || u_m ||^2 \\
& \geq \left( \frac{1}{2} - \frac{1}{\alpha} - \frac{\beta}{\alpha \lambda_1} \right)||\nabla u_m||^2 + \frac{1}{\alpha} I(u_m) \\
& \geq a ||\nabla u_m||^2,
\end{align*}
where $a = \frac{1}{2} - \frac{1}{\alpha} - \frac{\beta}{\alpha \lambda_1}$. Then from \eqref{eq-4.4} it follows that
\begin{equation}\label{eq-4.5}
\frac{1}{2} || u_{m}'||^2 + a||\nabla u_m||^2 < d, \,\,\, 0\leq t <\infty,
\end{equation}
for sufficiently large $m$. Then for $0\leq t <\infty $ \eqref{eq-4.5} implies the following estimates
\begin{align}
& || \nabla u_m ||^2 < \frac{d}{a}, \label{eq-4.6}\\
& || u_m||^2_{\gamma} \leq C_*^2|| \nabla u_m||^2 <C_*^2\frac{d }{a}, \label{eq-4.7} \\
& || u_{m}'||^2 < 2d , \label{eq-4.8}
\end{align}
and Lemma \ref{lem-1} and \eqref{eq-4.7} give
\begin{align}
|| f(u_m)||^q_q &\leq \int_{\Omega } [\gamma A |u_m|^{\gamma-1}]^q dx = \gamma^q A^q || u_m||^{\gamma}_{\gamma} \nonumber \\
& < \gamma^q A^q C_*^{\gamma} \left(\frac{d}{a}\right)^{\gamma/2},
\end{align}
where $q= \frac{\gamma}{\gamma-1}$.
Since a priori estimates \eqref{eq-4.6}-\eqref{eq-4.8}, we may extract the subsequence $\{u_l\}$ from sequence $\{u_m\}$ such that
\begin{align*}
u_l &\rightarrow u \,\, \text{ weakly star in } L^{\infty}(0,\infty;H_0^1(\Omega)) \,\, \text{ and a.e. } Q=\Omega \times [0,\infty) \\
u_l'&\rightarrow u_t \,\, \text{ weakly star in } L^{\infty}(0,\infty;L^2(\Omega)),\\
f(u_l)&\rightarrow \chi \,\, \text{ in } L^{\infty}(0,\infty;L^q(\Omega)) \,\,\, \text{ and a.e. } Q=\Omega \times [0,\infty).
\end{align*}
From Lemma \ref{lem-Lions} we have $\chi=f(u)$. We integrate equation \eqref{eq-4.1} with respect to time from $0$ to $t$ as follows
\begin{equation}\label{eq-4.10}
(u_m',w_s) -(u_m'(0),w_s) + \int_0^t(\nabla u_m,\nabla w_s) d\tau = \int_0^t (f(u_m),w_s) d\tau.
\end{equation}
For fixed $s$ in \eqref{eq-4.10} and taking $m=l\rightarrow \infty$ we get
\begin{equation*}
(u_t,w_s) -(u_1,w_s) + \int_0^t(\nabla u,\nabla w_s) d\tau = \int_0^t (f(u),w_s) d\tau, \,\,\, \forall s,
\end{equation*}
then
\begin{equation*}
(u_t,v) -(u_1,v) + \int_0^t(\nabla u,\nabla v) d\tau = \int_0^t (f(u),v) d\tau, \,\,\, \forall v \in H_0^1(\Omega), \,\, t>0.
\end{equation*}
Also \eqref{eq-4.2} implies that $u_0(x)$ in $H_0^1(\Omega)$. Therefore $u(x,t)$ is a global weak solution of problem \eqref{Wave-problem}.
\end{proof}
\begin{thm} Let $f(u)$ satisfy $(H)$.
Let $u_0(x)\in H_0^1(\Omega)$ and $u_1(x)\in L^2(\Omega)$. The problem \eqref{Wave-problem} admits a global weak solution $u(t)\in L^{\infty}(0,\infty;H_0^1(\Omega))$ with $u_t(t)\in L^{\infty}(0,\infty;L^2(\Omega)) $ and
\begin{itemize}
\item $u(t)\in W_{\delta}$ for $\delta_1<\delta<\delta_2$,
or
\item $|| \nabla u ||^2 \leq E(0)/d(\delta_1)$, $||u_t ||^2\leq 2E(0)$,
\end{itemize} if the following respective assumptions hold
\begin{itemize}
\item $0<E(0)<d$ and $I_{\delta_2}<0$ or
\item $0<E(0)<d$ and $|| \nabla u_0||<r(\delta_2)$.
\end{itemize}
\end{thm}
\subsection{Critical initial condition $E(0)=d$ and $I(u_0)\geq 0$} Here we consider the global existence of solution for problem \eqref{Wave-problem} with critical initial condition $E(0)=d$ and $I(u_0)\geq 0$.
\begin{thm}\label{thm-E=d}
Let $f(u)$ satisfy $(H)$. Let $u_0 \in H_0^1(\Omega)$ and $u_1(x)\in L^2(\Omega)$. Suppose that $E(0)=d$ and $I(u_0)\geq 0$. Then problem \eqref{Wave-problem} admits a hlobal weak solution $u(t) \in L^{\infty}(0,\infty;H_0^1(\Omega))$ with $u_t(t)\in L^{\infty}(0,\infty; L^2(\Omega))$ and $u(t)\in \overline W$ for $0\leq t <\infty$.
\end{thm}
\begin{proof}[Proof of Theorem \ref{thm-E=d}]
The proof is divided for two cases $|| \nabla u_0||\neq0$ and $|| \nabla u_0|| =0$.
In the case $|| \nabla u_0||\neq 0$, we take $\epsilon_m = 1 - 1/m$ and $u_{0m}=\epsilon_m u_0$ for $m=2,3,\ldots$. Then we consider problem \eqref{Wave-problem} with the following initial conditions
\begin{equation}\label{IC-m}
u(x,0) = u_{0m}(x) \,\, \text{ and } \,\, u_t(x,0) = u_1(x).
\end{equation}
The fact $\epsilon^*=\epsilon^*(u_0)\geq 1$ follows from $I(u_0)\geq 0$ and Lemma \ref{lem-3}. Then for $I(u_{0m})>0$ we have
\begin{equation}
J(u_{0m}) \geq \left(\frac{1}{2} -\frac{1}{\alpha} - \frac{\beta}{\alpha \lambda_1}\right)|| \nabla u_{0m}||^2 +\frac{1}{\alpha} I(u_{0m})>0,
\end{equation}
and $J(u_{m0})<J(u_0)$. Then
\begin{equation*}
0<E_m(0) = \frac{1}{2}|| u_1||^2 + J(u_{0m}) < \frac{1}{2}|| u_1||^2 + J(u_{0}) =E(0)=d.
\end{equation*}
Now it is easy to see from Theorem \ref{thm-global} that for each $m$ problem \eqref{Wave-problem} with initial conditions \eqref{IC-m} admits a global weak solution $u_m(t)\in L^{\infty}(0,\infty;H_0^1(\Omega))$ with $u_{mt}(t)\in L^{\infty}(0,\infty; L^2(\Omega))$ and $u_m(t)\in \overline W$ for $0\leq t<\infty$ satisfying
\begin{equation}\label{eq-5.2}
(u_{mt},v) + \int_0^t(\nabla u_m,\nabla v) d\tau = \int_0^t (f(u_m),v)d\tau +(u_1,v),
\end{equation}
for every $v \in H_0^1(\Omega)$, and
\begin{equation}\label{eq-5.3}
\frac{1}{2} || u_{mt} ||^2 + J(u_m) = E_m(0) <d.
\end{equation}
Then we obtain the following estimates
\begin{align*}
& || \nabla u_m ||^2 < \frac{d}{a},\\
& || u_m||^2_{\gamma} \leq C_*^2|| \nabla u_m||^2 <C_*^2\left(\frac{d}{a}\right), \\
& || u_{m}'||^2 < 2d ,
\end{align*}
and
\begin{align*}
|| f(u_m)||^q_q < \gamma^q A^q C_*^{\gamma} \left(\frac{d}{a}\right)^{\gamma/2},
\end{align*}
where $a = \frac{1}{2} -\frac{1}{\alpha} - \frac{\beta}{\alpha \lambda_1}$ and $q= \frac{\gamma}{\gamma-1}$.
From above priori estimates, we extract the subsequence $\{u_l\}$ from sequence $\{u_m\}$ such that as $l \rightarrow \infty$
\begin{align*}
u_l &\rightarrow u \,\, \text{ weakly star in } L^{\infty}(0,\infty;H_0^1(\Omega)) \,\, \text{ and a.e. } Q=\Omega \times [0,\infty) \\
u_l'&\rightarrow u_t \,\, \text{ weakly star in } L^{\infty}(0,\infty;L^2(\Omega)),\\
f(u_l)&\rightarrow f(u) \,\, \text{ in } L^{\infty}(0,\infty;L^q(\Omega)) \,\,\, \text{ and a.e. } Q=\Omega \times [0,\infty).
\end{align*}
Letting $m=l\rightarrow \infty$ in \eqref{eq-5.2} we arrive at
\begin{equation}
(u_{t},v) + \int_0^t(\nabla u,\nabla v) d\tau = \int_0^t (f(u),v)d\tau +(u_1,v),
\end{equation}
for every $v \in H_0^1(\Omega)$ and $0\leq t <\infty$, which implies that $u(x,t)$ is the global weak solution of problem \eqref{Wave-problem}.
In the case $|| \nabla u_0|| =0$, we have
\begin{equation*}
\frac{1}{2} || u_1||^2 + J(u_0) = \frac{1}{2} || u_1||^2 =E(0)=d.
\end{equation*}
Let $\epsilon_m = 1- 1/m$ such that $u_{1m}(x) = \epsilon_mu_1(x)$ for $m=2,3,\ldots$. We consider problem \eqref{Wave-problem} with the following initial conditions
\begin{equation}\label{IC-m1}
u(x,0) = u_0(x) \,\, \text{ and } \,\, u_t(x,0) = u_{1m}(x).
\end{equation}
From $|| \nabla u_0|| =0$, we get
\begin{equation*}
0<E_m(0)=\frac{1}{2} || u_{1m}||^2 + J(u_0) = \frac{1}{2} || u_{1m}||^2 <E(0)=d.
\end{equation*}
As the previous case from Theorem \ref{thm-global} follows that for each $m$ problem \eqref{Wave-problem} with initial conditions \eqref{IC-m1} admits a global weak solution $u_m(t)\in L^{\infty}(0,\infty;H_0^1(\Omega))$ with $u_{mt}(t)\in L^{\infty}(0,\infty; L^2(\Omega))$ and $u_m(t)\in \overline W$ for $0\leq t<\infty$ satisfying \eqref{eq-5.2} and \eqref{eq-5.3}. The rest of proof repeats the previous one.
\end{proof}
\section{Blow-up solutions of semilinear wave equation}\label{Sec5}
\subsection{Initial condition $E(0)<d$ and $I(u_0)<0$}
\begin{thm}\label{thm-blow}
Let $f$ satisfy $(H)$ with $2\beta < \lambda_1 (\alpha-2)$ and $\lambda_1$ is a first eigenvalue of Laplacian. Let $u_0(x)\in H_0^1(\Omega)$ and $u_1(x)\in L^2(\Omega)$. Suppose that energy $E(0)<d$ and $I(u_0)<0$. Then the solution of problem \eqref{Wave-problem} blows up such that
\begin{equation}
\lim_{t\rightarrow T} \int_{\Omega} u^2 dx = + \infty,
\end{equation}
in a finite time
\begin{equation}
T = \frac{\int_{\Omega} u_0^2(x) dx}{(\alpha+2) \int_{\Omega} u_0(x)u_1(x) dx}.
\end{equation}
\end{thm}
\begin{proof}[Proof of Theorem \ref{thm-blow}]
Define
\begin{align*}
M(t) &= \int_{\Omega} u^2 dx.
\end{align*}
Then we have
\begin{align}
M'(t)= 2 \int_{\Omega} uu_t dx \,\, \text{ and } \,\,M''(t)= 2\int_{\Omega} u_t^2 dx + 2\int_{\Omega} uu_{tt} dx.
\end{align}
Then by using $u_{tt} = \Delta u +f(u)$ and \eqref{cont_1} we have
\begin{align*}
M''(t) &= 2\int_{\Omega} u_t^2 dx + 2\int_{\Omega } [ \Delta u +f(u) ]u dx\\
& = 2\int_{\Omega} u_t^2 dx - 2I(u) \\
& \geq 2\int_{\Omega} u_t^2 dx +2\alpha \int_{\Omega } [F(u) - \sigma] dx - 2\beta \int_{\Omega } u^2 dx - 2\int_{\Omega } |\nabla u|^2dx.
\end{align*}
Then from the conservation of energy
\begin{equation*}
E(0) = \frac{1}{2} \int_{\Omega } u_t^2 dx + \frac{1}{2} \int_{\Omega } |\nabla u|^2 dx- \int_{\Omega } [F(u)-\sigma ]dx,
\end{equation*}
and the Poincar\'e inequality, we get
\begin{equation*}
M''(t) \geq (\alpha +2)\int_{\Omega} u_t^2 dx + \left(\lambda_1(\alpha -2)- 2\beta\right) M(t) - 2\alpha E(0),
\end{equation*}
where $\lambda_1$ is the first eigenvalue of Laplacian and $2\beta > \lambda_1(\alpha -2)$.
In the case $E(0)<0$ and $2\beta > \lambda_1(\alpha -2)$, then
\begin{equation}\label{eq-M}
M''(t) \geq (\alpha +2)\int_{\Omega} u_t^2 dx.
\end{equation}
In the case $0<E(0)<d$ and $2\beta < \lambda_1(\alpha -2)$, there exists sufficient large time $t_1$ such that
\begin{equation*}
\left(\lambda_1(\alpha -2)- 2\beta\right) M(t) - 2\alpha E(0)>0.
\end{equation*}
Since $M(t)$ is a convex function of $t$, then we can find a time $t_1$ such that $M'(t_1)>0$, then $M(t)$ is increasing for all $t>t_1$. Also this part of proof can be shown by using the family of potential wells $V_{\delta}$ and Lemma \ref{lem-4} (see \cite{LZh-06}).
We are able to prove that \eqref{eq-M} for two cases $E(0)<0$ and $0<E(0)<d$. Next we obtain the blow-up solutions by using concavity argument and \eqref{eq-M} as follows
\begin{equation*}
M''(t)M(t) - (\alpha+3) [M'(t)]^2 \geq (\alpha+3)\left[\left( \int_{\Omega }u^2dx\right) \left(\int_{\Omega }u_t^2 dx\right) - \left( \int_{\Omega } uu_tdx\right)^2 \right]\geq 0,
\end{equation*}
which gives for $t\geq 0$
\begin{equation*}
\frac{d}{dt} \left[ \frac{M'(t)}{M^{\alpha+3}(t)} \right] >0 \Rightarrow \begin{cases}
M'(t) \geq \left[ \frac{M'(0)}{M^{\alpha+3}(0)} \right] M^{\alpha+3}(t),\\
M(0)=\int_{\Omega} u_0^2 dx.
\end{cases}
\end{equation*}
Then we have
\begin{equation*}
M(t) \geq M(0)\left( 1-\frac{ (\alpha+2) M'(0) }{M(0)} t\right)^{-\frac{1}{\alpha+2}}.
\end{equation*}
Then the blow-up time $T$ satisfies
\begin{equation*}
0<T\leq \frac{M(0)}{(\alpha+2) M'(0)}.
\end{equation*}
This completes the proof.
\end{proof}
\subsection{Critical initial condition $E(0)=d$ and $I(u_0)<0$}
Next we prove the blow-up solution of problem \eqref{Wave-problem} with the critical initial conditions $E(0)=d$ and $I(u_0)<0$. First we need the following lemma.
\begin{lem}\label{lem-10}
Let $u_0(x) \in H_0^1(\Omega)$ and $u_1(x) \in L^2(\Omega)$. Suppose that $E(0)=d$ and $(u_0,u_1)>0$. Then set
\begin{equation*}
V' = \{ u \in H_0^1(\Omega)\, : \, I(u)<0 \}
\end{equation*}
is invariant under the flow of problem \eqref{Wave-problem}.
\end{lem}
\begin{proof}[Proof of Lemma \ref{lem-10}]
Let $u(t)$ be any weak solution of problem \eqref{Wave-problem} with $E(0)=d$, $I(u_0)<0$ and $\int_{\Omega }u_0u_1dx\geq 0$, and $T$ is the maximal existence time of $u(t)$. $I(u(t))<0$ for $t \in (0,T)$ is established by a contradiction. Let it be false, then there exists $t_0 \in (0,T)$ such that $I(u(t_0))=0$ and $I(u(t))$ for $t\in [0,t_0)$. By Lemma \ref{lem-4}, we have $|| \nabla u(t)||>r(1)$ for $t \in [0,t_0)$ and $|| \nabla u(t_0)||\geq r(1)>0$. Next we have $J(u(t_0))=d$ and $\int_{\Omega } u_t^2(t_0) dx=0$ from the definition of $d$ which implies $J(u(t_0))\geq d$ and
\begin{equation*}
\frac{1}{2} \int_{\Omega } u_t^2(t_0) dx + J(u(t_0)) \leq E(0) =d.
\end{equation*}
Let
\begin{equation*}
M(t) = \int_{\Omega } u^2(t)dx \,\, \text{ and } \,\, M'(t) = 2\int_{\Omega }u_t(x,t)dx,
\end{equation*}
with
\begin{align*}
M'(0) &= 2 \int_{\Omega } u_0(x)u_1(x) dx, \\
M''(t)&= 2 \int_{\Omega }u_t^2(t)dx - 2I(u(t)) > 0 \,\, \text{ for } \,\, 0\leq t<t_0.
\end{align*}
Hence $M'(t)$ is strictly increasing with respect to $t \in [0,t_0]$ along with $M'(0)\geq 0$ gives
\begin{equation*}
M'(t_0) = 2\int_{\Omega } u(t_0) u_t(t_0)dx > 0.
\end{equation*}
This clearly contradicts $\int_{\Omega } u_t^2(t_0) dx=0$. This completes the proof.
\end{proof}
\begin{thm}\label{thm-blow-up-E=d}
Let $f$ satisfy $(H)$ with $2\beta < \lambda_1 (\alpha-2)$ and $\lambda_1$ is a first eigenvalue of Laplacian. Let $u_0(x) \in H_0^1(\Omega)$ and $u_1(x) \in L^2(\Omega)$. Suppose that
\begin{align*}
E(0)=d, \,\, I(u_0) <0, \,\, \text{ and } \int_{\Omega } u_0(x)u_1(x)dx \geq 0.
\end{align*}
Then the weak solution of problem \eqref{Wave-problem} blows up in a finite time.
\end{thm}
\begin{proof}[Proof of Theorem \ref{thm-blow-up-E=d}]
Let
\begin{align*}
M(t) = \int_{\Omega } u^2dx, \,\, \text{ and }\,\, M'(t) = 2\int_{\Omega } uu_tdx.
\end{align*}
As in previous case we have
\begin{equation*}
M''(t) \geq (\alpha +2)\int_{\Omega} u_t^2 dx + \left(\lambda_1(\alpha -2)- 2\beta\right) M(t) - 2\alpha d,
\end{equation*}
and
\begin{equation}
M''(t) = 2 \int_{\Omega } u^2_t dx - 2I(u)>0,
\end{equation}
for $t \in [0,\infty)$ because of $\int_{\Omega } u_0(x)u_1(x)dx \geq 0$ and Lemma \ref{lem-10}. $M''(t) >0$ means that $M'(t)$ is strictly increasing for $t\in [0,\infty)$. Hence for any $t_0>0$ we have $M'(t)\geq M'(t_0)>0$ for $t\geq t_0$,
\begin{equation*}
M(t) \geq M'(t_0)(t-t_0) +M(t_0) \geq M'(t_0)(t-t_0).
\end{equation*}
Therefore there exists sufficiently large time $t$ such that
\begin{equation*}
\left(\lambda_1(\alpha -2)- 2\beta\right) M(t) > 2\alpha d,
\end{equation*}
for $2\beta<\lambda_1(\alpha -2)$. Then we have
\begin{equation*}
M''(t)\geq (\alpha+2)\int_{\Omega }u^2_tdx.
\end{equation*}
The rest of proof repeats the same computation as in Theorem \ref{thm-blow}.
\end{proof}
\section{Conclusion}\label{Sec6}
In conclusion, we arrive at the following results of global existence and blow-up of solutions for initial-boundary problem \eqref{Wave-problem}:
\begin{thm}
Let $f(u)$ satisfy $(H)$, $u_0(x)\in H_0^1(\Omega)$, $u_1(x)\in L^2(\Omega)$.
Then problem \eqref{Wave-problem} admits a global weak solution if the following assumptions hold:
\begin{itemize}
\item $E(0)<d$ and $I(u_0)\geq0$;
\item $E(0)=d$, $(u_0(x),u_1(x))\geq 0$, and $I(u_0)\geq0$.
\end{itemize}
The solution of problem \eqref{Wave-problem} blows up in finite time if the following assumption hold:
\begin{itemize}
\item $E(0)<d$ and $I(u_0)<0$;
\item $E(0)=d$, $(u_0(x),u_1(x))\geq 0$, and $I(u_0)<0$.
\end{itemize}
\end{thm}
|
1,116,691,497,742 | arxiv | \section{Introduction}
\label{sec:Introduction}
The Mott metal-insulator transition is one of the most striking manifestations of electronic correlations~\cite{Imada_RMP}. In a single-orbital Hubbard model at half filling (one particle per lattice site), carriers localize when the ratio between the Hubbard on-site repulsion $U$ and the hopping $t$ exceeds a certain critical value $U_c/t$.
In multiorbital models, this simple scenario is modified as $U$ is supplemented by interorbital terms, such as the Hund's exchange coupling, $J$, which favours high-spin configurations~\cite{Haule_Kotliar,De_Medici_Janus,De_Medici_multiorbital_correlations}.
For instance, an effective decoupling between the orbitals~\cite{De_Medici_out_lifting,De_Medici_multiorbital_correlations,De_Medici_Selective_Mott,
Yin2011,Fanfarillo_Electronic_correlations,Capone_Citro} opens the doors to orbital-dependent correlations and even to orbital-selective Mott phases where only some orbitals become Mott-localized~\cite{De_Medici_out_lifting,Vojta_Orbital_Selective,De_medici_orbital_selective,Ferrero_orbital_selective}, while the spin degrees of freedom are partially frozen~\cite{Spin-freezing}, anomalous responses are observed~\cite{Stadler_spin_orbital_separation}, and a peculiar spectral weight redistribution favours orbital-selective superconductivity~\cite{Fanfarillo2020}.
Turning back to the Mott transition, it has been shown that in a $N$-orbital Hubbard model at integer filling different from $N$, increasing $J/U$ (see below for the definition of the Hamiltonian) pushes $U_c/t$ to very high values~\cite{De_Medici_Janus} leading to a wide region of an interaction-resistant metal that has been labelled as a ``Hund's metal".
For the specific value $J/U=1/3$, the system remains metallic even in the limit of infinite $U$ and $J$. This scenario is particularly surprising, at least at first sight, because both $U$ and $J$ constrain the motion and are expected to localize the carriers.
In Ref.~\cite{Isidori_Charge_disproportionation}, the resilience of the metallic solution for large interactions has been described in terms of the competition between two strongly correlated solutions, a high-spin Mott state favoured by $U$ and a disproportionated Hund's insulator favoured by $J$.
In the last years, a new platform for the quantum simulation of multicomponent Hubbard-like models has been realized by means of ultracold $^{87}\mathrm{Sr}$~\cite{Strontium87}, $^{173}\mathrm{Yb}$~\cite{Ytterbium173}, and $^{6}\mathrm{Li}$ atoms~\cite{Ottenstein_Li_6,Huckans_Li_6}. In these atoms, the nuclear spin is essentially decoupled from the electronic degrees of freedom, thus providing one with $10$, $6$, or $3$ different spin flavors, and allowing for the quantum simulation of SU($N$)-symmetric Hubbard models, if loaded in deep optical lattices ~\cite{Gorshkov,Cazalilla,Pagano_1,Zhang,Pagano_2,Hofer,Cappellini,Scazza,Del_Re_Capone}.
Rather than a direct quantum simulation of the multiorbital Hubbard model with Hund's coupling, which is complicated by the need to define an orbital and a spin degree of freedom, we propose a different model which is expected to display a physics similar to the one of the Hund's metal, namely an SU($3$) system with a site-dependent single-particle potential characterized by a three-site pattern where two sites out of three have a lower energy with respect to the third. In this model the standard Mott insulator competes with a density wave pattern, similarly to the ionic Hubbard model on bipartite lattice.
In this work, we solve the two models by exact diagonalization in the smallest system that can capture the main physics of the two models, namely a three-site cluster, or ``trimer". The choice of this minimal lattice allows one to perform a full diagonalization of the Hamiltonian and to present thorough results for both the ground state and the thermal properties in an unbiased way. In particular, as we discuss in the following, the trimer is the smallest cluster which can host all the insulating solutions expected in the model.
For the multiorbital Hubbard model, we show that the trimer reproduces the most important features which have been uncovered in the thermodynamic limit using Dynamical Mean-Field Theory (DMFT)~\cite{Georges_Z_propto_D}, slave-spin mean-field theory~\cite{Capone_Citro}, and rotation-invariant slave bosons (RISB)~\cite{risb}.
This agreement represents a cross validation of different approaches with different limitations. In fact all the above methods are defined in the thermodynamic limit, but they treat dynamical correlations only at a local level, while the exact diagonalization of the trimer obviously includes also non-local (nearest-neighbor) correlations, but it suffers from finite-size effects.
The present results complement the previous literature with an analysis of the low-energy excitations and their relation with finite-temperature properties and with the investigation of nearest-neighbor correlations and their comparison with the on-site counterparts.
We highlight the similarity between the three-band Hubbard model and the three-flavor Hubbard model, which both display an interaction-resilient metal as a result of the competition between two different localization mechanisms. In both cases the metallic character is shown to be supported by the coexistence, within the same state, of atomic multiplets characteristic of the two competing localized solutions.
The outline of our work is the following. In Sec. \ref{sec:Kanamori_trimer} we investigate the three-site three-orbital Hubbard model with Kanamori interactions. In Sec. \ref{sec:Hubbard_trimer} we analyze the different physical regimes offered by a three-flavor ultracold fermionic gas with a patterned potential. Sec. \ref{sec:Concluding_Remarks} is devoted to some concluding remarks.
\section{The Hubbard-Kanamori trimer}
\label{sec:Kanamori_trimer}
In this section we introduce the three-orbital Hubbard-Kanamori model on a trimer
\begin{equation}
\label{eq:H_Kanamori}
H=-\sum_{ij,ab,\sigma} t_{ij}^{ab} d^\dagger_{ia\sigma}d_{jb\sigma} +\sum_j H_{\mathrm{int},j}
\end{equation}
where operator $d_{ia\sigma}^\dagger$ creates a spin-1/2 fermion with spin $\sigma=\uparrow,\downarrow$ in orbital $a=1,2,3,$ of site $i$. We assume three degenerate levels with orbital-diagonal hopping $t_{ij}^{ab}=t_{j,j+1}^a\delta_{ab}\delta_{ij}=:t\, \forall j,a$, i.e., we do not include hybridization between orbitals or crystal-field splitting. We expect our results to be robust with respect to terms breaking the symmetry between the orbitals, including different hopping, hybridization and crystal-field splitting as long as these terms are not comparable with the interaction terms. The robustness of the results is expected also in light of previous calculations for realistic electronic structures for iron-based superconductors \cite{YuSi,Lanata,De_Medici_Selective_Mott} which confirm the picture obtained for the symmetric model.
The local interaction reads
\begin{align}
\label{eq:H_int_Fock}
H_{\mathrm{int},j} & = U\sum_a n_{ja\uparrow}n_{ja\downarrow}
+ (U-3J)\sum_{a<b, \, \sigma} n_{ja\sigma}n_{jb\sigma} \nonumber\\
& + (U-2J)\sum_{a\neq b} n_{ja\uparrow}n_{jb\downarrow}
+ J\sum_{a\neq b} d^\dagger_{ja\uparrow}d^\dagger_{ja\downarrow}\,d_{jb\downarrow}d_{jb\uparrow} \nonumber\\
& - J \sum_{a\neq b} d^\dagger_{ja\uparrow}d_{ja\downarrow}\,d^\dagger_{jb\downarrow}d_{jb\uparrow},
\end{align}
where $U>0$ and $J>0$ are the standard Hubbard repulsion, and the Hund's exchange coupling, respectively.
The interaction term (\ref{eq:H_int_Fock}) can be recast in the form
\begin{equation}
\label{eq:H_int_n_L_S}
H_{\mathrm{int},j} = \frac{U-3J}{2}\hat{n}_j(\hat{n}_j-1) -J\left(2\mathbf{S}_j^2 + \frac{1}{2}\mathbf{L}_j^2 - \frac{5}{2} \hat{n}_j\right)
\end{equation}
where $\hat{n}_j = \sum_{a\sigma}d^\dagger_{ja\sigma} d_{ja\sigma}$ counts the total number of fermions at site $j$, ${\mathbf{S}_j} = \frac{1}{2} \sum_{a,\,\sigma\sigma'} d^\dagger_{ja\sigma} \hat{\bm{\sigma}}_{\sigma\sigma'} d_{ja\sigma'}$ and ${\mathbf{L}_j} = \sum_{ab,\,\sigma} d^\dagger_{ja\sigma} \hat{\bm{\ell}}_{ab} d_{jb\sigma}$ represent the local spin and orbital angular momentum operators, being $\hat{\bm{\sigma}}_{\sigma\sigma'}$ the Pauli matrices and $\hat{\bm{\ell}}^{(a)}_{bc} = - i \epsilon_{abc}$ the generators of group $O(3)$. The form (\ref{eq:H_int_n_L_S}) is rather instructive, as it highlights the presence of two contributions: the on-site repulsion ($\propto U-3J$) and the exchange mechanism ($\propto J$) which favours primarily high-spin states and, as a second condition, high-orbital-angular-momentum states, thus realizing the first two Hund's rules.
We consider a system with $L=3$ lattice sites (so that $i,j=1,2,3$) with periodic boundary conditions and a total of 6 particles, i.e. 2 particles on average per site.
This very small system has proven to be an effective minimal lattice in the investigation of a number of physical phenomena, for both fermionic~\cite{Shiba_Pincus_Specific_heat,Ullrich_Trimer,Schilling_trimer,Aligia_trimer} and bosonic~\cite{Richaud_PRA_supermixing,Richaud_Sci_Rep_1,Sci_Rep_Zenesini,Richaud_NJP,Las_Phys_2018} systems.
To our purposes, the trimer system is particularly useful because it can host all the insulating solutions expected in our model (and in the second model we discuss in Sec. \ref{sec:Hubbard_trimer}). Furthermore, small clusters can be used as building blocks for quantum cluster theories such as cluster perturbation theory \cite{Cluster-Perturbation-Theory}, variational cluster approximation \cite{Potthoff} or cluster extensions of DMFT \cite{CDMFT,DCA}.
\subsection{Atomic multiplets}
\label{sub:Rise_and_fall}
We begin reviewing known results about the atomic limit ($t=0$) of the Hamiltonian (\ref{eq:H_Kanamori}) which are useful to understand the results and to compare them with the three-component model we propose in Sec. \ref{sec:Hubbard_trimer}.
If $J/U$ is small, the system is in a Mott insulating (MI) state, which means that there are exactly 2 fermions on each site ($n_j = 2, \, \forall\, j$). Moreover, due to the presence of a non-zero $J$, on each site we have $S_j=1$ and $L_j=1$, and so the energy reads $E_{\mathrm{MI}}=3(U-3J)$. For larger values of $J/U$ the tendency to realize high-spin configurations is so strong that it becomes energetically convenient to have different numbers of fermions on different sites. In particular, for $1/3<J/U<3/4$, the ground state features two sites with $n_j=3$, $S_j=3/2$, and $L_j=0$, while one site is empty ($n_j=S_j=L_j=0$). This state is a disproportionated Hund's insulator (HI)~\cite{Isidori_Charge_disproportionation} with energy $E_{\mathrm{HI}}=6(U-3J)$. Thus the MI and the HI are degenerate, $E_{\mathrm{MI}}=E_{\mathrm{HI}}$, for $J/U=1/3$, as shown in Fig. \ref{fig:E_0_vs_J_U_Kanamori_3_with_inset}, where we compare these limiting results with the exact ground state energy $E_0 = \langle \psi_0| H|\psi_0\rangle$ computed for a finite $U/t =22$.
\begin{figure}[h!]
\centering
\includegraphics[width=1\linewidth]{E_0_vs_J_U_Kanamori_3_with_inset.pdf}
\caption{Ground state energy of a three-band Kanamori trimer hosting 6 particles as a function of $J/U$. Comparison between exact numerical diagonalization of Hamiltonian (\ref{eq:H_Kanamori}) for $t=1$ and $U=22$ (yellow dash-dotted line) and the atomic limit for $J/U\ll 1/3$ (blue solid line) and $J/U\gg 1/3$ (red dashed line). The gray vertical line has been drawn at $J/U=1/3$. The two sketches show the fermionic configurations for MI-like (left) and HI-like (right) states.}
\label{fig:E_0_vs_J_U_Kanamori_3_with_inset}
\end{figure}
Following Ref. ~\cite{Isidori_Charge_disproportionation}, in Fig. \ref{fig:Multiplets_populations_Kanamori_3} we show the evolution, as a function of $J/U$, of the population of the most relevant atomic multiplets for $U/t = 22$. The populations are simply the sum over degenerate atomic states of $|\langle n_j, S_j |\psi_0 \rangle|^2$ where $|n_j, S_j \rangle$ is an atomic state with the corresponding quantum numbers. The results do not depend on the site because of translation invariance. We only show the most relevant configurations, which are those with a high spin, because of the finite value of the Hund's coupling.
As expected, for small values of $J/U$, the ground state is a high-spin MI and the configuration $n=2$, $S=1$ is predominant. Conversely, for large values of $J/U$, we find the two local configurations with different values of $n$ characteristic of the HI: high-spin triplets $(n=3,\,S=3/2)$ with weight $\approx 2/3$ and empty sites with weight $\approx 1/3$.
\begin{figure}[h!]
\centering
\includegraphics[width=1\linewidth]{Multiplets_populations_Kanamori_3.pdf}
\caption{Multiplets' populations relevant to the ground state of Hamiltonian (\ref{eq:H_Kanamori}) as functions of the control parameter $J/U$. $t=1$, $U=22$ and 6 particles have been used. The gray vertical line has been drawn at $J/U=1/3$.}
\label{fig:Multiplets_populations_Kanamori_3}
\end{figure}
Following, in Fig. \ref{fig:Multiplets_populations_Kanamori_3}, the evolution of the multiplets as a function of $J/U$ at fixed and large $U$, one can notice that, for $J/U\approx 1/3$, the multiplet $n=2,\,S=1$ falls, while multiplets $n=3,\,S=3/2$ and $n=0$ rise. In this region, an additional multiplet, $n=1$, gets populated. This is understood from the atomic limit, where, for $J/U=1/3$, the energy $E_{\mathrm{MI}} = E_{\mathrm{HI}}$ coincides with that of the configuration $n_1=1,\,S_1=1/2,\,L_1=1$; $n_2=2,\,S_2=1,\,L_2=1$; $n_3=3,\,S_3=3/2,\,L_3=0$ (or any other permutation of the site indices). In this regime, different high-spin multiplets with different local occupation are selected. We notice that the sum of the probabilities of these multiplets is persistently very close to 1, signaling that all the other configurations with lower spin are essentially irrelevant. In Sec. \ref{sub:Entropy} we will show that the region where different multiplets are populated is linked to a well-defined maximum of the associated entropy.
Finally we notice that this scenario is determined essentially by the competition between local interaction terms. In this light, we expect it to be robust to the variation of single-particle terms, including orbital-dependent hoppings and energy splittings between the atomic energies of the various orbitals, at least as long as these changes are not so large to overcome the effect of the on-site terms. For instance, lifting one orbital by an energy $\Delta$ favors an uneven occupation between orbitals. Therefore a $\Delta \gg J$ would quench the effect of $J$ introducing qualitative changes, while a reasonably smaller $\Delta$ would only introduce minor corrections.
\subsection{Conduction Properties}
\label{sub:Probing_metal}
Even if our small system can not display real metal-insulator transitions, in this section we probe the conduction properties of the ground state to address its metallic nature in the region around $J/U = 1/3$. We follow the standard prescription~\cite{Kohn_Drude,Fye_Drude,Giamarchi_Drude,Scalapino_Drude} to compute the current by rotating the ring. In the rotating frame, the Coriolis force is formally equivalent to a magnetic flux $\Phi=2\pi m_e R^2 \Omega /\hbar $ threading the ring ($m_e$ is the fermion mass, and $R$ is the radius of the circumference where the ring-trimer is inscribed in, and $\Omega$ the angular rotation frequency) ~\cite{Arwas}. Accordingly, the hopping term of Hamiltonian (\ref{eq:H_Kanamori}) acquires a Peierls phase
\begin{equation}
\label{eq:H_t_rotante}
H_{\mathrm{hop}}=-t \sum_{j,a,\sigma} \left(e^{i\frac{\Phi}{L}}d_{j+1,a,\sigma}^\dagger d_{j,a,\sigma} +\mathrm{h.c.}\right),
\end{equation}
while the interaction term (\ref{eq:H_int_Fock}) is unchanged.
Denoting with $H'$ the new Hamiltonian, and with $\left| \psi_0^\prime \right\rangle$ the corresponding ground state, we consider the expectation value of the current operator $I=\left\langle \psi_0^\prime\left| -\frac{\partial H^\prime}{\partial \Phi}\right|\psi_0^\prime \right \rangle$ in this new ground state. For small fluxes, $I$ is proportional to the Drude weight ~\cite{Giamarchi_Drude}, which is, in turn, proportional to the (singular part of the) DC electrical conductivity. Rather than a numerically unstable calculation of the conductivity in the limit of zero flux, in Fig. \ref{fig:Piano_U_J} we plot the current for a fixed small flux $\Phi=0.1 \pi$.
The results are consistent with an insulating behavior in two regions of the plane $(U/t,J/t)$. The first region is the one for large values of $U/t$ and small values of $J/t$, while the second region is found at large values of $J/t$. As discussed in Sec. \ref{sub:Rise_and_fall}, these are the MI and HI respectively.
Importantly, one can appreciate the presence of a stripe centered about the line $J/U=1/3$ where the current $I$ is persistently rather large. This is the Hund's metal region, where the simultaneous presence of different atomic multiplets (see Fig. \ref{fig:Multiplets_populations_Kanamori_3}) connected by hopping processes ensures the motion of the carriers regardless of the large values of $U$ and $J$.
Our results follow the qualitative behaviour shown in Ref.~\cite{Isidori_Charge_disproportionation} demonstrating that the trimer captures the competition between solutions which is present in the thermodynamic limit.
The analysis we reported so far shows that the exact results for the trimer provide a very similar picture as those obtained using RISB \cite{Isidori_Charge_disproportionation} which are in turn consistent with slave-spin \cite{De_Medici_multiorbital_correlations,Capone_Citro} and DMFT \cite{De_Medici_Janus} results.
This agreement is far from trivial since all the above approaches are defined in the thermodynamic limit, but they include only on-site dynamical correlation effects, while the solution of the trimer is numerically exact, but it is obviously limited by important finite-size effects. In this regard, the agreement is a mutual validation of the two approaches which strengthens the evidence of a correlation-resistant Hund's metal.
\begin{figure}[h!]
\centering
\includegraphics[width=1\linewidth]{Piano_U_J.pdf}
\caption{Expectation value of the current for $\Phi=0.1 \pi$ as a function of $U/t$ and $J/t$ for $t=1$ and 6 particles. Yellow corresponds to large values of $I$, while dark-blue corresponds to vanishing small values of $I$. The red line corresponds to $J/U=1/3$.}
\label{fig:Piano_U_J}
\end{figure}
\subsection{Energy spectrum and specific heat}
\label{sub:Spectrum_collapse}
The exact numerical diagonalization of the system's Hamiltonian gives access to the full excitation spectrum. In Fig. \ref{fig:Energy_levels_Kanamori_3}, we plot the first $3.500$ energy levels $E_i=\langle \psi_i|H|\psi_i\rangle$ of Hamiltonian (\ref{eq:H_Kanamori}), as a function of the control parameter $J/U$. One can clearly notice the presence of different \textit{bundles} of energy levels, which come together or move apart upon varying $J/U$. Each bundle corresponds to a specific class of excitations. Importantly, at any given value of $J/U$, we can extract from these data valuable information about the hierarchical structure of the excitations, for example which degrees of freedom are active and which other are frozen.
As an example, the lowest bundle which is found in the MI at $J/U=0.1$ is composed by $729=9^3$ energy levels, which is easily understood because the degeneracy of the multiplet with $n_i=2$, $S_i=1$, and $L_i=1$ is indeed $9$ ~\cite{Isidori_Charge_disproportionation}.
All these states are degenerate in the atomic limit and they display small splittings for finite $t$ due to virtual hopping processes depending on the specific arrangement of the fermions in each state. The second lowest bundle is formed by $1215=9^2\cdot 5\cdot 3$ levels. This is the number of possible states such that \textit{one} of the three sites features one minimal violation of Hund rules, namely $n_i=2,S_i=0,L_i=2$ (the degeneracy of this single-site configuration is $5$). The energy gap between this bundle and the lowest one is $2J$ ($\approx 4.4$ for $J=0.1\,U$ and for the same model parameters used in Fig. \ref{fig:Energy_levels_Kanamori_3}). With a similar reasoning, one can verify that the third lowest bundle, which includes $675=9\cdot 5^2\cdot 3$ levels, corresponds to states where \textit{two} sites feature $n_i=2,S_i=0,L_i=2$ and it lies around an energy $4J$ ($\approx 8.8$ for $J=0.1\,U$ and for the same model parameters used in Fig. \ref{fig:Energy_levels_Kanamori_3}). It is only after another bundle including $243=9^2\cdot 1 \cdot 3$ levels (where $1$ is the degeneracy of single-site configuration $n_i=2$, $S_i=0$, $L_i=0$) and having energy $E_{\mathrm{MI}}+5J$, that we reach states in which the Mott condition $n_i=2$ is violated. The energy gap of these charge excitations is, for small values of $J/U$, $U-3J$. Notice that this gap depends on both $U$ and $J$, and closes exactly at $J/U=1/3$, where Hund's metallicity is found. This gap closure is clearly visible in the figure, and, indeed, it corresponds to the lowering of the energy of the charge excitation as $J/U$ grows.
\begin{figure}[h!]
\centering
\includegraphics[width=1\linewidth]{Energy_levels_Kanamori_3.pdf}
\caption{First 3.500 energy levels of Hamiltonian (\ref{eq:H_Kanamori}) as a function of $J/U$. $t=1$, $U=22$ and 6 particles have been used. The gray vertical line has been drawn at $J/U=1/3$.}
\label{fig:Energy_levels_Kanamori_3}
\end{figure}
In the opposite limit (right side of Fig. \ref{fig:Energy_levels_Kanamori_3}), the lowest bundle is made up of $48=4\cdot 4 \cdot 1 \cdot 3$ levels, where $4$ is the degeneracy of the single-site configuration $n_i=3$, $S_i=3/2$, $L_i=0$, and $1$ is the degeneracy of $n_i=0$, $S_i=0$, $L_i=0$. As opposed to the case of small Hund's coupling, here the second lowest bundle already involves \textit{charge} excitations. In fact, it includes $1296=9\cdot 4\cdot 6 \cdot 6$ levels, where one factor $6$ is the degeneracy of the single-site configuration $n_1=1$, $S_1=1/2$, $L_i=1$, and the other factor $6$ represents the number of possible permutations of site indices. Notice that the gap of this bundle is of the order of $6J-2U$ ($\approx 15.4$ for $J=0.45\,U$ and for the same model parameters used in Fig. \ref{fig:Energy_levels_Kanamori_3}) and that also this gap tends to close approaching $J/U=1/3$.
In spite of the rather complex dependence of the energy levels' structure on the control parameter $J/U$ (see Fig. \ref{fig:Energy_levels_Kanamori_3}), we have now a clear picture in which, approaching the limit $J/U = 1/3$, the charge gap collapses coming from both the MI and the HI, leading to the metallization.
The spectrum we have discussed is naturally reflected in the thermodynamic properties. We compute the specific heat $c=L^{-1}\partial \langle E\rangle/ \partial T$ where $\langle E \rangle = \frac{1}{Z}\sum_i E_i e^{-\frac{E_i}{k_BT}}$
is the thermal expectation value of the internal energy and $Z$ is the partition function. The result is shown in Fig. \ref{fig:Piano_J_T_Kanamori_3} in a $J/U$-$T$ plane (where the logarithmic scale for the temperature $T$ is used for graphical clarity). For a given value of $J/U$, the specific heat (regarded as a function of $T$) features peaks where a certain class of excitations unfreezes ~\cite{Shiba_Pincus_Specific_heat,Werner_specific_heat}. Therefore, the evolution of the different ``ridges" which we observe is directly connected with the evolution of the different bundles of energy levels in Fig. \ref{fig:Energy_levels_Kanamori_3}.
\begin{figure}[h!]
\centering
\includegraphics[width=0.9\linewidth]{Piano_J_T_Kanamori_3.pdf}
\caption{Specific heat as a function of $J/U$ and temperature $T$ (in logarithmic scale). Model parameters: $t=1$, $U=22$, $k_B=1$, and 6 particles. The two dashed red (white) lines correspond to simple analytical estimates of the specific heat contribution coming from charge excitations (local variations of quantum numbers $S_j$ and $L_j$) The vertical solid red line corresponds to $J/U=1/3$.}
\label{fig:Piano_J_T_Kanamori_3}
\end{figure}
\subsection{On-site and nearest-neighbors correlation properties}
\label{sub:Correlations}
In this section, we focus on charge and spin correlation functions of the Hubbard-Kanamori trimer. This analysis extends previous investigations based on DMFT and slave-particle mean-field approaches, which only focused on \textit{local} correlators.
We define the charge correlation functions between sites $i$ and $j$
\begin{equation}
\label{eq:C_jj_tot}
C_{i,j}^{\mathrm{tot}}= \langle \hat{n}_i\hat{n}_j \rangle -\langle \hat{n}_i \rangle\langle \hat{n}_j \rangle,
\end{equation}
where $\hat{n}_j = \sum_a\sum_\sigma \hat{n}_{j,a,\sigma}$ is the total density operator on site $j$.
We can decompose the correlations in an intra-orbital and an inter-orbital contribution according to $C_{i,j}^{\mathrm{tot}}=C_{i,j}^{\mathrm{intra}}+C_{i,j}^{\mathrm{inter}}$~\cite{Fanfarillo_Electronic_correlations}, where
\begin{equation}
\label{eq:C_jj_intra}
C_{i,j}^{\mathrm{intra}}= N \left( \langle \hat{n}_{i,a}\hat{n}_{j,a} \rangle -\langle \hat{n}_{i,a} \rangle \langle \hat{n}_{j,a} \rangle\right),
\end{equation}
\begin{equation}
\label{eq:C_jj_inter}
C_{i,j}^{\mathrm{inter}}= N(N-1)\left(\langle \hat{n}_{i,a} \hat{n}_{j,b} \rangle -\langle \hat{n}_{i,a}\rangle\langle \hat{n}_{j,b} \rangle\right).
\end{equation}
\begin{figure}[h!]
\centering
\includegraphics[width=0.9\linewidth]{Charge_correlations_Kanamori_3.pdf}
\caption{On-site (upper panel) and first-neighbors (lower panel) charge correlations as a function of $J/U$. $t=1$, $U=22$, and 6 particles. The vertical gray line corresponds to $J/U=1/3$.}
\label{fig:Charge_correlations_Kanamori_3}
\end{figure}
Before discussing the results for these observables, we notice that in our small cluster we have an important constraint. The trivial relation $\langle n^2\rangle = \sum_i \langle n_i^2\rangle + 2\sum_{i<j} \langle n_i n_j\rangle$ and the fact that $\langle n^2\rangle$ is a conserved quantity implies that the sum of the on-site correlators is directly connected with the sum of the nearest-neighbor ones, which are the only non-local contributions to the sum for a three-site system. Therefore, the information about the on-site and nearest-neighbor correlation functions are not independent in our cluster. Yet, they provide us with useful physical information that complements previous studies \cite{Nonlocal_DMFT}.
Fig.~\ref{fig:Charge_correlations_Kanamori_3} illustrates the behavior of on-site (upper panel) and nearest-neighbor (lower panel) charge correlations as a function of $J$ for fixed $U=22\,t$. For both quantities, we resolve intra- and inter-orbital contributions. The behavior of the on-site charge correlation function resembles the results obtained with local mean-fields~\cite{Fanfarillo_Electronic_correlations}. The total correlations are very small in the small-$J/U$ region, where the system is a Mott insulator, and they gradually increase as $J/U$ increases and the system reaches the Hund's insulator (passing through the Hund's-metal region). We notice, in particular, that charge fluctuations are not maximal in the metallic region, but they are even larger in the Hund's insulator, a circumstance which reflects the charge disproportionation. It is also clear from the figure that the evolution as a function of $J$ of the charge correlations is entirely due to the inter-orbital component, while the intra-orbital contribution is totally unaffected by $J$ and it only depends on $U/t$.
As expected, the results for the nearest-neighbor correlations follow a similar qualitative trend, with the total correlator vanishing in the MI and increasing (in absolute value) as $J/U$ grows. We remark that the inter-orbital correlations cross zero around the Hund's metal region for $J/U=1/3$ reflecting the decoupling between excitations in different orbitals, or orbital decoupling~\cite{De_Medici_out_lifting,De_Medici_multiorbital_correlations,Fanfarillo_Electronic_correlations,Capone_Citro}. We find therefore that the decoupling, which has been so far reported for on-site correlations, extends also to nearest-neighbor quantities, thereby strengthening its relevance.
Another important piece of information comes from the spin-spin correlation functions
\begin{equation}
\label{eq:M_jj_tot}
M_{i,j}^{\mathrm{tot}}= \langle \hat{\sigma}_i\hat{\sigma}_j \rangle -\langle \hat{\sigma}_i \rangle\langle \hat{\sigma}_j \rangle
\end{equation}
where $\hat{\sigma}_{j} = \sum_a \hat{\sigma}_{j,a}$, where $\hat{\sigma}_{j,a} = (n_{j,a,\uparrow} - n_{j,a,\downarrow})/2$.
Also in this case, the correlators can be expressed in terms of inter- and intra-orbital correlators as $ M_{i,j}^{\mathrm{tot}}=M_{i,j}^{\mathrm{intra}}+M_{i
,j}^{\mathrm{inter}}$, where
\begin{equation}
\label{eq:M_jj_intra}
M_{i,j}^{\mathrm{intra}}=N\left( \langle \hat{\sigma}_{i,a}\hat{\sigma}_{j,a} \rangle -\langle \hat{\sigma}_{i,a} \rangle \langle \hat{\sigma}_{j,a} \rangle\right),
\end{equation}
\begin{equation}
\label{eq:M_jj_inter}
M_{i,j}^{\mathrm{inter}}= N(N-1)\left(\langle \hat{\sigma}_{i,a} \hat{\sigma}_{j,b} \rangle -\langle \hat{\sigma}_{i,a}\rangle\langle \hat{\sigma}_{j,b} \rangle\right).
\end{equation}
Also in this case on-site and nearest-neighbors correlators are connected by $\langle \sigma_z^2\rangle = \sum_i \langle \sigma_{i}^2\rangle + 2\sum_{i<j} \langle \sigma_i \sigma_j\rangle$.
The functional dependence of these on-site magnetic correlations on $J/U$ is illustrated in the upper panel of Fig. \ref{fig:Spin_correlations_Kanamori_3} in the regime $U/t\gg 1$. On-site spin correlations are positive and they grow with $J/U$ signalling the increased on-site magnetic moment. Since intra-orbital spin fluctuations are constant throughout the whole explored range of $J/U$, the only contribution comes from inter-orbital spin alignment due to the Hund's coupling.
\begin{figure}[h!]
\centering
\includegraphics[width=1\linewidth]{Spin_correlations_Kanamori_3.pdf}
\caption{On-site (upper panel) and first-neighbours (lower panel) magnetic correlations with respect to the control parameter $J/U$. Model parameters $t=1$, $U=22$, and 6 particles have been used. The vertical gray line corresponds to $J/U=1/3$.}
\label{fig:Spin_correlations_Kanamori_3}
\end{figure}
Nearest-neighbor magnetic correlations are shown in the lower panel of Fig. \ref{fig:Spin_correlations_Kanamori_3}. The total correlator $M_{j,j+1}^{\mathrm{tot}}$ is negative, signalling antiferromagnetic spin correlations between the large local magnetic moments. The absolute value grows with $J/U$ without any anomaly when the Hund's metal region is reached and crossed. An antiferromagnetic ordering is found also in the Hund's insulator. In this case the intra-orbital correlations are negative and they have a mild dependence on $J/U$ which combines with the larger dependence of the inter-orbital terms to provide the final result.
\section{The SU(3) Hubbard trimer with patterned potential}
\label{sec:Hubbard_trimer}
In this section, we focus on a fermionic SU(3) Hubbard model with a suitable patterned potential which favours a charge-ordered state. The Hamiltonian reads
$$
H=-t\sum_{j=1}^L \sum_{a=1}^N\left(d^\dagger_{j,a}d_{j+1,a}+\mathrm{h.c.}\right)
$$
\begin{equation}
\label{eq:H_SU_3}
+ \frac{U}{2}\sum_{j=1}^L\hat{n}_j(\hat{n}_j-1) + \sum_{j=1}^L \mu_j \hat{n}_j,
\end{equation}
where operator $d^\dagger_{j,a}$ creates a fermion with flavor $a=1,2,3$ on site $j$, and operator $\hat{n}_j:=\sum_{a=1}^N \hat{n}_{j,a}$ counts the number of fermions at site $j$. Here, $N=3$ is the number of flavors, and $L=3$ is the number of sites, $t$ represents the hopping, $U$ is the Hubbard interaction and $\mu_j$ a site-dependent potential corresponding to the presence of a superlattice. We will assume two sites with the same energy and one at a higher energy $\mu_1=-\mu$, $\mu_2=0$, $\mu_3=-\mu$ ($\mu >0$). The physics of this model is ruled by the competition between $U$, which penalizes local occupancies different from the global average density, and $\mu$, which favours the occupation of the low-energy sites over the high-energy one. A similar pattern can also be realized in larger lattices formed by periodic repetitions of the three-site cluster. In this sense, also in this case, the trimer can be seen as the building block of a quantum cluster theory for a larger system.
The model can be seen as a three-flavour/three-site version of the ionic Hubbard model~\cite{Fabrizio_ionic,Wilkens_ionic,Japaridze_ionic,Manmana_ionic,Batista_ionic,Torio_ionic,Chattopadhyay_ionic,Garg_ionic,Paris_ionic}, where the local Hubbard repulsion competes with a staggered potential. This leads to a competition between Mott and charge-density-wave insulators, separated in the phase diagram by a narrow stripe where, depending on the dimensionality, a metal~\cite{Bouadim_metallic_ionic_2D} or a bond-order-wave phase have been reported, the latter being characterized by a staggered kinetic-energy on the bonds~\cite{Sengupta_BOW}.
The ionic Hubbard model has been recently realized in a Fermi gas loaded into a honeycomb optical lattice with a staggered energy offset~\cite{Messer_ionic_ultracold_atoms}. The two insulating phases were observed, but the bond-order-wave phase still eludes experimental detection~\cite{Loida_Bond_order_wave}.
This kind of system, where the three-site unit is periodically repeated in space, can be experimentally realized by means of a multicomponent fermionic quantum gas, for example $^{6}\mathrm{Li}$~\cite{Ottenstein_Li_6,Huckans_Li_6}, $^{87}\mathrm{Sr}$~\cite{Strontium87}, or $^{173}\mathrm{Yb}$~\cite{Ytterbium173} preparing a balanced mixture of three species with different nuclear spin. The second step is to introduce an optical lattice which can be partitioned in three sublattices such as the triangular~\cite{Triangular_lattice} or the Kagome lattice~\cite{Kagome_lattice} in two dimensions, or a simple one-dimensional chain \cite{mancini2015observation}, thus realizing a SU($N$) Hubbard model~\cite{Gorshkov}. If the laser is used to build one of these lattices, we would realize a model with uniform single-particle potential. In order to introduce the patterned potential where one site out of three has a higher energy, one can superimpose a second optical lattice with a larger wavelength which selects only one of the three sublattices~\cite{Messer_ionic_ultracold_atoms, Superlattice}.
\subsection{From a Mott-like insulator to a band-like insulator through metal-like states}
\label{sub:Fro_MI_to_BI}
We start our study from the atomic limit ($t=0$) of the Hamiltonian (\ref{eq:H_SU_3}) in the presence of $2L=6$ fermions. In contrast with the Hubbard-Kanamori model, we find three different regimes (see Fig. \ref{fig:E_0_vs_mu_U_SU_3_with_inset}). For $0<\mu/U<1$, the ground state describes a Mott insulator with
exactly two fermions per site (see the left part of the upper panel of Fig. \ref{fig:Multiplets_populations_SU_3}), which cannot hop due to the large Coulomb repulsion $U$. The energy of this configuration is $E_{\mathrm{MI}}=3U-4\mu$. On the other hand, for $\mu/U>2$, the non-uniform potential prevails over the Coulomb interaction and it is energetically convenient to pack all the fermions in the two low-energy sites, thus manifestly violating Mott's condition (see the right part of the upper panel of Fig. \ref{fig:Multiplets_populations_SU_3}). This configuration, whose energy is $E_{\mathrm{BI}}=6U-6\mu$ can be regarded as the atomic version of a band insulator (BI) (in analogy to the band-insulating phase of the well-known ionic Hubbard model ~\cite{Bouadim_metallic_ionic_2D}). Indeed, if we consider a large lattice with a finite hopping, the three atomic levels will broaden into bands. For small hopping amplitudes, the bands arising from the low-energy sites will be fully occupied while the band originating from the high-energy site will be empty. These two solutions are conceptually connected with the Mott and Hund's insulators found in the three-orbital Hubbard-Kanamori model.
\begin{figure}[h!]
\centering
\includegraphics[width=1\linewidth]{E_0_vs_mu_U_SU_3_with_inset.pdf}
\caption{Ground state energy of a three-flavor Hubbard trimer with staggered potential hosting 6 fermions for $t=1$, $U=22$ (purple dotted line). Atomic estimates for $\mu/U<1$ (blue solid line), $1<\mu/U<2$ (red dashed line), and $\mu/U>2$ (yellow dash-dotted line) are also shown. The three atomic configurations are sketched in the corresponding regions. The gray vertical lines have been drawn at $\mu/U=1$ and $\mu/U=2$.}
\label{fig:E_0_vs_mu_U_SU_3_with_inset}
\end{figure}
While the two solutions of the previous model become degenerate on a line, here we find a whole intermediate solution which is stable in the range $1<\mu/U<2$. Here, the competition between $U$ and $\mu$ results in a set of degenerate ground states such that the high-energy site hosts one fermion, while the low-energy sites host the remaining 5 fermions (see the central part of the upper panel of Fig. \ref{fig:Multiplets_populations_SU_3}).
The total energy of this intermediate configuration, in the atomic limit, reads $E_{\mathrm{INT}}=4U-5\mu$ and, as illustrated in Fig. \ref{fig:E_0_vs_mu_U_SU_3_with_inset}, one can verify that $E_{\mathrm{MI}}=E_{\mathrm{INT}}$ for $\mu/U=1$, and that $E_{\mathrm{INT}}=E_{\mathrm{BI}}$ for $\mu/U=2$.
In the following, we discuss the fate of these solutions once a finite hopping is included and their connection with the Hund's metal.
\begin{figure}[h!]
\centering
\includegraphics[width=1\linewidth]{Piano_U_mu.pdf}
\caption{Expectation value of the current operator, as a function of the control parameters $U/t$ and $\mu/t$ for 6 fermions and $\Phi=0.4\, \pi$. Yellow corresponds to large values of $I$, while dark-blue corresponds to a vanishing small value of $I$. The red lines correspond to $\mu/U=1$ and $\mu/U=2$.}
\label{fig:Piano_U_mu}
\end{figure}
\subsection{Conduction properties}
We begin the characterization of the model for finite value of the hopping by computing the current as described in Sec. \ref{sub:Probing_metal}.
The results are reported in Fig. \ref{fig:Piano_U_mu}. For $U/t\gg 1$ and small values of $\mu/t$, we recover the familiar Mott insulator, in which the current is suppressed in order to avoid creating triple occupancies. On the other hand, for $\mu/U \gg 2$, we have a band insulator with fully occupied low-energy sites and empty high-energy site, a configuration which inhibits the current as well.
Interestingly, and in contrast with the Hubbard-Kanamori system, the intermediate region between the two limiting lines $\mu=U$ and $\mu=2U$ does not host a metallic region. When we approach and cross the two lines $\mu=U$ and $\mu=2U$, we have an enhancement of the current, while the intermediate region between the two lines appears to host a metal only for small values of $U$ and $\mu$. These results suggest that an interaction-resistant metal is only realized close to the boundary lines found in the atomic limit, while the intermediate region appears to undergo a Mott transition as $U/t$ increases at fixed $\mu/U$. However, the stabilization of a metal for very large values of $U$ and $\mu$ along the two boundary lines emerges as the counterpart of the Hund's metal for the present SU(3) model. In the next section, we explore in more depth the connection between the two results.
\subsection{Local configurations in the SU(3) Hubbard model with patterned potential}
\label{sub:Shallower_vs_deeper}
In the upper panel of Fig.~\ref{fig:Multiplets_populations_SU_3}, we report the population of states with fixed number of fermions per site in the SU(3) Hubbard model with patterned potential in the strong-coupling regime. Given the nature of the atomic states of this model, we do not need to disentangle the contribution between different multiplets with the same local occupation. Our results clearly highlight that the solutions for finite $t$ are connected with the atomic solution. In particular, it is evident that we obtain a Mott insulator (where all the sites have $n=2$) for small values of $\mu/U$, and a band insulator [with two filled sites ($n=3$) and one empty site ($n=0$)] for large $\mu/U$. The intermediate region is instead characterized by a nearly uniform probability distribution of local configurations with $n=1$, $n=2$ and $n=3$, while the one with $n=0$ is suppressed. Interestingly, comparing the present results with the plot of the current of Fig. \ref{fig:Piano_U_mu}, the maximum value of the current is not obtained in the region where the local configurations are similar, but rather close to the boundaries of the intermediate region. This result may appear surprising since a metal is a state characterized by large density fluctuations.
We can understand this discrepancy noting that the present SU(3) Hubbard model is not translationally invariant because of the different local potential in the three sites. A more insightful picture of the local configurations and their relation with the metallic behavior can be obtained by showing the local occupation at each site, as we do in the lower panels of Fig. \ref{fig:Multiplets_populations_SU_3}. It is clear that, for large $\mu/U$, the low-energy sites 1 and 3 are completely filled, while the high-energy site 2 is practically empty (see Sec. \ref{sub:Fro_MI_to_BI}), while for small $\mu/U$ all the three sites host 2 fermions.
\begin{figure}[h!]
\centering
\includegraphics[width=1\linewidth]{Multiplets_populations_SU_3.pdf}
\includegraphics[width=1\linewidth]{Detailed_populations_SU_3.pdf}
\caption{The top panel shows the occupation of the atomic states in the ground state of the Hamiltonian (\ref{eq:H_SU_3}) as a function of $\mu/U$. $t=1$, $U=22$ and 6 particles have been used. The gray vertical lines have been drawn at $\mu/U=1$ and $\mu/U=2$.
The bottom panels show the site-resolved occupations for the three sites.}
\label{fig:Multiplets_populations_SU_3}
\end{figure}
On the other hand, in the middle of the intermediate region (consider, for example, $\mu \simeq 1.5\, U$), the high-energy site hosts $1$ fermion, while the low-energy sites are occupied by the remaining five fermions (each site with an equal probability to have two or three fermions). This configuration is not favourable for conduction because the hopping processes involving the high-energy site have a large energetic cost, so that the only allowed hopping processes are those connecting the low-energy sites.
When we approach the boundaries of the intermediate region, for example for $\mu \simeq U$, the low-energy sites have a finite amplitude to host two or three fermions, while the high-energy site can host one or two fermions. This pattern of local configurations allows instead for hopping processes as long as the hopping is finite, giving rise to a situation which closely resembles the Hund's metal.
This analysis clarifies the physical origin of the behavior of the current. In the intermediate region $1<\mu/U<2$, we certainly have hopping processes connecting the two low-energy sites, while the high-energy one remains disconnected as long as the hopping is not large enough to overcome the energy difference. According to the geometry and the dimensionality of the lattice, this may lead to metallic or insulating phases. In our small cluster, we clearly find a reduction of the conductivity as $U$ increases along all the lines with constant $\mu/U$ within the window $1<\mu/U<2$. The residual current is merely associated with the hopping between the low-energy sites.
Along the two lines in the $(U/t,\mu/t)$ plane corresponding to $\mu/U=1$ and to $\mu/U=2$, where the intermediate phase becomes degenerate with one of the two limiting solutions in the atomic limit, we find, instead, a metallic solution which exploits the competition between two different tendencies controlled by $U$ and $\mu$.
\subsection{Entropy and multiplet population}
\label{sub:Entropy}
We conclude our investigation of the analogies and differences between the two models by computing the entropy, which is in turn directly connected with the distribution of the local configurations reported in Fig. \ref{fig:Multiplets_populations_Kanamori_3} and Fig. \ref{fig:Multiplets_populations_SU_3}.
As anticipated in Sec. \ref{sub:Rise_and_fall}, it is possible to compute the entropy as
\begin{equation}
\label{eq:mathcal_S}
\mathcal{S}= -\sum_\ell p_\ell \log(p_\ell),
\end{equation}
where $p_\ell$ is the population of the $\ell$-th atomic state.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{Entropy_of_multiplets.pdf}
\caption{Comparison between the entropy associated with the occupation of atomic states $\mathcal{S}$ and the current $I$ for the Hubbard-Kanamori model (upper panel) and the SU(3) Hubbard model (lower panel). For both panels, model parameters $t=1$, $U=22$ have been used, and the data have been normalized to $1$ in order to allow for a direct comparison. }
\label{fig:Entropy_of_multiplets}
\end{figure}
In Fig. \ref{fig:Entropy_of_multiplets} we show, for both models, the evolution of the entropy and of the current along the usual cuts for $U/t =22$. In the case of the Hubbard-Kanamori trimer (upper panel), $\mathcal{S}$ and $I$ increase in a very similar way in the region $J/U < 1/3$ and they reach relatively close maxima in the Hund's metal region before dropping when the Hund's insulator regime is reached. In this case the best metallic behavior is found where various atomic states are democratically populated, so that the entropy is maximized. For $J/U\gtrsim 1/3$, $\mathcal{S}$ converges towards $0.64 \approx -2/3\log(2/3)-1/3\log(1/3)$, while $I$ drops towards $0$. This happens because the two multiplets which remain populated and determine a non-zero $\mathcal{S}$ are not compatible with hopping processes (and hence cannot support conductance).
For the SU(3) Hubbard trimer (lower panel of Fig. \ref{fig:Entropy_of_multiplets}), we find a similar behavior in the two external regions. Within the Mott insulator the entropy and the current increase at the same rate. In the intermediate region we find a different behavior, consistent with the previous observations. The current is maximized close to $\mu=U$ and $\mu=2U$ while the entropy remains large in the whole intermediate window. This is due to the fact that the atomic states selected in the middle of the intermediate region are not compatible with the conduction. The large $\mu/U$ behavior is similar to the large $J/U$ region of the Hubbard-Kanamori model with a finite entropy associated with a residual degeneracy which does not lead to a finite current because the degenerate states are not connected by hopping.
\section{Concluding remarks}
\label{sec:Concluding_Remarks}
In this manuscript, we have studied the general mechanism that stabilizes an interaction-resistant metal, i.e. a metal which survives to large values of repulsion at integer fillings. In particular, we have uncovered the existence of a similar metallic phase in two rather different models of strongly correlated fermions.
Namely, we studied and compared a three-orbital Hubbard model and a three-component SU(3) Hubbard model with a three-site patterned potential.
In both models, the standard Hubbard repulsion, which tends to stabilize a Mott insulator, competes with a term of the Hamiltonian which favours a different state with inhomogeneous density distribution, namely the Hund's coupling in the first model and a non-uniform single-particle potential in the second. In both cases, the competition between the two terms of the Hamiltonian leads to different insulating solutions which are separated in the respective phase diagrams by families of states which exhibit persistent metallicity, even in the presence of strong Coulomb repulsions, hence the name ``interaction-resilient metals". In the case of the Hubbard-Kanamori model, this metallic state has been recently labelled Hund's metal.
The exact results on the small cluster allow us to provide information on the nearest-neighbor correlation functions of the model, which have not been discussed using the above-mentioned approaches. In particular, we have demonstrated that, in the Hund's metal region, the nearest-neighbor inter-orbital charge correlations vanish just like the onsite components, strengthening the relation between the Hund's metal and the effective decoupling between orbitals. Moreover we have found that the nearest-neighbor spin correlators are always negative implying a tendency towards antiferromagnetic ordering in all the regions of the phase diagram, including the Hund's metal and the Hund's insulator.
We have supplemented previous studies with a detailed analysis of the fingerprint associated to the formation of the Hund's metal in the evolution of the many-body energy spectrum and in the temperature dependence of the specific heat.
In the second part of the manuscript, we have performed a similar analysis for the SU(3) Hubbard model with a three-site energy pattern (one site out of three with a higher energy).
Here, we find a slightly richer phase diagram, which is marked by two different lines along which metallic solutions outlive large interactions, in analogy with the Hund's metal. This result is understood by inspecting the probability distribution of different local configurations in the ground state, and is connected with the similar physical scenario emerging within the Hubbard-Kanamori model. An investigation of the entropy and its comparison with the conductivity strengthens the connection between the two models and the identification of the mechanism behind the stabilization of the discussed interaction-resilient metals.
We have therefore provided a strong evidence that an interaction-resistant metallic state is not peculiar of the Hund's physics, where it has been widely discussed, but it is a more general feature which is present for a wide class of models featuring competing insulating states associated with different local configurations. Tuning the parameters to make the insulating states degenerate or nearly degenerate, a correlation-resistant metal exists as long as the local configurations corresponding to the two insulators are connected by hopping processes.
In this work, we considered a minimal three-site cluster to
explore the differences and similarities between the two models. For the Hubbard-Kanamori model, our trimer reproduces the scenario obtained with other approaches including DMFT, RISB, and slave-spin mean-field. This agreement between different approaches represents a mutual validation of the different methods and confirms that the existence of the Hund's metal is an intrinsic feature of the multiorbital Hubbard model.
Moreover, exact result for small clusters can be used to reconstruct the properties of infinite lattices using quantum cluster methods such as cluster perturbation theory \cite{Cluster-Perturbation-Theory}, variational cluster approximation \cite{Potthoff}, or cluster extensions of DMFT \cite{CDMFT,DCA}. The discussed results represent, in this perspective, the basic building block from which the lattice physics can be built. These approximations are particularly accurate for the strongly correlated insulating solutions, which have an essentially local character, but they should not alter significantly the shape of the phase diagram including the region where the interaction-resistant metal is stable.
\section*{Acknowledgements}
\label{sec:Acknowledgements}
The authors would like to thank L. de' Medici, L.F. Tocchio, A. Isidori and L. Livi for fruitful discussions. We acknowledge financial support from MIUR through the PRIN 2017 (Prot. 20172H2SC4 005) programs and Horizon 2020 through the ERC project FIRSTORM (Grant Agreement 692670).
|
1,116,691,497,743 | arxiv |
\section{Discussion of Fig.~2 from the main manuscript when replotted on a semi-logarithmic scale}
\begin{figure*}[tbh!]
\centering
\resizebox{17.8cm}{!}{\includegraphics*{overscreening_logarithmic.pdf}} %
\caption{\label{fig:overscreening_logarithmic}$\ln\left(\left|\left<n(x)\right>-n_\infty\right|\right)$ for $n_{\infty} = 0.005$ and relative permittivities (a) $\varepsilon_\mathrm{r} =100$, (b) $\varepsilon_\mathrm{r}=10$, and (c) $\varepsilon_\mathrm{r}=1$.
For each set of simulation data we also plot the maximum likelihood exponential and oscillatory models.
\emph{Source:} Raw simulation data and scripts to generate this figure are available under CC BY 4.0 / MIT licences as Ref.~\onlinecite{Dean_dataset}.}
\end{figure*}
In Fig.~2 of the main manuscript we present the one-dimensional time-averaged mobile defect distributions for $n_{\infty} = 0.005$ and relative permittivities (a) $\varepsilon_\mathrm{r} =100$, (b) $\varepsilon_\mathrm{r}=10$, and (c) $\varepsilon_\mathrm{r}=1$, plotted on a linear scale.
For $\varepsilon_\mathrm{r} = 10$ and $\varepsilon_\mathrm{r}=1$ (Figs.~2b and 2c) the damped oscillatory decay model gives qualitatively better agreement with the simulation data than the purely exponential decay model.
For $\varepsilon_\mathrm{r} = 100$ (Fig.~2a) the two models give similar space-charge profiles, which both agree qualitatively with the simulation data.
To better show the differences between these two models, we have replotted the data from Fig.~2 on a semi-logarithmic scale at all three relative permittivities (Fig.~\ref{fig:overscreening_logarithmic}).
The oscillatory behaviour of the simulated space-charge profiles is now more easily visible, even at $\varepsilon_\mathrm{r}=100$.
This preferential agreement with the oscillatory model, even at this high relative permittivity, is also supported by the calculated Bayes factor for this simulation of $B=196$ (indicating strong support for the more complex damped oscillatory model).
In Figs.~\ref{fig:overscreening_logarithmic}b and Figs.~\ref{fig:overscreening_logarithmic}c the replotted data highlights that the long-ranged decay of these oscillatory space-charge profiles is still exponential (following a straight line on this semi-log plot).
\section{Discussion of the choice of non-dimensional coupling parameter used in Fig.~3 of the main manuscript.}
\begin{figure}[tb]
\centering
\resizebox{7.0cm}{!}{\includegraphics*{underscreening_ws.pdf}} %
\caption{\label{fig:a_ws} Log--log plot of scaled simulated decay length $\lambda_\mathrm{sys}/\lambda_\mathrm{D}=1/(\alpha\lambda_\mathrm{D})$ versus $a/\lambda_\mathrm{D}$ for simulations performed at four different defect concentrations.
The diagonal dashed line shows the maxiumum likelihood estimate for Eqn.~4 in the underscreening regime.
Open and closed circles correspond to space-charge profiles that best support the pure exponential decay model or the oscillatory decay model respectively, as determined through Bayesian model selection (see below for details).
Error bars show 95\% confidence intervals for $\lambda_\mathrm{sys}/\lambda_\mathrm{D}$.
\emph{Source:} Raw simulation data and scripts to generate this figure are available under CC BY 4.0 / MIT licences as Ref.~\cite{Dean_dataset}.}
\end{figure}
In Fig.~3 of the main manuscript we present the variation in simulated screening length, normalised with respect to the corresponding Debye length, as a function of the parameter $\Gamma=\lambda_\mathrm{B}/a$, where $\lambda_\mathrm{B}=e^2/(4 \pi \varepsilon_0 \varepsilon_\mathrm{r} k_\mathrm{B} T)$ is the Bjerrum length, and $a=\left[3/(4\pi n_\infty)\right]^{1/3}$ is a Wigner--Seitz defect-sphere radius \cite{Carley_PhysRev1963,BrushEtAl_JChemPhys1966}.
$\Gamma$ provides a non-dimensionalised measure of the strength of electrostatic coupling between mobile defects \cite{BausAndHansen_PhysRep1980}.
The choice of $\Gamma$ for our analysis is inspired by its historical use in studies of one-component plasmas \cite{BrushEtAl_JChemPhys1966,BausAndHansen_PhysRep1980}; these include a number of studies of one-component Coulomb lattice-gas models, which correspond to the bulk analogue of the grain boundary model we investigate here.
$\Gamma$ also provides an intuitive sense of how changes in model parameters---defect concentration, $n_\infty$, relative permittivity, $\varepsilon_\mathrm{r}$, or temperature, $T$---are expected to change the relative strength of electrostatic coupling.
The scaling of $\Gamma$ with respect to $n_\infty$, $\varepsilon_\mathrm{r}$, and $T$ is given by
\begin{equation}
\Gamma = \frac{\lambda_\mathrm{B}}{a} \propto \frac{n_\infty^{\frac{1}{3}}}{\varepsilon_\mathrm{r} T}.
\end{equation}
We therefore obtain the intuitive result that increasing defect concentration, or decreasing either the relative permittivity (increasing charge--charge screening) or the temperature will increase the electrostatic coupling strength.
The form of $\Gamma$ also highlights that we can expect to be in the weakly coupled regime if $a>>\lambda_\mathrm{B}$, i.e.~the typical defect--defect distance is larger than the distance at which defect--defect interactions are comparable to $k_\mathrm{B}T$, and the effect of Coulombic interactions can be treated as a mean-field perturbation \cite{BausAndHansen_PhysRep1980}.
Conversely, we can expect to be in the strongly coupled regime if $a<<\lambda_\mathrm{B}$, i.e.~the typical defect--defect distance is shorter than the distance at which the electrostatic defect--defect interactions are comparable to $k_\mathrm{B}T$.
We note that other choices of dimensionless scaling parameter with equivalent scaling behaviour would give the same phenomenological behaviour when plotted on a double-logarithmic scale.
For example, one analogous scaling parameter that might be considered to be closer in spirit to $d/\lambda_\mathrm{D}$, where $d$ is an ion diameter, used to describe underscreening in ionic liquids and liquid electrolytes, is $a/\lambda_\mathrm{D}$.
This scaling parameter scales with respect to $n_\infty$, $\varepsilon_\mathrm{r}$, and $T$ as
\begin{equation}
\frac{a}{\lambda_{D}} \propto \frac{n^{\frac{1}{6}}}{(\varepsilon_\mathrm{r} T)^\frac{1}{2}}.
\end{equation}
i.e.
\begin{equation}
\Gamma \propto \left(\frac{a}{\lambda_\mathrm{D}}\right)^2.
\end{equation}
A log--log plot of $\lambda_\mathrm{sys}/\lambda_\mathrm{D}$ as a function of $a/\lambda_\mathrm{D}$ (Fig.~\ref{fig:a_ws}) therefore shows the same universal scaling behaviour as plotting against $\Gamma$.
As expected from the scaling analysis above, the scaling parameter $\nu$ obtained when using $a/\lambda_\mathrm{D}$ is related to the parameter obtained when using $\Gamma$ by a factor of $2$.
The gradient of the linear region in Fig.~\ref{fig:a_ws}, calculated using the same Bayesian analysis procedure used for Fig.~3, is $2.17\pm0.01$ (95\% \textrm{CI}), twice that of the gradient calculated in the main text where $\Gamma$ is used.
\section{Further details of the Bayesian analysis of the simulated space-charge profiles and subsequent parameter estimation}
The statistical modelling of space charge profiles (Fig.~2 and Fig.~\ref{fig:overscreening_logarithmic}) and the scaling law (Fig.~3 and Fig.~\ref{fig:a_ws}) in the main paper was performed using Bayesian analysis.
For the data presented in Fig.~2 and Fig.~\ref{fig:overscreening_logarithmic}, we have performed maximum likelihood sampling to obtain posterior distributions for the model parameters (the use of a uniform prior probability means that the maximum likelihood parameters and the maximum \emph{a posteriori} values are equivalent). We have then performed nested sampling to estimate the Bayesian evidence $B$ for the more complex damped oscillatory model versus the simpler purely exponential model. To estimate the scaling parameters for the data presented in Fig.~3 and Fig.~\ref{fig:a_ws} we used maximum likelihood sampling.
{\renewcommand{\arraystretch}{1.5}
\begin{table*}[b]
\caption{Bounds used in Bayesian analysis calculations for each of the four concentrations considered. Curves were fit to distributions of average number of charges in a given plane, not the average fractional site-occupation number $\langle n(x) \rangle$. }
\label{tab: bounds}
\footnotesize
\centering
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{} & \multicolumn{2}{c|}{$\alpha$} & \multicolumn{2}{c|}{$A$} & \multicolumn{2}{c|}{$n_{\infty}$} & \multicolumn{2}{c|}{$\xi$} & \multicolumn{2}{c|}{$\theta$} \\ \cline{2-11}
& lower & upper & lower & upper & lower & upper & lower & upper & lower & upper \\
\hline
$n_{\infty} = 0.00025$ & $10^{6} $ & $10^{10} $ & 0 & 10 & 0.9 & 1.9 & $10^{6} $ & $10^{10} $ & 0 & 2$\pi$ \\
\hline
$n_{\infty} = 0.0005$ & $10^{6} $ & $10^{10} $ & 0 & 25 & 2.3 & 3.3 & $10^{6} $ & $10^{10} $ & 0 & 2$\pi$ \\
\hline
$n_{\infty} = 0.001$ & $10^{6} $ & $10^{10} $ & 0 & 35 & 5.1 & 6.1 & $10^{6} $ & $10^{10} $ & 0 & 2$\pi$ \\
\hline
$n_{\infty} = 0.005$ & $10^{6} $ & $10^{10} $ & 0 & 75 & 26 & 30 & $10^{6} $ & $10^{10} $ & 0 & 2$\pi$ \\
\hline
\end{tabular}
\end{table*}}
\subsection{Space charge profile modelling}
The Bayesian analysis of the exponential and oscillatory models used uniform prior parameter distributions.
These bounding values for these distributions are given in Table~\ref{tab: bounds} for each bulk defect concentration.
The prior distributions used for both models are identical for $\alpha$, $A$, and $n_{\infty}$ at a given concentration.
Additionally, we have calculated Bayes factors for the comparative evidence supporting the two proposed space-charge models (damped oscillatory versus purely exponential), where the prior probability of both models was \num{0.5}.
Because of the size of our simulations is limited by computational cost, the time-average space-charge profiles, $\left<n(x)\right>$, are not guaranteed to relax to the exact nominal bulk defect concentration, $n_\infty$.
When fitting the exponential and oscillatory model functions to our simulation data, we therefore consider $n_\infty$ as a free fitting parameter.
The layers immediately adjacent to the grain-boundary $xy$ plane do not follow the same asymptotic behaviour as the space-charge profile. We therefore exclude three layers for the $n_\infty = 0.00025$, and $n_\infty=0.0005$ distributions and two layers for the $n_{\infty} = 0.001$, and $n_\infty= 0.005$ distributions from our fit to get a more accurate description of the long-range behaviour.
The use of nested sampling estimates the Bayesian evidence, $\mathcal{Z}$, for a given model with respect to the data.
This evidence can then be used to calculate the Bayes factor, $B_{01}$, between the null hypothesis model $0$ and the more complex model $1$,
\begin{equation}
2\ln{B_{01}} = 2(\ln{\mathcal{Z}_{x}} - \ln{\mathcal{Z}_{y}}).
\end{equation}
Individual Bayes factors were interpreted using the classification given by Kass and Raftery \cite{KassAndRaftery_JAmerStatAssoc1995}, where $2\ln{B_{01}} > 10$ (defined as ``very strong'' support for the more complex model) was considered sufficient use of the damped oscillatory model for subsquently estimating $\alpha$.
The evidence values and resulting Bayes factors for each of the space charge profiles are provided in the supporting dataset as Ref.~\onlinecite{Dean_dataset}.
\subsection{Scaling law modelling}
Uniform prior parameter distributions were also used in the maximum likelihood sampling of the scaling law.
The constant of proportionality was bounded by $(-1, +1)$ and $\nu$ was bounded by $(0, +5)$.
\bibliographystyle{ieeetr}
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1,116,691,497,744 | arxiv | \section*{\refname\@mkboth{\refname}{\refname}}%
\newcommand \online{online}
\section{Introduction}\label{section:introduction}
The multi-armed bandit (MAB) problem~\cite{robbins1985some} has been extensively studied as a fundamental framework for online optimization problems with partial observations.
In the MAB, a player chooses an arm (choice) from a set of possible arms.
Then, the player incurs a cost and obtains feedback according to the selected arm.
The aim of the player is to minimize the cumulative cost by exploring possible arms and exploiting those with low costs.
There have been many studies on MAB\ applications, e.g., clinical trials~\cite{kuleshov2014algorithms},
and recommendation systems~\cite{li2010contextual}.
\newcommand{OSP}{OSP}
\newcommand{DST}{DST}
\newcommand{CG}{CG}
In many real-world problems, each possible choice that the player can make is not expressed as a single arm but as a {\it super arm}, which is a set of arms that satisfies certain combinatorial constraints;
a full set of super arms is called a {\it decision set}.
This problem is called the {\it combinatorial multi-armed bandit} (CMAB) problem,
and
the CMAB\ is said to be {\it adversarial} if the cost of each arm is arbitrarily changed by an adversary.
Examples of the adversarial CMAB\
include various important problems on networks such as
the {\it online shortest path} (OSP) problem~\cite{awerbuch2004adaptive,gyorgy2007online}, the {\it dynamic Steiner tree} (DST) problem~\cite{imase1991dynamic}, and the {\it congestion game} (CG)~\cite{rosenthal1973games};
although
the original CG\ is a resource allocation problem over multiple players, it can be formulated as an adversarial CMAB\ if a player considers the other players to be adversaries.
For instance,
in the OSP\ on a traffic network, an arm corresponds to an edge (road) of a given network,
a super arm is an $s$-$t$ path that connects the current point $s$ and the destination $t$,
and the decision set is a set of all $s$-$t$ paths.
Furthermore, in the OSP,
the cost of an arm (road) represents the traveling time on the road, and it dynamically changes due to the time-varying amount of traffic or accidents (e.g., cyber attacks in the case of the OSP\ on communication networks). In this paper, we focus on the adversarial CMAB.
The main difficulty with the adversarial CMAB\ is that the size of the decision set is generally exponential in the number of arms.
To handle huge decision sets, existing methods for this problem assume that the decision set has certain properties.
One such method is
{C{\scriptsize OMB}EXP}~\cite{combes2015combinatorial}, which
can cope with the difficulty if the decision set consists of,
for example,
sets of arms satisfying a size constraint,
or matchings on a given network.
However, it has been hard to design practical algorithms for adversarial CMAB\ instances with complex decision sets defined on networks;
for example, the OSP, DST, and CG\ on undirected networks.
In this paper, we develop a practical and theoretically guaranteed algorithm for the adversarial CMAB,
which is particularly effective for
network-based adversarial CMAB\
instances.
We first propose {C{\scriptsize OMB}WM}\ ({C{\scriptsize OM}B{\scriptsize AND}}~\cite{cesa2012comband} with Weight Modification), which is theoretically
guaranteed to achieve either $O(T^{2/3})$ {\it regret} with high probability or $O(\sqrt{T})$ {\it expected regret}, where $T$ is the number of rounds.
The above bounds are {\it any-time guarantees}~\cite{braun2016efficient}, and we can choose which regret value
{C{\scriptsize OMB}WM}\ actually achieves by setting its hyper parameter at an appropriate value.
We then show that our {C{\scriptsize OMB}WM}\ can be performed on a {\it compressed} decision set;
we assume that a decision set is given as a {\it zero-suppressed decision diagram} (ZDD)~\cite{minato1993zero},
which is a compact graph representation of a family of sets.
The time and space complexities of {C{\scriptsize OMB}WM}\ with a ZDD
are linear in the size of the ZDD,
whereas those of the naive {C{\scriptsize OMB}WM}\ is proportional to the size of a decision set.
It is known that a ZDD tends to be small if
it represents a set of subnetworks such as $s$-$t$ paths or Steiner trees~\cite{kawahara2014frontier}.
Thus our algorithm is effective for network-based adversarial CMAB\ instances including the OSP, DST, and
CG.
Experimental results on OSP, DST, and CG\ instances show that our algorithm is more scalable than naive algorithms that directly deal with decision sets.
To the best of our knowledge, this is the first work to implement algorithms for the adversarial CMAB\ and provide experimental results, thus revealing the practical usefulness of adversarial CMAB\ algorithms.
\section{Related work}
Many studies have considered the adversarial CMAB\ with specific decision sets, e.g., $m$-sets~\cite{kale2010slate} and permutations~\cite{ailon2014permutation}.
In particular, the OSP, which is a CMAB\ problem on a network with an $s$-$t$\ path constraint, has been extensively studied~\cite{awerbuch2004adaptive,gyorgy2007online} due to its practical importance.
Whereas the previous studies have focused on the OSP\ on directed networks,
our algorithm is also applicable to the OSP\ on undirected networks.
The adversarial CMAB\ with general decision sets has been also extensively studied in~\cite{audibert2014regret,braun2016efficient,bartlett2008high,cesa2012towards,cesa2012comband,combes2015combinatorial,uchiya2010adversarial}.
One of the best known algorithms for this problem is {C{\scriptsize OM}B{\scriptsize AND}}~\cite{cesa2012comband}, which has been proved to achieve $O(\sqrt{T})$ expected regret.
Recently the algorithm has been also proved to achieve $O(T^{2/3})$ {\it regret} with high probability in~\cite{braun2016efficient};\footnote{
The proof
seems to include some mistakes.
However,
their techniques for the proof are still useful,
and so we prove the $O(T^{2/3})$ high-probability regret bound of our algorithm by partially modifying their proof;
the modified parts are the description of the algorithm and Lemma~\ref{lem:highprob1} in the supplementary materials.
}
more precisely, the regret of {C{\scriptsize OM}B{\scriptsize AND}}\ is bounded by $O(t^{2/3})$ with high probability in any $t$-th round $(t=1,\dots,T)$, which is called an any-time guarantee.
Although {C{\scriptsize OM}B{\scriptsize AND}}\ has the strong theoretical results, its time complexity generally depends on the size of decision sets, which can be prohibitively large in practice.
To avoid such expensive computation, {C{\scriptsize OMB}EXP}~\cite{combes2015combinatorial} scales up {C{\scriptsize OM}B{\scriptsize AND}}\ by employing a projection onto the convex hull of the decision set via KL-divergence.
For some decision sets for which the projection can be done efficiently (e.g., $m$-sets or a set of matchings), {C{\scriptsize OMB}EXP}\
runs faster than {C{\scriptsize OM}B{\scriptsize AND}}, achieving the same theoretical guarantees.
However, it is difficult to perform the projection for other decision sets (e.g., $s$-$t$\ paths or Steiner trees); actually it is NP-hard to do the projection in the case of the OSP\ and DST\ on undirected networks.
On the other hand, thanks to recent advances in constructing {\it decision diagrams} (DDs), optimization techniques using DDs have been attracting much attention~\cite{bergman2016decision,coudert1997solving,morrison2016solving}.
Those techniques are advantageous in that DDs can efficiently store all solutions satisfying some complex constraints; for example, constraints that are hard to represent as a set of inequalities.
The ZDD~\cite{minato1993zero}, which we use in our algorithm, is a kind of DD that is known to
be suitable for storing specific network substructures (e.g., $s$-$t$\ paths or Steiner trees).
Thus our algorithm with ZDDs runs fast in many CMAB\ instances defined on networks, including the OSP, DST, and CG.
\input{section_preliminaries.tex}
\section{Proposed algorithm for adversarial CMAB}
\label{section:algorithm}
We here propose {C{\scriptsize OMB}WM}\ ({C{\scriptsize OM}B{\scriptsize AND}}\ with Weight Modification), which is an algorithm for designing the player's strategy $p_t(\Xsub{t})$ with strong theoretical guarantees as described later.
Algorithm~\ref{alg:comband} gives the details of {C{\scriptsize OMB}WM}.
In what follows,
we define $L:=\max_{X\in\mathcal{S}} \norm{\mbox{\boldmath $1$}_X}$ for any given $\mathcal{S}\subseteq 2^E$, where $\norm{\cdot}$ is the Euclidian norm.
We also define $\lambda$ as the smallest non-zero eigenvalue of $\mathbf{E}_{X\sim u}[\mbox{\boldmath $1$}_X\mbox{\boldmath $1$}_X^\top]$,
where $u$ is the uniform distribution over $\mathcal{S}$.
\begin{algorithm}
\caption{{C{\scriptsize OMB}WM}$(\alpha, \mathcal{S})$}\label{alg:comband}
\begin{algorithmic}[1]
\STATE $\tlweight{1}{i}\gets1$ ($i\in E$)
\FOR{$t=1,\dots,T$}
\STATE $\gamma_t\gets \frac{t^{-1/\alpha}}{2}$, $\eta_t\gets\frac{\lambda t^{-1/\alpha}}{2L^2}$, $\eta_{t+1}\gets\frac{\lambda (t+1)^{-1/\alpha}}{2L^2}$
\STATE $X_t \sim p_t$
\STATE $c_t\gets\mbox{\boldmath $\ell$}_t^\top\mbox{\boldmath $1$}_{\widetilde X}$ ($\mbox{\boldmath $\ell$}_t$ is unobservable)
\STATE $P_t(i,j)\gets \sum_{X\in\mathcal{S} : i,j\in X} p_t(X)$ ($i,j\in[d])$
\STATE $\mbox{{\boldmath $\hat \ell$}}_t\gets c_tP_t^+\mbox{\boldmath $1$}_{\widetilde X}$
\STATE $\tlweight{t+1}{i}\gets \tlweight{t}{i}^{\eta_{t+1}/\eta_{t}}
\exp\big(-\eta_{t+1}\hloss{t}{i}\big)$ ($i\in E$)
\ENDFOR
\RETURN $\{X_t \mid t \in [T]\}$
\end{algorithmic}
\end{algorithm}
Given an arbitrary non-negative vector $\mbox{\boldmath $w$}=(w_1,\dots,w_d)^\top \in \mathbb{R}^d$ and a decision set $\mathcal{S} \subseteq 2^{E}$, we define the {\it constrained distribution} $p(X ; \mbox{\boldmath $w$}, \mathcal{S})$ over $\mathcal{S}$ as follows:
\begin{align}
p(X ; \mbox{\boldmath $w$}, \mathcal{S}) &:= \frac{w(X)}{Z(\mbox{\boldmath $w$}, \mathcal{S})},
&
Z(\mbox{\boldmath $w$}, \mathcal{S}) &:= \sum_{X \in \mathcal{S}} w(X),
&
w(X) &:= \prod_{i \in X} w_i.
\label{eq:constrained_distribution}
\end{align}
Using the above,
we define the player's strategy $p_t(\Xsub{t})$,
which appears in Step~4, as follows:
\begin{align}
p_t(\Xsub{t}):=(1-\gamma_t)p(\Xsub{t} ; \mbox{{\boldmath $\widetilde w$}}_t, \mathcal{S})+\gamma_t p(\Xsub{t} ; \mbox{\boldmath $1$}_E, \mathcal{S}), \label{def:dists}
\end{align}
where $\mbox{{\boldmath $\widetilde w$}}_t=(\tlweight{t}{1},\dots,\tlweight{t}{d})^\top$ is the weight vector defined in Step~8, and $\gamma_t$ is the parameter defined in Step~3; we note that $p(\Xsub{t};\mbox{\boldmath $1$}_E,\mathcal{S})$ is the uniform distribution over $\mathcal{S}$.
Thus $p_t$ is a mixture of two constrained distributions with the mixture rate $\gamma_t$.
Given a distribution $p$ over $\mathcal{S}$,
a matrix $P$ is called a
{\it co-occurrence probability matrix} (CPM)
if its $(i,j)$ entry $P(i,j)$ is given by the
{\it co-occurence probability} $p(i\in X,j\in X):=\sum_{X\in\mathcal{S}:i,j\in X} p(X)$.
The matrix $P_t$ computed in Step~6 is the CPM of $p_t$,
and $P_t^+$ used in Step~7 is the pseudo-inverse of $P_t$.
From Eq.~\eqref{def:dists}, the following equation holds:
\begin{equation}\label{eq:cmppt}
P_t(i,j)=(1-\gamma_t)p(i\in X, j\in X ; \mbox{{\boldmath $\widetilde w$}}_t, \mathcal{S})+\gamma_t p(i\in X, j\in X ; \mbox{\boldmath $1$}_E, \mathcal{S}).
\end{equation}
The above {C{\scriptsize OMB}WM}\ is based on
{C{\scriptsize OM}B{\scriptsize AND}}~\cite{cesa2012comband};
if we replace Step~8 of {C{\scriptsize OMB}WM}\ with $\tlweight{t+1}{i}\gets\tlweight{t}{i}\exp(-\eta_t\hloss{t}{i})$, {C{\scriptsize OMB}WM}\ corresponds perfectly to the original {C{\scriptsize OM}B{\scriptsize AND}}.
Hence the one and only one difference is the weight modification in Step~8.
However, introducing this weight modification gives us the following theoretical guarantees
(for proofs, see the supplementary materials):
\begin{thm}\label{thm:highprob}
For any $\mathcal{S}$, {C{\scriptsize OMB}WM}$(\alpha=3, \mathcal{S})$ achieves $\regret{T}\le O\Big(\Big(\frac{d\lambda}{L^2} + \sqrt{\frac{L^2}{ \lambda} \ln\frac{|\mathcal{S}|+2}{\delta} }\Big)T^{2/3}\Big)$
with probability at least $1-\delta$.
\end{thm}
\begin{thm}\label{thm:expectation}
For any $\mathcal{S}$, {C{\scriptsize OMB}WM}$(\alpha = 2, \mathcal{S})$ achieves
$\eregret{T}\le
O\Big(
\Big(
\frac{d\lambda}{L^2}
+
\frac{L^2\ln |\mathcal{S}|}{\lambda}
\Big)
\sqrt{T}\Big)$.
\end{thm}
In other words, {C{\scriptsize OMB}WM}\ achieves either $O(T^{2/3})$ regret with high probability or
$O(\sqrt{T})$ expected regret
as an any-time guarantee
by choosing the hyper parameter $\alpha$ appropriately.
There are two difficulties when it comes to performing {C{\scriptsize OMB}WM};
the first is sampling from the player's strategy $p_t(X_t)$ (Step~4), and the second is computing the CPM $P_t$ (Step~6).
Naive methods for sampling from $p_t$ and computing $P_t$ require $O(|\mathcal{S}|)$ and $O(d^2|\mathcal{S}|)$ computation times, respectively, where $|\mathcal{S}|$ is generally exponential in $d$, and so are the time complexities.
In the following section, we propose efficient methods for sampling from any given constrained distribution $p(X ; \mbox{\boldmath $w$}, \mathcal{S})$ and for computing the CPM of $p(X ; \mbox{\boldmath $w$}, \mathcal{S})$.
Because $p_t$ is a mixture of two constrained distributions, we can efficiently sample from $p_t$ and compute the CPM of $p_t$ using the proposed methods.
\input{section_zdd}
\section{Experiments}
We applied our {C{\scriptsize OMB}WM}\ with ZDDs to three network-based CMAB\ problems: the OSP, DST, and CG.
In the OSP\ and DST, we used artificial networks to observe the scalability of our algorithm.
In the CG, we used two real-world networks to show the practical utility of our algorithm.
We implemented our algorithm in the C programming language and used Graphillion~\cite{inoue2016graphillion} to obtain the ZDDs.
We note that constructing ZDDs with the
software is not a drawback;
in all of the following instances
a ZDD was obtained within at most several seconds.
\subsection{OSP\ and DST\ on artificial networks}
\newcommand{u(0,1)}{u(0,1)}
\newcommand{\mbox{\boldmath $\mu$}}{\mbox{\boldmath $\mu$}}
{\bf Experimental Setting}:
We applied our {C{\scriptsize OMB}WM}\ with ZDDs to the OSP\ and DST\ instances on artificial networks,
which are undirected grid networks with $3\times m$ nodes ($m = 3,\dots,10$).
In both problems, an arm corresponds to an edge of the given network.
In the OSP, a decision set $\mathcal{S}$ is a set of all $s$-$t$ paths from the starting node $s$ to the goal node $t$ that are placed on diagonal corners of the given grid.
In the DST, $\mathcal{S}$ is a set of all Steiner trees that contains the four corners of the grid.
The aim of the player is to minimize the cumulative cost of the selected subnetworks over some time horizon.
In this experiment, we define the loss vector $\mbox{\boldmath $\ell$}_t$ as follows:
We first uniformly sample $\mbox{\boldmath $\mu$}_0$ from $[0,1]^d$.
In the $t$-th round, we set $\mbox{\boldmath $\mu$}_t=\mbox{\boldmath $\mu$}_{t-1}$ with probability $0.9$ or draw a new $\mbox{\boldmath $\mu$}_t$ uniformly from $[0,1]^d$ with probability $0.1$.
Then, for each $i \in E$, we draw $h_i\sim\Ber{\mu_{t,i}}$ and set $\loss{t}{i} = 1/d$ if $h_t = 1$ otherwise $-1/d$.
This setting is a stochastic CMAB\ with distributions $\Ber{\mu_{t,i}}$ in the short run, but the adversary secretly reset $\mbox{\boldmath $\mu$}_{t}$ with probability $0.1$ in each round to foil the player.
{\bf Compression Power}:
We first assess the compression power of ZDDs
constructed for the decision sets of the OSP\ and DST\ instances.
Table~\ref{table:zddsize} shows the sizes of decision sets $\mathcal{S}$ and those of the corresponding ZDDs $\zdd{\mathcal{S}}$.
In both problems, the ZDD size, $|V|$, grows much more slowly than $|\mathcal{S}|$.
In particular, with the DST\ on the $3 \times 10$ grid, we see that $|V|$ is five orders of magnitude smaller than $|\mathcal{S}|$.
In such cases, our {C{\scriptsize OMB}WM}, which only deals with a ZDD $\zdd{\mathcal{S}}$,
is much more scalable than the naive method that directly deals with $\mathcal{S}$.
{\bf Empirical Regret}:
We next show that the empirical regrets of our {C{\scriptsize OMB}WM}\ and {C{\scriptsize OM}B{\scriptsize AND}}\ actually grow sublinearly,
where {C{\scriptsize OM}B{\scriptsize AND}}\ is also performed on ZDDs.
We applied these algorithms to the OSP\ and DST\ on a $3 \times 10$ grid and computed their empirical regrets over a time horizon.
Figures~\ref{fig:experiments}~(a) and~(b) summarize their regrets for the OSP\ and DST, respectively.
We see that all of the algorithms achieved more or less the same sublinear regrets.
It was confirmed that all of the regret values were lower than those of the theoretical bounds stated in Theorem~\ref{thm:highprob} and Theorem~\ref{thm:expectation};
the precise values of the bounds are provided in the supplementary materials.
\begin{table}[htb]
\caption{
The sizes of decision sets $\mathcal{S}$ and the corresponding ZDDs $\zdd{\mathcal{S}}$ for the OSP\ and DST\
(numbers with more than six digits are rounded to three significant digits).
}
\label{table:zddsize}
\vspace{-10pt}
\begin{small}
\begin{center}
\begin{tabular}{ccrrrrrrrr}
& $m$ & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
\multirow{2}{*}{OSP}
& $|\mathcal{S}|$
& 12
& 38
& 125
& 414
& 1,369
& 4,522
& 14,934
& 49,322
\\
& $|V|$
& 31
& 76
& 183
& 451
& 1,039
& 2,287
& 4,991
& 11,071
\\
\hline
\multirow{2}{*}{DST}
& $|\mathcal{S}|$
& 266
& 4,285
& 69,814
& 1.14$\times 10^6$
& 1.86$\times 10^7$
& 3.04$\times 10^8$
& 4.97$\times 10^9$
& 8.12$\times 10^{10}$
\\
& $|V|$
& 80
& 304
& 1,147
& 4,616
& 18,032
& 67,484
& 238,364
& 933,394
\end{tabular}
\end{center}
\end{small}
\end{table}
\iffigure
\begin{figure}[htb]
\begin{minipage}[b]{0.35\hsize}
\begin{center}
\includegraphics[width=1.00\textwidth, bb=0 0 576 432]{./figures/path_3_10_100000_regret.pdf}
\subcaption{OSP\ on the $3\times 10$ grid}
\end{center}
\begin{center}
\includegraphics[width=1.00\textwidth, bb=0 0 576 432]{./figures/steiner_3_10_1000_regret.pdf}
\subcaption{DST\ on the $3\times 10$ grid}
\end{center}
\end{minipage}
\begin{minipage}[b]{0.35\hsize}
\begin{center}
\includegraphics[width=1.0\textwidth, bb=0 0 576 432]{./figures/mci_3_1.pdf}
\subcaption{CG on MCI}
\end{center}
\begin{center}
\includegraphics[width=1.0\textwidth, bb=0 0 576 432]{./figures/att_3_1.pdf}
\subcaption{CG on ATT}
\end{center}
\end{minipage}
\begin{minipage}[b]{0.28\hsize}
\begin{center}
\includegraphics[width=\textwidth, bb=0 0 300 377]{./figures/mcitop2.pdf}
\subcaption{MCI}
\end{center}
\vspace{-52pt}
\begin{center}
\includegraphics[width=\textwidth, bb=0 0 300 379]{./figures/atttop2.pdf
\subcaption{ATT}
\end{center}
\end{minipage}
\caption{
(a) and (b) show the regret values for the OSP\ and DST, respectively.
The regret values are averaged over 100 trials and the error bars indicate the standard deviations.
(c) and (d) show the regret values of each player for the CG\ on the MCI and ATT, respectively.
(e) and (f) are the topologies of the two networks;
the triangles are the starting nodes and the squares are the goal nodes.
The red (blue) paths indicate the top two paths most frequently chosen by player 1 (2).
}
\label{fig:experiments}
\end{figure}
\fi
\subsection{CG on real-world networks}
\defn_{\text{start}}{n_{\text{start}}}
\defn_{\text{end}}{n_{\text{end}}}
{\bf Experimental Setting}:
We applied our {C{\scriptsize OMB}WM}\ with ZDDs to the CG, which is a multi-player version of the OSP, on two real-world networks.
The CG\ is described as follows:
Given $m$ players and an undirected network with a starting node $s$ and a goal node $t$, the players concurrently send a message from $s$ to $t$.
The aim of each player is to minimize the cumulative time needed to send $T$ messages.
In this problem an arm corresponds to an edge of
a given network, and a super arm is an $s$-$t$ path.
The loss value of an arm is the transmission time required when using the edge,
and the cost of a super arm is the total transmission time needed to send a message along the selected $s$-$t$ path.
In the experiments, we assume that the loss of each edge increases with the number of players who use the same edge at the same time;
therefore, a player regards the other players as adversaries.
We use $X_t^k \in \mathcal{S}$ ($k \in [m]$) to denote the $k$-th player's choice in the $t$-th round and use $X_{t,i}^k \in \{0, 1\}$ to denote the $i$-th element of $\mbox{\boldmath $1$}_{X_t^k}$.
We also use $\ell_{t,i}^k$ to denote the transmission time that the $k$-th player consumes when sending
a message using the $i$-th edge
at the $t$-th round.
We here define
$\loss{t}{i}^k := \beta_i \kappa^{N_{t,i}^{-k}}$, where $\beta_i \in \mathbb{R}$ is the length of the edge, $\kappa$ is an overhead constant, and $N_{t,i}^{-k} := \sum_{k^\prime \not= k} X_{t,i}^{k^\prime}$ is the number of adversaries who also choose the $i$-th edge at the $t$-th round.
Namely, we assume that the transmission time of each edge increases exponentially with the number of players using the same edge at the same time.
Consequently, to reduce the total transmission time, the players should adaptively avoid contending with each other.
Note that this setting violates the assumption $|c_t| < 1$; however, in practice, this violation barely matters.
In the experiments, we set $m = 2$ and $\kappa = 10$.
We use two real-world communication networks in the Internet topology zoo~\cite{knight2011internet}: the InternetMCI network (MCI) and the ATT North America network (ATT).
Figure~\ref{fig:experiments} (e) and (f) illustrate the topologies of the MCI and ATT, respectively.
Both networks correspond to the U.S. map and we choose Los Angeles as the starting point $s$ and New York as the goal $t$.
The statistics for each network are shown in Table~\ref{tbl:info}.
\begin{table}[tbh]
\caption{Statistics for two real-world communication networks.}
\label{tbl:info}
\vspace{-10pt}
\begin{small}
\begin{center}
\begin{tabular}{crrrr}
Network & \# nodes & \# edges & \# $s$-$t$ paths $|\mathcal{S}|$ &ZDD size $|V|$ \\
\hline
MCI & 19 & 33 & 1,444 & 756 \\
ATT & 25 & 56 & 213,971 & 37,776 \\
\end{tabular}
\end{center}
\end{small}
\end{table}
{\bf Experimental Results}:
Figures~\ref{fig:experiments}~(c) and~(d) show the regret values of each player for the MCI and ATT, respectively.
The figure shows that each player attained sublinear regrets.
Figures~\ref{fig:experiments}~(e) and~(f) show the top two most frequently selected paths for each player.
We see that each player successfully avoided congestion.
In the {\it full information setting} where the players can observe the costs of all $s$-$t$ paths after choosing the current path, it is known that the Hedge algorithm~\cite{freund1999hedge} can achieve the Nash equilibria~\cite{kleinberg2009congestion} on the CG.
In this experiment,
even though we employed the {\it bandit setting}
where each player can only observe the cost of the selected path, the players successfully found almost optimal strategies on
both networks.
To conclude, the experimental results suggest that our algorithm is useful for adaptive routing problems on real-world networks.
\section{Conclusion}
We proposed {C{\scriptsize OMB}WM}\ with ZDDs, which is a practical and theoretically guaranteed algorithm for the adversarial CMAB.
We also
showed that our algorithm is effective for network-based
adversarial CMAB\ instances,
which include various important problems
such as the OSP, DST, and CG.
The efficiency of our algorithm is thanks to
the compression of the decision sets via ZDDs,
and its time and space complexities are
linear in the size of ZDDs;
more precisely, they are $O(d|V|)$.
We showed experimentally that the ZDDs for the OSP, DST, and CG\ are much smaller in size than original decision sets.
Our algorithm is also theoretically guaranteed to achieve either $O(T^{2/3})$ regret with high probability or $O(\sqrt{T})$ expected regret as an any-time guarantee;
we experimentally confirmed that our algorithm attained sublinear regrets.
The results on CG\ showed that our algorithm is useful for adaptive routing problems on real-world networks.
\newpage
{\small
\bibliographystyle{plain}
\section{Adversarial CMAB}\label{section:background}
\newcommand{\regret}[1]{\mathrm{R}_{{#1}}}
\newcommand{\eregret}[1]{\overline{\mathrm{R}}_{{#1}}}
\newcommand{\Xsub}[1]{{X_{{#1}}}}
\newcommand{\lsub}[1]{{{\text{\boldmath{$\ell$}}}_{{#1}}}}
\newcommand{\csub}[1]{{c_{{#1}}}}
We here define the adversarial CMAB, which is a sequential decision problem consisting of $T$ rounds.
Let $[m] := \{1,\dots,m\}$ for any $m \in \mathbb{N}$.
We use $E = [d]$ to denote a set of arms and also use $\mathcal{S} \subseteq 2^E$ to denote a {decision set}, where $X \in \mathcal{S}$ is a {super arm}.
At each $t$-th round ($t\in[T]$),
an {adversary} secretly defines a {\it loss vector} $\mbox{\boldmath $\ell$}_t:=(\losst{1},\dots,\losst{i})^\top\in\mathbb{R}^d$
and
a {player} chooses a super arm $X_t\in\mathcal{S}$.
Then, the player incurs and observes the cost $c_t=\mbox{\boldmath $\ell$}^\top_t\mbox{\boldmath $1$}_\widetilde X$,
where $\mbox{\boldmath $1$}_\widetilde X \in \{0, 1\}^d$ is an indicator vector such that its $i$-th element is $1$ if $i \in \widetilde X$ and $0$ otherwise.
Note that the player {\it cannot} observe $\mbox{\boldmath $\ell$}_t$.
The aim of the player is to minimize the {regret} $\regret{T}$ defined as follows:
\begin{align*}
\regret{T}:=
\sum_{t=1}^T
\mbox{\boldmath $\ell$}_t^\top\mbox{\boldmath $1$}_\widetilde X - \min_{X\in\mathcal{S}}\sum_{t=1}^T\mbox{\boldmath $\ell$}_t^\top\mbox{\boldmath $1$}_X.
\end{align*}
The first term is the cumulative cost and the second term is the total cost of the best single super arm selected with hindsight.
Namely, $\regret{T}$ expresses the extra cost that the player incurs against the best single super arm.\footnote{
If the adversary behaves adaptively,
the above interpretation of the regret is somewhat inappropriate;
in such cases using the {\it policy regret}~\cite{arora2012online}
is considered to be more suitable.
However, we here focus on
the above regret and expected regret,
leaving an analysis based on the policy regret for future work.}
As is customary,
we assume $\max_{t\in[T], X\in\mathcal{S}}|\mbox{\boldmath $\ell$}^\top_t\mbox{\boldmath $1$}_X|\le 1$.
If the adversary and/or the player choose $\lsub{t}$ and $\Xsub{t}$ in a stochastic manner, then $\regret{t}$ is a random variable of a joint distribution $p(\lsub{1:t}, \Xsub{1:t})$,
where $\Xsub{1:t} = \{X_1,\dots,X_t\}$ and $\lsub{1:t} = \{\mbox{\boldmath $\ell$}_1,\dots,\mbox{\boldmath $\ell$}_t\}$.
In the adversarial CMAB, $p$ is assumed to satisfy the following conditional independence:
$p(\lsub{1:T},\Xsub{1:T}) = \prod_{t \in [T]} p(\Xsub{t} \mid \lsub{1:t-1}, \Xsub{1:t-1})p(\lsub{t} \mid \lsub{1:t-1},\Xsub{1:t-1})$, where $\Xsub{1:0} = \lsub{1:0} = \{\}$.
$p(\lsub{t} \mid \Xsub{1:t-1}, \lsub{1:t-1})$ corresponds to the adversary's strategy and $p(\Xsub{t} \mid \Xsub{1:t-1}, \lsub{1:t-1})$ corresponds to the player's strategy.
Since the player cannot directly observe $\lsub{1:t}$, the player's strategy must satisfy $p(\Xsub{t} \mid \Xsub{1:t-1}, \lsub{1:t-1}) = p(\Xsub{t} \mid \Xsub{1:t-1}, \csub{1:t-1})$.
Using the joint distribution $p$, the expected regret $\eregret{T}$ is defined as follows:
\[
\eregret{T}:=\mathbf{E}_{\lsub{1:T},\Xsub{1:T}\sim p} [\regret{T}].
\]
The objective of the adversarial CMAB\ is to design the player's strategy
$p(\Xsub{t} \mid \Xsub{1:t-1}, \csub{1:t-1})$ so that it minimizes $\regret{T}$ or $\eregret{T}$.
In this paper, we use $p_t(X_t)$ as shorthand for $p(\Xsub{t} \mid \Xsub{1:t-1}, \csub{1:t-1})$.
\section{{C{\scriptsize OMB}WM}\ on compressed decision sets}
\label{section:implementation}
As shown above,
{C{\scriptsize OMB}WM}\ requires sampling from constrained distributions and computing CPMs as its building blocks, which generally require $O(|\mathcal{S}|)$ and $O(d^2|\mathcal{S}|)$ computation times, respectively.
Moreover, computing $L$ can also require $O(|\mathcal{S}|)$ time.
Those computation costs can be prohibitively expensive since $|\mathcal{S}|$ is generally exponential in $d$.
In this section, we present efficient algorithms for the building blocks that are based on dynamic programming (DP) on a ZDD, which is a compressed representation of $\mathcal{S}$.
We first briefly describe ZDDs and then propose two DP methods
for sampling and computing CPMs.
$L$ can also be computed in a DP manner on a ZDD.
\subsection{Zero-suppressed binary decision diagrams (ZDDs)}
\label{subsection:ZDD}
A ZDD \cite{minato1993zero} is a compact graph representation of a family of sets.
Given $\mathcal{S} \subseteq 2^{E}$, a ZDD for $\mathcal{S}$ is a directed acyclic graph (DAG) denoted by $\zdd{\mathcal{S}} = (V, A)$, where $V = \{0,1,\dots,|V|-1\}$ is a set of vertices and $A \subseteq V \times V$ is a set of directed arcs.
$\zdd{\mathcal{S}}$ contains one root vertex $r\in V$ and two terminal vertices: 1-terminal and 0-terminal.
Without loss of generality, we assume that $V=\{0,\dots,|V|-1\}$ is arranged in a topological order of $\zdd{\mathcal{S}}$;
$r=|V|-1$ holds and
the $b$-terminal ($b \in \{0,1\}$) is
denoted simply by $b\in V$.
Each non-terminal vertex $v \in V \backslash \{0,1\}$ is labeled by an integer in $E$ and has exactly two outgoing arcs: 1-arc and 0-arc.
A vertex pointed by the $b$-arc of $v$ is called the $b$-child of $v$.
We use $\lbl{v}$, $\arc{v}{b}$, and $\child{v}{b}$ to denote $v$'s label, $b$-arc, and $b$-child, respectively.
Consequently, $\arc{v}{b} = (v, \child{v}{b})$ holds.
We use $\Rt{v}{u}$ ($v,u \in V$) to denote a set of routes (directed paths) from $v$ to $u$ on $\zdd{\mathcal{S}}$, where a route $R \in \mathcal{R}_{v,u}$ is a set of directed arcs: $R \subseteq A$.
Given $R \in \Rt{v}{u}$, we define $X(R) \subseteq E$ as $X(R) := \{\lbl{ {v^{\prime}} } \mid (v^{\prime}, \child{ v^{\prime} }{1}) \in R \}$.
Then, $\zdd{\mathcal{S}}$ satisfies
\begin{align}
\mathcal{S} = \{X(R) \mid R \in \mathcal{R}_{r,1} \}.
\label{eq:zdd}
\end{align}
Therefore, $\zdd{\mathcal{S}}$ represents the decision set $\mathcal{S}$ as a set of all routes from its root $r$ to the 1-terminal.
Note that once $\zdd{\mathcal{S}}$ is obtained, $L=\max_{X\in\mathcal{S}} \sqrt{|X|}$ is easily computed by
a DP method to find $R \in \mathcal{R}_{r,1}$ that maximizes $|X(R)|$.
In general, a ZDD is assumed to be {\it ordered} and {\it reduced}.
$\zdd{\mathcal{S}}$ is said to be {\it ordered} if $v > u \Rightarrow \lbl{v} < \lbl{u}$ holds for all $v, u \in V \backslash \{0, 1\}$.
A non-terminal vertex $v$ is said to be {\it redundant} if $\child{v}{1} = 0$:
its 1-arc directly points to the 0-terminal.
A redundant vertex $v$ can be removed by
replacing all $(u, v) \in A$ with $(u, \child{v}{0})$
without loss of the property~\eqref{eq:zdd}.
A non-terminal vertex $v$ is said to be {\it sharable} if there exists another vertex $v^\prime$ such that $\lbl{v} = \lbl{v^\prime}$ and $\child{v}{b} = \child{v^\prime}{b}$ ($b \in \{0,1\})$:
$v$ and $v^\prime$ have the same label and children.
A sharable vertex $v$ can be removed by replacing $(u, v) \in A$ with $(u, v^\prime)$.
$\zdd{\mathcal{S}}$ is said to be {\it reduced} if no vertex is redundant or sharable.
In this paper, we assume that $\zdd{\mathcal{S}}$ is ordered and reduced.
We show an example of a ZDD in Figure~\ref{fig:zddfigs}.
\iffigure
\begin{wrapfigure}[22]{r}{0.4\linewidth}
\begin{minipage}[b]{0.49\linewidth}
\begin{center}
\includegraphics[width=0.9\linewidth, bb=0 0 540 720]{./figures/graph.pdf}
\subcaption{}
\end{center}
\end{minipage}
\begin{minipage}[b]{0.49\linewidth}
\begin{center}
\includegraphics[width=0.9\linewidth, bb=0 0 540 720]{./figures/zdd.pdf}
\subcaption{}
\end{center}
\end{minipage}
\caption{
(a) An example network with an edge set $E=\{1,\dots,5\}$,
and (b) a ZDD that stores all paths from the start to the goal;
each non-terminal vertex $v$ is labeled $\lbl{v} \in E$,
and
$0$-arc s and $1$-arc s are indicated by dashed and solid lines,
respectively.
Note that we have $\mathcal{S} = \{X(R) \mid R \in \mathcal{R}_{r,1} \}=\{ \{1,4\}, \{2,5\}, \{1,3,5\}, \{2,3,4\} \}$. }
\label{fig:zddfigs}
\end{wrapfigure}
\fi
The ZDDs are known to store various families of sets
compactly in many applications.
In particular, if a decision set is a set of specific network substructures
(e.g, a set of $s$-$t$\ paths or Steiner trees),
the ZDD representing the decision set
tends to be small.
As we will see later,
the time complexity of {C{\scriptsize OMB}WM}\ with a ZDD $\zdd{\mathcal{S}}=(V,A)$
is $O(d|V|)$,
and so it runs fast if the ZDD
is small.
In theory,
if $\mathcal{S}$ is a set of specific network substructures,
then $|V|$ is bounded by a value that is exponential
in the {\it pathwidth}~\cite{inoue2016acceleration}.
Thus, even if a network-based decision set
is exponentially large in $d$,
the time complexity of our algorithm in each round can be polynomial in $d$ if the pathwidth of the network
is bounded by a small constant.
The {\it frontier-based search}~\cite{kawahara2014frontier},
which is based on Knuth's {\it Simpath} algorithm~\cite{knuth2011art1},
has recently received much attention as a fast top-down construction algorithm for ZDDs that represent a family of
subnetworks.
In practice ZDDs are easily obtained via existing software~\cite{inoue2016graphillion} for various network-based constraints.
In this paper, we omit the details of ZDD construction and assume that a
decision set $\mathcal{S}$
is represented by a ZDD $\zdd{\mathcal{S}}$ rather than by the explicit enumeration of the components of $\mathcal{S}$.
\subsection{Sampling from constrained distributions}
We here propose an efficient algorithm for sampling from a constrained distribution $p(X ; \mbox{\boldmath $w$}, \mathcal{S})$.
We first introduce the following {\it forward weight} (FW) $\F{v}$ and {\it backward weight} (BW) $\B{v}$ ($v \in V$):
\begin{align}
\F{v} &:= \sum_{R \in \Rt{r}{v}} w(R),
&
\B{v} &:= \sum_{R \in \Rt{v}{1}} w(R),
\label{eq:backward_forward_weight}
\end{align}
where $w(R)$ is an abbreviation of $w(X(R))=\prod_{i\in X(R)}w_i$.
By combining Eq.~\eqref{eq:constrained_distribution}, \eqref{eq:zdd}, and \eqref{eq:backward_forward_weight}, we obtain $Z(\mbox{\boldmath $w$}, \mathcal{S}) = \B{r} = \F{1}$.
$\B{} := \{\B{0},\dots,\B{r}\}$ and $\F{} := \{\F{0},\dots,\F{r}\}$ can be efficiently computed in a dynamic programming manner on $\zdd{\mathcal{S}}$ as shown in Algorithm FW($\zdd{\mathcal{S}}, \mbox{\boldmath $w$}$) and BW($\zdd{\mathcal{S}}, \mbox{\boldmath $w$}$).
Once we obtain $\B{}$, we can draw a sample from $p(X ; \mbox{\boldmath $w$}, \mathcal{S})$ by top-down sampling on $\zdd{\mathcal{S}}$
without rejections
as shown in Algorithm Draw($\zdd{\mathcal{S}}, \mbox{\boldmath $w$}, \B{}$),
where $\Ber{\theta}$ is the Bernoulli distribution
with the parameter $\theta\in[0,1]$.
The space and time complexity when computing $\F{}$ and $\B{}$ is proportional to $|V|$.
This constrained sampling is based on the same idea as that used in logic-based probabilistic modeling \cite{Ishihata08,Ishihata11}.
\begin{table}[htb]
\centering
\begin{tabular}[t]{ccc}
\begin{minipage}[t]{0.3\textwidth}
\label{alg:FW}
\begin{algorithmic}[1]
\STATE {\bf Algorithm} FW($\zdd{\mathcal{S}}, \mbox{\boldmath $w$}$)
\STATE $\F{r} \leftarrow 1$
\STATE $\F{v} \leftarrow 0$ ($\forall v \in V \backslash \{r\}$)
\FOR{$v = r,\dots,2$}
\STATE $\F{ \child{v}{0} } +\!= \F{v}$
\STATE $\F{ \child{v}{1} } +\!= w_{\lbl{v}} \F{v}$
\ENDFOR
\STATE $\F{} := \{\F{0},\dots,\F{r}\}$
\RETURN $\F{}$
\end{algorithmic}
\end{minipage}
&
\begin{minipage}[t]{0.3\textwidth}
\begin{algorithmic}[1]
\label{alg:BW}
\STATE {\bf Algorithm} BW($\zdd{\mathcal{S}}, \mbox{\boldmath $w$}$)
\STATE $\B{1} \leftarrow 1$
\STATE $\B{v} \leftarrow 0$ ($\forall v \in V\backslash\{1\}$)
\FOR{$v = 2,\dots,r$}
\STATE $\B{v} \leftarrow \B{ \child{v}{0} } + w_{\lbl{v}} \B{ \child{v}{1} }$
\ENDFOR
\STATE $\B{} := \{\B{0},\dots,\B{r}\}$
\RETURN $\B{}$
\end{algorithmic}
\end{minipage}
&
\begin{minipage}[t]{0.35\textwidth}
\begin{algorithmic}[1]
\STATE {\bf Algorithm} Draw($\zdd{\mathcal{S}}, \mbox{\boldmath $w$}, \B{}$)
\STATE $X \leftarrow \{\}$, $v \leftarrow r$
\WHILE{$v > 1$}
\STATE $\theta \leftarrow w_{\lbl{v}} \B{ \child{v}{1} } / \B{v}$
\STATE $b \sim \Ber{\theta}$
\STATE $X \leftarrow X \cup \{ \lbl{v} \} $ ~if~ $b = 1$
\STATE $v \leftarrow \child{v}{b}$
\ENDWHILE
\RETURN $X$
\end{algorithmic}
\end{minipage}
\end{tabular}
\end{table}
\subsection{Computing co-occurrence probabilities}
Given a constrained distribution $p(X ; \mbox{\boldmath $w$}, \mathcal{S})$,
we define $\CP{i}{j} := p(i\in X, j \in X ; \mbox{\boldmath $w$}, \mathcal{S})$ as
the co-occurrence probability of $i$ and $j$ ($i,j \in E$).
We here propose an efficient algorithm for computing $\CP{i}{j}$ ($i\le j$),
which suffices for obtaining $\CP{i}{j}$ for all $i,j\in[d]$ since $\CP{i}{j}=\CP{j}{i}$.
Using Eq.~\eqref{eq:constrained_distribution}
and the notion of $\zdd{\mathcal{S}}$, $\CP{i}{j}$ can be written as follows:
\begin{align}
\CP{i}{j}
&=
\sum_{R \in \Rt{r}{1} : i,j \in X(R)}
\frac{w(R)}{Z(\mbox{\boldmath $w$}, \mathcal{S})}.
\label{eq:CP}
\end{align}
We first consider $\CP{i}{i}$ as a special case of $\CP{i}{j}$.
By combining Eq.~\eqref{eq:backward_forward_weight} and \eqref{eq:CP}, we obtain
\begin{align*}
\CP{i}{i}
&=
\sum_{R \in \Rt{r}{1} : i \in X(R)}
\frac{w(R)}{Z(\mbox{\boldmath $w$}, \mathcal{S})}
=
\sum_{v \in V : \lbl{v}=i}
\sum_{
\substack{
R^\prime \in \Rt{r}{v} \\
R^{\prime\prime} \in \Rt{\child{v}{1}}{1}
}
}
\frac{ w( R^\prime \cup \{i\} \cup R^{\prime\prime} ) }{\B{r}}
=
\sum_{v \in V : \lbl{v}=i}
\frac{\F{v} w_i \B{ \child{v}{1} }}{\B{r}}.
\end{align*}
Next, to compute $\CP{i}{j}$ $(i < j)$,
we rewrite the right hand side of Eq.~\eqref{eq:CP}
using the {\it backward weighted co-occurrence} (BWC) $\BC{v}{j}$ ($j \geq \lbl{v}$) as follows:
\begin{align*}
\CP{i}{j}
=
\sum_{v \in V : \lbl{v}=i}
\frac{\F{v} w_i \BC{ \child{v}{1} }{j}}{ \B{r} },
&&
\BC{v}{j}
&:=
\sum_{R \in \Rt{v}{1} : j \in X(R)}
w(R).
\end{align*}
Because $\BC{v}{j}$ is a variant of $\B{v}$, $\mathrm{C} := \{ \BC{v}{j} \mid v \in V, j \geq \lbl{v} \}$ can be computed in a similar manner to $\B{}$ as shown in Algorithm BWC($\zdd{\mathcal{S}}, \mbox{\boldmath $w$}, \B{}$).
To conclude, $\CP{i}{j}$ can be computed by Algorithm CPM($\zdd{\mathcal{S}}, \mbox{\boldmath $w$}, \F{}, \B{}, \mathrm{C}$).
The total space and time complexity of computing $\mathrm{P} := \{\CP{i}{j} \mid i \leq j\}$ is $O(d|V|)$.
\begin{table}[htb]
\centering
\begin{tabular}[t]{cc}
\begin{minipage}[t]{0.45\textwidth}
\begin{algorithmic}[1]
\STATE {\bf Algorithm} BWC($\zdd{\mathcal{S}}, \mbox{\boldmath $w$}, \B{}$)
\STATE $\BC{v}{j} \leftarrow 0$ ($\forall v \in V$, $\forall j \in E$)
\FOR{$v = 2,\dots,r$}
\STATE $\BC{v}{\lbl{v}} \leftarrow w_{\lbl{v}} \B{ \child{v}{1} }$
\FOR{$j = \lbl{v}+1,\dots,d$}
\STATE $\BC{v}{j} \leftarrow \BC{ \child{v}{0} }{j} + w_{\lbl{v}} \BC{ \child{v}{1} }{j}$
\ENDFOR
\ENDFOR
\STATE $\mathrm{C} := \{\BC{v}{j} \mid v \in V, j \geq \lbl{v}\}$
\RETURN $\mathrm{C}$
\end{algorithmic}
\end{minipage}
&
\begin{minipage}[t]{0.45\textwidth}
\begin{algorithmic}[1]
\STATE {\bf Algorithm} CPM($\zdd{\mathcal{S}}, \mbox{\boldmath $w$}, \B{}, \F{}, \mathrm{C}$)
\STATE $\CP{i}{j} \leftarrow 0$ ($\forall i,j \in E$)
\FOR{$v = 2,\dots,r$}
\STATE $i \leftarrow \lbl{v}$
\STATE $\CP{i}{i} +\!= \F{v} w_i \B{ \child{v}{1} } / \B{r}$
\FOR{$j = i+1,\dots,d$}
\STATE $\CP{i}{j} +\!= \F{v} w_i \BC{ \child{v}{1} }{j} / \B{r}$
\ENDFOR
\ENDFOR
\STATE $\mathrm{P} := \{\CP{i}{j} \mid i,j \in [d], i \leq j\}$
\RETURN $\mathrm{P}$
\end{algorithmic}
\end{minipage}
\end{tabular}
\end{table}
|
1,116,691,497,745 | arxiv | \section{\label{sec:level1} Introduction}
There has been overwhelming evidence that coherent structures such as vortices, streamers and zonal flows ($m=n=0$, where $m$ and $n$ are the poloidal and toroidal modenumbers respectively) play a critical role in determining the overall transport in magnetically confined plasmas.\cite{diamond2005,terry2000} Some of these coherent structures, so called streamers, are radially elongated structures that cause intermittent, bursty events, which can mediate significant transport of heat and particles, for instance, imposing a large heat load on container walls. Zonal flows on the other hand may impede transport by shear decorrelation.\cite{diamond2005,terry2000} The Geodesic Acoustic Mode (GAM)\cite{winsor1968, itoh2005, miki2007, chak2007, miki2010, hallatschek2012, mckee2008, fujisawa2007, conway2009, dong2009} is the oscillatory counterpart of the zonal flow ($m=n=0$ in the potential perturbation, $m=1$, $n=0$ in the perturbations in density, temperature and parallel velocity) and thus a much weaker effect on turbulence is expected. Nevertheless experimental studies suggest that GAMs ($n=0$, $m=1$) are related to the L-H transition and transport barriers. The GAMs are weakly damped by Landau resonances and moreover this damping effect is weaker at the edge suggesting that GAMs are more prominent in the region where transport barriers are expected. \cite{mckee2008}
The electron-temperature-gradient (ETG) mode driven by a combination of electron temperature gradients and field line curvature effects is a likely candidate for driving electron heat transport.\cite{liu1971, horton1988, jenko2000, singh2001, tangri2005, nevins2006} The ETG driven electron heat transport is determined by short scale fluctuations that do not influence ion heat transport and is largely unaffected by the large scale flows stabilizing ion-temperature-gradient (ITG) modes.
In this work the first demonstration of a high frequency branch of the geodesic acoustic mode (GAM) driven by electron temperature gradient (ETG) modes is presented. We have utilized a fluid model for the ETG mode based on the Braghinskii equations with non-adiabatic ions including impurities and finite $\beta$ - effects. \cite{singh2001, tangri2005} A new saturation mechanism for ETG turbulence through the interaction with high frequency GAMs, balanced by Landau damping, is found, resulting in a significantly enhanced ETG turbulence saturation level compared to the mixing length estimate.
The remainder of the paper is organized as follows: In Section II the linear ETG mode including the ion impurity dynamics is presented. The linear high frequency GAM is presented and the non-linear effects are discussed in Section III, whereas the saturation mechanism for the ETG turbulence is treated in Section IV. The paper is concluded in Section VI.
\section{\label{sec:level2} The linear Electron Temperature Gradient Mode}
In this section we will describe the preliminaries of the electron-temperature-gradient (ETG) mode which is present under the following restrictions on real frequency and wave length: $\Omega_i \leq \omega \sim \omega_{\star} << \Omega_e$, $k_{\perp} c_i > \omega > k_{||} c_e$. Here $\Omega_j$ are the respective cyclotron frequencies, $\rho_j$ the Larmor radii and $c_j = \sqrt{T_j/m_j}$ the thermal velocities. The diamagnetic frequency is $\omega_{\star} \sim k_{\theta} \rho_e c_e / L_n$, $k_{\perp}$ and $k_{||}$ are the perpendicular and the parallel wavevectors. The ETG model consists of a combination of an ion and electron fluid dynamics coupled through the quasineutrality including finite $\beta$-effects. \cite{singh2001, tangri2005}
\subsection{\label{sec:level3} Ion and impurity dynamics}
In this section, we will start by describing the ion fluid dynamics in the ETG mode description, in the limit $\omega > k_{\parallel} c_e$ the ions are stationary along the mean magnetic field $\vec{B}$ (where $\vec{B} = B_0 \hat{e}_{\parallel}$) whereas in the limit $k_{\perp} c_i >> \omega$, $k_{\perp} \rho_i >> 1$ the ions are unmagnetized. We note that the adiabatic ion response follows from the perpendicular ion momentum equation by balancing the linear parts of,
\begin{eqnarray} \label{eq:0.1}
-en_i \nabla \phi = T_i \nabla n_i,
\end{eqnarray}
and we find
\begin{eqnarray} \label{eq:0.2}
\tilde{n}_i = - \tau \tilde{\phi}.
\end{eqnarray}
In this paper we will use the non-adabatic responses in the limits $\omega < k_{\perp} c_I < k_{\perp} c_i$, $\tau_I = \sqrt{\frac{T_I}{m_I}}$ and $\Omega_i < \omega < \Omega_e$ are fulfilled for the ions and impurities. In the ETG mode description we can utilize the ion and impurity continuity and momentum equations of the form,
\begin{eqnarray} \label{eq:1.1}
\frac{\partial \tilde{n}_j}{\partial t} + n_j \nabla \cdot \vec{v}_j & = & 0, \ \ \mbox{and} \\
m_j n_j \frac{\partial v_j}{\partial t} + e n_j \nabla \phi + T_j \nabla n_j & = & 0,
\end{eqnarray}
where $j=i$ for ions and $j=I$ for impurities. Now, we derive the non-adiabatic ion response
with $\tau_i = T_e/T_i$ and impurity response with with $\tau_I = T_e/T_I$, respectively. We have thus,
\begin{eqnarray}
\tilde{n}_j = - \left( \frac{z \tau_j}{1 - \omega^2/(k_{\perp}^2 c_j^2)}\right) \tilde{\phi}, \label{eq:1.2}
\end{eqnarray}
Here $T_j$ and $n_j$ are the mean temperature and density of species ($j=e,i,I$), where $\tilde{n}_i = \delta n/ n_i$, $\tilde{n}_I = \delta n_I/ n_I$ and $\tilde{\phi} = e \phi/T_e$ are the normalized ion density, impurity density and potential fluctuations. We also define $z_{eff} = \sum z_k \tilde{n}_k/\sum z_k \tilde{n}_k \approx (\tilde{n}_i + z^2 \tilde{n}_I)/n_e$. Next we present the electron dynamics and the linear dispersion relation.
\subsection{\label{sec:level4} The electron model}
The electron dynamics for the toroidal ETG mode are governed by the continuity, parallel momentum and energy equations adapted from the Braghinskii's fluid equations. The electron equations are analogous to the ion fluid equations used for the toroidal ITG mode,
\begin{eqnarray}
\frac{\partial n_{e}}{\partial t} +\nabla \cdot \left( n_{e} \vec{v}_{E} + n_{e} \vec{v}_{\star e} \right) + \nabla \cdot \left( n_{e} \vec{v}_{pe} + n_{e} \vec{v}_{\pi e} \right) + \nabla \cdot (n_e v_{|| e}) & = & 0 \label{eq:1.4} \\
\frac{3}{2} n_{e} \frac{d T_{e}}{dt} + n_{e} T_{e} \nabla \cdot \vec{v}_{e} + \nabla \cdot \vec{q}_{e} & = & 0. \label{eq:1.5}
\end{eqnarray}
Here we used the definitions $\vec{q}_e = - (5 p_e/2m_e \Omega_e) e_{||} \times \nabla T_e$ as the diamagnetic heat flux, $v_{E}$ is the $\vec{E} \times \vec{B}$ drift, $v_{\star e}$ is the electron diamagnetic drift velocity, $\vec{v}_{Pe}$ is the polarization drift velocity, $\vec{v}_{\pi}$ is the stress tensor drift velocity, and the derivative is defined as $d/dt = \partial/\partial t + \rho_e c_e \hat{e} \times \nabla \phi \cdot \nabla$. A relation between the parallel current density and the parallel component of the vector potential ($J_{\parallel}$) can be found using Amp\`{e}re's law,
\begin{eqnarray}\label{eq:1.6}
\nabla^2_{\perp} \tilde{A}_{\parallel} = - \frac{4 \pi}{c} J_{\parallel},
\end{eqnarray}
Simplifying and linearizing Eqs. (\ref{eq:1.4}, \ref{eq:1.5} and \ref{eq:1.6}) we find the evolution equations for the electron density and the electron temperature in normalized form,
\begin{eqnarray}
- \frac{\partial \tilde{n}_e}{\partial t} - \nabla_{\perp}^2 \frac{\partial}{\partial t} \tilde{\phi} - \left( 1 + (1 + \eta_e) \nabla_{\perp}^2\right)\nabla_{\theta} \tilde{\phi} - \nabla_{||} \nabla_{\perp}^2 \tilde{A}_{||} & + & \nonumber \\ \epsilon_n \left( \cos \theta \frac{1}{r}\frac{\partial}{\partial \theta} + \sin \theta \frac{\partial}{\partial r} \right)(\tilde{\phi} - \tilde{n}_e - \tilde{T}_e) & = & 0, \label{eq:1.101} \\
\left((\beta_e/2 - \nabla_{\perp}^2) \frac{\partial}{\partial t} + (1+\eta_e)(\beta_e/2)\nabla_{\theta}\right)\tilde{A}_{||} + \nabla_{||} (\tilde{\phi} - \tilde{n}_e - \tilde{T}_e) & = & 0, \label{eq:1.102}\\
\frac{\partial}{\partial t}\tilde{T}_e + \frac{5}{3} \epsilon_n \left( \cos \theta \frac{1}{r}\frac{\partial}{\partial \theta} + \sin \theta \frac{\partial}{\partial r} \right) \frac{1}{r}\frac{\partial}{\partial \theta} \tilde{T}_e + (\eta_e - \frac{2}{3}) \frac{1}{r}\frac{\partial}{\partial \theta} \tilde{\phi} - \frac{2}{3} \frac{\partial}{\partial t} \tilde{n}_e & = & 0 \label{eq:1.103}.
\end{eqnarray}
The variables are normalized according to
\begin{eqnarray}
(\tilde{\phi}, \tilde{n}, \tilde{T}_e) & = & (L_n/\rho_e)(e \delta \phi/T_{eo}, \delta n_e/n_0, \delta T_e/T_{e0}), \\ \label{eq:1.11}
\tilde{A}_{||} & = & (2 c_e L_n/\beta_e c \rho_e) e A_{||}/T_{e0}, \\ \label{eq:1.12}
\beta_e & = & 8 \pi n T_e/B_0^2. \label{eq:1.13}
\end{eqnarray}
Using the Poisson equation in combination with (\ref{eq:1.2}) we then find
\begin{eqnarray}
\tilde{n}_e = - \left( \frac{\tau_i \tilde{n}_i/n_e}{1 - \omega^2/k_{\perp}^2 c_i^2} + \frac{(Z^2\tilde{n}_I/n_e) \tau_I}{1 - \omega^2/(k_{\perp}^2 c_I^2)} + k_{\perp}^2 \lambda_{De}^2\right) \tilde{\phi}. \label{eq:1.3}
\end{eqnarray}
First we will consider the linear dynamical equations (\ref{eq:1.101}, \ref{eq:1.102} and \ref{eq:1.103}) and utilizing Eq. (\ref{eq:1.5}) in the same manner as in Refs \onlinecite{singh2001, tangri2005} and we find a semi-local dispersion relation as follows,
\begin{eqnarray} \label{eq:1.14}
\left[ \omega^2 \left( \Lambda_e + \frac{\beta_e }{2} (1 + \Lambda_e ) \right) + \left( 1 - \bar{\epsilon}_n (1 + \Lambda_e) \right) \omega_{\star} \right. & + & \nonumber \\
\left. k_{\perp}^2 \rho_e^2 \left( \omega - (1 + \eta_e) \omega_{\star} \right) \right] \left( \omega - \frac{5}{3} \bar{\epsilon}_n \omega_{\star} \right) & + & \nonumber \\
\left( \bar{\epsilon}_n \omega_{\star} - \frac{\beta_e}{2} \omega\right) \left( (\eta_e - \frac{2}{3})\omega_{\star} + \frac{2}{3} \omega \Lambda_e \right) & = & \nonumber \\
c_e^2k_{||}^2 k_{\perp}^2 \rho_e^2 \left( \frac{(1 + \Lambda_e) \left( \omega - \frac{5}{3}\bar{\epsilon}_n \omega_{\star}\right) - \left( \eta_e - \frac{2}{3}\right)\omega_{\star} - \frac{2}{3} \omega \Lambda_e}{\omega \left( \frac{\beta_e}{2} + k_{\perp}^2 \rho_e^2\right) - \frac{\beta_e}{2} (1 + \eta_e) \omega_{\star}} \right)
\end{eqnarray}
In the following we will use the notation $\Lambda_e = \tau_i (n_i/n_e)/(1 - \omega^2/k_{\perp}^2 c_i^2) + \tau_I (Z_{eff} n_I/n_e)/(1 - \omega^2/k_{\perp}^2 c_I^2) + k_{\perp}^2 \lambda_{De}^2$. Note that in the limit $T_i = T_e$, $\omega<k_{\perp} c_i$, $k_{\perp} \lambda_{De} < k_{\perp} \rho_e \leq 1$ and in the absence of impurity ions, $\Lambda_e \approx 1$ and the ions follow the Boltzmann relation in the standard ETG mode dynamics. Here $\lambda_{De} = \sqrt{T_c/(4 \pi n_e e^2)}$ is the Debye length, the Debye shielding effect is important for $\lambda_{De}/\rho_e > 1$ \cite{singh2001}. The dispersion relation Eq. (\ref{eq:1.14}) is analogous to the toroidal ion-temperature-gradient mode dispersion relation ecept that the ion quantities are exchanged to their electron counterparts. In Eq. (\ref{eq:1.14}), the geometrical quantities will be determined using a semi-local analysis by assuming an approximate eigenfunction while averageing the geometry dependent quantities along the field line. The form of the eigenfunction is assumed to be\cite{jarmen1998},
\begin{eqnarray}\label{eq:1.15}
\Psi(\theta) = \frac{1}{\sqrt{3 \pi}}(1 + \cos \theta) \;\;\;\;\; \mbox{with} \;\;\;\;\; |\theta| < \pi.
\end{eqnarray}
In the dispersion relation we will replace $k_{\parallel} = \langle k_{\parallel} \rangle$, $k_{\perp} = \langle k_{\perp} \rangle$ and $\omega_D = \langle \omega_D \rangle$ by the averages defined through the integrals,
\begin{eqnarray}
\langle k_{\perp}^2 \rangle & = & \frac{1}{N(\Psi)}\int_{-\pi}^{\pi} d \theta \Psi k_{\perp}^2 \Psi = k_{\theta}^2 \left( 1 + \frac{s^2}{3} (\pi^2 - 7.5) - \frac{10}{9} s \alpha + \frac{5}{12} \alpha^2 \right), \label{eq:1.16} \\
\langle k_{||}^2 \rangle & = & \frac{1}{N(\Psi)} \int_{-\pi}^{\pi} d \theta \Psi k_{||}^2 \Psi = \frac{1}{3 q^2 R^2}, \label{eq:1.17}\\
\langle \omega_D \rangle & = & \frac{1}{N(\Psi)} \int_{-\pi}^{\pi} d \theta \Psi \omega_D \Psi = \epsilon_n \omega_{\star} \left( \frac{2}{3} + \frac{5}{9}s - \frac{5}{12} \alpha \right), \label{eq:1.17} \\
\langle k_{\parallel} k_{\perp}^2 k_{\parallel} \rangle & = & \frac{1}{N(\Psi)} \int_{-\pi}^{\pi} d \theta \Psi k_{\parallel} k_{\perp}^2 k_{\parallel} \Psi = \frac{k_{\theta}^2}{3 (qR)^2} \left( 1 + s^2 (\frac{\pi^2}{3} - 0.5) - \frac{8}{3}s \alpha + \frac{3}{4} \alpha^2 \right). \label{eq:1.18} \\
N(\Psi) & = & \int_{-\pi}^{\pi} d \theta \Psi^2.
\end{eqnarray}
Here $\alpha = \beta q^2 R \left(1 + \eta_e + (1 + \eta_i) \right)/(2 L_n)$ and $\beta = 8 \pi n_o (T_e + T_i)/B^2$ is the plasma $\beta$, $q$ is the safety factor and $s = r q^{\prime}/q$ is the magnetic shear. The $\alpha$-dependent term above (in Eq. \ref{eq:1.16}) represents the effects of Shafranov shift.
\section{\label{sec:level6} Modeling High Frequency Geodesic Acoustic modes}
The Geodesic Acoustic Modes are the $m=n=0$, $k_r \neq 0$ perturbation of the potential field and the $n=0$, $m=1$, $k_r \neq 0$ perturbation in the density, temperatures and the magnetic field perturbations. The high frequency GAM ($q, \Omega$) induced by ETG modes ($k,\omega$) is prevailing under the conditions when the ETG mode real frequence satisfies ($\Omega_e > \omega > \Omega_i$) at the scale ($k_{\perp } \rho_i < 1$) and the real frequence of the GAM fulfills ($\Omega_e > \Omega > \Omega_i$) at the scale ($q_x < k_{x}$).
\subsection{Linear Geodesic Acoustic Modes}
We start by deriving the linear electron GAM dispersion relation, by writing the $m=1$ equations for the density, parallel component of the vector potential, temperature and the $m=0$ of the electrostatic potential
\begin{eqnarray}
- \tau_i \frac{\partial \tilde{n}^{(1)}_{eG}}{\partial t} + \epsilon_n \sin \theta \frac{\partial}{\partial r} \tilde{\phi}^{(0)}_G - \nabla_{||} \nabla_{\perp}^2 \tilde{A}_{|| G}^{(1)} & = & 0, \label{eq:3.101} \\
\left((\beta_e/2 - \nabla_{\perp}^2) \frac{\partial}{\partial t} + (1+\eta_e)(\beta_e/2)\nabla_{\theta}\right)\tilde{A}^{(1)}_{|| G} - \nabla_{||} (\tilde{n}^{(1)}_{eG} + \tilde{T}^{(1)}_{eG}) & = & 0, \label{eq:3.102} \\
\frac{\partial}{\partial t}\tilde{T}^{(1)}_{eG} - \frac{2}{3} \frac{\partial}{\partial t} \tilde{n}^{(1)}_{eG} & = & 0, \label{eq:3.103} \\
- \nabla_{\perp}^2 \frac{\partial}{\partial t} \tilde{\phi}^{(0)}_G - \epsilon_n \sin \theta \frac{\partial}{\partial r} (\tilde{n}^{(1)}_{eG} + \tilde{T}^{(1)}_{eG}) & = & 0. \label{eq:3.104}
\end{eqnarray}
First we will derive the linear GAM frequency assuming electrostatic GAMs ($\beta_e \rightarrow 0$) this yields a relation between the parallel component of the vector potential and the density and electron perturbations using Eq. (\ref{eq:3.102}) as,
\begin{eqnarray} \label{eq:3.3}
- \nabla_{\perp}^2 \frac{\partial \tilde{A}_{|| G}^{(1)}}{\partial t} - \nabla_{||} (\tilde{n}_G^{(1)} + \tilde{T}_{e G}^{(0)}) = 0.
\end{eqnarray}
The $m=1$ component of the electron density can be eliminated by taking a time derivative of Eq. (\ref{eq:3.104}) and using Eq. (\ref{eq:3.101}) and we get,
\begin{eqnarray} \label{eq:3.8}
\rho_e^2 \frac{\partial^2}{\partial t^2} \nabla_{\perp}^2 \tilde{\phi}^{(0)} + \epsilon_n v_{\star} \langle \sin \theta \frac{\partial}{\partial r} \left( \epsilon_n v_{\star} \sin \theta \frac{\partial \tilde{\phi}^{(0)}}{\partial r} + \nabla_{||} \frac{J_{||}^{(1)}}{e n_0} \right) \rangle = 0.
\end{eqnarray}
Here $\langle \cdots \rangle$ is the average over the poloidal angle $\theta$. In the simplest case this leads to the dispersion relation,
\begin{eqnarray} \label{eq:3.9}
\Omega^2 = \frac{c_e^2}{R^2} \left( \frac{10}{3} + \frac{1}{q^2}\right).
\end{eqnarray}
Note that the linear electron GAM is purely oscillating analogously to its ion counterpart. In previous section we computed the linear dispersion relation for the GAM now we will study the non-linear contributions through a modulational instability analysis.
\subsection{\label{sec:level7} The Non-linearly Driven Geodesic Acoustic modes}
We will now study the system including the non-linear terms and derive the electron GAM growth rate. The non-linear extension to the evolution equations presented previously in Eqs (\ref{eq:1.101}) - (\ref{eq:1.103}) are
\begin{eqnarray}
- \frac{\partial \tilde{n}_e}{\partial t} - \nabla_{\perp}^2 \frac{\partial}{\partial t} \tilde{\phi} - \left( 1 + (1 + \eta_e) \nabla_{\perp}^2\right)\nabla_y \tilde{\phi} - \nabla_{||} \nabla_{\perp}^2 \tilde{A}_{||} & + & \nonumber \\ \epsilon_n \left( \cos \theta \frac{1}{r}\frac{\partial}{\partial \theta} + \sin \theta \frac{\partial}{\partial r} \right)(\tilde{\phi} - \tilde{n}_e - \tilde{T}_e) & = & \nonumber \\
-(\beta_e/2) [\tilde{A}_{||},\nabla_{||}^2 \tilde{A}_{||}] + [ \tilde{\phi}, \nabla^2 \tilde{\phi}], \label{eq:7.101}\\
\left((\beta_e/2 - \nabla_{\perp}^2) \frac{\partial}{\partial t} + (1+\eta_e)(\beta_e/2)\nabla_y\right)\tilde{A}_{||} + \nabla_{||} (\tilde{\phi} - \tilde{n}_e - ) & = & - (\beta_e/2) [\tilde{\phi} - \tilde{n}_e, \tilde{A}_{||}] \nonumber \\
+ (\beta_e/2)[\tilde{T}_e,\tilde{A}_{||}] + [\tilde{\phi}, \nabla_{\perp}^2 \tilde{A}_{||}], \label{eq:7.102}\\
\frac{\partial}{\partial t}\tilde{T}_e + \frac{5}{3} \epsilon_n \left( \cos \theta \frac{1}{r}\frac{\partial}{\partial \theta} + \sin \theta \frac{\partial}{\partial r} \right) \frac{1}{r}\frac{\partial}{\partial \theta} \tilde{T}_e + (\eta_e - \frac{2}{3}) \frac{1}{r}\frac{\partial}{\partial \theta} \tilde{\phi} - \frac{2}{3} \frac{\partial}{\partial t} \tilde{n}_e & = & -[\tilde{\phi},\tilde{T}_e]. \label{eq:7.103}
\end{eqnarray}
In order to find the relevant equations for the electron GAM dynamics we consider the $m=1$ component of Eqs (\ref{eq:7.101}) - (\ref{eq:7.103}) and we find,
\begin{eqnarray} \label{eq:3.1}
- \tau_i \frac{\partial \tilde{n}^{(1)}_{eG}}{\partial t} + \epsilon_n \sin \theta \frac{\partial}{\partial r} \tilde{\phi}^{(0)}_G - \nabla_{||} \nabla_{\perp}^2 \tilde{A}_{|| G}^{(1)} = \langle [\tilde{\phi}_k, \nabla^2 \tilde{\phi}_k] \rangle^{(1)} & - &(\beta_e/2) \langle [\tilde{A}_{|| k},\nabla_{||}^2 \tilde{A}_{|| k}]\rangle^{(1)}, \nonumber \\ \\
\left((\beta_e/2 - \nabla_{\perp}^2) \frac{\partial}{\partial t} + (1+\eta_e)(\beta_e/2)\nabla_{\theta}\right)\tilde{A}^{(1)}_{|| G} - \nabla_{||} (\tilde{n}^{(1)}_{eG} + \tilde{T}^{(1)}_{eG}) & = & - (\beta_e/2) \langle [\tilde{\phi}_k - \tilde{n}_{e k}, \tilde{A}_{|| k}] \rangle^{(1)} \nonumber \\
+ (\beta_e/2) \langle [\tilde{T}_{e k},\tilde{A}_{|| k}] \rangle^{(1)} + \langle [\tilde{\phi}_k, \nabla_{\perp}^2 \tilde{A}_{|| k}] \rangle^{(1)}, \\
\frac{\partial}{\partial t}\tilde{T}^{(1)}_{eG} - \frac{2}{3} \frac{\partial}{\partial t} \tilde{n}^{(1)}_{eG} & = & -\langle [\tilde{\phi}_k,\tilde{T}_{ek}]\rangle^{(1)},
\end{eqnarray}
where superscript (1) over the fluctuating quantities denotes the $m=1$ poloidal mode number and $\langle \cdots \rangle $ is the average over the fast time and spatial scale of the ETG turbulence and that non-linear terms associated with parallel dynamics are small since $\frac{1}{q^2} << 1$. We now study the $m=0$ potential perturbations,
\begin{eqnarray} \label{eq:3.2}
- \nabla_{\perp}^2 \frac{\partial}{\partial t} \tilde{\phi}^{(0)}_G - \epsilon_n \sin \theta \frac{\partial}{\partial r} (\tilde{n}^{(1)}_{eG} + \tilde{T}^{(1)}_{eG}) = \langle [\tilde{\phi}_k, \nabla^2 \tilde{\phi}_k] \rangle^{(0)} & - & (\beta_e/2) \langle [\tilde{A}_{|| k},\nabla_{||}^2 \tilde{A}_{||} k] \rangle^{0}.
\end{eqnarray}
We will now neglect the effects of Debye shielding and make use of quasi-neutrality in the plasma of the form ($\tilde{n}_i = \tilde{n}_e = \tilde{n}$) and subtracting Eq. (\ref{eq:1.101}) from Eq. (\ref{eq:1.102}) we find,
\begin{eqnarray} \label{eq:3.5}
\rho_e^2 \frac{\partial}{\partial t} \nabla_{\perp}^2 \tilde{\phi} + \frac{v_{\star}}{r} \frac{\partial \tilde{\phi}}{\partial \theta} - \epsilon_n v_{\star} \left( \cos \theta \frac{1}{r} \frac{\partial}{\partial \theta} + \sin \theta \frac{\partial}{\partial r}\right) (\tilde{\phi} - \tilde{n}_e - \tilde{T}_e) - \nabla_{||} \tilde{J}_{||} = N_1.
\end{eqnarray}
Here we have defined the non-linear term on the RHS as $N_1 = \rho_e^3 c_e \hat{z} \times \nabla \tilde{\phi} \cdot \nabla \nabla_{\perp}^2 \tilde{\phi} + \frac{\delta B_r}{B} \cdot \nabla \frac{\tilde{J}_{||}}{e n_0}$. For the GAM we find the ($n=0, m=1$) component of Eq. (\ref{eq:1.101}) as,
\begin{eqnarray} \label{eq:3.6}
\frac{\partial \tilde{T}^{(1)}_e}{\partial t} - \frac{2}{3} \frac{\partial \tilde{n}^{(1)}_e}{\partial t} = N^{(1)}_2.
\end{eqnarray}
This can be written $\tilde{T}_e = \frac{2}{3} \tilde{n}_e^{(1)} + N_2^{1}$, where the $m=1$ component is determined by an integral of the convective non-linear term as $N_2^{1} = - \int dt \rho_s c_s \hat{z} \times \nabla \tilde{\phi}^{(0)} \cdot \nabla \tilde{T}^{(1)}_e$. This leads to a relation between the $m=1$ component of the density and temperature fluctuations modified by a non-linear term. We continue by taking the $m=1$ component of Eq. (\ref{eq:1.102}) and $m=0$ of Eq. (\ref{eq:1.103}),
\begin{eqnarray}
\frac{\partial \tilde{n}^{(1)}_e}{\partial t} - \frac{\nabla_{||} \tilde{J}_{||}^{(1)}}{e n_0} - \epsilon_n v_{\star} \sin \theta \frac{\partial \tilde{\phi}^{(0)}}{\partial r} = N_1^{(1)}, \label{eq:3.71}\\
\rho_e^2 \frac{\partial}{\partial t} \nabla_{\perp}^2 \tilde{\phi}^{(0)} + \epsilon_n v_{\star} \langle \sin \theta \frac{\partial}{\partial r} \left( \frac{5}{3} \tilde{n}_e^{(1)} + N_2^{1}\right) \rangle = N_1^{(0)}. \label{eq:3.72}
\end{eqnarray}
Similar to the operations performed to find the linear electron GAM frequency we eliminate the $m=1$ component of the electron density by taking a time derivative of Eq. (\ref{eq:3.72}) this yields,
\begin{eqnarray} \label{eq:3.8}
\rho_e^2 \frac{\partial^2}{\partial t^2} \nabla_{\perp}^2 \tilde{\phi}^{(0)} + \epsilon_n v_{\star} \langle \sin \theta \frac{\partial}{\partial r} \left( \epsilon_n v_{\star} \sin \theta \frac{\partial \tilde{\phi}^{(0)}}{\partial r} + \nabla_{||} \frac{J_{||}^{(1)}}{e n_0} + N_2^{(1)} + \frac{\partial}{\partial t} N_2^{(1)}\right) \rangle = \frac{\partial}{\partial t}N_1^{(0)}.
\end{eqnarray}
Note that this will be modified by the effects of the parallel current density ($\tilde{J}_{||}$) and the non-linear terms, however we see by inspection that on average the term $N_2^{(1)}$ does not contribute whereas the $N_1^{(0)}$ non-linearity may drive the GAM unstable.
We will use the wave kinetic equation\cite{diamond2005, chak2007, smol2000, smol2002, krommes2000, anderson2002, anderson2006, anderson2007, hallatschek2012} to describe the background short scale ETG turbulence for $(\Omega, \vec{q}) < (\omega, \vec{k})$, where the action density $N_k = E_k/|\omega_r| \approx \epsilon_0 |\phi_k|^2/\omega_r$. Here $\epsilon_0 |\phi_k|^2$, is the total energy in the ETG mode with mode number $k$ where $\epsilon_0 = \tau + k_{\perp}^2 + \frac{\eta_e^2 k_\theta^2}{|\omega|^2}$. In describing the large scale plasma flow dynamics it is assumed that there is a sufficient spectral gap between the small scale ETG turbulent fluctuations and the large scale GAM flow. The electrostatic potential is represented as a sum of fluctuating and mean quantities
\begin{eqnarray} \label{eq:4.1}
\phi(X,x,T,t) = \Phi(X,T) + \tilde{\phi}(x,t)
\end{eqnarray}
where $\Phi(X,T)$ is the mean flow potential. The coordinates $\left( X, T\right)$, $\left( x,t \right)$ are the spatial and time coordinates for the mean flows and small scale fluctuations, respectively. The wave kinetic equation can be written as,
\begin{eqnarray} \label{eq:4.2}
\frac{\partial }{\partial t} N_k(x,t) & + & \frac{\partial }{\partial k_x} \left( \omega_k + \vec{k} \cdot \vec{v}_g \right)\frac{\partial N_k(x,t)}{\partial x} - \frac{\partial }{\partial x} \left( \vec{k} \cdot\vec{v}_g\right) \frac{\partial N_k(x,t)}{\partial k_x} \nonumber \\
& = & \gamma_k N_k(x,t) - \Delta\omega N_k(x,t)^2.
\end{eqnarray}
We will solve Equation (\ref{eq:4.2}) by assuming a small perturbation ($\delta N_k$) driven by a slow variation for the GAM compared to the mean ($N_{k0}$) such that $N_k = N_{k 0} + \delta N_k$. The relevant non-linear terms can be approximated in the following form\cite{singh2001},
\begin{eqnarray}
\langle [\tilde{\phi}_k, \nabla_{\perp}^2 \tilde{\phi}_k] \rangle & \approx & q_x^2 \sum_k k_x k_y \frac{|\omega_r|}{\epsilon_0} \delta N_k(\vec{q},\Omega), \\ \ \label{eq:4.3}
\langle [ \tilde{A}_{|| k}, \nabla_{\perp}^2 \tilde{A}_{|| k}] \rangle & \approx & q_x^2 \sum_k k_x k_y \lambda_0 \frac{|\omega_r|}{\epsilon_0} \delta N_k(\vec{q},\Omega), \\ \ \label{eq:4.4}
\langle [\tilde{\phi}_k, \tilde{T}_{e k}] \rangle & \approx & -i q_x \eta_e \sum_k \frac{k_y}{\gamma_k} \frac{|\omega_r|}{\epsilon_0} \delta N_k(\vec{q},\Omega). \label{eq:4.5}
\end{eqnarray}
For all GAMs we have $q_x > q_y$, with the following relation between $\delta N_k$ and $\partial N_{k 0} / \partial k_x$,
\begin{eqnarray} \label{eq:4.6}
\delta N_k = -i q_x^2 k_y \phi^0_q R \frac{\partial N_{0k}}{\partial k_x} + \frac{k_y q_x T_q^{(1)} N_{0k}}{\tau_i \sqrt{(\eta_e - \eta_{eth})}},
\end{eqnarray}
where we have used $\delta \omega_q = k\cdot v_{Eq} \approx i (k_y q_x - k_x q_y) \phi^0_q$ in the wave kinetic equation and the definition $R = \frac{1}{\Omega_q - q_x v_{gx} + i \gamma_k}$. Using the results from the wave-kinetic treatment we can compute the non-linear contributions to be of the form,
\begin{eqnarray}
\langle \phi, \nabla_{\perp}^2 \phi \rangle & = & -i q_x^4 \sum k_x k_y^2 \frac{|\omega_r|}{\epsilon_0} R \frac{\partial N_k}{\partial k_x} \phi_q^{(0)} + \frac{2}{3} q_x^3 \sum k_x k_y \frac{|\omega_r|}{\epsilon_0} \frac{R N_0}{\tau (\eta_e - \eta_{th e})^{1/2}} n_q^{(1)}, \\ \label{eq:4.6}
\langle \phi, T_e \rangle & = & q_x^3 \sum k_y^2 \frac{\eta_e \gamma }{|\omega_r |^2} \frac{|\omega_r|}{\epsilon_0} R \frac{ \partial N_0}{\partial k_x} \phi_q^{(0)} + i q_x^2 \sum \frac{2}{3} \frac{k_y^3 \eta_e \gamma }{|\omega_r|^2} \frac{|\omega_r|}{\epsilon_0} \frac{R N_0}{\tau (\eta_e - \eta_{th e})} n_q^{(1)}. \label{eq:4.7}
\end{eqnarray}
In order to find the non-linear growth rate of the electron GAM we need to find relations between the variables $n_G^{(1)}$, $T_G^{(1)}$ and $\phi_G^{(0)}$,
\begin{eqnarray}
n_G^{(1)} & = & - \frac{\epsilon_n q_x \sin \theta }{\Omega - q_{\parallel}^2/\Omega} \phi_G^{0}, \label{eq:4.8} \\
T_G^{(1)} & = & \frac{2}{3} n_G^{(1)} - \frac{2}{3} q_x^2\sum \frac{k_y^3 \eta_e \gamma}{|\omega|^2} \frac{|\omega_r|}{\epsilon_0} \frac{R N_0}{\tau (\eta_e - \eta_{th e})^{1/2}} n_q^{(1)}. \label{eq:4.9}
\end{eqnarray}
Using Eqs. (\ref{eq:4.8}) and (\ref{eq:4.9}) in the Fourier representation of Eq. (\ref{eq:3.8}) resulting in
\begin{eqnarray} \label{eq:4.10}
\Omega q_x^2 \phi_G^{(0)} + \epsilon_n q_x \sin \theta (n_G^{(1)} + T_G^{(1)}) = i \langle \phi, \nabla_{\perp}^2 \phi \rangle^{(0)},
\end{eqnarray}
and we finally find
\begin{eqnarray} \label{eq:4.11}
\Omega^2 - k_{\parallel}^2 - \frac{5}{6} \epsilon_n^2 & = & - \frac{1}{3} \epsilon_n^2 q_x^2 \sum \frac{k_y^3 \eta_e \gamma}{|\omega|} \frac{|\omega_r|}{\epsilon_0} \frac{R N_0}{\tau (\eta-\eta_{th e})^{1/2}} \nonumber
\\ & + & \left( \frac{\Omega^2 - q_{\parallel}^2}{\Omega} \right) q_x^2 \sum k_x k_y^2 \frac{|\omega_r|}{\epsilon_0} R \frac{\partial N_k}{\partial k_x}.
\end{eqnarray}
Eq. (\ref{eq:4.11}) is the sought dispersion relation for the electron GAM and we solve it pertubatively by assuming $\Omega = \Omega_0 + \Omega_1$ where $\Omega_0$ is the solution to the linear part c.f. Eq. (\ref{eq:3.9}). Now we find the perturbation $\Omega_1 = i \gamma_q$ which will determine the growth rate of the GAM as,
\begin{eqnarray}
\frac{\gamma_q}{c_e/L_n} & = & i \frac{\epsilon_n^2}{6 \Omega_0} q_x^2 \rho_e^2 \sum \frac{k_y^3 \rho_e^3 \eta_e \gamma}{|\omega|} \frac{|\omega_r|}{\epsilon_0} \frac{N_0}{i \gamma} \frac{1}{(\eta_e - \eta_{th e})^{1/2}} - i \frac{5}{12} \frac{\epsilon_n^2}{\Omega_0^2} q_x^2 \rho_e^2 \sum k_x k_y^2 \rho_e^3 \frac{|\omega_r|}{\epsilon_0} \frac{1}{i \gamma} \left| \frac{\partial N_0}{\partial k_x} \right| \nonumber \\
& \approx & \frac{5}{12} \frac{q_x^2 \rho_e^2 k_y \rho_e}{\sqrt{\epsilon_n} (\eta_e - \eta_{th e})^{1/2}} \left| \phi_k^2 \right|. \label{eq:4.14}
\end{eqnarray}
In the last expression we have assumed that the GAM frequency ($\Omega_0$) can be approximated by $\Omega_0 \approx 2 c_e/R$, i.e. the linear GAM is purely oscillatory as found in Eq. (\ref{eq:3.9}). The non-linearly driven electron GAM is unstable with a growth rate depending on the saturation level $\left| \phi_k^2 \right|$ of the ETG mode turbulence.
\section{\label{sec:level9} Saturation mechanism}
In this section we will estimate a new saturation level for the ETG turbulent electrostatic potential ($\phi_k$) by using the Landau damping in competition with the non-linear growth rate of the GAM in a constant background of ETG mode turbulence, according to the well known predator-prey models used\cite{malkov2001}, c.f. Eq. (4) in Ref (\onlinecite{miki2007}) and as well as Ref. (\onlinecite{miki2010}),
\begin{eqnarray} \label{eq:5.1}
\frac{\partial N_k}{\partial t} & = & \gamma_k N_k - \Delta \omega N_k^2 - \gamma_1 U_G N_k \\
\frac{\partial U_G}{\partial t} & = & \gamma_q U_G - \gamma_L U_G.
\end{eqnarray}
Here we have represented the ETG mode turbulence as $N_k = |\phi_k|^2 \frac{L_n^2}{\rho_e^2}$ and $U_G = \langle \frac{ e \phi_G^{(0)}}{T_e} \frac{L_n}{\rho_e} \sin \theta \rangle$ with the following parameters $\gamma$ is the ETG mode growth rate, $\gamma_{NL}$ is the non-linear damping and $\gamma_1$ is the coupling between the ETG mode and the GAM. The Landau damping rate $\left( \gamma_L = \frac{4 \sqrt{2}}{3 \sqrt{\pi}} \frac{c_e}{qR} \right)$ is assumed to be balanced by GAM growth rate Eq. (\ref{eq:4.14}) in stationary state $ \frac{\partial N}{\partial t} \rightarrow 0$ and $ \frac{\partial U_G}{\partial t} \rightarrow 0$. In steady state find the saturation level for the ETG turbulent intensity as ($\gamma_q = \gamma_L$),
\begin{eqnarray} \label{eq:5.2}
\left|\frac{e \phi_k}{T_e} \frac{L_n}{\rho_e} \right|^2 = \frac{48 \sqrt{2}}{3 \sqrt{\pi}} \left( \frac{L_n}{q R} \right) \left( \frac{L_n}{\rho_e} \right)^2 \frac{\sqrt{\epsilon_n (\eta_e - \eta_{th e})}}{(q_x \rho_e)^2 (k_y \rho_e)}.
\end{eqnarray}
Note that this saturation level is significantly enhanced compared to the mixing length estimate,
\begin{eqnarray} \label{eq:5.3}
\left|\frac{e \phi_k}{T_e} \frac{L_n}{\rho_e} \right| \sim 10.
\end{eqnarray}
Here in this estimation we have used $L_n = 0.05$, $q = 3.0$, $R = 4$, $\epsilon_n = 0.025$, $q_x \rho_e = k_y \rho_e = 0.3$ and $\eta_e - \eta_{e th} \sim 1$. Note that the result found using a mixing length estimate is $\left| \frac{e \phi}{T_e} \frac{L_n}{\rho_e}\right| \sim 1$ significantly smaller.
\section{\label{sec:level10} Conclusion}
In this paper we have presented the first derivation of a high frequency branch of the Geodesic Acoustic Mode (GAM). The linear dispersion relation of the high frequency GAM showed that the new branch is purely oscillatory with a frequency $\Omega \sim \frac{c_e}{R}$. To estimate the GAM growth rate, a non-linear treatment based on the wave-kinetic approach was applied. The resulting non-linear dispersion relation showed that the high frequency GAM is excited in the presence of ETG modes with a growth rate depending on the fluctuation level of the ETG mode turbulence. An analytical expression for the resulting GAM growth rate was obtained. To estimate the ETG mode fluctuation level and GAM growth, a predator-prey model was used to describe the coupling between the GAMs and small scale ETG turbulence. The stationary point of the coupled system implies that the ETG turbulent saturation level $\phi_k$ can be drastically enhanced by a new saturation mechanism, stemming from a balance between the Landau damping and the GAM growth rate. This may result in highly elevated particle and electron heat transport, relevant for the edge pedestal region of H-mode plasmas.
The present work was based on a fluid description of ETG mode turbulence, including finite beta electromagnetic effects and retaining non-adiabatic ions. A more accurate treatment based on quasi-linear and nonlinear gyrokinetic simulations is left for future work.
|
1,116,691,497,746 | arxiv | \section*{Introduction}
Letting $\KK$ be an algebraically closed field of characteristic zero,
we let $\KK^{[n]}=\KK[x_1,\ldots,x_n]$ be the polynomial algebra in $n$
variables, and $\AF^n=\spec \KK^{[n]}$ be the affine space. The
Cremona group $\Aut(\AF^n)$ is the group of automorphisms of $\AF^n$,
or equivalently, the group of $\KK$-automorphisms of $\KK^{[n]}$. We
define $\Aut^*(\AF^n)$ as the subgroup of volume preserving
automorphisms i.e.,
$$\Aut^*(\AF^n)=\left\{\gamma\in \Aut(\AF^n) \mid \det\left(
\frac{\partial}{\partial
x_i}\gamma(x_j)\right)_{i,j}=1\right\}\,.$$ %
The groups $\Aut(\AF^n)$ and $\Aut^*(\AF^n)$ are infinite
dimensional algebraic groups \cite{Sha66,Kam79}.
It follows from \cite{Bia66,Bia67} that the maximal dimension of an
algebraic torus contained in $\Aut^*(\AF^n)$ is $n-1$. Moreover, every
algebraic torus of dimension $n-1$ contained in $\Aut^*(\AF^n)$ is
conjugated to the diagonal torus
$$\TD=\left\{\gamma=\diag(\gamma_1,\ldots,\gamma_n)\in \Aut^*(\AF^n)
\mid \gamma_1\cdots \gamma_n=1\right\}\,.$$
A $\KK$-derivation $\partial$ on an algebra $A$ is called locally
nilpotent (LND for short) if for every $a\in A$ there exists $k\in
\ZZ_{\geq 0}$ such that $\partial^k(a)=0$. If
$\partial:\KK^{[n]}\rightarrow \KK^{[n]}$ is an LND on the polynomial
algebra, then $\exp(t\partial)\in \Aut^*(\AF^n)$, for all $t\in \KK$
\cite{Fre06}. Hence, $\partial$ belongs to the Lie algebra
$\Lie(\Aut^*(\AF^n))$.
In analogy with the notion of root from the theory of algebraic groups
\cite{Spr98}, Popov introduced the following definition. A non-zero
locally nilpotent derivation $\partial$ on $\KK^{[n]}$ is called a
root vector of $\Aut^*(\AF^n)$ with respect to the diagonal torus
$\TD$ if there exists a character $\chi$ of $\TD$ such that
$$\gamma\circ\partial\circ\gamma^{-1}=\chi(\gamma)\cdot\partial,\quad\mbox{for
all}\quad \gamma\in\TD\,.$$ %
The character $\chi$ is called the root of $\Aut^*(\AF^n)$ with
respect to $\TD$ corresponding to $\partial$.
Letting $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_n)\in \ZZ_{\geq
0}^n$, we let $\mathbf{x}^{\boldsymbol{\alpha}}$ be the monomial
$x_1^{\alpha_1}\cdots x_n^{\alpha_n}$. In this note we apply the
results in \cite{Lie10} to prove the following theorem. This answers a
question due to Popov \cite{FrRu05}.
\begin{theorem} \label{theo}
If $\partial$ is a root vectors of $\Aut^*(\AF^n)$ with respect to
the diagonal torus $\TD$, then
$$\partial=\lambda\cdot\mathbf{x}^{\boldsymbol{\alpha}}
\cdot\frac{\partial}{\partial x_i}\,,$$ for some $\lambda\in \KK^*$,
some $i\in \{1,\ldots,n\}$, and some
$\boldsymbol{\alpha}\in\ZZ_{\geq 0}^n$
such that $\alpha_i=0$. The corresponding root is the character
$$\chi:\TD\rightarrow \KK^*,\quad
\gamma=\diag(\gamma_1,\ldots,\gamma_n)\mapsto
\gamma_i^{-1}\prod_{j=1}^n\gamma_j^{\alpha_j}\,.$$
\end{theorem}
\section{Proof of the Theorem}
It is well known that the set $\chara(\TD)$ of characters of $\TD$
forms a lattice isomorphic to $M=\ZZ^{n-1}$. It is customary to fix an
isomorphism $M\simeq\chara(\TD)$ and to denote the character
corresponding to $m$ by $\chi^m$. The natural $\TD$-action on $\AF^n$
gives rise to an $M$-grading on $\KK^{[n]}$ given by
$$\KK^{[n]}=\bigoplus_{m\in M}B_m,\quad\mbox{where}\quad
B_m=\left\{f\in\KK^{[n]}\mid \gamma(f)=\chi^m(\gamma)f,\, \forall
\gamma\in \TD\right\}\,.$$
An LND $\partial$ on $\KK^{[n]}$ is called homogeneous if it send
homogeneous elements into homogeneous elements. Let $f\in
\KK^{[n]}\setminus \ker\partial$ homogeneous. We define the degree of
$\partial$ as $\deg\partial=\deg(\partial(f))-\deg(f)\in M$
\cite[Section 1.2]{Lie10}.
\begin{lemma}\label{root-hom}
An LND on $\KK^{[n]}$ is a root vectors of $\Aut^*(\AF^n)$ with
respect to the diagonal torus $\TD$ if and only if $\partial$ is
homogeneous with respect to the $M$-grading on $\KK^{[n]}$ given by
$\TD$. Furthermore, the corresponding root is the character
$\chi^{\deg\partial}$.
\end{lemma}
\begin{proof}
Let $\partial$ be a root vector of $\Aut^*(\AF^n)$ with root
$\chi^e$, so that
$\partial=\chi^{-e}(\gamma)\cdot\gamma\circ\partial\circ\gamma^{-1}$,
$\forall\gamma\in\TD$. We consider a homogeneous element $f\in
B_{m'}$ and we let $\partial(f)=\sum_{m\in M} g_m$, where $g_m$ is
homogeneous, so that
$$\sum_{m\in M}g_m=\partial(f)=\chi^{-e}(\gamma)\cdot\gamma\circ
\partial\circ\gamma^{-1}(f)=\chi^{-e-m'}(\gamma)\sum_{m\in
M}\chi^m(\gamma)\cdot g_m,\quad\forall\gamma\in\TD\,.$$ %
This equality holds if and only if $g_m=0$ for all but one $m\in
M$ i.e., if $\partial$ is homogeneous. In this case,
$\partial(f)=g_m=\chi^{-e-m'+m}(\gamma)\cdot \partial(f)$, and so
$e=m-m'=\deg(\partial(f))-\deg(f)=\deg\partial$.
\end{proof}
In \cite{AlHa06}, a combinatorial description of a normal affine
$M$-graded domain $A$ is given in terms of polyhedral divisors, and in
\cite{Lie10} a description of the homogeneous LNDs on $A$ is given in
terms of these combinatorial data in the case where $\trdeg A=\rank
M+1$. In the following we apply these results to compute the
homogeneous LNDs on the $M$-graded algebra $\KK^{[n]}$. First, we give
a short presentation of the combinatorial description in \cite{AlHa06}
in the case where $\trdeg A=\rank M+1$. For a more detailed treatment
see \cite[Section 1.1]{Lie10}.
Let $N$ be the dual lattice of $M$. The combinatorial description in
\cite{AlHa06} deals with the following data: A pointed polyhedral cone
$\sigma\in N_\QQ:=N\otimes \QQ$ dual to the weight cone
$\sigma^\vee\subseteq M_\QQ:=M\otimes\QQ$ of the $M$-grading; a smooth
curve $C$; and a divisor $\DD=\sum_{z\in C}\Delta_z\cdot z$ on $C$
whose coefficients $\Delta_z$ are polyhedra in $N_\QQ$ having tail
cone $\sigma$. Furthermore, if $C$ is projective we ask for the
polyhedron $\sum_{z\in C} \Delta_z$ to be a proper subset of
$\sigma$. For every $m\in \sigma^\vee\cap M$ the evaluation of $\DD$
at $m$ is the $\QQ$-divisor given by
$$\DD(m)=\sum_{z\in C}\min_{p\in \Delta_z}p(m)\,.$$
We define the $M$-graded algebra
\begin{align} \label{AH}
A[\DD]=\bigoplus_{m\in \sigma^\vee\cap
M}A_m\chi^m,\quad\mbox{where}\quad A_m=H^0(C,\OO_C(\DD(m)))\,,
\end{align}
and $\chi^m$ is the corresponding character of the torus $\spec\KK[M]$
seen as a rational function on $\spec A$ via the embedding $\fract
\KK[M]\hookrightarrow \fract A[\DD]=\fract \KK(C)[M]$.
It follows from \cite{AlHa06} that $A[\DD]$ is an normal affine domain
and that every normal affine $M$-graded domain $A$ with $\trdeg
A=\rank M+1$ is equivariantly isomorphic to $A[\DD]$ for some
polyhedral divisor on a smooth curve, see also \cite[Theorem
1.4]{Lie10}.
\medskip
We turn back now to our particular case where we deal with the
polynomial algebra $\KK^{[n]}$ graded by $\chara(\TD)$. First, we need
to fix an isomorphism $M\simeq \chara(\TD)$. Let
$\{\mu_1,\ldots,\mu_{n-1}\}$ be the canonical basis of $M$. We define
the isomorphism $\mu_i\mapsto \chi^{\mu_i}$, $i\in\{1,\ldots,n-1\}$,
where
$$\chi^{\mu_i}:\TD\rightarrow \KK^*,\quad
\gamma=\diag(\gamma_1,\ldots,\gamma_n)\mapsto \gamma_i\,.$$ %
Since for every $\gamma\in \TD$ we have
$\gamma_n=\gamma_1^{-1}\cdots\gamma_{n-1}^{-1}$, the character mapping
$\diag(\gamma_1,\ldots,\gamma_n)\mapsto \gamma_n$ is given by
$\chi^{-\mathds{1}}$, where
$\mathds{1}:=\mu_1+\ldots+\mu_{n-1}$. Under this isomorphism, the
algebra $\KK^{[n]}$ is graded by $M$ via $\deg x_i=\mu_i$, for all
$i\in\{1,\ldots,n-1\}$ and $\deg x_n=-\mathds{1}$.
Let now $\{\nu_1,\ldots,\nu_{n-1}\}$ be a basis of $N$ dual to the
basis $\{\mu_1,\ldots,\mu_{n-1}\}$ of $M$. We also let $\Delta$ be the
standard $(n-1)$-simplex in $N_\QQ$ i.e., the convex hull of the set
$\{\nu_1,\ldots,\nu_{n-1},\bar{0}\}$.
\begin{lemma} \label{isom}
The $M$-graded algebra $\KK^{[n]}$ is equivariantly isomorphic to
$A[\DD]$, where $\DD$ is the polyhedral divisor $\DD=\Delta\cdot[0]$
on $\AF^1$.
\end{lemma}
\begin{proof}
By \cite{AlHa06}, the $M$-graded algebra $\KK^{[n]}$ is isomorphic
to $A[\DD]$ for some polyhedral divisor $\DD$ on a smooth curve
$C$. Since the weight cone $\sigma^\vee$ of $\KK^{[n]}$ is $M_\QQ$,
the coefficients of $\DD$ are just bounded polyhedra in $N_\QQ$.
Since $\AF^n$ is a toric variety and the torus $\TD$ is a subtorus
of the big torus, we can apply the method in \cite[Section
11]{AlHa06}. In particular, $C$ is a toric curve. Thus $C=\AF^1$ or
$C=\PP^1$. Furthermore, the graded piece $B_{\bar{0}}\supsetneq
\KK$ and so $C$ is not projective by \eqref{AH}. Hence $C=\AF^1$.
The only divisor in $\AF^1$ invariant by the big torus is $[0]$, so
$\DD=\Delta\cdot[0]$ for some bounded polyhedron $\Delta$ in
$N_\QQ$. Finally, applying the second equation in \cite[Section
11]{AlHa06}, a routine computation shows that $\Delta$ can be chosen
as the standard $(n-1)$-simplex.
\end{proof}
\begin{remark} \label{rem-isom}
Letting $\AF^1=\spec \KK[t]$, it is easily seen that the isomorphism
$\KK^{[n]}\simeq A[\DD]$ is given by $x_i=\chi^{\mu_i}$, for all
$i\in \{1,\ldots,n-1\}$, and $x_n=t\chi^{-\mathds{1}}$.
\end{remark}
In \cite{Lie10} the homogeneous LNDs on an normal affine $M$-graded
domain are classified into 2 types: fiber type and horizontal type. In
the case where the weight cone is $M_\QQ$, there are no LNDs of fiber
type. Thus, $\KK^{[n]}$ only admits homogeneous LNDs of horizontal
type. The homogeneous LNDs of horizontal type are described in
\cite[Theorem 3.28]{Lie10}. In the following, we specialize this
result to the particular case of $A[\DD]\simeq \KK^{[n]}$.
Let $v_i=\nu_i$, $i\in\{1,\ldots,n-1\}$ and $v_n=\bar{0}$, so that
$\{v_1,\ldots,v_n\}$ is the set of vertices of $\Delta$. For every
$\lambda\in \KK^*$, $i\in \{1,\ldots,n\}$, and $e\in M$ we let
$\partial_{\lambda,i,e}:\fract A[\DD]\rightarrow \fract A[\DD]$ be the
derivation given by
$$\partial_{\lambda,i,e}(t^r\cdot\chi^m)=\lambda(r+v_i(m))\cdot
t^{r-v_i(e)-1}\cdot\chi^{m+e},\quad \forall (m,r)\in M\times \ZZ\,.$$
\begin{lemma}[{\cite[Theorem 3.28]{Lie10}}] \label{lnd} %
If $\partial$ is a non-zero homogeneous LND of $A[\DD]$, then
$\partial=\partial_{\lambda,i,e}|_{A[\DD]}$ for some $\lambda\in
\KK^*$, some $i\in \{1,\ldots,n\}$, and some $e\in M$ satisfying
$v_j(e)\geq v_i(e)+1,\ \forall j\neq i\,.$ Furthermore, $e$ is the
degree $\deg\partial$.
\end{lemma}
\begin{proof}[Proof of Theorem~\ref{theo}]
By Lemma~\ref{root-hom} the root vectors of $\KK^{[n]}$ correspond
to the homogeneous LNDs in the $M$-graded algebra $\KK^{[n]}$. But
the homogeneous LNDs on $A[\DD]\simeq \KK^{[n]}$ are given in
Lemma~\ref{lnd}, so we only need to translate the homogeneous LND
$\partial=\partial_{\lambda,i,e}|_{A[\DD]}$ in Lemma~\ref{lnd} in
terms of the explicit isomorphism given in Remark~\ref{rem-isom}.
Let $e=(e_1,\ldots,e_{n-1})$ and $i\in \{1,\ldots,n-1\}$, so that
$v_i=\mu_i$. The condition $v_j(e)\geq v_i(e)+1$ yields $e_i\leq -1$
and $e_j\geq e_i+1$, $\forall j\neq i$. Furthermore,
$\partial(x_k)=\partial(\chi^{\mu_k})=0$, for all $k\neq i$,
$k\in\{1,\ldots,n-1\}$,
$\partial(x_n)=\partial(t\chi^{-\mathds{1}})=0$, and
$$\partial(x_i)=\partial(\chi^{\mu_i})=\lambda
t^{-e_i-1}\chi^{e+\mu_i}= \lambda
t^{-e_i-1}\chi^{(e_i+1)\mathds{1}}\chi^{e+\mu_i-
(e_i+1)\mathds{1}}=\lambda\mathbf{x}^{\boldsymbol{\alpha}}\,,$$
where $\alpha_i=0$, $\alpha_n=-e_i-1\geq 0$, and
$\alpha_k=e_k-e_i-1\geq 0$, for all $k\neq i$,
$k\in\{1,\ldots,n-1\}$. Hence,
$\partial=\lambda\cdot\mathbf{x}^{\boldsymbol{\alpha}}
\cdot\tfrac{\partial}{\partial x_i}$, for some $\lambda\in \KK^*$,
some $i\in \{1,\ldots,n\}$, and some
$\boldsymbol{\alpha}\in\ZZ_{\geq 0}^n$ such that $\alpha_i=0$.
Let now $e=(e_1,\ldots,e_{n-1})$ and $i=n$, so that $v_n=0$. The
condition $v_j(e)\geq v_n(e)+1$ yields $e_j\geq 1,\, \forall
j\in\{1,\ldots,n-1\}$. Furthermore,
$\partial(x_k)=\partial(\chi^{\mu_k})=0$, $k\in\{1,\ldots,n-1\}$, and
$$\partial(x_n)=\partial(t\chi^{-\mathds{1}})=\lambda
\chi^{e-\mathds{1}}=\lambda\mathbf{x}^{\boldsymbol{\alpha}}\,,$$
where $\alpha_n=0$, and $\alpha_k=e_k-1\geq 0$, for all
$k\in\{1,\ldots,n-1\}$. Hence,
$\partial=\lambda\cdot\mathbf{x}^{\boldsymbol{\alpha}}
\cdot\tfrac{\partial}{\partial x_n}$, for some $\lambda\in \KK^*$,
some $i\in \{1,\ldots,n\}$, and some
$\boldsymbol{\alpha}\in\ZZ_{\geq 0}^n$ such that $\alpha_n=0$.
The last assertion of the theorem follows easily from the fact that
the root corresponding to the homogeneous LND $\partial$ is the
character $\chi^{\deg\partial}$.
\end{proof}
Finally, we describe the characters that appear as a root of
$\Aut^*(\AF^n)$.
\begin{corollary}
The character $\chi\in\chara(\TD)$ given by
$\diag(\gamma_1,\ldots,\gamma_n)\mapsto
\gamma_1^{\beta_1}\cdots\gamma_n^{\beta_n}$ is a root of
$\Aut^*(\AF^n)$ with respect to the diagonal torus $\TD$ if and only
if the minimum of the set $\{\beta_1,\ldots,\beta_n\}$ is achieved
by one and only one of the $\beta_i$.
\end{corollary}
\begin{proof}
By Theorem~\ref{theo}, the roots of $\Aut^*(\AF^n)$ are the
characters $\diag(\gamma_1,\ldots,\gamma_n)\mapsto
\gamma_1^{\beta_1}\cdots\gamma_n^{\beta_n}$, where $\beta_i=-1$ for
some $i\in\{1,\ldots,n\}$ and $\beta_j\geq 0$ $\forall j\neq i$. The
corollary follows from the fact that $\gamma_1\cdots\gamma_n=1$.
\end{proof}
\section*{Acknowledgment}
The author is grateful to Ivan Arzhantsev who pointed out this problem
to us.
\bibliographystyle{alpha}
|
1,116,691,497,747 | arxiv | \section{Motivation}
The appearance of instantons (topologically nontrivial Euclidean configurations
minimizing the action) is a very common feature of many different field
theories
{}~\cite{Raj}. Of particular interest are the instantons in QCD~\cite{BPTS}
which
are so beautiful that a serious hope existed~\cite{CDG} and still exists~\cite
{Shur} that, working with instantons, one can perceive the essential features
of QCD vacuum state.
However, soon after discovery of instantons, it has been
understood that the {\em naive} instanton calculations meet problems. The
matter is that the quasiclassical approximation on which these calculations are
based just does not work in QCD---everything depends on the instantons of
large sizes where quantum corrections are 100\%~ important. It is impossible
to calculate analytically these quantum corrections in QCD for $\rho \sim
\Lambda_{QCD}^{-1}$ (though for smaller $\rho$ where the quantum effects are
still
under control and can be considered as perturbation, it is possible~\cite
{vain}).
Generally, the situation is much better, however, at high temperatures $T \gg
\Lambda_{QCD}$. The high-$T$ instanton action $S^I = 8\pi^2/g^2(T)$ is large
and quasiclassical approximation works: the amplitude of quantum fluctuations
is small compared to the amplitude of classical field. As the temperature
grows up, the instantons cool down! That allows one to perform some explicit
instanton calculations in high-$T$ QCD which {\em are} under control. E.g., the
fermion condensate in high-$T$ QCD with {\em one} quark flavour can be found
{}~\cite{Yung}.
The {\em pure} Yang-Mills theory involves, besides instantons, also planar
topologically nontrivial Euclidean field configurations. They appear due to
nontrivial $\pi_1[{\cal G}] = Z_N$ where the true gauge group ${\cal G}$ for
the
pure Yang-Mills theory is $SU(N)/Z_N$ rather than just $SU(N)$ (gluon fields
belong to the adjoint representation and are not transormed under the action of
the elements of the center---see Ref.~\cite{bub} for the detailed discussion).
The action of these configurations involves the large area factor $\sim L^2$
where L is the size of the box, but, if the size of the box is finite (as is
always so in practical numerical calculations in QCD ), these topologically
nontrivial sectors (known as 't Hooft fluxes~\cite{flux}) do contribute in the
partition function.
At low temperatures, quasiclassical approximation does not work, and little can
be said about properties of these configurations. But at high temperatures,
quasiclassical approximation works and the characteristic field configurations
in the path integral present the classical wall-like solutions with the width
of order of Debye screening length $\sim (gT)^{-1}$ and the surface action
density\footnote{The result (~\ref{surf}) has been derived in Ref.~\cite{pisa}.
The authors of that work tried to calculate also the next-to-leading term
$\propto gT^2$ in the action density, but it has been shown in Ref.~\cite{bub1}
that this calculation is not infrared stable. }
\begin{equation}
\label{surf}
\frac{S^{SU(N)}}{{\cal A}} = \frac{4\pi^2(N-1)T^2}{3\sqrt{3N}g}
\end{equation}
, and quantum fluctuations are relatively small.
These high-temperature wall-like solutions turn out to be static (by the
simple reason that, when the size of the Euclidean cylinder $\beta = 1/T$
on which the theory is considered is small, higher Fourier harmonics are not
excited). Originally, they have been interpreted as real Minkowski space
domain walls separating distinct $Z_N$ high-$T$ states. However, there are
serious reasons to believe that these solutions have relevance only for the
Euclidean path integral and {\em cannot} be interpreted as real physical
objects
in the Minkowski space~\cite{bub,bub1}. That means that the common assertion
about spontaneous breaking of $Z_N$--symmetry in high-$T$ ~Yang-Mills theory
is misleading---there is no symmetry breaking in the physical meaning of this
word as the physical domain walls separating the distinct phases do not appear.
Note that the Euclidean quasiclassical wall-like solutions exist also in
high-$T$ ~QED. Their surface action density can be found by the same token as
in nonabelian case~\cite{bub1}.
We want to emphasize that the narrow wall-like solutions appear only as the
solutions for the {\em effective} action (after adding the logarithm of
one-loop
determinant). At the pure classical level walls do not appear---the nontrivial
't Hooft fluxes still exist, but they are delocalized~\cite{flux,toron}.
\section{Schwinger model}
\setcounter{equation}{0}
Our main remark here is that the full scope of questions (of the relevance
of the instanton vacuum picture, of applicability of quasiclassical
approximation
and of the physical meaning of the high-$T$ wall-like solutions) can be
effectively studied in the Schwinger model (2-dimensional massless QED). SM is
exactly soluble so that every reasonable question can be given an exact and
exhaustive answer and, from the other hand, resembles QCD in all gross
features. The action of the model reads
\begin{equation}
\label{SMAct}
S = \int d^2x [-\frac{1}{4}F_{\mu\nu}^2 + i\bar{\psi}{\cal D}_{\mu}\gamma_{\mu}
\psi]
\end{equation}
where ${\cal D}_{\mu} - igA_{\mu}$ and $g$ is the coupling constant having the
dimension of mass.
SM involves confinement---like in QCD, the spectrum involves only `mesons' with
the mass
\begin{equation}
\label{mu}
\mu = \frac{g}{\sqrt{\pi}}
\end{equation}
but not free fermions and photons. The axial current is anomalous:
\begin{equation}
\label{anom}
\partial_{\mu}\bar{\psi}\gamma_{\mu}\gamma^5\psi = \frac{1}{2\pi}\epsilon
_{\alpha\beta}F_{\alpha\beta}
\end{equation}
And what is most important for us here~---~it involves topologically nontrivial
Euclidean gauge field configurations with integer topological charge
\begin{equation}
\label{nu}
\nu = \frac{g}{2\pi} \int E(x)d^2x
\end{equation}
where $E = F_{01}$ ($\nu$ is the 2-dimensional analog of the Pontryagin class),
and hence the instantons---configurations with $\nu = \pm1$ realizing the
minimum of the action.
What is the explicit form of these configurations? If proceeding along the
same lines as people usually do in QCD and looking or the configuration which
minimizes the pure {\em bosonic} Euclidean action
\begin{equation}
\label{sbos}
S_B = \frac{1}{2}\int d^2x ~E^2
\end{equation}
with the constraint $\nu = 1$, we are led to the {\em constant field strength}
solution
\[E = \frac{2\pi}{g{\cal A}}\]
where ${\cal A}$ is the total area of the Euclidean manifold on which the
theory
is defined. Introducing the compact manifold is neccessary in this approach to
provide for the infrared regularization of the theory.
Different choices for
this manifold are possible. In Ref.~\cite{indus}, the Euclidean functional
integral on the 2-dimensional sphere in different topological sectors has been
calculated. But, bearing in mind the parallels with 4-dimensional theories and
also the generalization of the results for finite temperature case, it is more
instructive to consider the theory on the 2-dimensional torus with the spatial
size $L$ and the imaginary time size $\beta$ ~\cite{Wipf}. Here we always
assume
that $L\gg\mu^{-1}$. When also $\beta\gg\mu^{-1}$, the boundary effects become
irrelevant for physical quantities
\footnote{On the large torus, the boundary effects are always exponentially
suppressed
{}~\cite{Lee,cond,LS}. On the large 2-dim sphere with $R\gg\mu^{-1}$,
they are suppressed only as a power due to finite curvature of the manifold
{}~\cite{cond}. }.
When $\beta \leq \mu^{-1}$, they are relevant and should be interpreted as the
effects due to finite temperature $T = \beta^{-1}$.
The constant field strength solution presents a fiber bundle on the torus.
Bearing in mind the subsequent discussion of the high-T case and the parallels
with 4-dim gauge theories we are going to draw, it is convenient to choose
the gauge
$A_1 = 0$. The solution then takes the form
\begin{equation}
\label{trinst}
A_0(x,\tau) = -\frac{2\pi}{gL\beta}x
\end{equation}
It satisfies the twisted boundary conditions
\begin{eqnarray}
\label{bc}
A_0(x,\beta) = A_0(x,0)
\nonumber \\
A_0(L,\tau) = A_0(0,\tau) - \frac{i}{g}\Omega^{\dagger}\partial_0\Omega
\end{eqnarray}
where
\begin{equation}
\label{Omega}
\Omega(\tau) = \exp(-2\pi i \tau/\beta)
\end{equation}
is the gauge transformation matrix. In finite temperature applications, the
corresponding b.c. for fermion fields in imaginary time directions are
antiperiodic, and the b.c. in spatial direction involve an extra gauge
transformation
\[\psi(L,\tau) = \Omega(\tau)\psi(0,\tau)\]
The transition matrix (~\ref{Omega}) satisfies self-consistency condition
$\Omega(\beta) = \Omega(0)$ which {\em is} the reason for the topological
charge (~\ref{nu}) to be quantized.
The solution (~\ref{trinst}) is the direct analog of the 4-dimensional 't Hooft
toron solutions~\cite{toron}.
The configurations with $\nu = \pm 1$ are responsible for the formation of the
fermion condensate in the Schwinger model
\begin{equation}
\label{cond}
<\bar{\psi}\psi>_{vac.} = - \frac{\mu}{2\pi}e^{\gamma}
\end{equation}
where $\gamma$ is the Euler constant. There are many ways to get this result.
The easiest way is to employ the bosonization technique~\cite{Hetrick,cond}.
But bosonization rules are specific for two dimensions, and it is more
instructive to extract the condensate from the Euclidean path integral in the
sectors with $\nu = \pm 1$. This has been done in Refs.~\cite{indus,Wipf}. In
particular, Sachs and Wipf~\cite{Wipf} considered SM on 2-dimensional torus
and calculated the functional integral over the quantum fluctuations around
the constant strength solution (~\ref{trinst}). The amplitude of these quantum
fluctuations turns out to be large (the characteristic field configurations
contributing to the path integral are rather far in Hilbert space from the
classical bosonic solution (~\ref{trinst}), and the quasiclassical {\em
picture}
does not work). The calculation is still possible, however, due to the fact
that the functional integral is {\em exactly} Gaussian.
\section{Path integrals and their saddle points.}
\setcounter{equation}{0}
Any field in the topological sector $\nu = 1$ can be decomposed as
\begin{equation}
\label{decomp}
A_\mu = A_{\mu}^{cl.i.} + A_{\mu}^{(0)} - \epsilon_{\mu\nu}\partial_\nu \phi
+ \partial_\mu \chi
\end{equation}
where $A_\mu^{(0)}$ is the constant part of the potential, $A_\mu^{cl.i.}$ is
the classical instanton solution (~\ref{trinst}), ~$\partial_\mu \chi$ is the
gauge part, and the part $-\epsilon_{\mu\nu}\partial_\nu\phi$ carries
nontrivial
dynamic information. After calculating the fermion determinant $\det\parallel
i{\cal D} - m \parallel$ ($m$ is the small fermion mass $m \ll g$), the
partition function can be written as ~\cite{Wipf}
\begin{equation}
\label{Z1}
Z_1 \propto m\int \prod d\phi \int d^2x e^{-2g\phi(x)} \exp\left\{-\frac{1}{2}
\int \phi(\Delta^2 - \mu^2\Delta)\phi d^2y\right\}
\end{equation}
Here $\phi\Delta^2\phi/2 = (\Delta\phi)^2/2 = E^2/2$ is the classical part of
the action density, the term $\propto \mu^2\phi\Delta\phi$ in the effective
action is the
local part of the fermion determinant and gives mass to photon, and the factor
$m\int d^2x e^{-2g\phi(x)}$ comes from the fermion zero mode:~ $m$ is the
eigenvalue and the integral $\int d^2x e^{-2g\phi(x)}$ is the normalization
factor of the fermion zero mode
\begin{equation}
\label{zero}
\psi_L \propto \exp\{-g\phi(x)\}
\end{equation}
Let us find now the saddle points of the Gaussian functional integral \\
$\int\prod d\phi\exp\{-S^{eff}[\phi]\}$. To this end, we first substitute
\[ \int d^2x \exp\{-2g\phi(x)\} \rightarrow {\cal A}\exp\{-2g\phi(x_0)\} \]
where ${\cal A}$ is the total area of our Euclidean manifold [obviously, this
substitution does not change the value of the functional integral (~\ref{Z1})].
The stationary point is the solution to the equation
\begin{equation}
\label{eqmot}
(\Delta^2 - \mu^2\Delta)\phi^{eff.inst.}(x) = -2g\delta(x-x_0)
\end{equation}
Thus, $\phi^{e.i.}$ is just the Green's function of the operator
${\cal O} = \Delta^2 - \mu^2\Delta$. It has the form
\footnote{
The notion of effective (or `induced') instanton in SM has been introduced
long ago ~\cite{schroer}, but seems to be very well forgotten since that time.}
\begin{equation}
\label{inst}
\phi^{e.i.}(x) = \frac{1}{g} [ K_0(\mu|x-x_0|) + \ln(\mu|x-x_0|)
] + const
\end{equation}
[The possible linear in $x$ part of $\phi^{e.i.}(x)$ can be absorbed in the
constant component of the gauge field in the decomposition (\ref{decomp})].
The solution (~\ref{inst}) is regular at zero.
The electric field
\begin{equation}
\label{field}
E^{e.i.}(x) = \Delta\phi^{e.i.}(x) = \frac{g}{\pi}K_0(\mu|x-x_0|)
\end{equation}
falls down exponentially at large distances. The topological charge (~\ref
{nu}) of this solution is equal to 1 as it should be.
We see that, in contrast to the delocalized classical instanton (~\ref{trinst})
, the effective instanton (~\ref{inst}) {\em is} localized---quantum effects
changed the properties of the solution drastically. The parameter $x_0$ should
be thought of as the collective coordinate of the center of the instanton.
Let us forget for the moment about a not fixed yet constant in the RHS in Eq.
(~\ref{inst}) and find the action of this instanton. Setting $const = 0$, we
get
\begin{equation}
\label{inac}
S^{e.i.} = g\phi^{e.i.}(0) = \ln2 - \gamma
\end{equation}
Then the {\em exact} result for the partition function (~\ref{Z1}) in the
sectors with $\nu = \pm1$ can be presented as
\begin{equation}
\label{zres}
Z_1 = Z_{-1} = m{\cal A} \frac{\mu}{2\pi}\exp\{-S^{e.i.}\}Z_0
\end{equation}
The factor $m$ comes from the fermion zero mode, the factor ${\cal A}$ arises
due to integration over collective coordinates $d^2x_0$, the factor $\mu$
appears by the dimensional reasons. The hard (in this
approach) part of the problem is to determine correctly the numerical factor
$(2\pi)^{-1}$.
To do it, one should proceed more accurately. First of all, the exact
proportionality coefficient in Eq.(~\ref{Z1}) should be found~---~more exactly,
the {\em ratio} of this coefficient to the corresponding coefficient in the
functional integral for the partition function in the topologically trivial
sector $\nu = 0$. This ratio depends on the particular way of the infrared
regularization and thereby on the size of our torus. Second, the equation
(~\ref{eqmot}) has no solution at all on a compact manifold. The correct
procedure is to project out the zero modes of the operator $\Delta^2 -
\mu^2\Delta$ by substituting $\delta(x)\rightarrow\delta(x) - 1/{\cal A}$
and imposing the constraint
\begin{equation}
\label{constr}
\int \phi(x) d^2x = 0
\end{equation}
Thereby the constant in RHS of Eq.(~\ref{inst}) is fixed.
This constant (and
hence the instanton action) also depends on the size of the torus.
\footnote{To be quite precise, the fixing of this constant requires also taking
into account the modification of the solution to the equation (~\ref{eqmot})
due to boundary effects which are essential when $x \sim L, \tau \sim \beta$.
The solution has the simple form (~\ref{inst}) only at the
vicinity of the center of the instanton and far away from the boundaries.}
For large
$L,\beta$, this dependence cancels exactly the similar dependendence of the
normalization constant, and the finite result (~\ref{zres}) is obtained
{}~\cite{Wipf}.
We cannot suggest in this respect anything new compared to the
calculation by Sachs and Wipf, but the final result (~\ref{zres}) looks so
suspiciously simple
that one is tempted to guess that an easier way to derive it may exist.
Differentiating Eq.(~\ref{zres}) over the fermion mass and adding the equal
contribution from the sector $\nu=-1$, we reproduce the result (~\ref{cond}):
\begin{equation}
\label{cond1}
<\bar{\psi}\psi> = -\frac{1}{{\cal A}Z_0} \frac{\partial}{\partial m}[Z_1
+ Z_{-1}] =
- \frac{\mu}{2\pi}e^{\gamma}
\end{equation}
There is, however, a simple {\em indirect} way to fix the coefficient in
Eq.(~\ref{cond}). To this end, one should consider the correlator $<\bar{\psi
}\psi(x)~ \bar{\psi}\psi(0)>$ in the topologically trivial sector $\nu=0$
{}~\cite
{baaq}. It can be calculated rather straightforwardly using the {\em exact}
expression for the fermion Green's function in {\em any} gauge field background
\begin{equation}
\label{Green}
S_\phi(x,y) = \exp\{-g\gamma^5\phi(x)\} S_0(x-y) \exp\{-g\gamma^5\phi(y)\}
\end{equation}
where $S_0(x-y)$ is the free fermion Green's function.
\footnote{The result (\ref{Green}) is specific for SM. Unfortunately, no
similar simple formula is known for 4-dimensional theories.}
We have
\begin{eqnarray}
\label{corr}
<\bar{\psi}\psi(x) ~\bar{\psi}\psi(0)>_{\nu=0}\ =\ Z_0^{-1} \int \prod d\phi
\exp\left\{-\frac{1}{2}
\int \phi(\Delta^2 - \mu^2\Delta)\phi ~d^2y\right\} \nonumber\\
= \frac{1}{2\pi^2x^2} \int \prod d\phi \exp\left\{-\frac{1}{2}
\int \phi(\Delta^2 - \mu^2\Delta)\phi ~d^2y\right\}
\exp\{2g[\phi(x)-\phi(0)]\}
\end{eqnarray}
where we substituted $\cosh\{\ldots\}\rightarrow \exp\{\ldots\}$ as odd powers
of $\phi(x) - \phi(0)$ give zero after integration.
The integral is Gaussian. Its stationary point is the solution to the equation
\begin{equation}
\label{eqmot1}
(\Delta^2 - \mu^2\Delta)\phi^{stat.}(y) = 2g\left[\delta(y-x) -
\delta(y)\right]
\end{equation}
which is
\begin{equation}
\label{IA}
\phi^{stat.}(y) = \phi^{e.i.}(y) - \phi^{e.i.}(y-x)
\end{equation}
with $\phi^{e.i.}(y)$ being taken from Eq.(~\ref{inst}). $\phi^{stat.}(y)$
presents an {\em instanton-antiinstanton configuration}. Note that the free
constant in the RHS of Eq.(~\ref{inst}) cancels out completely in the
difference.
\footnote
{The {\em antiinstanton-instanton configuration} $-\phi^{stat.}(y)$ plays
exactly the same role (we could substitute $\cosh\{\ldots\} \rightarrow
\exp\{-\ldots\}$ with an equal ease). A quite precise way would be to present
$\cosh\{\ldots\}$ as the sum of two exponentials and split the path integral
(\ref{corr}) in two equal parts. The stationary point of one of them
(which describes
the correlator $<\bar{\psi}_L \psi_R(x)~\bar{\psi}_R \psi_L(0)>$ ) is
$\phi^{stat.}(y)$ while the stationary point of the other corresponding to
$<\bar{\psi}_R \psi_L(x)~\bar{\psi}_L \psi_R(0)>$ is $-\phi^{stat.}(y)$.}
The calculation is standard. Introduce a new integration variable
\begin{equation}
\label{diff}
\varphi(y) = \phi(y) - \phi^{stat.}(y)
\end{equation}
We have
\begin{eqnarray}
\label{corr1}
<\bar{\psi}\psi(x)~\bar{\psi}\psi(0)>_{\nu=0}~~ =~
{}~ Z_0^{-1} \frac{1}{2\pi^2 x^2}
\exp\{2g[\phi^{stat.}(x)-\phi^{stat.}(0)]\} \nonumber\\
\int \prod d\varphi
\exp\left\{-\frac{1}{2}
\int \varphi(\Delta^2 - \mu^2\Delta)\varphi ~d^2y\right\}
\end{eqnarray}
The integral $\int \prod d\varphi \exp\{\ldots\}$ exactly cancels out the
identical functional integral for $Z_0$ and the result is
\begin{equation}
\label{corr2}
<\bar{\psi}\psi(x)~\bar{\psi}\psi(0)>_{\nu=0} = \frac{\mu^2}{8\pi^2}
e^{2\gamma} \exp\{-2K_0(\mu x)\}
\end{equation}
At large $x$, this correlator tends to a constant. One may be tempted to
extract square root of this constant and call it the fermion condensate, but
that would not be quite correct. Only the full correlator (the contributions
of all topological sectors being summed over) enjoys the cluster decomposition
property, and also the configurations with topological charge $\nu=\pm2$
contribute to the large $x$ asymptotics of the correlator. One can derive that
\footnote
{The result (\ref{nu2}) follows from Ward identities which dictate the
particular form of $\theta$-dependence of the partition function. It holds
both in SM ~\cite{baaq} and in QCD ~\cite{LS}. In SM, it has been confirmed
by explicit calculations ~\cite{indus,Wipf}.}
\begin{eqnarray}
\label{nu2}
\lim_{x \rightarrow \infty}<\bar{\psi}\psi(x) ~\bar{\psi}\psi(0)>_{\nu=2}\ =\
\lim_{x \rightarrow \infty}<\bar{\psi}\psi(x) ~\bar{\psi}\psi(0)>_{\nu=-2}
\nonumber \\ =\
\frac{1}{2}\lim_{x \rightarrow \infty}<\bar{\psi}\psi(x) ~\bar{\psi}\psi(0)>_
{\nu=0}
\end{eqnarray}
Adding together all contributions and extracting square root, we arrive at
the result (~\ref{cond}).
There is an important point which we want to emphasize here. In
zero-temperature SM as well as in zero-temperature QCD, the quasiclassical
picture
does not really work---the quantum fluctuations are large and the
characteristic
field configurations in the path integral {\em have nothing to do} with either
classical (~\ref{trinst}) or effective (~\ref{inst}) instanton solutions. We
have shown in Ref.\cite{vac} that all essential properties of the
characteristic
vacuum fields in SM are well reproduced in the model of vortex-antivortex
liquid. The basic ingredient of this model was a vortex configuration
\begin{equation}
\label{vort}
\phi^{vort.}(y) = \frac{1}{2g}\ln[(y-x_0)^2 + \rho^2]
\end{equation}
It carries unit topological charge (~\ref{nu}) but does not minimize neither
classical nor effective action. The full field in our model presented a
stochastic superposition of vortices and antivortices with certain correlation
properties. All results of Ref.\cite{vac} can be, however, rederived using the
solution (~\ref{inst}) rather than (~\ref{vort}) as a basic ingredient. The
form
of the instanton is anyway distorted by quantum fluctuations and, for all
practical purposes, the configurations (~\ref{inst}) and (~\ref{vort}) are
equivalent. The really essential feature of characteristic vacuum fields is
fluctuating {\em local density} $E(x)$ of the topological charge (\ref{nu}).
Both vortex (\ref{vort}) and effective instanton (\ref{inst}) configurations
model such local fluctuations well.
We think also that all spectacular results of the
instanton-antiinstanton liquid model in QCD ~\cite{Shur} can be reproduced
if choosing as a basic ingredient not BPST instanton but any other localized
field configuration carrying topological charge.
At large temperatures, however, the situation simplifies a lot and it is very
instructive to analyze SM in $T \gg \mu$ region.
\section{High temperatures}
\setcounter{equation}{0}
Let us repeat all derivations of the previous section for the case when the
imaginary time extension of our torus $\beta = 1/T$ is small compared to the
massive photon Compton wavelength $\mu^{-1}$ (and the spatial size is still
large $L \gg \mu^{-1}$). The partition function in the sector $\nu = 1$ is
given again by Eq.(~\ref{Z1}), only the integral $\int d^2x$ extends now over
the narrow torus. If $\beta\mu \ll 1$, one can assume the field $\phi(x,\tau)$
to be static (the contribution of higher Fourier harmonics $\propto \exp
\{2\pi ik\tau/\beta\}$ in the path integral is suppressed), and we can
substitute $\int d^2x \rightarrow \beta \int dx,\ \Delta \rightarrow \partial^2
/\partial x^2$ (from now on $x$ will always be assumed to be spatial). The
effective instanton solution satisfies the equation
\begin{equation}
\label{eqT}
\left( \frac{\partial^4}{\partial x^4} - \mu^2 \frac{\partial^2}{\partial x^2}
\right) \phi^{e.i.}_{high~ T}(x)~ = ~-2g\delta(x-x_0)
\end{equation}
It has the form
\begin{equation}
\label{instT}
\phi^{e.i.}_{high~ T}(x) = \frac{\pi T}{g} \left[
\frac{1}{\mu}\exp\{-\mu|x-x_0|
\} + |x-x_0| \right] + const
\end{equation}
The corresponding gauge field is
\begin{eqnarray}
\label{AinstT}
A_0^{e.i.}(x)\mid_{high~T} = -\frac{\partial}{\partial x} \phi^{e.i.}_{high~T}
(x) - \frac{\pi T}{g} =
\nonumber \\
\frac{\pi T}{g} \ sign(x-x_0) \left[\exp\{-\mu|x-x_0|\}
-1 \right] - \frac{\pi T}{g}
\end{eqnarray}
(The gauge $A_1 = 0$ is chosen; the term $-\pi T/g$ comes from the properly
fixed constant term $A^{(0)}_\mu$ in the decomposition (\ref{decomp})
{}~\cite{Wipf}). When going from $x = -\infty$ to $x = \infty$,
$A_0$ changes from zero to $-2\pi T/g$ which corresponds to unit net
topological
charge
\[\nu = \frac{g\beta}{2\pi}\int E(x) dx = -\frac{g}{2\pi T}[A_0(\infty)
-A_0(-\infty)] \ =~1\]
But the electric field
\begin{equation}
\label{ET}
E^{e.i.}_{high~T} = -\frac{\partial}{\partial x} A_0^{e.i.}\mid_{high~T} =
T\sqrt{\pi} \exp\{-\mu|x-x_0|\}
\end{equation}
is localized at $x \sim x_0$ in contrast to the classical solution
(~\ref{trinst}).
As we have already noticed, the notion of instanton has more physical content
at high T where quantum fluctuations are relatively small. To see this, let
us make a simple estimate ~\cite{bub1}. Let us expand the integrand in the path
integral around the classical solution (\ref{instT}--\ref{ET}) and express the
fluctuating part in terms of $a_0 = -\partial/\partial x\ [\phi(x) -
\phi^{e.i.}_{high~T}(x)]$ :
\begin{equation}
\label{Z0}
Z \propto \int \prod da_0 \exp\left\{-\frac{\beta}{2}\int dx
[(\partial_x a_0)^2 + \mu^2 a_0^2]\right\}
\end{equation}
Let us expand $a_0(x)$ in the Fourier series
\[a_0(x) = \sum_{n=-\infty}^{\infty} c_n \exp\{2\pi inx/L\},~~~~
{}~~~~~~~~~~~~~~~~~~~~c_{-n}=c_n^* \]
and rewrite the functional integral as the integral over $\prod_{n} dc_n$.
It is easy to see that
\[ <|c_n|^2>_{char} \ \sim\ \frac{1}{\beta L[\mu^2 + 4\pi^2n^2/L^2]} \]
Assuming stochastic phases $\alpha_n$ for $c_{n>0}$, we may estimate
\begin{eqnarray}
\label{fluc}
a_0^{fluct.}(x) \ \sim\ \sqrt{LT} \sum_{n=1}^\infty \frac{\cos(\alpha_n )}
{\sqrt{4\pi^2n^2 + \mu^2 L^2}} \ \sim \nonumber \\
\sqrt{LT} \left[ \sum_{n=1}^\infty \frac{1}
{4\pi^2n^2 + \mu^2 L^2} \right]^{1/2} \ \sim \sqrt{\frac{T}{\mu}}
\end{eqnarray}
which is much less than the amplitude of the slassical solution (\ref{AinstT})
$\sim T/\mu$. If setting $const = 0$ in the RHS of Eq.(\ref{instT}), the
action of the instanton is
\begin{equation}
\label{actT}
S^{e.i.}_{high~T} = g\phi^{e.i.}_{high~T}(0)~ = ~\frac{\pi T}{\mu}
\end{equation}
The exact result for the partition function in the sectors $\nu = \pm1$
at high temperatures can be written as
\begin{equation}
\label{TZ1}
Z_1 = Z_{-1} = mL\exp\{-S^{e.i.}_{high~T}\}Z_0
\end{equation}
The factor $m$ comes from the fermion zero mode and the factor $L$~---~from the
integral over the collective coordinate $x_0$ describing the spatial instanton
position. The numerical coefficient is just 1. From this, one can easily derive
\begin{equation}
\label{Tcond}
<\bar{\psi}\psi>_{T\gg\mu} = -\frac{1}{\beta LZ_0} \frac{\partial}{\partial m}
\left[Z_1 + Z_{-1}\right] = -2T\exp\{-\pi T/\mu\}
\end{equation}
Again, the results (\ref{TZ1},\ \ref{Tcond}) look suspiciously simple and
again we
cannot suggest a simple direct way to derive them (other than to use the whole
machinery of the quantization in the box as in Ref.\cite{Wipf}). And again, the
simplest way to fix the exact coefficient in $<\bar{\psi}\psi>_{T\gg\mu}$ we
know of is to study the correlator $<\bar{\psi}\psi(x)~\bar{\psi}\psi(0)>_
{T\gg\mu}$ in the topologically trivial sector at large spatial $x$. The
stationary
point of the corresponding path integral is the instanton-antiinstanton
configuration
\begin{equation}
\label{IAT}
\phi^{stat.}_{high~T}(y) = \frac{\pi T}{g}\left[\frac{1}{\mu}\exp\{-\mu|y|\}
+ |y| - \frac{1}{\mu}\exp\{-\mu|y-x|\}
- |y-x| \right]
\end{equation}
The integral over fluctuations cancels out with the same integral in $Z_0$, and
we get
\begin{equation}
\label{corrT}
<\bar{\psi}\psi(x)~\bar{\psi}\psi(0)>_
{T\gg\mu} = <\bar{\psi}\psi(x)~\bar{\psi}\psi(0)>_
{free} \exp\{2g[\phi^{stat.}_{high~T}(x) - \phi^{stat.}_{high~T}(0)]\}
\end{equation}
where the free correlator on the cylinder is
\begin{equation}
\label{free}
<\bar{\psi}\psi(x)~\bar{\psi}\psi(0)>_
{T\gg\mu}\ =\ 2 T^2\ \left[\sum_{n=0}^\infty \exp\{-\pi T(2n+1)x\}\right]^2 =
\frac{T^2}{2\sinh^2(\pi Tx)}
\end{equation}
It falls down exponentially at large distances, the exponent being given
by the lowest fermion Matsubara frequency $\omega_{min} = \pi T$.
The factor $\exp\{\ldots\}$ in the RHS of Eq.(\ref{corrT}), however, rises
exponentially at large $x$ with the same exponent so that
\begin{equation}
\label{asym}
\lim_{x\rightarrow\infty} <\bar{\psi}\psi(x)~\bar{\psi}\psi(0)>_
{T\gg\mu}\ = \ 2T^2\exp\{-2\pi T/\mu\}
\end{equation}
Adding the equal contribution from the sectors $\nu = \pm2$ (cf.
Eq.(\ref{nu2}))
and taking the square root, we arrive at the result (\ref{Tcond}).
Let us look now at the solution (\ref{AinstT}) more intently. As has been
explained in details in \cite{bub}, it may be thought of as the configuration
interpolating between two adjacent minima of the high-T effective potential
$V^{eff}(A_0)$ in the constant $A_0$ background. The form of the potential is
\begin{equation}
\label{pot}
V^{eff}(A_0)\ =\ \frac{\mu^2}{2}\left[\left(A_0 + \frac{\pi T}{g}\right)_{mod.
{}~2\pi T/g} - \frac{\pi T}{g}\right]^2
\end{equation}
, and it has minima exactly at $A_0^{min} = 2\pi nT/g$.
Thus, the solution (\ref{AinstT}) is much analogous to the `walls' which appear
in Euclidean path integral in 4-dimensional gauge theories
{}~\cite{pisa,bub,bub1}
--- the planar static field configurations which interpolate between different
minima of effective potential $V^{eff}(A_0)$. In Ref.\cite{pisa}, they have
been interpreted as real walls living in Minkowski space separating different
thermal $Z_N$-phases. We argued in Refs.\cite{bub,bub1} that such domain walls
do not
actually exist in Minkowski space and there is only {\em one} physical phase
both in QED and pure Yang-Mills theory at high temperature.
One of the argument comes from SM analysis---the static solutions
(\ref{AinstT})
can (or cannot) be interpreted as real Minkowski space `solitons' with the
mass
\begin{equation}
\label{sol}
M^{sol.?}\ =\ TS^{e.i.}_{high T}\ =\ \frac{\pi T^2}{\mu}
\end{equation}
exactly by the same token as the planar 4-dim Euclidean static solitons can (or
cannot) be interpreted as real walls.
Here we want to present an additional argument why they {\em can}not. If the
solitons with the mass (\ref{sol}) would really exist in some reasonable sense,
this new mass scale should have displayed itself somehow in the physical
correlators, in particular---in the correlator $<\bar{\psi}\psi(x)~\bar{\psi}
\psi(0)>$.
\footnote
{We have seen that the operator $\bar{\psi}\psi$ is `intimately connected' with
the solution (\ref{instT}--\ref{ET}). Its expectation value is proportional to
$\exp\{-S^{e.i.}_{high T}\}\ =\ \exp\{-M^{sol.?}/T\}$, and, if $\bar{\psi}\psi$
cannot be interpreted as the creation operator for this soliton, nothing can.}
But it does not. The correlator (\ref{corrT}) has three asymptotic regions.
\begin{enumerate}
\item At very small $x\ll T^{-1}$, it does not feel the boundaries of the box
and behaves as $\sim 1/2\pi^2x^2$.
\item At $T^{-1}\ll x\ll\mu^{-1}$, it feels boundary but does not feel yet
fluctuating gauge fields, and is given by the expression (\ref{free}).
\item At $x~>~\mu^{-1}$, it starts to feel the gauge field dynamics and
rapidly
levels off on a constant (\ref{asym}), the preasymptotic terms being of order
$\exp\{-\mu x\}$. The scale $\propto T^2/\mu$ is absent.
\end{enumerate}
The full correlator
\begin{equation}
\label{full}
<\bar{\psi}\psi(x)\bar{\psi}\psi(0)>_T\ =\ <\bar{\psi}\psi>^{~~2}_T
\cosh\left\{
\int dk e^{ikx} \frac{1}{\sqrt{k^2+\mu^2}} \coth\frac{\sqrt{k^2+\mu^2}}{2T}
\right\}
\end{equation}
(the contributions from all topological sectors being summed over) exhibits
the same qualitative behaviour.
\footnote
{The derivation of the result (\ref{full}) is given in the Appendix.}
\section{Partition function and characteristic field configurations at
high T.}
\setcounter{equation}{0}
As we have seen, instantons display themselves in the path integral for
$<\bar{\psi}\psi>$ and $<\bar{\psi}\psi(x)~\bar{\psi}\psi(0)>$. But it is
important to note that the path integral for the partition function itself
knows
nothing about instantons in the strictly massless Schwinger model. Really, at
$m=0$, only topologically trivial sector $\nu=0$ contributes (all
$Z_{\nu\neq0}$
involve the factor $m^{|\nu|}$ due to $|\nu|$ fermion zero modes and vanish in
the massless limit). The path integral for $Z_0$ has a trivial form (\ref{Z0})
and no instantons are seen there. To understand better what actually happens
and why instantons reappear in the path integral for the correlator
$<\bar{\psi}\psi(x)~\bar{\psi}\psi(0)>_{T \gg \mu}$ in the same topologically
trivial sector
, let us estimate the contribution of the instanton-antiinstanton pair
(\ref{IAT}) in the path integral (\ref{Z0}). The configuration $a_0^{~IA}(y)$
is depicted in Fig.1. Obviously, the corresponding contribution to $Z_0$ is
\begin{equation}
\label{Z0IA}
Z_0^{~IA} ~\propto~\exp\left\{-\frac{\beta x}{2}\mu^2 \left(\frac{2\pi T}{g}
\right)^2\right\}
{}~=~\exp\{-2\pi Tx\}
\end{equation}
where $x$ is the separation between instanton and antiinstanton (we assumed
$x\gg\mu^{-1}$).
Now, the suppression (\ref{Z0IA}) can be interpreted as being
due to {\em quasizero} modes in the fermion determinant.
Really, an individual instanton (\ref{instT}) (the field configuration with
$\nu=1$) involves an exact fermion zero mode
\begin{equation}
\label{zeroT}
\psi_L(y)~=~ \exp\{-g\phi(y)\} ~\propto~ \exp\{-\pi T|y|\}
\end{equation}
(where we changed the notations ~$x \rightarrow y$ and put $x_0 = 0$ ).
For instanton-antiinstanton configuration (\ref{IAT}),
the former zero mode (\ref{zeroT}) and
its counterpart for the antiinstanton located at $y=x$
\[~~~~~~~~~~~~~~~~~~~~~~~~~ \psi_R(y) = \exp\{-\pi T|y-x|\}
{}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(5.2a)\]
are no longer exact eigenfunctions of the Dirac operator. But, if instanton
and antiinstanton are well separated, they are {\em almost} the solutions. The
true solutions and their eigenvalues can be found by solving the secular
equation taking the functions (\ref{zeroT}) and (\ref{zeroT}a) as basis and
regarding the
effects due to finite IA separation as perturbation. As a result, two
quasi-zero
modes with $\lambda\propto\exp\{-\pi Tx\}$ appear. Their product in $\det
\parallel i{\cal D} \parallel$ brings about the suppression (\ref{Z0IA}).
It is clear now why the IA configuration displayed itself in the correlator \\
$<\bar{\psi}\psi(x)~\bar{\psi}\psi(0)>$~---~the fermion operators `absorbed'
the zero modes from the fermion determinant, and the answer is finite in the
limit $x\rightarrow\infty$ (see Eq.(\ref{asym})). The finite result
(\ref{Tcond}) for the
fermion condensate $<\bar{\psi}\psi>_{T \gg \mu}$ being determined by the
path integral
in the sectors $\nu=\pm1$ is obtained exactly by the same reason.
However, the instantons {\em reappear} even in the path integral for the
partition function in high-T SM if we allow for a small but nonzero fermion
mass. It is clear why~---~the determinant factor is now $\det\parallel i
{\cal D}-m\parallel$ rather than just $\det\parallel i{\cal D}\parallel$ ,
and the contribution of the quasizero modes remains finite $\propto m^2$ even
at large $IA$ separation.
If one chooses, one may speak of the {\em confinement} of instantons in the
strictly massless case and their {\em liberation} for any small nonzero mass.
The suppression (\ref{Z0IA}) can be interpreted as being due to the linearly
rising `potential' between an instanton and antiinstanton. At $m\neq 0$, the
potential levels off at a constant at large separations. Freely moving
instantons bring about a large enthropy factor, and the contribution of the IA
configuration to $Z_0$ is
\begin{equation}
\label{ZIAm}
Z_0^{~IA}~=~ \left[mL\exp\{-\pi T/\mu\}\right]^2~Z_{m=0}
\end{equation}
If the spatial box is large enough,
\begin{equation}
\label{kappa}
\kappa = 2mL\exp\{-\pi T/\mu\} = m|<\bar{\psi}\psi>_{T}|\beta L \gg 1
\end{equation}
, the contribution of IA configuration to $Z_0$ {\em dominates} over purely
perturbative contribution $Z_{m=0}$ given by Eq.(\ref{Z0}).
Let us find now the contribution due to $N$ instantons and $N$ antiinstantons
(in the sector $\nu=0$, their number should be the same). It is
\begin{equation}
\label{ZNIA}
Z_0^{~N(IA)}~=~ \frac{1}{(N!)^2}\left[mL\exp\{-\pi T/\mu\} \right]^{2N}~Z_{m=0}
\end{equation}
The factor $(N!)^{-2}$ appeared due to undistinguishability of instantons (and,
separately, antiinstantons). Summing over all $N$, we get
\begin{equation}
\label{Z0m}
Z_0 = \sum_{N=0}^\infty Z_0^{~N(IA)} =~~ Z_{m=0}~I_0(\kappa)
\end{equation}
($I_0$ is the exponentially rising modified Bessel function). The sum
(\ref{Z0m})
is saturated at $N_{char}\sim \kappa$, i.e., when the condition (\ref{kappa})
is
satisfied, characteristic field configurations involve {\em many} (of order
$\kappa$) instanton-antiinstanton pairs.
The partition function in the topologically nontrivial sectors $\nu\neq 0$ can
be calculated in the same way. Assume for definiteness $\nu > 0$. Then the
contribution of $N+\nu$ instantons and $N$ antiinstantons in the partition
function is
\begin{equation}
\label{ZnuNIA}
Z_\nu\left(N_I~=~N+\nu\ ,\ N_A~=~ N\right)~=~ \frac{1}{N!(N+\nu)!}
\left(\frac{\kappa}{2}\right)^{2N+\nu}~Z_{m=0}
\end{equation}
Summing over $N$, we get
\begin{equation}
\label{Znum}
Z_\nu~=~ Z_{m=0}~I_\nu(\kappa)
\end{equation}
The full partition function is very simple (and could, of course, be written
right from the beginning from the most general premises):
\begin{equation}
\label{Zm}
Z~=~\sum_{\nu=-\infty}^{\infty}Z_\nu~=~Z_{m=0}\ e^{\kappa}~=~ Z_{m=0}\exp\{
-m<\bar{\psi}\psi>_T \beta L\}
\end{equation}
The sum (\ref{Zm}) is saturated at $\nu_{char} \sim \sqrt{\kappa}$. That can
be rather easily understood. When no constraint on the net topological charge
is put, we have two independent Poisson distributions for instantons and
antiinstantons with central values of order $\kappa$. The dispersion of these
distributions and the average mean square deviation $<(N_+ -N_- )^2>^{1/2}$
are of order $\kappa^{1/2}$.
The results (\ref{Z0m},\ \ref{Znum}) are very general, they hold not only in
high-T SM, but also in QCD~\cite{LS}. They can be rigourously derived by
studying $\theta$-dependence of the partition function~\cite{LS}. It is very
instructive to see, however, how these results could be rederived here in
another way
in the case where the characteristic field configurations in the path integral
are known exactly and explicitly.
The last remark concerns topological susceptibility. It is defined as
\begin{equation}
\label{hidef}
\chi~=~\left(\frac{g}{2\pi}\right)^2~\int d^2x~<E(x)E(0)>~=~\frac{1}{V}<\nu^2>
\end{equation}
where $V=L\beta$ is the total Euclidean volume. As has been mentioned,
$\nu_{char}\sim \sqrt{\kappa}$. The exact result is
\begin{equation}
\label{hi}
<\nu^2>~=~V\chi~=~m|<\bar{\psi}\psi>|V
\end{equation}
It can be derived either from Ward identities \cite{Crewther} or directly
from the explicit result (\ref{Znum}) and holds universally at any temperature.
But the {\em mechanism} providing the suppression $\propto m$ in $\chi$ is
much different at high temperatures compared to that at low temperatures.
At $T \gg\mu$, the characteristic field configurations present a noninteracting
instanton-antiinstanton gas. The density of the instantons is low~---~it
involves the fermion mass factor $m$ (on top of the factor $\exp\{
-S^{e.i.}_{high~T}\} = \exp\{-\pi T/\mu\}$). This factor exhibits itself in the
topological susceptibility. From the other hand, at low temperatures, the
characteristic vacuum fields present the dense strongly interacting
instanton-antiinstanton liquid \cite{vac} (with large quantum fluctuations
distorting the shape of individual instantons). The suppression in $\chi$
appears in that case due to strong correlations in this liquid providing
the effective screening of topological charge.
It is worthwhile to mention that the whole analysis can be transferred without
essential change to QCD with {\em one} nearly massless quark. The fermion
condensate survives in this theory even at very large temperatures~\cite{Yung}
and, if the quark mass is small but nonzero and the spatial volume is large
enough, characteristic field configurations include of order
\begin{equation}
\label{kapQCD}
\kappa^{QCD}~~=~~m|<\bar{\psi}\psi>_T^{~QCD}|\beta V^{(3)}
\end{equation}
instantons and about the same number of antiinstantons which do not interact
with each other (vs. dense strongly correlated instanton-antiinstanton liquid
at T=0).
\section{Acknowledgements.}
The paper was written at the Institute of Theoretical
Physics, Santa Barbara, during the research program {\em Strong Interactions at
Finite
Temperatures}. The author expresses gratitude for the warm hospitality extended
there and acknowledges the support of the National Science Foundation under
Grant No. PHY89--04035. I am indebted to E.V.Shuryak for illuminating
discussions.
|
1,116,691,497,748 | arxiv | \section{Introduction}
Predicting the orientation of thin fibers suspended in fluid flows with low Reynolds number finds many industrial applications, for example, when creating parts using injection molded plastics. One method that has been widely used is to start with the assumption that Jeffery's equation \cite[][]{jeffery:23}, or some variation of it, is a good predictor of the orientation of the fibers.
However, Jeffery's equation assumes that we are dealing with a single fiber in a fluid that extends to infinity in all directions. Thus unless the volume ratio of fibers to fluid is extremely small, it is reasonable to suppose that Jeffery's equation will require some modification. And indeed experiments performed for large numbers of fibers in shear flows show that the prediction from Jeffery's equation is somewhat inaccurate:
\begin{enumerate}
\item The eventual steady state disagrees with the prediction from the Jeffery's equation. Jeffery's equation only predicts a steady state if the aspect ratio of the fibers is infinite ($\lambda=1$), but a steady state seems to be observed experimentally even if the aspect ratio is finite.
\item Jeffery's equation for fibers with finite aspect ratio ($|\lambda|<1$) predicts that the fiber orientation is periodic in time (this is referred to as Jeffery's `tumbling'), but this is not seen in experiments \cite[see, for example,][]{anczurowski:67}.
\item Thirdly, the rate of any alignment is much slower than predicted.
\end{enumerate}
The first and second points were successfully countered by adding diffusion terms to Jeffery's equation \cite[][]{bird:87b}, for example, the Folgar-Tucker equation \cite[][]{folgar:84}. The diffusion term is meant to simulate an effect similar to Brownian motion, which is assumed to arise from the fibers colliding with each other. But experimental evidence shows that the diffusion terms fail to account for the delay in alignment, and the observed times for alignment are many times longer than the theoretically obtained values \cite[see, for example,][]{nguyen:08,sepehr:04,wang:08}.
We also bring attention to the work of \cite{stover:92}. They experimentally measured other parameters for fibers in fluids in shear flow, such as the distribution of the so called `orbit constants' for the fibers. This provides more evidence that when there are many fibers present, experiments deviate from the predictions obtained from Jeffery's equation.
One obvious suggestion is that the deviation of experimental data from theory is caused by hydrodynamic interactions between the fibers. That is, if a fiber is pushed by the fluid, the fiber will in turn effect the fluid. And in turn this will cause the fluid to push other fibers differently than if the first fiber were not present. These hydrodynamic interactions between the fibers are likely to be extremely complex. For example, we might see some groups of fibers get close to each other and form groups that tend to move as one rather than separately. This could, for example, be simulated by assuming that the viscosity of the suspension is higher within these groups than it is on the outside of these groups. This alignment inside these groups is likely to be far slower. Such a situation is illustrated in Figure~\ref{shear}. Other examples of these complex interactions in colloids or suspensions are described, for example, in \cite{villermaux:09} and \cite{wagner:09}. Furthermore, \cite{wagner:09} state that this kind of chaotic behavior takes place at fluid flows at much slower rates than required for the inertia of the underlying liquid to play an important factor, that is, the Reynolds number is much lower than is required for traditional turbulence to take place.
\begin{figure}
\begin{center}
\begin{tikzpicture}\begin{scope}[scale=0.5]
\draw[thick,->] (-4,4)--(4,4);
\draw (0,2) node{Low viscosity};
\draw (0,-2) node{Low viscosity};
\draw (0,0) node{High viscosity} ellipse(3cm and 1cm);
\draw (3.4641,2) [dashed,->]arc(30:-30:4cm);
\draw (-3.4641,-2) [dashed,->]arc(30:-30:-4cm);
\draw[thick,<-] (-4,-4)--(4,-4);
\end{scope}\end{tikzpicture}
\caption{A fluid of varying viscosity reacting to a shear flow.}
\label{shear}
\end{center}
\end{figure}
At first sight it would seem that one would have to set up large numerical experiments, which model large numbers of fibers in a Newtonian fluid, to observe this kind of effect. However, in this paper we propose that at a good approximation of this effect might be observed without losing the assumption that the suspension is modeled well by a continuum, that is, the fluid is modeled by a simple Stokes equation. The fibers are assumed to be smaller than the observed variations in the behavior of the suspension, so that the orientation of the fibers can be modeled by a probability distribution function.
Thus the form of Jeffery's equation we use solves for $\psi$, the probability distribution of the orientation of fibers, at each point in time and space. This is a function $\psi(\bx,\bp,t)$ of the three variables space $\bx \in \mathbb R^3$, time $t\ge0$, and orientation $\bp\in S$, where $S = \{\bp = (p_1,p_2,p_3):|\bp|^2 = p_1^2+p_2^2+p_3^2 = 1\}$ is the two dimensional sphere. The equations involve the velocity field $\bu = (u_1,u_2,u_3)$, which is a function of space $\bx$ and time $t$. Associated with the velocity field $\bu$ are the Jacobian matrix $\bnabla\bu = \left({\partial u_i}/{\partial x_j}\right)_{1\le i,j\le 3}$, the deformation matrix or rate of strain tensor $\mathsf \Gamma = \bnabla\bu + (\bnabla\bu)^T$, and the vorticity matrix $\mathsf \Omega = \bnabla\bu - (\bnabla\bu)^T$. Jeffery's equation is
\begin{gather}
\label{psi}
\frac{\partial \psi}{\partial t} + \bu \cdot\bnabla\psi = - \tfrac12\pnabla\cdot((\mathsf \Omega\cdot \bp + \lambda(\mathsf \Gamma\cdot\bp - \mathsf \Gamma:\bp\bp\bp))\psi) \\
\label{boundary-psi}
\psi = \tfrac1{4\pi} \text{ at } t=0
\end{gather}
Here $\pnabla$ denotes the gradient on the sphere $S$.
Our assertion is that if the initial state of $\psi$ is not completely isotropic, then small perturbations grow, and after a short amount of time dramatically effect the solution. We do not produce a formula that predicts how the fibers orient, but we do cast doubt upon Jeffery's equation being able to produce a good prediction of large numbers of fibers in a Newtonian fluid.
\begin{figure}
\begin{center}
\begin{tikzpicture}\begin{scope}[scale=0.3]
\foreach \y in {-2,2} {
\begin{scope}[yshift=\y cm]
\foreach \x in {-8,4} {
\begin{scope}[xshift=\x cm]
\draw [->] (0,1.5) -- (0,1.1);
\draw [->] (0,-1.5) -- (0,-1.1);
\draw [->] (-1.1,0) -- (-1.5,0);
\draw [->] (1.1,0) -- (1.5,0);
\end{scope}
}
\foreach \x in {-4,8} {
\begin{scope}[xshift=\x cm]
\draw [->] (-0.7,1.3) -- (0.7,1.3);
\draw [->] (0.7,-1.3) -- (-0.7,-1.3);
\end{scope}
}
\end{scope}
}
\begin{scope}[yshift=2 cm]
\draw (0,0) [xshift=-8cm,rotate=0] ellipse (0.7 and 0.1);
\draw (0,0) [xshift=-4cm,rotate=45] ellipse (0.7 and 0.1);
\draw (0,0) [xshift=4cm,rotate=45] ellipse (0.7 and 0.1);
\draw (0,0) [xshift=8cm,rotate=0] ellipse (0.7 and 0.1);
\end{scope}
\begin{scope}[yshift=-2 cm]
\draw (0,0) [xshift=-8cm,rotate=90] ellipse (0.7 and 0.1);
\draw (0,0) [xshift=-4cm,rotate=-45] ellipse (0.7 and 0.1);
\draw (0,0) [xshift=4cm,rotate=-45] ellipse (0.7 and 0.1);
\draw (0,0) [xshift=8cm,rotate=90] ellipse (0.7 and 0.1);
\end{scope}
\draw [dashed] (0,6) -- (0,-4) ;
\draw (-6,6) node {High resistance} ;
\draw (6,6) node {Low resistance} ;
\end{scope}\end{tikzpicture}
\caption{How slender fibers act as `stiffeners' in elongation and shear flows.}
\label{stiffener}
\end{center}
\end{figure}
The orientation of fibers at each point effects the rheology of the suspension. A slender fiber that is oriented parallel or perpendicular to the principle axes of the rate of strain tensor is going to act like a `stiffener' to the fluid, whereas a fiber that is oriented along a null direction of the rate of strain tensor (that is, if $\mathsf \Gamma\cdot\bp = 0$) does not hinder the flow in any way. This is illustrated in Figure~\ref{stiffener}. It is stated in \cite{batchelor:71,shaqfeh:90a} that if the underlying fluid is Newtonian, then the stress-strain relation for slender fibers is
\begin{equation}
\mathsf \sigma = \nu(\beta (\mathbb A:\mathsf \Gamma - \tfrac13 \mathsf I (\mathsf A:\mathsf \Gamma)) + \mathsf \Gamma) - p \mathsf I \\
\end{equation}
Here $\mathsf \sigma$ is the stress tensor, $\mathsf A$ and $\mathbb A$ are respectively the the 2nd and 4th moment tensors
\begin{gather}
\mathsf A = \int_S \bp\bp \, \psi \, d\bp \\
\label{A}
\mathbb A = \int_S \bp\bp\bp\bp \, \psi \, d\bp
\end{gather}
$\nu$ is the Newtonian viscosity that the underlying fluid would have if the fibers were absent (without loss of generality we set $\nu = 1$), $p$ is the pressure, and $\beta$ is a dimensionless quantity that is related to the volume fraction of the fibers in the fluid. The quantity $\beta$ represents the extent to which fibers act as `stiffeners' to the fluid motion. We refer to the case $\beta = 0$ as the uncoupled Jeffery's equation. The paper \cite{sepehr:04} suggests that the order of magnitude of $\beta$ could easily be as large as 50 or 100.
We assume that Reynolds number is close to zero, and so we neglect inertial terms in computing the flow of the suspension. We also assume the suspension is incompressible. Then the velocity field obeys the following Stokes' equation:
\begin{gather}
\label{stokes}
\bnabla \cdot \mathsf \sigma = 0 \\
\label{incompressible}
\bnabla \cdot \bu = 0
\end{gather}
Since the fluid is incompressible, the pressure $p$ is obtained implicitly, and hence without changing any of the results, we can replace $\frac13 \beta \mathsf A:\Gamma + p$ by a single scalar $q$, so that the stress-strain equation becomes
\begin{equation}
\label{stress}
\mathsf \sigma = \beta \mathbb A:\mathsf \Gamma + \mathsf \Gamma - q \mathsf I \\
\end{equation}
Next, to simplify the mathematics, we assume that the fluid occupies the whole of three dimensional space, and that there is an `ambient' velocity gradient $\mathsf U$, a three by three matrix with trace zero, so that
\begin{equation}
\label{stokes-boundary}
\bnabla\bu \to \mathsf U \text{ as } \bx \to \infty
\end{equation}
Thus we are really attempting to model the case when the fibers are much smaller than the characteristic length and width of the flow. And we are also assuming that the perturbations whose growth in time we are calculating should have a short characteristic wavelength. In fact, we show that the wavelength of the perturbations has no effect on the growth rate, only the direction in which oscillations take place is important to the growth rate.
\section{A heuristic argument for why alignment might be slowed down}
We propose that large changes in fiber orientation are created by initially small perturbations in the fiber orientation, which in a short amount of time are greatly enlarged. An example of a growth of perturbations is a kind of `buckling' effect, illustrated in Figure~\ref{buckling}. This shows an elongation flow where the fluid is squeezed along the $y$ axis, is expanded along the $x$ axis, and no elongation takes place along the $z$ axis, that is, $\bnabla\bu=\left[\begin{smallmatrix}G&0&0\\0&-G&0\\0&0&0\end{smallmatrix}\right]$ for some $G>0$. The small ellipsoids represent the orientation of the perturbation, and show a small perturbation of $\psi$ of the form $\epsilon\hat\psi \sin(2\pi y/L)$. It is reasonable to expect that this perturbation causes the fibers to `buckle,' that is, create perturbations to $\bu$ of the form $\epsilon\left[\begin{smallmatrix}0\\0\\\hat u_3\end{smallmatrix}\right] \cos(2\pi y/L)$.
This could be seen as an effect similar to placing a large number of fibers end to end, and then pushing in from both ends. One would expect the fibers to `buckle,' but not in the $x$-direction, where they are being `guided' by the elongation flow. Obviously this intuition in of itself is not terribly convincing, but we will show later that the mathematics does predict this effect, and indeed that this `buckling' can take place in any direction in the $yz$-plane.
This `buckling' then feeds back into the perturbation of $\psi$ causing it to grow exponentially. We should expect similar behavior from a shear flow, because shear flow is an elongation flow at $45^\circ$ to the $x$ and $y$ axes, combined with a rotation.
\begin{figure}
\begin{center}
\begin{tikzpicture}\begin{scope}[scale=0.3]
\draw[thick,->] (0,6)--(0,4);
\foreach \x in {-4,-2,0,2,4} {
\draw (0,0) [xshift=\x cm,yshift=3cm,rotate=-60] ellipse (0.7 and 0.1);
\draw (0,0) [xshift=\x cm,yshift=1cm,rotate=60] ellipse (0.7 and 0.1);
\draw (0,0) [xshift=\x cm,yshift=-1cm,rotate=-60] ellipse (0.7 and 0.1);
\draw (0,0) [xshift=\x cm,yshift=-3cm,rotate=60] ellipse (0.7 and 0.1);
}
\foreach \x in {-6,6} {
\draw [xshift=\x cm,->] (-0.5,2) -- (0.5,2);
\draw [xshift=\x cm,<-] (-0.5,0) -- (0.5,0);
\draw [xshift=\x cm,->] (-0.5,-2) -- (0.5,-2);
}
\draw[thick,->] (0,-6)--(0,-4);
\begin{scope}[xshift=5cm,yshift=-6cm]
\draw[->] (-0.1,0)--(1.5,0) node[right]{$z$};
\draw[->] (0,-0.1)--(0,1) node[above]{$y$};
\end{scope}
\end{scope}\end{tikzpicture}
\caption{`Buckling' caused by elongation flow applied to a non-constant fiber orientation distribution.}
\label{buckling}
\end{center}
\end{figure}
However `buckling' in of itself is insufficient to cause small perturbations to become very large. That is, over a short amount of time, buckling might cause perturbations to grow maybe two or three times. But to really expect the perturbations to play a large role, they really need to grow ten fold or even hundred fold.
To see why this much larger growth in perturbations is plausible, let us consider that we have applied a shear flow for a while, so that the fiber orientation is no longer isotropic. When the fibers are about $45^\circ$ angle to the $x$ and $y$ axis, those fibers that are a little ahead in the alignment will have a little less stiffening effect on the fluid, and hence the shear strain will be larger on these fibers. This will cause a positive feedback where those fibers that are a little ahead of the others will become even further ahead. This self-reinforcing shear-banding is illustrated in Figure~\ref{shear-banding}.
\begin{figure}
\begin{center}
\begin{tikzpicture}\begin{scope}[scale=0.3]
\draw[thick,->] (-4,4)--(4,4);
\foreach \x in {-4,-2,0,2,4} {
\draw (0,0) [xshift=\x cm,yshift=2cm,rotate=45] ellipse (0.7 and 0.1);
\draw (0,0) [xshift=\x cm,yshift=0cm,rotate=25] ellipse (0.7 and 0.1);
\draw (0,0) [xshift=\x cm,yshift=-2cm,rotate=45] ellipse (0.7 and 0.1);
}
\foreach \x in {-6.5,6.5} {
\draw [xshift=\x cm,->] (0,2.5) -- (1.7,2.5);
\draw [xshift=\x cm,->] (0,2) -- (1.6,2);
\draw [xshift=\x cm,->] (0,1.5) -- (1.5,1.5);
\draw [xshift=\x cm,->] (0,1) -- (1,1);
\draw [xshift=\x cm,->] (0,0.5) -- (0.5,0.5);
\draw [xshift=\x cm,->] (0,-0.5) -- (-0.5,-0.5);
\draw [xshift=\x cm,->] (0,-1) -- (-1,-1);
\draw [xshift=\x cm,->] (0,-1.5) -- (-1.5,-1.5);
\draw [xshift=\x cm,->] (0,-2) -- (-1.6,-2);
\draw [xshift=\x cm,->] (0,-2.5) -- (-1.7,-2.5);
}
\draw[thick,->] (4,-4)--(-4,-4);
\end{scope}\end{tikzpicture}
\caption{Shear-banding caused by shear flow applied to a non-constant fiber orientation distribution.}
\label{shear-banding}
\end{center}
\end{figure}
We suspect that this simple model of shear banding, although plausible, is perhaps not enough to explain the very large growth of perturbations. Rather something more complex will take place, and it seems to require all three dimensions, not just the $x$ and $y$ directions, to produce chaotic behavior.
\section{Solutions to Jeffery's equation}
The coupled system of equations we propose is a partial differential equation in six variables (three space variables, two fiber orientation variables, and the time variable). Our task will be much easier if we can reduce this to a finite dimensional ordinary differential equation.
By \cite{lipscomb:88,dinh:84,szeri:96,montgomery-smith:09b}, it is known that if $\psi$ was ever isotropic at some time in the past, then solution to equation~\eqref{psi} is
\begin{equation}
\label{psi-B}
\psi(\bp) = \psi_{\mathsf B}(\bp) = \frac1{4\pi (\mathsf B:\bp\bp)^{3/2}}
\end{equation}
where
\begin{equation}
\label{BC}
\mathsf B = \mathsf C^T\cdot \mathsf C
\end{equation}
the matrix $\mathsf C$ satisfies the equation
\begin{equation}
\label{C-full}
\frac{\partial \mathsf C}{\partial t} + \bu\cdot\bnabla \mathsf C = - \tfrac12\mathsf C \cdot(\mathsf \Omega+\lambda \mathsf \Gamma)
\end{equation}
and equation~\eqref{boundary-psi} becomes
\begin{equation}
\label{C0}
\mathsf C = \mathsf I \text{ at } t=0
\end{equation}
Equivalently, $\mathsf B$ is a symmetric positive definite matrix with determinant one satisfying
\begin{equation}
\label{B-full}
\frac{\partial \mathsf B}{\partial t} + \bu\cdot\bnabla \mathsf B = - \tfrac12 \mathsf B \cdot(\mathsf \Omega+\lambda \mathsf \Gamma) - \tfrac12 (- \mathsf \Omega+\lambda \mathsf \Gamma) \cdot \mathsf B
\end{equation}
and equation~\eqref{boundary-psi} becomes
\begin{equation}
\label{B0}
\mathsf B = \mathsf I \text{ at } t=0
\end{equation}
Thus from now on, we assume that fiber orientation always has the form $\psi = \psi_{\mathsf B}$, given by equation~\eqref{psi-B}, for some positive definite matrix $\mathsf B$ whose determinant is one. And perturbations of $\psi_{\mathsf B}$ are equivalent to perturbations of $\mathsf B$.
Furthermore, the 4th order moment tensor $\mathbb A$ can be calculated directly from $\mathsf B$ using elliptic integrals \cite[]{montgomery-smith:09b} \cite[see also][]{verleye:93,verweyst:98}
\begin{equation}
\label{A from B}
\mathbb A = \mathbb A(\mathsf B) = \tfrac34 \int_0^\infty \frac{s\,\mathcal S((\mathsf B+s \mathsf I)^{-1}\otimes(\mathsf B+s \mathsf I)^{-1}) \, ds}{\sqrt{\text{det}(\mathsf B+s \mathsf I)}}
\end{equation}
where $\mathcal S$ is the symmetrization of a tensor, that is, if $\mathbb B$ is a rank $n$ tensor, then $\mathcal S(\mathbb B)_{i_1\dots i_n}$ is the average of $\mathbb B_{j_1\dots j_n}$ over all permutations $(j_1,\dots,j_n)$ of $(i_1,\dots,i_n)$.
\section{Solution of the unperturbed Jeffery's equation}
The unperturbed problem is to assume that the initial data is isotropic (equation~\eqref{B0}), or at least that $\psi_{\mathsf B}$, and hence $\mathsf B$, at time $t=0$ does not depend upon $\bx$. If $\mathsf B$ does not depend upon $\bx$, then from equation~\eqref{A} or~\eqref{A from B}, it follows that $\mathbb A$ does not depend upon $\bx$, and it becomes apparent that the solution to equations~\eqref{stokes}, \eqref{incompressible}, \eqref{stress} and~\eqref{stokes-boundary} is given by $\bnabla\bu = \mathsf U$, which does not depend upon $\bx$. Hence from equation~\eqref{psi} or~\eqref{B-full}, it follows that $\partial\psi_{\mathsf B}/\partial t$, or equivalently $\mathsf B$, does not depend upon $\bx$. Thus the solution to the unperturbed problem does not depend upon $\bx$ for all $t>0$.
Thus the terms $\bu\cdot\bnabla\psi$, $\bu\cdot\bnabla \mathsf B$ and $\bu\cdot\bnabla \mathsf C$ are zero, and
\begin{gather}
\label{Gamma}
\mathsf \Gamma = \mathsf U+\mathsf U^T \\
\label{Omega}
\mathsf \Omega = \mathsf U-\mathsf U^T
\end{gather}
In particular, if $\mathsf U$ is independent of $t$, then $\mathsf C$ can be computed easily by
\begin{equation}
\label{C-exp}
\mathsf C = \exp\left(-\tfrac12 t (\mathsf \Omega + \lambda\mathsf \Gamma)\right)
\end{equation}
\section{The Mathematical nature of the perturbations}
The perturbation we propose is to apply the replacement
\begin{equation}
\psi \to \psi + \epsilon \tilde\psi
\end{equation}
and assume that both $\psi$ and its perturbation satisfy equations~\eqref{psi}, \eqref{A}, \eqref{stokes}, \eqref{incompressible}, \eqref{stress} and~\eqref{stokes-boundary}. The perturbation $\tilde\psi$ depends upon $\bx$, even though the unperturbed $\psi$ does not. The idea is that if $\epsilon$ is small, then one assumes that terms of order $\epsilon^2$ and higher can be ignored, and from the terms of order $\epsilon$ we form a differential equation in $\tilde\psi$. Furthermore, this differential equation is linear in $\tilde\psi$, even though the coefficients of the differential equation may depend upon $\psi$ in a possibly highly non-linear manner. The thinking is that if the linearized perturbations $\tilde\psi$ grow by a large amount, that one might expect the terms of order $\epsilon^2$ and higher to have a large effect on the solution. At this point the linearization is no longer valid, and it is reasonable to assume that after this point the solution becomes chaotic, or at least differs significantly from the unperturbed solution.
We can write the perturbation as a Fourier transform
\begin{equation}
\label{fourier-transform}
\tilde\psi(\bx,\bp,t) = {\int\!\!\int\!\!\int} \hat\psi(\bkappa,\bp,t) e^{i \bkappa\cdot\bx} \, d\bkappa
\end{equation}
It can be seen that if we consider only terms of order $\epsilon$ and lower, the different terms of the integrand do not interact. As long as one assumes that $\bkappa$ evolves according to equation~\eqref{bkappa} (see below), without loss of generality, we can assume that the perturbation is of the form
\begin{equation}
\psi \to \psi + \epsilon\hat\psi e^{i \bkappa\cdot\bx}
\end{equation}
for some wave number $\bkappa$. While this solution involves complex numbers, by considering linear combinations, we see that $e^{i \bkappa\cdot\bx}$ is simply `code' for $\sin(\bkappa\cdot\bx)$, and similarly $i e^{i \bkappa\cdot\bx}$ is the same solution that is $90^\circ$ out of phase, that is, $\cos(\bkappa\cdot\bx)$. Note that this kind of perturbed quantity does not allow the boundary condition~\eqref{stokes-boundary} for $\bu$, but this is not a real problem, since we know that ultimately these terms come from equation~\eqref{fourier-transform}, and that $\tilde\psi$ does satisfy this boundary condition. (From a more mathematical point of view, we are calculating the continuous spectrum of our linear operators rather than the point spectrum.)
In order to satisfy Jeffery's equation, we require that $\bkappa$ satisfy
\begin{equation}
\label{bkappa-evolves-so-that}
\left(\frac\partial{\partial t} + \bu\cdot\bnabla\right)(\hat\psi e^{i \bkappa\cdot\bx}) = \frac{\partial \hat \psi}{\partial t} \, e^{i\bkappa\cdot\bx}+O(\epsilon)
\end{equation}
To make this happen, we evolve $\bkappa$ according to the equation
\begin{gather}
\label{bkappa}
\frac\partial{\partial t}\bkappa = - \mathsf U^T \cdot \bkappa \\
\label{bkappa0}
\bkappa = \bkappa_0 \text{ at } t=0
\end{gather}
Equation~\eqref{bkappa-evolves-so-that} then follows because
\begin{equation}
\frac\partial{\partial t} e^{i\bkappa\cdot\bx}
=
-i[(\mathsf U^T\cdot\bkappa)\cdot\bx] e^{i\bkappa\cdot\bx}
=
-(\mathsf U\cdot\bx)\cdot(i\bkappa) e^{i\bkappa\cdot\bx}
=
-\bu\cdot\bnabla [e^{i\bkappa\cdot\bx}]
+O(\epsilon)
\end{equation}
Note that if $\mathsf U$ does not depend upon $t$, then equations~\eqref{bkappa} and~\eqref{bkappa0} are solved by
\begin{equation}
\label{bkappa-exp}
\bkappa = e^{-t \mathsf U^T}\bkappa_0
\end{equation}
We illustrate how a wave number $\bkappa$ is effected by a shear flow in Figure~\ref{U kappa}.
\begin{figure}
\begin{center}
\begin{tikzpicture}\begin{scope}[scale=0.3]
\draw[thick,->] (-4,5.5) -- node [above] {Shear flow} (4,5.5);
\draw[dashed] (-8.6,3) -- (7.4,5);
\draw[dashed] (-8.3,1) -- (7.7,3);
\draw[dashed] (-8,-1) -- node [above,sloped] {level curves of $\sin(\bkappa\cdot\bx)$} (8,1);
\draw[dashed] (-7.7,-3) -- (8.3,-1);
\draw[dashed] (-7.4,-5) -- (8.6,-3);
\draw [->,very thick](0.55,-4.4) -- (-0.55,4.4) node [above]{$\bkappa$} ;
\draw[thick,<-] (-4,-5.5) -- (4,-5.5) ;
\end{scope}\end{tikzpicture}
\end{center}
\caption{A diagram illustrating how $\bkappa$ is effected by a shear flow. The level curves of $\sin(\bkappa\cdot\bx)$ will converge to the horizontal, and the wave number $\bkappa$ will converge to the vertical.}
\label{U kappa}
\end{figure}
Now the perturbation in $\psi = \psi_{\mathsf B}$ is more easily described by a perturbation in $\mathsf B$:
\begin{equation}
\label{perturb B}
\mathsf B \to \mathsf B + \epsilon \hat{\mathsf B} e^{i\bkappa\cdot\bx}
\end{equation}
However, if we are going to measure the size of the growth of the perturbation, then $\epsilon\hat{\mathsf B}$ is not the correct thing to measure. This is because $\epsilon \hat{\mathsf B}$ is, in some sense, an absolute error for $\mathsf B$. Thus if $\epsilon\hat{\mathsf B}$ becomes large, this might simply reflect that $\mathsf B$ is becoming large.
For this reason, we instead measure a relative error, $\epsilon \tilde{\mathsf B}$, where
\begin{equation}
\label{tilde B}
\tilde{\mathsf B} = (\mathsf C^T)^{-1} \cdot \hat{\mathsf B} \cdot \mathsf C^{-1}
\end{equation}
This also simplifies calculations, because $\text{Tr}(\tilde{\mathsf B}) = 0$, a consequence that the perturbation needs to preserve $\text{det}(\mathsf B)=1$.
The goal, then, is to create an equation
\begin{equation}
\label{matrix-ode}
\frac{\partial \tilde{\mathsf B}}{\partial t} = M(\bkappa,t)[\tilde{\mathsf B}]
\end{equation}
where $M(\bkappa,t)$ is linear operator acting upon symmetric, trace zero matrices, depending (amongst other things) upon $\bkappa$.
The solution to equation~\eqref{matrix-ode} is described by the linear operator $L(\bkappa_0,t)$, where
\begin{equation}
\label{matrix-ode-soln}
\tilde{\mathsf B}(t) = L(\bkappa_0,t) [\tilde{\mathsf B}(0)]
\end{equation}
Here $\bkappa_0$ is as defined in equation~\eqref{bkappa0}, and we assume that $\bkappa$ in equation~\eqref{matrix-ode} satisfies equation~\eqref{bkappa}.
The largeness of the operator $L(\bkappa_0,t)$ can be found in two possible ways, by computing its spectral norm (that is the largest singular value), or by computing its spectral radius (that is the largest absolute value of its eigenvalues). The second approach always gives a smaller or equal answer than the first approach, and so we can regard the second approach as the more conservative. We use both approaches in our calculations, and we will see that in our specific situations that the answers are not significantly different.
\section{The Linearized Equations}
We make the additional replacements
\begin{gather}
\bu \to \mathsf U\cdot\bx+\epsilon \hat \bu e^{i\bkappa\cdot\bx} \\
\label{perturb Gamma}
\mathsf \Gamma \to \mathsf \Gamma + \epsilon\hat{\mathsf \Gamma} e^{i\bkappa\cdot\bx} \\
\label{perturb Omega}
\mathsf \Omega \to \mathsf \Omega+\epsilon \hat{\mathsf \Omega} e^{i\bkappa\cdot\bx} \\
\mathsf \sigma \to \mathsf \sigma+\epsilon \hat{\mathsf \sigma} e^{i\bkappa\cdot\bx} \\
q \to q+\epsilon \hat q e^{i\bkappa\cdot\bx} \\
\mathbb A \to \mathbb A+\epsilon \hat{\mathbb A} e^{i\bkappa\cdot\bx}
\end{gather}
The linearized equations can be shown to be
\begin{gather}
\label{fourier-psi}
\frac{\partial \tilde{\mathsf B}}{\partial t} = - \tfrac12\mathsf C\cdot(\hat{\mathsf \Omega}+\lambda\hat{\mathsf \Gamma})\cdot \mathsf C^{-1} - \tfrac12(\mathsf C^T)^{-1} \cdot(- \hat{\mathsf \Omega}+\lambda\hat{\mathsf \Gamma})\cdot \mathsf C^T \\
\label{fourier-A}
\hat{\mathbb A} = \frac{\partial \mathbb A}{\partial \mathsf B} : (\mathsf C^T\cdot\tilde{\mathsf B}\cdot \mathsf C) \\
\label{fourier-gamma}
\hat{\mathsf \Gamma} = i(\bkappa \hat\bu + \hat\bu \bkappa) \\
\label{fourier-omega}
\hat{\mathsf \Omega} = i(\bkappa \hat\bu - \hat\bu \bkappa) \\
\label{fourier-stress-def}
\hat{\mathsf \sigma} = \beta \mathbb A:\hat{\mathsf \Gamma} + \beta \hat{\mathbb A}:\mathsf \Gamma + \hat{\mathsf \Gamma} -\hat q \mathsf I \\
\label{fourier-stress-equ}
\bkappa \cdot \hat{\mathsf \sigma} = 0 \\
\label{fourier-div-free}
\bkappa \cdot \hat\bu = 0 \\
\label{dA/dB}
\frac{\partial \mathbb A}{\partial \mathsf B}
= - \tfrac{15}8 \int_0^\infty \frac{s\,\mathcal S[(\mathsf B+s \mathsf I)^{-1}\otimes(\mathsf B+s \mathsf I)^{-1}\otimes(\mathsf B+s \mathsf I)^{-1}] \, ds}{\sqrt{\text{det}(\mathsf B+s \mathsf I)}}
\end{gather}
noting here that $\frac{\partial \mathbb A}{\partial \mathsf B}$ is a rank 6 tensor.
Let us provide some details of the derivation of equation~\eqref{fourier-psi}. Substituting equations~\eqref{perturb B}, \eqref{perturb Gamma} and~\eqref{perturb Omega} into equation~\eqref{B-full}, and retaining only the terms of order $\epsilon$, we obtain
\begin{equation}
\begin{split}
\frac\partial{\partial t} (\hat{\mathsf B}\, e^{i\bkappa\cdot\bx})
+
\bu\cdot\bnabla(\hat{\mathsf B}\, e^{i\bkappa\cdot\bx})
=
& - \tfrac12\hat{\mathsf B} \cdot(\mathsf \Omega+\lambda \mathsf \Gamma) \, e^{i\bkappa\cdot\bx}
- \tfrac12(-\mathsf \Omega+\lambda \mathsf \Gamma) \cdot \hat{\mathsf B} \, e^{i\bkappa\cdot\bx} \\
& - \tfrac12 \mathsf B \cdot(\hat{\mathsf \Omega}+\lambda \hat{\mathsf \Gamma}) \, e^{i\bkappa\cdot\bx}
- \tfrac12 (-\hat{\mathsf \Omega}+\lambda \hat{\mathsf \Gamma})\cdot \mathsf B \, e^{i\bkappa\cdot\bx}
\end{split}
\end{equation}
Applying equations~\eqref{bkappa-evolves-so-that}, \eqref{C-full} and~\eqref{tilde B}, dividing by $e^{i\bkappa\cdot\bx}$, and neglecting terms of order $\epsilon$ and higher, we obtain
\begin{equation}
\mathsf C^T \cdot \frac{\partial \tilde{\mathsf B}}{\partial t} \cdot \mathsf C = - \tfrac12 \mathsf B \cdot(\hat{\mathsf \Omega}+\lambda \hat{\mathsf \Gamma}) - \tfrac12 (-\hat{\mathsf \Omega}+\lambda \hat{\mathsf \Gamma}) \cdot \mathsf B
\end{equation}
and equation~\eqref{fourier-psi} is established.
Next, setting
\begin{equation}
\mathsf N = |\bkappa|^2 \mathsf I + 2 \beta \mathbb A:\bkappa\bkappa
\end{equation}
we have that equations~\eqref{fourier-stress-def}, \eqref{fourier-stress-equ} and~\eqref{fourier-div-free} can be replaced by
\begin{equation}
\label{fourier-bu}
\hat\bu = i \beta \left(\mathsf I - \frac{\mathsf N^{-1} \cdot \bkappa\bkappa}{\mathsf N^{-1}:\bkappa\bkappa} \right)\cdot \mathsf N^{-1} \cdot (\hat{\mathbb A}:\mathsf \Gamma)\cdot\bkappa
\end{equation}
To see this, note first that $\mathsf N$ is positive definite, and hence invertible. Substitute equations~\eqref{fourier-gamma} and~\eqref{fourier-stress-def} into \eqref{fourier-stress-equ}, and apply equation~\eqref{fourier-div-free}, to obtain
\begin{equation}
\label{app2}
\hat\bu + i \mathsf N^{-1} \cdot \bkappa \hat q = i \beta \mathsf N^{-1} \cdot (\hat{\mathbb A}:\mathsf \Gamma)\cdot\bkappa
\end{equation}
Dot producting both sides with $\bkappa$, and again applying equation~\eqref{fourier-div-free}, yields
\begin{equation}
\hat q = \beta \frac{\bkappa\cdot \mathsf N^{-1} \cdot (\hat{\mathbb A}:\mathsf \Gamma)\cdot\bkappa}{\mathsf N^{-1}:\bkappa\bkappa}
\end{equation}
Substituting this into equation~\eqref{app2} gives equation~\eqref{fourier-bu}.
Finally, it is important to note that in equation~\eqref{matrix-ode}, that $M(\bkappa,t)$ only depends on the direction of $\bkappa$, not on its length.
\section{Linear perturbations of the fourth moment tensor}
Equation~\eqref{dA/dB} follows, since by differentiating equation~\eqref{A from B} we have for any symmetric matrix $\hat{\mathsf B}$
\begin{equation}
\begin{split}
\frac{\partial \mathbb A}{\partial \mathsf B} : \hat{\mathsf B}
= {} &
\frac{d}{dt} \left[\mathbb A(\mathsf B+t \hat{\mathsf B})\right] \Big|_{t=0} \\
= {} & - \tfrac32 \int_0^\infty \frac{s\,\mathcal S[((\mathsf B+s \mathsf I)^{-1}\cdot \hat{\mathsf B}\cdot(\mathsf B+s \mathsf I)^{-1})\otimes(\mathsf B+s \mathsf I)^{-1}] \, ds}{\sqrt{\text{det}(\mathsf B+s \mathsf I)}} \\
& - \tfrac38 \int_0^\infty \frac{s\, \text{Tr}[(\mathsf B+s \mathsf I)^{-1}\cdot \hat{\mathsf B}]\,\mathcal S[(\mathsf B+s \mathsf I)^{-1}\otimes(\mathsf B+s \mathsf I)^{-1}]\, ds}{\sqrt{\text{det}(\mathsf B+s \mathsf I)}}
\end{split}
\end{equation}
and noting that for any symmetric matrix $\mathsf K$
\begin{equation}
\mathcal S(\mathsf K\otimes \mathsf K\otimes \mathsf K):\hat{\mathsf B}
=
\tfrac45 \mathcal S((\mathsf K\cdot \hat{\mathsf B}\cdot \mathsf K)\otimes \mathsf K)
+
\tfrac15 \text{Tr}(\mathsf K\cdot \hat{\mathsf B})\,\mathcal S(\mathsf K\otimes \mathsf K)
\end{equation}
To compute $\frac{\partial \mathbb A}{\partial \mathsf B}$ in practice, one first diagonalizes $\mathsf B$ using an orthogonal similarity matrix, so that $\mathsf B=\text{diagonal}(b_1,b_2,b_3)$. In this case, the following kinds of quantities appear
\begin{equation}
\mathcal E_{m_1,m_2,m_3}^n =
\int_0^\infty \frac{s^n \, ds}{(b_1+s)^{m_1+\frac12}(b_2+s)^{m_2+\frac12}(b_3+s)^{m_3+\frac12}}
\end{equation}
where $m_1$, $m_2$, $m_3$ and $n$ are non-negative integers satisfying $n \le m_1+m_2+m_3$. For example, it is shown in \cite{montgomery-smith:09b} that the 4th moment tensor is given by the formulae (here $i\ne j\ne k$)
\begin{gather}
\mathbb A_{iiii} = \tfrac34\mathcal E^1_{2\delta_{i1},2\delta_{i2},2\delta_{i3}} \\
\mathbb A_{iijj} = \tfrac14\mathcal E^1_{\delta_{i1}+\delta_{j1},\delta_{i2}+\delta_{j2},\delta_{i3}+\delta_{j3}} \\
\mathbb A_{ijkk} = 0
\end{gather}
where $\delta_{ij}$ denotes the Kronecker delta symbol. To compute $\mathcal E_{m_1,m_2,m_3}^n$, they are Carlson forms of elliptic integrals \cite[][]{carlson:95} if $n=0$ and $m_1+m_2+m_3=0$ or $1$. Furthermore, if $b_1$, $b_2$ and $b_3$ are distinct, then the other quantities can be calculated using the following types of relations (the only mildly difficult equality is the first, which requires a single application of integration by parts):
\begin{gather}
(m_1+\tfrac12)\mathcal E^{n+1}_{m_1+1,m_2,m_3} + (m_2+\tfrac12)\mathcal E^{n+1}_{m_1,m_2+1,m_3} + (m_3+\tfrac12)\mathcal E^{n+1}_{m_1,m_2,m_3+1} = (n+1)\mathcal E^n_{m_1,m_2,m_3} \\
\mathcal E^n_{m_1,m_2+1,m_3+1} = (b_2-b_3)^{-1}(\mathcal E^n_{m_1,m_2,m_3+1}-\mathcal E^n_{m_1,m_2+1,m_3}) \\
\mathcal E^n_{m_1+1,m_2,m_3+1} = (b_3-b_1)^{-1}(\mathcal E^n_{m_1+1,m_2,m_3}-\mathcal E^n_{m_1,m_2,m_3+1}) \\
\mathcal E^n_{m_1+1,m_2+1,m_3} = (b_1-b_2)^{-1}(\mathcal E^n_{m_1,m_2+1,m_3}-\mathcal E^n_{m_1+1,m_2,m_3}) \\
\mathcal E^{n+1}_{m_1+1,m_2,m_3} = \mathcal E^n_{m_1,m_2,m_3} - b_1\mathcal E^n_{m_1+1,m_2,m_3} \\
\mathcal E^{n+1}_{m_1,m_2+1,m_3} = \mathcal E^n_{m_1,m_2,m_3} - b_2\mathcal E^n_{m_1,m_2+1,m_3} \\
\mathcal E^{n+1}_{m_1,m_2,m_3+1} = \mathcal E^n_{m_1,m_2,m_3} - b_3\mathcal E^n_{m_1,m_2,m_3+1}
\end{gather}
From a numerical perspective, good approximations to these elliptic integrals can also be found when the eigenvalues are not distinct by artificially adding a small term to some of the eigenvalues to make them distinct. To avoid excessive floating point errors, the eigenvalues need to be quite far from each other, for example, using IEEE double precision arithmetic we found that the eigenvalues have to be at least $10^{-6}$ from each other to get reasonable results when $m_1+m_2+m_3 \le 3$.
If $i\ne j\ne k \ne i$, we have the formulae
\begin{gather}
\label{dA/dB iiiiii}
\left(\frac{\partial \mathbb A}{\partial \mathsf B}\right)_{iiiiii}
= -\tfrac{15}8 \mathcal E^1_{3\delta_{i1},3\delta_{i2},3\delta_{i3}}
\\
\label{dA/dB iiiijj}
\left(\frac{\partial \mathbb A}{\partial \mathsf B}\right)_{iiiijj}
=
-\tfrac38 \mathcal E^1_{2\delta_{i1}+\delta_{j1},2\delta_{i2}+\delta_{j1},2\delta_{i3}+\delta_{j1}} \\
\label{dA/dB iijjkk}
\left(\frac{\partial \mathbb A}{\partial \mathsf B}\right)_{iijjkk}
=
-\tfrac18 \mathcal E^1_{\delta_{i1}+\delta_{j1}+\delta_{k1},\delta_{i2}+\delta_{j2}+\delta_{k2},\delta_{i3}+\delta_{j3}+\delta_{k3}}
\end{gather}
and all terms which have any index appearing precisely an odd number of times are 0.
If the eigenvalues are all the same, that is, $\mathsf B=\mathsf I$, then equation~\eqref{dA/dB} simplifies to the formula
\begin{equation}
\label{dA/dB 0}
\frac{\partial \mathbb A}{\partial \mathsf B} = - \tfrac3{14} \mathcal S(\mathsf I\otimes \mathsf I\otimes \mathsf I)
\end{equation}
\section{Calculating the growth of perturbations}
The linear operator $M(\bkappa,t)$ in equation~\eqref{matrix-ode} can be computed by combining the
equations~\eqref{BC},
\eqref{C-full},
\eqref{C0},
\eqref{A from B},
\eqref{Gamma},
\eqref{Omega},
\eqref{fourier-psi},
\eqref{fourier-A},
\eqref{fourier-gamma},
\eqref{fourier-omega},
\eqref{dA/dB},
\eqref{fourier-bu},
\eqref{dA/dB iiiiii},
\eqref{dA/dB iiiijj},
and~\eqref{dA/dB iijjkk}.
If we only want to compute $M(\bkappa,0)$, then we can use equation~\eqref{dA/dB 0} in place of \eqref{dA/dB iiiiii}, \eqref{dA/dB iiiijj}, and~\eqref{dA/dB iijjkk}. Solving equation~\eqref{matrix-ode} to give $L(\bkappa_0,t)$, as described in equation~\eqref{matrix-ode-soln} requires the additional equations~\eqref{bkappa} and~\eqref{bkappa0}.
The operators $M(\bkappa,t)$ and $L(\bkappa_0,t)$ act on the space of symmetric, trace zero, $3\times 3$ matrices. This is a five dimensional space, and with respect to the Frobenius norm, has an orthogonal basis
\begin{equation}
\begin{split}
\mathsf T_1 &= \tfrac1{\sqrt2}\left[\begin{smallmatrix}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 0
\end{smallmatrix}\right]
,\quad
\mathsf T_2 = \tfrac1{\sqrt2}\left[\begin{smallmatrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{smallmatrix}\right]
,\quad
\mathsf T_3 = \tfrac1{\sqrt6}\left[\begin{smallmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -2
\end{smallmatrix}\right]
,\\
\mathsf T_4 &= \tfrac1{\sqrt2}\left[\begin{smallmatrix}
0 & 0 & 1 \\
0 & 0 & 0 \\
1 & 0 & 0
\end{smallmatrix}\right]
,\quad
\mathsf T_5 = \tfrac1{\sqrt2}\left[\begin{smallmatrix}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{smallmatrix}\right]
\end{split}
\end{equation}
Thus $M(\bkappa,t)$ and $L(\bkappa_0,t)$ with respect to this basis are given by the five by five matrices $\mathcal M(\bkappa,t)$ and $\mathcal L(\bkappa_0,t)$ where
\begin{gather}
\mathcal M(\bkappa,t)_{i,j} = \text{Tr}(\mathsf T_i \cdot (M(\bkappa,t)[\mathsf T_j]))
\\
\mathcal L(\bkappa_0,t)_{i,j} = \text{Tr}(\mathsf T_i \cdot (L(\bkappa_0,t)[\mathsf T_j]))
\end{gather}
Thus we obtain the ordinary differential equation
\begin{gather}
\frac{\partial}{\partial t} \mathcal L(\bkappa_0,t) = \mathcal M(\bkappa,t) \cdot \mathcal L(\bkappa_0,t) \\
\mathcal L(\bkappa_0,0) = I
\end{gather}
The growth of $\mathcal L(\bkappa_0,t)$ is measured by either computing $n(t)$, the maximum over all $\bkappa_0 \in S$ of the spectral norm of $\mathcal L(\bkappa_0,t)$, or if we want to be more conservative, $r(t)$, the maximum over all $\bkappa_0 \in S$ of the spectral radius of $\mathcal L(\bkappa_0,t)$.
The calculations were performed with a C++ program, making use of the \emph{Newmat} software package \cite[][]{newmat} for the matrix calculations, and the \emph{GSL} software package \cite[][]{gsl} for the elliptic integrals. Also, because the differential equation turned out to be rather stiff, we used the implicit Runge-Kutta method of order 5 (Radau IIA) by \cite{hairer:96b,hairer:96a} for solving the ordinary differential equation. The C++ programs may be found at \url{http://www.math.missouri.edu/~stephen/software/jeff-stokes}.
\section{Demonstration of `buckling'}
\label{section buckling}
Here the idea is to see how fast $\mathcal L(\bkappa_0,t)$ grows for small $t\ge0$. This can be performed by computing the eigenvalues of $\mathcal M(\bkappa,0)$ for various values of $\bkappa$, and to see if any of them have positive real part. Having computed the eigenvalues of $M(\bkappa,0)$, we feed them into equation~\eqref{fourier-bu} to see if they produce a significant perturbation to $\bu$, and to see in what direction these perturbations take place.
In Figure~\ref{fig1} we display $m(\theta,\phi)$, the largest real part of the eigenvalues of $\mathcal M(\bkappa,0)$, with elongation flow $\bnabla\bu=\left[\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&0\end{smallmatrix}\right]$ and $\lambda=1$ and $\beta=10$. This was computed using the software product \emph{Mathematica} by Wolfram Research, Inc., and independently verified by the software described in the next section. We only need to consider $\bkappa$ on the unit sphere, that is, when $\bkappa = (\cos(\phi)\sin(\theta),\sin(\phi)\sin(\theta),\cos(\theta))$, with $0<\phi<\pi$ and $0<\theta<\pi/2$.
\begin{figure}
\begin{center}
\psfrag{phi}{$\phi$ in degrees}
\psfrag{theta}{$\theta$ in degrees}
\psfrag{m(theta,phi)}{$m(\theta,\phi)$}
\includegraphics[trim=4cm 3cm 4cm 2.5cm,clip]{plot-buckle.eps}
\caption{Graph of largest real part of eigenvalue of $\mathcal M(\bkappa,0)$ under elongation flow.}
\label{fig1}
\end{center}
\end{figure}
We see that this becomes particularly large, with value $24/49$, when $\phi = 90^\circ$. For example, if $(\theta,\phi)=(90^\circ,90^\circ)$, then equation~\eqref{fourier-bu} gives $\hat\bu = (0,0, 24/49)$. This is exactly the kind of buckling illustrated in Figure~\ref{buckling}. When $\theta=0$, we find that $\hat\bu = \pm(0,24/49, 0)$, that is, the fibers slide past each other in the $xy$-plane in a direction parallel to the $y$-axis.
Notice that the perturbations use all three dimensions. Simulations of the coupled system have usually been performed in only two dimensions \cite[][]{chinesta:02,chinesta:03}, or at least with some assumption of symmetry that effectively limits the spacial degrees of freedom to two dimensions \cite[][]{verweyst:02}. So we also performed the computation assuming that the fluid has no freedom to move in the $z$-direction. This is achieved by limiting $\hat{\mathsf C}$ so that its non-zero components are only in the top left two by two submatrix (equivalently considering only the basis elements $\mathsf T_1$ and $\mathsf T_2$, or only considering the top left two by two submatrix of $\mathcal M(\bkappa,0)$), and setting $\theta=90^\circ$. Computations reveal that $M(\bkappa,0) = 0$. This suggests that to obtain a true sense of how instabilities effect the flow, simulations should allow freedom in all three dimensions, with no symmetry assumptions.
\section{Analysis of growth of perturbations}
\label{section analysis of growth}
We denote by $n(t)$ the maximum of the spectral norm of $\mathcal L(\bkappa_0,t)$ over all $\bkappa_0$ in the two dimensional sphere, and by $r(t)$, the maximum over all $\bkappa_0$ of the spectral radius of $\mathcal L(\bkappa_0,t)$. Again we are looking at the case with shear flow $\bnabla\bu=\left[\begin{smallmatrix}0&1&0\\0&0&0\\0&0&0\end{smallmatrix}\right]$ and $\lambda=1$ and $\beta=10$, $100$ and $1000$. The results are shown in Table~\ref{growth-beta}. Since we are only interested in orders of magnitude, the spectral norms are only calculated to about two digits of accuracy. It can be seen that it takes quite a while ($t \approx 6$) before the growths even begin to become large enough that we might expect significant difference from the unperturbed solution. It may also be observed that the values of the spectral radius are of the same magnitude as the values of the spectral norm.
\begin{table}
\begin{center}
\begin{tabular}{@{}ccccccc@{}}
&
\multicolumn{2}{c}{$\beta=10$} &
\multicolumn{2}{c}{$\beta=100$} &
\multicolumn{2}{c}{$\beta=1000$} \\
$t$
& $n(t)$ & $r(t)$
& $n(t)$ & $r(t)$
& $n(t)$ & $r(t)$ \\
1 & 1.2 & 1.2 & 1.3 & 1.3 & 1.3 & 1.3 \\
2 & 1.3 & 1.1 & 1.6 & 1.2 & 1.7 & 1.2 \\
3 & 2.1 & 1.3 & 2.7 & 1.6 & 2.7 & 1.6 \\
4 & 4 & 2.1 & 6 & 2.8 & 6.4 & 2.8 \\
5 & 6.5 & 3.7 & 12 & 6.4 & 13 & 6.9 \\
6 & 9 & 5.9 & 19 & 13 & 22 & 15 \\
7 & 11 & 8.1 & 30 & 24 & 38 & 30 \\
8 & 12 & 9.8 & 48 & 42 & 68 & 59 \\
9 & 12 & 11 & 83 & 70 & 140 & 110 \\
10 & 11 & 11 & 150 & 120 & 320 & 240 \\
\end{tabular}
\end{center}
\
\caption{Maximal growth of spectral norm and spectral radius of linearized perturbations starting with isotropic data.}
\label{growth-beta}
\end{table}
One can actually follow where the large growths take place, by analyzing the singular decomposition of $\mathcal L(\bkappa_0,t)$. For example, for the case $t=10$ and $\beta=1000$, most of the growth takes place around $\bkappa$ in the direction $(\theta,\phi) = (90^\circ,-78.4^\circ)$ (having coming from $\bkappa_0$ directed $(\theta,\phi) = (90^\circ,79^\circ)$), and $\hat\bu$ is directed parallel to the $z$-axis. This looks like an extreme form of the `buckling' effect described above.
\section{Different Initial Conditions}
\label{section different initial}
The actual physical experiments can be performed by pouring the suspension upon a plate, and then running another plate along the top. It is to be expected (and has been reported) that the initial fiber distribution is not isotropic, but rather the effect of the suspension being squeezed by the pouring action is that most of the fibers are aligned in the $xz$-plane. Squeezing the fluid $\rho>1$ times can be simulated by a fluid gradient $\bnabla \bu = \left[\begin{smallmatrix}1/2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1/2\end{smallmatrix}\right]$ running for a time $\log(\rho)$. Thus, this squeezing can be modeled by replacing equation~\eqref{C0} by
\begin{equation}
\mathsf C = \mathsf C_\rho \text{ at } t=0
\end{equation}
where
\begin{equation}
\mathsf C_\rho = \left[\begin{matrix}\rho^{-\lambda/2} & 0 & \\ 0 & \rho^\lambda & 0 \\ 0 & 0 & \rho^{-\lambda/2}\end{matrix}\right]
\end{equation}
and in the case that $\mathsf U$ is independent of $t$, equation~\eqref{C-exp} is replaced by
\begin{equation}
\mathsf C = \mathsf C_\rho \cdot \exp\left(-\tfrac12 t (\mathsf \Omega + \lambda\mathsf \Gamma)\right)
\end{equation}
In Table~\ref{growth-s} we show growth of perturbations for the shear flow $\bnabla\bu=\left[\begin{smallmatrix}0&1&0\\0&0&0\\0&0&0\end{smallmatrix}\right]$ and $\lambda=1$ and $\beta = 100$ and $\rho=1$, $2$, $3$ and $4$.
\begin{table}
\begin{center}
\begin{tabular}{@{}lcccccccc@{}}
&
\multicolumn{2}{c}{$\rho=1$} &
\multicolumn{2}{c}{$\rho=2$} &
\multicolumn{2}{c}{$\rho=3$} &
\multicolumn{2}{c}{$\rho=4$} \\
$t$
& $n(t)$ & $r(t)$
& $n(t)$ & $r(t)$
& $n(t)$ & $r(t)$
& $n(t)$ & $r(t)$ \\
0.5 & 1.2 & 1.2 & 1.7 & 1.4 & 3 & 1.9 & 4.6 & 2.6 \\
1 & 1.3 & 1.3 & 2.6 & 1.8 & 6.9 & 3.6 & 14 & 6.7 \\
1.5 & 1.4 & 1.3 & 3.9 & 2.3 & 14 & 6.2 & 39 & 16 \\
2 & 1.6 & 1.2 & 5.4 & 2.7 & 27 & 10 & 100 & 38 \\
2.5 & 1.9 & 1.4 & 7.2 & 3 & 49 & 16 & 240 & 84 \\
3 & 2.7 & 1.6 & 9.3 & 3.2 & 84 & 25 & 570 & 180 \\
3.5 & 4 & 1.8 & 12 & 3.2 & 140 & 36 & 1300 & 360 \\
4 & 6 & 2.8 & 14 & 3.4 & 210 & 51 & 2700 & 680 \\
4.5 & 8.5 & 4.2 & 17 & 4.7 & 310 & 67 & 5400 & 1300 \\
5 & 12 & 6.4 & 19 & 6.7 & 430 & 81 & 10000 & 2200 \\
\end{tabular}
\end{center}
\
\caption{Growth of spectral properties with initial data created by squeezing.}
\label{growth-s}
\end{table}
Another way the experiment can be performed (see \cite{wang:08}) is to first apply a reverse shear to the suspension until it achieves its steady state, and then to use this as the initial condition. Running a shear in the opposite direction using the Folgar-Tucker equation \cite[][]{folgar:84} with $C_I=0.01$ gives a second order tensor matrix
\begin{equation}
\mathsf A = \left[\begin{matrix}
0.727123 & -0.074765 & 0 \\
-0.074765 & 0.0923099 & 0 \\
0 & 0 & 0.180567 \end{matrix}\right]
\end{equation}
(this was computed using spherical harmonics \cite[][]{spherical,montgomery-smith:09a}), which by inverting the elliptic integrals \cite[see][]{montgomery-smith:09b}, can be shown to correspond to
\begin{equation}
\mathsf C = \left[\begin{matrix}
0.372914 & 0.216212 & 0 \\
0.216212 & 2.20872 & 0 \\
0 & 0 & 1.28714
\end{matrix}\right]
\end{equation}
(Note that $\mathsf C$ is uniquely determined from $\mathsf B$ up to pre-multiplication by an orthogonal matrix, and pre-multiplying $\mathsf C$ by an orthogonal matrix has the effect of changing $\tilde{\mathsf B}$ by an orthogonal change of basis, which does not change its spectral norm.) Using this value of $\mathsf C$ as the initial data, and applying the above shear with $\lambda=1$ and $\beta=100$ gave the results shown in Table~\ref{growth-preshear}.
\begin{table}
\begin{center}
\begin{tabular}{@{}lcc@{}}
$t$ & $n(t)$
& $r(t)$ \\
0.5 & 4.7 & 2.9 \\
1 & 14 & 6.8 \\
1.5 & 37 & 13 \\
2 & 81 & 22 \\
2.5 & 160 & 29 \\
3 & 280 & 43 \\
3.5 & 470 & 130 \\
4 & 720 & 290 \\
4.5 & 1100 & 540 \\
5 & 1500 & 900 \\
\end{tabular}
\end{center}
\
\caption{Growth of spectral properties with initial data created by a reverse shear.}
\label{growth-preshear}
\end{table}
It can be seen that both sets of initial conditions illustrated here give rise to growths of perturbations that greatly exceed the growth with isotropic initial conditions.
\section{Conclusion}
\label{section conclusion}
The results from Tables~\ref{growth-beta}, \ref{growth-s} and~\ref{growth-preshear} show that the size of $\tilde{\mathsf B}$ increases by a fairly substantial amount. If the original perturbations are reasonably large, one can see that even after a short amount of time, that large deviations from the unperturbed solution are very likely. We propose that this is what leads to the greatly reduced rate of alignment of the fibers.
We suspect that the perturbations take place at small length scales, in both numerical and physical experiments. The reason we think this is as follows. The rate of growth of the perturbations does not depend upon the size of the wave number. One of our assumptions is that the medium is of infinite extent. If we have the more realistic situation where there are boundaries to the medium, it is reasonable to think that if the wave number is smaller (that is, the length scale of the oscillations of the perturbations is larger), that the boundary has a greater effect, and that most likely this effect slows down the growth. Thus we think that the perturbations take place with large wave numbers. The largest realistic wave number for physical experiments is probably the reciprocal of the length of the fibers, and in numerical experiments is probably the reciprocal of the size of the spacial grid or elements used. For this reason, we suspect that the growth of perturbations may be quite difficult to measure in physical experiments, and in numerical experiments might be mistaken for problems with the numerical method. Indeed, when one is operating at these kinds of small length scales, one begins to doubt even the assumption that discrete fibers can be well represented by the continuum. This perhaps undermines the original assumption of this paper, that the complex hydrodynamic interactions between the fibers can be somehow modeled by continuum equations. Nevertheless, we still feel that there is a good chance that the coupled continuum equations might accurately describe the evolution of flows of fibers in fluids.
For future work, we would like to see the result of a full numerical simulation of the coupled Jeffery-Stokes equation, that is,
equations~\eqref{stokes},
\eqref{incompressible},
\eqref{stress},
\eqref{B-full}
and~\eqref{A from B},
replacing equation~\eqref{B0} with $\mathsf B$ initially set to a small perturbation of $\mathsf I$. We note that some numerical simulations of the coupled equations have already been performed \cite{chinesta:02,chinesta:03,verweyst:02}, but we believe that more detailed computations, that allow complete freedom in all three spacial dimensions, are necessary to observe the effects predicted in this paper.
\section{Acknowledgments}
The author gratefully acknowledges support from N.S.F.\ grant C.M.M.I.\ 0727399. The author also is grateful for comments from the reviewers of the paper, which were extremely helpful.
\bibliographystyle{jfm}
|
1,116,691,497,749 | arxiv | \section{Introduction}
Given a sequence of probability measures $\{\mu_i\}_{i=1}^\infty$ on a homogeneous space $X$, it is natural to ask what we can say about weak$^*$ limits of this sequence? Often one is interested in measures that are invariant under a transformation $T$ acting on $X$, and in this case weak$^*$ limits are clearly also invariant under $T$. If $X$ is non-compact, maybe the next question to ask is whether any weak$^*$ limit is a probability measure.
If $T$ acts on $X=\Gamma\backslash G$ by a unipotent element where $G$ is a Lie group and $\Gamma$ is a lattice,
then it is known that $\mu$ is either the zero measure or a probability measure
\cite{MozSha}. This fact relies on the quantitative non-divergences estimates for
unipotents due to works of S.~G.~Dani~\cite{Dani} (further refined by G.~A.~Margulis and D.~Kleinbock~\cite{MarKle}). On the other hand, if $T$ acts on $X=\operatorname{SL}_{d}( \mathbb{Z}) \backslash\operatorname{SL}_d(\mathbb R)$ by a diagonal element, then $\mu(X)$ can be any value in the interval $[0,1]$
due to softness of Anosov-flows, see for instance \cite{Const}. However, as we will see there are constraints on $\mu(X)$ if we have additional
information about the entropies $h_{\mu_i}(\operatorname{T})$. This has been observed in \cite{TO} for the action of the geodesic flow on $\operatorname{SL}_2( \mathbb{Z}) \backslash \operatorname{SL}_2(\mathbb R)$, see Theorem~\ref{thm:torusorbit}.
In this paper we will generalize this theorem to the space $\operatorname{SL}_{3}( \mathbb{Z}) \backslash \operatorname{SL}_{3} (\mathbb R)$ with the action of a
particular diagonal element.
We identify $X=\operatorname{SL}_{d}( \mathbb{Z}) \backslash \operatorname{SL}_{d} (\mathbb R)$ with the space of unimodular lattices in $\mathbb R^{d}$, see $\S$~\ref{sec:UnModLat}.
Using this identification we can define for $d=3$ the height function $\operatorname{ht}(x)$ of a lattice $x \in X$ as follows.
\begin{mydef} For any 3-lattice $x \in \operatorname{SL}_3( \mathbb{Z}) \backslash \operatorname{SL}_3(\mathbb R) $ we define the {\rm height} $\operatorname{ht}(x)$ to be the inverse of the minimum of the length of the shortest nonzero vector in $x$ and the smallest covolume of planes w.r.t. $x$.
\end{mydef} Here, the length of a vector is given in terms of the Euclidean norm on $\mathbb R^{d}$. Also, if $d=2$ then we consider the height $\operatorname{ht}(x)$ to be the inverse of the length of the shortest nonzero vector in $x$. Let
$$X_{\le M}:=\{x \in X \, | \,\operatorname{ht}(x)\le M\} \text{ and } X_{\geq M}:=\{x \in X \, | \, \operatorname{ht}(x)\geq M\}.$$
By Mahler's compactness criterion (see Theorem~\ref{thm:mahler})
$X_{\leq M}$ is compact and any compact subset of $X$ is
contained in some $X_{\leq M}$.
In \cite{TO}, M.~E., E.~Lindenstrauss, Ph.~Michel, and A.~Venkatesh give the following theorem:
\begin{thm}
\label{thm:torusorbit}
Let $X$ be the homogeneous space $\operatorname{SL}(2, \mathbb{Z})\backslash\operatorname{SL}(2,\mathbb R)$, let $T$ be the time-one-map for the geodesic flow, and $\mu$ be a $T$ invariant probability measure on $X$. Then, there exists $M_0,$ such that
$$
h_\mu(T)\leq 1+\frac{\log\log M}{\log M}-\frac{\mu(X_{\geq M})}{2}
$$
for any $M\geq M_0$.
In particular, for a sequence of $T$-invariant probability measures $\mu_i$ with entropies $h_{\mu_i}(T)\geq c$
we have that any weak$^*$ limit $\mu$ has at least $\mu(X)\geq 2c-1$ mass left.
\end{thm}
Here, $\mu$ is a weak$^*$ limit of the sequence $\{\mu_i\}_{i=1}^\infty$ if for some subsequence $i_k$ and for all
$f \in C_c(X)$ we have $$\lim_{k\to \infty}\int_X f d\mu_{i_k} \to \int_X f d \mu.$$
The proof of Theorem \ref{thm:torusorbit} in \cite{TO} makes use
of the geometry of the upper half plane $\mathbb H$.
From now on we let $X=\operatorname{SL}_{3}( \mathbb{Z}) \backslash
\operatorname{SL}_{3} (\mathbb R)$ and let
$$\alpha=\left( \begin{array}{ccc} e^{1/2} & & \\
& e^{1/2}&\\
& & e^{-1} \end{array} \right) \in \operatorname{SL}_{3}(\mathbb R).$$
We define the transformation $\operatorname{T}:X \to X$ via $\operatorname{T}(x)=x\alpha$. We now state the main theorem of this paper.
\begin{thm}
\label{thm:main}
Let $X$ and $\operatorname{T}$ be as defined above. Then there exists a function $\varphi(M)$ (which is given explicitly), with $\varphi(M) \to_{M\to \infty} 0,$ and $M_0$ such that for any $\operatorname{T}$-invariant probability measure $\mu$ on $X$, and any $M>M_0$, one has
$$h_{\mu}(\operatorname{T})\leq 3-\mu(X_{\geq M })+\varphi(M).$$
\end{thm}
In this context we note that the maximal measure theoretic entropy, the entropy of $\operatorname{T}$ with respect to Haar measure on $X$, is $3$. This follows e.g.\ from \cite[Prop. 9.2]{MarTom}. We will see later that $\varphi(M)=O(\frac{\log \log M}{\log M})$.
As a consequence of Theorem~\ref{thm:main} we have:
\begin{cor}
\label{cor:lim} A sequence of $\operatorname{T}$-invariant probability measures
$\{\mu_i\}_{i=1}^\infty$ with entropy $h_{\mu_i}(\operatorname{T}) \geq c$ satisfies that any
weak$^*$ limit $\mu$ has at least $\mu(X) \geq c-2$ mass left.
\end{cor}
This result is sharp in the following sense. For any $c \in (2,3)$ one can construct a sequence of probability measures $\mu_i$ with $h_{\mu_i}(\operatorname{T})\to c $ as $i\to \infty $ such that any weak$^*$ limit $\mu$ has precisely $c-2$ mass left, see \cite{Const}.
Another interesting application of our method arises when we do not assume $\operatorname{T}$-invariance of the measures we consider. In this case, instead of entropy consideration we assume that our measures have high dimension and study the behaviour of the measure under iterates of $T$.
Let us consider the following subgroups of $G$
\begin{equation}
\label{eqn:U^+}U^+=\{g\in G:\alpha^{-n}g\alpha^n\to 1 \text{ as } n\to -\infty\},
\end{equation}
\begin{equation}
\label{eqn:U^-}
U^-=\{g\in G:\alpha^{-n}g\alpha^n\to 1 \text{ as } n\to \infty\},
\end{equation}
\begin{equation}
\label{eqn:C}
C=\{g\in G : g \alpha=\alpha g\}.
\end{equation}
For any $\epsilon >0$, group $H$, and $g \in H$ we write $B_{\epsilon}^H(g)$ for the $\epsilon$-ball in $H$ around $g$, see also \S~\ref{inj}. Throughout this paper we write $A \ll B$ if there exits a constant $c>0$ such that $A \le c B$. If the constant $c$ depends on $M$, then we write $A \ll_M B.$
\begin{mydef}For a probability measure $\nu$ on $X$ we say that $\nu$ has \emph{dimension at least $d$ in the unstable direction} if for any $\delta>0$ there exists $\kappa>0$ such that for any $\epsilon\in (0,\kappa)$ and for any $\eta \in (0,\kappa)$ we have
\begin{equation}
\label{eqn:unstable}
\nu(xB_{\epsilon}^{U^+}B_\eta^{U^-C})\ll_\delta \epsilon^{d-\delta} \text{ for any } x\in X.
\end{equation}
\end{mydef}
Note that the maximum value for $d$ in the definition is 2 since $U^+$ is two dimensional. The most interesting case of this definition concerns a measure $\nu$ supported on a compact subset, say $\overline{x_0B_1^{U^+}}$, of an orbit $x_0U^+$ under the unstable subgroup. In this case, \eqref{eqn:unstable} is equivalent to $\nu(x_0 u B_\epsilon^{U^+})\ll \epsilon^{d-\delta}$ for all $u \in U^+$ (which is one of the inequalities of the notion of Ahlfors regularity of dimension $d-\delta$) and for any $\delta>0$. See \cite[Chaps. 4-6]{Mat} for more information on Ahlfors regularity.
Let us consider the following sequence of measures $\mu_n$ defined by
$$\mu_n=\frac{1}{n}\sum_{i=0}^{n-1}\operatorname{T}^i_*\nu$$
where $\operatorname{T}^i_*\nu$ is the push-forward of $\nu$ under $\operatorname{T}^i$. We have
\begin{thm}
\label{thm:dim}
For a fixed $d$, let $\nu$ be a probability measure of dimension at least $d$ in the unstable direction, and let $\mu_n$ be as above. Let $\mu$ be a weak$^*$ limit of the sequence $(\mu_n)_{n\ge 1}$. Then $\mu(X) \geq \frac{3}{2}(d-\frac{4}{3})$. In other words, at least $\frac{3}{2}(d-\frac{4}{3})$ of the mass is left.
\end{thm}
In particular, if $d=2$ then the limit $\mu$ is a probability measure. In this case with a minor additional assumption on $\nu$ one in fact obtains the equidistribution result, that is, the limit measure $\mu$ is the Haar measure \cite{Shi}.
Another application of Theorem~\ref{thm:dim} is that it gives the sharp upper bound for the Hausdorff dimension of singular pairs. The exact calculation of Hausdorff dimension of singular pairs was achieved in \cite{Che}. We say that ${\bf r} \in \mathbb R^2$ is \emph{singular} if for every $\delta>0$ there exists $N_0>0$ such that for any $N>N_0$ the inequality
$$\|q {\bf r} - {\bf p}\|<\frac{\delta}{N^{1/2}}$$
admits an integer solution for $\bf p\in \mathbb{Z}^2$ and for $q\in \mathbb{Z}$ with $0<q<N$. From our results we obtain the precise upper bound for the Hausdorff dimension of the set of singular pairs; namely this dimension is at most $\frac{4}{3}.$ This gives an independent proof for this fact which was proved in \cite{Che}.
Let $x\in\operatorname{SL}_3( \mathbb{Z})\backslash\operatorname{SL}_3(\mathbb R)$. Then we say $x$ is divergent if $T^n(x)$ diverges in $\operatorname{SL}_3( \mathbb{Z})\backslash\operatorname{SL}_3(\mathbb R)$.
We recall (e.g.\ from \cite{Che}) that ${\bf r}$ is singular if and only if
$$
x_{\bf r}=\operatorname{SL}_3( \mathbb{Z})\begin{pmatrix}1&&\\&1&\\r_1&r_2&1\end{pmatrix}
$$
is divergent. An equivalent formulation\footnote{Roughly speaking the additional 6 dimensions corresponding to $U^-C,$ are not as important as the 2 directions in the unstable horospherical subgroup $U^+$. The latter is parametrized by the unipotent matrix as in the definition of $x_{\bf r}$.} of the above Hausdorff dimension result (see \cite{Che}) is that the set of divergent points in $\operatorname{SL}_3( \mathbb{Z})\backslash\operatorname{SL}_3(\mathbb R)$ has Hausdorff dimension $8-\frac23=\frac43+6$.
However, we can also strengthen this observation as follows.
A weaker requirement on points (giving rise to a larger set) would be divergence on average, which we define as follows. A point $x$ is {\em divergent on average} (under T) if the sequence of measures
\[
\frac1N\sum_{n=0}^{N-1}\delta_{T^n(x)}
\]
converges to zero in the weak$^*$ topology, i.e.\ if the mass of the orbit --- but not necessarily the orbit itself --- escapes to infinity.
\begin{cor}
\label{cor:sing2}
The Hausdorff dimension of the set of points that are divergent on average is also $\frac{4}{3}+6$.
\end{cor}
We finally note that the nondivergence result \cite[Theorem 3.3]{KleLinWei} is related to Theorem~\ref{thm:dim}. In fact, \cite[Theorem 3.3]{KleLinWei} implies that $\mu$ as in Theorem~\ref{thm:dim} is a probability measure if $\nu$ has the additional regularity property; namely if $\nu$ is assumed to be friendly. However, to our knowledge these additional assumptions make it impossible to derive e.g. Corollary~\ref{cor:sing2}.
The next section below has some basic definitions and facts. In
\S~\ref{sec:marked}, we characterize what it means for a
trajectory of a lattice to be above height $M$ in some time
interval. Using this we prove Theorem~\ref{thm:main} in
\S~\ref{sec:mainprop}-\ref{sec:mainthm}. Theorem~\ref{thm:dim} and its corollary are discussed in \S~\ref{sec:locdim}.
{\bf Acknowledgements:} We would like to thank Jim Tseng for discussions and for pointing out the reference
to \cite{Fal}. We also thank the anonymous referee for his detailed report and his suggestions.
\section{Preliminaries}
\label{sec:prelim}
\subsection{The space of unimodular lattices}
\label{sec:UnModLat}
In this section we will give a brief introduction to the space of
unimodular lattices in $\mathbb R^{3}$.
\begin{mydef}
$\Lambda \subset \mathbb R^{3}$ is a \emph{lattice}
if it is a discrete subgroup and the quotient
$\mathbb R^{3}/\Lambda$ is compact.
\end{mydef}
Note that this is equivalent to saying that
$\Lambda=\langle v_1,v_2,v_{3}\rangle_{ \mathbb{Z}}$ where $v_1,v_2,v_{3}$ are
linearly independent vectors over $\mathbb R$.
\begin{mydef}
A lattice $\Lambda=\langle v_1,v_2,v_{3}\rangle_{ \mathbb{Z}}$ is said to be $\emph{unimodular}$ if it has covolume equal
to $1,$ where the \emph{covolume} is the absolute value of the
determinant of the matrix with row vectors $v_1,v_2,v_{3}$.
\end{mydef}
We identify a point $\operatorname{SL}_{3}( \mathbb{Z})g \in X$ with the unimodular lattice in
$\mathbb R^{3}$ generated by the row vectors of $g \in G$. We leave it as an exercise for the reader to convince himself that this correspondence is well defined and a bijection.
We now state Mahler's compactness criterion which motivates the definition of the height function in the introduction.
\begin{thm}[Mahler's compactness criterion]
\label{thm:mahler}
A closed subset $K\subset X$ is compact if and
only if there exists $\delta >0$ such that no lattice in $K$ contains a nonzero vector of length less that $\delta$.
\end{thm}
For the proof the reader can refer to \cite[Corollary 10.9]{Rag}. We now deduce Corollary~\ref{cor:lim} from Theorem~\ref{thm:main}.
\begin{proof}
We need to approximate $1_{X_{\le M}}$ by functions of compact
support. So, let $f \in C_c(X)$ be such that
$$
f(x) =
\begin{cases}
1 & \text{for }x\in X_{\leq M} \\
0 & \text{for }x \in X_{\geq (M+1) }
\end{cases}
$$
and $0\leq f(x)\leq 1$ otherwise. Such $f$ exists by Urysohn's
Lemma. Hence, $$\int f \,d\mu_i \geq \int 1_{X_{\le M}}\,
d\mu_i=\mu_i(X_{\le M})\ge c-2-\varphi(M)$$
Let $\mu$ be a weak$^*$ limit, then we have
$$\lim_{i_k \to \infty}\int f \,d\mu_k =\int f\, d\mu$$
and hence we deduce that
$$ \int f \,d\mu \ge c-2-\varphi(M).$$
Now, by definition of $f$ we get $\int f \,d\mu \le \mu(X_{<(M+1)})$. Thus,
$$\mu(X_{<(M+1)})\ge c-2-\varphi(M).$$
This is true for any $M\ge M_0$, so letting $M \to \infty$ finally we have
$$\mu(X)\ge c-2$$
which completes the proof.
\end{proof}
\subsection{Riemannian metric on $ X$}
\label{inj}
Let $G=\operatorname{SL}_{3}(\mathbb R)$ and $\Gamma=\operatorname{SL}_{3}( \mathbb{Z})$. We fix a left-invariant Riemannian metric $d_G$ (or simply $d$) on $G$ and for any $x_1=\Gamma g_1,x_2=\Gamma g_2 \in X$ we define
$$d_X(x_1,x_2)=\inf_{\gamma \in \Gamma} d_{G}(g_1,\gamma g_2)$$
which gives a metric $d_X$ on $X=\Gamma \backslash G.$ For more information about the Riemannian metric, we refer to \cite[Chp. 2]{RM}.
For a given subgroup $H$ of $G$ we let $B_r^H(g):=\{h \in H\,|\,d_G(h,g)<r\}$. It makes sense to abbreviate and write $B_r^H=B_r^H(1)$, where we write 1 for the identity in $G$.
\begin{mydef}
We say that $r>0$ is an injectivity radius of $x \in X$ if the map $g \mapsto xg$ from $B_r^G \to B_r^X(x)$ is an isometry.
\end{mydef}
\begin{lem}
For any $x\in X$ there exists $r>0$ which is an injectivity radius of $x$.
\end{lem}
Note that since $X_{\le M}$ is compact, we can choose $r>0$ which is an injectivity radius for every point in $X_{\le M}$. In this case, $r$ is called \textit{an injectivity radius of $X_{\le M}$}. We refer to Proposition 9.14 in \cite{EinWar} for a proof of these claims.
\subsubsection{Operator norms}
\label{sec:metrics}
We endow $\mathbb R^3$ with the standard euclidean metric, writing $|u|$ for the norm of $u \in \mathbb R^3$. Rescaling the Riemannian metric if necessary we may assume that there exists some $\eta_0>0$ such that $|u-ug|<|u| d_G(1,g)$ for any $u \in \mathbb R^3$ and $g \in B_{\eta_0}^G$.
\subsubsection{Metric on $U^+$}
\label{sec:metric on U^+}
We may identify $U^+$ with $\mathbb R^2$ using the parametrization
$$(t_1,t_2)\in \mathbb R^2 \to \left(\begin{array}{ccc} 1&&\\&1&\\t_1&t_2&1\end{array}\right).$$
It will be convenient to work with the maximum norm on $\mathbb R^2$. We will write $D_{\eta}^{U^+}=\{\left(\begin{array}{ccc} 1&&\\&1&\\t_1&t_2&1\end{array}\right):|t_1|,|t_2|<\eta\}$ for a ball in $U^+$ of radius $\eta$ centred at the identity. Rescaling the maximum norm on $\mathbb R^2$ if necessary we will assume that
$$D_\epsilon^{U^+} \subset B_\epsilon^{U^+}.$$
\subsection{Entropy}
\label{sec:entropy}
Instead of giving here the formal definition of the ergodic theoretic entropy $h_\mu(\operatorname{T})$ we will state only a well-known and important lemma that will enter our arguments later. We refer to \cite[\S~4]{WB} for a complete definition.
Fix $\eta>0$ small enough so that $B_{\eta}^{\operatorname{SL}_{3}(\mathbb R)}$ is an injective image under the exponential map of a neighborhood of $0$ in the Lie algebra. Define a Bowen
$N$-ball to be the translate $xB_N$ for some $x \in X$ of
$$B_N=\bigcap_{n=-N}^N \alpha^{-n} B_{\eta}^{\operatorname{SL}_{3}(\mathbb R)}\alpha^{n}.$$
Roughly speaking the Bowen $N$-ball $x B_N$ consists of all $y $ near $x$ which have the property that the trajectories from time $-N$ to time $N$ of $x$ and $y$ are $\eta$-close to each other.
The following lemma gives an upper bound for entropy in terms of covers of Bowen balls.
\begin{lem}\label{entropy} Let $\mu$ be a $\operatorname{T}$-invariant probability measure on $X$. For any $N\geq
1$ and $\epsilon >0$ let $BC(N,\epsilon)$ be the minimal number of
Bowen $N$-balls needed to cover any subset of $X$ of measure
bigger that $1-\epsilon$. Then
$$h_{\mu}(\operatorname{T})\leq \lim_{\epsilon \to 0}\liminf_{N \to \infty}\frac{\log
BC(N,\epsilon)}{2N}.$$
\end{lem}
We omit the proof which is very similar to \cite[Lemma 5.2]{TO} and goes back to \cite{KatBri}.
\section{Sets of labeled marked times}
\label{sec:marked}
Let $N,M>0$ be given. In this section we define for every $x \in \operatorname{T}^{N}(X_{\le M})$ the set of labeled marked times. Each configuration of such markings will correspond to a particular element of a partition of $X$, and we will estimate the cardinality of this partition (which is desirable due to the link of entropy and the logarithmic growth of covers as in Lemma~\ref{entropy}). This marking has the property that it will tell whether the lattice $\operatorname{T}^n(x)$ is above or below height $M$, without having to know $x$. However, we do not want to consider all vectors (or planes) of $x$ that become short at some point - it is likely that a partitioning of $X$ that uses all such vectors (or planes) will be too large to be of use.
Rather whenever there are two linearly independent primitive $1/M$-short vectors, our
strategy is to consider a plane in $x$ that contains both vectors.
So, for a given lattice $x$ we would like to associate a set of
labeled marked times in $[-N,N]$ which tells us when
a vector or a plane is getting resp. stops being $1/M$-short. Choosing the vectors and planes of $x$ carefully in the following construction we obtain a family $\mathcal{M}_N$ of sets of labeled marked times. This will give rise to a partition of $X$, which will be helpful in the main estimates given in \S~\ref{sec:mainprop}.
\subsection{Short lines and planes}
\label{sec:T-actiononplane}
Let $u,v \in \mathbb R^3$ be linearly independent. We recall that the covolume of the two-dimensional lattice $ \mathbb{Z} u+ \mathbb{Z} v$ in the plane $\mathbb R u + \mathbb R v$ equals $|u \wedge v|$. Here, $u \wedge v = (u_1,u_3,u_3)\wedge (v_1,v_2,v_3)=(u_2v_3-u_3v_2,u_3v_1-u_1v_3,u_1v_2-u_2v_1)$. Below, $u,v \in \mathbb R^3$ will always be such that $ \mathbb{Z} u+ \mathbb{Z} v= x \cap (\mathbb R u + \mathbb R v)$ for a lattice $x$. In this case we call $\mathbb R u + \mathbb R v$ {\it rational} w.r.t. $x$ and will call $|u \wedge v|$ {\it the covolume of the plane} $\mathbb R u + \mathbb R v$ w.r.t. $x$. We sometimes write {\it a plane $P$ in $x$} to mean the plane $P=\mathbb R u+\mathbb R v$ rational w.r.t. $x$.
We also note that the action of $\operatorname{T}$ extends to $\bigwedge^2\mathbb R^2$ via
\begin{multline}
\label{eqn:exterior}
\operatorname{T}(u \wedge v)=(u_1e^{1/2},u_2e^{1/2},u_3e^{-1}) \wedge (v_1e^{1/2},v_2e^{1/2},v_3e^{-1})\\
=((u_2v_3-u_3v_2)e^{-1/2},(u_3v_1-u_1v_3)e^{-1/2},(u_1v_2-u_2v_1)e^1).
\end{multline}
For a plane $P=\mathbb R u+\mathbb R v$ as above, we sometimes write $\operatorname{T}(P)$ for $\operatorname{T}(u \wedge v)$. For a vector $v=(v_1,v_2,v_3) \in \mathbb R^3$ we let $\operatorname{T}(v):=v \alpha=(v_1e^{1/2},v_2e^{1/2},v_3 e^{-1}).$
Let $\epsilon > 0$ be given. Fix $x \in X$, a vector $v$ in $x$ is {\it $\epsilon$-short at time $n$} if $|\operatorname{T}^n(v)| \le \epsilon$. Similarly for plane $P \subset \mathbb R^3$ we say that it is {\it $\epsilon$-short at time $n$} (w.r.t. $x$) if $\operatorname{T}^n(P)$ is rational w.r.t. $\operatorname{T}^n(x)$ and its covolume is $\le \epsilon$.
\subsection{(Labeled) Marked Times}
\label{sec:lebMar}
For a positive number $N$ and a lattice $x \in \operatorname{T}^N(X_{\le M})$ we explain which times will be marked in $[-N,N]$ and how they are labeled. The following lemma which is special to $\operatorname{SL}_3( \mathbb{Z}) \backslash \operatorname{SL}_3(\mathbb R)$ is crucial.
\begin{lem}[Minkowski]
\label{lem:minkowski}
Let $\epsilon_1,\epsilon_2 \in(0,1)$ be given. If there are two linearly independent $\epsilon_1$-short and $\epsilon_2$-short vectors in a unimodular lattice in $x$, then there is a unique rational plane in $x$ with covolume less than $1$ which in fact is $\epsilon_1 \epsilon_2$-short.
If there are two different rational planes of covolumes $\epsilon_1$ and $\epsilon_2$ in a unimodular lattice $x$, then there is a unique primitive vector of length less than $1$ which in fact is $\epsilon_1 \epsilon_2$-short. In this case, the unique $\epsilon_1 \epsilon_2$-short vector lies in the intersection of the two short planes.
\end{lem}
The first part of the lemma follows quickly from the assumption that $x$ is unimodular. The second follows by considering the dual lattice to $x$.
We will use these facts to mark and label certain times in an efficient manner so as to keep the total number of configurations as low as possible.
\subsubsection{Some observations}
\label{sec:observations}
Let us explain how we will use Lemma~\ref{lem:minkowski}. Assume that we have the following situation:
There are two linearly independent primitive vectors $u,v$ in a unimodular lattice such that
$$|u| \le 1/M \text{ and } |\operatorname{T}(v)| \le 1/M.$$
Let $u=(u_1,u_2,u_3)$. It is easy to see that
$$|\operatorname{T}(u)|=|(e^{1/2}u_1,e^{1/2}u_2,e^{-1}u_3)| \le \frac{e^{1/2}}{ M}.$$
Assume $M\ge e^{1/2}$. From Lemma~\ref{lem:minkowski} we have that the plane containing both $\operatorname{T}(u),\operatorname{T}(v)$ has covolume at most $\frac{e^{1/2}}{M^2}\le \frac{1}{M}$, and it is unique with this property.
The similar situation arises when we have two different planes $P,P'$ which are rational for a unimodular lattice such that
$$|P| \le 1/M \text{ and } |\operatorname{T}(P')| \le 1/M$$
where $| \cdot |$ means the covolume. Assume $M \ge e$. One can see that $|\operatorname{T}(P)| \le \frac{e}{M}.$ Thus, we conclude from Lemma~\ref{lem:minkowski} that there is a unique vector of length at most $\frac{e}{M^2}\le \frac{1}{M}$ contained in both planes $\operatorname{T}(P)$ and $\operatorname{T}(P')$.
\subsubsection{Marked times}
\label{sec:markedtimes}
Let $V_{N,x}=\{i \in [-N,N]: \operatorname{T}^i(x) \not \in X_{\le M}\}$. $V_{N,x}$ is a disjoint union of maximal intervals and let $V=[a,b]$ be one them.
\begin{enumerate}
\item[(a)] either $a=-N$ (and so $\operatorname{ht}(T^a(x))\leq M$) or $a>-N$ and $\operatorname{ht}(\operatorname{T}^{a-1}(x)) < M$,
\item[(b)] either $b=N$ or $\operatorname{ht}(T^{b+1}(x))<M$, and
\item[(c)] $\operatorname{ht}(\operatorname{T}^n(x)) \ge M$ for all $n \in V$.
\end{enumerate}
We first show how one should inductively pick the marked times for this interval $V$:
We will successively choose vectors and planes in $x$ and mark the time instances with particular labels when these vectors and planes get $1/M$-short on $V$ and when they become big again. At time $a$ we know that there is either a unique plane or a unique vector getting $1/M$-short. Here, uniqueness of either follows from Lemma~\ref{lem:minkowski}. Moreover, we cannot have two $1/M$-short vectors ($1/M$-short planes) as otherwise there is a $1/M^2$-short plane (or vector) which contradicts the assumption that $V=[a,b]$ has $a$ as a left endpoint. If we have both a unique $1/M$-short plane and vector then we consider whichever stays $1/M$-short longer (say with preference to vectors if again this gives no decision). Assume that we have a unique plane. The case where we start with a unique vector is similar. Mark $a$ by $p_1$ which is the time when the plane is getting $1/M$-short, and also mark by $p_1'$ the last time in $[a,b]$ when the same plane is still $1/M$-short. If $p_1'=b$ we stop marking. If not, then there is again by Lemma~\ref{lem:minkowski} a unique $1/M$-short plane or vector at $p_1'+1$. If it is a $1/M$-short plane then at time $p_1'+1$ we must have a unique $1/M$-short vector by the discussions in $\S$~\ref{sec:observations}. In either case, we have a unique $1/M$-short vector at time $p_1'+1$. Let us mark by $l_1$ the instance in $[a,p_1'+1]$ when this vector is getting $1/M$-short. Also, mark by $l_1',$ the last time in $[p_1'+1,b]$ for which this vector is still $1/M$-short. If $l_1'=b$ we stop, otherwise at time $l_1'+1$ there must be a unique $1/M$-short plane or vector. If it is a short vector then we know that there must be a unique plane of covolume at most $1/M$ by the discussions in $\S$~\ref{sec:observations}. So, in either case there is a unique $1/M$-short plane at time $l_1'+1$. So, there is an instance in $[a,l_1'+1]$ which we mark by $p_2$ when for the first time this plane is $1/M$-short. Also, mark by $p_2',$ the last instance of time in $[l_1'+1,b],$ for which the plane is $1/M$-short. If $p_2'=b$ we stop here, otherwise we repeat the arguments above and keep marking the time instances in $V$ by $l_i,l_i',p_j,p_j'$ until we hit time $b$.
Given a positive number $N$ and a lattice $x \in \operatorname{T}^N(X_{\le M})$ we first consider the disjoint intervals $V_i$ of maximum length with the property as $V$ above. Now start labeling some elements of the sets $V_i$ as explained earlier starting with $V_1$ and continuing with $V_2$ etc. always increasing the indices of $l_i,l_i',p_i,p_i'$.
For any lattice $x$ as above we construct in this way a set of labeled marked times in $[-N,N]$. We denote this set by
$$\mathcal{N}(x)=\mathcal{N}_{[-N,N]}(x)=(\mathcal{L,L',P,P'}).$$
Here $\mathcal{L}=\mathcal{L}(x),\mathcal{L'}=\mathcal{L'}(x),\mathcal{P}=\mathcal{P}(x),\mathcal{P'}=\mathcal{P'}(x)$ are subsets in $[-N,N]$ that contain all the labeled marked times $l_i,l_i',p_j,p_j'$ for $x$ respectively. Finally, we let
$$\mathcal{M}_N=\{\mathcal{N}(x)\,:\, x \in \operatorname{T}^N(X_{\le M})\}$$
be the family of all sets of labeled marked times on the interval $[-N,N]$.
\subsubsection{The Estimates}
\begin{lem}[Noninclusion of marked intervals]
\label{lem:properties}
Let $(\mathcal{L,L',P,P'})\in \mathcal{M}_N$ be given. For any $q$ in $\mathcal{L}$ or in $\mathcal{P}$ there is no $r$ in $\mathcal{L}$ or in $\mathcal{P}$ with $q \le r \le r' \le q'$.
\end{lem}
\begin{proof}
We have four cases to consider. Let us start with the case that $r=p_i, r'=p_i'$ and $q=p_j, q'=p_j'$ (where $j>i$ as it is in our construction only possible for a later marked interval $[q, q']$ to contain an earlier one). However, by construction the plane $P_i$ that is $1/M$-short at that time we introduce the marked interval $[p_i,p_i']$ (which is either the beginning of the interval $V$ or is the time the earlier short vector stops to be short) is the unique short plane at that time. Hence, it is impossible to have the stated inclusion as the plane $P_j$ (responsible for $[p_j,p_j']$) would otherwise also be short at that time. The case of two lines is completely similar.
Consider now the case $q=p_j \in \mathcal P$ and $r=l_i \in \mathcal L$ with $p_j \le l_i \le l_i' \le p_j'$. If $l_i=a$ (and so also $l_i=p_j=a$) is the left end point of interval $V=[a,b]$ in the construction, then we would have marked either $l_i,l_i'$ or $p_j,p_j'$ but not both as we agreed to start by marking the end points of the longer interval (if there is a choice). Hence, we may assume $l_i >a$ and that times $l_i, l_i'$ have been introduced after consideration of a plane with marked times $p_k, p_k'$ satisfying $l_i \le p_k'+1 \le l_i'$, in particular $j \not = k$. We now treat two cases depending on whether $p_k \ge l_i$ or not. If $p_k \ge l_i$ then $p_j \le p_k\le p_k' \le p_j'$ which is impossible by the first case. So, assume $p_k < l_i$ then we have two different planes that are $1/M$-short at time $l_i$. This implies that the vector responsible for the interval $[l_i,l_i']$ is $1/M^2$-short by Lemma~\ref{lem:minkowski}. However, this shows that the same vector is also $1/M$-short at time $l_i-1$ for $M \ge e$, which contradicts the choice of $l_i$. The case of $q=l_i \in \mathcal L$ and $r=p_j \in \mathcal P$ is similar.
\end{proof}
We would like to know that the cardinality of $\mathcal{M}_N$ can be made small (important in Lemma~\ref{entropy}) with
$M$ large. In other words, for $M$ large we would like to say that
$\lim_{N \to \infty}\frac{\log \#\mathcal{M}_N}{2N}$
can be made close to zero. The proof is based on the geometric facts in Lemma~\ref{lem:minkowski}.
Let $\mathcal{N}=(\mathcal{L,L',P,P'}) \in \mathcal{M}_N$ and let $\mathcal{L}=\{l_1,l_2,...,l_m\}$ and $\mathcal{P}=\{p_1,p_2,...,p_n\}$ be as in the construction of marked times. It is clear from the construction that $l_i'<l_{i+1}'$ for $l_i',l_{i+1}' \in \mathcal L'$. Thus from Lemma~\ref{lem:properties} we conclude that $l_i \le l_{i+1}$. Hence we have $\mathcal{L}=\{l_1\le l_2\le ...\le l_m\}$. Similarly, we must have $\mathcal{P}=\{p_1\le p_2 \le...\le p_n\}$. In fact, we have the following.
\begin{lem}[Separation of intervals]
\label{lem:card}
For any $i=1,2,...,m-1$ and for any $j=1,2,...,n-1$ we have
$$l_{i+1}-l_i > \lfloor \log M\rfloor \text{ and } p_{j+1}-p_j > \lfloor \log M\rfloor .$$
Also,
$$l_{i+1}'-l_i' > \lfloor \log M\rfloor \text{ and } p_{j+1}'-p_j' > \lfloor \log M\rfloor .$$
\end{lem}
\begin{proof}
For $1/M$-short vectors in $\mathbb R^{3}$, considering their forward trajectories under the action of the diagonal flow $(e^{t/2},e^{t/2},e^{-t})$, we would like to know the minimum possible amount of time needed for the vector to reach size $\ge 1$. Let $v=(v_1,v_2,v_{3})$ be a vector of size $\le 1/M$ which is of size $\ge 1$ at time $t \ge 0$. We have
$$1 \le v_1^2 e^t+v_2^2 e^t+v_3^2 e^{-2t} \le (v_1^2+v_2^2+v_3^2)e^t \le \frac{e^t}{M^2}.$$
So, we have
$$t \ge \log M^2.$$
Hence, it takes more than $2\lfloor \log M\rfloor$ steps for the vector to reach size $\ge 1$. Similarly, for a vector $v=(v_1,v_2,v_{3})$ of size $\ge 1$, we calculate a lower bound for the time $t \ge 0$ when its trajectory reaches size $\le 1/M$. We have
$$\frac{1}{M^2} \ge v_1^2 e^t+v_2^2 e^t+v_3^2 e^{-2t} \ge (v_1^2+v_2^2+v_3^2)e^{-2t} \ge e^{-2t}.$$
So, we must have $t \ge \log M$ and hence it takes at least $t=\lfloor\log M\rfloor$ steps for the vector to have size $\le 1/M$.
Now, assume that $l_{i+1}-l_i \le \lfloor \log M\rfloor.$ Let $u,v$ be the vectors in $x$ that are responsible for $l_i, l_{i+1}$ respectively. That is, $u, v$ are $1/M$-short at times $l_i,l_{i+1}$ respectively but not before. Then the above arguments imply that
\begin{equation*}
|\operatorname{T}^{l_i}(v)|\le 1 \text{ and } |\operatorname{T}^{l_{i+1}}(u)|\le 1
\end{equation*}
so the plane $P$ containing both $u$ and $v$ is $1/M$-short at times $l_i$ and $l_{i+1}$.
The covolume of $\operatorname{T}^n(P)$ w.r.t. $\operatorname{T}^n(x)$ is $\sqrt{a_1e^{n}+a_2 e^{-n/2} }$ for some nonnegative $a_1$ and $a_2$. In particular, it is a concave function of $n$ and hence the plane $P$ is $1/M$-short in $[l_i, l_{i+1}]$ (and so $l_i, l_{i+1}$ are constructed using the same $V$). From our construction we know that $l_i' < l_{i+1}'$. By Lemma~\ref{lem:minkowski} the same plane $P$ is $1/M^2$-short on $[l_i,l_i'] \cap [l_{i+1}, l_{i+1}']$. If this intersection is non-empty, then $P$ is also $e/M^2$-short at time $l_i'+1$. As $M \ge e$ this shows that it is the unique plane that is used to mark points, say $p_k,p_k'$, after marking $l_i,l_i'$. If on the other hand $l_i' < l_{i+1}$, then we already know that $P$ is also $1/M$-short at time $l_i'+1 \in [l_i,l_{i+1}]$ and get the same conclusion as before. Therefore, $p_k \le l_i \le l_i' \le p_k'$ which is a contradiction to Lemma~\ref{lem:properties}.
The proof of the remaining three cases are very similar to the arguments above and are left to the reader.
\end{proof}
Let us consider the marked points of $\mathcal{L}$ in a subinterval of length $\lfloor \log M\rfloor$ then there could be at most 1 of them. Varying $x$ while restricting ourselves to this interval of length $\lfloor \log M\rfloor$ we see that the number of possibilities to set the marked points in this interval is no more than
$\lfloor \log M\rfloor + 1.$
For $M$ large, say $M \ge e^4$, we have
$$=\lfloor \log M\rfloor +1 \le \lfloor \log M\rfloor^{1.25}.$$
Therefore, there are
\begin{equation*}
\le \lfloor \log M\rfloor^{1.25(\left \lfloor \frac{2N}{\lfloor \log M\rfloor} \right \rfloor + 1)} \ll_M e^{\frac {2.5 N \log \lfloor \log M\rfloor}{\lfloor \log M\rfloor}}
\end{equation*}
possible ways of choosing labeled marked points for $\mathcal{L}$ in $[-N,N]$. The same is true for $\mathcal{L',P,P'}$. Thus we have shown the following.
\begin{lem}[Estimate of $\mathcal{M}_N$] For $M \ge e^4$ we have
\label{lem:marked}
$$\#\mathcal{M}_N \ll_M e^{\frac{10 N\log \lfloor \log M\rfloor}{\lfloor \log M\rfloor}}.$$
\end{lem}
\subsection{Configurations}
Before we end this section, we need to point out another technical detail.
For our purposes, we want to study a partition element in $X_{\le M}$ corresponding to a particular set of
labeled marked times. Since $X_{\le M}$ is compact, it is sufficient for us to study an $\eta$-neighborhood of some $x_0$
in this partition. These are the close-by lattices which have the same set of labeled marked times. We shall see that the fact that $N(x)=N(x_0)$, for $x$ in $x_0 B_\eta^G$, gives rise to restrictions on the position of $x$ with respect to $x_0$ (see \S
\ref{sec:restrictions}). However, just knowing that $\mathcal{N}(x_0)=\mathcal{N}(x)$ will not be sufficient for the later argument.
Hence, we need to calculate how many possible ways (in terms of vectors and planes) we can have the same labeled marked times. For this purpose, we consider the following configurations.
\subsubsection{Vectors}
Let $l$ be a marked time in the first component $\mathcal L$ of the marking $\mathcal{N}(x_0)$. Let $v_0$ be the vector in $x_0$ that is responsible for $l$ in the construction of marked times for $x_0$. Let $y=\operatorname{T}^{l-1}(x)$ be in $\operatorname{T}^{l-1}(x_0) B_\eta^{\operatorname{SL}_3(\mathbb R)}$ with $\mathcal{N}(x)=\mathcal{N}(x_0)$ and $v$ in $x$ that is responsible for $l$ in the construction of marked times for $x$. Let $v' \in x_0$ be such that $\operatorname{T}^{l-1}(v')g=\operatorname{T}^{l-1}(v) $ for some $g \in B_\eta^{\operatorname{SL}_3(\mathbb R)}$ with $y=\operatorname{T}^{l-1}(x_0) g$. We want to know how many choices for $v'$ are realized by the various choices of $x$ as above.
\begin{lem}
\label{lem:vectorconfig}
Let $\mathcal{N}(x_0)$ be given. Also, let $l \in \mathcal L=\mathcal L(x_0)$ and $v_0 \in x_0$ be the vector which is responsible for $l$. There are two possibilities:
\begin{enumerate}
\item[(1)] If $l$ is the end point of a maximal interval $V$ in $V_{x_0}$, then for any $x$ with $\mathcal{N}(x)=\mathcal{N}(x_0)$ and $\operatorname{T}^{l-1}(x)=\operatorname{T}^{l-1}(x_0)g,$ with $g \in B_\eta^G$, the vector $\pm v_0\alpha^{l-1}g\alpha^{-(l-1)}$ is responsible for $l$ in $\mathcal{L}(x)$.
\item[(2)] If not, then there are $p,p'$ in $\mathcal{P}(x_0),\mathcal{P}'(x_0)$ respectively, with $p\le l-1\le p'$, and a set $W \subset x_0$, of size $\ll \min\{e^{p'-l},e^{(l-p)/2}\}$, such that if $x$ is a lattice such that $\mathcal{N}(x)=\mathcal{N}(x_0)$, and $\operatorname{T}^{l-1}(x)=\operatorname{T}^{l-1}(x_0)g$, with $g \in B_\eta^G$, then for some $w\in W$, $w\alpha^{l-1}g\alpha^{-(l-1)}$, is the vector responsible for $l$ in $\mathcal{L}(x)$.
\end{enumerate}
\end{lem}
\begin{proof}
To simplify the notation below we set $w_0=\operatorname{T}^{l-1}(v_0) \in \operatorname{T}^{l-1}(x_0)$, $w=\operatorname{T}^{l-1}(v) \in y$, and $w'=\operatorname{T}^{l-1}(v')=w g \in \operatorname{T}^{l-1}(x_0)$.
We have
$$\frac{1}{M}\le |w| \le \frac{e}{M},$$
and so
\begin{align*}
|w'| & \le |w'-w|+|w|\\
& \le |w| d(g^{-1},1)+ |w|\\
& \le e(1+ \eta)/M.
\end{align*}
Also,
\begin{align*}
|w'| &\ge |w|-|w-w'|\\
&\ge (1-\eta)/M.
\end{align*}
Together
\begin{equation}
\label{eqn:w'}
\frac{1- \eta }{M} \le |w'| \le \frac{e(1+ \eta)}{M}.
\end{equation}
Assume first that $l=a$ is the left end point of the interval $V=[a,b]$ in the construction of marked times. In this case, $w'$ and $w_0$ lie in the same line in $\mathbb R^3$. Otherwise, if they were linearly independent then the plane containing both would be $e^2 (1+\eta)/M^2$-short by Lemma~\ref{lem:minkowski}. For $M \ge 3e^2$ this is a contradiction to the assumption that $l=a$. Since we only consider primitive vectors we only have the choice of $w'=\pm w_0$.
Now, assume that $l$ is not the left end point of the interval $V$. Then, there is a plane $P$ in $x_0$ responsible for $p,p'$ with $p \le l-1 \le p' $ such that
\begin{align*}
&|\operatorname{T}^{p-1}(P)|\ge 1/M \text{ and }|\operatorname{T}^{p'+1}(P)|>1/M\\
&|\operatorname{T}^k(P)| \le 1/M \text{ for } k \in [p,p'].
\end{align*}
Let us calculate how many possibilities there are for $w'\in \operatorname{T}^{l-1}(x_0)$. By \eqref{eqn:w'} $w'$ is in the plane $\operatorname{T}^{l-1}(P)$ of covolume $<1$ w.r.t. $\operatorname{T}^{l-1}(x_0)$ since $\operatorname{T}^{l-1}(x_0)$ is unimodular. Since
$$\frac{1}{M} < |\operatorname{T}^{p'+1}(P)| \text{ and } \frac{1}{M} \le |\operatorname{T}^{p-1}(P)|, $$
we get
$$ \max\left \{\frac{e^{-(p'-l+2)}}{ M},\frac{e^{-(l-p)/2}}{M}\right \}\le |\operatorname{T}^{l-1}(P)|$$
(see \S~\ref{sec:T-actiononplane} for the action of $\operatorname{T}$ on planes). We note that the ball of radius $r$ contains at most $\ll \max\{\frac{r^2}{A},1\}$ primitive vectors of a lattice in $\mathbb R^2$ of covolume $A$. This follows since in the case of $r$ being smaller than the second successive minima we have at most 2 primitive vectors, and if $r$ is bigger, then area considerations give $\ll \frac{r^2}{A}$ many lattice points in the $r$-ball.
We apply this for $A=|\operatorname{T}^{l-1}(P)|\ge \max\ \left \{\frac{e^{-(p'-l+2)}}{ M},\frac{e^{-(l-p)/2}}{M}\right \}$ and $r=\frac{(1+\eta)e}{M}$ where
$$\frac{r^2}{A}= \frac{(1+ \eta)^2e^2/M^2 }{\max\left \{\frac{e^{-(p'-l+2)}}{ M},\frac{e^{-(l-p)/2}}{M}\right \}}\ll \min \{e^{(p'-l)},e^{(l-p)/2} \},$$
which proves the lemma.
\end{proof}
\subsubsection{Planes}
Let $p$ be a marked time in the third component $\mathcal P$ of the marking $\mathcal{N}(x_0)$. Let $P_0$ be a plane in $T^{p-1}(x_0)$ that is responsible for $p$ in the construction of marked times for $x_0$. Let $y=\operatorname{T}^{p-1}(x)$ be in $\operatorname{T}^{p-1}(x_0) B_\eta^{\operatorname{SL}_3(\mathbb R)}$ with $\mathcal{N}(x)=\mathcal{N}(x_0)$ and $P$ in $x$ that is responsible for $p$ in the construction of marked times for $x$. Let $P'$ be a plane that is rational w.r.t. $x_0$ such that $\operatorname{T}^{p-1}(P')g=\operatorname{T}^{p-1}(P) $ for some $g \in B_\eta^{\operatorname{SL}_3(\mathbb R)}$ with $y=\operatorname{T}^{p-1}(x_0) g$. We want to know how many choices for $P'$ are realized by the various choices of $x$ as above. We have two cases.
\begin{lem}
\label{lem:planeconfig}
Let $\mathcal{N}(x_0)$ be given. Also, let $p \in \mathcal P=\mathcal P(x_0)$ and $P_0$ in $x_0$ be the plane which is responsible for $p$. There are two possibilities:
\begin{enumerate}
\item[(1)] If $p$ is the end point of a maximal interval $V$ in $V_{x_0}$, then for any $x$ with $\mathcal{N}(x)=\mathcal{N}(x_0)$ and $\operatorname{T}^{p-1}(x)=\operatorname{T}^{p-1}(x_0)g,$ with $g \in B_\eta^G$, the plane $P_0 \alpha^{p-1}g\alpha^{-(p-1)}$ is responsible for $p$ in $\mathcal{P}(x)$.
\item[(2)] If not, then there are $l,l'$ in $\mathcal{L}(x_0),\mathcal{L}'(x_0)$ respectively, with $l\le p-1\le l'$, and a set of planes $W \subset x_0$, of size $\ll \min\{e^{(l'-p)/2},e^{p-l}\}$, such that if $x$ is a lattice such that $\mathcal{N}(x)=\mathcal{N}(x_0)$, and $\operatorname{T}^{p-1}(x)=\operatorname{T}^{p-1}(x_0)g$, with $g \in B_\eta^G$, then for some $P\in W$, $P \alpha^{p-1}g\alpha^{-(p-1)}$, is the plane responsible for $p$ in $\mathcal{P}(x)$.
\end{enumerate}
\end{lem}
We will not prove the lemma since a similar argument to that giving Lemma~\ref{lem:vectorconfig} gives this lemma.
\section{Main Proposition and Restrictions}
\label{sec:mainprop}
Fix a height $M \geq 1$. Let $N \geq 1$ and consider $\mathcal{N}=\mathcal{N}(x_0)\in \mathcal{M}_N$. Let $V=V_{x_0} \subset [-N,N]$ be as before so that for any $n\in [-N,N]$, $n \in V_{x_0}$ if and only if there is a $1/M$-short plane or a $1/M$-short vector at time $n$. Define the set
$$Z_{\le M}(\mathcal{N}):= \{x \in \operatorname{T}^{N}( X_{\le M}) \,|\, \mathcal{N}(x)=\mathcal{N}\}.$$
Now, we state the main proposition.
\begin{prop}\label{prop:main}
There exists a constant $c_0>0$, independent of $M$, such that the set $Z(\mathcal{N})$ can be covered by
$\ll_M e^{6N-|V|}c_0^{\frac{18N}{\lfloor \log M\rfloor}}$ Bowen $N$-balls.
\end{prop}
In the proof of Theorem \ref{thm:main} we will consider
$$\lim_{N \to \infty}\frac{\log \#Z(\mathcal{N})}{2N}.$$
Thus, in this limit, the term arising from $c_0^{\frac{18N}{\lfloor \log M\rfloor}}$ can be made small for $M$ large since $c_0$ does not depend on $M$. So, our main consideration is the $e^{6N-|V|}$ factor. On the other hand, it is easy to see that the set $Z(\mathcal{N})$ can be covered by $\ll e^{6N}$ many Bowen $N$-balls. But this does not give any meaningful conclusion. Therefore, $e^{-|V|}$ is the factor appearing in Proposition \ref{prop:main} that leads to the conclusion of Theorem \ref{thm:main}.
In proving Proposition \ref{prop:main}, we will make use of the lemmas below which give the restrictions needed in order to get the drop in the number of Bowen $N$-balls to cover the set $Z(\mathcal{N})$.
\subsection{Restrictions of perturbations}
\label{sec:restrictions}
\subsubsection{Perturbations of vectors}
\label{sec:Perturbations of vectors}
Let $v=(v_1,v_2,v_3)$ be a vector in $\mathbb R^{3}$.
\begin{lem}\label{lem:rest}For a vector $v$ of size $\ge 1/M$, if its trajectory under the action of $\operatorname{T}$ stays $1/M$-short in the time
interval $[1,S]$ then we must have $\frac{v_1^2+v_2^2}{v_3^2}< 2e^{-S}.$
\end{lem}
\begin{proof}
We will prove a slightly stronger statement. For this let $\lambda_1 \ge 1$ and $\lambda_2 \le 1$ and assume that
$$\lambda_1(v_1^2+v_2^2+v_{3}^2)\ge \frac{1}{M^2}\geq \lambda_2(v_1^2e^{S}+v_2^2e^{S}+v_{3}^2e^{-2S}).$$
This simplifies to
$$\lambda_1 v_{3}^2>(v_1^2+v_2^2)(\lambda_2 e^{S}-\lambda_1).$$
Assuming $\lambda_1,\lambda_2$ are close to 1, we must have $v_{3}\neq 0$ and
$$\frac{v_1^2+v_2^2}{v_3^2}\le \frac{\lambda_1}{\lambda_2e^S-\lambda_1}.$$
Assuming again that $\lambda_1,\lambda_2$ are close to 1 the last expression is bounded by $2e^{-S}$.
\end{proof}
We would like to get restrictions for the vectors which are close to the vector $v$ and whose trajectories behave as $v$ on the time interval $[0,S]$ . So, let $u=(u_1,u_2,u_3)$ be a vector in $ \mathbb R^{3}$ with $u=v g$ for some $g \in B_\eta^{\operatorname{SL}_3(\mathbb R)}$ such that $|u|\ge 1/M$ and that its forward trajectory stays $1/M$-short in the time interval $[1,S]$.
Let us first assume $g=\left( \begin{array}{ccc} 1& & \\
& 1&\\
-t_1 &-t_2 &1 \end{array} \right) \in
B_{\eta}^{U^+}$ so that
\begin{equation*}
\left( \begin{array}{ccc}
u_1&u_2 &u_{3} \end{array} \right)=
\left( \begin{array}{ccc}
v_1&v_2 &v_3 \end{array}\right)\left( \begin{array}{ccc} 1& & \\
&1&\\
-t_1 &-t_2 &1 \end{array} \right) .
\end{equation*}
From Lemma~\ref{lem:rest} we know that $\frac{u_1^2+u_2^2}{u_3^2}< 2e^{-S}.$ So,
$$\frac{(v_1-v_3t_1)^2+(v_2 -v_3t_2)^2}{v_3^2} < 2e^{-S}.$$
We are interested in possible restrictions on $t_j$'s since they belong to the unstable horospherical subgroup of $\operatorname{SL}_{3}(\mathbb R)$ under conjugation by $\alpha=\text{diag}(e^{1/2},e^{1/2},e^{-1})$. Simplifying the left hand side, we obtain
$$(\frac{v_1}{v_3}-t_1)^2+(\frac{v_2}{v_3}-t_2)^2 < 2e^{-S}.$$
We also know $\frac{v_1^2}{v_3^2}+\frac{v_2^2}{v_3^2}<2e^{-S}.$ Together with the triangular inequality, we get
$$t_1^2+t_2^2<(\sqrt{2 e^{-S}}+\sqrt{2 e^{-S}})^2=8 e^{-S}.$$
In general, we have
$$g=\left( \begin{array}{ccc} 1& & \\
& 1&\\
-t_1 &-t_2 &1 \end{array} \right) \left( \begin{array}{ccc} a_{11}&a_{12} &a_{13} \\
a_{21}&a_{22}&a_{23}\\
0 & 0&a_{33} \end{array} \right) \in B_{\eta}^{\operatorname{SL}_{3}(\mathbb R)}.$$
In this case, we still claim that $$t_1^2+t_2^2< 8 e^{-S}.$$
Let
$$w=\left( \begin{array}{ccc}
w_1&w_2 &w_{3} \end{array} \right)=
\left( \begin{array}{ccc}
v_1&v_2 &v_3 \end{array}\right)\left( \begin{array}{ccc} 1& & \\
&1&\\
-t_1 &-t_2 &1 \end{array} \right) $$
so that
\begin{equation}
\label{eqn:u=vg}
u=v g=w \left( \begin{array}{ccc} a_{11}&a_{12} &a_{13} \\
a_{21}&a_{22}&a_{23}\\
0 & 0&a_{33} \end{array} \right).
\end{equation}
We observe
$$
\operatorname{T}^S(u)=\operatorname{T}^S(w) \left( \begin{array}{ccc} a_{11}&a_{12} &a_{13} e^{-3S/2}\\
a_{21}&a_{22}&a_{23}e^{-3S/2}\\
0&0&a_{33}\end{array}\right) ,
$$
so that $\operatorname{T}^S(u) \in \operatorname{T}^S(w) B_{\eta}^{\operatorname{SL}_3(\mathbb R)}$ and $|\operatorname{T}^S(u)-\operatorname{T}^S(w)|<\eta |\operatorname{T}^S(u)|$ by the discussion in \S~\ref{sec:metrics}.
Hence, $|\operatorname{T}^S(u)|<1/M$ implies
\begin{equation}
\label{eqn:|w alpha^S|}
|\operatorname{T}^S(w)| \le |\operatorname{T}^S(u)|+|\operatorname{T}^S(u)-\operatorname{T}^S(w)|<\frac{1+\eta}{M}.
\end{equation}
On the other hand, since $g\in B_{\eta}^{\operatorname{SL}_3(\mathbb R)}$ we have
\begin{equation}
\label{eqn:|w|}
|w| \ge |u|-|u-w| > \frac{1-\eta}{M}
\end{equation}
Combining \eqref{eqn:|w alpha^S|} and \eqref{eqn:|w|} we get
$$\frac{|w|}{1-\eta}>\frac{1}{M}>\frac{|\operatorname{T}^S(w)|}{1+\eta}.$$
Now, the proof of Lemma~\ref{lem:rest}, for sufficiently small $\eta>0$ implies
$$\frac{w_1^2+w_2^2}{w_3^2}<2 e^{-S}.$$
Hence, we are in the previous case with $u$ replaced by $w$. So, we have $t_1^2+t_2^2<8 e^{-S}$ which proves the claim. We have shown the following.
\begin{lem}
\label{lem:uv}
There exists a sufficiently small $\eta>0$, such that for any $M,S >0$ the following holds. Let $v,u$ be vectors in $\mathbb R^{3}$ with sizes $\ge 1/M$ whose trajectories in $[1,S]$ stay $1/M$-short. Assume that $u=vg$ with $g \in B_\eta^{\operatorname{SL}_3(\mathbb R)}$ and that the notation is as in \eqref{eqn:u=vg}. Then
$$t_1^2+t_2^2\leq 8 e^{-S}.$$
\end{lem}
\begin{lem}
\label{lem:volv}
Let $\eta>0$ be given. For any $S,S' > 0$, let us divide $[-2\eta,2\eta]^2$ into
small squares of side length $\frac{1}{2}\eta e^{-3S'/2}$. Then there exists a constant $c>0$ such that there are $\ll \max \{1,e^{3S'-S}\} $ small squares that intersect with the ball $t_1^2+t_2^2 \leq 8 e^{-S}$ on $[-2\eta,2\eta]^2$.
\end{lem}
\begin{proof}
Note that $t_1^2+t_2^2 \leq 8 e^{-S}$ defines a ball with diameter $2\sqrt{8}e^{-S/2}$. If $\frac{1}{2}\eta e^{-3S'/2} \ge 2 \sqrt{8} e^{-S/2}$ then there are 4 squares that intersects the ball. Otherwise (which makes $3S'-S$ bounded below), there can be at most $ \ll \frac{(e^{-S/2})^2}{(e^{-3S'/2})^2}=e^{3S'-S}$
small squares that intersect with the given ball.
\end{proof}
What Lemma \ref{lem:uv} and Lemma \ref{lem:volv} say is the following:
Consider a neighborhood ${\it O}=x_0 B_{\eta/2}^{U^+}B_{\eta/2}^{U^-C}$ of $x_0$ in $X$ where as before $U^+,U^-,$ and $C$ are the unstable, stable, and centralizer subgroups of $\operatorname{SL}_3(\mathbb R)$ with respect to $\alpha$, respectively. If we partition the square with side length $2\eta$ in $B_{\eta/2}^{U^+}$ into small squares with side lengths $\eta e^{-3S'/2}$, then we have $\ll \lceil \frac{2 \eta}{\eta e^{-3S'/2}} \rceil^2 \ll \lceil e^{3S'/2} \rceil^2$ many elements in this partition. Now, assume that there is a vector $v \in x_0$ with $|v|\ge 1/M$ that stays $1/M$-short in $[1, S]$ and consider the set of lattices $x=x_0 g$ in ${\it O}$ with the property that the vector $w=vg$ in $x$ behaves as $v$ in $[0,S]$. Then the above two lemmas say that this set is contained in $\leq c_0e^{3S'-S}$ many partition elements (small squares). Hence, in the proof of Proposition \ref{prop:main}, instead of $\leq c_0 \lceil e^{3S'/2} \rceil^2$ many Bowen balls we will only consider $\leq c_0 e^{3S'-S}$ many of them and this (together with the case below) will give us the drop in the exponent as appeared in Proposition~\ref{prop:main}.
\subsubsection{Perturbations of planes}
Assume that for a lattice $x \in X$ there is a rational plane $P$ w.r.t. $x$ with
$$|P|\ge 1/M \text{ and } |\operatorname{T}^k(P)| \le 1/M \text{ for } k \in [1,S].$$
Let $u,v$ be generators of $P$ with $|P|=|u \wedge v|$. So we have
$$|u\wedge v|\ge 1/M \ge |\operatorname{T}^{S}(u\wedge v)|.$$
Thus, substituting $a=u_2v_3-u_3v_2,b=u_3v_1-u_1v_3,c=u_1v_2-u_2v_1$ (cf.~\ref{eqn:exterior}) we obtain
$$a^2+b^2+c^2 \ge a^2e^{-S}+b^2e^{-S}+c^2e^{2S},$$
which gives
\begin{equation*}
\frac{c^2}{a^2+b^2}\le \frac{1-e^{-S}}{e^{2S}-1}=e^{-2S}\frac{1-e^{-S}}{1-e^{-2S}}=e^{-2S}\frac{1}{1+e^{-S}}<e^{-2S}.
\end{equation*}
Assume $x'=xg$ for some $g\in B_\eta^{\operatorname{SL}_3(\mathbb R)}$. For now, let us assume that
$$g=\left( \begin{array}{ccc} 1&& \\
&1&\\
t_1&t_2&1 \end{array} \right).$$
Let $u',v' \in x'$ be such that
\begin{align*}
\left( \begin{array}{c} u' \\
v' \end{array} \right)=\left( \begin{array}{ccc} u_1'&u_2'&u_3' \\
v_1'&v_2'&v_3' \end{array} \right)&=\left( \begin{array}{ccc} u_1&u_2&u_3 \\
v_1&v_2&v_3 \end{array} \right)\left( \begin{array}{ccc} 1&& \\
&1&\\
t_1&t_2&1 \end{array} \right)\\
&=\left( \begin{array}{ccc} u_1+t_1u_3&u_2+t_2u_3&u_3 \\
v_1+t_1v_3&v_2+t_2v_3&v_3 \end{array} \right).
\end{align*}
We let $a'=u_2' v_3' -u_3' v_2'=(u_2+t_2 u_3)v_3-u_3 (v_2+t_2 v_3)$ and hence $a'=a$. Similarly, $b'=u_3' v_1'-u_1' v_3'=b$ and let
\begin{multline*}
c'=u_1' v_2' - u_2'v_1'=\\
(u_1+t_1 u_3)(v_2+t_2 v_3)-(u_2+t_2 u_3)(v_1+t_1 v_3)=c-a t_1 -b t_2.
\end{multline*}
Now, assume that
\begin{align*}
|u'\wedge v'|\ge 1/M \text{ and } |\operatorname{T}^k(u'\wedge v')| \le 1/M \text{ for } k \in [1,S]
\end{align*}
which by the above implies
$$\frac{c'^2}{a'^2+b'^2}=\frac{(c-at_1-bt_2)^2}{a^2+b^2}<e^{-2S}.$$
For a general $g \in B_\eta^{\operatorname{SL}_3(\mathbb R)}$ we would like to obtain a similar equation. Let us write $g$ as
\begin{equation}
\label{eqn:g}
g=\left( \begin{array}{ccc} 1&& \\
&1&\\
t_1&t_2&1 \end{array} \right)\left( \begin{array}{ccc} g_{11}&g_{12}&g_{13} \\
g_{21}&g_{22}&g_{23}\\
0&0&g_{33} \end{array} \right).
\end{equation}
Then we have
$$\operatorname{T}^l(x')=\operatorname{T}^l(xg)=\operatorname{T}^l\left(x\left( \begin{array}{ccc} 1&& \\
&1&\\
t_1&t_2&1 \end{array} \right)\right)\left( \begin{array}{ccc} g_{11}&g_{12}&g_{13}e^{-\frac{3}{2}l} \\
g_{21}&g_{22}&g_{23}e^{-\frac{3}{2}l}\\
0&0&g_{33} \end{array} \right).$$
Hence the forward trajectories of $x'$ and $x\left( \begin{array}{ccc} 1&& \\
&1&\\
t_1&t_2&1 \end{array} \right)$ stay $\ll \eta$ close.
Thus, we have $$\frac{(c-at_1-bt_2)^2}{a^2+b^2}\ll e^{-2S}.$$
From the triangular inequality we obtain
$$\frac{(at_1+bt_2)^2}{a^2+b^2}\ll e^{-2S}.$$
Let $C>0$ be the constant that appeared in the last inequality.
\begin{lem}
\label{lem:LL'}
Let $P,P'$ be two dimensional lattices in $\mathbb R^{3}$ of covolume $\ge 1/M$ whose trajectories in $[1,S]$ stay $1/M$-short and assume that $P'=Pg$ for some $g\in B_{\eta}^{\operatorname{SL}_3(\mathbb R)}$, then for some $a,b$ (dependent on $P$) we must have in the notation of \eqref{eqn:g} that
$$\frac{(at_1+bt_2)^2}{a^2+b^2}\leq C e^{-2S}.$$
\end{lem}
We note that the inequality above describes a neighborhood of the line in $\mathbb R^2$ defined by the normal vector $(a,b)$ of width $2\sqrt{C}e^{-s}.$
\begin{lem}
\label{lem:volp}
Consider the set defined by $\frac{(at_1+bt_2)^2}{a^2+b^2}\leq C e^{-2S}$ on
$[-2\eta,2\eta]^2$ and let us divide $[-2\eta,2\eta]^2$ into
small squares of side length $\frac{1}{2}\eta e^{-3S'/2}$. Then there are $\ll \max \{e^{3S'/2},e^{3S'-S}\} $ small squares that intersect with the region $\frac{(at_1+bt_2)^2}{a^2+b^2}\le C e^{-2S}.$
\end{lem}
\begin{proof}
The type of estimate depends on whether the side length $\frac{1}{2}\eta e^{-3S'/2}$ of the squares is smaller or bigger than the width $2\sqrt{C} e^{-S}$ of the neighborhood.
We need to calculate the length and the area of the region $R$ given by $$|at_1+bt_2|\leq \sqrt{C(a^2+b^2)} e^{-S}$$ restricted to $[-2\eta,2\eta]^2.$ As mentioned earlier, the inequality above describes a $\sqrt{C} e^{-S}$-neighborhood of the line $at_1+bt_2=0.$ The length of the segment of this line in $[-2\eta,2\eta]^2$ is at most $4\sqrt{2}\eta$, so that the area of $R$ is $\le 4\sqrt{2C} \eta e^{-S}$.
If $\sqrt{C} e^{-S} \le \frac{1}{2}\eta e^{-3S'/2}$ then there are $\ll \frac{\eta}{\eta e^{-3S'/2}}= e^{3S'/2}$ many intersections. Otherwise, there are at most
$$\ll \frac{\sqrt{C} \eta e^{-S}}{\eta^2 e^{-3S'}} \ll e^{3S'-S}$$
small squares that intersect the region $R$.
\end{proof}
\subsection{Proof of Main Proposition}
\begin{proof}[Proof of Proposition~\ref{prop:main}]
By taking the images under a positive power of $\operatorname{T}$ it suffices to consider forward trajectories and the following reformulated problem:
Let $V \subset [0,N-1]$ and $x_0 \in X_{\le M}$ be such that
$$n \in V \text{ if and only if } \operatorname{T}^n(x_0) \in X_{\ge M}.$$
Also let $\mathcal{N}=\mathcal{N}_{[0,N-1]}(x_0)$ be the marked times for $x_0$ (defined similarly to $\mathcal N_{[-N,N]}$ as in \S~\ref{sec:markedtimes}).
We claim that
$$Z_{\le M}^+=\{x \in X_{\le M} : \mathcal{N}_{[0,N-1]}(x)=\mathcal{N}\}$$
can be covered by $\ll_M e^{3N-|V|} c_0^{\frac{9N}{\lfloor \log M \rfloor}}$ forward Bowen $N$-balls $x B_N^+$ defined by
$$B_N^+=\bigcap_{n=0}^{N-1} \alpha^{n} B_{\eta}^{\operatorname{SL}_{3}(\mathbb R)}\alpha^{-n}.$$
Since $X_{\le M}$ is compact and since we allow the implicit constant above to depend on $M$ it suffices to prove the following:
Let $U^+,U^-$, and $C$ be the subgroups of $G$ introduced in \eqref{eqn:U^+}, \eqref{eqn:U^-}, and \eqref{eqn:C} respectively. Given $x_0 \in X_{\le M}$ and a neighborhood
$${\it O}=x_0 D_{\eta/2}^{U^+}B_{\eta/2}^{U^-C}$$
of $x_0$ where as before $D_{\eta/2}^{U^+}$ is the $\eta/2$-neighborhood of $1$ in $U^+$ (identified with $\mathbb R^2$) w.r.t. maximum norm. Then we claim that the set
$$Z_{\emph O}^+=\{x \in {\it O} : \mathcal{N}_{[0,N-1]}(x)=\mathcal{N} \}$$
can be covered by $\ll e^{3N-|V|} c_0^{\frac{9N}{\lfloor \log M \rfloor}}$ forward Bowen $N$-balls.
If we apply $\operatorname{T}^n$ to ${\it O}$ we get a neighborhood of
$\operatorname{T}^n(x_0)$ for which the $U^+$-part is stretched by the factor $e^{3n/2}$, while the
second part is still in $B_{\eta/2}^{U^-C}$. By breaking the
$U^+$-part into $\lceil e^{3n/2} \rceil^2$ sets of the form $u_i^+
D_{\eta/2}^{U^+}$ for various $u_i^+ \in U^+$ we can write $\operatorname{T}^n(\emph
O)$ as a union of $\lceil e^{3n/2} \rceil^2$ sets of the form
$$\operatorname{T}^n(x_0)u_i^+D_{\eta/2}^{U^+}\alpha^{-n} B_{\eta/2}^{U^-C}\alpha^{n}.$$
Hence we got similar neighborhoods as before. If we take the
pre-image under $\operatorname{T}^n$ of this set, we obtain the set
$$\operatorname{T}^{-n}(\operatorname{T}^n(x_0)u_i^+)\alpha^nD_{\eta/2}^{U^+}\alpha^{-n} B_{\eta/2}^{U^-C}.$$
Notice that $\operatorname{T}^{-n}(\operatorname{T}^n(x_0)u_i^+)\alpha^nD_{\eta/2}^{U^+}\alpha^{-n} B_{\eta/2}^{U^-C}$ is contained in the forward Bowen $n$-ball $\operatorname{T}^{-n}(\operatorname{T}^n(x_0)u_i^+)B_n^+.$ Indeed by assumption on the metrics (see \S~\ref{sec:metric on U^+}) we have $D_\epsilon \subset B_\epsilon$ and so for $0\le k < n$ we have
$$
\alpha^{-k}(\alpha^n D_{\eta/2}^{U^+} \alpha^{-n})\alpha^k
\subset \alpha^{n-k} B_{\eta/2}^{U^+} \alpha^{-(n-k)} \alpha^{-k}B_{\eta/2}^{U^- C} \alpha^k \subset B_{\eta/2}^{U^+}B_{\eta/2}^{U^-C}\subset B_{\eta}^{\operatorname{SL}_3(\mathbb R)}.
$$
We would like to reduce the number of $u_i^+$'s, so that we do not have to
use all $\lceil e^{3n/2} \rceil^2$ forward Bowen $n$-balls to cover the set $Z_{\it O}^{+}$.
We can decompose $V$ into maximal intervals $V_1,V_2,\dots,V_m$ for some $m$. We note here that $m \le |\mathcal{L}|+|\mathcal{P}|$ so that from Lemma~\ref{lem:card} we obtain
\begin{equation}
\label{eqn:boundform}
m \le \frac{2N}{\lfloor \log M \rfloor}+2
\end{equation}
Now, write $[0, N-1]\setminus V = W_1 \cup W_2 \cup ... \cup W_l$ where $W_i$'s
are maximal intervals. A bound similar to \eqref{eqn:boundform} also holds for $l$.
We will consider intervals $V_j$ and $W_i$ in their respective
order in $[0, N-1]$. At each stage we will divide any of the sets
obtained earlier into $\lceil e^{3|V_j|/2}\rceil^2$- or $\lceil e^{3|W_i|/2}\rceil^2$- many
sets, and in the case of $V_j$ show that we do not have to keep
all of them. We inductively prove the following:
For $K \leq N$ such that $[0,K]=V_1 \cup V_2 \cup ... \cup V_n \cup W_1 \cup W_2 \cup ... \cup W_{n'} $ the set $Z_{{\it O}}^+$ can be covered by
$\ll e^{3K}e^{-(|V_1|+...+|V_{n}|)} c_0^{4\frac{|V_1|+...+|V_n|}{\lfloor \log M\rfloor}+4n+n'}$
many pre-images under $\operatorname{T}^K$ of sets of the form
$$\operatorname{T}^K(x_0)u^+D_{\eta/2}^{U^+}\alpha^{-K} B_{\eta/2}^{U^-C}\alpha^K$$
and hence can be covered by $\ll e^{3K}e^{-(|V_1|+...+|V_{n}|)}c_0^{4\frac{|V_1|+...+|V_n|}{\lfloor \log M\rfloor}+4n+n'}$ many forward Bowen K-balls. When $K=N$ we obtain the proposition.
For the inductive step, if the next interval is $W_{n'+1}$ then after dividing the set $\operatorname{T}^K(x_0)u^+B_{\eta/2}^{U^+}\alpha^{-K} B_{\eta/2}^{U^-C}\alpha^K$ into $\lceil e^{3|W_{n'+1}|/2}\rceil^2\le 4e^{3|W_{n'+1}|} $ many sets of the form
$$\operatorname{T}^{K+|W_{n'+1}|}(x_0)u^+B_{\eta/2}^{U^+}\alpha^{-K-|W_{n'+1}|} B_{\eta/2}^{U^-C}(1)\alpha^{K+|W_{n'+1}|}$$
we just consider all of them, and hence have that $Z_{\emph O}^+$ can be covered by
$$\ll e^{3(K+|W_{n'+1}|)}e^{-(|V_1|+...+|V_{n}|)} c_0^{4\frac{|V_1|+...+|V_n|}{\lfloor \log M\rfloor}+4n+n'+1}$$
many forward Bowen $K+|W_{n'+1}|$-balls (assuming $c_0 \ge 4$).
So, assume that the next time interval is
$V_{n+1}=[K+1,K+R]$. Pick one of the sets obtained in an earlier step
and denote it by
$$Y=\operatorname{T}^K(x_0)u^+B_{\eta/2}^{U^+}\alpha^{-K} B_{\eta/2}^{U^-C}\alpha^K.$$
We are interested in lattices $x \text{ in } Y\cap X_{\le M}$ such that
$$\mathcal{N}_{[0,R]}(x)=\mathcal{N}_{[0,R]}(\operatorname{T}^K(x_0))=\{\mathcal{L,L',P,P'}\}.$$
We have
$$\mathcal{L}=\{l_1<l_2<...<l_k\},\,\, \mathcal{L'}=\{l_1'<l_2'<...<l_k'\}$$
and
$$\mathcal{P}=\{p_1<p_2<...<p_{k'}\}, \,\,\mathcal{P'}=\{p_1'<p_2'<...<p_{k'}'\}$$
for some $k,k' \ge 0$. For simplicity of notation assume that $K+1=l_1.$ We note that
$$K+1=l_1< p_1<l_2<p_2<...<\min\{l_k,p_{k'}\}<\max\{l_k,p_{k'}\}.$$
This easily follows from the construction of labeled marked times together with Lemma~\ref{lem:properties}. So, we can divide the interval $V_{n+1}$ into subintervals
$$[l_1,p_1],[p_1,l_2],...,[\min\{l_k,p_{k'}\},\max\{l_k,p_{k'}\}],[\max\{l_k,p_{k'}\},K+R].$$
We consider each of the (overlapping) intervals in their respective order.
Let us define $c_0$ to be the maximum of the implicit constants that appeared in the conclusions of Lemma~\ref{lem:vectorconfig}, Lemma~\ref{lem:planeconfig}, Lemma~\ref{lem:volv}, and Lemma~\ref{lem:volp}.
We would like to apply Lemma~\ref{lem:volv} and Lemma~\ref{lem:volp} to obtain a smaller number of forward Bowen $K+|V_{n+1}|$-balls to cover the set $\operatorname{T}^{-K}(Y)$. Assume for example that there is a vector $v$ in a lattice $x$ that is getting $1/M$-short and staying short in some time interval, also assume that there is a vector $u$ in a lattice $xg$ for some $g\in B_{\eta}^{\operatorname{SL}_3(\mathbb R)}$ which behaves the same as $v$. However, we can apply Lemma~\ref{lem:volv} only if we know that $u=vg.$ Thus, it is necessary to know how many vectors $w'$ there are in $x$ for which $u=w'g$ for some $g$. This is handled by Lemma~\ref{lem:vectorconfig}. Similar situation arises when we want to apply Lemma~\ref{lem:volp}, and this case we first need to use Lemma~\ref{lem:planeconfig}.
Let us start with the interval $[l_1,p_1]$. Let us divide the set $Y \cap X_{\le M}$ into $\lceil e^{3(p_1-l_1)/2}\rceil^2$ small sets by partitioning the set $D_{\eta/2}^{U^+}$ in the definition of $Y$ as we did before. Since $l_1$ is the left end point of $V_{n+1}$ we see that the assumptions of Lemma~\ref{lem:uv} are satisfied in the sense that if there is a lattice $\operatorname{T}^{l_1-1}(x_0)g$ which has the same set of marked times as $\operatorname{T}^{l_1-1}(x_0)$ for some $g\in B_{\eta}^{\operatorname{SL}_3(\mathbb R)},$ then there are unique vectors $v\in \operatorname{T}^{l_1-1}(x_0)$ and $u=v g \in \operatorname{T}^{l_1-1}(x_0)g$ which are of size $\ge 1/M$ and stay $1/M$-short in $[l_1,l_1']$. (cf. Lemma~\ref{lem:vectorconfig}). Now, from Lemma~\ref{lem:uv} and Lemma~\ref{lem:volv} with $S'=p_1-l_1$ and $S=l_1'-l_1$ we see that we only need to consider
\begin{equation}
\label{eqn:N_1}
\le c_0\max\{1,e^{3(p_1-l_1)-(l_1'-l_1)}\}=:N_1
\end{equation}
of these $\lceil e^{3(p_1-l_1)/2}\rceil^2$ sets (see the discussion at the end of \S~\ref{sec:Perturbations of vectors}). Thus, we obtain sets of the form
$$\operatorname{T}^{p_1}(x_0)u^+D_{\eta/2}^{U^+}\alpha^{-p_1} B_{\eta/2}^{U^-C}\alpha^{p_1}.$$
Now, let us consider the next interval $[p_1, l_2]$. Divide the sets obtained earlier into $\lceil e^{3(l_2-p_1)/2}\rceil^2$ subsets for which the $U^+$-component is of the from $u^+D_{e^{-3(l_2-p_1)/2}\eta/2}^{U^+}$. We would like to apply Lemma~\ref{lem:volp}. However, Lemma~\ref{lem:volp} concerns itself with the restrictions on $g$ arising from common behaviors of two planes $P,P'=Pg$ and we only know the common behavior of the lattices. Moreover, if $P_0$ (resp. $P$) is the plane that is rational w.r.t. $\operatorname{T}^{p_1}(x_0)$ (resp. $\operatorname{T}^{p_1}(x_0)g$) which is responsible for the marking of $[p_1,p_1']$ then we do not necessarily know that $P=P_0 g$. On the other hand, we see from Lemma~\ref{lem:planeconfig} that there are $\le c_0 \min\{e^{(l_1'-p_1)/2},e^{p_1-l_1}\}$ choices of planes $P'$ that are rational w.r.t. $\operatorname{T}^{p_1}(x_0)$ for which we could possibly have $P=P'g.$ For each choice we can apply Lemma~\ref{lem:volp} with $S'=l_2-p_1$ and $S=p_1'-p_1$. Thus, for each choice we need to consider only $\le c_0 \max\{e^{3(l_2-p_1)/2},e^{3(l_2-p_1)-(p_1'-p_1)}\}$ of the $\lceil e^{3(l_2-p_1)/2}\rceil^2$ subsets. Thus, in total, we need to consider only
\begin{equation}
\label{eqn:N_2}
\le c_0^2 \min\{e^{(l_1'-p_1)/2},e^{p_1-l_1}\} \max\{e^{3(l_2-p_1)/2},e^{3(l_2-p_1)-(p_1'-p_1)}\}=:N_2
\end{equation}
of these subsets.
Taking the images of these sets under $\operatorname{T}^{l_2-p_1}$ we obtain sets of the form
$$\operatorname{T}^{l_2}(x_0)u^+D_{\eta/2}^{U^+}\alpha^{-l_2} B_{\eta/2}^{U^-C}\alpha^{l_2}.$$
Now, let us consider the interval $[l_2,p_2]$ and let us divide the sets obtained earlier into $\lceil e^{3(p_2-l_2)/2}\rceil^2$ subsets of the form
$$\operatorname{T}^{p_2}(x_0)u^+D_{\eta/2}^{U^+}\alpha^{-p_2} B_{\eta/2}^{U^-C}\alpha^{p_2}.$$
From Lemma~\ref{lem:vectorconfig} we know that there are $\le c_0 \min \{e^{p_1'-l_2},e^{(l_2-p_1)/2}\}$ many configurations and for each of them we can apply Lemma~\ref{lem:volv} with $S'=p_2-l_2$ and $S=l_2'-l_2$. So, for each configuration we need only $\le c_0\max\{1,e^{3(p_2-l_2)-(l_2'-l_2)}\}$ many of the subsets. Thus, we need
\begin{equation}
\label{eqn:N_3}
\le c_0^2 \min \{e^{p_1'-l_2},e^{(l_2-p_1)/2}\} \max\{1,e^{3(p_2-l_2)-(l_2'-l_2)}\}=:N_3
\end{equation}
many of these subsets. Continuing in this way at the end of the inductive step we consider the interval $[\max\{l_k,p_{k'}\},K+R]$. Assume that $\max\{l_k,p_{k'}\}=l_k$ so that $l_k'=K+R$ and $k'=k-1$ (the other case is similar and left to the reader). We have the sets of the form
$$\operatorname{T}^{l_k}(x_0)u^+D_{\eta/2}^{U^+}\alpha^{-l_k} B_{\eta/2}^{U^-C}\alpha^{l_k}$$
that are obtained in the previous step. Let us divide them into $\lceil e^{3(l_k'-l_k)/2} \rceil^2$ small sets. By Lemma~\ref{lem:vectorconfig} we have $\le c_0 \min \{e^{p_{k-1}'-l_k},e^{(l_k-p_{k-1})/2}\}$ configurations and for each we apply Lemma~\ref{lem:volv} with $S'=S=l_k'-l_k$. Hence, we need to consider only
\begin{equation}
\label{eqn:N_{2k-1}}
\le c_0^2\min \{e^{p_{k-1}'-l_k},e^{(l_k-p_{k-1})/2}\} e^{3(l_k'-l_l)-(l_k'-l_k)}=:N_{2k-1}
\end{equation}
of them. Thus, in the inductive step we divided the sets obtained earlier into
$$\lceil e^{3(p_1-l_1)/2}\rceil^2 \lceil e^{3(l_2-p_1)/2}\rceil^2 \cdots \lceil e^{3(l_k'-l_k)/2}\rceil^2$$
many parts and deduced that we only need to
take
\begin{equation}
\le N_1 N_2 N_3 \cdots N_{2k-1}
\end{equation}
many of them where each set is of the form
$$\operatorname{T}^{K+R}(x_0)u^+D_{\eta/2}^{U^+}\alpha^{-K-R} B_{\eta/2}^{U^-C}\alpha^{K+R}.$$
On the other hand, let us multiply the $\max$ term of \eqref{eqn:N_1} with the $\min$ term of \eqref{eqn:N_2} to get
$$\max\{1,e^{3(p_1-l_1)-(l_1'-l_1)}\} \min\{e^{(l_1'-p_1)/2},e^{p_1-l_1}\}.$$
If $\max\{1,e^{3(p_1-l_1)-(l_1'-l_1)}\}=e^{3(p_1-l_1)-(l_1'-l_1)}$ then clearly the multiplication above is $\le e^{3(p_1-l_1)-(l_1'-l_1)}e^{(l_1'-p_1)/2}\le e^{2(p_1-l_1)}. $ Otherwise, it is $\le e^{p_1-l_1}$. Thus, in either case we have
$$\le e^{2(p_1-l_1)}.$$
Similarly, let us multiply the $\max$ term of \eqref{eqn:N_2} with the $\min$ term of \eqref{eqn:N_3}
$$\max\{e^{3(l_2-p_1)/2},e^{3(l_2-p_1)-(p_1'-p_1)} \}\min \{e^{p_1'-l_2},e^{(l_2-p_1)/2}\}. $$
If $\max\{e^{3(l_2-p_1)/2},e^{3(l_2-p_1)-(p_1'-p_1)}\}=e^{3(l_2-p_1)-(p_1'-p_1)}$ then the above multiplication is $\le e^{3(l_2-p_1)-(p_1'-p_1)} e^{p_1'-l_2}=e^{2(l_2-p_1)}.$ Otherwise, it is
$$\le e^{3(l_2-p_1)/2} e^{(l_2-p_1)/2} =e^{2(l_2-p_1)}.$$
Hence, in either case we have that the product is $\le e^{2(l_2-p_1)}.$
We continue in this way until we have considered all $\max$ and $\min$ terms. Thus, we obtain that
\begin{align*}
N_1 N_2 N_3 \cdots N_{2k-1} &\le c_0^{4k} e^{2(p_1-l_1)}e^{2(l_2-p_1)}\cdots e^{2(p_{k-1}-l_{k-1})}e^{2(l_k'-l_k)}\\
&=c_0^{4k} e^{2(p_1-l_1)+2(l_2-p_1)+\cdots +2(l_k'-l_k)}\\
&=c_0^{4k} e^{2|V_{n+1}|}
\end{align*}
We know that $k$ is the number of elements of $\mathcal{L}$ restricted to the interval $V_{n+1}$. From Lemma~\ref{lem:card} we have that $k \leq \frac{|V_{n+1}|}{\lfloor \log M\rfloor}+1$. Therefore, for the inductive step $K+|V_{n+1}|$, we get that the set $Z_{\emph O}^+(V)$ can be covered by
\begin{align*}
&\ll e^{3K}e^{-(|V_1|+...+|V_{n}|)}c_0^{4\frac{|V_1|+...+|V_{n}|}{\lfloor \log M\rfloor}+4n+n'}e^{2|V_{n+1}|}c_0^{4\frac{|V_{n+1}|}{\lfloor \log M\rfloor}+4}\\
&=e^{3(K+|V_{n+1})|}e^{-(|V_1|+...+|V_{n+1}|)}c_0^{4\frac{|V_1|+...+|V_{n+1}|}{\lfloor \log M\rfloor}+4(n+1)+n'}
\end{align*}
many forward Bowen $K+|V_{n+1}|$-balls.
Hence, letting $K=N$ together with \eqref{eqn:boundform} we see that the set $Z_{\emph O}^+(V)$ can be covered by
$\leq e^{3N-|V|}c_0^{\frac{4|V|}{\lfloor \log M\rfloor}+\frac{5N}{\lfloor \log M\rfloor}}\leq e^{3N-|V|}c_0^{\frac{9N}{\lfloor \log M\rfloor}}$ many forward Bowen N-balls.
\end{proof}
\section{Proof of Theorem~\ref{thm:main}}
\label{sec:mainthm}
Our main tool in proving Theorem~\ref{thm:main} will be Lemma~\ref{entropy}.
\begin{proof}[Proof of the Theorem~\ref{thm:main}] Note first that it is sufficient to
consider ergodic measures. For if $\mu$ is not ergodic, we can
write $\mu$ as an integral of its ergodic components $\mu=\int
\mu_t d\tau(t)$ for some probability space $(E,\tau)$, see for example \cite[Theorem~6.2]{EinWar}. Therefore,
we have $\mu(X_{\ge M})=\int \mu_t(X_{\ge M})d\tau(t)$, but also
$h_{\mu}(\operatorname{T})=\int h_{\mu_t}(\operatorname{T}) d\tau(t)$, see for example \cite[Theorem~8.4]{WB}, so that the
desired estimate follows from the ergodic case.
Suppose that $\mu$ is ergodic. We would like to apply
Lemma~\ref{entropy}. For this we need to find an upper bound for
covering $\mu$-most of the space $X$ by Bowen $N$-balls. So, let $M\ge 100$ be such that $\mu(X_{\le M})>0$.
Thus, ergodicity of $\mu$ implies that $\mu(\bigcup_{k=0}^{\infty} \operatorname{T}^{-k}
X_{\le M})=1$. Hence, for every $\epsilon>0$ there is a constant $K \geq 1$
such that $Y=\displaystyle\bigcup_{k=0}^{K-1} \operatorname{T}^{-k} (X_{\le M})$
satisfies
$\mu(Y)>1-\epsilon.$\\
Moreover, the pointwise ergodic theorem implies
\[
\frac{1}{2N-1}\sum_{n=-N+1}^{N-1}1_{X_{\geq M}}(\operatorname{T}^n(x))\to \mu(X_{\geq M})
\]
as $N \to \infty$ for a.e. $x \in X$. Thus, given $\epsilon
>0$, there exists $N_0$ such that for $N>N_0$ the average on the
left will be bigger than $\mu(X_{\geq M})-\epsilon$ for any $x \in
X_1$ for some $X_1 \subset X$ with measure $\mu(X_1)> 1-\epsilon$.
Clearly, for any $N$ we have $\mu(Z)>1-2\epsilon$ where
\[Z=X_1\cap\operatorname{T}^NY.\]
Now, we would like to find an upper bound for the number of Bowen
$N$-balls needed to cover the set $Z$. Here $N \rightarrow \infty$
while $\epsilon$ and hence $K$ are fixed. Since
$Y=\displaystyle\bigcup_{k=0}^{K-1} \operatorname{T}^{-k} X_{\le M}$, we can
decompose $Z$ into $K$ sets of the form
\[Z'=X_1\cap\operatorname{T}^{N-k}X_{\le M}\]
but since $K$ is fixed, it suffices to find an upper bound for the
number of Bowen $N$-balls needed to cover one of these. Consider the set
$Z'$ which we split into the sets
$Z(\mathcal{N})$ as in Proposition~\ref{prop:main} (applied to the parameter $N-k$ instead of $N$) for the
various subsets $\mathcal{N} \in \mathcal{M}_{N-k}$. By
Lemma~\ref{lem:marked} we know that we need $\ll_M e^{\frac{10N\log \lfloor \log M\rfloor}{\lfloor \log M\rfloor}}$ many of these under the assumption that $M\ge 100>e^4$. Moreover, by our assumption on $X_1$ we only need to look at sets $V_x \subset
[-N+k+1,N-k-1]$ with $|V_x| \geq (\mu(X_{\geq M })-2\epsilon)(2N-1)$ (where we assume that $N$ is sufficiently large).
On the other hand, Proposition~\ref{prop:main}
gives that each of those sets $Z(\mathcal{N})$ can be covered by
$\leq e^{6N-|V_x|}c_0^{\frac{18N}{\lfloor \log M\rfloor}}$ Bowen $(N-k)$-balls for some constant $c_0>0$ that does not depend on $M$. It is easy to see from the definition that a Bowen $(N-k)$-ball can be covered by at most $c_1^k$ many Bowen $N$-balls. Together we see that $Z$ can be covered by
$$\ll_{M,K} e^{\frac{10N\log \lfloor \log M\rfloor}{\lfloor \log M\rfloor}}c_0^{\frac{18N}{\lfloor \log M\rfloor}}e^{6N-(\mu(X_{\geq M })-2\epsilon)(2N-1)}$$ many Bowen
$N$-balls. Applying Lemma~\ref{entropy} we arrive at
\begin{eqnarray*}
h_{\mu}(\operatorname{T})&\leq
&\lim_{\epsilon \to 0}\liminf_{N \rightarrow
\infty}\frac{\log BC(N,\epsilon)}{2N} \\
&\leq &\lim_{\epsilon \to 0} (3-(\mu(X_{\geq M })-2\epsilon)+O(\frac{\log \log M}{ \log M})\\
&\le & 3-\mu(X_{\geq M })+O(\frac{\log \log M}{ \log M})
\end{eqnarray*}
which completes the proof for any sufficiently large $M$ with $\mu(X_{\le M})>0$.
However, we claim that the same conclusion holds for any sufficiently large $M$ independent of $\mu$ (which e.g.\ is crucial for proving Corollary~\ref{cor:lim}).
If $\mu(X_{\leq100})>0$ then the claim is true by the above discussion. So, assume that $\mu(X_{\leq 100})=0$ and let
$$M_\mu=\inf\{M >100 : \mu(X_{\leq M})>0\}.$$
Since $\mu(X_{\le M})>0$ for any $M > M_\mu \ge 100$ we have by the discussion above
\begin{equation}
\label{eqn:h}
h_{\mu}(\operatorname{T}) \leq 3-\mu(X_{\ge M })+O(\frac{\log \log M}{\log M}).
\end{equation}
If $\mu(X_{\le M_{\mu}})>0$ then $\eqref{eqn:h}$ also holds for $M=M_{\mu}$ by the above. If on the other hand, $\mu(X_{\le M_{\mu}})=0$ then $\lim_{n \to \infty}\mu(X_{\ge M_{\mu}+\frac{1}{n}})=\mu(X_{>M_{\mu}})=\mu(X_{\ge M_{\mu}})$
and \eqref{eqn:h} for $M=M_{\mu}$ follows from \eqref{eqn:h} for $M=M_{\mu}+\frac{1}{n}.$ Since $\mu(X_{\ge M_{\mu}})=1$ this simplifies to
$$h_{\mu}(\operatorname{T})\le 2+O(\frac{\log \log M}{ \log M}).$$
Since $\frac{\log \log M}{ \log M}$ is a decreasing function for $M\ge 100$ and $\mu(X_{\ge M})=1$ for $M \le M_{\mu}$ we obtain that \eqref{eqn:h} trivially also holds for any $M\in [100,M_{\mu}).$
\end{proof}
\section{Limits of measures with high dimension}
\label{sec:locdim}
In this section we prove Theorem~\ref{thm:dim} and Corollary~\ref{cor:sing2}. Our main tool is a version of Proposition~\ref{prop:main}. Let $N,M>0$ be given. For any $x$ we define $V_{x}\in [0,N-1]$ to be the set of times $n \in [0,N-1]$ for which $\operatorname{T}^n(x) \in X_{\ge M}.$
Now, Proposition~\ref{prop:main} can be rephrased as follows.
\begin{prop}
\label{prop:main'}
For a fixed set $\mathcal{N}=\mathcal{N}_{[0,N-1]}(x_0)$ of labeled marked times in $[0,N-1]$ we have that the set
$$Z^+(\mathcal{N})=\{x \in X_{\le M} : \mathcal{N}_{[0,N-1]}(x)=\mathcal{N}_{[0,N-1]} \}$$
can be covered by $\ll_M e^{3N-|V_{x_0}|}c_0^{\frac{9N}{\lfloor \log M \rfloor}}$ many sets of the form
$$\operatorname{T}^{-N}(\operatorname{T}^N(x)u^+)D_{\frac{\eta}{2}e^{-3N/2}}^{U^+} B_{\frac{\eta}{2}}^{U^-C}.$$
\end{prop}
\begin{proof}
In the proof of Proposition~\ref{prop:main} we inductively proved that the set
$$Z_{\it O}^+=\{x \in {\it O} : \mathcal{N}_{[0,N-1]}(x)=\mathcal{N}_{[0,N-1]} \}$$ can be covered by
$e^{3N-|V_{x_0}|} c_0^{\frac{9N}{\lfloor \log M \rfloor}}$ many pre-images under $\operatorname{T}^N$ of sets of the form
$$\operatorname{T}^N(x_0)u^+D_{\eta/2}^{U^+}\alpha^{-N} B_{\eta/2}^{U^-C}\alpha^N.$$
So, $Z_{\it O}^+$ can be covered by the sets of the form
$$\operatorname{T}^{-N}(\operatorname{T}^N(x_0)u^+)\alpha^N D_{\eta/2}^{U^+}\alpha^{-N} B_{\eta/2}^{U^-C}.$$
This completes the proof since we have $\alpha^N D_{\eta/2}^{U^+}\alpha^{-N}=D_{\frac{\eta}{2}e^{-3N/2}}^{U^+}$ and since $X_{\le M}$ is compact.
\end{proof}
In the following let $\nu$ be a probability measure on $X$ which has a dimension at least $d$ in the unstable direction (see \eqref{eqn:unstable}). We wish to prove Theorem~\ref{thm:dim}.
For any $\kappa>0$ small we are interested in the upper estimate for
$$\nu(\{x \in X_{<M} : |V_x|>\kappa N\}).$$
Proposition~\ref{prop:main'} together with Lemma~\ref{lem:marked} gives the following.
\begin{lem}
\label{lem:kappa}
For any $N>0$ large, we have
$$\nu(\{x \in X_{\le M} : |V_x|>\kappa N\}) \ll_{M,\delta} e^{\frac{6-2\kappa-3d+3\delta}{2}N+ \frac{9N\log (c_0\log M)}{ \log M }}.$$
\end{lem}
\begin{proof}
From Lemma~\ref{lem:marked} we know that the set $X_{<M}$ can be decomposed into
$$\ll_M e^{\frac{5 N\log \lfloor \log M\rfloor}{\lfloor \log M\rfloor}}$$
many sets of the form $Z^+(\mathcal{N})$. We are only interested in those sets of marked times $\mathcal{N}_{[0,N-1]}(x)$ for which $|V_x|>\kappa N$. On the other hand, from Proposition~\ref{prop:main'} we know that such sets can be covered by
$e^{(3-\kappa)N} c_0^{\frac{9N}{\lfloor \log M \rfloor}}$
many sets of the form
$$\operatorname{T}^{-N}(\operatorname{T}^N(x)u^+)D_{\frac{\eta}{2}e^{-3N/2}}^{U^+} B_{\frac{\eta}{2}}^{U^-C}.$$
However, from the assumption on dimension of the measure $\nu$ we have
$$\nu(\operatorname{T}^{-N}(\operatorname{T}^N(x)u^+)D_{\frac{\eta}{2}e^{-3N/2}}^{U^+} B_{\frac{\eta}{2}}^{U^-C}) \ll_\delta (\frac{\eta}{2}e^{-3N/2})^{d-\delta}$$
once $N$ is sufficiently large. Thus,
$$ \nu(\{x \in X_{\le M} : |V_x|>\kappa N\}) \ll_{M,\delta} e^{\frac{5 N\log \lfloor \log M\rfloor}{\lfloor \log M\rfloor}}e^{(3-\kappa)N} c_0^{\frac{9N}{\lfloor \log M \rfloor}}(\frac{\eta}{2}e^{-3N/2})^{d-\delta}.$$
This simplifies to
$$\nu(\{x \in X_{\le M} : |V_x|>\kappa N\}) \ll_{M,\delta} e^{\frac{6-2\kappa-3d+3\delta}{2}N+ \frac{9N\log (c_0\log M)}{ \log M }}.$$
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:dim}]
Note that for $d \le 4/3$ the conclusion in the theorem is trivial. Hence we assume that $d >4/3$. In order to prove Theorem~\ref{thm:dim} we need to estimate an upper bound for $\mu_N(X_{\ge M})$ for $M, N$ large. Let us recall that
$$\mu_N=\frac{1}{N}\sum_{i=0}^{N-1}\operatorname{T}^i_*\nu.$$
Hence,
\begin{align*}
\mu_N(X_{\geq M})&=\frac{1}{N}\sum_{n=0}^{N-1}\nu(\operatorname{T}^{-n}(X_{\geq M}))\\
&=\frac{1}{N}\sum_{n=0}^{N-1}\nu(X_{\le M} \cap \operatorname{T}^{-n}(X_{\geq M}))+\frac{1}{N}\sum_{n=0}^{N-1}\nu(X_{> M} \cap \operatorname{T}^{-n}(X_{\geq M})).
\end{align*}
However, we have $\nu(X_{> M})<\epsilon(M)$ where $\epsilon(M) \to 0$ as $M \to \infty$.
Hence,
\begin{equation}
\label{eqn:muN}
\mu_N(X_{\geq M}) \leq \epsilon(M) + \frac{1}{N}\sum_{n=0}^{N-1}\nu(X_{\le M} \cap \operatorname{T}^{-n}(X_{\geq M})).
\end{equation}
Thus, all we need to estimate is $\frac{1}{N}\sum_{n=0}^{N-1}\nu(X_{\le M} \cap \operatorname{T}^{-n}(X_{\geq M}))$.
Now, recalling that $V_x=\{n \in [0,N-1] : \operatorname{T}^n(x) \in X_{\ge M}\}$ we note that
\begin{multline*}
\frac{1}{N}\sum_{n=0}^{N-1}\nu(X_{\le M} \cap \operatorname{T}^{-n}(X_{\geq M}))\\
=\frac{1}{N}\sum_{n=0}^{N-1} \sum_{W \subset [0,N]} \nu(\{x \in X_{\le M}: V_x=W\}\cap \operatorname{T}^{-n}(X_{\ge M})),
\end{multline*}
where $\nu(\{x \in X_{\le M}: V_x=W\}\cap \operatorname{T}^{-n}(X_{\ge M}))$ is either 0 or $\nu(\{x \in X_{\le M}: V_x=W\})$. Therefore, we switch the order of summation and get
\begin{align*}
&=\frac{1}{N}\sum_{W \subset [0,N-1]}|W|\nu(\{x \in X_{\le M}: V_x=W\})\\
&=\frac{1}{N}\sum_{i=1}^{N}i\nu(\{x \in X_{\le M}: |V_x|=i\})\\
&= \frac{1}{N}\sum_{i=1}^{\lfloor \kappa N \rfloor}i\nu(\{x \in X_{\le M}: |V_x|=i\})+\frac{1}{N}\sum_{i=\lceil \kappa N \rceil }^{N}i \nu(\{x \in X_{\le M}: |V_x|=i\})\\
&\le \frac{1}{N}\lfloor \kappa N \rfloor\nu(X_{\le M})+\frac{1}{N}N\nu(\{x \in X_{\le M}: |V_x|>\kappa N\})
\end{align*}
Let $K(M,\delta)>0$ be the implicit constant that appeared in Lemma~\ref{lem:kappa}. Then using Lemma~\ref{lem:kappa} we obtain
$$\frac{1}{N}\sum_{n=0}^{N-1}\nu(X_{<M} \cap \operatorname{T}^{-n}(X_{\geq M}))\le \kappa+K(M,\delta)e^{\frac{6-2\kappa-3d+3\delta}{2}N+ \frac{9N\log (c_0\log M)}{ \log M }}.$$
Thus, together with \eqref{eqn:muN} we get
\begin{equation}
\mu_N(X_{\ge M})\le \epsilon(M)+\kappa+K(M,\delta)e^{(\frac{6-2\kappa-3d+3\delta}{2}+ \frac{9\log (c_0\log M)}{ \log M })N}.
\end{equation}
By assumption we have $d>\frac{4}{3}$. Let $\kappa > \frac{6-3d}{2}$ (which we will later choose to approach $\frac{6-3d}{2}$). Now, we let $\delta>0$ to be small enough so that
$$6-2\kappa-3d+3\delta<0.$$
Let $\epsilon>0$ be given. For $M$ sufficiently large we can make sure that $\epsilon(M)<\epsilon/2$ and that $\frac{6-2\kappa-3d+3\delta}{2}+ \frac{9\log (c_0\log M)}{ \log M }<0.$
Thus,
$$K(M,\delta)e^{(\frac{6-2\kappa-3d+3\delta}{2}+ \frac{9\log (c_0\log M)}{ \log M })N} \to 0$$
as $N\to \infty$. So, we conclude that for $N$ large enough we get
$$\mu_N(X_{\ge M})\le \kappa+\epsilon$$
which gives in the limit that $\mu(X)>1- \kappa.$ This is true for any $\kappa >\frac{6-3d}{2}$. Thus,
$$\mu(X)\ge 1-\frac{6-3d}{2}=\frac{3d-4}{2}.$$
\end{proof}
Next, we prove Corollary~\ref{cor:sing2}. We need the following Corollary 4.12 from \cite{Fal}.
\begin{thm}
\label{thm:Fal}
Let $F$ be a Borel subset of $\mathbb R^n$ with $0<\mathcal{H}^s(F) \le \infty.$ Then there is a compact set $E \subset F$ such that $0<\mathcal{H}^s(E)<\infty$ and a constant $b$ such that
$$\mathcal{H}^s(E\cap B_\delta({\bf r}))\le b \delta^s$$
for all ${\bf r} \in \mathbb R^n$ and $\delta>0.$
\end{thm}
\begin{proof}[Proof of Corollary~\ref{cor:sing2}]
As any divergent point is also divergent on average, we get from \cite[Corollary 1.2]{Che} that the set of points $F_0\subset X$ that are divergent on average has at least dimension $\frac43+6$.
So assume now that the Hausdorff dimension of $F_0$ is greater than $\frac{4}{3}+6.$ Then, by the behavior of Hausdorff dimension under countable unions, there is some subset $F\subset F_0$ with compact closure and small diameter for which the Hausdorff dimension is also bigger than $\frac43+6$. Here we may assume that $F=F_0\cap (x_0D_\eta B_\eta^{U^-C})$ and that $x_0D_\eta B_\eta^{U^-C}$ is the injective image of the corresponding set in $\operatorname{SL}_3(\mathbb R)$. It then follows that $F=x_0 D' B_\eta^{U^-C}$
and that $D'$ has Hausdorff dimension bigger than $\frac{4}3$. Thus, for sufficiently small $\epsilon>0$ we have that $\mathcal{H}^{\frac{4}{3}+\epsilon}(D')=\infty.$ We may identify $U^+$ with $\mathbb R^2$ and apply Theorem~\ref{thm:Fal}. Therefore, there exists a compact set $E \subset D'$ such that $0<\mathcal{H}^{\frac{4}{3}+\epsilon}(E)<\infty$ and a constant $b$ such that
$$\mathcal{H}^{\frac{4}{3}+\epsilon}(E\cap B_\delta({\bf r}))\le b \delta^{\frac{4}{3}+\epsilon}$$
for all ${\bf r} \in \mathbb R^2$ and $\delta>0.$ We define $\nu_0=\frac{1}{\mathcal{H}^{\frac{4}{3}+\epsilon}(E)}\mathcal{H}^{\frac{4}{3}+\epsilon} _{|_E}$ so that $\nu_0(U^+)=1$.
Let $\tau$ be the map from $U^+$ to $X$ defined by $\tau(u)=x_0u.$ Now, we let $\nu=\tau_*\nu_0$ to be the push-forward of the measure $\nu_0$ under the map $\tau$. It follows that for any $\delta>0$ and for any $x\in X$ we have
$$\nu(xB_{\delta}^{U^+}B_\eta^{U^-C})\ll \delta^{\frac{4}{3}+\epsilon}. $$
Now, if we define $\mu_N$ as before then Theorem~\ref{thm:dim} implies that
the limit measure $\mu$ has at least $\frac{3}{2}(\frac{4}{3}+\epsilon-\frac{4}{3})\frac{3\epsilon}{2}>0$ mass left. However,
the assumption on $F_0$ and dominated convergence applied to
\[
\mu_N(X_{\le M})=\int\frac1N\sum_{n=0}^{N-1}\chi_{T^{-n}X_{\le M}} d\nu
\]
implies that $\mu_N(X_{\le M})\to 0$ as $N\to\infty$ for any fixed $M$. This gives a contradiction and the corollary.
\end{proof}
\bibliographystyle{plain}
|
1,116,691,497,750 | arxiv | \section{Introduction} \label{sec0}
\begin{convention*}\label{conv110211a}
In this paper, $R$ is a commutative noetherian
ring.
In this section, assume that $(R,\ideal{m},k)$ is local.
\end{convention*}
A ``semidualizing'' $R$-complex is a homologically finite $R$-complex $X$ such that
the natural morphism $R\to\Rhom XX$ is an isomorphism in the derived category $\mathcal{D}(R)$.
In particular, if $X$ is a finitely generated $R$-module, then it is semidualizing if $\Hom XX\cong R$
and $\Ext{\geq 1}XX=0$. These notions were introduced by Foxby~\cite{foxby:gmarm} and Christensen~\cite{christensen:scatac}
and, as special cases, recover Grothendieck's dualizing complexes. For some indications of their usefulness,
see, e.g., \cite{avramov:rhafgd, beck:sgidgca, sather:cidfc, sather:bnsc}.
In~\cite{nasseh:lrfsdc} we show that $R$ has only finitely many semidualizing complexes up to shift-isomorphism,
answering a question of Vasconcelos~\cite{vasconcelos:dtmc}. The next natural question is:
how many semidualizing complexes does a given ring have up to shift-isomorphism?
Progress on this question is limited, see, e.g., \cite{celikbas:ncdfcmlr, sather:divisor, sather:lbnsc}.
As further progress, the main result of this paper is the following, which we prove in~\ref{proof131230a}.
In the statement, the \emph{embedding codepth} of $R$ is
$\operatorname{ecodepth}(R)=\operatorname{edim}(R)-\depth(R)$.
\begin{intthm}\label{cor130816a}
Let $R$ be a local ring that is Golod or such that $\operatorname{ecodepth}(R)\leq 3$.
Then $R$ has at most two distinct semidualizing complexes up to shift isomorphism,
namely, $R$ itself and a dualizing $R$-complex if one exists.
\end{intthm}
The proof uses differential graded algebra techniques, as pioneered by Avramov and his collaborators.
It is worth noting that
one can prove the Golod case of this result using a result of Jorgensen~\cite[Theorem 3.1]{jorgensen:gabf}.
However, our approach is different and addresses both cases simultaneously.
\subsection*{Summary}
Section~\ref{sec131230a} consists of foundational material about semidualizing objects in the DG setting.
Section~\ref{sec130815b} contains versions of results from~\cite{nasseh:oeire} for trivial extensions of DG algebras,
including Theorem~\ref{thm130816b} which is key for our proof of Theorem~\ref{cor130816a} and may be of independent interest.
Section~\ref{sec131230b} is mainly concerned with the proof of Theorem~\ref{cor130816a}.
\subsection*{Background}
We assume that the reader is familiar with many notions from the world of
differential graded (DG) algebra. References on the subject include~\cite{apassov:hddgr, avramov:ifr, avramov:lgh,
avramov:dgha, beck:sgidgca, felix:rht, frankild:ddgmgdga, nasseh:lrfsdc, nasseh:ldgm}.
We most closely follow the conventions from~\cite{nasseh:lrfsdc}.
For the reader's convenience, we specify some terminology and notation.
Complexes of $R$-modules are indexed homologically. This includes DG algebras and DG modules.
Also, our DG algebras are all non-negatively graded.
Given a DG $R$-algebra $A$, the underlying graded algebra for $A$ is denoted $A^\natural$,
and $A$ is \emph{homologically degree-wise noetherian}
if $\operatorname{H}_0(A)$ is noetherian and the $\operatorname{H}_0(A)$-module $\operatorname{H}_i(A)$ is finitely generated for all $i\geq 0$.
When $(R,\ideal{m})$ is local,
we say that $A$ is \emph{local} if it is homologically degree-wise noetherian and
the ring $\operatorname{H}_0(A)$ is a local $R$-algebra; in this case, the ``augmentation ideal'' of $A$ is denoted $\ideal{m}_A$.
The derived category of DG $A$-modules is denoted $\mathcal{D}(A)$,
and $\Lotimes[A]--$ and $\Rhom[A]--$ are the derived functors of $\Otimes[A]--$ and $\Hom[A]--$.
Given an integer $i$ and DG $A$-modules $X$ and $Y$,
we set $\Tor[A]iXY:=\operatorname{H}_i(\Lotimes[A]XY)$, and we let $\mathsf{\Sigma}^iX$ denote the $i$th \emph{shift}
(or \emph{suspension}) of $X$.
Isomorphisms in $\mathcal{D}(A)$ are identified by the symbol $\simeq$, and the DG modules $X$ and $Y$ are ``shift-isomorphic'',
denoted $X\sim Y$, if $X\simeq\mathsf{\Sigma}^i Y$ for some $i\in\mathbb{Z}$.
We write
$\id_A(X)<\infty$ when $X$ has a bounded semi-injective resolution;
and $\pd_A(X)<\infty$ means that $X$ has a bounded semi-free resolution.
When $A$ is local and $X$ is homologically finite, the \emph{Poincar\'e series} of $X$ is
$P^A_X(t):=\sum_{i\in\mathbb{Z}}\operatorname{len}_k(\Tor[A]ikX)t^i$
where $k=A/\ideal{m}_A$.
\section{Semidualizing DG Modules}\label{sec131230a}
\begin{convention*}
In this section, $A$ is a homologically degree-wise noetherian DG $R$-algebra.
\end{convention*}
This section contains some useful DG variations of standard results for semidualizing complexes over rings.
We begin with the following definitions from~\cite{christensen:dvke,frankild:ddgmgdga}.
\begin{defn}\label{defn131230a}
A \emph{semidualizing} DG $A$-module is a homologically finite DG $A$-module $X$ such that
the natural homothety morphism $R\xrightarrow{\chi^X_A}\Rhom[A]XX$ is an isomorphism in $\mathcal{D}(A)$.
When $A$ is a ring concentrated in degree 0, these are ``semidualizing $A$-complexes''.
The set of shift-isomorphism classes (in $\mathcal{D}(A)$) of semidualizing DG $A$-modules is denoted $\mathfrak{S}(A)$.
A \emph{dualizing} DG $A$-module is a semidualizing DG $A$-module $D$ such that for every homologically finite
DG $A$-module $M$
the complex $\Rhom[A]{M}{D}$ is homologically finite, and the natural morphisms
$M\to\Rhom[A]{\Rhom[A]MD}D$
and
$\Lotimes[A]DM\to \Rhom[A]{\Rhom[A]MR}D$
are isomorphisms in $\mathcal{D}(A)$. The DG algebra $A$ is \emph{Gorenstein} if $A$ is dualizing.
\end{defn}
Next, we summarize some facts about dualizing DG $A$-modules.
\begin{fact}\label{lem130821a}
Assume that $R$ has a dualizing complex $D^R$ and $A^\natural$ is finitely generated as an $R$-module.
Then $\Rhom[R]{A}{D^R}$ is a dualizing DG $A$-module by~\cite[Proposition 2.7]{frankild:ddgmgdga}.
Since $\id_R(D^R)<\infty$, we also have $\id_A(\Rhom[R]{A}{D^R})<\infty$.
In particular, every finite dimensional DG algebra over a field has a dualizing DG module of finite injective dimension.
\end{fact}
If $A$ is concentrated in degree 0, the next result is well known, and our proof is not surprising; see, e.g.,
\cite[(2.12) Corollary]{christensen:scatac},
\cite[2.9.1-2]{frankild:rbsc},
and~\cite[Corollary 3.3]{gerko:sdc}.
\begin{lem}\label{lem130821b}
Assume that $A$ has a dualizing DG module $D^A$ of finite injective dimension, and let $X$ be a semidualizing DG $A$-module.
Then $\Rhom[A]X{D^A}$ is a semidualizing DG $A$-module such that $\Lotimes[A]{X}{\Rhom[A]X{D^A}}\simeq D^A$ in $\mathcal{D}(A)$.
If $R$ and $A$ are local and either $\pd_A(X)<\infty$ or $\id_A(A)<\infty$, then $X\sim A$.
\end{lem}
\begin{proof}
By~\cite[Theorem 1]{apassov:hddgr}, we have
\begin{align*}
\Lotimes[A]{X}{\Rhom[A]X{D^A}}
&\simeq \Rhom[A]{\Rhom[A]XX}{D^A}
\simeq \Rhom[A]{A}{D^A}
\simeq D^A.
\end{align*}
Next, we have the following commutative diagram of DG $A$-module morphisms.
$$\xymatrix@C=15mm{
A\ar[d]_-{\chi^{D^A}_A}^-\simeq\ar[r]^-{\chi^{\Rhom[A]X{D^A}}_A}&\Rhom[A]{\Rhom[A]X{D^A}}{\Rhom[A]X{D^A}}\ar[d]^-\simeq
\\
\Rhom[A]{D^A}{D^A}\ar[r]_-{\simeq}^-{\Rhom[A]{\zeta}{D^A}}&\Rhom[A]{\Lotimes[A]{X}{\Rhom[A]X{D^A}}}{D^A}
}
$$
The morphism $\zeta$ is the isomorphism from the beginning of this proof.
It follows that $\chi^{\Rhom[A]X{D^A}}_A$ is an isomorphism, so $\Rhom[A]X{D^A}$ is a semidualizing.
Assume that $R$ and $A$ are local and either $\pd_A(X)<\infty$ or $\id_A(A)<\infty$.
Then by~\cite[Theorem 1]{apassov:hddgr} we have the following isomorphisms in $\mathcal{D}(A)$.
$$
\Lotimes[A]{X}{\Rhom[A]XA}\simeq \Rhom[A]{\Rhom[A]XX}{A}\simeq A.
$$
It follows that $P^A_X(t)P^A_{\Rhom[A]XA}(t)=1$, so we have $P^A_X(t)=t^d$ for some $d\in\mathbb{Z}$.
From this, we conclude that the minimal semi-free resolution $F\xrightarrow\simeq X$ has $F^{\natural}\simeq\mathsf{\Sigma}^d A^\natural$.
From the Leibniz rule on $F$, one concludes that $X\simeq F\simeq\mathsf{\Sigma}^dA$.
\end{proof}
The next result compares to~\cite[A.3. Lemma (a)]{christensen:dvke}.
\begin{lem}\label{lem130825a}
Let $\underline t=t_1,\ldots,t_n$ be a sequence in the Jacobson radical of $R$,
and set $K=K^R(\underline t)$.
Let $C$ be a DG $R$-algebra such that $C^\natural$ is finitely generated over $R$, and set $B=\Otimes KC$.
Then for each homologically finite DG $C$-module $X$ we have $X\in \mathfrak{S}(C)$ if and only if $\Lotimes[C]BX\in \mathfrak{S}(B)$.
\end{lem}
\begin{proof}
Consider the following commutative diagram of chain maps.
$$\xymatrix{
B\ar[r]^-=\ar[d]_-{\chi^{\Lotimes[C]BX}_B}
&\Otimes KC\ar[d]^-{\Otimes[]K{\chi^X_C}}\\
\Rhom[B]{\Lotimes[C]BX}{\Lotimes[C]BX}\ar[d]_-\simeq
&\Otimes K\Rhom[C]XX\ar[dd]^-{\simeq}\\
\Rhom[C]X{\Lotimes[C] BX}\ar[d]_-= \\
\Rhom[C]X{\Lotimes[C]{(\Otimes KC)}X}\ar[r]^-\simeq
&\Rhom[C]X{\Lotimes KX}}$$
It follows that $\chi^{\Lotimes[C]BX}_B$ is an isomorphism
if and only if $\Otimes[]K{\chi^X_C}$ is an isomorphism.
Since $C$ and $\Rhom[C]XX$ are homologically degree-wise finite over $R$, we conclude (say, from a routine mapping cone
argument) that $\Otimes[]K{\chi^X_C}$ is an isomorphism
if and only if
$\chi^X_C$ is an isomorphism.
\end{proof}
\section{Vanishing of Tor over Trivial Extensions of DG Algebras}\label{sec130815b}
\begin{convention*}
In this section, $(R,\ideal{m},k)$ is a local ring.
\end{convention*}
The point of this section is to prove that Tor-vanishing over certain trivial extensions of DG algebras implies finite projective dimension;
see Theorem~\ref{thm130816b}. We begin with a useful construction.
\begin{prop}\label{prop130820a}
Let $(A,A_+)$ be a local DG $R$-algebra.
Let $X$ be a homologically finite DG $A$-module. Then there exists a short exact sequence
$$0\to X'\xrightarrow\alpha L\xrightarrow\pi \widetilde X\to 0$$
of morphisms of DG $A$-modules such that $L$ is semi-free with a finite semibasis and we have $\widetilde X\simeq X$
and $\operatorname{Im}(\alpha)\subseteq A_+ L$.
\end{prop}
\begin{proof}
Let $F\xrightarrow{\simeq} X$ be a minimal semi-free resolution of $X$, and let $E$ be a semi-basis for $F$.
Let $F^{(p)}$ be the semi-free DG $A$-submodule of $F$ spanned by $E_{\leq p}:=\bigcup_{m\leq p}E_m$.
To be clear, ${F^{(p)}}^{\natural}$ is the graded submodule of $F^{\natural}$ spanned by $E_{\leq p}$. Note that $\partial^F({F^{(p)}}^{\natural})\subseteq {F^{(p)}}^{\natural}$. To see this, by the Leibniz rule we have $\partial^F_{i+j}(te)=\partial^A_{i}(t)e+(-1)^{i}t\partial^F_{j}(e)$ for each $t\in A$ of degree $i$ and each $e\in F$ of degree $j$. If $j\leq p$, then the term $\partial^A_{i}(t)e$ is
in $\operatorname{Span}_A(E_{\leq p})$ by assumption, and the term $(-1)^{i}t\partial^F_{j}(e)$ is
in $\operatorname{Span}_A(E_{\leq p})$ by a degree argument.
Let $s=\sup(X)$, and set $\widetilde X=\tau_{\leq s}(F)$, the ``soft truncation'' of $F$. Note that the natural morphism
$F\to\widetilde X$ is a surjective quasiisomorphism of DG $A$-modules, so we have $\widetilde X\simeq F\simeq X$.
Next, set $L=F^{(s)}$, which is semi-free with a finite semibasis $E_{\leq s}$.
Furthermore, the composition $\pi$ of natural morphisms $L=F^{(s)}\to F\to \widetilde X$ is surjective
because the morphism $F\to\widetilde X$ is surjective, the morphism $L\to F$ is surjective in degrees $\leq s$,
and we have $\widetilde X_i=0$ for all $i>s$. Thus, it remains to show that $X':=\ker(\pi)\subseteq A_+L$.
By construction, for $i\leq s$, the map $\pi_s$ is an isomorphism, so $X'_i=0\subseteq A_+L$, as desired. In degree $s+1$, we have
$X_{s+1}'=\operatorname{Im}(\partial^F_{s+1})$, which is contained in $(A_+F)_s$ by construction.
Also, since $A_+$ consists of elements of positive degree, we have
$$(A_+F)_s\subseteq A_+\operatorname{Span}_A(E_{<s})\subseteq A_+F^{(s)}.$$
Thus, we have $X_{s+1}'\subseteq A_+F^{(s)}$.
Lastly, for $i>s$, we have $X'_i=F^{(s)}_i=(A_+F^{(s)})_i$ by a degree argument, since $F^{(s)}$ is
generated over $A$ in degrees $\leq s$.
\end{proof}
\begin{cor}\label{cor130820a}
Let $(A,A_+)$ be a bounded local DG $R$-algebra.
Let $X$ and $Y$ be homologically finite DG $A$-modules of infinite projective dimension over $A$.
Assume that $\Tor[A]{i\gg 0}{X}{Y}=0$.
Then there are homologically finite DG $A$-modules $X',Y'$ of infinite projective dimension over $A$
such that $\operatorname{Ann}_A(A_+)X'=0=\operatorname{Ann}_A(A_+)Y'$
and $\Tor[A]{i\gg 0}{X'}{Y'}=0$.
\end{cor}
\begin{proof}
By Proposition~\ref{prop130820a}, there are short exact sequences
\begin{gather*}
0\to X'\xrightarrow\alpha L\to \widetilde X\to 0 \\
0\to Y'\xrightarrow\beta M\to \widetilde Y\to 0
\end{gather*}
of morphisms of DG $A$-modules such that $L$ and $M$ are semi-free with a finite semibases, and $\widetilde X\simeq X$,
$\widetilde Y\simeq Y$,
$\operatorname{Im}(\alpha)\subseteq A_+ L$,
and $\operatorname{Im}(\beta)\subseteq A_+ M$.
The condition $\operatorname{Im}(\alpha)\subseteq A_+ L$ implies that
$$\operatorname{Ann}_A(A_+)X'\cong \operatorname{Ann}_A(A_+)\operatorname{Im}(\alpha)\subseteq \operatorname{Ann}_A(A_+)A_+ L=0$$
and similarly $\operatorname{Ann}_A(A_+)Y'=0$.
By assumption, $\pd_A(L)<\infty$, so the condition $\pd_A(\widetilde X)=\pd_A(X)=\infty$ implies that $\pd_A(X')=\infty$,
because of the exact sequence $0\to X'\to L\to \widetilde X\to 0$.
And similarly $\pd_A(Y')=\infty$.
Finally, since $A$ is bounded and $\pd_A(L)$ is finite, the fact that $Y$ is homologically bounded implies that
$\Tor[A]{i\gg 0}{L}{Y}=0$.
By assumption, we have $\Tor[A]{i\gg 0}{\widetilde X}{Y}\cong\Tor[A]{i\gg 0}{X}{Y}=0$
so the above exact sequence implies that $\Tor[A]{i\gg 0}{X'}{Y}=0$.
Similarly, we deduce $\Tor[A]{i\gg 0}{X'}{Y'}=0$, as desired.
\end{proof}
Compare the next two results to~\cite[Lemma 3.2 and Theorem 3.1]{nasseh:oeire}.
\begin{lem}\label{lem130815a}
Let
$(B,\ideal{m}_B,k)$ be a local DG $R$-algebra.
Set $A=B\ltimes \mathsf{\Sigma}^n k$ for some $n\geq 0$,
and let $\ideal{m}_A$ be the augmentation ideal of $A$. Let $x$ be a generator for the DG ideal $0\oplus \mathsf{\Sigma}^n k\subseteq A$.
Let $X$, $Y$ be DG $B$-modules, i.e., DG $A$-modules such that $xX=0=xY$.
Then for all $i\in\mathbb{Z}$ we have $R$-module isomorphisms
$$\Tor[A]{i}XY\cong \Tor[B]{i}XY\bigoplus\left(\bigoplus_{p+q=i-n-1}\Otimes[k]{\Tor[A]{p}Xk}{\Tor[B]{q}kY}\right).$$
\end{lem}
\begin{proof}
Let $s\colon L\xra{\simeq} B$ be a semi-free resolution over $A$.
Let $p\colon A\to B$ be the natural surjection, and let $\widetilde p\colon A\to L$ be a lift of $p$.
Hence the following diagram of morphisms of DG $A$-modules
$$\xymatrix{
A\ar[r]^-{\widetilde p}\ar[rd]_-p &L\ar[d]^<<<s_<<<\simeq \\
&B}$$
commutes up to homotopy.
Apply $\Otimes[A]{X}{-}$ to obtain the next diagram of morphisms of DG $B$-modules
$$\xymatrix{
\Otimes[A]{X}{A}\ar[r]^-{\Otimes[]{X}{\widetilde p}}\ar[rd]_-{\Otimes[]{X}{p}}^-\cong &\Otimes[A]{X}{L}\ar[d]^{\Otimes[]{X}{s}} \\
&\Otimes[A]{X}{B}}$$
that commutes up to homotopy. Note that $\Otimes[]{X}{p}$ is an isomorphism since $xX=0$.
Also, the chain map $\Otimes[A]{X}{\widetilde p}$ represents the morphism $\Lotimes[A]{X}{p}\colon\Lotimes[A]{X}{A}\to\Lotimes[A]{X}{B}$
in $\mathcal{D}(B)$. It follows that $\Lotimes[A]{X}{p}$ has a left-inverse in $\mathcal{D}(B)$,
so we have the first isomorphism (in $\mathcal{D}(B)$) in the next sequence:
\begin{align*}
\Lotimes[A]{X}{B}
&\simeq\left(\Lotimes[A]{X}{A}\right)\bigoplus\left(\Lotimes[A]{X}{(\mathsf{\Sigma} xA)}\right) \\
&\simeq\left(\Lotimes[A]{X}{A}\right)\bigoplus\left(\Lotimes[A]{X}{(\mathsf{\Sigma}^{n+1}k)}\right)\\
&\simeq\left(\Lotimes[A]{X}{A}\right)\bigoplus\left(\mathsf{\Sigma}^{n+1}\Lotimes[A]{X}{k}\right).
\end{align*}
The second isomorphism is from the assumption $\ideal{m}_Ax=0$.
(Note that this isomorphism is in $\mathcal{D}(A)$, hence also in $\mathcal{D}(B)$ via the split injection $B\to A$.)
Now, we apply $\Lotimes[B]{-}{Y}$ to these isomorphisms to conclude that
\begin{align*}
\Lotimes[A]{X}{Y}
&\simeq\Lotimes[B]{(\Lotimes[A]{X}{B})}{Y}\\
&\simeq\left(\Lotimes[B]{(\Lotimes[A]{X}{A})}{Y}\right)\bigoplus\left(\mathsf{\Sigma}^{n+1}\Lotimes[B]{(\Lotimes[A]{X}{k})}{Y}\right)\\
&\simeq\left(\Lotimes[B]{X}{Y}\right)\bigoplus\left(\mathsf{\Sigma}^{n+1}\Lotimes[k]{(\Lotimes[A]{X}{k})}{(\Lotimes[B]{k}{Y})}\right).
\end{align*}
Apply $\operatorname{H}_i(-)$ to obtain the first isomorphism in the next sequence:
\begin{align*}
\Tor[A]{i}{X}{Y}
&\cong\Tor[B]{i}{X}{Y}\bigoplus\operatorname{H}_i\left(\mathsf{\Sigma}^{n+1}\Lotimes[k]{(\Lotimes[A]{X}{k})}{(\Lotimes[B]{k}{Y})}\right)\\
&\cong\Tor[B]{i}{X}{Y}\bigoplus\operatorname{H}_{i-n-1}\left(\Lotimes[k]{(\Lotimes[A]{X}{k})}{(\Lotimes[B]{k}{Y})}\right)\\
&\cong\Tor[B]{i}{X}{Y}\bigoplus\left(\bigoplus_{p+q=i-n-1}\Otimes[k]{\operatorname{H}_p(\Lotimes[A]{X}{k})}{\operatorname{H}_q(\Lotimes[B]{k}{Y})}\right)\\
&\cong\Tor[B]{i}{X}{Y}\bigoplus\left(\bigoplus_{p+q=i-n-1}\Otimes[k]{\Tor[A]p{X}{k}}{\Tor[B]q{k}{Y}}\right).
\end{align*}
The third isomorphism is from the K\"unneth formula.
\end{proof}
\begin{thm}\label{thm130816b}
Let
$(B,B_+,k)$ be a bounded local DG $R$-algebra.
Set $A=B\ltimes \mathsf{\Sigma}^n k$ for some $n\geq 1$. Let $x$ be a generator for the DG ideal $0\oplus \mathsf{\Sigma}^n k\subseteq A$.
Let $X$ and $Y$ be non-zero homologically finite
DG $A$-modules such that $\Tor[A]{i\gg 0}XY=0$. Then $\pd_A(X)<\infty$ or $\pd_A(Y)<\infty$.
\end{thm}
\begin{proof}
Assume by way of contradiction that $\pd_A(X)=\infty=\pd_A(Y)$.
Thus, Corollary~\ref{cor130820a}
provides homologically finite DG $A$-modules $X',Y'$ of infinite projective dimension
such that $\operatorname{Ann}_A(A_+)X'=0=\operatorname{Ann}_A(A_+)Y'$
and such that $\Tor[A]{i\gg 0}{X'}{Y'}=0$.
Thus, we may replace $X$ and $Y$ by $X'$ and $Y'$ to assume that $\operatorname{Ann}_A(A_+)X=0=\operatorname{Ann}_A(A_+)Y$.
In particular, we have $xX=0=xY$.
Since $\Tor[A]{i\gg 0}XY=0$,
Lemma~\ref{lem130815a} implies that
\begin{equation}\label{eq131227a}
\bigoplus_{p+q=i-n-1}\Otimes[k]{\Tor[A]{p}Xk}{\Tor[B]{q}kY}=0
\end{equation}
for all $i\gg 0$.
The fact that $\pd_A(Y)$ is infinite implies that $Y\not\simeq 0$.
Since $Y$ is homologically finite, Nakayama's Lemma implies that there is an integer $q_0$ such that
$\Tor[B]{q_0}kY\neq 0$.
Thus, equation~\eqref{eq131227a} for $i\gg 0$ implies that $\Tor[A]{p\gg 0}Xk=0$,
contradicting the assumption $\pd_A(X)=\infty$.
\end{proof}
\section{The Number Of Semidualizing Complexes For Embedding Codepth 3}
\label{sec131230b}
\begin{convention*}
In this section, $(R,\ideal{m},k)$ is a local ring.
\end{convention*}
The purpose of this section is to prove Theorem~\ref{cor130816a} from the introduction, which we do in~\ref{proof131230a}.
Our main tool for this is the following result.
\begin{thm}\label{cor130821a}
Assume that $R$ admits a dualizing complex, and let
$(B,B_+,k)$ be a bounded homologically finite local DG $R$-algebra.
Set $A=B\ltimes W$ for some
non-zero finitely generated positively graded $k$-vector space $W$.
Then $|\mathfrak{S}(A)|\leq 2$.
\end{thm}
\begin{proof}
Fact~\ref{lem130821a} implies that $A$ has a dualizing DG module $D^A$ of finite injective dimension.
Set $(-)^{\dagger}=\Rhom[A]{-}{D^A}$.
It suffices to show that for every semidualizing DG $A$-module $X$ we have $X\sim A$ or $X\sim D^A$.
Case 1: $W=\mathsf{\Sigma}^n k$ for some $n\geq 1$. The isomorphism $\Lotimes[A]{X}{X^{\dagger}}\simeq D^A$
from Lemma~\ref{lem130821b} implies that $\Tor[A]{i\gg 0}{X}{X^{\dagger}}=0$. By Theorem~\ref{thm130816b} either $\pd_A(X)<\infty$ or $\pd_A(X^{\dagger})<\infty$. And by Lemma~\ref{lem130821b} we have $X\sim A$ or $X^{\dagger}\sim A$. If $X^{\dagger}\sim A$, then by
definition of $D^A$
we have
$X\simeq X^{\dagger \dagger}\sim A^{\dagger}\simeq D^A$, as desired.
Case 2: General case.
Since $W\neq 0$, write $W = W' \oplus \mathsf{\Sigma}^n k$ for some $n\geq 1$, and set $B' := B \ltimes W'$. It follows that $A \cong B' \ltimes \mathsf{\Sigma}^n k$,
so the assertion follows from the previous case.
\end{proof}
\begin{disc}\label{disc131221a}
Let $\underline{t}=t_1,\cdots,t_n$ be a minimal generating set for $\mathfrak m$,
and consider the Koszul complex $K=K^R(\underline t)$.
From~\cite[(1.2)]{avramov:cslrec3} (or~\cite[(2.8)]{avramov:lgh}) there is a finite-dimensional DG $k$-algebra $A$ that is
linked to $K$ by a sequence of quasiisomorphisms of DG algebras.
As in~\cite[5.4 (Proof of Theorem A)]{nasseh:lrfsdc}, one has an injection $\mathfrak{S}(R)\hookrightarrow\mathfrak{S}(\comp R)$
and bijections $\mathfrak{S}(\comp R)\xrightarrow\cong\mathfrak{S}(K)\xrightarrow\cong\mathfrak{S}(A)$.
By definition
(or by a result of Golod~\cite{golod:hslr}), $R$ is Golod if and only if $A$ is of the form $k\ltimes W$ for some
finitely generated positively graded $k$-vector space $W$.
Assume for the rest of this remark that $c=\operatorname{ecodepth}(R)\leq 3$.
Up to isomorphism,
the algebra $A$ has $\partial^A=0$ and is in one of the classes described in the next table, copied from~\cite[1.3]{avramov:cslrec3}.
\begin{center}
\begin{tabular}{lllll}
\hline
Class & $A$&$B$&$C$&$D$ \\
\hline
$\mathbf{C}(c)$&$B$&$\bigwedge_k\mathsf{\Sigma} k^c$\\
$\mathbf{S}$&$B\ltimes W$&$k$\\
$\mathbf{T}$&$B\ltimes W$&$C\ltimes\mathsf{\Sigma}(C/C_{\geq 2})$&$\bigwedge_k\mathsf{\Sigma} k^2$\\
$\mathbf{B}$&$B\ltimes W$&$C\ltimes\mathsf{\Sigma} C_+$&$\bigwedge_k\mathsf{\Sigma} k^2$\\
$\mathbf{G}(r)$&$B\ltimes W$&$C\ltimes\Hom[k]{C}{\mathsf{\Sigma}^3k}$&$k\ltimes\mathsf{\Sigma} k^r$\\
$\mathbf{H}(p,q)$&$B\ltimes W$&$\Otimes[k]CD$&$k\ltimes(\mathsf{\Sigma} k^p\oplus\mathsf{\Sigma}^2k^q)$&$k\ltimes\mathsf{\Sigma} k$
\end{tabular}
\end{center}
\
\noindent Here, $W$ is a finitely generated positively graded $k$-vector space such that $B_+W=0$.
Note that if $R$ is not regular, then $\sum_i(-1)^i\operatorname{len}_k(A_i)=\sum_i(-1)^i\operatorname{len}_k(\operatorname{H}_i(K))=0$.
\end{disc}
\begin{para}[Proof of Theorem~\ref{cor130816a}] \label{proof131230a}
Continue with the notation from Remark~\ref{disc131221a}.
Recall that $\comp R$ has a dualizing complex $D^{\comp R}$, and that $R$ is Gorenstein if and only if $D^{\comp R}\sim \comp R$.
Hence, it suffices to show that $|\mathfrak{S}(A)|\leq 2$.
Assume for this paragraph that $R$ is Golod. Then we have
$A\cong k\ltimes W$ for some
finitely generated positively graded $k$-vector space $W$.
If $W=0$, then $A\cong k$ which is a commutative local Gorenstein ring, hence $|\mathfrak{S}(A)|=1$ in this case.
If $W\neq 0$, then $|\mathfrak{S}(A)|\leq 2$ by Theorem~\ref{cor130821a}.
Assume for the rest of the proof that $c=\operatorname{ecodepth}(R)\leq 3$. We analyze the classes from Remark~\ref{disc131221a}.
Note that if $R$ is Gorenstein, then $|\mathfrak{S}(R)|=1$.
Also, if $W\neq 0$, then the conclusion $|\mathfrak{S}(A)|\leq 2$ follows from Theorem~\ref{cor130821a}.
Thus, we assume for the rest of the proof that $W=0$.
If $R$ is in the class $\mathbf{C}(c)$, then $R$ is a complete intersection (hence Gorenstein), which has already been treated.
If $R$ is in the class $\mathbf{S}$, then $R$ is Golod, which has also already been treated.
(Class \textbf{T})
In this case the algebra $C$ has the form $0\to k\to k^2\to k\to 0$. In particular, $C_{\geq 2}=\mathsf{\Sigma}^2k$.
It follows that
$$\sum_i(-1)^i\operatorname{len}_k(B_i)=\sum_{i}(-1)^i\operatorname{len}_k(C_i)+\sum_{i<2}(-1)^{i-1}\operatorname{len}_k(C_i)= 1\neq 0.$$
Since we know that
\begin{align*}
0&=\sum_i(-1)^i\operatorname{len}_k(A_i)\\
&=\sum_i(-1)^i\operatorname{len}_k(B_i)+\sum_i(-1)^i\operatorname{len}_k(W_i)\\
&=1+\sum_i(-1)^i\operatorname{len}_k(W_i)
\end{align*}
we have $W\neq 0$, which is a case we have already treated.
(Class \textbf{B})
As in the previous case, one has $W\neq 0$, which has already been~treated.
(Class \textbf{G}(r))
The DG $C$-module $\hom[k]{C}{\mathsf{\Sigma}^3k}$ is dualizing, so the algebra
$$A\cong B=C\ltimes\Hom[k]{C}{\mathsf{\Sigma}^3k}$$
is Gorenstein
by~\cite[Theorem 2.2]{jorgensen:rdc}.
It follows from~\cite[Theorem III]{frankild:ddgmgdga} that $A\simeq \Hom[k]{A}{\mathsf{\Sigma}^3k}$.
Since $\Hom[k]{A}{\mathsf{\Sigma}^3k}$ is bounded and semi-injective over $A$, we conclude that $\id_A(A)<\infty$.
Thus, we have $|\mathfrak{S}(A)|=1$ by Lemma~\ref{lem130821b}.
(Class \textbf{H}(p,q))
We first note that $|\mathfrak{S}(C)|\leq 2$. Indeed, if $p=q=0$, then we have $C=k$ so $|\mathfrak{S}(C)|=1$, as above;
if $p\neq 0$ or $\ideal{q}\neq 0$, then this follows from Theorem~\ref{cor130821a}.
Hence, it remains to show that $|\mathfrak{S}(B)|=|\mathfrak{S}(C)|$.
For this,
consider the map $\mathfrak{S}(C)\to \mathfrak{S}(B)$ defined by $X\mapsto\Lotimes[C]BX$. This is well-defined by Lemma~\ref{lem130825a},
and it is bijective by~\cite[Theorems 3.4 and 3.11]{nasseh:ldgm}.
(This uses the fact that $D$ is isomorphic to the trivial Koszul complex $K^k(0)$.)
\qed
\end{para}
\section*{Acknowledgments}
We are grateful to Lucho Avramov for helpful conversations about this material.
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1,116,691,497,751 | arxiv | \section{The $(k,\ell)$-rainbow index for complete bipartite\\ graphs}
This section is devoted to the $(k,\ell)$-rainbow index for complete
bipartite graphs. We start with a lemma about the regular complete
bipartite graphs $K_{n,n}$.
\begin{lemma}\label{lem1}
For every pair of positive integers $k,\ell$ with $k\geq3$, there
exists a positive integer $N=N(k,\ell)$, such that
$rx_{k,\ell}(K_{n,n})\leq k+1$ for every integer $n \geq N$.
\end{lemma}
\begin{proof}
Let $C=\{1,2,\cdots,k+1\}$ be a set of $k+1$ different colors. We
color the edges of $K_{n,n}$ with the colors from $C$ randomly and
independently. For $S\subseteq V(K_{n,n})$ with $|S|=k$, define
$A(S)$ as the event that there exist at least $\ell$ internally
disjoint rainbow $S$-trees. If Pr[ $\bigcap\limits_{S}A(S)$ ]$>0$,
then there exists a required $(k+1)$-edge-coloring, which implies
$rx_{k,\ell}(K_{n,n})\leq k+1$.
Assume that $K_{n,n}=G[U,V]$, where
$U={\{u_{1},u_{2},\cdots,u_{n}\}}$ and
$V={\{v_{1},v_{2},\cdots,v_{n}\}}$. We distinguish the following
three cases.
\emph{Case 1: $S\subseteq U$.}
Without loss of generality, we suppose $S=\{u_{1}, u_{2}, \cdots,
u_{k}\}$. For any vertex $v_{i}\in V$, let $T(v_{i})$ denote the
star with $v_{i}$ as its center and $E(T(v_{i}))=\{u_{1}v_{i},
u_{2}v_{i},\cdots, u_{k}v_{i}\}$. Clearly, $T(v_{i})$ is an
$S$-tree. Moreover, for $v_{i}, v_{j} \in V$ and $v_{i}\neq v_{j}$,
$T(v_{i})$ and $T(v_{j})$ are two internally disjoint $S$-trees. Let
$\mathcal{T}_{1}=\{T(v_{i})|v_{i} \in V, 1\leq i\leq n\}$. Then
$\mathcal{T}_{1}$ is a set of $n$ internally disjoint $S$-trees. It
is easy to see that $p_{1}$:= Pr[$T\in \mathcal{T}_{1}$ is a rainbow
$S$-tree]$=(k+1)!/(k+1)^{k}$. Define $X_{1}$ as the number of
rainbow S-trees in $\mathcal{T}_{1}$. Then we have
\begin{center}
$Pr[\overline{A(S)}]\leq Pr[X_{1}\leq\ell-1]\leq {n \choose \ell-1}(1-p_{1})^{n-\ell+1}\leq n^{\ell-1}(1-p_{1})^{n-\ell+1}$
\end{center}
\emph{Case 2: $S\subseteq V$.}
Similar to $Case\ 1$, we get $Pr[\overline{A(S)}]\leq
n^{\ell-1}(1-p_{1})^{n-\ell+1}.$
\emph{Case 3: $S\cap U \neq\emptyset$, $S\cap V \neq\emptyset$.}
Assume that $U^{\prime}=S\cap
U=\{u_{x_{1}},u_{x_{2}},\cdots,u_{x_{a}}\}$ and $V^{\prime}=S\cap
V=\{v_{y_{1}},v_{y_{2}},\cdots,v_{y_{b}}\}$, where $x_{i},y_{i}\in
\{1,2,\cdots, n\}$, $a\geq1$, $b\geq1$ and $a+b=k$. For every pair
$\{u_{i},v_{i}\}$ of vertices with $u_{i}\in U\setminus U^{\prime}$
and $v_{i}\in V\setminus V^{\prime}$, let $T(u_{i}v_{i})$ denote the
S-tree, where $V(T(u_{i}v_{i}))=S \cup \{u_{i},v_{i}\}$ and
$E(T(u_{i}v_{i}))=\{u_{i}v_{i},u_{i}v_{y_{1}},u_{i}v_{y_{2}},
\cdots, u_{i}v_{y_{b}}, v_{i}u_{x_{1}},v_{i}u_{x_{2}}, \cdots,
v_{i}u_{x_{a}}\}$. Clearly, for $i\neq j$, $T(u_{i}v_{i})$ and
$T(u_{j}v_{j})$ are two internally disjoint $S$-trees. Let
$\mathcal{T}_{2}=\{T(u_{i}v_{i})|u_{i}\in U\setminus U^{\prime},
v_{i}\in V\setminus V^{\prime}\}$. Then $\mathcal{T}_{2}$ is a set
of $n-d$ $(max\{a,b\}\leq d \leq k)$ internally disjoint $S$-trees.
It is easy to see that $p_{2}:=$Pr[T$\in \mathcal{T}_{2}$ is a
rainbow tree]=$(k+1)!/(k+1)^{k+1}$. Define $X_{2}$ as the number of
rainbow S-trees in $\mathcal{T}_{2}$. Then we have
\begin{center}
$Pr[\overline{A(S)}]\leq Pr[X_{2}\leq\ell-1]\leq {n-d \choose
\ell-1} (1-p_{2})^{n-d-\ell+1}\leq
n^{\ell-1}(1-p_{2})^{n-k-\ell+1}$.
\end{center}
Comparing the above three cases, we get $Pr[\overline{A(S)}]\leq
n^{\ell-1}(1-p_{2})^{n-k-\ell+1}$ for every set $S$ of $k$ vertices
in $K_{n,n}$. From the union bound, it follows that
\begin{eqnarray*}
Pr[\ \bigcap\limits_{S}A(S)\ ] &=& 1-Pr[\ \bigcup\limits_{S}\overline{A(S)}\ ] \\
&\geq & 1- \sum\limits_{S}Pr[\overline{A(S)}] \\
&\geq & 1- {2n \choose k}n^{\ell-1}(1-p_{2})^{n-k-\ell+1} \\
&\geq & 1-2^{k}n^{k+\ell-1}(1-p_{2})^{n-k-\ell+1}
\end{eqnarray*}
Since $\lim\limits_{n\rightarrow
\infty}1-2^{k}n^{k+\ell-1}(1-p_{2})^{n-k-\ell+1}=1$, there exists a
positive integer $N=N(k,\ell)$ such that $Pr[\
\bigcap\limits_{S}A(S)\ ]>0$ for all integers $n\geq N$, and thus
$rx_{k,\ell}(K_{n,n})\leq k+1$.
\end{proof}
We proceed with the definition of bipartite Ramsey number, which is
used to derive a lower bound for $rx_{k,\ell}(K_{n,n})$. Classical
Ramsey number involves coloring the edges of a complete graph to
avoid monochromatic cliques, while bipartite Ramsey number involves
coloring the edges of a complete bipartite graph to avoid
monochromatic bicliques. In \cite{Hattingh}, Hattingh and Henning
defined the {\it bipartite Ramsey number} $b(t,s)$ as the least
positive integer $n$ such that in any red-blue coloring of the edges
of $K(n, n)$, there exists a red $K(t,t)$ (that is, a copy of
$K(t,t)$ with all edges red) or a blue $K(s,s)$. More generally, one
may define the bipartite Ramsey number $b(t_{1},t_{2}, \cdots
,t_{k})$ as the least positive integer $n$ such that any coloring of
the edges of $K(n, n)$ with $k$ colors will result in a copy of
$K(t_{i},t_{i})$ in the $i$th color for some $i$. If $t_{i} = t$ for
all $i$, we denote the number by $b_{k}(t)$. The existence of all
such numbers follows from a result first proved by Erd\H{o}s and
Rado \cite{erdos}, and later by Chv$\acute{a}$tal \cite{Chvatal}.
The known bounds for $b(t,t)$ are
$(1+o(1))\frac{t}{e}(\sqrt{2})^{t+1}\leq b(t,t) \leq
(1+o(1))2^{t+1}\log t$, where the $\log$ is taken to the base 2. The
proof of the lower bound \cite{Hattingh} is an application of the
\emph{Lov$\acute{a}$sz Local Lemma}, while the upper bound
\cite{Conlon} is proved upon the observation that, in order for a
two-colored bipartite graph $K_{m,n}$ to necessarily contain a
monochromatic $K_{k,k}$, it is only necessary that one of $m$ and
$n$ be very large. With similar arguments, we can obtain the bounds
for $b_{k}(t)$ as $(1+o(1))\frac{t}{e}(\sqrt{k})^{t+1}\leq b_{k}(t)
\leq (1+o(1))k^{t+1}\log_{k}t$.
\begin{lemma}\label{lem2}
For every pair of positive integers $k,\ell$ with $k\geq4$, if $n
\geq b_{k}(k)$, then $rx_{k,\ell}(K_{n,n})\geq k+1$.
\end{lemma}
\begin{proof}
By contradiction. Suppose $\textbf{\textit{c}}$ is a
$k$-edge-coloring of $K_{n,n}$ such that for any set $S$ of $k$
vertices in $K_{n,n}$, there exist $\ell$ internally disjoint
rainbow $S$-trees. From the definition of bipartite Ramsey number,
we know that if $n \geq b_{k}(k)$, then in this $k$-edge-coloring
$\textbf{\textit{c}}$, one will find a monochromatic subgraph
$K_{k,k}$. Let $K_{n,n}=G[U,V]$ and $U^{\prime}$, $V^{\prime}$ be
the bipartition of the monochromatic $K_{k,k}$, where
$U^{\prime}\subset U$, $V^{\prime}\subset V$ and
$|U^{\prime}|=|V^{\prime}|=k$. Now take $S$ as follows: two vertices
are from $V^{\prime}$ and the other $k-2 \ (\geq2)$ vertices are
from $U^{\prime}$. Assume that $T$ is one of the $\ell$ internally
disjoint rainbow $S$-trees. Since there are $k$ different colors,
the rainbow tree $T$ contains at most $k+1$ vertices (i.e., at most
one vertex in $V\setminus S$). It is easy to see that $T$ is in the
possession of at least two edges from the subgraph induced by $S$,
which share the same color. It contradicts the fact that $T$ is a
rainbow tree. Thus $rx_{k,\ell}(K_{n,n})\geq k+1$ when $k\geq4$ and
$n\geq b_{k}(k)$.
\end{proof}
From Lemmas \ref{lem1} and \ref{lem2}, we get that if $k\geq4$,
$rx_{k,\ell}(K_{n,n})=k+1$ for sufficiently large $n$. What remains
to deal with is the $(3,\ell)$-rainbow index of $K_{n,n}$.
\begin{lemma}\label{lem3}
For every integer $\ell\geq 1$, there exists a positive integer $N=N(\ell)$ such that
\begin{displaymath}
rx_{3,\ell}(K_{n,n}) = \left\{ \begin{array}{ll}
3 & \textrm{if $\ell=1,2 $},\\
4 & \textrm{if $\ell\geq3$\ \ \ \ }
\end{array} \right.
\end{displaymath}
for every integer $n\geq N$.
\end{lemma}
\begin{proof}
Assume that $K_{n,n}=G[U,V]$ and $|U|=|V|=n$.
\emph{\textbf{Claim $1$:}} If $\ell\geq1$,
$rx_{3,\ell}(K_{n,n})\geq3$; furthermore, if $\ell\geq3$,
$rx_{3,\ell}(K_{n,n})\geq4$.
\indent If $S=\{x,y,z\}\subseteq U$, then the size of $S$-trees is
at least three, which implies that $rx_{3,\ell}(K_{n,n})\geq3$ for
all integers $\ell\geq1$. If $S=\{x,y,z\}, \{x, y\}\subseteq U, z\in
V$, then the number of internally disjoint $S$-trees of size two or
three is no more than two. Thus we have $rx_{3,\ell}(K_{n,n})\geq4$
for all integers $\ell\geq3$.
Combing with $Lemma\ \ref{lem1}$, we get that if $\ell \geq3$, there
exists a positive integer $N=N(\ell)$ such that
$rx_{3,\ell}(K_{n,n})=4$ for every integer $n\geq N$.
\emph{\textbf{Claim $2$:}} If $\ell=1,2$, then there exists a
positive integer $N=N(\ell)$ such that $rx_{3,\ell}(K_{n,n})=3$ for
every integer $n\geq N$.
Note that $3\leq rx_{3,1}(K_{n,n})\leq rx_{3,2}(K_{n,n})$. So it
suffices to prove $rx_{3,2}(K_{n,n})\leq 3$. In other words, we need
to find a $3$-edge-coloring $c:E(K_{n,n})\rightarrow \{1,2,3\}$ such
that for any $S\subseteq V(K_{n,n})$ with $|S|=3$, there are at
least two internally disjoint rainbow $S$-trees. We color the edges
of $K_{n,n}$ with colors $1,2,3$ randomly and independently. For
$S=\{x,y,z\}\subseteq V(K_{n,n})$, define $B(S)$ as the event that
there exist at least $two$ internally disjoint rainbow $S$-trees.
Similar to the proof in $Lemma$ \ref{lem1}, we only need to prove
$Pr[\ \overline{B(S)}\ ]=o(n^{-3})$, since then
Pr[$\bigcap\limits_{S}B(S)$ ]$>0$ and $rx_{3,2}(K_{n,n})\leq 3$ for
sufficiently large $n$. We distinguish the following two cases.
\emph{Case 1}: x, y, z are in the same vertex class.
Without loss of generality, assume that $\{x,y,z\} \subseteq U$. For
any vertex $v\in V$, let $T(v)$ denote the star with $v$ as its
center and $E(T(v))=\{xv, yv,zv\}$. Clearly,
$\mathcal{T}_{3}=\{T(v)|v \in V\}$ is a set of $n$ internally
disjoint $S$-trees and Pr[T$\in \mathcal{T}_{3}$ is a rainbow
tree]=$\frac{3\times2\times1}{3\times3\times3}=\frac{2}{9}$. Define
$X_{3}$ as the number of internally disjoint rainbow S-trees in
$\mathcal{T}_{3}$. Then we have
\begin{center}
$Pr[\overline{B(S)}]=Pr[X_{3}\leq1]={n\choose
1}\frac{2}{9}(1-\frac{2}{9})^{n-1}+(1-\frac{2}{9})^{n}=o(n^{-3}).$
\end{center}
\emph{Case 2}: $x,y,z$ are in two vertex classes. Without loss of
generality, assume that $\{x,y\} \subseteq U$ and $z\in V$.
\emph{Subcase 2.1}: The edges $xz,yz$ share the same color. Without
loss of generality, we assume $c(xz)=c(yz)=1$. For any vertex $v\in
V\setminus \{z\}$, if $\{c(xv), c(yv)\}=\{2,3\}$, then $\{xz, xv,
yv\}$ or $\{yz, xv, yv\}$ induces a rainbow $S$-tree. So $Pr[\{xz,
xv, yv\}$ induces a rainbow $S$-tree$]=Pr[\{yz, xv, yv\}$ induces a
rainbow $S$-tree$]=\frac{2}{9}$. If there do not exist two
internally disjoint rainbow $S$-trees, then we can find at most one
vertex $v\in V\setminus \{z\}$ satisfying $\{c(xv),
c(yv)\}=\{2,3\}$. Thus
\begin{eqnarray*}
Pr[\ \overline{B(S)}|c(xz)=c(yz)\ ]&=&{n-1 \choose 1}\cdot\frac{2}{9}\cdot(1-\frac{2}{9})^{n-2}+(1-\frac{2}{9})^{n-1}\\
&=&\frac{2n+5}{9}(\frac{7}{9})^{n-2}.
\end{eqnarray*}
\emph{Subcase 2.2}: The edges $xz,yz$ have distinct colors. Without
loss of generality, we assume $c(xz)=1, c(yz)=2$. For any vertex
$v\in V\setminus \{z\}$, if $\{c(xv), c(yv)\}=\{2,3\}$, then $\{xz,
xv, yv\}$ induces a rainbow $S$-tree, and so $Pr[\{xz, xv, yv\}$
induces a rainbow $S$-tree$]=\frac{2}{9}$. Moreover, if $\{c(xv),
c(yv)\}=\{1,3\}$, then $\{yz, xv, yv\}$ induces a rainbow $S$-tree,
and so $Pr[\{yz, xv, yv\}$ induces a rainbow
$S$-tree$]=\frac{2}{9}$. If there do not exist two internally
disjoint rainbow $S$-trees, then we can not find two vertices
$v,v^{\prime}\in V\setminus \{z\}$ satisfying $\{c(xv),
c(yv)\}=\{2,3\}$ and $\{c(xv^{\prime}), c(yv^{\prime})\}=\{1,3\}$.
Thus
\begin{eqnarray*}
Pr[\ \overline{B(S)}|c(xz)\neq c(yz)\ ]&=&(1-\frac{2}{9}-\frac{2}{9})^{n-1}+2\sum_{i=1}^{n-1}{n-1\choose i}(\frac{2}{9})^{i}(1-\frac{2}{9}-\frac{2}{9})^{n-1-i}\\
&=&2(\frac{7}{9})^{n-1}-(\frac{5}{9})^{n-1}.
\end{eqnarray*}
From the law of total probability, we have
\begin{eqnarray*}
Pr[\ \overline{B(S)}\ ]&=&Pr[\ c(xz)= c(yz)\ ]\cdot Pr[\ \overline{B(S)}|c(xz)= c(yz)\ ]\\
&\ \ &+Pr[\ c(xz)\neq c(yz)\ ]\cdot Pr[\ \overline{B(S)}|c(xz)\neq c(yz)\ ]\\
&=&\frac{1}{3}\cdot\frac{2n+5}{9}(\frac{7}{9})^{n-2}
+\frac{2}{3}\cdot[2(\frac{7}{9})^{n-1}-(\frac{5}{9})^{n-1}]\\
&\leq& 2n(\frac{7}{9})^{n-2}=o(n^{-3}).
\end{eqnarray*}
Thus, there exists a positive integer $N=N(\ell)$ such that
$rx_{3,2}(K_{n,n})\leq3$, and then
$rx_{3,1}(K_{n,n})=rx_{3,2}(K_{n,n})=3$ for all integers $n\geq N$.
The proof is thus complete.
\end{proof}
By Lemmas \ref{lem1}, \ref{lem2} and \ref{lem3}, we come to the
following conclusion.
\begin{theorem}\label{1}
For every pair of positive integers $k,\ell$ with $k\geq3$, there
exists a positive integer $N=N(k,\ell)$, such that
\begin{displaymath}
rx_{k,\ell}(K_{n,n}) = \left\{ \begin{array}{ll}
3 & \textrm{if $k=3, \ell=1,2$}\\
4 & \textrm{if $k=3, \ell\geq3$}\\
k+1 & \textrm{if $k \geq4 $\ \ \ \ }
\end{array} \right.
\end{displaymath}
for every integer $n \geq N$.
\end{theorem}
With similar arguments, we can expand this result to more general
complete bipartite graphs $K_{m,n}$, where $m=O(n^{\alpha})$ (i.e.,
$m\leq Cn^{\alpha}$ for some positive constant $C$), $\alpha\in
\mathbb{R}$ and $\alpha\geq1$.
\begin{corollary}\label{2}
Let $m,n$ be two positive integers with $m=O(n^{\alpha})$,
$\alpha\in \mathbb{R}$ and $\alpha\geq1$. For every pair of positive
integers $k,\ell$ with $k\geq3$, there exists a positive integer
$N=N(k,\ell)$, such that
\begin{displaymath}
rx_{k,\ell}(K_{m,n}) = \left\{ \begin{array}{ll}
3 & \textrm{if $k=3, \ell=1,2$}\\
4 & \textrm{if $k=3, \ell\geq3$}\\
k+1 & \textrm{if $k \geq4 $\ \ \ \ }
\end{array} \right.
\end{displaymath}
for every integer $n \geq N$.
\end{corollary}
\section{The $(k,\ell)$-rainbow index for complete multipartite graphs}
In this section, we focus on the $(k,\ell)$-rainbow index for
complete multipartite graphs. Let $K_{r\times n}$ denote the
complete multipartite graph with $r\geq 3$ vertex classes of the
same size $n$. We obtain the following results about
$rx_{k,\ell}(K_{r\times n})$:
\begin{theorem}\label{thm2}
For every triple of positive integers $k,\ell,r$ with $k\geq 3$ and
$r\geq 3$, there exists a positive integer $N=N(k,\ell,r)$ such that
\begin{displaymath}
rx_{k,\ell}(K_{r\times n}) = \left\{ \begin{array}{ll}
k & \textrm{if $k< r$}\\
k \ or \ k+1 & \textrm{if $k\geq r, \ell\leq\frac{{r\choose2}\lceil\frac{k}{r}\rceil^{2}}{\lfloor\frac{k}{r}\rfloor}$} \\
k+1 & \textrm{if $k\geq r, \ell>\frac{{r\choose2}\lceil\frac{k}{r}\rceil^{2}}{\lfloor\frac{k}{r}\rfloor}$}\ \ \ \
\end{array} \right.
\end{displaymath}
for every integer $n\geq N$.
\end{theorem}
\begin{proof}
Assume that $K_{r\times n}=G[V_{1},V_{2},\ldots,V_{r}]$ and
$V_{i}=\{v_{i1},v_{i2},\cdots,v_{in}\}$, $1\leq i\leq r$. For
$S\subseteq V(K_{r\times n})$ with $|S|=k$, define $C(S)$ as the
event that there exist at least $\ell$ internally disjoint rainbow
$S$-trees.
\emph{\textbf{Claim $1$:}} $rx_{k,\ell}(K_{r\times n})\geq k$;
furthermore, if $k\geq r$,
$\ell>\frac{{r\choose2}\lceil\frac{k}{r}\rceil^{2}}{\lfloor\frac{k}{r}\rfloor}
$, then $rx_{k,\ell}(K_{r\times n})\geq k+1$.
\indent Let $S\subseteq V_{1}$. Then the size of $S$-trees is at
least $k$, which implies that $rx_{k,\ell}(K_{r\times n})\geq k$ for
all integers $\ell\geq1$. If $k\geq r$, we can take a set
$S^{\prime}$ of $k$ vertices such that $|S^{\prime}\cap
V_{i}|=\lfloor\frac{k}{r}\rfloor$ or $\lceil\frac{k}{r}\rceil$ for
$1\leq i\leq r$. Let $H$ denote the subgraph induced by
$S^{\prime}$. We know $|E(H)|\leq
{r\choose2}\lceil\frac{k}{r}\rceil^{2}$. Let
$\mathcal{T}=\{T_{1},T_{2},\cdots, T_{c}\}$ be the set of internally
disjoint $S^{\prime}$-trees with $k$ or $k-1$ edges. Clearly, if
$T\in \mathcal{T}$ is an $S^{\prime}$-tree with $k-1$ edges, then
$|E(T)\cap E(H)|=k-1$; if $T\in \mathcal{T}$ is an $S^{\prime}$-tree
with $k$ edges, then $|E(T)\cap E(H)|\geq
\lfloor\frac{k}{r}\rfloor$. Therefore,
$c\leq\frac{{r\choose2}\lceil\frac{k}{r}\rceil^{2}}{\lfloor\frac{k}{r}\rfloor}$
(note that the bound is sharp for $r=3$, $k=3$). Thus we have
$rx_{k,\ell}(K_{r\times n})\geq k+1$ for $k\geq r$ and
$\ell>\frac{{r\choose2}\lceil\frac{k}{r}\rceil^{2}}{\lfloor\frac{k}{r}\rfloor}$.
\emph{\textbf{Claim $2$:}} If $k<r$, then there exists an integer
$N=N(k,\ell,r)$ such that $rx_{k,\ell}(K_{r\times n})\leq k$ for
every integer $n\geq N$.
We color the edges of $K_{r\times n}$ with $k$ colors randomly and
independently. Since $k<r$, no matter how $S$ is taken, we can
always find a set $V_{i}$ satisfying $V_{i}\cap S=\emptyset$. For
any $v_{ij}\in V_{i}$, let $T(v_{ij})$ denote the star with $v_{ij}$
as its center and $E(T(v_{ij}))=\{v_{ij}s|s\in S\}$. Clearly,
$\mathcal{T}_{4}=\{T(v_{ij})|v_{ij} \in V_{i}\}$ is a set of $n$
internally disjoint $S$-trees. Similar to the proof of Case 1 in
Lemma \ref{lem1}, we get $Pr[\overline{C(S)}]=o(n^{-k})$. So there
exists a positive integer $N=N(k,\ell,r)$ such that $Pr[\
\bigcap\limits_{S}C(S)\ ]>0$, and thus $rx_{k,\ell}(K_{n,n})\leq k$
for all integers $n\geq N$.
It follows from $Claims\ 1$ and $2$ that if $k< r$, there exists a
positive integer $N=N(k,\ell,r)$ such that $rx_{k,\ell}(K_{r\times
n})= k$ for every integer $n\geq N$.
\emph{\textbf{Claim $3$:}} If $k\geq \ell$, then there exists a
positive integer $N=N(k,\ell,r)$ such that $rx_{k,\ell}(K_{r\times
n})\leq k+1$ for every integer $n\geq N$.
Color the edges of $K_{r\times n}$ with $k+1$ colors randomly and
independently. We distinguish the following two cases.
\emph{Case 1}: $S\cap V_{i} =\emptyset$ for some $i$.
We follow the notation $T(v_{ij})$ and $\mathcal{T}_{4}$ in $Claim\
2$. Similarly, $\mathcal{T}_{4}$ is a set of $n$ internally disjoint
$S$-trees and $Pr[\overline{C(S)}]=o(n^{-k})$.
\emph{Case 2}: $S\cap V_{i}\neq\emptyset$ for $1\leq i\leq r$.
We pick up two vertex classes $V_{1}, V_{2}$. Suppose
$V_{1}^{\prime}=S\cap
V_{1}=\{v_{1x_{1}},v_{1x_{2}},\cdots,v_{1x_{a}}\}$ and
$V_{2}^{\prime}=S\cap
V_{2}=\{v_{2y_{1}},v_{2y_{2}},\cdots,v_{2y_{b}}\}$, where
$x_{i},y_{i}\in \{1,2,\cdots, n\}$, $a\geq1$, $b\geq 1$. Note that
$a+b\leq k-r+2$. For every pair $\{v_{1i},v_{2i}\}$ of vertices with
$v_{1i}\in V_{1}\setminus V_{1}^{\prime}$ and $v_{2i}\in
V_{2}\setminus V_{2}^{\prime}$, let $T(v_{1i}v_{2i})$ denote the
S-tree, where $V(T(v_{1i}v_{2i}))=S \cup \{v_{1i},v_{2i}\}$ and
$E(T(v_{1i}v_{2i}))=\{v_{1i}v_{2i}\}\cup\{v_{1i}v|v\in
V_{2}^{\prime}\}\cup\{v_{2i}v|v\in S\backslash V_{2}^{\prime} \}$.
Clearly, for $i\neq j$, $T(v_{1i}v_{2i})$ and $T(v_{1j}v_{2j})$ are
two internally disjoint $S$-trees. Let
$\mathcal{T}_{5}=\{T(v_{1i}v_{2i})|v_{1i}\in V_{1}\setminus
V_{1}^{\prime}, v_{2i}\in V_{2}\setminus V_{2}^{\prime}\}$. Then
$\mathcal{T}_{5}$ is a set of $n-d$ $(max\{a,b\}\leq d \leq a+b \leq
k-r+2)$ internally disjoint $S$-trees. Similar to the proof of Case
3 in Lemma \ref{lem1}, we get $Pr[\overline{C(S)}]=o(n^{-k})$.
Therefore we conclude that there exists a positive integer
$N=N(k,\ell,r)$ such that\\ $Pr[\ \bigcap\limits_{S}C(S)\ ]>0$, and
thus $rx_{k,\ell}(K_{r\times n})\leq k+1$ for all integers $n\geq
N$.
It follows from $Claims$ $1$ and $3$ that if $k\geq r,
\ell>\frac{{r\choose2}\lceil\frac{k}{r}\rceil^{2}}{\lfloor\frac{k}{r}\rfloor}$,
$rx_{k,\ell}(K_{r\times n})=k+1$ and if $k\geq r,
\ell\leq\frac{{r\choose2}\lceil\frac{k}{r}\rceil^{2}}{\lfloor\frac{k}{r}\rfloor}$,
$rx_{k,\ell}(K_{r\times n})=k$ or $k+1$ for sufficiently large $n$.
The proof is thus complete.
\end{proof}
Note that if $k\geq r\geq 3,
\ell\leq\frac{{r\choose2}\lceil\frac{k}{r}\rceil^{2}}{\lfloor\frac{k}{r}\rfloor}$,
we cannot tell $rx_{k,\ell}(K_{r\times n})=k$ or $k+1$ from Theorem
\ref{thm2}. However, the next lemma shows that when $k=r=3$,
$\ell\leq\frac{{r\choose2}\lceil\frac{k}{r}\rceil^{2}}{\lfloor\frac{k}{r}\rfloor}=3$,
$rx_{3,\ell}(K_{n,n,n})=3$ for sufficiently large $n$.
\begin{lemma}\label{lem4}
For $\ell\leq 3$, there exists a positive integer $N=N(\ell)$ such that
$rx_{3,\ell}(K_{n,n,n})=3$ for every integer $n\geq N$.
\end{lemma}
\begin{proof} Assume that $K_{n,n,n}=G[V_{1},V_{2},V_{3}]$ and
$|V_{1}|=|V_{2}|=|V_{3}|=n$. Note that $3\leq
rx_{3,1}(K_{n,n,n})\leq rx_{3,2}(K_{n,n,n})\leq
rx_{3,3}(K_{n,n,n})$. So it suffices to show
$rx_{3,3}(K_{n,n,n})\leq 3$. In other words, we need to find a
$3$-edge-coloring $c: $ $E(K_{n,n,n})\rightarrow \{1,2,3\}$ such
that for any $S=\{u,v,w\}\subseteq V(K_{n,n,n})$, there are at least
three internally disjoint rainbow $S$-trees. We color the edges of
$K_{n,n,n}$ with the colors $1,2,3$ randomly and independently.
Define $D(S)$ as the event that there exist at least $three$
internally disjoint rainbow $S$-trees. We only need to prove
$Pr[\overline{D(S)}]=o(n^{-3})$, since then Pr[
$\bigcap\limits_{S}D(S)$ ]$>0$ and $rx_{3,3}(K_{n,n,n})\leq 3$ for
sufficiently large $n$. We distinguish the following three cases.
\emph{Case 1}: $u,v,w$ are in the same vertex class.
Without loss of generality, assume that $\{u,v,w \}\subseteq V_{1}$.
For any vertex $z\in V_{2}\cup V_{3}$, let $T_{1}(z)$ denote the
star with $z$ as its center and $E(T_{1}(z))=\{zu, zv, zw\}$.
Obviously, $\mathcal{T}_{6}=\{T_{1}(z)|z\in V_{2}\cup V_{3}\}$ is a
set of $2n$ internally disjoint $S$-trees and Pr[T$\in
\mathcal{T}_{6}$ is a rainbow tree]=$\frac{2}{9}$. So,
\begin{center}
$Pr[\ \overline{D(S)}\ ]\leq {2n \choose 2}(1-\frac{2}{9})^{2n-2}\leq4n^{2}(\frac{7}{9})^{2n-2}=o(n^{-3})$
\end{center}
\emph{Case 2}: $u,v,w$ are in two vertex classes.
Without loss of generality, assume that $\{u,v\} \subseteq V_{1}$
and $w\in V_{2}$. For any vertex $z\in V_{3}$, let $T_{2}(z)$ denote
a star with $z$ as its center and $E(T_{2}(z))=\{zu, zv, zw\}$.
Clearly, $\mathcal{T}_{7}=\{T_{2}(z)|z\in V_{3}\}$ is a set of $n$
internally disjoint $S$-trees and Pr[T$\in \mathcal{T}_{7}$ is a
rainbow tree]=$\frac{2}{9}$. So,
\begin{center}
$Pr[\overline{D(S)}]\leq {n \choose 2} (1-\frac{2}{9})^{n-2}\leq n^{2}(\frac{7}{9})^{n-2}=o(n^{-3})$
\end{center}
\emph{Case 3}: $u,v,w$ are in three vertex classes.
Assume that $u\in V_{1}$, $v\in V_{2}$ and $w\in V_{3}$. For
$e\in\{uv, vw, wu\}$, define $E_{e}$ as the event that $e$ is not
used to construct a rainbow $S$-tree. Clearly,
$Pr[\overline{D(S)}]\leq Pr[ E_{uv}]+Pr[E_{vw}]+Pr[E_{wu}]$.
\emph{Subcase 3.1}: The edges $uv,vw,wu$ receive distinct colors.
Let $p_{e}=Pr[E_{e}|uv,vw,wu$ receive distinct colors] for $e\in
\{uv, vw, wu\}$. Without loss of generality, we assume $c(uv)=1,\
c(vw)=2,\ c(wu)=3$. If $uv$ is not used to construct a rainbow
$S$-tree, then for any vertex $u'\in V_{1}\setminus \{u\}$, $v'\in
V_{2}\setminus \{v\}$, $\{c(u'v), c(u'w)\}\neq\{2,3\}$ and
$\{c(v'u), c(v'w)\}\neq\{2,3\}$. So
$p_{uv}\leq(1-\frac{2}{9})^{2n-2}$. Similarly
$p_{vw}\leq(1-\frac{2}{9})^{2n-2}$,
$p_{wu}\leq(1-\frac{2}{9})^{2n-2}$. Thus
\begin{center}
$Pr[\overline{D(S)}|uv,vw,wu$ receive distinct colors$]\leq p_{uv}+
p_{vw} +p_{wu}\leq3 (\frac{7}{9})^{2n-2}$
\end{center}
\emph{Subcase 3.2}: the edges $uv,vw,wu$ receive two colors.
Let $p'_{e}=Pr[E_{e}|uv,vw,wu$ receive two colors] for $e\in \{uv,
vw, wu\}$. Without loss of generality, we assume $c(uv)= c(vw)=1,\
c(wu)=2$. If $uv$ is not used to construct a rainbow $S$-tree, then
either
$\bullet$ for any vertex $u'\in V_{1}\setminus \{u\}$, $v'\in
V_{2}\setminus \{v\}$, $\{c(u'v), c(u'w)\}\neq\{2,3\}$ and
$\{c(v'u), c(v'w)\}$ $\neq\{2,3\}$; or
$\bullet$ there exists exactly one vertex $v'\in V_{2}\setminus
\{v\}$ satisfying $\{c(v'u), c(v'w)\}=\{2,3\}$, and at the same
time, for any vertex $u'\in V_{1}\setminus \{u\}$, $w'\in
V_{3}\setminus \{w\}$, $\{c(u'v), c(u'w)\}\neq\{2,3\}$ and
$\{c(w'u), c(w'v)\}\neq\{2,3\}$.
So, $p'_{uv}\leq(\frac{7}{9})^{2n-2}+{n-1 \choose
1}\frac{2}{9}(\frac{7}{9})^{n-2}(\frac{7}{9})^{2n-2}$. Similarly,
$p'_{vw}\leq(\frac{7}{9})^{2n-2}+{n-1 \choose
1}\frac{2}{9}(\frac{7}{9})^{n-2}(\frac{7}{9})^{2n-2}$. Similar to
$p_{wu}$, we have $p'_{wu}\leq(1-\frac{2}{9})^{2n-2}$. Thus
\begin{center}
$Pr[\overline{D(S)}|uv,vw,wu$ receive two colors$] \leq p'_{uv}+
p'_{vw} +p'_{wu}\leq(2n+1) (\frac{7}{9})^{2n-2}$
\end{center}
\emph{Subcase 3.3}: The edges $uv,vw,wu$ receive the same color.
Let $p''_{e}=Pr[E_{e}|uv,vw,wu$ receive the same color] for $e\in
\{uv, vw, wu\}$. Without loss of generality, we assume
$c(uv)=c(vw)=c(wu)=1$. If $uv$ is not used to construct a rainbow
$S$-tree, then one of the three situations below must occur:
$\bullet$ for any vertex $w'\in V_{3}\setminus\{w\}$,
$\{c(w'v),c(w'u)\}\neq \{2,3\}$, and at the same time, there exists
at most one vertex $u'\in V_{1}\setminus\{u\}$ satisfying
$\{c(u'v),c(u'w)\}= \{2,3\}$ and at most one vertex $v'\in
V_{2}\setminus\{v\}$ satisfying $\{c(v'u),c(v'w)\}= \{2,3\}$.
$\bullet$ there exists exactly one vertex $w'\in
V_{3}\setminus\{w\}$ satisfying $\{c(w'v),c(w'u)\}=\{2,3\}$, and at
the same time, there exists at most one vertex $s\in
(V_{1}\setminus\{u\})\cup (V_{2}\setminus\{v\} )$ satisfying
$\{c(sv),c(sw)\}= \{2,3\}$ or $\{c(su),c(sw)\}= \{2,3\}$.
$\bullet$ there exist at least two vertices $w_{1},w_{2}$ in
$V_{3}\setminus\{w\}$ satisfying $\{c(w_{i}u),c(w_{i}v)\}= \{2,3\}$
for i=1,2, and at the same time, for any vertex $u^{\prime}\in
V_{1}\setminus\{u\}$, $v^{\prime}\in V_{2}\setminus\{v\}$,
$\{c(u^{\prime}v),c(u^{\prime}w)\}\neq\{2,3\}$ and
$\{c(v^{\prime}u),c(v^{\prime}w)\}\neq \{2,3\}$.
So, $p''_{uv}\leq (\frac{7}{9})^{n-1}{n-1 \choose 1}{n-1 \choose
1}(\frac{7}{9})^{2n-4}+{n-1 \choose
1}\frac{2}{9}(\frac{7}{9})^{n-2}{2n-2 \choose
1}(\frac{7}{9})^{2n-3}+[1-(\frac{7}{9})^{n-1}- {n-1 \choose 1}$
$\frac{2}{9}(\frac{7}{9})^{n-2}]\cdot(\frac{7}{9})^{2n-2} $.
Obviously, $p''_{vw}=p''_{wu}=p''_{uv}$. Thus,
\begin{center}
$Pr[\overline{D(S)}|uv,vw,wu$ receive the same color$]\leq p''_{uv}+
p''_{vw} +p''_{wu}\leq3(3n^{2}+1)(\frac{7}{9})^{2n-2}.$
\end{center}
By the law of total probability, we obtain
\begin{eqnarray*}
Pr[\overline{D(S)}]&=&\frac{2}{9}\cdot Pr[\overline{D(S)}|uv,vw,wu\ have\ distinct\ colors]\\
&\ \ &+\frac{2}{3}\cdot Pr[\overline{D(S)}|uv,vw,wu\ receive\ two\ colors]\\
&\ \ &+\frac{1}{9}\cdot Pr[\overline{D(S)}|uv,vw,wu\ share\ the\
same\ color ]\\
&\leq&\frac{2}{9}\cdot 3\cdot (\frac{7}{9})^{2n-2}+\frac{2}{3}(2n+1) (\frac{7}{9})^{2n-2}+\frac{1}{9}\cdot3(3n^{2}+1) (\frac{7}{9})^{2n-2}\\
&=&o(n^{-3}).
\end{eqnarray*}
Therefore, there exists a positive integer $N=N(\ell,r)$ such that
$Pr[\ \bigcap\limits_{S}D(S)\ ]>0$ for all integers $n\geq N$, which
implies $rx_{3,3}(K_{n,n, n})\leq 3$ for $n\geq N$.
\end{proof}
From Theorem \ref{thm2} and Lemma \ref{lem4}, the $(3,\ell)-$rainbow
index of $K_{r\times n}$ is totally determined for sufficiently
large $n$.
\begin{theorem}\label{thm3}
For every pair of positive integers $\ell,r$ with $r\geq3$, there
exists a positive integer $N=N(\ell,r)$ such that
\begin{displaymath}
rx_{3,\ell}(K_{r\times n}) = \left\{ \begin{array}{ll}
3 & \textrm{if $r=3, \ell=1,2,3 $}\\
4 & \textrm{if $r=3, \ell\geq4$} \\
3 & \textrm{if $r\geq4$}\ \ \ \
\end{array} \right.
\end{displaymath}
for every integer $n\geq N$.
\end{theorem}
With similar arguments, we can expand these results to more general
complete multipartite graphs $K_{n_{1},n_{2},\cdots,n_{r}}$ with
$n_{1}\leq n_{2}\leq\cdots\leq n_{r}$ and $n_{r}=O(n_{1}^{\alpha})$,
where $\alpha\in \mathbb{R}$ and $\alpha\geq1$.
\begin{theorem}\label{thm4}
Let $n_{1}\leq n_{2}\leq\cdots\leq n_{r}$ be $r$ positive integers
with $n_{r}=O(n_{1}^{\alpha})$, $\alpha\in \mathbb{R}$ and
$\alpha\geq1$. For every triple of positive integers $k,\ell,r$
with $k\geq 3$ and $r\geq 3$, there exists a positive integer
$N=N(k,\ell,r)$ such that
\begin{displaymath}
rx_{k,\ell}(K_{n_{1},n_{2},\cdots,n_{r}}) = \left\{ \begin{array}{ll}
k & \textrm{if $k< r$}\\
k \ or \ k+1 & \textrm{if $k\geq r, \ell\leq\frac{{r\choose2}\lceil\frac{k}{r}\rceil^{2}}{\lfloor\frac{k}{r}\rfloor}$} \\
k+1 & \textrm{if $k\geq r, \ell>\frac{{r\choose2}\lceil\frac{k}{r}\rceil^{2}}{\lfloor\frac{k}{r}\rfloor}$}\ \ \ \
\end{array} \right.
\end{displaymath}
for every integer $n_{1}\geq N$.
Moreover, when $k=r=3$, $\ell\leq\frac{{r\choose2}\lceil\frac{k}{r}\rceil^{2}}{\lfloor\frac{k}{r}\rfloor}=3$,
there exists a positive integer $N=N(\ell)$ such that $rx_{3,\ell}(K_{n_{1},n_{2},n_{3}})=3$ for every integer $n_{1}\geq N$.
\end{theorem}
\section{Concluding remarks}
In this paper, we determine the $(k,\ell)$-rainbow index for some
complete bipartite graphs $K_{m,n}$ with $m=O(n^{\alpha})$,
$\alpha\in \mathbb{R}$ and $\alpha\geq1$. But, we have no idea of
the $(k,\ell)$-rainbow index for general complete bipartite graphs,
e.g., when $m=O(2^{n})$, is $rx_{k,\ell}(K_{m,n})$ equal to $k+1$ or
$k+2$ ? The question seems not easy, since even for the simplest
case $k=2$, $rc_{\ell}(K_{m,n})=3$ or $4$ for sufficiently large
$m,n$ is still an open problem; see \cite{Fujita}.
It is also noteworthy that we use the bipartite Ramsey number in
Lemma \ref{lem2} to show that $rx_{k,\ell}\geq k+1$ for sufficiently
large $n$. But, unfortunately the multipartite Ramsey number does
not always exist; see \cite{David}. Instead, we analyze the
structure of $S$-trees in complete multipartite graphs and give some
lower bounds for $rx(K_{r\times n})$. Since these bounds are weak
(they do not involve coloring), we can not tell
$rx_{k,\ell}(K_{r\times n})=k$ or $k+1$ when $k\geq r$ and
$\ell\leq\frac{{r\choose2}\lceil\frac{k}{r}\rceil^{2}}{\lfloor\frac{k}{r}\rfloor}$,
except for the simple case $k=3$; see Lemma \ref{lem4}. An answer to
this question would be interesting.
|
1,116,691,497,752 | arxiv | \section{Introduction}
Automatic invoice processing systems gain significant interest of
large companies who deal with enormous numbers of invoices each day,
due to not only their legal value requiring them to be stored for
years but also for economic reasons. Cristani et
al in~\cite{cristani2018future} offer a comparison of \EUR{9} per manually
processed invoice and \EUR{2} per automated processing of one invoice
based on surveys in 2004 and 2003 respectively. A 2016 report by the
Institute of Finance and Management~\cite{iofm2016} suggested that the
average cost to process an invoice was \$12.90.
Several challenges on recognition and extraction of key texts from
scanned receipts and invoices have been organized recently, e.g.\ the
Robust Reading Challenge on Scanned Receipt OCR and Information
Extraction (SROIE) at ICDAR 2019~\cite{jaume2019funsd} or the Mobile-Captured
Image Document Recognition for Vietnamese Receipts at
RIVF2021~\cite{vu2021mc}.
Still, annotated benchmark invoice data\-sets are not generally
available due to confidential information, and the published papers do
not offer detailed dataset descriptions and error analyses of the
content. Moreover, although receipts and invoices have some common
attributes, their analyses differ vastly due to complex graphical
layouts and richer content of the required pieces of information used
in invoices.
In their experiments with recently published Kleister
datasets~\cite{stanislawek2021kleister} that include financial
reports of charity organizations and non-disclosure agreements,
Stanis{\l}awek et al.\ found that results using state-of-the-art models
drop vastly with long, complex layouts in comparison to SROIE.
In 2006, Lewis et al.~\cite{10.1145/1148170.1148307} published the IIT
Complex Document Information Processing Test Collection (IIT-CDIP)
based on the Legacy Tobacco Documents Library, containing roughly
40~millions scanned pages for evaluation of document information
processing tasks. In subsequent works, Harley et
al.~\cite{harley2015evaluation} took random samples of 16 categories
in IIT-CDIP, each with 25,000 images, for document image classification
evaluation. Nevertheless, none of those has benchmark results or gold
standard data for the information extraction task.
In this article, we present the latest development of the OCRMiner
project aiming at automated processing of (semi-)structured business
documents, such as contracts and invoices, based solely on the
analysis of OCR processing of the document pages. The current system
significantly extends the original pipeline~\cite{ha2018recognition}
for information extraction. We detail the improvements in each module,
i.e.\ more precise page segmentation as well as Long short-term
memory (LSTM) based OCR engine in text recognition, coping with OCR
errors in keywords using an approximate string matching algorithm, or
recognizing named entities using BERT based models. The presented
approach boosted the results for the English invoice dataset and
adapted well for Czech invoices.
The OCRMiner system
represents the documents as a graph of hierarchical text blocks with
automatic modular feature annotations based on keywords, text
structures, named entity processing and location parser. With these
features, a rule-based method is built to extract the desired
information. The core idea behind OCRMiner lies in the human-like
approach to the document content analysis. Different from other works
overviewed in the next section, OCRMiner employs rich annotation from
several text analysis modules in combination with text positional
attributes. It does not only consider the absolute position of the text in the page
presented by the bounding box, but also
the relation with other groups of text.
OCRMiner is also able to cope with OCR errors using
a similarity search keyword matching algorithm. The system is unique
in distinguishing both local (company name, address, ...) and global
information such as buyer/seller identification. The higher-level
modules of OCRMiner (invoice page classification, block type
detection, ...) are language independent and the system is thus
portable to new languages.
The next section brings a detailed overview of the latest results in
invoice information extraction. Section~\ref{sec:ocrminer} details the
insides of OCRMiner and its modules. In Section~\ref{sec:eval}, we offer
a detailed evaluation of the system based on two datasets containing
invoices in the English and Czech languages from tens of different
suppliers.
\section{Related works}
Previous works aiming at automated processing of OCR
invoices,\footnote{We denote as ``OCR invoices'' all kinds of invoice
documents processed as an image by an Optical Character Recognition
tool. This includes both scanned paper documents as well as digital
born documents such as PDF.}
rely on predefined sets of layout of text data associated with
a specific invoice format. One of the first
systems~\cite{bayer1997generic} used a specific programming language
for the description of the frame representation language for
structured documents (FRESCO). For both text blocks and tables,
keywords are located first, then the data items are extracted
according to the programmed rules. The system did not deal with other
items without a unique syntax. The recognition rate was between
40--60\%.
Adaptive rule-based systems that are based on layout information and
text keywords usually divide the rules into the \emph{vendor
dependent} and \emph{vendor independent}
knowledge~\cite{cesarini1997conceptual,cesarini2003analysis}. Cesarini
et al.\ in~\cite{cesarini2003analysis} introduces a multi-class
invoice analysis system (IAS) that interconnects the \emph{layout
structure} with the \emph{logical structure} of the analysed document
based on rules divided into the above mentioned two levels of
knowledge. A comparable approach is presented in the German system
named SmartFix~\cite{klein2004smartfix,schulz2009seizing}. In this
system, the invoices are classified into known and unknown classes
based on layout graph using case-base reasoning technique (similar
to~\cite{hamza2007case}).
Those systems need a high amount of annotated graph samples
to reach an acceptable result.
A template-based approach is proposed
in~\cite{bart2010information,esser2014few}.
In~\cite{bart2010information}, users have to provide one entry of
a table or a list in a document, then the system tries to find
repeated structures which have the same or similar format. These
candidates are evaluated by the overall match quality in term of
perceptual coherence features such as alignment, height, width,
separation and gaps. In~\cite{esser2014few}, their objectives are
towards small companies or single practitioners who cannot afford
a large training set nor
a customized system. The key idea is grouping documents generated by
the same template, then generating extraction rules for each group
based on at least a minimal number of similar training examples. The
final result is a combination of results from different indexers for
fields with a fixed value (e.g.\ the sender), a fixed position (e.g.\
the invoice number), a variant position and value (e.g. the total)
etc. User feedback is allowed to add annotated documents as a new
training example. For at least one training document per template, the
system reaches 78\% F1 score.
One of the main problems of processing invoices is the huge
variety of invoice layout formats used in practice. Although a known
set of metadata (i.e.\ the invoice number, date, the total, seller and
buyer) is obligatory in most cases, their layout and positions are
unconstrained. Therefore, many invoice analysis systems consider
a particular layout as a class, usually connected to a particular
vendor. If an invoice is assigned the correct class, then this will
boost the extraction performance. However, the system must manage
a large number of classes which may grow over time due to new
suppliers or to layout updates. Trying to avoid classifying invoice
into known and unknown classes which could lead to errors by
misclassification, Aslan et al.~\cite{aslan2016part} apply the
part-based modeling (PBM) approach from computer vision to invoice
processing based on deformable compositions of invoice parts
candidates obtained from machine learning based classification
techniques. The presented evaluation of invoice block detection ranges
from 69\% to 87\% with the average of 71\% accuracy.
Applying the results
from the deep learning area, the CloudScan
system~\cite{palm2017cloudscan} uses recurrent neural networks
(specifically the bidirectional long short-term memory
network~\cite{hochreiter1997long}) to learn the item dependencies
based on end-user provided feedback. The system reaches an F1 score of
84--89\% when trained on more than 300,000 invoices. A follow-up
system, named InvoiceNet~\cite{palm2019attend}, introduced a specific
deep network architecture denoted as Attend, Copy, Parse. The
architecture does not use word-level labels for the end-to-end
information extraction task, but it relies on experimentally designed
set of dilated convolution blocks for each of the operations. The
results improved the CloudScan results for four of the seven evaluated
item types.
Riba at al.~\cite{riba2019table} propose a method using Graph Neural
Networks (GNNs) for tabular layout detection based on the
classification of graph node embeddings. The method does not rely on
the document content although a commercial OCR tool is used for
encoding classifying entity textual attributes into numeric,
alphabetic, or symbol. The feature vector for each detected entity
(node of the graph) includes its bounding box position and a histogram
of textual attributes. An edge is created between two nodes if and
only if a straight horizontal or vertical line exists between bounding
boxes of the two nodes without crossing any other, excluding long
edges covering more than a quarter of the page. The best accuracy of
classifying 8~regions including address, table, or header lies between
82.7\% and 84.5\%.
Regarding the task of scanned invoice classification, in~\cite{tarawneh2019invoice} the authors proposed a method to
classify invoices into three categories: handwritten, machine-printed
and receipts, for the best selection of OCR approach in latter
processing. The features are extracted using a pre-trained Deep
Convolution Neural Network (CNN) Alexnet. The experiments with 1,380
documents showed that the K-nearest neighbors method achieved the best
result when compared to Random Forest (RF) and Naive Bayes (NB) with
98.4\% total accuracy. In a similar task, Jadli and
Hain~\cite{jadli2020automatic} test different combinations of
pre-trained CNN models (including VGGNet, Inception V3, DenseNet, and
MobileNet-V2), dimensional reduction techniques (Principal Components
Analysis, Linear Discriminant Analysis), and classification algorithms
with a small dataset of 200 samples involving electronic invoices,
hand written invoices, bank checks and receipts. The best result is
96.1\% using full features extracted using VGG19 and Logistic
Regression classifier. Similar techniques are used for document image
source classification with the results ranging from 93\% to 98\%
depending on the dataset
granularity~\cite{sharad2019first,tsai2019deep}.
CUTIE (Convolution Universal Text Information Extractor)~\cite{zhao2019cutie} applied
convolution neural networks to texts converted to semantic embeddings
in a grid-like structure.
In this work, position features are taken into account, however, the exact positional relations are only implied.
In~\cite{xu2020layoutlm,xu2020layoutlmv2}, the authors proposed LayoutLM(v2)
for pre-training text, layout, and image.
The idea is inspired by Bidirectional Encoder Representations from Transformers (BERT)~\cite{devlin2018bert}.
Although the image feature is used and the model works well on downstream tasks
such as form understanding~\cite{jaume2019funsd}, receipt understanding~\cite{huang2019icdar2019},
or document image classification~\cite{harley2015evaluation},
same as in CUTIE system, the potential relationship between text segments
has not been taken into consideration. In addition,
sufficient data and time are required to pre-train the model.
A very similar idea on using text along with its position
based on transformer architecture,
particularly BERT, is presented in~\cite{garncarek2020lambert}.
These systems need large datasets for pre-training models,
and smaller labelled datasets for fine-tuning. Such datasets
are not available by now for non-mainstream languages
like the Czech language.
Experiments by Hamdi et al~\cite{hamdi2021information} with invoice
information extraction of document type and number, dates, amounts,
and currency show that word classification approach (similar
to~\cite{palm2017cloudscan}) outperforms sequence labelling
methods~\cite{garncarek2020lambert} on most fields.
Recently, graph-based models have been proposed
either using Graph Convolutional Networks~\cite{lohani2018invoice,yu2020pick}
or Graph Attention Networks~\cite{liu2019graph,krieger2021information}
for sequence labelling
or node classification to identify the key items.
The graph-based document representations include positional relations between
text segments but they are limited to the direct
neighbours only. Furthermore, training the graph embeddings
needs a large amount of manually labelled document graphs.
\begin{figure}[t]
\centering
\includegraphics[height = 0.5\textheight]{pipeline_general_1.png}%
\caption{The processing pipeline}
\label{fig:pipe}
\end{figure}
A similar idea to the OCRMiner method that combines information
from multiple sources of the data types and textual
content in relative positions is found in~\cite{majumder2020representation}.
In that system, for each field to be extracted,
the generated candidates (e.g.\ dates, codes or numbers)
are scored by their neighbourhood word and position embeddings.
In contrast, OCRMiner begins with key phrases which drive
the search for the data types in the visual surroundings
inspired by the human-like reading of complex layouts.
A disadvantage of the other method lies in limiting
the neighbourhood to a fixed window.
Since the layouts of invoices vary a lot, this approach
can create unnecessary neighbours if the text is dense,
and miss neighbours where the keyword and data are
on the same line but distant. Moreover, OCRMiner not only
deals with key texts matching fixed patterns,
i.e.\ (invoice-, due-) dates or (invoice-, order-) numbers,
but identifies also fields such as company names, contact
persons, or general addresses.
\section{The OCRMiner pipeline}
\label{sec:ocrminer}
A natural approach that humans use to extract
specific information from a document involves
2 steps: scanning the document to locate the
position of the information and then extract it
in the desired format. Some types of information
have unique structure defined by human rules, e.g.\ a VAT number.
Other types need context to differentiate them from the same or other types of
information, for instance, a date is distinctly identified as the invoice date or the due
date by the keyword which goes with it. And others may need heuristic knowledge,
for example the seller's name or address. The OCRMiner pipeline is
generally inspired by this human approach.
An overall schema of the OCRMiner processing pipeline is illustrated
in Figure~\ref{fig:pipe}. The system is made of interconnected
modules that allow to add any kind of partial annotations to the
analysed document. First, the invoice image is processed by an OCR
tool.$\!$\footnote{See~\cite{ha2017recognition} for
details of the OCR setup.} When OCRMiner
needs to deal with multilingual documents,
detecting languages used in the document is essential, not only for
setting up OCR tool, but also for latter annotations. The
main language of a document is defined as the language used by the title and data
field names. For instance, in invoices, these are ``invoice'', ``date'',
``order number'', ``payment method'', etc. Therefore, given a dictionary
of those terms for each possible language, if the document contains
the terms of a specific language, then this language is assigned as
the main language of the document. Otherwise, the language of the
document is defined based on text language distribution detected by the OCR
tool in the first run with a multiple languages setup.
Then, the physical document layout is built
based on the word and character positional information from OCR
(Layout analysis module). The Page classification\footnote{The
OCRMiner page classification is described in detail in
Section~\ref{sec:first_page}.} is performed with the information
obtained by this step in the pipeline.
From
that point, a series of annotations using different natural language
processing (NLP) techniques is applied to the content in the form of
XML tagging, involving title, keywords, data types, addresses and
named entities. These annotations provide semantic information and
along with positional information, they create the context for further
analysis.
Based on these annotations, the logical structure, in
which each text block is assigned an informational type, is extracted
using a rule-based model. The final module extracts the desired
metadata information.
\begin{table*}[t]
\centering
\caption{Evaluation of the invoice first page detection module.}
\label{tab:first_page}
\begin{tabular}{r|c|c|c|c|c}
Model & freq\_words & title+& freq\_words+ & annotation+ & all \\
& &page\# &title+page\# & title+page\# &features \\
\hline
Naive Bayes Classifier:& & & & \\
Precision & 93\% & 89\% & 95\% & 93\% & 95\%\\
Recall & 94\% &96\% &96\% &\textbf{99\%} &98\%\\
F1 score & 94\% &92\% &95\% &96\% &96\%\\
\hline
Logistic Regression:& & & & \\
Precision & 97\% &95\% &\textbf{98\%} &95\% &97\%\\
Recall & 97\% &96\% &97\% &97\% &98\%\\
F1 score &97\% &95\% &\textbf{98\%} &96\% &\textbf{98\%}\\
\hline
Linear SVC:& & & & \\
Precision &96\% &96\% &93\% &90\% &94\%\\
Recall & 96\% &96\% &87\% &85\% &92\%\\
F1 score &96\% &96\% &88\% &86\% &93\%\\
\end{tabular}
\end{table*}
\subsection{Document physical layout analysis}
The analysis of document structure can be divided into the
\emph{layout structure} (also called physical structure or geometric
layout structure) and the \emph{logical structure}.
The layout structure analysis (or page segmentation) refers to the
process of segmenting a document page into groups of text lines, text
blocks, and/or graphical elements. In this paper, the physical layout
structure is built using a bottom-up approach to obtain the document
layout structure from the output of an OCR tool.
The invoice image is first processed by the OCR engine\footnote{The
open source Tesseract-OCR~\cite{smith2007overview} version 4.1.0 is used now.},
which computes the positions (bounding boxes) of each recognized word.
Within the analysis, the words are grouped into lines based on three
criteria: alignment, style, and distance. If two words have similar
alignment and style, and the distance
between them is less than a threshold then they are in the same line.
The threshold was derived from the histogram of distances between each
couple of adjacent words in 215 invoice images
and currently corresponds to the height of the first word in the line.
The identified lines form (tabular) text blocks
in a similar process, i.e.\ if two lines are
vertically aligned, they have nearly the same font size,
and the distance between them is less than a threshold,
then they are grouped into a block.
However, while the distances between words in a line usually
correspond to the space character, the distances between lines in
a block vary a lot depending on the graphical format.
The best block-line threshold was set as twice as the height of
previous line in the block.
After deciding the document physical layout, each block receives an
identification of its named positions in the form of block attributes,
including the absolute position in the page and the relative position
of the block to other blocks. The former divides a page into
header, top, middle, bottom, and footer in the
vertical direction, and left, right parts in the
horizontal direction whereas the latter searches
for the block's neighbors in the top, bottom,
left, right and bottom-right positions respectively.
\subsection{Page classification}
\label{sec:first_page}
\begin{figure*}
\centering
\includegraphics[width=.85\textwidth]{first_page.png}
\caption{Evaluation of invoice first page classifiers using
different features.}
\label{fig:first_page}
\end{figure*}
The OCRMiner pipeline includes two classification tasks to answer the
questions whether the document is actually an invoice or not and if
the page is the first page. The reason is that business documents such
as invoices are often scanned and processed in batches. OCRMiner thus
contains the detection of an invoice first page. Similar
to~\cite{wu2018visual}, the classification uses the available pieces
of the semantic content of the document. The extent of the set of the
available features depends on the module's position in the pipeline.
Usually, this classification step goes between the layout analysis and
the logical structure analysis. At this position,
the features include binary features for the most frequent
words obtained from the document collection for each
language excluding entity names and stop words,$\!$\footnote{In the
experiment, these binary features (True/False meaning the word
appears/does not appear in the text) are created for 111 words that
resulted from choosing 150 most frequent words and eliminating entity
names and stop words.}
the (identified) page title
(text, its position in the page and its size), and the (identified)
page number. In case when the classification module goes further in
the pipeline, the features are enriched with keywords and data type
annotations as well as block types. While the \emph{frequent words}
feature considers only individual words, the annotations bring the
logical meaning information. As an example, a part of the feature
vector from an invoice first page looks as follows:\footnote{The
prefix \texttt{u""} stands for a \emph{frequent word}, \texttt{"K\_"}
for a \emph{keyword} annotation, \texttt{"D\_"} for a \emph{data type}
annotation. The presented frequent words are ``dodavatel -- seller'',
``odběratel -- buyer'', ``faktura -- invoice'', ``doklad --
document'', ``splatnosti -- due date'', ``zdanitelného -- taxable''.}
\begin{quote}
\begin{tabbing}
\small\tt \{ \= \kill
\small\tt \{ \>\+ \small\tt u"dodavatel":~True, u"odběratel":~True,\\
\small\tt u"faktura":~False, u"doklad":~True, \\
\small\tt u"bankovní":~True, u"splatnosti":~True, \\
\small\tt u"zdanitelného":~False, \ldots{}, \\
\small\tt "title":~True, "top": 292, "height": 61, \\
\small\tt "page\#": 1, "K\_BUYER":~True, \\
\small\tt "K\_SELLER":~True, "K\_INVOICE NUMBER":~True, \\
\small\tt "K\_PAY\-MENT METHOD":~True, \+\\
\small\tt "K\_ORDER NUMBER":~False, \ldots{}, \\
\small\tt "D\_ORGANIZATION":~True, "D\_DATE":~True, \\
\small\tt "general info":~True, "buyer info":~True, \\
\small\tt "seller info":~True~\}
\end{tabbing}
\end{quote}
The classification output label is 1~if the page is a first page of an
invoice, and 0~for others. The latter means the page can be second or
further page of an invoice or not an invoice at all.
The first page classification module was evaluated with a Czech
dataset containing 1,150 pages (650 first invoice pages and 500 other
pages\footnote{The data set is a bit different from the one used
in~\cite{ha2017recognition}. In the current experiment, pages which
are not in Czech and extremely bad quality scans were removed and new
data was added.}) using 10-fold cross validation.
The experiments employed several fast classifiers,
see Table~\ref{tab:first_page} for the average
precision, recall and F1 score of Support
Vector Machines (Linear SVC), Logistic Regression, and Naive Bayes.
The effect of the non-word features is clear in the group of SVC
classifiers whereas the two other groups are not affected to such
extent. Surprisingly, just the identified title and the page number
can serve for 96\% precision and recall. Moreover, by adding the
annotations only (without the \emph{frequent words}), the module can
recognize up to 99\% of the invoice first page. Among the
classifiers, Naive Bayes reaches the highest recall of 99\% while
the Logistic Regression attains the best balance
between the precision and the recall with the highest F1 score of
\textbf{98\%} (this is in accordance with the results
of~\cite{jadli2020automatic}).
Other combinations of features and the visualization of the result are
illustrated in Figure~\ref{fig:first_page}.
\subsection{The annotation modules}
The annotation modules process the enriched OCR document and
supplement its content with valuable pieces of information based on
the output of natural language processing tasks. This part corresponds
to the human reader procedure, who identifies e.g.\ a name of a city
in various parts of the document and uses this information in the
global-level rules.
The first three modules in the series of task-oriented annotators operate
over plain text of recognised block lines to identify specific
keywords and structured data such as dates, prices, VAT number, and
IBAN.
Keywords include words or phrases which are the signposts
to find the desired piece of information. For instance,
keywords for the invoice date are ``invoice date'',
``date issued'', ``issued date'', ``billed on'', or
``date'' at the beginning of a line. The set of
keywords (per language) is obtained by analysing
invoices in the development set. These modules
can also cope with some OCR character-level errors.
The subsequent modules provide higher-level information such as named
entities (personal names, organizations, locations) and formatted
address specifications.
\begin{figure*}[t]\hfuzz=8pt
\begin{verbatim}
seller info -> block_annot.data in [ORGANIZATION, PERSON, LOCATION, CITY,
COUNTRY, EMAIL, PHONE] and SELLER in top_blocks.block_annot.keyword
and top_blocks.num_lines == 1
\end{verbatim}
\caption{Block type rule example: if annotations in the block contains
more than at least one label in the set {\tt \{ORGANIZATION, PERSON, LOCATION, CITY,
COUNTRY, EMAIL, PHONE\}} and the top block has only one line marking
the SELLER keyword, then the block type is {\tt seller info}.}
\label{fig:block_types}
\end{figure*}
\subsubsection{Named entity recognition}
The invoice content analysis relies heavily on the ability to identify
the relevant entities (typically person name, place name,
organization, or a product name, artwork, date, time) mentioned
in the tabular formatted texts.
The task of named entity recognition (NER) consists of two steps:
named entity identification (including named entity boundaries
detection in case of multi-word NER) and the entity classification.
OCRMiner currently employs pre-trained BERT NER model~\cite{bertlarge}
based on the Google BERT-LARGE model~\cite{devlin2018bert} for English
and the SLAVIC-BERT model~\cite{arkhipov2019tuning} for the Czech language.
Although current state-of-the-art results in NER look promising for
both English and Czech, the text style of invoices causes the general
models to miss important context properties. First of all, the text
chunks are rather short in invoice blocks when compared to the full
sentences in existing models. The next challenge is that uppercase
formatting (which is an important feature in the NER models for Czech
and English) is used more frequently in this type of documents, for
example, for headings or company names. The third problem is strong
multilingual characteristics of many invoices, e.g.\ in English
invoices, the names of organizations or streets can be in a different
(local) language.
\subsubsection{Location names recognition}
Beside the location entities detected by NER, OCRMiner uses a global
address parser \texttt{libpostal}~\cite{libpostal2020}, a statistical
model based on conditional random fields trained on Open Street
Maps\footnote{\url{http://openstreetmap.org/}} and Open
Addresses.$\!$\footnote{\url{http://openaddresses.io/}} The reasons
for this doubling reflect the importance of addresses in the invoice
metadata and the need to be able to solve possible conflicts in the
name classification. Each of the two modules has positive matches on
different data. For example, street names such as
\emph{Mahenova 9/181}
are well recognized by \texttt{libpostal}
while the NER module recognizes \emph{Mahenova} as
person name. On the other hand, the location names recognition matches
\emph{Konica Minolta Business} as an office building in the U.S. while
the NER module annotates it (correctly) as an organization name.
\subsection{Block type detection}
\label{sec:block_type}
After the annotation modules add all relevant local pieces of
information to the analyzed content, each block is assigned
possible block types. The invoice text blocks are categorized into:
\begin{itemize}
\item general information containing invoice number, order number,
date;
\item seller/buyer information including company name, address, vat
number, contact detail;
\item delivery information such as delivery address, date, method,
code and cost;
\item bank information covering the bank name, address, swift code,
account number;
\item the invoice title and page number.
\end{itemize}
Blocks that do not belong to any of these categories are assigned an
empty label.
The OCRMiner block type detection technique is based on a set of
logical rules that combine information obtained in the preceding pipeline
steps. The rules are in human readable and easy to edit form (see an
example in Figure~\ref{fig:block_types}). Each rule is applied to each
block in the invoice document. If a block meets the rule's condition,
then the label is added to the block type.
A detailed evaluation of this module with a prototype English dataset
is provided in~\cite{ha2018recognition}.
\subsection{Information extraction}
\begin{figure*}[t]
\centering
\includegraphics[width = .6\textwidth]{extract_diagram.png}
\caption{The information extraction workflow process.}
\label{fig:ext_diagram}
\end{figure*}
The extracted data can be a single piece of information such as the
invoice number, invoice date, the order number, the order date, the
due date, the payment date, the payment method, IBAN, or the swift
code, or a group of information such as the buyer information, seller
information and the delivery address. The first group can be divided
into smaller categories: ``DATE'', ``PRICE'', ``NUMBER'', and
``GENERAL''. The first two clusters usually attach keywords with data
types whereas the two others often have no rule for the data. Unique
structured information such as VAT number can be revealed without
keywords.
Each type of information has a confidence score attached
distinguishing the situations when only the keyword was found, the
data type matches, or both.
\filbreak
\noindent
For instance, with the VAT number as an
example:
\begin{verbatim}
{"key": VAT NUMBER,
"data": VAT NUMBER,
"key_conf": 0.7,
"data_conf": 0.8,
"combine_conf": 1.0}
\end{verbatim}
If both the keyword and the data type are found, then the confidence
score is set to 1.0. Otherwise, if only the keyword or the data type
is found for some reason (e.g.\ OCR errors), then the confidence is
lowered to 0.7, or 0.8 respectively. The confidence score values are
based on the development set evaluation.
The extraction workflow process is illustrated in
Figure~\ref{fig:ext_diagram}. First, the system searches for the
appropriate keywords in the annotations. This returns the block and the
line containing the keyword if the keyword is found. Then, if the line
does not contain the expected data, a weighted function is applied to
the right block and bottom/bottom-right block to see
if the data lies on the right of the keyword or under the keyword.
This function sets a high score to the correct data type, a lower score to
any other data found in the annotations, and a penalty for other
keywords founds. The reason is that the information is often organized in
columns where keywords are in the left column and data on its right.
The higher weighted block is more likely to contain the data. If the
keyword is not found, then the system will look at the data type in
case it is structured information. The annotation steps are able to
cope with some OCR errors, so, the validation step checks and corrects
the data. For example, the Czech VAT number begins with ``\texttt{CZ}''
and not ``\texttt{C2}'' or the email symbol is ``\texttt{@}'' and not
``©''.
\begin{figure}[b]
\centering
\begin{minipage}{.90\columnwidth}
\begin{verbatim}
If block is vertically aligned or horizontally
aligned with SELLER block and
has no different content in [COMPANY,
ADDRESS, ID, VAT NUMBER]
compared to SELLER,
then label is SELLER.
\end{verbatim}
\end{minipage}
\caption{A buyer/seller classification rule example}
\label{fig:selbuy_rule}
\end{figure}
\begin{figure*}[t]
\centering
\begin{tabular}{ll}
~~a) & ~~c) \\
\setlength{\fboxsep}{1pt}%
\fbox{\includegraphics[width = .25\textwidth,valign=t]{DesignThinker_img.png}}%
& \includegraphics[width = .69\textwidth,valign=t]{Design_extract.png}\\
~~b)&\\
\multicolumn{2}{l}{\includegraphics[width = 0.97\textwidth]{Design_annot.png}}
\end{tabular}
\caption{An example of an address block analysis: a) the input
image, b) after annotation, and c) the extracted information.}
\label{fig:pipeline_eg}
\end{figure*}
The blocks with buyer and seller information are not always
accompanied by keywords and their information may even be spread into
several blocks. For those blocks, the information is classified into
buyer's or seller's using a set of logical rules. These rules consider
the block position in the page (e.g.\ prevalently, the seller's
contact information is located at the bottom of the page), and
alignment and content comparison with the found seller/buyer blocks.
Finally, global rules based on the observation that the seller block
often comes before the buyer block in vertical or horizontal direction
are applied to blocks unprocessed in previous steps. Currently,
OCRMiner uses 15 classification rules and 5 global rules written in
descending confidence order. If a block satisfies a rule, then the
corresponding label is returned and the system ignores the remaining
rules. A classification rule example can be found in
Figure~\ref{fig:selbuy_rule}.
Different from other types of information which are mostly expressed
in a text line, addresses usually spread in more than one line, in
particular cases even with other text in between. Therefore, the
address extraction does not simply extract an annotated
\emph{location} entity. In this step, the Libpostal address parser is
employed with detailed address parts recognition.
The extraction step takes ``road'' (street name) and ``house\_number'' as
important signpost to start an address. The labels are sorted in the
order of ``country'', ``city'', ``city\_district'', ``suburb'', ``postcode'',
``road'', ``house\_number'', ``house''. The labels at the beginning of the
list have higher rank than the labels at the end. Then, the text line
containing the highest ranked label is considered as the end of the
address. If the extracted address contains a label at the beginning of
the list, i.e.\ a country or a city then the confidence number is
higher than for the other labels (the address may miss some part of
information). A LOC entity annotation also increases the confidence
number. Example of an address block analysis is presented in
Figure~\ref{fig:pipeline_eg}.
\section{Experiments and evaluation}
\label{sec:eval}
\subsection{The invoice dataset}
The experimental invoice dataset is collected in cooperation with
a renowned copy machine producer. Due to the sensitiveness of the
invoice content, the dataset is not publicly available. The collection
consists of two main parts: English invoices and Czech invoices, see
Figure~\ref{fig:example}. The English group consists of invoices from
more than 50 suppliers all over the world whereas the Czech data comes
from over 100 vendors in the Czech republic.
\begin{figure*}
\centering
\begin{tabular}{ll}
\includegraphics[width=0.4\textwidth, valign=t]{161130071_fa_192-1_mod_shorten.png}
& \includegraphics[width=0.4\textwidth, valign=t]{161130071_fa_192-1_mod_annotation_with_label.png}\\
a) Invoice with the desired information
& b) Invoice with annotations\\
\end{tabular}
\caption{The input invoice dataset examples}
\label{fig:example}
\end{figure*}
In each data set, for developing and testing purposes 60 invoices were
randomly selected. Those 120 invoices are from 76 vendors.
Each vendor has a different layout, even the layouts of
invoices from the same vendor change over time.
Ten out of the 60 are used as a development set.
For instance, the ten English development invoices
are by nine different suppliers in
Austria, Poland, US, UK, the Netherlands, Germany, and Italy. More
than a half of the 50 testing invoices are from 18 other suppliers
which have not appeared in the development set.
The annotated XML files expose detailed positional information of
7,602 text blocks,
18,672 lines, and 9,096 automated annotations of 64 annotation types.
The files have also been manually annotated for the metadata items
containing altogether 1,523 values of 24 types for the information
extraction process.
A detailed overview of the information items is displayed in
Table~\ref{tab:item_types}.
\subsection{Evaluation}
In the system evaluation,$\!$\footnote{The information extraction task
is evaluated independently from the page classification
module which is fully described and evaluated in
Section~\ref{sec:first_page}.} the extracted field values are classified
into 3 result classes. Each piece of information is evaluated as
a \emph{match}, a \emph{partial match}, or a \emph{mismatch}. For
other data than company name and address, the \emph{match} means
always an exact match. With company name and address, the
\emph{match} classification allows to ignore differences in
unimportant pieces of the gold standard values. A \emph{match} is
assigned if the extracted data contains the main information, even
though some words are missing or surplus.
For example, a company name of:
\begin{quote}
\begin{description}
\item[ground truth:]~\\\raggedright
\texttt{Konica Minolta Business Solution Czech spol, s.r.o.}
\item[extracted:]~\\\raggedright
\texttt{Konica Minolta Business Solution Czech spol, s.r.0}
\end{description}
\end{quote}
are considered a \emph{match}. In this example,
the letter ``o'' in ``s.r.o.'' was recognized as ``0'' (zero)
by the OCR engine. In this context, such OCR errors
are easy to identify and fix as the ``tail'' of the
name contains the known organization type (``s.r.o.''
in Czech corresponds to ``Ltd.'' in English).
Or the address:
\begin{quote}
\begin{description}
\item[ground truth:]~\\\raggedright
\texttt{Level 6, 341 George St, Sydney NSW 2000, Australia}
\item[extracted:]~\\\raggedright
\texttt{Atlassian Pty Ltd, Level 6, 341 George St, Sydney NSW
2000, Australia}
\end{description}
\end{quote}
are also regarded as \emph{match}ing.
\begin{table*}[t]
\centering
\caption{The most common item types in the development and test sets}
\label{tab:item_types}
\begin{tabular}{l|r|r|r|r}
Type of info & En dev set & En test set & Cz dev set & Cz test set \\
\hline
Invoice number & 10~~ & 45~~ & 10~~ & 49~~ \\
Order number & 6~~ & 12~~ & 6~~ & 8~~ \\
Swift & 6~~ & 8~~ & 6~~ & 20~~ \\
Account number & 8~~ & 9~~ & 8~~ & 45~~ \\
Page number & 4~~ & 7~~ & 4~~ & 16~~ \\
Invoice date & 10~~ & 47~~ & 10~~ & 46~~ \\
Due date & 7~~ & 23~~ & 8~~ & 45~~ \\
Payment date & 8~~ & 10~~ & 8~~ & 37~~ \\
Total due & 9~~ & 27~~ & 9~~ & 25~~ \\
IBAN & 7~~ & 9~~ & 7~~ & 22~~ \\
Payment method & 8~~ & 18~~ & 8~~ & 34~~ \\
Company name & 23~~ & 101~~ & 23~~ & 114~~ \\
Contacts & 6~~ & 24~~ & 6~~ & 21~~ \\
Address & 25~~ & 101~~ & 25~~ & 119~~ \\
Vat number & 18~~ & 59~~ & 18~~ & 91~~ \\
Email & 8~~ & 24~~ & 8~~ & 35~~ \\
Website & 3~~ & 25~~ & 3~~ & 23~~ \\
Phone number & 15~~ & 17~~ & 15~~ & 40~~ \\
Company id & 0~~ & 0~~ & 20~~ & 99~~ \\
Amount paid & 0~~ & 21~~ & 0~~ & 0~~ \\
\hline
\end{tabular}
\end{table*}
\begin{table}[b]
\begin{center}
\caption{Information extraction evaluation results for the English set}
\label{tab:ex_eval_en}
\begin{tabular}{ l|r|r|r|r| }
& \multicolumn{2}{c}{regular expressions} & \multicolumn{2}{|c|}{similarity search}\\
\cline{2-5}
& match & in \% & match & in \% \\
\hline
Match &523 &84.22 & 527 & \textbf{84.86} \\
Partial match &35 & 5.64 & 35 & \textbf{5.64} \\
Mismatch &63 & 10.14 & 59 & \textbf{9.50} \\
\hline
Total &621 &100.00 & 621 & 100.00 \\
\end{tabular}
\end{center}
\end{table}
A \emph{partial match} is assigned if the extracted information
contains correct information, but some part is missing or some
character is incorrectly recognized by the OCR tool. For instance, if
the OCR tool recognized ``\texttt{0}'' character as ``\texttt{O}'' in the
order number ``\texttt{BXOEF24CA4E2}'' instead of ``\texttt{BX0EF24CA4E2}'', or
an address misses the street name:
\begin{quote}
\begin{description}
\item[ground truth:]~\\\raggedright
\texttt{Zarosicka 4395/13, Brno, Jihomoravsky kraj 62800,
Czech Republic}
\item[extracted:]~\\\raggedright
\texttt{Brno, Jihomoravsky kraj 62800, Czech Republic}
\end{description}
\end{quote}
The \emph{partial match} is ``correctable'' in the sense of finding
regular OCR character mismatches or via the possibility to search for
the expanded value in the identified block.
The comparison between the ground truth and the extracted information
in the evaluation is first preprocessed automatically using the
Levenshtein distance. The result returns a \emph{match} if the
distance is zero, a \emph{partial match} if the edit distance is less
than a threshold (set to~2 characters in the experiment) and
a \emph{mismatch} otherwise. Then, due to the complexity of address
and company name blocks, the evaluation is double-checked manually.
For the group-type information such as seller and buyer, the result
counts are counted for each subfield, i.e.\ the company name, the
address, and so on. This means that even if all information including
company name, address, company id, vat number are correctly extracted
but the group is wrongly classified as seller or buyer, then the
evaluation module will count all the subfields as mismatches.
\begin{table}[t]
\begin{center}
\caption{Information extraction evaluation results with the Czech set}
\label{tab:ex_eval_cs}
\begin{tabular}{ l|r|r|r|r| }
& \multicolumn{2}{c}{regular expressions} & \multicolumn{2}{|c|}{similarity search} \\
\cline{2-5}
& match & in \% & match & in \% \\
\hline
Match &679 & 75.28 & 756 & \textbf{83.81} \\
Partial match &30 & 3.33 & 40 & \textbf{4.43} \\
Mismatch &193 & 21.40 & 106 & \textbf{11.75} \\
\hline
Total &902 &100.00 & 902 & 100.00 \\
\end{tabular}
\end{center}
\end{table}
As can be seen in the evaluation of the page classification module,
the keyword features are the most important in the correct
identification of the block types. To be able to cope with OCR
character misclassifications, OCRMiner contains two different modules
to search for keywords. The first module uses regular expressions
patterns and the second module is based on an approximate
similarity search. Within the similarity search, the text substrings
are evaluated with weighted edit distance
(Levenshtein distance) measure which includes
a small cost (0.1 in the experiments) for a substitution of common
errors (such as `\texttt{l}', `\texttt{t}' and `\texttt{f}', or
`\texttt{u}' and `\texttt{v}') and insertion or deletion of
punctuation or white space, while any other edits have the cost of
1.0\footnote{See~\cite{ha2019approximate} for more details.}.
The substring of the text line which has the weighted edit distance
less than a threshold (15\% of the length of the keyword) is marked as
the keyword. The results of these two modules for the English and
Czech evaluation sets are presented in Tables~\ref{tab:ex_eval_en}
and~\ref{tab:ex_eval_cs}. The similarity search module adapts better
to OCR errors, which leads to slight improvements in both the
languages when compared to the method using regular expressions match
-- 0.64\% or 3.22\% of mismatches are solved or partially solved in
the English and Czech datasets respectively.
In general, the whole system works better for the English set where
above 90\% of ground truth items are correctly identified (with exact
or partial match) compared to about 88\% of the Czech invoice items.
\begin{figure*}[t]
\centering
\begin{tabular}{ll}
~~a) & ~~b) \\
\includegraphics[width = .47\textwidth,valign=t]{en_ind_bert_perc.png}%
& \includegraphics[width = .47\textwidth,valign=t]{cs_ind_bert_perc.png}%
\end{tabular}
\caption{Evaluation of individual fields for a) the English
invoices, and b) the Czech invoices.}
\label{fig:w_his}
\label{fig:ind_eval}
\end{figure*}
A detailed analysis of the extraction accuracy for each field is
illustrated in Figure~\ref{fig:ind_eval}.
Both datasets have high match ratios at the page number, the invoice
date, the due date, and the payment method. The extraction works
better for the English invoices with items such as the invoice number,
the VAT number, IBAN, the total due amount, or the website.
In contrast,
the Czech dataset has better extraction results with the order
number, the account number, the payment date, and the contact
information. There is an exceptional low match ratio for the payment
date in the English dataset due to OCR errors in a subset of English
invoices from a particular provider.
English invoices also witness a high partial match ratio
with the order number. Different from Czech invoices,
where the order number is a digital series, in the English dataset,
the order number is often a mixture of numbers and letters, for example
``AP0CDF89CC10''. The first number ``0'' is between letters,
and the OCR engine often misrecognises it as the letter ``O'',
causing the high partial match.
Another high mismatch is the contact person item which
is caused by the situation when NER misrecognized a person as
an organization due to capitalized letters.
A detailed error analysis is given in Section~\ref{sec:error_analysis}.
\hbadness=1500
For a performance comparison, the OCRMiner dataset has been evaluated
with the state-of-the-art InvoiceNet
system\footnote{\url{https://github.com/naiveHobo/InvoiceNet}} which is
based on the work of Palm et al.~\cite{palm2019attend}. InvoiceNet
solves the end-to-end information extraction task using deep neural
network models.
Six common invoice fields were selected for the evaluation: the invoice number, the
invoice date, the seller's name and address and the buyer's name and
address. Each field requires a separate model.
Since the original pre-trained
models are not publicly available, InvoiceNet has now been trained in
three possible setups. First, the InvoiceNet models have been trained
with the SROIE datasets~\cite{jaume2019funsd} including 727~receipts
(denoted as InvNet1). The second system (InvNet2) has been trained
with 208 English invoices of the OCRMiner dataset and the last one has
been trained with 500 Czech invoices from the same dataset (InvNet3).
InvNet1 and InvNet2 are evaluated with the English test set
whereas InvNet3 is evaluated with the Czech test set. The
results are summarised in Table~\ref{tab:invnet}.
\begin{table*}[t]
\caption{Performance comparison on specific fields in \%}
\label{tab:invnet}
\centering
\begin{tabular}{l|l|c|c|c||c|c|}
& & \multicolumn{3}{c||}{English data} & \multicolumn{2}{c|}{Czech data} \\
field & measure & InvNet1 & InvNet2 & OCRMiner & InvNet3 & OCRMiner \\
\hline
inv\_number & match & - & 46.67 & \textbf{97.78} & 83.67 & \textbf{87.76}\\
& partial match & - & 0.00 & 0.00 & 4.08 & 0.00\\
& mismatch & - & 53.33 & 2.22 & 12.24 & 12.24\\
\hline
inv\_date & match & 0.00 & 0.00 & \textbf{100.0} & 0.00 & \textbf{95.65}\\
& partial match & 0.00 & 0.00 & 0.00 & 0.00 & 2.17\\
& mismatch & 100.0 & 100.0 & 0.00 & 100.0 & 2.17\\
\hline
seller\_name & match & 12.24 & 44.90 & \textbf{83.67} & 77.27 & \textbf{79.55}\\
& partial match & 0.0 & 2.04 & 0.00 & 9.09 & 9.09\\
& mismatch & 87.76 & 53.06 & 16.33 & 13.46 & 11.36\\
\hline
seller\_addr & match & 4.08 & 6.25 & \textbf{81.63} & 2.00 & \textbf{86.00}\\
& partial match & 16.33 & 87.50 & 14.29 & 82.00 & 4.00\\
& mismatch & 79.59 & 6.25 & 4.08 & 16.00 & 10.00\\
\hline
buyer\_name & match & - & 71.11 & \textbf{86.67} & 82.00 & \textbf{84.00}\\
& partial match & - & 2.22 & 8.89 & 8.00 & 2.00\\
& mismatch & - & 26.67 & 4.44 & 10.00 & 14.00\\
\hline
buyer\_addr & match & - & 2.22 & \textbf{66.67} & 0.00 & \textbf{80.00}\\
& partial match & - & \textbf{93.33} & 20.00 & 30.00 & 8.00\\
& mismatch & - & 4.44 & 13.33 & 70.00 & 12.00\\
\end{tabular}
\end{table*}
Overall, OCRMiner performs better than In\-voice\-Net in all fields with
both the English and the Czech dataset. InvoiceNet locates the
position of addresses in English invoices better than OCRMiner but
fails to extract the full data. In multiple lines, multi-word text data
(e.g.\ an address or a company name) are identified only partially
with important parts missing. In most cases, only the street name and
house number are extracted. The results also show that a system
trained with receipts cannot be generalised to invoices.
\begin{table}
\caption{Evaluation of location detection by the combination of
Libpostal and Named entity recognition (NER) modules.}
\label{tab:loc_detect}
\centering
\begin{tabular}{ l|r|r|r|r }
English set: & \multicolumn{2}{l|}{Libpostal} & \multicolumn{2}{l}{NER} \\
\hline
both modules & 287 & 25\% & 287 & 83\% \\
one module only & 876 & 75\% & 57 & 17\% \\
~--~true positives & 230 & 20\% & 23 & 7\% \\
~--~false positives & 646 & 55\% & 34 & 10\% \\
\hline
\end{tabular}
\smallskip
\begin{tabular}{ l|r|r|r|r }
Czech set: & \multicolumn{2}{l|}{Libpostal} & \multicolumn{2}{l}{NER} \\
\hline
both modules & 165 & 12\% & 165 & 59\% \\
one module only & 1223 & 88\% & 117 & 41\% \\
~--~true positives & 292 & 21\% & 116 & 41\% \\
~--~false positives & 931 & 67\% & 1 & 0\% \\
\hline
\end{tabular}
\end{table}
\begin{table}[t]
\caption{Ablation study with the English dataset}
\label{tab:ablation}
\centering
\begin{tabular}{l|c|c|c}
Features & match & partial match & mismatch\\
\hline
full model & 84.86\% & 5.64\% & ~9.50\% \\
without keywords & 61.06\% & 3.04\% & 35.90\% \\
without data & 49.20\% & 1.12\% & 49.68\% \\
\hline
\end{tabular}
\end{table}
\begin{table*}[t]
\caption{Detailed analysis of the error types}
\label{tab:error_types}
\begin{center}
\renewcommand{\tabcolsep}{4pt}
\begin{minipage}[t]{.35\textwidth}
\begin{tabular}{|r||r|r|r|r|}
\hline
\multicolumn{5}{|c|}{Mismatch} \\
\hline
error types & \multicolumn{2}{c|}{En set} & \multicolumn{2}{c|}{Cz set}\\
\hline
Misclassified role & 27\% & 16 & \textbf{33\%} & 35\\
OCR errors & 10\% & 6 & 10\% & 11\\
Misclassified NE & \textbf{41\%} & 24 & 7\% & 7\\
New keywords & 12\% & 7 & 12\% & 13\\
Layout & 3\% & 2 & 17\% & 18\\
Other & 7\% & 4 & 21\% & 22\\
\hline
\textbf{Total} & 100\% & 59 & 100\% & 106\\
\hline
\end{tabular}
\end{minipage}\hspace{3em}%
\begin{minipage}[t]{.3\textwidth}
\begin{tabular}{|r||r|r|r|r|}
\hline
\multicolumn{5}{|c|}{Partial matches}\\
\hline
error types & \multicolumn{2}{c|}{En set} & \multicolumn{2}{c|}{Cz set}\\
\hline
OCR errors & 34\% & 12 & \textbf{66}\% & 29\\
Addresses & \textbf{46\%} & 16 & 14\% & 6\\
Other & 20\% & 7 & 20\% & 9\\
\hline
\textbf{Total} & 100\% & 35 & 100\% & 44\\
\hline
\end{tabular}
\end{minipage}
\end{center}
\end{table*}
Table~\ref{tab:loc_detect} displays an evaluation of the location
detection by the two modules employed in the process -- Libpostal and
the named entity recognition (NER) module.
Both modules mostly agree on city and country names.
Libpostal offers detailed annotation types of ``road'',
``postcode'', ``suburb'', ``city\_district'', ``city'',
``state district'', ``state'', and ``country'' labels. Of these,
the postcode and street are the most important
regarding the final scoring.
Consequently, Libpostal module has a higher recall
than the NER model,
while the NER module offers better precision.
Libpostal is also not so sensitive to uppercasing,
e.g.\ BRNO (a city), or CZECH REPUBLIC.
This is why an ensemble module is used in the system which allows to
take advantage of each model's capabilities.
The system also requires a combined
information support of at least two detected address items in the
analysed text block to eliminate most of false positive errors.
To better understand the contribution of keyword and data annotations
to the extraction, a set of ablation test were run for OCRMiner. In
each test, one of those two annotation types is left out and the
information extraction is based on the remaining annotations. The
extraction process can still identify some fields but with a decreased
value of the confidence score.
The overall results evaluated with the English test set are shown in
Table~\ref{tab:ablation}. As expected, when keywords are missing, the
match ratio has fallen off for fields relying on the keyword only,
such as the invoice number, the SWIFT code, or the account number with
the drop of 97.78\%, 87.75\%, and 77.78\% respectively. The textual
fields such as the company name or address and the contact person are
slightly affected with the drop of 0.99\%, 4.16\%, and 6.93\%
respectively. In contrast, these fields' \emph{match} ratios drop
quickly (by 62.38\%, 58.33\%, and 73.27\%) when the data annotations
are absent.
\subsection{Error analysis}
\label{sec:error_analysis}
A detailed analysis of errors in the system output was evaluated based
on the results of the similarity search module. The causes of errors can
be categorized into five groups, but their ratios in each dataset are not
the same.
Different from other pieces of extracted information, the seller/buyer/delivery
addresses are groups of information. They are
often not denoted by specific keywords, fragmented into several
blocks and their exact distinction is left to the reader.
To extract this information, the system needs to classify the
specified blocks into the seller, the buyer and the delivery address
based on their position and content.
27\% of errors in the English invoices and 35\% of errors in
the Czech ones
are correctly extracted but misclassified in these three categories.
The combination of reasons for this type of errors includes errors in
detecting the addresses and named entities (false positives),
insufficient rule details, and an extra variety in the layout formats
(e.g.\ the position relation between the seller and the buyer without
a \emph{keyword} distinction) that makes the global rules conflicting
in the difficult cases.
While the misclassification causes the largest portion of errors
in the Czech invoices (33\%), incorrect identification of entities
forms the main reason of mismatches in the English invoices
(41\% of errors) including persons classified as
organizations (``PETR GOTTHARD'', ``EVA LYSONKA'',...)
or undetected organizations and locations.
OCR errors and unknown keywords cause almost
the same portion of errors in each dataset with 10\% and 12\% respectively.
The Czech invoices display a broad range of layout formats which is
projected to 17\% of layout errors compared to only 3\% in the English set.
Most of these errors happen with special cases when the keyword and the data
are in an unexpected relative position, e.g.\ the keyword below the
data, or one line of a company info in a distant stand-alone block.
The error analysis of partial matches revealed that
the OCR errors cause 64\% of errors for English, and 34\% for the Czech invoice items.
A significant portion of these errors happens
in email addresses where the ``at'' symbol ``\texttt{@}'' is not
recognized correctly in the Czech invoices with the output being
recognized as ``\texttt{\&}'', ``\texttt{\&\&}'', ``\texttt{Q}'', or ``©''. Another
typical OCR error is a duplicated number. For example, in the VAT
number ``\texttt{CZ\underline{00}176150}'' the ``\texttt{0}'' is
doubled to ``\texttt{CZ\underline{000}176150}''.
Besides OCR errors, a missing part of address is the reason for
46\% of partial matches within the English invoices and for 16\% in the Czech invoices
respectively. The extraction module only looks for (parts of) addresses in continuous lines.
Therefore, in cases where some other pieces of information are inserted in those lines, e.g.\
a VAT number, a part of the address is misidentified.
An overview of all
the error types is presented in Table~\ref{tab:error_types}.
\section{Conclusions and future directions}
In this paper, we have presented the current results of the OCRMiner
system for extracting information from semi-structured scanned
business documents. OCRMiner is designed to process the document in
the same way as a human reader employing a combination of text
analysis techniques and positional layout features.
The system has been evaluated with two invoice datasets (English and
Czech) containing data from more than 150 different vendors from all
over the world. Using an open source OCR, with a small development
set consisting of only 20 invoices, the system is able to detect
\textbf{90.5\%} items in the English invoices and \textbf{88.2\%} in
the Czech invoices with 93--96\% of these item values being
{correct} matches. OCRMiner is also able to detect whether an image is
(the first page of) an invoice or not with \textbf{98\%} accuracy.
Such result is fully comparable to the
state-of-the-art systems which are, however, trained on much larger
datasets. The main advantages of OCRMiner lie in employing rich text
analysis annotations combined with the positional properties of each
text block. With underlying language dependent modules (OCR, keywords
and named entities), the higher-level modules are language independent
and offer a straightforward application to a new language without
a need of huge annotated invoice datasets. The system is also able to
cope with OCR errors using similarity search keyword detection.
The system tries to mimic the extraction procedure used by humans
when reading an invoice. Even when the layout changes,
the principle that the data is located around keywords
(often on the right or under, or in some cases, above)
should not change. This allows OCRMiner to adapt to most
possible new layout varieties.
Although, as is mentioned in the error analysis,
the rules have limited effect in distinguishing the buyer
and seller blocks when there are no keywords attached
to them. In this case, the rules cover the most
frequent possibilities, but with some invoices they may
even conflict with the designer's decision. For example,
the seller mostly appears on the left of the buyer but,
in some rare cases, the seller and buyer position are swapped.
In the future version, an ensemble seller/buyer distinguishing model
with a supervised learning method will be tested as preliminary
experiments show promising results.
Another possibility to improve the performance lies in
the layout analysis step. A portion of errors are
caused by block fragmentations where the desired data is
disconnected from the keywords and the surrounding context. Identification
of such remote pieces of information as e.g.\ parts of the seller or
buyer address can further boost the accuracy of extracted information.
One possible direction is taking the advantage of
drawn tabular lines that separate invoice parts but that are ignored
by the OCR process.
In the ongoing research, OCRMiner is currently adapted to the
information extraction task from other types of business documents
such as contracts, reports, orders, or meeting minutes.
\begin{acknowledgements}
This work has been partly supported by the Ministry of Education of CR
within the LINDAT/\discretionary{}{}{}CLARIAH-CZ research
infrastructure LM2018101 and by Konica Minolta Business Solution Czech
within the OCRMiner project.
\end{acknowledgements}
\hbadness=10000
\bibliographystyle{splncs}
|
1,116,691,497,753 | arxiv | \section{Introduction}
Multicore optical fibers are devices of great importance.
%
In telecommunications, they increase transmission bandwidth via space-division multiplexing \cite{Richardson2013} by parallel transmission of, either, signals though different uncoupled cores \cite{Zhu2010,Takara2012,Hu2018}, or normal modes arising from coupled cores \cite{Xia2011,Arik2013}.
In sensing, they provide a powerful platform for detection due to the high sensibility of the inter-core couplings to external conditions of the fiber; for example, sensors for curvature, refractive index and temperature have been experimentally demonstrated \cite{Kim2009,Moore2012,SalcedaDelgado2015,Zhao2016,MayArrioja2017a,MayArrioja2017b}.
Both in telecommunication and sensing applications, it is crucial to describe the inter-core cross-talk as accurately as possible.
In multicore fibers, when cores are not excessively close to each other, different cores are coupled by the evanescent tails of their individual modes.
This allows the use of coupled-mode theory to approximate electromagnetic field propagation by a linear combination of the individual core modes \cite{Snyder1972,Snyder1988,Huang1994},
\begin{equation}
-i \frac{\mathrm{d}}{\mathrm{d}z} \vec{\mathcal{E}}(z) = M \cdot \vec{\mathcal{E}}(z),
\end{equation}
where the vector $\vec{\mathcal{E}}(z)$ collects the amplitudes for each individual core mode and $M$ is the mode-coupling matrix that stores the information about effective propagation constanst for the individual core modes in the diagonal and the effective couplings between these modes in the off-diagonal elements.
The normal modes of the multicore structure are determined by the eigenvectors of the mode-coupling matrix
and their propagation constants are the corresponding eigenvalues.
\begin{figure}[htbp]
\centering
\fbox{\includegraphics{Fig1}}
\caption{
(a) Array of $n$ identical cores distributed in a circle of radius $R$ around a, possibly-different, central core.
The cores in the circle have radius $r_e$ and the central core has radius $r_c$.
(b)--(d) show the array when the circle has five, six, and seven cores respectively.
}
\label{Fig:1:Structure}
\end{figure}
We focus on a multicore optical fiber with an underlying cyclic symmetry.
It is composed by $n$ identical cores evenly distributed on a ring around a possibly-different central core, Fig.~\ref{Fig:1:Structure}.
This structure, both with and without the central core, has been studied before using nearest-neighbor coupled-mode theory.
In a seminal paper \cite{Snyder1972}, Snyder reduced the problem to that of two coupled effective modes, one with nonzero amplitudes in each of the outer cores and another with nonzero amplitude in just the central core.
Yamashita \textit{et al.} used symmetry considerations and Maxwell's equations to study the circular array without a central core \cite{Yamashita1985}.
Kishi \textit{et al.} followed this approach to study particular cases of the circular array with a central core \cite{Kishi1986,Kishi1988}.
The nearest-neighbor coupled-mode theory solution for the circular array is well-known \cite{Schmidt1991,Hudgings2000}.
It has been used to study the commensurability of propagation constants \cite{Rubenchik2013} and the stability of nonlinear generalizations \cite{Hizanidis2004,Hizanidis2006,Radosavljevi2015}.
In particular, this approximation shows good agreement between theory and experiment for the seven-core array \cite{Mortimore1991,Takenaga2010,Koshiba2011,Chan2012,Chekhovskoy2017,Hossain2017}.
However, longer propagation distances and higher sensing precision requires going beyond nearest-neighbor interactions \cite{Kishi1988}.
Group-theoretical techniques are a powerful tool to describe electromagnetic field mode amplitude propagation in waveguide arrays with underlying symmetries \cite{Vance1996,RodriguezLara2015,Vergara2015,Huerta2016,NodalStevens2018,RodriguezLara2018}.
Here, we use the cyclic symmetry of this annular multicore fiber to account for the effective coupling between any pair of cores.
We focus on the normal modes with vertical polarization but the results can be extended for other polarization modes.
In the following, we will assume that each core is a single-mode optical fiber supporting an $LP_{01}$ mode.
We first find the normal modes for the outer cores without the central core using the discrete Fourier matrix.
Then, we include the central core and show that only one of the discrete Fourier modes couples to it.
Finally, we find the normal modes of the whole fiber and provide their effective propagation constants.
For the sake of comparison, we use finite element simulations to confirm our higher-order neighbor results.
\section{Outer ring cores}
First, we focus on the outer ring composed by $n$ identical single-mode fiber cores evenly placed in a circular array.
We label the cores from $1$ to $n$ in clockwise order, Fig. \ref{Fig:1:Structure}.
Since all the cores are identical, we label the effective propagation constant for every single core mode as $\beta$.
We call the effective $k$-th neigbor coupling $g_{k}$ with $k=1,2,\ldots, \lfloor n/2 \rfloor$; the function $\lfloor x \rfloor$ yields the greatest integer less than or equal to the argument $x$.
The coupling between a pair of cores decreases with the distance, $g_1 > g_2 > \ldots > g_{\lfloor n/2 \rfloor}$.
We can write the elements of the mode-coupling matrix in the following manner,
\begin{eqnarray}
\left[ M \right]_{p,q} = \left\{ \begin{array}{lll}
g_{k} & q = p + n-k, & k =1,2, \ldots, \lfloor \frac{n-1}{2} \rfloor, \\
g_{k} & q = p + k, & k =1,2, \ldots, \lfloor \frac{n}{2} \rfloor, \\
\beta & q = p, & \\
g_{k} & p = q + k, & k =1,2, \ldots, \lfloor \frac{n}{2} \rfloor, \\
g_{k} & p = q +n-k, & k =1,2, \ldots, \lfloor \frac{n-1}{2} \rfloor.
\end{array} \right.
\end{eqnarray}
The first and last upper and lower diagonals in the mode-coupling matrix contain all the first neighbor couplings, $g_{1}$, and so on.
The cyclic symmetry of the system produces shifts in the columns of the coupled-mode matrix.
This symmetry is embodied by the discrete cyclic group with $n$ elements, $\mathbb{Z}_{n}$.
%
We have used in the past the fact that both the representation of the generator of $\mathbb{Z}_{n}$ and the coupled-mode matrix are diagonalized by the Fourier matrix \cite{NodalStevens2018},
\begin{eqnarray}\label{eq:FourierMatrix}
\left[ F \right]_{p,q} = \frac{1}{\sqrt{n}} e^{i \frac{2\pi}{n} (p-1) (q-1) }.
\end{eqnarray}
%
This approach yields the propagation constants for the normal modes of the circular array,
\begin{eqnarray}\label{Eq:CircularArray:PropagationConstants}
\lambda_j
= \beta +
\left\{ \begin{array}{ll}
2 \sum\limits_{k=1}^{m-1} \left\{ g_{k} \, \cos\left[ \frac{\pi}{m} (j-1) k \right] \right\} + g_{m} (-1)^{j-1}, & n= 2m, \\
2 \sum\limits_{k=1}^{m} \left\{ g_{k} \, \cos\left[ \frac{2\pi}{2m+1} (j-1) k \right] \right\}, & n=2m+1. \\
\end{array} \right.
\end{eqnarray}
In the first-neighbors approximation, where $g_k \to 0$ for $k \geq 2$, these values match previous results \cite{Schmidt1991,Rubenchik2013}.
The first mode propagation constant, $\lambda_1$, is always the largest value because all the arguments in the cosines are zero.
Its corresponding supermode has equal field amplitudes in all the cores.
Additionally, there is multiplicity in the propagation constants.
The second propagation constant is equal to the $n$-th one, $\lambda_2=\lambda_n$, the third is equal to the $(n-1)$-th one, $\lambda_3=\lambda_{n-1}$, and so on.
When $n$ is even, there are $n/2 - 1$ duplicated propagation constants and two non-duplicated ones, $\lambda_1$ and $\lambda_{n/2 + 1}$.
When $n$ is odd, there are $(n-1)/2$ duplicated and a single non-duplicated, $\lambda_1$.
The normal modes, or supermodes, are given by the action of the discrete Fourier matrix on the standard orthonormal basis \cite{NodalStevens2018}.
The mode corresponding to the $j$-th effective propagation constant is the $j$-th column of the conjugate transpose of the discrete Fourier matrix,
\begin{equation}
\vec{u}_j = F^{\dagger} \cdot \hat{e}_j,
\end{equation}
where $\hat{e}_j$ is $j$-th vector of the standard orthonormal basis.
This allows us to realize that the normal modes are independent of the coupling and propagation constants, Eq.~(\ref{eq:FourierMatrix}).
They form a complete orthonormal basis, thus the duplicity in propagation constants is not a degeneracy.
\section{Full multicore fiber}
Now, we add the central core by assigning an effective propagation constant $\beta_c$ to it and an effective coupling $g_c$ between it and each other core in the outer ring.
The mode-coupling matrix for the whole system can be written as a block matrix,
\begin{equation}
M_c
=
\left(
\begin{array}{c|c}
M
& \begin{array}{c} g_c \\ \vdots \\ g_c \end{array}
\\
\hline
\begin{array}{ccc} g_c & \cdots & g_c \end{array}
& \beta_c
\end{array}
\right).
\end{equation}
This matrix can be diagonalized by extending our procedure above.
This leads to an effective coupling, with strength $\sqrt{n} g_c$, between the central single core mode
and the first supermode of the external cores,
\begin{equation}
D_{c}
=
\left(
\begin{array}{cccc|c}
\lambda_1 & 0 & \ldots & 0 & g_c \sqrt{n} \\
0 & \lambda_2 & \ddots & \vdots & 0 \\
\vdots & \ddots & \ddots & 0 & \vdots \\
0 & \ldots & 0 & \lambda_n & 0 \\
\hline
g_c \sqrt{n} & 0 & \cdots & 0 & \beta_c
\end{array}
\right).
\end{equation}
Snyder discussed the coupling between these two modes \cite{Snyder1972}.
The rest of the outer ring supermodes do not couple to the central core and, therefore, remain normal modes when the central core is added.
It is straightforward to diagonalize the effective two-by-two real-symmetric mode-coupling matrix $D_{c}$ and obtain the propagation constants,
\begin{eqnarray}\label{Eq:PropagationConstantsWithCentral}
&&\lambda_{\pm} = \left( \lambda_1 + \beta_c \pm \sqrt{ \left( \lambda_1 - \beta_c \right)^2 + 4 \, g_c^2 \, n }, \right)/2, \nonumber \\
&&\lambda_2, \ldots, \lambda_n,
\end{eqnarray}
for the supermodes of the whole multicore fiber.
The propagation constant $\lambda_{+}$ is the largest, but $\lambda_{-}$ is not necessarily the smallest.
The supermodes with propagation constants $\lambda_{+}$ and $\lambda_{-}$ have nonzero complex field amplitudes in the central core.
The mode associated with $\lambda_{+}$ has field amplitudes with identical phases in all cores.
That associated with $\lambda_{-}$ has a $\pi$ phase difference between the central core and the outer field amplitudes.
We will call these symmetric and anti-symmetric phase modes,
\begin{eqnarray}
\vec{v}_{+} &=&
\left(\begin{array}{c} \vec{u}_{1} \\ 0 \end{array}\right) \cos \theta
+\left(\begin{array}{c} \vec{0}_{n} \\ 1 \end{array}\right) \sin \theta, \nonumber \\
\vec{v}_{-} &=&
- \left(\begin{array}{c} \vec{u}_{1} \\ 0 \end{array}\right) \sin \theta
+\left(\begin{array}{c} \vec{0}_{n} \\ 1 \end{array}\right) \cos \theta,
\end{eqnarray}
where the $n$-dimensional vectors $\vec{u}_{1}$ and $\vec{0}_{n}$ are the first supermode of the external ring and the zero vector, in that order.
The mixing angle,
\begin{equation}
\tan \theta = \frac{2\sqrt{n}g_{c}}{\lambda_{1}-\beta_{c}+\sqrt{(\lambda_{1}-\beta_{c})^2+4n g_{c}^2}},
\end{equation}
allows us to control the total field intensity distribution in the outer ring and the central core.
For example, if the system is designed to fulfill $\lambda_{1} = \beta_{c}$, the total power in the outer cores will be identical to the power in the central core.
\begin{figure}[htbp]
\centering
\fbox{\includegraphics{Fig2}}
\caption{
(a)--(g) Comparison of analytic coupled-mode theory (small light orange dots) versus numerical finite element (large dark green dots) normal modes for a multicore fiber composed by seven identical cores; fiber parameters can be found in the text.
(a)--(e) Normal modes that do not mix with the central core and (f)--(g) modes that mix with the central core.
The propagation constants associated to the normal modes shown in (f) and (g) are $\lambda_{+}$ and $\lambda_{-}$, respectively.
In these figures, the horizontal axis is the core number, where $c$ labels the central core and the vertical axis is the real part of the average electric field in each core.
Figures (h) and (i) display the real part of the electric field in a cross-section of the waveguide, where the electric field is normalized to its maximum value in the array, for normal modes associated to $\lambda_{+}$ and $\lambda_{-}$, in that order.
}
\label{Fig:2:Modes}
\end{figure}
We use numerical finite element simulation to validate our coupled-mode theory analysis.
Figure \ref{Fig:2:Modes} shows the results for a multicore fiber composed by identical cores and cladding with refractive indices $n_{e} = n_{c} = 1.4479$ and $n_{cl} = 1.4440$, in that order.
The radii of all cores are identical too, $r_{e} = r_{c} = 4.5~\mu\mathrm{m}$, as are the distances between each external and central core, $R=15~\mu\mathrm{m}$.
These parameters are typical in Silica multicore fibers at telecomm wavelength, $\lambda = 1550~\mathrm{nm}$ \cite{MayArrioja2017a}.
We assume a cladding radius of $35~\mu\mathrm{m}$.
The cores support a $\mathrm{LP}_{01}$ mode that can be approximated to a Gaussian with waist $\omega_{0} = 5.5564~\mu\mathrm{m}$.
Coupled-mode theory translates these parameters into the following effective propagation constants and couplings, $\beta = \beta_{c} \approx 5.8606 \times 10^{6} ~\mathrm{rad}/\mathrm{m}$ and $g_{1} \approx g_{c} = 310.00 ~\mathrm{rad}/\mathrm{m}$, respectively.
We use our analytic result and the finite element simulation to fit and recover numerical values of $\beta = 5.8598\times 10^{6}~ \mathrm{rad}/\mathrm{m}$ and $g_{1} = g_{c} = 312.44~ \textrm{rad}/ \mathrm{m}$.
That is a $0.0137 \%$ and $0.78 \%$ relative difference with respect to the numerically fitted values, in that order.
For the sake of curiousity, we study a continuous case where a ring core surrounds the central core.
A type of this concentric ring fiber has been used to experimentaly measure the Abraham force of light over a liquid contained in a central hollow core \cite{Choi2017}.
We expected the symmetric and anti-symmetric phase modes to survive with effective propagation constants in the outer ring and central core given by $\beta_{R}$ and $\beta_{c}$, in that order, and a finite coupling where $g_{c} \sqrt{n} \rightarrow g_{R}$.
This yields an effective two-mode coupled array with propagation constants $\lambda_{(R,\pm)} = \left( \beta_{R} + \beta_{c} \pm \sqrt{ \left( \beta_{R} - \beta_{c} \right)^{2} + 4 g_{R}^2} \right)/2$ to good agreement with the finite element simulation providing $\lambda_{(R,+)} = 5.8651\times 10^{6}~ \mathrm{rad}/\mathrm{m}$ and $\lambda_{(R,-)}=5.8598\times 10^{6}~ \mathrm{rad}/\mathrm{m}$, Fig.~\ref{Fig:3:RingModes}.
We fit these numerical results to the analytic approximation to obtain an effective coupling between outer ring and central core, $g_{R}=1401~\mathrm{rad}/\mathrm{m}$.
This fit uses an effective propagation constant for the outer ring mode that comes from a finite element simulation of the ring alone,
$\beta_{R} = 5.8651 \times 10^{6} ~\mathrm{rad}/\mathrm{m}$,
and the propagation constant for the central core that is identical with the discrete case, $\beta_{c} = 5.8606 \times 10^{6} ~\mathrm{rad}/\mathrm{m}$.
\begin{figure}[htbp]
\centering
\fbox{\includegraphics{Fig3}}
\caption{
Mode profiles for the multicore structure with a continuous ring around a central core with parameters identical to those in Fig. \ref{Fig:2:Modes}.
}
\label{Fig:3:RingModes}
\end{figure}
\section{Conclusions}
In summary, we provided a symmetry-based analysis for a multicore fiber composed by $n$ evenly spaced cores around a central core.
The cyclic group $\mathbb{Z}_{n}$ allowed us to find the normal modes and their propagation constants considering all inter-core couplings.
First, we used the discrete Fourier matrix to resolve the normal modes of the outer cores and their propagation constants; none of these supermodes depend on the parameters of the system.
Then, we studied how these outer core modes couple to the central core and confirmed Snyder's seminal result \cite{Snyder1972}:
only the supermode composed by equal complex field amplitudes in all the outer cores couples to the central core.
The rest $n-1$ outer core supermodes, whose field distribution does not depend on the parameters of the system, remain uncoupled to the single central core mode.
We used finite element analysis to verify our theoretical results for a fiber where all cores are identical to a good agreement; the relative differences between analytic and numerical results were of the order of $0.01 \%$ and $0.78 \%$ of the numerical value for the effective propagation constant of each core and the coupling between cores, in that order.
Our symmetry-based analysis provides a description that might help improve the prediction of crosstalk through propagation in multicore fibers for telecomm and sensing applications.
\textbf{Funding.}
CONACYT, C\'{a}tedra Grupal \#551;
CONACYT, Consorcio en \'Optica Aplicada, FORDECYT \#296355.
Academia Mexicana de Ciencias through Verano de la Investigac\'on Cient\'ifica.
|
1,116,691,497,754 | arxiv | \section{Introduction}
Registration is a critical technology to establish correspondences between medical images \cite{jour/mia/heinrich2012mind}.
The study of registration algorithm enables tumor monitoring \cite{jour/bone/seeley2014co}, image-guided intervention \cite{jour/bbe/alam2018medical}, and treatment planning \cite{jour/eo/giesel2009image}.
Multi-modality images, such as CT, MR and US, capture different anatomical information.
Alignment of multi-modality images can help a clinician to improve the disease diagnosis and treatment \cite{jour/pm/fu2020deep}. For instance, Zhuang \cite{jour/pami/zhuang2018multivariate} registered multi-modality myocardium MR images to fuse complementary information for myocardial segmentation and scar quantification.
Heinrich {et al. } \cite{conf/miccai/heinrich2013towards} performed registration of intra-operative US to pre-operative MR, which could aid image-guided neurosurgery.
Over the last decades, various methods have been proposed to perform multi-modality image registration.
The most common methods are based on statistical similarity metric, such as mutual information (MI) \cite{jour/tmi/maes1997multimodality}, normalized MI (NMI) \cite{jour/pr/studholme1999overlap} and spatial-encoded MI (SEMI) \cite{jour/tmi/zhuang2011nonrigid}.
Registrations are performed by maximizing these similarity metrics between the moved and fixed images.
However, these metrics usually suffer from the loss of spatial information \cite{conf/miccai/qin2019unsupervised}.
Other common methods are based on invariant representation.
Wachinger {et al. } \cite{jour/mia/wachinger2012entropy} presented the entropy and Laplacian image which are invariant structural representations across the multi-modality image, and registrations were achieved by minimizing the difference between the invariant representations.
Zhuang {et al. } \cite{conf/iccis/zhuang2005medical} proposed the normal vector information of intensity image for registration, which obtained comparable performance to the MI and NMI.
Furthermore, Heinrich {et al. } \cite{jour/mia/heinrich2012mind} designed a handcraft modality independent neighborhood descriptor to extract structure information for registrations.
Nevertheless, these conventional methods solved the registration problem by iterative optimizing, which is not applicable for time-sensitive scenarios.
Recently, several registration networks, which could efficiently achieved registrations in an one-step fashion, have been widely investigated.
Hu {et al. } \cite{jour/mia/hu2018weakly} proposed a weakly supervised registration neural network for multi-modality images by utilizing anatomical labels as the criteria for network training.
Similarly, Balakrishnan {et al. } \cite{jour/tmi/balakrishnan2019voxelmorph} proposed a learning-based framework for image registration.
The framework could extend to multi-modality images when the anatomical label is provided during the training.
Furthermore, Luo {et al. } \cite{conf/miccai/luo2020mvmm} proposed a group-wise registration network, which could jointly register multiple atlases to the target image.
Nevertheless, these methods required extensive anatomical labels for network training, which prevents them from unlabeled datasets.
At present, based on image-to-image translation generative adversarial network (GAN) \cite{conf/eccv/huang2018multimodal}, several unsupervised registration networks had been proposed.
Qin {et al. } \cite{conf/miccai/qin2019unsupervised} disentangled a shape representation from multi-modality images via GAN, then a convenient mono-modality similarity metric could be applied on the shape representation for registration network training.
Arar {et al. } \cite{conf/cvpr/arar2020unsupervised} connected registration network with a style translator.
The network could jointly perform spatial and style transformation on a moving image, and was trained by minimizing the difference between the transformed moving image and fixed image.
The basic idea of these GAN-based methods is converting the multi-modality registration problem into a mono-modality one. Unfortunately, GAN methods easily cause geometrically distortions and intensity artifacts during image translation \cite{conf/cvpr/zhang2018translating}, which may lead to unrealistic registration results.
In this work, we propose an end-to-end multi-modality 3D registration network (MMRegNet).
The main contributions are:
(1) We present a spatially encoded gradient information (SEGI), which can provide a similarity criteria to train the registration network in an unsupervised manner.
(2) We evaluated our method on multi-modality cardiac left ventricle and liver registration tasks and obtained promising performance on both applications.
\section{Method}
\paragraph{\textbf{Registration Network:}}
Let $I_m$ and $I_f$ be a moving and fix image, respectively. Here, $I_m$ and $I_f$ are acquired via different imaging protocols, and are defined in a 3-D spatial domain $\Omega$. We construct the MMRegNet based on a U-shape convolution neural network \cite{conf/miccai/ding2020cross}, which takes a pair of $I_m$ and $I_f$ images as input, and predicts forward $U$ and backward $V$ dense displacement fields (DDFs) between them simultaneously. Therefore, MMRegNet is formulated as follows,
\begin{equation}
(U,V)=f_\theta(I_m,I_f)
\end{equation}
where ${\theta}$ is the parameter of MMRegNet. Each voxel $x \in \Omega$ in $I_m$ and $I_f$ can be transformed by $U$ and $V$ as follows,
\begin{equation}
({I}_m\circ U)(x)=I_m (x+U(x)),
\end{equation}
\begin{equation}
({I}_f \circ V)(x)=I_f (x+V(x)),
\end{equation}
where $\circ$ is spatial transformation operation, and ${I}_m \circ U$, ${I}_f \circ V$ denote the moved image of $I_m$, $I_f$, respectively.
\paragraph{\textbf{Spatially Encoded Gradient Information:}}
Generally, the parameter of MMRegNet could be optimized by minimizing the intensity-based criteria, such as mean square error of intensity between the moved image and fixed image. However, such metrics are ill-posed when applied in multi-modality scenarios. This is because the intensity distribution of an anatomy usually varies across different modalities of images. Normalized gradient information (NGI) \cite{conf/miccai/haber2006intensity} is widely explored for conventional multi-modality registration methods.
\begin{figure}[htb]
\centering
\includegraphics[width=1.0\textwidth]{img/2.pdf}
\caption{A visual demonstration of the SEGI.
}
\label{fig:2}
\end{figure}
The basic idea of NGI is based on the assumption that image structures can be defined by intensity changes. Let $G_{I}$ be the NGI of an intensity image $I$, each element of $G_{I}$ is calculated as follows,
\begin{equation}
G_{I}(x)={\frac{\nabla I(x)}{\|\nabla I(x)\|_2}},
\end{equation}
where $x\in \Omega$, and $\nabla I$ refers to the gradient of image $I$. Ideally, MMRegNet can be trained by minimizing the difference between $G_{I_m \circ U}$ and $G_{I_f}$. However, such a criteria is sensitive to the noises or artifacts of intensity images. It is still error-prone to train a registration network via NGI criteria in our practical experiments. To overcome this, we extend NGI to SEGI. Figure \ref{fig:2} illustrates the idea of SEGI, it is achieved by introducing a set of spatial variables ${\Sigma}=\{\sigma_1,\sigma_2\cdots,\sigma_K\}$ to the standard NGI ($G_I$). For each spatial variable $\sigma_k$, we compute its associated SEGI ($SG_I^{\sigma_k}$) as follows,
\begin{equation}
SG_{I}^{\sigma_k}(x)=\sum_{p\in \Omega}{\mathcal{N}(p|x,\sigma_k^2) \frac{\nabla I(p)}{\|\nabla I(p)\|_2}},
\end{equation}
where $x \in \Omega$, and $\mathcal{N}(p|x,\sigma_k^2)$ denotes Gaussian distribution. Notably, we accumulate gradient information around $x$ for a more robust representation of the intensity change.
Finally, given a set of spatial variables $\Sigma$, the SEGI of an intensity image $I$ is defined as,
\begin{equation}
SG_I=\{SG_{I}^{\sigma_1},SG_{I}^{\sigma_2},\cdots,SG_{I}^{\sigma_K}\}.
\label{eq:sg_variable}
\end{equation}
\paragraph{\textbf{Loss Function:}}
We train the network by minimizing the cosine distance between the SEGI of moved ($SG_{I_m \circ U}$) and fixed ($SG_{I_f}$) images,
\begin{equation}
\mathcal{L}_{SG}=\frac{1}{K}\sum_{k=1}^{K}\mathcal{D}({SG}_{I_m \circ U}^{\sigma_k},SG_{I_f}^{\sigma_k}),
\end{equation}
\begin{equation}
\mathcal{D}({SG}_{I_m \circ U }^{\sigma_k},SG_{I_f}^{\sigma_k})=\frac{-1}{|\Omega|}\sum_{x \in \Omega}cos({SG}_{I_m \circ U}^{\sigma_k}(x),SG_{I_f}^{\sigma_k}(x)),
\label{eq:sg}
\end{equation}
where $|\Omega|$ counts the number of voxels in an image, and $cos(\bm{A},\bm{B})$ calculates the cosine distance between vector $\bm{A}$ and $\bm{B}$.
Meanwhile, MMRegNet is designed to simultaneously predict $U$ and $V$ for each pair of $I_m$ and $I_f$. Normally, $U$ and $V$ should be inverse of each other. Hence, we employ the cycle consistent constraint \cite{conf/miccai/ding2020cross} for the DDFs such that each $I_m$ can be restored to its original one after transforming by $U$ and $V$ in succession,
\begin{equation}
\mathcal{L}_{CC}=\frac{1}{|\Omega|}\sum_{x \in \Omega}\|{I}_{m} \circ U \circ V (x)-I_m(x)\|_1.
\end{equation}
Finally, the total trainable loss of the registration network is defined as follows,
\begin{equation}
\label{equ:loss}
\mathcal{L}=\mathcal{L}_{SG}+\lambda_1 \mathcal{L}_{CC}+{\lambda_2}\{\Psi(U)+\Psi(V)\},
\end{equation}
where $\Psi(U)$ and $\Psi(V)$ are smoothness regularization terms for DDFs, and $\lambda_1$, $\lambda_2$ are the hyper-parameters.
\section{Experiments and Results}
\paragraph{\textbf{Experimental Setups:}}
MMRegNet was implemented by the TensorFlow on an NVIDIA P100. We tested it on two public datasets, i.e., the MM-WHS\footnote{www.sdspeople.fudan. edu.cn/zhuangxiahai/0/mmwhs/} \cite{jour/mia/zhuang2019evaluation} and CHAOS\footnote{https://chaos.grand-challenge.org/} \cite{jour/mia/kavur2021chaos}.
\begin{itemize}
\item MM-WHS: MM-WHS contains multi-modality (CT, MR) cardiac medical images. We utilized 20 MR and 20 CT images for left ventricle registration task. MMRegNet was trained to perform the registration of MR to CT images.
\item CHAOS: CHAOS contains multi-modality abdominal images from healthy volunteers. For each volunteer, the dataset includes their T1, T2 and CT images. We adopted 20 T1 MR, 20 T2 MR and 20 CT images for liver registration.
\end{itemize}
During the training phase, we employed ADAM optimizer to optimize the network parameters for 5000 epochs.
The spatial variables $\Sigma$ were given to $\{1,1.5,3\}$ practically, aiming to capture multi-scale of robust gradient information for registration.
Meanwhile, we tested $\lambda_1 $ and $\lambda_2$ with four different weighting values, i.e., 0.01, 0.1, 1, 10. According to the corresponding results of different setups, we set $\{\lambda_1 = 0.1, \lambda_2 = 10 \}$ and $\{\lambda_1 = 0.1, \lambda_2 = 1 \}$ for MM-WHS and CHAOS dataset, respectively.
To evaluate the performance of MMRegNet, we computed the Dice (DS) and average symmetric surface distance (ASD) between the corresponding label of moved and fix images. All experimental results were reported by 4-fold cross-validation.
\paragraph{\textbf{Results:}}
We compared our registration method with three state-of-the-art multi-modality registration methods.
\begin{itemize}
\item Sy-NCC: The conventional affine + deformable registration, which is based on the symmetric image normalization method with normalized cross-correlation (NCC) as optimization metric \cite{journal/inj/avants2009advanced}. We implemented it based on the popular ANTs software package\footnote{https://github.com/ANTsX/ANTsPy} .
\item Sy-MI: The Sy-NCC method which uses the MI instead of the NCC as optimization metric.
\item VM-NCC: The state-of-the-art registration network \cite{jour/tmi/balakrishnan2019voxelmorph}, which was trained by using the NCC as training criteria. We adopted their official online implementation\footnote{https://github.com/voxelmorph/voxelmorph}.
\end{itemize}
\begin{table*}[tb]
\centering
\caption{The performance of different multi-modality registration methods on MM-WHS dataset.}
\begin{tabular}{lllll}
\hline
\multicolumn{1}{l|}{\multirow{2}{*}{Method}} & \multicolumn{2}{c}{LVC (MR$\rightarrow$CT)} & \multicolumn{2}{c}{Myo (MR$\rightarrow$CT)} \\
\cline{2-5}
\multicolumn{1}{l|}{} & \multicolumn{1}{c}{DS (\%)$\uparrow$} & \multicolumn{1}{c|}{ASD (mm)$\downarrow$} & \multicolumn{1}{c}{DS (\%)$\uparrow$} &
\multicolumn{1}{c}{ASD (mm)$\downarrow$} \\
\cline{1-5}
\multicolumn{1}{l|}{Sy-NCC \cite{journal/inj/avants2009advanced}} &70.07$\pm$16.57& \multicolumn{1}{l|}{4.51$\pm$2.67} & 50.66$\pm$16.02 & \multicolumn{1}{l}{4.10$\pm$1.77} \\
\multicolumn{1}{l|}{Sy-MI \cite{journal/inj/avants2009advanced}} &69.16$\pm$15.25 & \multicolumn{1}{l|}{4.66$\pm$2.54} & 49.00$\pm$16.21 & \multicolumn{1}{l}{4.34$\pm$2.04} \\
\multicolumn{1}{l|}{VM-NCC \cite{jour/tmi/balakrishnan2019voxelmorph}} &79.46$\pm$8.73 & \multicolumn{1}{l|}{\textbf{2.81$\pm$1.05}} & 62.77$\pm$9.51 & \multicolumn{1}{l}{\textbf{2.49$\pm$0.61}} \\
\multicolumn{1}{l|}{MMRegNet} & \textbf{80.28$\pm$7.22} & \multicolumn{1}{l|}{{3.46$\pm$1.30}} & \textbf{62.92$\pm$8.62}& \multicolumn{1}{l}{{3.01$\pm$0.74}} \\
\hline
\end{tabular}
\label{tab:mmwhs}
\end{table*}
\begin{table*}[tb]
\centering
\caption{The performance of different multi-modality registration methods on CHAOS dataset.}
\begin{tabular}{lllll}
\hline
\multicolumn{1}{l|}{\multirow{2}{*}{Method}} & \multicolumn{2}{c}{Liver (T1$\rightarrow$CT)} & \multicolumn{2}{c}{Liver (T2$\rightarrow$CT)} \\
\cline{2-5}
\multicolumn{1}{l|}{} & \multicolumn{1}{c}{DS (\%)$\uparrow$} & \multicolumn{1}{c|}{ASD (mm)$\downarrow$} & \multicolumn{1}{c}{DS (\%)$\uparrow$} &
\multicolumn{1}{c}{ASD (mm)$\downarrow$} \\
\cline{1-5}
\multicolumn{1}{l|}{Sy-NCC \cite{journal/inj/avants2009advanced}} &74.94$\pm$11.05 & \multicolumn{1}{l|}{8.46$\pm$4.10} & 75.46$\pm$9.42 & \multicolumn{1}{l}{8.41$\pm$3.86} \\
\multicolumn{1}{l|}{Sy-MI \cite{journal/inj/avants2009advanced}} &73.88$\pm$10.08 & \multicolumn{1}{l|}{8.84$\pm$3.70} & 75.82$\pm$7.23 & \multicolumn{1}{l}{8.32$\pm$2.73} \\
\multicolumn{1}{l|}{VM-NCC \cite{jour/tmi/balakrishnan2019voxelmorph}} &74.63$\pm$6.54 & \multicolumn{1}{l|}{8.25$\pm$2.17} & 71.10$\pm$6.09 & \multicolumn{1}{l}{9.30$\pm$2.01} \\
\multicolumn{1}{l|}{MMRegNet} & \textbf{79.00$\pm$8.06} & \multicolumn{1}{l|}{\textbf{7.03$\pm$2.55}} & \textbf{76.71$\pm$8.80}& \multicolumn{1}{l}{\textbf{7.87$\pm$1.75}} \\
\hline
\end{tabular}
\label{tab:chaos}
\end{table*}
\begin{figure}[htb]
\centering
\includegraphics[width=1.0\textwidth]{img/3.pdf}
\caption{Visualization of different methods on MM-WHS and CHAOS datasets. The showed images are the representative cases in terms of DS by MMRegNet. The blue contours are the gold standard label of the fixed images, while the red contours delineate the label of moving or moved images. We indicate the advantage of MMRegNet via yellow arrows. Moreover, the last column presents the moved images of MMRegNet. (The reader is referred to the online version of this article)}
\label{fig:mmwhse_chaos}
\end{figure}
Table \ref{tab:mmwhs} shows the results on MM-WHS dataset.
Compared with the conventional methods (Sy-NCC and Sy-MI), MMRegNet could achieve better performance on both left ventricle cavity (LVC) and left ventricle myocardium (Myo). Notably, compared to the state-of-the-art registration network, i.e., VM-NCC, MMRegNet obtained comparable results in terms of DS and ASD.
This reveals that MMRegNet is applicable for multi-modality registration tasks, and the proposed SEGI could serve as another efficient metric, such as MI and NCC, for multi-modality registration.
Table \ref{tab:chaos} shows the results on CHAOS dataset.
We independently reported the registration result of T1 or T2 to CT images. MMRegNet achieved comparable accuracy to the state-of-the-art conventional methods, i.e., Sy-MI and Sy-NCC. Meanwhile, compared to VM-NCC, MMRegNet obtained average 4.99\% (T1$\rightarrow$CT: 4.37\%, T2$\rightarrow$CT: 5.61\%) and 1.33 mm (T1$\rightarrow$CT: 1.22 mm, T2$\rightarrow$CT: 1.43 mm) improvements in terms of DS and ASD, respectively. This indicates that MMRegNet could achieve promising performance for multi-modality registration tasks.
Additionally, Figure \ref{fig:mmwhse_chaos} visualizes four representative cases from the two datasets. On MM-WHS dataset, one can observe that both VM-NCC and MMRegNet achieved better visual results than Sy-MI and Sy-NCC, which is consistent with the quantitative results in Table \ref{tab:mmwhs}. On CHAOS dataset, the yellow arrows highlight where MMRegNet could obtain relative reasonable results than other methods.
\section{Conclusion}
In this paper, we present an end-to-end network for multi-modality registration.
The network is both applicable for heart and liver registration tasks.
Meanwhile, we propose SEGI to obtain a robust structural representation for multi-modality images, and then applied it as the loss function for unsupervised registration network training.
The results showed that MMRegNet could achieve promising performance when comparing with the state-of-the-art registration methods. Further work will extend MMRegNet to other multi-modality datasets.
\bibliographystyle{splncs04}
|
1,116,691,497,755 | arxiv | \section{Introduction}
Modern wireless communication systems heavily rely on multi-carrier wave-forms. Orthogonal Frequency Division Multiplexing (OFDM) \cite{chang1966synthesis}, Generalized Frequency Division Multiplexing (GFDM) \cite{fettweis2009gfdm}, Filter Bank Multi-Carrier (FBMC) and Universal Filtered Multi-Carrier (UFMC) \cite{schaich2014waveform} are examples for multi-carrier wave-forms. Such variability in wave-forms, brings up the need for using unified receiver architectures that are flexible enough to be used for different multiple wave-forms. For example, a receiver that can be used for both OFDM and GFDM systems would be a great use for wireless communication systems. Deep learning (DL) based receiver solutions are such emerging potential solutions in receiver design and there are already deep learning based solutions proposed in the literature as in \cite{turhan2019deep,van2019deep}.
Among many deep learning architectures, Convolutional Neural Networks (CNNs) have revolutionized multiple research fields (as in computer vision and natural language processing). That is due to the fact that CNNs provide significant performance improvements, when compared to the classical approaches in many applications. However, their use in the receiver design for wireless communication remains limited. The literature for the receiver design mainly focused on designing deep architectures using fully-connected (dense) layers, however, there are not many works using 2D CNNs in the relevant literature. CNNs showed their value on spatial (2D) datasets already. In communication systems, it is essential to deal with complex numbers and complex numbers can also be represented and considered as 2D (spatial) data. Consequently, a 2D CNN dealing with spatial relations among the data points can be useful in the receiver design.
In this paper, we mainly analyze and report the performance of 2D CNNs in receiver design for wireless systems using multi-carrier wave-forms. Among multiple carrier types, we focus on studying the performance of 2D CNNs on OFDM and GFDM systems.
Classical GFDM receivers are known with their heavy computational requirements \cite{kislalsurvey}. While deep learning architectures can be considered as alternatives for receiver design, as mentioned above, they also introduce their own complexities. The existing architectures using fully connected layers typically require heavier computational complexities, when compared to the use of typical 2D CNNs. Using 2D convolutional layers typically provides a significant reduction in computational complexity when they replace the larger sized fully connected layers containing the largest number of neurons in the earlier layers; thus they can help reduce the complexities of the used deep architectures.
\addtolength{\topmargin}{+.075in}
In this paper, our contributions are two-fold: (i) To our best knowledge, this is the first work that introduces data detector without channel equalization using only a deep 2D CNN architecture by eliminating the need for using a coarse detector for multicarrier systems; and (ii) we introduce using CNNs to reduce the required complexities in deep architectures and provide an analysis on parameter computation for various deep architectures.
\begin{figure*}[!ht]
\centering
\vspace{0.1in}
\includegraphics[width=0.7\linewidth]{mc.pdf}
\caption{An overview of the proposed deep receiver architecture.}
\label{fig:mc}
\end{figure*}
\section{Related Work}
Deep learning-based architectures have been studied in receiver design in the recent literature, however, their properties yet to be fully exploited. While the researchers started to focus on including deep network architectures in wireless communication systems, the main focus has been on utilizing fully-connected (dense) networks in various communication systems. However, using convolutional networks (2D or 3D) can reduce the computational cost when compared to dense networks, while providing similar or better performance to that of dense networks as shown in this paper (see Table \ref{tab:trainpar}). Dense networks are also known as multi-layer perceptron (MLP) in some literature and in this paper, we will use both terms interchangeably to refer to the same network type.
Fully connected (dense) networks were used in \cite{dorner2017deep} as a part of a communication system. In \cite{ye2017power}, the authors proposed using dense networks to design receivers for OFDM-based systems. In \cite{he2018model,zhang2019artificial} the authors proposed an OAMP based algorithm that uses training to learn the OAMP parameters in MIMO systems. In \cite{van2019deep}, a dense network was proposed as detector. In \cite{balevi2019one}, the authors proposed using a dense network for OFDM receivers under the constraint of one-bit complex quantization. Long-short term memory (LSTM) based networks were also used in communication systems. For example, in \cite{gao2018comnet}, a deep architecture using LSTM and fully connected layers for OFDM systems was proposed. In \cite{fang2017deep}, deep belief networks and auto-encoders were proposed. Auto-encoders were also proposed in \cite{kim2018deep}. Similarly, in \cite{o2017deep}, a fully-connected auto-encoder structure was also proposed.
\addtolength{\topmargin}{+.075in}
Most of the above-mentioned work focused on utilizing dense networks in different architectures. In \cite{farsad2017detection}, the authors studied the performance of dense networks, convolutional networks and recurrent neural networks for chemical (molecular) communication systems. In \cite{zhao2018deep}, the authors proposed an auto-encoder architecture utilizing 2D convolutions for OFDM.
The closest work to ours in the literature is the work in \cite{turhandeep,turhan2019deep} as they both design a deep receiver for GFDM system using 2D CNNs. The receiver design in \cite{turhandeep,turhan2019deep} contains two detectors: a coarse detector and a fine detector where the coarse detector uses one of the classical methods (e.g., zero force or Minimum Mean Square Estimator) first and then a 2D CNN is used to further improve the detection performance.
In this work, we investigate on utilizing only a deep architecture that combines both coarse and fine detectors for multi-carrier wave-forms. By representing a complex number as 2D number, we can utilize 2D CNNs in the receiver. Therefore, in this paper, we study the performance of 1D and 2D CNNs and compare that to dense networks for communication systems utilizing multi-carrier wave-forms. Furthermore, while the literature using deep architectures mostly focused on OFDM wave-forms, in this work, we study the performance of using various deep architectures for multiple systems including OFDM and GFDM.
\section{System Architecture: Overview}
In this section, we provide an overview of a wireless system using multi-carrier wave-forms. Fig.~\ref{fig:mc} shows the block diagram of such wireless system \cite{anwarperformance}.
Binary data vector $\textbf{b}$ is generated by a data source. Coding and $2^\phi$-valued complex constellation mapping is performed to obtain symbol vector $\textbf{d}$, where $\phi$ is the modulation order. The resulting vector $\textbf{d}$ has block-based structure. Thus, it can be decomposed in frequency and time space into $M$ subsymbols and $K$ subcarriers, wherein $K$ is the total number of subcarriers and $M$ is the number of symbols in one block according to $\textbf{d}=(\textbf{d}_0,\dots,\textbf{d}_{M-1}^{\; T})^T$ and $\textbf{d}_m=(\textbf{d}_{0,m},\dots,\textbf{d}_{K-1,m})^T$. The total number of symbols ($N$) in one multi-carrier wave-form symbol becomes $N = KM$. $N$ dimensional $\textbf{x}$ time vector is created by modulating matrix $\textbf{d}$ with a desired multi-carrier wave-form (e.g., OFDM, GFDM, SC-FDM, SC-FDE or OTFS) where $x(n)$ is defined as:
\begin{equation}
x(n)=\sum_{k=0}^{K-1}\sum_{m=0}^{M-1}d_{k,m}g_{k,m}(n),\quad n = 0,\dots,N-1.
\end{equation}
The model of $g_{k,m}(n)$ vector differs according to the preferred wave-form type. For example, in GFDM, the $g_{k,m}(n)$ is defined as in Eq.~\ref{eq:gkmn} where, operations such as pulse shaping, upsampling and frequency shifting are applied on vector \textbf{d}.
\begin{equation}
g_{k,m}(n)=g\big((n-mK){modN}\big)exp\Big(j2\pi\frac{k}{K}n\Big),
\label{eq:gkmn}
\end{equation}
where $g_{k,m}(n)$ is formed with the pulse shaping filter $g(n)$. The linear relationship between the vectors $\textbf{x}$ and $\textbf{d}$ is given as follows: $\textbf{x}=\textbf{A}\textbf{d}$. The $NxN$ modulation matrix $\textbf{A}$ is created as shown in Eq. \ref{eq:modmat} with $g_{k,m}(n)$ which defines the columns of $\textbf{A}$. The mapped bits in $\textbf{d}$ are modulated by the modulation matrix $\textbf{A}$. Here, the modulation matrix is selected according to the chosen modulation type:
\begin{equation}
\textbf{A}=[g_{0,0},\ldots,g_{K-1,0},g_{0,1},\ldots,g_{K-1,1},\ldots,g_{K-1,M-1}]
\label{eq:modmat}
\end{equation}
Cyclic prefix is added to the resulting modulated signal to cope with multipath channel effects and adding cyclic prefix yields the time vector $\Tilde{\textbf{x}}$ to be transmitted. The received signal is exposed to a frequency selective Rayleigh fading channel as considered in Eq.~\ref{eq:received}:
\begin{equation}
\Tilde{\textbf{y}}=\Tilde{\textbf{H}}\Tilde{\textbf{x}} + \Tilde{\textbf{w}}
\label{eq:received}
\end{equation}
where, we assume that the channel length is shorter than the CP and perfect synchronization is ensured. After removing the cyclic prefix, Eq.~\ref{eq:received} can we re-written as in Eq.~\ref{eq:rec}:
\begin{equation}
\textbf{y}=\textbf{H}\textbf{x} + \textbf{w}
\label{eq:rec}
\end{equation}
where, $\textbf{y}$ is $N$ dimensional and the received matrix $\textbf{H}$ is the $NxN$ circular convolution matrix constructed from the channel impulse response coefficients given by $\textbf{h} = [h(1),h(2),\dots,h(N_{ch})]^T $,
and $\textbf{w}$ is $N$ dimensional additive white Gaussian noise (AWGN) vector. The elements of $\textbf{h}$ and $\textbf{w}$ follow $\mathcal{C}\mathcal{N}(0,1)$ and $\mathcal{C}\mathcal{N}(0,\,\sigma_w^{2})$ distributions respectively, where $\mathcal{C}\mathcal{N}(\mu,\,\sigma^{2})$ shows the distribution of a circularly symmetric complex Gaussian random variable with mean $\mu$ and variance $\sigma^2$. Combining Eq.~\ref{eq:rec} with the linear relation of $\textbf{x}=\textbf{A}\textbf{d}$ yields to Eq.~\ref{eq:r1}:
\begin{equation}
\textbf{y}=\textbf{H}\textbf{A}\textbf{d} + \textbf{w}
\label{eq:r1}
\end{equation}
After zero forced (ZF) channel equalization we obtain the following equation:
\begin{equation}
\textbf{z}=\textbf{H}^{-1}\textbf{H}\textbf{A}\textbf{d} + \textbf{H}^{-1}\textbf{w}
\label{eq:r2}
\end{equation}
Linear demodulation of the signal can be
expressed as:
\begin{equation}
\hat{\textbf{d}}=\textbf{B}\textbf{z}
\label{eq:receiver}
\end{equation}
where $\textbf{B}$ is $NxN$ dimensional receiver matrix. Different types of linear detectors, e.g. zero forced receiver $\textbf{B}_{\textbf{ZF}}=\textbf{A}^{-1}$,matched filter (MF) receiver
$\textbf{B}_{\textbf{MF}}=\textbf{A}^{\textbf{H}}$ and minimum mean square error (MMSE) receiver $\textbf{B}_{\textbf{MMSE}}=(\textbf{R}_{\textbf{w}}+\textbf{A}^{\textbf{H}}\textbf{A})^{-1}\textbf{A}^{\textbf{H}}$
can be used to detect the data symbols from the equalized observation signal, where $\textbf{R}_{\textbf{w}}$ denotes the covariance matrix of the noise.
For neural network-based detection, the received signal $\textbf{y}$ is fed as an input to a deep neural network. After the estimated symbol vector $\hat{\textbf{d}}$ is taken from the output of the deep receiver, decoding and constellation demapping operations are performed. The resulting vector $\textbf{b}$ is sent to the binary sink.
Modulation matrix performs different operations for each wave-forms. OFDM modulation matrix performs $N$-point IFFT operation which means frequency shifting. GFDM modulation matrix performs upsampling, pulse shaping and frequency shifting. Furthermore, it has circular structure and because of that, it allows the use of cyclic prefix to make frequency domain equalization possible. The magnitude of the GFDM modulation matrix is depicted in Fig. \ref{fig:modmatrix}.
Next section focuses on the details of utilizing a deep neural network in the receiver part.
\begin{figure}[t!]
\centering
\vspace{0.1in}
\includegraphics[width=1.00\linewidth]{modmatrix.pdf}
\caption{Visualization of the GFDM Modulation Matrix ( K=8, M=5, RRC Filter (\emph{a}=0.1))}
\label{fig:modmatrix}
\end{figure}
\section{Deep Receiver with CNNs}
In wireless communication, signal detection can be considered as a classification process of recovering the transmitted signal from the (distorted) received signal where a deep architecture can be used for classification in the deep receiver.
Wireless channel can be evaluated unchanged at small intervals. If the channel is known in small intervals, the transmitted symbols can be recovered. In classical receiver architecture, pilot signals are often used and the channel is estimated by using those pilot signals. Once the channel is estimated, channel equalization is performed to eliminate multipath effects. The deep learning technique has the same framework. Deep receiver works in two different modes: Training and Testing. Training mode can be considered as using the pilot signals which can relate to the training data. Using the training data, the model weights are obtained in the neural network to recover the symbols. Testing does not require any pilot symbol and works for recovering messages to be transmitted by using weights obtained in the training.
A typical CNN may contain convolutional layers and fully connected (FC) layers. Our proposed CNN architectures are given in Fig. \ref{fig:mod} for both OFDM and GFDM. In our proposed deep receiver, we use multiple convolutional layers followed by FC layers. Based on the number of used convolutional layers and the number of FC layers, the performance of the deep receiver changes (see Section \ref{Simulations}). Complex input data is represented as a 2D vector (where each of real and imaginary parts of a complex number forms one dimension). 2D convolution is applied to the input data and deep features are extracted. At the end of the fully connected layers, the symbols are estimated. The vector $\textbf{d}$ mapped and coded is used as ground truth while the network is being trained. We used Adam optimizer to train all the networks. Mean Squared Error (MSE) loss function is used in our network during the training as in Eq. \ref{eq:loss}:
\begin{equation}
\mathcal{L}=\frac{1}{N}(\hat{\textbf{d}}-\textbf{d})^{T}(\hat{\textbf{d}}-\textbf{d})
\label{eq:loss}
\end{equation}
At the end of the process, it is expected that underlying relationship formed by wireless channel is learned and tried to be eliminated from the signal.
\section{Simulations} \label{Simulations}
In this section, we simulate the wireless system given in Fig. \ref{fig:mc} and study the performance of the deep receiver for multi-carrier wave-forms. In particular, we study the performance for both OFDM and GFDM wave-forms separately. We compared the deep receiver performance with the traditional methods in terms of bit-error rates (BERs) under different signal-to-noise ratios (SNRs). For comparison, we compared deep receiver's performance to classical methods including matched receiver with channel equalization, zero forced receiver without channel equalization and MMSE receiver with channel equalization \cite{michailow2014generalized}.
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth]{nn-diagram-conv2d.pdf}
\caption{Model Summaries for both OFDM and GFDM}
\label{fig:mod}
\end{figure}
\addtolength{\topmargin}{+.075in}
In our experiments, we use 10 tap Rayleigh fading with Extended Pedestrian A (EPA) channel model \cite{EPA} and that is fixed for both training and testing steps. In our all simulations, we used BPSK modulation and omitted channel coding and its corresponding decoding processes. In the following subsections, we first present the experimental results for the OFDM wave-form and then present the results for the GFDM wave-form.
\begin{table}[t]
\caption{Simulation Parameters for Both OFDM and GFDM }
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\multicolumn{4}{|c|}{\textbf{OFDM \& GFDM Parameters used in the Simulations}} \\
\cline{1-4}
\textbf{\textit{Description}}& \textbf{\textit{Parameter}}& \textbf{\textit{OFDM Value}}& \textbf{\textit{GFDM Value}} \\
\hline
Number of subcarriers&K & 64&32 \\
\hline
Number of subsymbols&M& 1&3 \\
\hline
Pulse shaping filter&g& -&RRC\\
\hline
Roll-off factor&$a$& -& 0.1 \\
\hline
Length of cyclic prefix&Ncp&16&24\\
\hline
Channel Taps (Fixed)&Nch&10&10\\
\hline
\end{tabular}
\label{gfdm}
\end{center}
\end{table}
\vspace{0.14in}
For both OFDM and GFDM experiments, we performed an empirical study by changing various hyperparameters of the deep architecture used in the receiver. Such as, number of filters (16, 32, 64, 128, 256, 512, 1024) for each layer, filter size (1-8), activation function (ReLU, Sigmoid and Tanh) of each layer except the output layer and the stride value (1, 2, 3). We choose the best configuration found in those experiments for the final parameters of the network and choose hyperbolic tangent (tanh) function as the activation function in the last layer as its output range is between -1 and 1. We also perform a search on the total number of needed convolutional layers by varying the total number of convolutional layers between 2 and 5. After the convolutional layers, we used a fully connected (FC) layer as the output layer. We named those models as 2+1, 3+1, 4+1 and 5+1, where 2+1 means that 2 convolutional layers and one FC layer as output is used. Similarly, 3+1 means that 3 convolutional layers and one FC layer is used in that network and so on. A summary of those networks is given in Fig. \ref{fig:mod} for CNNs where the given values in each convolution block represent the kernel size, the output size and the activation function respectively as used in that layer. All four models (2+1, 3+1, 4+1 and 5+1 models) are shown in the same figure. For instance, for the 3+1 model, the output after the first 3 convolution layers is connected to the flattening layer. Training step for each model is carried out separately at each SNR level and the networks trained up to 30 epochs with early stopping criteria.
Real and imaginary components of receiving signal are concatenated serially and given to the input of MLP and 1D CNN network. 5 different layers were chosen for the MLP. Layers have the neuron number 256, 128, 64, 32 and 16 respectively. The output layer consists of 192 neurons for GFDM 128 neurons for OFDM. Likewise, a structure consisting of 5 different layers is designed for 1D CNN. Convolution layers consist of 16, 32, 64, 128, 256 filters of size 3 respectively. The output layer consists of 192 neurons for GFDM 128 neurons for OFDM.
\begin{figure*}[t]
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=\linewidth]{ofdm2D.pdf}
\vspace{-0.24in}
\caption{\small OFDM results for 2D-CNNs}
\label{fig:ofdm}
\end{subfigure}
~
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=\linewidth]{ofdm1D.pdf}
\vspace{-0.24in}
\caption{\small OFDM results for 1D-CNNs}
\label{fig:OFDM-1Dconv}
\end{subfigure}%
~
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=\linewidth]{ofdmmlp.pdf}
\vspace{-0.24in}
\caption{\small OFDM results for MLP}
\label{fig:ofdmmlp}
\end{subfigure}
\\
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=\linewidth]{gfdm2D.pdf}
\vspace{-0.24in}
\caption{\small GFDM results for 2D-CNNs}
\label{fig:gfdm}
\end{subfigure}
~
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=\linewidth]{gfdm1D.pdf}
\vspace{-0.24in}
\caption{\small GFDM results for 1D-CNNs}
\label{fig:GFDM-1Dconv}
\end{subfigure}
~
\begin{subfigure}{.33\linewidth}
\centering
\includegraphics[width=\linewidth]{gfdmmlp.pdf}
\vspace{-0.24in}
\caption{\small GFDM results for MLP}
\label{fig:gfdmmlp}
\end{subfigure}
\vspace{-0.1in}
\caption{\small BER vs. SnR values are shown for both OFDM and GFDM wave-forms using various networks.}
\label{fig:BERFigures}
\end{figure*}
\subsection{Results for OFDM Modulation}
OFDM parameters used in the simulations are given in Table~\ref{gfdm}. First 640000 bits are generated to map 10000 OFDM symbols and splitted over 64 subcarriers ($K$). Cyclic prefix (CP) is inserted at a length of 1 to 4 of the generated time signal. Training data consist of 10000 OFDM symbols and testing data consist of 10000 OFDM symbols. Note that each OFDM symbol carries 64 bits.
We demonstrate how the BER changes with respect to Eb/No (signal to noise ratio) at different configurations. First, we compare the performance of a 2D-CNN at different layer numbers (where the total number of convolutional layers varies between 2 and 5). The results are summarized in Fig.~\ref{fig:ofdm}. In the figure, the best performance is obtained with the 2+1 2D-CNN. We also demonstrate the performance of another type of convolutional networks: 1D-CNN. Fig.~\ref{fig:OFDM-1Dconv} summarizes the results obtained with different 1D-CNNs. The best result is obtained at 2+1 1D-CNN.
Finally, to compare CNN performance to that of MLP, we study the performance of various MLP networks. The results are shown in Fig.~\ref{fig:ofdmmlp}. The best performance is obtained with 2-layers MLP.
\subsection{Results for GFDM }
GFDM parameters used in the simulations are given in Table \ref{gfdm}. First 960000 bits are generated to map 10000 GFDM symbol and splitted over 32 subcarriers $(K)$ and 3 subsymbols $(M)$. These 32 subcarriers and 3 subsymbols located in time-frequency space forms one GFDM symbol. The chosen pulse shape for the GFDM prototype filter is the root raised cosine (RRC) filter which is widely used in practice with a roll-off factor ($a$) of 0.1. Cyclic prefix (CP) is inserted at a length of 1 to 4 of the generated time signal. Training data consist of 10000 GFDM symbols and testing data consist of 10000 GFDM symbols. Note that each GFDM symbol carries 96 bits.
In the deep receiver, received complex signal with dimensions of 10000x96 is splitted into real and imaginary parts and a represented as 2D data forming 10000x96x2 dimensional 10000 training symbols. The output dimension is 10000x192x1 real vector.
\addtolength{\topmargin}{+.075in}
We demonstrate how the BER changes with respect to Eb/No (signal to noise ratio) for different configurations for GFDM. Similar to OFDM experiments, we first compare the performance of a 2D-CNN at different layer numbers (where the total number of convolutional layers varies between 2 and 5). The results are summarized in Fig.~\ref{gfdm}. In the figure, the best performance is obtained with the 2+1 2D-CNN. 1D-CNN results are summarized in Fig.~\ref{fig:GFDM-1Dconv}. In the figure, the best result is obtained at 2+1 1D-CNN. As shown in in Fig.~\ref{fig:gfdmmlp}, the best MLP results obtained with 2-layers MLP.
For both OFDM and GFDM simulations, learning rate of Adam optimizer is set to 0.0001. Dropout layer is added to the output of each layer (dropout parameter of 0.1) to avoid overfitting.
When there are some changes in channel taps, whole processing is not fully changed. In classical methods all the processing is re-evaluated. In the deep learning technique, this re-evaluation process can be mitigated with the help of transfer learning. It is seen that it converges even at 1 epoch. Therefore adaptation to the channel is provided faster.
\begin{table}[!t]
\vspace{0.1in}
\caption{Comparison of Trainable Parameters}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\multicolumn{4}{|c|}{\textbf{Numbers of Trainable Parameters}} \\
\cline{1-4}
\textbf{Method} & \textbf{\textit{Model Name}}& \textbf{\textit{OFDM}}& \textbf{\textit{GFDM}} \\
\hline
& 2+1 & 132832 & 296736 \\ \cline{2-4}
Conv1D & 3+1 & 139040 & 302944 \\ \cline{2-4}
& 4+1 & 163744 & 327648 \\ \cline{2-4}
& 5+1 & 262304 & 426208 \\ \cline{1-4}
& 2+1 & 134416 & 298320 \\ \cline{2-4}
Conv2D & 3+1 & 146768 & 310672 \\ \cline{2-4}
& 4+1 & 196048 & 359952 \\ \cline{2-4}
& 5+1 & 392912 & 556816 \\ \cline{1-4}
& 2 & 2130688 & 4752192 \\ \cline{2-4}
MLP & 3 & 1090368 & 2401152 \\ \cline{2-4}
& 4 & 568160 & 1223584 \\ \cline{2-4}
& 5 & 306544 & 634288 \\ \cline{2-4}
\hline
\end{tabular}
\label{tab:trainpar}
\end{center}
\end{table}
For the CNN based receiver result, our method approaches the scenario where the channel is fully known. Note that the deep receiver gives results without channel estimation and equalization.
\subsection{Comparison of Parameters}
We compare the total number of parameters for the models used in this paper for 1D-CNNs, 2D-CNNs and MLPs in Table~\ref{tab:trainpar}. Each entry in the table shows the sum of trainable weights and biases used in that particular model. As shown in the table, MLPs typically require more parameters to be trained than CNNs. However, keep in mind that those numbers vary with respect to the total number of filters used in CNNs and the total number of neurons used in each layer in FC layers in MLPs.
\section{Conclusion}
In this paper, we present a deep receiver architecture for multi-carrier wireless systems.
To our best knowledge, this is the first work that introduces data detector without channel equalization using only a deep 2D CNN architecture by eliminating the need for using a coarse detector for multi-carrier systems comparing both OFDM and GFDM.
\addtolength{\topmargin}{+.075in}
In our deep receiver, we analyzed the performance of various neural network architectures including 1D-CNNs, 2D-CNNs and MLPs. Furthermore, we compared those networks' performance to the classical techniques. In our preliminary simulations, we tested networks containing different layers with different hyper-parameters and as shown in our results, shallow networks yielded the best performance (while the difference was not much different, when compared to the deeper architectures). For example, for both OFDM and GFDM experiments, 2+1 networks yielded the best BER among CNN architectures.
While MLPs can yield slightly better performance for OFDM, they may require significantly more parameters to be kept in the memory. A further study can focus on analyzing the performance of various network architectures on a larger set of multi-carrier forms with larger sets of data.
\bibliographystyle{IEEEtran.bst}
|
1,116,691,497,756 | arxiv | \section{Introduction}
Transport in normal metal-superconductor-normal metal (N$_a$SN$_b$)
three-terminal hybrid nanostructures has received a special attention, because
those structures allow in principle to produce split pairs of spin-entangled
electrons from a superconductor, acting as a Cooper pair beam splitter\cite{recher,lesovikmartin}. This is possible
when the size of the region separating the N$_a$S and the N$_b$S interface
becomes comparable to the superconducting coherence length, allowing coherent
processes involving two quasi-particles, each simultaneously crossing one of
the two interfaces
\cite{allsopp,byers,torres,deutscher,falci,melin1,melin2}. Much effort has
been devoted to the theoretical understanding and to the experimental
observation of such a Cooper pair splitting effect. In a transport experiment
where electrons are difficult to measure one by one --- contrarily to the
similar production of entangled photons ---, one relies on steady transport
measurement, e.\,g. the current-voltage characteristics (conductance) and the
cumulants of the current fluctuations (non-equilibrium current noise
\cite{antibunching,bignon} and its counting statistics
\cite{fazio,morten}). In practice, the conductance and the second-order
cumulant (the shot noise and the current cross-correlations between the two
current terminals N$_a$, N$_b$) are the quantities to be extracted from
experiments. Indeed, due to Fermi statistics, the ``partition`` noise
correlations at a three-terminal crossing of normal metal contacts are
negative~\cite{antibunching,AD1996}, manifesting the antibunching properties
of individual electrons. If instead one contact is made superconducting, the
cross-correlations may become positive, suggesting the splitting of Cooper
pairs~\cite{torres}.
Two elementary nonlocal processes occur at a double NSN interface: Crossed
Andreev Reflection (CAR) which alone leads to Cooper pair splitting into
separated electrons bearing opposite spins (for a spin singlet
superconductor), and Elastic Cotunneling (EC) which alone leads to
(spin-conserving) quasi-particle transmission between the normal contacts,
across the superconducting gap \cite{falci}. For tunnel contacts, at lowest
order in the barrier transparencies, those two processes are decoupled and
simply related to the conductance, leading to positive (resp. negative)
conductance and current cross-correlations for CAR (resp. EC)
processes. Indeed Bignon \textit{et\,al.}~\cite{bignon} showed that for tunnel
contacts the linear dependence of the current cross-correlation on the
voltages applied to the contacts N$_a$, N$_b$ allows to separately track the
amplitudes of CAR and EC. Due to the expected compensation of the opposite CAR
and EC conductance components at low transparencies
\cite{falci,brinkmangolubov} , ferromagnetic contacts are required to detect
CAR and EC from the conductance with tunnel contacts
\cite{deutscher,falci,sanchez}. Yet, such polarizations are not easily
achievable, moreover, if one is interested in producing spin-entangled
electrons in a nonlocal singlet state, one should of course not spin polarize
the contacts.
As regards experiments, the situation for (extended) tunnel barriers looks
more complicated than given by a simple tunnel model
\cite{DelftExpt,Levy-Yeyati}.
Zero-frequency noise measurements can be carried out in low impedance sample
(current noise), or at high impedance (voltage noise). The positive
current-current cross correlations discussed here at high transparency may be
measured in the setup of Ref.~\onlinecite{Lefloch} using three SQUIDs as
current amplifiers. It has been found theoretically that, at high transparency
\cite{melin2,melin3,duhotmelin,brataas,kalenkovzaikin,FFM2010}, the
nonlocal conductance is negative, which leaves the current cross-correlations
as the only possible probe of Cooper pair splitting processes, provided one
controls the voltages on both contacts. Contrarily to conductance measurements
with metals \cite{exptconductance} or quantum dots \cite{exptdots},
cross-correlations have led to few experimental results
\cite{exptcrossnoise,das2012}. At the theoretical level, the dependence of the
cross-correlations on the contact transparency is not yet fully understood.
In view of the current experiments on metallic structures, the main question
is therefore: Can the cross-correlations be positive, and if the answer is
yes, is this a signature of Cooper pair splitting ? Previous work on a NSN
structure~\cite{melinbenjamin} showed that the cross-correlations can indeed
be positive at large transparencies, although the nonlocal conductance is
negative. The origin of this somewhat surprising result was not fully
elucidated. Further work~\cite{FFM2010} showed that the sign of the
cross-correlations indeed changes with the transparency of the interfaces,
being positive at low transparency, negative at intermediate transparency and
positive again at high transparency. While the positive sign at low
transparency is clearly ascribed to Cooper pair splitting, it was shown that
the positive sign at high transparency should not be interpreted in the same
way. Indeed, at high transparency, CAR processes do not dominate either in the
conductance or in the noise. Instead, the positive cross-correlations should
be ascribed to local Andreev reflection (AR) on one side, and the opposite
process on the other side, a process equivalent to exchanging a pair of
electrons between the two normal contacts. In
Ref. \onlinecite{ZaikinPRBnoise}, a quasiclassical analysis using a
perturbative expansion in the nonlocal Green's function connecting the two
interfaces led to positive noise correlations at high transparencies, and the
authors conclude that it is due to CAR. This interpretation looks surprising,
given the domination of EC in the conductance in the same regime, and the
total absence of CAR at high transparency in the transmission coefficient. We
insist that it is of importance for the community, before embarking into
experimental developments, to state clearly that positive cross-correlations
should not be interpreted in terms of CAR at high transparencies.
To show that contacts with high transparency, or with effectively high
transparency, are not suitable as a source of Cooper pair splitting, we also
investigate how the localizing effects of disorder influence the nonlocal
conductance and the cross-correlations in a NSN structure. At a single NS
interface, it was shown that disorder in the N region, or multiple scattering
at a clean NN$_l$S double interface, can strongly enhance Andreev reflection,
by a mechanism nicknamed ''reflectionless tunneling''
\cite{reflectionlesstunnexpt,reflectionlesstunntheo,MB1994}. With a disordered
$N_l$ region, it leads to a zero-bias anomaly below the Thouless energy, where
the mutual dephasing of electrons and Andreev-reflected holes is
negligible. In the case of a clean double NN$_l$S interface, maximum Andreev
transmission is obtained by balancing the transparencies T$_{nn}$ and T$_{ns}$
of the NN$_l$ and N$_l$S interfaces, such that the N$_l$ region acts as a
resonant cavity. Melsen and Beenakker \cite{MB1994} performed an average on
the modes inside N$_l$ in order to mimick a disordered region. We ask here the
question of whether a similar mechanism can enhance nonlocal Andreev
reflection e.g. boost CAR compared to AR and EC. A zero-bias anomaly was
obtained in Ref.~\onlinecite{ZaikinPRL}, within a quasiclassical analysis. Its
sign reveals an amplification of quasi-particle transmission. It is however
not mentionned in this reference whether one should also expect boosting of
the CAR channel, which is a requirement for experimental observation of Cooper
pair splitting. We demonstrate on the contrary that reflectionless tunneling
in the nonlocal conductance is not accompagnied by reflectionless tunneling in
the CAR channel.
The needed clarification, both for conductance and noise, comes from a model
which is exactly solvable and where all local and nonlocal amplitudes can be
clearly distinguished. This is an advantage over the quasiclassical approach
used in Ref.~\onlinecite{ZaikinPRL}, which does not produce separate
expressions for the CAR and for the EC contributions to the conductance. Thus,
we use the scattering approach \cite{BTK1982} for a set-up
N$_a$N$_l$SN$_r$N$_b$ with a quadruple interface. The scattering theory is
performed in a one-dimensional geometry, varying the transparencies of the
barriers and the width of the superconductor. It is known to reproduce the
main qualitative features of realistic devices, and allows to account for any
barrier transparency and any distance $d$ between the interfaces. It does not
rely on any expansion in the nonlocal scattering matrix elements (or Green's
functions). This is especially important if noticing that close to the gap
edges, the relevant length scale for the penetration of evanescent
quasiparticles (thus for the Andreev reflection) diverges as $\xi(\omega) =
\xi_0 / \sqrt{1-\frac{\omega^2}{\Delta^2}}$. The scattering method only
assumes a sharp variation of the order parameter at the interface, which is
strictly valid for contact sizes smaller than $\xi_0$. It should be modified
to take into account self-consistency if $d \sim \xi$, or to describe
non-equilibrium effects. One advantage of the scattering approach is the
precise bookkeeping of the scattering amplitudes associated to the
various (AR, CAR, EC) processes. This allows an unambiguous diagnosis of
Cooper pair splitting in either the conductance or the noise, as obtained by
simple expressions of these scattering amplitudes.
If averaging independently the modes in the left and the right regions N$_{l,r}$,
no boosting of the CAR process is obtained. Indeed, ``reflectionless
tunneling`` is a quantum coherent process which demands that the
Andreev-reflected hole retrace the path of the electron, by scattering on the
same impurities. On the contrary, with nonlocal Andreev reflection, the
electron and the transmitted hole sample different disorders and no coherence
is obtained. This result does not contradict Ref. \onlinecite{ZaikinPRL} which
states that the total crossed conductance is enhanced. Again, the
scattering technique allows to track the different contributions, and the
zero-bias anomaly is here due to the enhancement of the direct Andreev
reflection, not to CAR.
Section~\ref{sec:paperNSN:model} presents the model and
section~\ref{paperNNSNN:sec:comp} the scattering theory of the
N$_a$N$_l$SN$_r$N$_b$ system. Section~\ref{paperNNSNN:sec:AS} provides the
results obtained by averaging over channels in N$_l$ and N$_r$, in the spirit
of Ref.~\onlinecite{MB1994}.
\section{The model\label{sec:paperNSN:model}}
We study a one-dimensional model of a symmetrical three-terminal normal
metal-superconductor-normal metal hybrid structure depicted in
Fig.~\ref{paperNNSNN:fig:model}. The central superconducting electrode is
grounded, the normal terminals are biased with voltages $V_a$ and $V_b$. The
length $R$ of the superconducting electrode can be comparable to the
superconducting coherence length. The interfaces between the normal metal and
the superconducting electrodes are modeled by barriers with transparencies
$T_{lns}$ and $T_{rsn}$. In both normal metal electrodes there is an
additional barrier at a distance $L_l$ (respectivly $L_r$) from the normal
metal superconductor interface with transparency $T_{lnn}$ (respectivly
$T_{rnn}$).\\
%
\begin{figure}
\includegraphics[width=0.45\textwidth]{./FIGURE1}
\caption{\label{paperNNSNN:fig:model} Schematic of the model.}
\end{figure}
The system can be described by a $4\times4$ scattering matrix
$s_{ij}^{\alpha\beta}$ where Latin indices run over the normal electrodes $a$
and $b$ and Greek indices over electrons $e$ and holes $h$. The scattering
theory assumes that the superconductor is a reservoir of Cooper pairs, so the
structure is implicitly a three-terminal one and the superconducting electrode
is taken as grounded. The transformation of quasi-particles into Cooper pairs
is taken into account by the correlation length $\xi(\omega)$ which sets the
scale of the damping of the electron and hole wavefunctions in the
superconductor.
The elements of the scattering matrix are evaluated from the BTK
approach\cite{BTK1982} (see Appendix~\ref{paperNSN:app:BTK}). Here we do not
use the Andreev approximation valid in the limit of zero energy where the
electron and hole wavevectors are set to the Fermi wavelength, but instead
keep the full expressions for the wavevectors. The calculation of the average
current $I$ and current cross-correlations $S$ rely on the formulas derived by
Anantram and Datta in Ref.~\onlinecite{AD1996}:
\begin{align}
&I_i=\frac{e}{h}\sum_{k\in \{a,b\}}\sum_{\alpha,\beta\in\{e,h\}} \mathrm{sgn}(\alpha)\label{paperNSN:eq:current}\\
&\times\int dE\phantom{0} \left[\delta_{ik}\delta_{\alpha\beta}-\left|s_{ik}^{\alpha\beta}\right|^2\right]f_{k\beta}(E)\notag\\
&S_{ij}=\frac{2e^2}{h}\sum_{k,l\in \{a,b\}}\sum_{\alpha,\beta,\gamma,\delta\in\{e,h\}} \mathrm{sgn}(\alpha)\mathrm{sgn}(\beta)\\
&\times\int dE\phantom{0}A_{k\gamma,l\delta}(i\alpha,E)A_{l\delta,k\gamma}(i\alpha,E)f_{k\gamma}(E)\left[1-f_{l\delta}(E)\right]\notag
\end{align}
with $ A_{k\gamma,l\delta}(i\alpha,E)=\delta_{ik}\delta_{il}\delta_{\alpha\gamma}\delta_{\alpha\delta}-s_{ik}^{\alpha\gamma\dagger}s_{il}^{\alpha\delta}$, $\mathrm{sgn}(\alpha=e)=1$ $\mathrm{sgn}(\alpha=h)=-1$,
$f_{i\alpha}$ the occupancy factors for the electron and hole states in
electrode $i$, given by the Fermi function where the chemical potential are the
applied voltages
$f_{ie}(E)=\left[1+\exp\left(\frac{E-V_i}{k_BT}\right)\right]^{-1}\xrightarrow[T\to0]{}\theta(-E+V_i)$,
$
f_{ih}(E)=\left[1+\exp\left(\frac{E+V_i}{k_BT}\right)\right]^{-1}\xrightarrow[
T\to0]{}\theta(-E-V_i)$.
In this one-dimensional model, both current and noise are highly sensitive to
the distances $L_l$, $R$, $L_r$ between the barriers: They oscillate as a
function of these distances with a period equal to the Fermi wavelength
$\lambda_F\ll L_l, R, L_r$. In a higher-dimensional system with more than
one transmission mode, the oscillations in the different modes are
independent and are thus averaged out. Multidimensional behavior can be
simulated qualitatively with a one-dimensional system by averaging all
quantities over one oscillation period:
\begin{align}
\label{PaperNSN:eq:av}
&\overline{X}(L_l, R, L_r)\\ &= \frac{1}{\lambda_F^3}
\int_{L_l-\frac{\lambda_F}{2}}^{L_l+\frac{\lambda_F}{2}}
dl_l\int_{R-\frac{\lambda_F}{2}}^{R+\frac{\lambda_F}{2}} dr
\int_{L_r-\frac{\lambda_F}{2}}^{L_r+\frac{\lambda_F}{2}} dr\kern2pt
X(l_l,r,l_r).\notag
\end{align}
This procedure is appropriate to describe metallic systems. These averaged
quantities are studied in section \ref{paperNNSNN:sec:AS}.
\section{Components of the Differential Conductance and the Differential
Current Cross-Correlations\label{paperNNSNN:sec:comp}}
An electron, arriving from one of the normal metal reservoirs at the interface
to the superconductor, can be: i) reflected as an electron (normal reflection
(NR)), or ii) reflected as a hole (Andreev reflection (AR)), or iii)
transmitted as an electron (elastic cotunneling (EC)) or iv) transmitted as a
hole (crossed Andreev reflection (CAR)), and similarly for holes. The
corresponding elements of the scattering matrix are for NR: $s_{aa}^{ee}$,
$s_{aa}^{hh}$, $s_{bb}^{ee}$, $s_{bb}^{hh}$, AR: $s_{aa}^{eh}$, $s_{aa}^{he}$,
$s_{bb}^{eh}$, $s_{bb}^{he}$, EC: $s_{ab}^{ee}$, $s_{ab}^{hh}$, $s_{ba}^{ee}$,
$s_{ba}^{hh}$, and CAR: $s_{ab}^{eh}$, $s_{ab}^{he}$, $s_{ba}^{eh}$,
$s_{ba}^{he}$.
The current in electrode $N_a$ given by Eq.~\eqref{paperNSN:eq:current} can
naturally be divided into AR, CAR and EC contributions (the unitarity of the
scattering matrix has been used):
\begin{widetext}
\begin{align}
I_a=\frac{|e|}{h}\int \mathrm{d} E
&\underbrace{\left[\left(|s_{aa}^{eh}(E)|^2+|s_{aa}^{he}(E)|^2\right)(f_{ae}(E)
- f_{ah}(E))\right.}_{\text{local Andreev
reflection}}\notag\\ +&\underbrace{|s_{ab}^{ee}(E)|^2(f_{ae}(E) -
f_{be}(E))+|s_{ab}^{hh}(E)|^2(f_{bh}(E)-f_{ah}(E))}_{\text{elastic
cotunneling}}\notag\\ +&\underbrace{\left.|s_{ab}^{eh}(E)|^2(f_{ae}(E)
-
f_{bh}(E))+|s_{ab}^{he}(E)|^2(f_{be}(E)-f_{ah}(E))\right]}_{\text{crossed
Andreev reflection}}.
\end{align}
In the following, we focus on i) the differential conductance in the
symmetrical case where $V_a=V_b=V$ and the current $I_a$ is differentiated
with respect to $V$, and ii) the differential nonlocal conductance in the
asymmetrical case where $V_a=0$ and the current $I_a$ is differentiated with
respect to $V_b$. In the zero temperature limit, only the nonlocal processes,
CAR and EC, contribute to the nonlocal conductance:
\begin{align}
\left.\frac{\partial I_a}{\partial V_b}\right|_{V_a=0}
=\underbrace{-\frac{e^2}{h}\left[
|s_{ab}^{ee}(|e|V_b)|^2+|s_{ab}^{hh}(-|e|V_b)|^2\right]}_{\text{elastic
cotunneling}} +\underbrace{\frac{e^2}{h}\left[|s_{ab}^{eh}(-|e|V_b)|^2 +
|s_{ab}^{he}(|e|V_b)|^2\right]}_{\text{crossed Andreev reflection}},
\end{align}
while the symmetric case contains local Andreev reflection and crossed
Andreev reflection:
\begin{align}
\left.\frac{\partial I_a}{\partial V}\right|_{V_a=V_b=V} =
&\underbrace{\frac{e^2}{h}\left[\left(|s_{aa}^{eh}(|e|V)|^2 +
|s_{aa}^{he}(|e|V)|^2\right) + \left(|s_{aa}^{eh}(-|e|V)|^2 +
|s_{aa}^{he}(-|e|V)|^2\right)\right]}_{\text{local Andreev
reflection}}\notag
\\ +&\underbrace{\frac{e^2}{h}\left[\left(|s_{ab}^{eh}(|e|V)|^2 +
|s_{ab}^{he}(|e|V)|^2\right) + \left(|s_{ab}^{eh}(-|e|V)|^2 +
|s_{ab}^{he}(-|e|V)|^2\right)\right]}_{\text{crossed Andreev reflection}}
\end{align}
Let us now perform a similar analysis for the current cross-correlations. We
study only the zero temperature limit, where
$f_{k\gamma}(E)[1-f_{l\delta}(E)]$ is zero if
$k=l$ and $\gamma=\delta$ and the current
cross-correlations are:
\begin{equation}
S_{ab}(T=0)=\frac{2e^2}{h}\sum_{k,l\in
\{a,b\}}\sum_{\alpha,\beta,\gamma,\delta\in\{e,h\}}\mathrm{sgn}(\alpha)\mathrm{sgn}(\beta)\int
dE
s_{ak}^{\alpha\gamma\dagger}s_{al}^{\alpha\delta}s_{bl}^{\beta\delta\dagger}
s_{bk}^{\beta\gamma}f_{k\gamma}(E)[1-f_{l\delta}(E)]
\end{equation}
\end{widetext}
Every summand in $S_{ab}$ contains the product of four elements of the
scattering matrix. As pointed out in Refs. \onlinecite{ML1992, Buettiker1990},
in difference to the situation for the current, it is impossible to combine
those matrix elements to absolute squares. Let us now sort out and classify the
contributions of the noise as we did above for the current. We find that no
summand consists of only one kind of elements of the scattering matrix. Every
element consists of two local elements (NR or AR ) and two nonlocal elements
(CAR or EC) (see Appendix~\ref{paperNNSNN:app:Noise}). Either the two local
elements and the two nonlocal elements are identical, that gives the
components EC-NR, CAR-NR, EC-AR, CAR-AR, or all four matrix elements belong to
different categories and we will call these summands MIXED. Sometimes, it is
useful to divide MIXED further as a function of its voltage dependence. As
the formulas for the current cross correlations are lengthy, they are
relegated into Appendix~\ref{paperNNSNN:app:Noise}.
Examination of these expressions allows an interpretation of the various
components. First, EC-NR does not involve any Andreev scattering and
corresponds to quasiparticle fluctuations across the double $NSN$
barrier. Second, CAR-NR involves two amplitudes for electron-hole scattering across $NSN$
(Crossed Andreev) and two normal scattering amplitudes. This process tracks fluctuations
of the current of split Cooper pairs emitted in- or absorbed by $S$. Third,
EC-AR involves two local Andreev scattering amplitudes and propagation of a
pair of quasiparticles in $S$. It thus reflects the fluctuations of pairs back
and forth across the $NSN$ double interface. Fourth, CAR-AR involves two
crossed Andreev and two normal Andreev amplitudes. This process
which amounts to splitting two pairs from $S$ is usually weak. Fifth, mixed processes can be
analyzed in the same fashion, they involve a combination of split
pair and quasiparticle crossing fluctuations.
For the interpretation of current cross-correlations, the global sign plays an
important role. Later on, the differential cross-correlation $\partial
S_{ab}/\partial V_{a,b}$ will be plotted. For positive applied bias voltages,
this quantity has the same sign as the current cross-correlations. For
negative applied voltages current, cross-correlations and differential current
cross-correlations have opposite signs. To avoid confusion, we only show
pictures of the differential current cross-correlations calculated for
positive bias voltages (and thus negative energies $E=-|e|V$). Due to the
electron-hole symmetry of the model, differential current cross-correlations
calculated for negative bias voltages are up to a global sign identical to the
ones calculated at positive bias voltage. For small bias voltages, current
cross-correlations depend linearly on voltage. Thus, current
cross-correlations and differential current cross-correlations show the same
qualitative behavior if studied as a function of the interface transparency or
the distance between barriers.
In Ref.~\onlinecite{FFM2010}, we performed a similar analysis of current
cross-correlations in terms of Green's functions for a NSN structure. For the
relations between these two classifications see
Appendix~\ref{PaperNSN:app:Green}. Bignon \textit{et\,al.}~\cite{bignon} have
studied current cross-correlations in the tunneling limit. They find that
noise measurements in the tunneling limit can give access to the CAR and EC
contribution of the current. We have just seen, that at least two processes
are involved in every component of noise, but the contributions of noise they
calculate fall into the categories EC-NR and CAR-NR. In the tunneling limit,
the NR contribution is very close to one, therefore what
remains is very similar to the current contributions. In what follows, more
generally, the CAR-NR component provides the diagnosis of Cooper pair
splitting.
\section{Results \label{paperNNSNN:sec:AS}}
\subsection{Positive Cross-Correlations without CAR}
If a CAR process is interpreted as the splitting of a Cooper pair into two
electrons leaving the superconductor in different electrodes, positive
cross-correlations are its logical consequence. However, the CAR process is
not the only one which can lead to positive cross-correlations. Let us
investigate in more detail the influence of the different processes on the
current cross-correlations in a NSN-system, i.\,e. in a system without the
additional barriers in the normal conducting electrodes.
\begin{figure}
\includegraphics[width=\columnwidth]{FIGURE2}
\caption{\label{paperNSN:fig:ecnr}Averaged differential current
cross-correlations for a symmetrical biased ($V=V_a=V_b\ll\Delta/|e|$)
NSN-system as a function of the transparency of the interfaces
$T_{lns}=T_{rsn}$. The positive cross-correlations at high interface
transparency are due to the EC-AR process, represented by a dotted line.}
\end{figure}
The black line in Fig.~\ref{paperNSN:fig:ecnr} shows the averaged differential
current cross-correlations for symmetric bias ($V=V_a=V_b$). The total current
cross-correlations have already been published in Ref.~\onlinecite{FFM2010},
but here, Fig.~\ref{paperNSN:fig:ecnr} shows in addition the different parts
which contribute to the total current cross-correlations. The total
cross-correlations are positive for high interface transparencies and for low
interface transparencies. As we have already argued in
Ref.~\onlinecite{FFM2010}, the positive cross-correlations at high interface
transparencies are not due to CAR : only processes which conserve momentum can
occur, since there are no barriers which can absorb momentum. CAR processes
do not conserve momentum: if e.g. an electron arrives from the left-hand side
carrying momentum $k_F$, the hole that leaves at the right hand side carries
momentum $-k_F$. As the cross-correlations do not tend to zero even for very
high transparencies, they cannot be due do CAR. Indeed, if we plot the
different components of the noise introduced in the last section separately,
we see that the positive current cross-correlations at high interface
transparencies have a different origin: a large positive EC-AR
contribution, thus correlated pair fluctuations without pair splitting. But positive current cross-correlations at low interface
transparencies are a consequence of a large CAR-NR component and therefore a
consequence of CAR processes.
We can put the contributions to the current cross-correlations into two
categories with respect to their sign, which is independent of the interface
transparency. EC-NR, CAR-AR, MIXED2 and MIXED4 carry a negative sign, CAR-NR,
EC-AR, MIXED1 and MIXED3 carry a positive sign. The current can either be
carried by electrons $I^e$ or by holes $I^h$. The sign of the different
contributions to the current cross-correlations depends on whether only
currents of the same carrier type are correlated~\cite{AD1996} ($\langle
\Delta\hat I_a^e\Delta\hat I_b^e\rangle$+$\langle \Delta\hat I_a^h\Delta\hat
I_b^h\rangle+a\leftrightarrow b$), which is the case for EC-NR, CAR-AR, MIXED2
and MIXED4 and leads to a negative sign; or whether electron currents are
correlated with hole currents ($\langle \Delta\hat I_a^e\Delta\hat
I_b^h\rangle$+$\langle \Delta\hat I_a^h\Delta\hat
I_b^e\rangle+a\leftrightarrow b$), which is the case for CAR-NR, EC-AR, MIXED1
and MIXED3 and leads to a positive sign. In purely normal conducting systems,
the electron and hole currents are uncorrelated, only correlations of the same
carrier type contribute to the current cross-correlation and lead to a
negative sign. The sign of the total current cross-correlations is a
consequence of the relative strength of the different parts of the current
cross-correlations, which depends on the interface transparency.
\subsection{Multiple Barriers}
In the last paragraph, we showed that positive cross-correlations due to CAR
can only be found in the tunneling regime, where the signals are quite weak.
The conductance over an NS-tunnel junction can be amplified for a ``dirty``
normal conductor containing a large number of non-magnetic impurities, where
transport is diffusive, by an effect called reflectionless tunneling
\cite{reflectionlesstunnexpt,reflectionlesstunntheo}, yielding an excess of
conductance at low energy. It is thus natural to ask if a similar effect
could also enhance conductance and current cross-correlations in a three
terminal NSN-structure. To answer this question within the scattering
approach, we use the model of Melsen and Beenakker \cite{MB1994} where the
disordered normal conductor is replaced by a normal conductor with an
additional tunnel barrier leading to an NNS structure. Duhot and M\'elin
\cite{duhotmelin} have studied the influence of additional barriers on the
nonlocal conductance in three terminal NSN-structures. They indeed find that
two symmetric additional barriers enhance the nonlocal conductance. A similar
result is obtained by quasiclassical methods in Ref. \onlinecite{ZaikinPRL}.
First, let us get a deeper understanding of the result of
Ref. \onlinecite{duhotmelin} by calculating the AR, CAR and EC components of
the current separately. Afterwards, we will study the influence of additional
barriers on the current cross-correlations.
\begin{figure}
a)\hspace{-0.5cm}\includegraphics[width=\columnwidth]{./FIGURE3a}
b)\hspace{-0.5cm}\includegraphics[width=\columnwidth]{./FIGURE3b}
\caption{\label{paperNNSNN:fig:AvCond}Averaged differential conductance in the
limit of zero energy in a) the symmetrical bias situation
$V_a=V_b\ll\Delta/|e|$ and b) in the asymmetrical case $V_a=0$,
$V_b\ll\Delta/|e|$ for a superconducting electrode much shorter than the
coherence length ($R=0.25\xi$) as a function of the transparencies of the
additional barriers $T_{lnn}=T_{rnn}$. The barriers next to the
superconductor are in the tunnel regime ($T_{lns}=T_{rsn}=0.01$).}
\end{figure}
\begin{figure}
a)\hspace{-0.5cm}\includegraphics[width=\columnwidth]{./FIGURE4a}
b)\hspace{-0.5cm}\includegraphics[width=\columnwidth]{./FIGURE4b}
\caption{\label{paperNSN:fig:EnDepAvCond} Total averaged differential conductance, as in Fig.~\ref{paperNNSNN:fig:AvCond} but at different energies.}
\end{figure}
Fig.~\ref{paperNNSNN:fig:AvCond} shows the averaged conductance in the
symmetrical bias situation $V_a=V_b\ll\Delta/|e|$ and in the asymmetrical
voltage case $V_a=0$, $V_b\ll\Delta/|e|$ for a superconducting electrode much
shorter than the coherence length ($R=0.25\xi$). The sum of the AR, CAR and EC
components, traced in black, features in both cases an extremum. Yet,
examining at the behavior of these components, we see that they arise from
different mechanisms. Let us first have a look at the symmetrically biased
case. Without the additional barriers, i.e. in the limit
$T_{lnn}=T_{rnn}\to1$, the contributions of AR and CAR are similar in
magnitude. The EC component is completely suppressed, since it is proportional
to the difference of the applied voltages. The introduction of two additional
barriers increases the AR component up to a factor 30. The shape of the curves
is similar to the one of the NNS structure derived analytically by Melsen and
Beenakker \cite{MB1994}, which can be exactly recovered by increasing the
length of the superconducting electrode far beyond the coherence
length. However, the CAR curve stays almost constant over a wide range of
values of barrier strength of the additional barriers, and it eventually
vanishes when the transparencies go to zero.
In the asymmetrical voltage case $V_a=0$, the AR component
is zero as it is proportional to the
local voltage $V_a$. Like in the first case, the additional barriers have little influence on the CAR component,
except for the fact that it obviously tends to zero for vanishing transparency. Over a wide range of
barrier strength values, lowest order tunneling prevails, and the
EC component is identical in amplitude, but opposite in sign to the CAR
component \cite{falci}. For small $T_{lnn}=T_{rnn}$ values the EC
component displays a small extremum, but it is much less pronounced than the
maximum of the AR component of the
first case. The CAR component, on the other hand, does not show any extremum.
In the limit $T_{lnn}=T_{rnn}\to1$, the EC component tends more slowly to zero
than the CAR component which as a consequence yields a maximum in the absolute
value of the total conductance, {\it dominated by EC}. The fact that the
conductance maxima in the symmetrical bias case and in the asymmetrical bias
case have different origins can also be illustrated by studying their energy
dependence, depicted in Fig.~\ref{paperNSN:fig:EnDepAvCond}: The enhancement
of the AR component of the conductance in the symmetrical biased case
disappears completely with increasing bias voltage, as expected for a
zero-bias anomaly. On the contrary, the extremum of the conductance in the
asymmetrical biased case decreases slightly with increasing bias voltage, but
only up to a certain voltage value, then it saturates.
Why is the AR component enhanced by the additional barriers, but not the EC or
CAR components? Reflectionless tunneling occurs because the electrons and
holes resonate inside the double barrier and have therefore a higher
probability to enter the superconductor at low energy, despite
phase-averaging. In the AR case the incoming electron and the leaving hole may
encounter the same scattering path. On the contrary the EC and CAR process,
the incoming particle and the leaving particle encounter different scattering
path. The energy dependence of the conductance enhancement of the AR component
is consistent with reflectionless tunneling which occurs at low bias
voltage. At higher bias voltage, electrons and holes have different
wavevectors and the reflectionless tunneling peak disappears. The integrals
over the phases between the additional barriers on the left- and on the
right-hand side have, of course, been taken independently. There is no reason
to think that the channel mixing, which is emulated by the integrals, on the
left- and on the right-hand side are coupled. To verify this scenario, let us
couple the two integrals in an gedankenexperiment. We set the distance $L_l$
between the two left-hand side barriers to be equal to the distance $L_r$
between the two right-hand side barriers and perform only one integral over
$L=L_l=L_r$. The result is shown in Fig.~\ref{paperNSN:fig:Coupled}. Now, the
CAR and the EC component are also enhanced by a large factor. Yet, the
increase of the EC component is larger than the one of the CAR component, and
EC still dominates the nonlocal conductance, like for a transparent $NSN$
structure.
To sum up the above analysis of the conductance enhancement by disorder, we
showed that the crossed processes CAR and EC cannot be amplified but by an
irrealistic correlation between disorder on the two sides of the NSN structure.
Comparing qualitatively with the different approach of Ref.
\onlinecite{ZaikinPRL}, we also find an enhancement of the crossed conductance
(Fig. 3b), which is not due to any marked maximum in CAR or EC components.
Having the sign of EC, it cannot be interpreted in terms of enhanced Cooper pair
splitting.
\begin{figure}
\includegraphics[width=\columnwidth]{./FIGURE5}
\caption{\label{paperNSN:fig:Coupled}Effect of correlated averaging in the
asymmetrical case with coupled integrals $T_{lns}=T_{rsn}=0.01$,
$R=0.25\xi$: Now, also the EC and the CAR component are enhanced by
reflectionless tunneling.}
\end{figure}
Let us turn back to independent averaging and consider the current
cross-correlations. The results are shown in Fig.~\ref{paperNSN:fig:SE-10}. In
the symmetrical bias case, similarly to Fig. \ref{paperNNSNN:fig:AvCond}a, the
additional barriers do not lead to an enhancement of the signal. The noise is
dominated by the CAR-NR component, featuring Cooper pair splitting and, as we
have seen above, CAR is not influenced by reflectionless tunneling. The EC-AR
component, on the other hand, is amplified by the additional barriers, because
the AR amplitude describing a local process is amplified. This leads to a
small shoulder in the total cross-correlations. But since we are in the tunnel
regime and the leading order of CAR-NR is $T^2$ while the leading order of
EC-AR is $T^4$, the influence of the EC-AR-component is too small to lead to a
global maximum.
In the asymmetrical bias case $V_a=0,V_b\ll\Delta/|e|$, on the other hand, the
additional barriers weakly enhance the signal. But the cross-correlations are
dominated by EC-NR and are therefore negative, and they do not feature Cooper
pair splitting. In conclusion, in a phase-averaged system, additional
barriers only enhance the AR-component, a local process. It cannot help to
amplify nonlocal signals. Again, positive cross-correlations signalling Cooper
pair splitting are only encountered in the tunneling regime. This conclusion
is contrary to the interpretation of the quasiclassical theory given in
Ref. \onlinecite{ZaikinPRBnoise}.
\begin{figure}
a)\hspace{-0.5cm}\includegraphics[width=\columnwidth]{./FIGURE6a}\\
b)\hspace{-0.5cm}\includegraphics[width=\columnwidth]{./FIGURE6b}
\caption{\label{paperNSN:fig:SE-10} Averaged differential current
cross-correlations in a) the symmetrical bias situation
$V_a=V_b\ll\Delta/|e|$ and b) the asymmetrical bias case $V_a=0$,
$V_b\ll\Delta/|e|$ for a superconducting electrode shorter than the
coherence length ($R=0.25\xi$) as a function of the transparencies of the
additional barriers $T_{lnn}=T_{rnn}$. The barriers next to the
superconductor are in the tunnel regime ($T_{lns}=T_{rsn}=0.01$).}
\end{figure}
\section{Conclusion}
We have found that at high transparency, crossed processes are dominated
by electron transmission and that positive cross-correlations in this range of
interface transparency are not due to Cooper pair splitting. Instead, for symmetrical voltages,
they originate from correlated fluctuations of Cooper pairs from the superconductor
to both metallic contacts and vice-versa. Cooper pair
splitting in the tunnel regime cannot be enhanced with additional barriers by
a process similar to reflectionless tunneling, if an average (here, over the
interbarrier lengths) has to be performed, mimicking disorder landscapes which
are uncorrelated on the two sides of the set-up. In analogy, one expects that
the same conclusion holds if one uses diffusive normal metals. These
conclusions are important for settling future experimental programs. Positive
cross-correlations might well be observed at high transparency, but they are
not a signature of Cooper pair splitting. They are not related to spatially
separated spin-entangled pairs.
Conductance and cross-correlation measurements with controlled and tunable interface transparencies would be very useful, and
may be attempted for instance in carbon nanotubes junctions.
Finally, a cross-over to negative current cross-correlations for a highly
transparent NSN junction biased above the gap is expected. Its theoretical
description is more involved, because it requires taking nonequilibrium
effects such as charge imbalance \cite{imb1,imb2,imb3} into account. A
starting point for those calculations can be found in
Ref.~\onlinecite{Melin-Bergeret-Yeyati}.
\section*{Acknowledgements}
The authors have benefited from several fruitful discussions with
B. Dou\c{c}ot. Part of this work was supported by ANR Project "Elec-EPR".
|
1,116,691,497,757 | arxiv | \section{Introduction}
The advent of model predictive control (MPC) has enabled control processes to take advantage of the rapid advancement in optimization techniques and the computational power of modern hardware \cite{qin2003survey}. The ability to optimize over dynamic models and incorporate various constraints has bestowed model predictive controllers with feasibility and stability guarantees for complex control tasks \cite{borrelli2017predictive}. However, identifying accurate dynamic models remains a central challenge to MPC. Building accurate first-principle models often requires domain expertise and deep physical insights into the systems. Even for experts, modeling can be a daunting and tedious task when the system or its operating environment is complex.
Recent advances in machine learning algorithms have enabled efficient discovery of coherent patterns in complex data. Neural networks (NNs) have been used for various tasks in computer vision, natural language processing, and recommender systems. Most recently, a new family of neural networks -- neural ordinary differential equations (NODE), has been shown as an effective tool for extracting dynamic models from data. NODE approximates a continuous-depth neural network which is used to directly model differential equations. Knowledge-based NODE (KNODE) was proposed to leverage NODE's compatibility with first principles. It combines first-principle models and NNs into hybrid models to improve the open-loop prediction accuracy of nonlinear dynamic models \cite{jiahao2021knowledgebased}. However, it is unclear whether such hybrid models could improve the closed-loop performance of robotic systems in the real world.
In this work, we use KNODE to model the dynamics of a quadrotor. In particular, we utilize the neural network in the hybrid model to account for residual and uncertain dynamics within the system. This hybrid model is then incorporated into a model predictive control framework, known as KNODE-MPC. A schematic of the proposed framework is depicted in Fig. \ref{fig:cover_image}. We demonstrate that the integrated framework not only provides better predictions of future states, but also improves the closed-loop performance in both simulations and real-world experiments.
\begin{figure}
\centering
\includegraphics[scale=0.355]{Cover_image.png}
\caption{Schematic of the KNODE-MPC framework, applied to a quadrotor system. The NN accounts for residual or uncertain dynamics in the system, and is combined with a model derived from first principles to form a KNODE model. The KNODE model is then incorporated into a MPC framework, where optimal control commands are generated and applied to the quadrotor.}
\label{fig:cover_image}
\end{figure}
\section{Related Work}
In recent years, increasing data availability has fueled rapid development in data-driven system identification. One body of work aims to extract the governing equations of dynamic models directly from data \cite{SINDy, EntropicRegression, Champion22445}. A major contribution uses sparse regression to reconstruct dynamic models from a predetermined library of basis functions \cite{SINDy}. This method has been shown as an effective alternative to system identification \cite{Kaiser_2018}. However, a fundamental challenge to this method lies in coming up with the library of functions. If the correct basis functions are missing from the library, the sparse regression fails to identify the correct dynamic model. Such methods also do not scale well to high-dimensional systems as the library of functions can become unreasonably large. This prevents deployment on real-world systems, which are very often high-dimensional.
On the other hand, neural networks have recently been shown as an effective parameterization of dynamic models. Stochastic neural networks were used to model residual physics in contact models \cite{ajay2018augmenting}, and subsequently ContactNets was proposed to model frictional contact behaviors \cite{pfrommer2020contactnets}. Recurrent neural networks (RNNs) have been used to model the dynamics of high-dimensional flows \cite{qraitem2020bridging} and spatio-temporally chaotic systems \cite{OttRescomp}. NODE was developed for similar purposes but differs from RNNs by explicitly modeling differential equations from data \cite{conf_nips_ChenRBD18}. Notably, NODE has enabled various deep learning tools to facilitate system identification. For example, convolutional neural networks have been used to reduce the dimensionality of high-dimensional systems \cite{jiahao2021knowledgebased}. In particular, KNODE has demonstrated neural network's generality by modeling a variety of systems with nonlinear and chaotic dynamics, and has also shown NODE's compatibility with first-principle dynamic models \cite{jiahao2021knowledgebased}. These developments have opened up new avenues for system identification for MPC.
There are a number of works in the literature that integrate data-driven models into a predictive control framework. One approach to synthesizing models for MPC is through Gaussian processes (GP) regression. In \cite{kabzan2019learning}, GP regression is used to learn the residual errors between the true and nominal dynamics of a race car, accounting for prior knowledge of the system. The authors in \cite{torrente2021data} extend this framework to model residual dynamics in quadrotors. However, these methods assume that the relationship between the residual and true dynamics is known and this limits the nature of uncertainties that these models are able to accommodate. Another drawback of GP regression is computational complexity. This implies the need to select a small number of training data points that best represent the dynamics. Since the dynamics are either unknown or uncertain, it is difficult to select these points in practice. Another increasingly popular approach that incorporates learning into MPC is model-based reinforcement learning (MBRL). In \cite{williams2017information}, neural networks are used to learn a dynamics model, which is then applied to a MPC framework to solve iterative tasks. The authors in \cite{lambert2019} use MBRL to design low-level controllers using sensors on-board the quadrotor. Compared to the approaches that use GP regression and our proposed framework, these methods do not account for prior knowledge of the system dynamics and are therefore less sample-efficient.
Our contributions in this work are three-fold. First, we employ KNODE to develop a high-fidelity quadrotor model capable of capturing poorly understood uncertainties and residual dynamics. The KNODE model combines prior knowledge of the system dynamics with a neural ODE, and provides a better representation of the quadrotor dynamics. Second, we show that the KNODE model improves the accuracy of state predictions significantly, over both the nominal model, as well as a state-of-the-art GP-based prediction model, which is similar to those described in \cite{kabzan2019learning} and \cite{torrente2021data}. In particular, we compare our framework against a more general variant of these GP models that does not impose any structural assumptions between the system uncertainties and the true dynamics. Third, the hybrid model is integrated into a novel model predictive control framework, known as KNODE-MPC. Simulations and physical experiments are conducted with KNODE-MPC to evaluate the improvement in closed-loop performance. Empirical results show that the framework outperforms a nominal model predictive controller with a first-principle model.
\section{Methodology}
\subsection{Quadrotor Dynamics}
In this work, we use a six degrees-of-freedom quadrotor system to illustrate our proposed KNODE-MPC framework. The model used to describe this system is similar to that in \cite{mellinger2011}. Denoting the mass and moments of inertia as $m$ and $J := \text{diag}(J_{xx},\,J_{yy},\,J_{zz})$, the dynamics of the quadrotor, governed by its equations of motion, can be described as:
\begin{equation} \label{eq:forces}
\begin{bmatrix} \ddot{x}\\\ddot{y}\\\ddot{z} \end{bmatrix} := \ddot{\mathbf{r}} = \frac{1}{m} \left( \begin{bmatrix} 0\\0\\-mg \end{bmatrix} + R_B^W \begin{bmatrix} 0\\0\\u_1 \end{bmatrix} \right),\\
\end{equation}
\begin{equation} \label{eq:moments}
\begin{bmatrix} \dot{p}\\\dot{q}\\\dot{r} \end{bmatrix} := \dot{\boldsymbol{\omega}} = J^{-1} \left( \mathbf{u}_2 - \begin{bmatrix} p\\q\\r \end{bmatrix} \times J \begin{bmatrix} p\\q\\r \end{bmatrix} \right),
\end{equation}
where $R_B^W$ is the transformation matrix between the quadrotor body frame and the world frame. $u_1$ is the summation of the motor forces, \textit{i.e.}, $u_1 := \sum_{i=1}^4 F_i$ and $\mathbf{u}_2$ is the moment vector that is related to the motor forces by
\begin{equation} \label{eq:motor_moments}
\mathbf{u_2} = \begin{bmatrix} 0 & L & 0 & -L\\ -L & 0 & L & 0\\ \gamma & -\gamma & \gamma & -\gamma \end{bmatrix} \begin{bmatrix} F_1 \\ F_2 \\ F_3 \\ F_4 \end{bmatrix},
\end{equation}
where $L$ is the arm length of the quadrotor and $\gamma$ is the ratio between the moment and thrust coefficient of the motors. The orientation in terms of Euler angles velocities is related to the angular velocities by
\begin{equation} \label{eq:orientation}
\boldsymbol{\omega} = \begin{bmatrix} cos(\theta) &0 &-\cos(\phi)\sin(\theta) \\0 &1 &\sin(\phi) \\
\sin(\theta) & 0 & \cos(\phi)\cos(\theta) \end{bmatrix} \begin{bmatrix} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{bmatrix}.
\end{equation}
By defining the state as $\mathbf{x} := [\mathbf{r}\;\dot{\mathbf{r}}\;\phi\;\theta\;\psi\;\boldsymbol{\omega}] \in \mathbb{R}^n$, we have the following compact representation of the quadrotor system,
\begin{equation} \label{eq:nominal}
\dot{\mathbf{x}} = f(\mathbf{x},\mathbf{u}),
\end{equation}
with $\mathbf{u} = [u_1\; \mathbf{u}_2]^T \in \mathbb{R}^m$ and $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$ denotes a mapping given by \eqref{eq:forces}, \eqref{eq:moments} and \eqref{eq:orientation}. In this work, both quaternions and Euler angles are used interchangeably to represent orientation.
\subsection{Knowledge-based Neural ODEs} \label{KNODE_section}
Although the nominal quadrotor model in \eqref{eq:nominal} represents the system with certain fidelity, it is generally not sufficient for applications where a more accurate model is required. For instance, the quadrotor experiences aerodynamic and environmental disturbances in flight, and these are not captured by the nominal model. In addition, system parameters such as the moments of inertia are inherently difficult to measure or estimate. These uncertainties cause deviations between the true dynamics of the system and those given by the nominal model. The KNODE framework tackles this challenge by learning these uncertainties and residual dynamics using a data-driven approach. By combining prior knowledge of the system with the learned dynamics, this hybrid model not only improves accuracy, but is also more sample efficient.
Consider the following representation of the true system dynamics
\begin{equation} \label{eqn:uncertainDynamics}
\dot{\mathbf{x}} = f_t(\mathbf{x},\mathbf{u},\, \Delta(\mathbf{x},\mathbf{u})),
\end{equation}
where $f_t : \mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R}^p \rightarrow \mathbb{R}^n$ denotes the true dynamics. The subscript $t$ denotes the true system. $\Delta : \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^p$ is a mapping from the state-input space to the uncertainty space. Next, we parameterize the uncertainty $\Delta$ using an NN with parameters $\boldsymbol{\theta}$. This represents part of the true dynamics that are absent in the nominal model. With this parameterization, the hybrid KNODE model (subscript $h$) can be written as
\begin{equation} \label{eqn:hybrid_model_coupled}
\dot{\mathbf{x}} = f_h(\mathbf{x},\mathbf{u},\, \Delta_{\boldsymbol{\theta}}(\mathbf{x},\mathbf{u})),
\end{equation}
where $\Delta_{\boldsymbol{\theta}}$ is a neural network, and $f_h$ includes both the nominal model and its coupling with the neural network. This framework is general enough to allow coupling between different components of the state and input, and it accommodates uncertain forces and moments readily. Furthermore, compared to non-parametric methods like Gaussian process (GP) regression, this parametric approach allows training on large amounts of data while keeping the size of the uncertainty model constant, and can potentially be computationally faster during inference. Hence, by modeling uncertainty in the dynamics, this hybrid model gives a more accurate representation of the true system.
\subsection{Model Predictive Control}
An MPC framework uses an optimization-based approach to generate a sequence of optimal control inputs in a receding horizon manner. In its formulation, a model of the system, together with state and input constraints, are considered within an optimization problem of the following form,
\begin{equation} \label{eq:mpc}
\begin{split}
\underset{\mathbf{u}}{\textnormal{minimize}}\quad\; & \sum_{i=1}^{N-1} \left( x_i^T Q x_i + u_i^T R u_i \right) + x_N^T P x_N\\
\text{subject to}\quad\; &x_{i+1} = f_{d}(x_i, u_i), \quad \forall\, i=0,\dotsc,N-1,\\
\; &x_i \in \mathcal{X}, \quad u_i \in \mathcal{U}, \quad \forall\, i=0,\dotsc,N-1,\\
\; &x_0 = x(k),
\end{split}
\end{equation}
where $\mathcal{X}$ and $\mathcal{U}$ are sets defining the state and input constraints and $x(k)$ is the initial state to the optimization problem at time step $k$. $Q$, $R$ and $P$ are weighting matrices for the stage, input and terminal costs respectively. At each time step, \eqref{eq:mpc} is solved to obtain a sequence of control inputs $\mathbf{u}:=[u_0,\dotsc,u_{N-1}]^T$ and $u_0$ is applied to the system. The discrete-time model $f_d : \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$ is derived from a continuous-time counterpart using a numerical solver. An explicit fourth-order Runge-Kutta method is applied to obtain the predicted states $x_{i+1},\,i=0,\dotsc,N-1$. In the ideal case, the true dynamics in \eqref{eqn:uncertainDynamics} are discretized, but since the uncertainty is unknown, a discrete-time version of the nominal model described in \eqref{eq:nominal} is commonly used in this framework.
There exist robust and stochastic MPC methods in the literature that account for uncertainty by assuming that the uncertainty is either bounded \cite{bemporad1999robust,mayne2011tube} or characterized by a probability distribution \cite{mesbah2016a,mesbah2016b}. However, these approaches require additional assumptions on the nature of the uncertainty, which can be challenging to ascertain in practice. In most cases, the closed-loop performance is sensitive to the accuracy of the model and hence, using the KNODE model in \eqref{eqn:hybrid_model_coupled} is expected to yield better closed-loop system performance than a nominal model. This motivates the KNODE-MPC framework, where a hybrid knowledge-based NODE model is integrated into the MPC framework.
\subsection{KNODE Training}
While the KNODE framework is applicable to the general case shown in \eqref{eqn:uncertainDynamics}, we shall focus on a more practical scenario. Assuming that the uncertainty is separable from the nominal dynamics, the true dynamics can then be expressed as
\begin{equation}
\dot{\mathbf{x}} = f_t(\mathbf{x},\mathbf{u},\,\Delta(\mathbf{x},\mathbf{u})) = f(\mathbf{x},\mathbf{u}) + \Delta(\mathbf{x},\mathbf{u}),
\end{equation}
with the first component given by the nominal dynamics in \eqref{eq:nominal}. Following the parameterization described in \ref{KNODE_section}, the hybrid model that consists of both the nominal model and neural network is given by
\begin{equation} \label{eqn:hybrid_model}
\dot{\mathbf{x}} = f_h(\mathbf{x},\mathbf{u},\Delta_{\boldsymbol{\theta}}(\mathbf{x},\mathbf{u})) = f(\mathbf{x},\mathbf{u}) + \Delta_{\boldsymbol{\theta}}(\mathbf{x},\mathbf{u}),
\end{equation}
where $\Delta_{\boldsymbol \theta}$ denotes the neural network. Training of $\Delta_{\boldsymbol \theta}$ is done with the trajectory data collected from closed-loop simulations or experiments. The training process in our work differs from \cite{jiahao2021knowledgebased} in that control inputs in the training data need to be injected into the model to make the one-step-ahead predictions. Similar to \cite{jiahao2021knowledgebased}, we first divide up trajectory data from each time step, and then use the state at each of these time steps as initial conditions to make one-step-ahead predictions using \eqref{eqn:hybrid_model}. Specifically, we consider a set of state observations and control inputs $\mathcal{O} = [(\mathbf{x}(t_1), \mathbf{u}(t_1)), (\mathbf{x}(t_2), \mathbf{u}(t_2)), \cdots, (\mathbf{x}(t_N), \mathbf{u}(t_N))]^T$, sampled at the times $T = \{t_1, t_2,\cdots,t_N\}$. For a time step $t_i$, we compute the one-step-ahead prediction given by
\begin{equation}
\hat{\mathbf{x}}(t_{i+1}) = \mathbf{x}(t_i) + \int^{t_{i+1}}_{t_i}f_h(\mathbf{x}(t_i), \mathbf{u}(t_i)) dt,
\label{eqn: cont one-step-ahead}
\end{equation}
where $\hat{\mathbf{x}}(t_{i+1})$ denotes the state prediction at time $t_{i+1}$. The learning framework in this work performs the integration in \eqref{eqn: cont one-step-ahead} numerically using the explicit fourth order Runge-Kutta method. Next, to compute the mean squared error (MSE) between the predictions and the ground truth, we define a loss function as
\begin{equation}
L(\boldsymbol{\theta}) := \frac{1}{N-1}\sum^{N}_{i=2}\left(\hat{\mathbf{x}}(t_i) - \mathbf{x}(t_i)\right),
\label{eqn: ave mse}
\end{equation}
and the optimization problem is therefore given by
\begin{equation}
\begin{aligned}
\min_{\boldsymbol{\theta}} \quad & L(\boldsymbol{\theta}) \\
\textrm{s.t.} \quad & \dot{\mathbf{x}} = f_h(\mathbf{x}, \mathbf{u},\Delta_{\boldsymbol{\theta}}(\mathbf{x},\mathbf{u})),
\end{aligned}
\label{eqn: optimization}
\end{equation}
where the constraint comprises of the dynamic model. To train the hybrid model on multiple trajectories, we can sum up the losses computed on each trajectory. Similar to \cite{jiahao2021knowledgebased, conf_nips_ChenRBD18}, we solve the optimization task in \eqref{eqn: optimization} using the adjoint sensitivity method which propagates gradients from the loss function to the neural network parameters. The parameters are then iteratively updated based on the gradients using readily available optimizers like Adam \cite{Kingma2015AdamAM}. The adjoint sensitivity method has been noted as a memory-efficient alternative to the conventional backpropagation \cite{conf_nips_ChenRBD18}, and its proof can be found in \cite{Cao2003AdjointSA}.
\subsection{Evaluation and Verification}
In our experimental design and evaluation process, we set out to answer the following questions about our proposed KNODE-MPC framework. (a) How accurate are the state predictions given by the KNODE model, as compared to state-of-the-art data-driven models in \cite{kabzan2019learning} and \cite{torrente2021data}? (b) How well does the KNODE model generalize beyond training data? (c) How much improvement does the KNODE-MPC framework provide, in terms of closed-loop trajectory tracking, compared to a nominal MPC framework?
To evaluate the accuracy of state predictions and to verify the generalization capability of the KNODE model, a generalized variant of the model in \cite{torrente2021data} is used as a benchmark. In \cite{torrente2021data}, it is assumed that the relationship between the uncertainties and equations of motion are known and this is incorporated as user-defined selection matrices within the model (see Section III.C of \cite{torrente2021data}). The model also assumes that the uncertainties are decoupled and only allows for a mapping between the body velocities and acceleration disturbances (see Section III.E of \cite{torrente2021data}), without accounting for any rotational disturbances. These assumptions are often difficult to ascertain in practice.
In this work, we implement a general version of this GP model that removes these assumptions, which can be compactly represented as
\begin{equation} \label{eq:gp_model}
\dot{\mathbf{x}}_{\textbf{gp}} = f_{\textbf{gp}}(\mathbf{x},\mathbf{u}) = f(\mathbf{x},\mathbf{u}) + \boldsymbol \mu \left(\begin{bmatrix}\mathbf{x} \\\mathbf{u}\end{bmatrix}\right),
\end{equation}
where $f(\mathbf{x},\mathbf{u})$ represents the nominal model in \eqref{eq:nominal} and $\boldsymbol{\mu}$ denotes the mean posterior of a GP. A nominal model that does not account for uncertainty is also included in our tests as a baseline. To demonstrate the effectiveness of both data-driven models, we incorporate nonlinear components into the true dynamics and compare the prediction accuracy across three models; the nominal, KNODE and GP models from \eqref{eq:nominal}, \eqref{eqn:hybrid_model} and \eqref{eq:gp_model} respectively. Quantitatively, we want to measure the spatial similarity between the true and predicted trajectories and this can be evaluated using a distance metric generated by the dynamic time warping (DTW) algorithm \cite{sakoe1978}. Next, for evaluation of the trajectory tracking performance of KNODE-MPC, we compare it against a nominal MPC framework, both in simulations and in physical experiments. The nominal MPC framework uses a model that does not consider uncertainty such as aerodynamic forces or environmental disturbances in its formulation. The closed-loop trajectories obtained from both frameworks are compared against the planned trajectory using the same distance metric.
\section{Experiments and Results}
\subsection{Setup}
\textit{Simulations}: A nominal quadrotor model is constructed using the equations of motion given in \eqref{eq:nominal}. An explicit 5$^{\text{th}}$ order Runge-Kutta method (RK45) with a sampling interval of 2 milliseconds is used for numerical integration to generate dynamic responses of the quadrotor. We assume that the model predictive controller has access to perfect measurements of the quadrotor dynamics. The MPC architecture is implemented in CasADi \cite{Andersson2019} and the optimization routine generates optimal thrust and moment commands. These commands act as inputs to the quadrotor model, which simulates the closed-loop responses. Two classes of trajectories are considered in the simulations; circular and lemniscate trajectories. Both the GP and KNODE models are trained on circular trajectories of radii 3m and 6m, and a circular trajectory of radius 4m is used as validation data. To model uncertainty within the system, nonlinear aerodynamic drag effects from the fuselage and rotors are incorporated into the true quadrotor dynamics. These uncertain dynamics are absent in the nominal model and are to be learnt by the GP and KNODE models.
The KNODE model uses a two layer neural network architecture with 64 and 16 hidden units each. A hyperbolic tangent function is used as the activation function. Each of the two trajectories in the training data has a duration of 8s and 4000 data points. The GP model is trained using the scikit-learn package \cite{scikit-learn} and uses a kernel consisting of the product of constant and radial basis function (RBF) kernels. Eighty data points, sampled at regular intervals, are used for training. The inputs for the training data are data samples containing the state-input vector $[\mathbf{x}\;\mathbf{u}]^T$, while the training labels are the differences between the true and nominal state derivatives, \textit{i.e.}, $\Delta\dot{x} := f_{\textbf{gp}}(\mathbf{x},\mathbf{u})-f(\mathbf{x},\mathbf{u})$. With the trained models, simulations are conducted along both types of trajectories to evaluate the prediction accuracy of the KNODE model against the nominal and GP models. To assess generalization performance, the models are tested on trajectories beyond the training data. These include circular trajectories of different radii, as well as lemniscate trajectories. Lastly, to test closed-loop performance, the KNODE-MPC framework is simulated on circular trajectories of various radii and compared against a nominal model predictive controller.
\begin{figure}
\centering
\includegraphics[scale=0.35]{Hardware_Schematic.PNG}
\caption{Schematic of the experimental setup: The Crazyflie experimental platform operating with a Vicon motion capture system. KNODE-MPC publishes control commands through the Crazyflie ROS server and into the commander module on-board Crazyflie.}
\label{fig:crazyflie}
\end{figure}
\textit{Physical Experiments}: We use the open-source Crazyflie 2.1 quadrotor \cite{Bitcraze} as the experimental platform. An image of a Crazyflie is shown in Fig. \ref{fig:crazyflie}. The Crazyflie, together with motion trackers, weighs 32 g and has a size of 9 cm$^2$. The software architecture is developed using the CrazyROS wrapper \cite{crazyflieROS}. A laptop running on Intel i7 CPU acts as the base station and communication with the Crazyflie is done via the Crazyradio PA at a rate of 250 Hz. To obtain pose information of the quadrotor, we use a Vicon motion capture system which communicates with the base station via a Vicon bridge and publishes at an approximate rate of 88 Hz. Linear velocities are estimated from the positions obtained from the Vicon, while angular velocities are measured from the gyroscope sensors on-board the quadrotor. This experimental setup is depicted in Fig. \ref{fig:crazyflie}.
To train and validate the KNODE model, data is collected by flying the Crazyflie with the nominal MPC framework along circular trajectories of radii 0.5 and 1m, each for a duration of 76 seconds and 40000 data points. We preprocess the velocity data using a fifth order Butterworth low pass filter before training. Similar to the simulation setup, both the nominal and KNODE-MPC frameworks are implemented in CasADi and run on the base station. The KNODE in physical experiments uses a one-layer neural network with 32 hidden units and the hyperbolic tangent activation function. The frameworks generate three-dimensional acceleration commands and run on top of a geometric controller \cite{mellinger2011}, as well as attitude and thrust controllers within the Crazyflie firmware. For each of the two frameworks, we conduct five test runs along circular trajectories of radii 0.5m and 1.25m to evaluate closed-loop trajectory tracking performance.
We refer the reader to the video in the supplemental materials for a better understanding of the physical experiments.
\subsection{Simulation Results} \label{section:simulations}
To illustrate the accuracy of state predictions, the 3D DTW position errors between the predicted trajectories and the true trajectory are plotted in Fig. \ref{fig:circular_dtw}. Both the KNODE and GP models are able to account for the nonlinear aerodynamic drag effects included in the true dynamics and provide accurate predictions, outperforming the nominal model by more than a factor of 10. Moreover, the accuracy of the KNODE model is observed to be substantially higher than that of the GP model across all trajectories. In particular, by contrasting the DTW errors of both models, an average improvement of 82\% was observed for the KNODE model under circular trajectories. It is also noted that although the GP model provide reasonable predictions on the training data (circular trajectories of radii 3m and 6m), there are cases in which it does not generalize. For instance, the accuracy of the GP model under a circular trajectory of radius 2m is comparable to that of a nominal model, as shown in Fig. \ref{fig:circular_dtw}. On the other hand, the accuracy of the KNODE model is consistent across all trajectories, which implies that the KNODE model is capable of both interpolating within and extrapolating outside of its training domain. For lemniscate trajectories, KNODE outperforms the GP model by an average of 63\% and this is shown in Fig. \ref{fig:lemniscate_dtw}. These results for lemniscate trajectories ascertain the generalization capability of the KNODE model.
\begin{figure}
\centering
\includegraphics[scale=0.25]{dtw_circular_sim.PNG}
\caption{DTW prediction errors for the nominal, GP and KNODE models, tested with circular trajectories of different radii.}
\label{fig:circular_dtw}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.25]{dtw_lemni_sim.PNG}
\caption{DTW prediction errors for the nominal, GP and KNODE models, tested with lemniscate trajectories of different radii.}
\label{fig:lemniscate_dtw}
\end{figure}
Closed-loop simulation results for both the nominal and KNODE-MPC frameworks are plotted in Fig. \ref{fig:closed_loop_sim}. Compared to nominal MPC, KNODE-MPC reduces the 3D trajectory tracking DTW errors by 73\% on average. This demonstrates the effectiveness of the integrated framework in the presence of uncertain dynamics. Fig. \ref{fig:closed_loop_traj} provides a comparison between the closed-loop trajectories under the two frameworks. Since the model in the nominal MPC framework does not account for residual or uncertain dynamics, the quadrotor trajectory deviates from the planned trajectory, with position errors accumulating over time. On the other hand, the KNODE model compensates for the residual dynamics and allows the quadrotor to follow the planned trajectory more closely.
\begin{figure}
\centering
\includegraphics[scale=0.26]{closed_loop_sim.png}
\caption{Trajectory tracking errors for the nominal and KNODE models, under the nominal and KNODE MPC frameworks, along circular trajectories of different radii.}
\label{fig:closed_loop_sim}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.26]{closed_loop_traj.png}
\caption{3D trajectories for the nominal and KNODE models, under the nominal and KNODE MPC frameworks, under a circular trajectory of radius = 2m.}
\label{fig:closed_loop_traj}
\end{figure}
\subsection{Physical Experimental Results} \label{section:experiments}
The system and planned trajectories of the quadrotor with both nominal MPC and KNODE-MPC under a circular trajectory of radius 0.5m are shown in Fig \ref{fig:xy_plot_0p5}. The boundaries of the red annulus indicate the maximum deviation of the system trajectories under the nominal MPC framework. The same annulus is overlaid on the right subplot and it provides a qualitative assessment of the improvement in accuracy with the KNODE-MPC framework. Notably, the trajectories under KNODE-MPC do not exceed the boundaries of the annulus, which implies there are smaller deviations from the planned trajectory. By taking the average across five runs, the nominal MPC yields an DTW x-y position error of 4.69, while an error of 2.54 is obtained with KNODE-MPC. These indicate an improvement of over 45\%. In the z axis, the average DTW errors between the frameworks are comparable, valued at 11.59 and 12.11 respectively.
Next, the generalization ability of KNODE-MPC is verified using trajectories beyond its training dataset. In this scenario, the planned trajectory has a radius of 1.25m, which differs from training trajectories, which have radii of 0.5 and 1m. Fig. \ref{fig:xy_plot_1p25} depicts the system and planned trajectories under both frameworks. Here, the boundaries of the red annulus also indicate the maximum deviation of the system trajectories under the nominal MPC and it is overlaid on the right subplot. The trajectories generated by KNODE-MPC lie strictly within the red annulus, which implies the closed-loop tracking performance of KNODE-MPC is more accurate than nominal MPC. Quantitatively, the average DTW x-y position error across five test runs for both the nominal and KNODE-MPC are 10.3 and 8.81 respectively, which yields a 14.5\% improvement in trajectory tracking for KNODE-MPC. The differences in performance improvement between simulation and physical experimental results can be attributed to the different control architectures between the two setups. In particular, the KNODE-MPC framework in simulations generate control commands to actuate the quadrotor directly, while in physical experiments, it runs on top of the geometric controller and low-level controllers within the firmware, and it does not have direct control over the quadrotor dynamics.
\begin{figure}
\centering
\includegraphics[scale=0.26]{xy_plot_0p5m.PNG}
\caption{Top view: Quadrotor trajectories under both nominal MPC and KNODE-MPC, with a planned trajectory of radius 0.5m.}
\label{fig:xy_plot_0p5}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.26]{xy_plot_1p25m.PNG}
\caption{Top view: Quadrotor trajectories under both nominal MPC and KNODE-MPC, with a planned trajectory of radius 1.25m.}
\label{fig:xy_plot_1p25}
\end{figure}
\section{Conclusion and Future Work}
In this work, we presented the KNODE-MPC framework with an application to quadrotor control. The knowledge-based, data-driven model, KNODE, provides a more accurate representation of the system dynamics, as compared to a state-of-the-art GP model. By incorporating KNODE into the MPC framework, the predictive controller generates optimal control commands that yield better closed-loop performance. This framework is tested extensively in simulations and physical experiments. Results show significant improvements in terms of state predictions, as well as closed-loop trajectory tracking. Given its flexibility, the framework can be applied to robotic systems operating in complex and uncertain environments such as marine robots or robotic teams. In the near future, we would like to extend this framework to an online learning setting, where the KNODE model can be updated and refined in real-time, which can further improve the performance of closed-loop control tasks.
\bibliographystyle{IEEEtran}
|
1,116,691,497,758 | arxiv |
\section{Approach}
In general, a patch stack (also known as patch set) is defined as a set of
patches (commits) that are developed and maintained independently of the base
project.
Well-known examples include the Preempt-RT{} Linux realtime extension, the Linux LTSI
(Long Term Support Initiative) kernel, and vendor-specific Android stacks needed
to port the system to a particular hardware.
In many cases, patch stacks are applied on top of individual releases
of an upstream version, but they do not necessarily have to be
developed in a linear way~\cite{bird09}.
The commits of the patched version of a base project are identified as the set of
commit hashes that do not occur in the mainline project.
Our analysis is based on the following assumptions:
\begin{itemize}
\item Mainline \emph{upstream} development takes place in one single branch.
\item Every release of the patch stack is represented by a separate branch.
\end{itemize}
The work flow of \texttt{PaStA}{} consists of the following steps:
(1) Set up a repository containing all releases of the patch stacks.
(2) Identify and group similar patches across different versions of the patch
stacks.
(3) Compare representatives of those groups against mainline.
(4) Use statistical methods to draw conclusions on the development and evolution
of the patch stacks.
A \emph{commit hash} provides a unique identifier for every commit: In
the following, $U$ is the set of all commit hashes of the base project,
while $P_i$ is the set of the commit hashes of a release $i$ of the patch stacks.
$P\equiv \bigcup_i P_i$ denotes all commit hashes on the patch stacks.
Note that $P\cap U = \emptyset$. Let $H\equiv P \cup U$ be the set of
all commit hashes of interest. A semi-automatic classification
function $\text{comp}: P \times H \rightarrow \{\text{True},
\text{False}\}$ decides whether two patches are similar or not. A
detailed description of the function $\text{comp}$ can be found in
Section~\ref{sec:detect}.
In the implementation, \texttt{PaStA}{} mines git repositories.
Without loss on generality, we focus on
this particular version control system because it is widely employed
in current OSS development.
\subsection{Grouping Similar Patches}
Patch stacks change as they are being aligned with the changes in base
project and additionally integrate or loose functionalities. New
patches are pushed on top of the stack, existing patches may be
amended to follow up with API changes, or patches are dropped.
Because of the rapid dynamics and growth of Open Source
projects~\cite{osgrowth}, a significant amount of patches must manually be
ported from one release of the base project to the next. Since the
base project changes over time, it is necessary to
continuously adapt the details of individual patches. Those
adaptations can be classified in textual and higher-order
conflicts~\cite{brun}. Textual conflicts can be solved by manually
porting the patch to the next version. In a series of patches,
patches may depend on each other, so that textual conflicts in one
patch lead to follow-up conflicts in further patches. Higher-order
conflicts occur when a patch obtains a new (erroneous) semantic
meaning after changes in the base project diverged, despite a lack of
textual conflicts. Both types are known to induce high maintenance
cost~\cite{ltsi}.
Even if the semantics of patches remain invariant over time (e.g., a
patch introduces identical functional modifications in subsequent
revisions of the patch), their textual content can change considerably
over time.
To track patches with unchanged semantics over time, we introduce the classifier
function $\text{comp}$ that places similar patches into equivalence classes
$R_j$, so that $P=\bigcup_j R_j$. If $\text{comp}$ were able to track the exact
semantics of patches, it would hold that
$\text{comp}(a,b) = \text{yes} \Leftrightarrow a \sim b$.
But as $\text{comp}$ can only compare textual changes, it follows that
$\text{comp}(a,b) = \text{yes} \Rightarrow a \sim b$.
This results from the fact that two similar patches between two successive
versions usually have less textual changes than the first and last occurrence of
the same patch. We approximate $P \approx \bigcup_j \hat{R_j}$.
\subsection{Comparing Groups Against Mainline}
After grouping all patches on the stacks in equivalence classes
$\hat{R_j}$, a complete representative system $\mathcal{R} \subseteq
P$ is chosen and compared against the commits in the base project.
As representative of an equivalence class, we choose the patch with the latest
version.
$Q = \{(r, u) | r\in \mathcal{R}, u \in U, \text{comp}(r,u)=1\}$ denotes the set
of all patches that are found in the base project.
\newpage
\subsection{Detecting Similar Patches}
\label{sec:detect}
To group patches into equivalence classes and find them in the base project, it
is necessary to detect similar commits.
Generally, a commit consists of a unique hash, a descriptive message that
informally summarises the modifications, and so called \emph{diffs}~\cite{diff}
that describe the actual changes of the code.
Existing work on detecting similar code fragments primarily targets on detecting
code duplicates~\cite{codedup} or on revealing code plagiarism.
Possible approaches include language-dependent lexical analysis, code
fingerprinting~\cite{smith-horwitz}, or the comparison of abstract syntax
trees~\cite{deckard}.
However, all these approaches concentrate on the comparison of code fragments
and not on the comparison of \emph{similar diffs} or commits, as required in our
case.
A diff of a file consists of a sequence of \emph{hunks} that describe the
changes at a textual level.
Every hunk $h$ is introduced by a range information that determines the location
of the changes within a file and contains a section heading $h_\text{head}$.
Section headings display ``the nearest unchanged line that precedes each
hunk''~\cite{diff} and are determined by a regular expression.
Range information is followed by the actual changes: lines $h^+$ that are added
to the new resulting file are preceded by '$+$', lines $h^-$ that are removed
from the original file are preceded by '$-$' and lines $h^\circ$ that did not
change are preceded by a whitespace '\textvisiblespace'.
For the projects considered in the case study, we observed the following
properties:
\begin{itemize}
\item Commit messages of upstream patches tend to be more verbose, but still
are similar to those on patch stacks.
\item Variable and identifier names do not significantly change between
different versions.
\item Range information of similar hunks changes between different releases.
\item Section headings tend to stay similar between different releases.
\end{itemize}
In contrast to the detection of code plagiarism or the detection of code
duplicates, in our case the the textual content of diffs between successive
releases of the patch stack tends to stay very close.
For this case, string or edit distances provide an easy but powerful language
independent method for detecting similar code fragments.
Comparing $n$ diffs against each other requires $\mathcal{O}(n^2)$ comparison
operations.
As the necessary string operations are computationally intensive, we employ a
coarse-grained pre-evaluation that serves as a filter:
Two commits can only be similar if both touch at least one common file.
If the intersection of touched files is disjoint the two commits are
automatically considered to be not similar.
Our algorithm calculates a rating for the similarity of the commit message and a
rating for the similarity of the diff.
When comparing diffs, only similar hunks of commonly changes files are compared.
Insertions and deletions are compared independently.
Algorithm~\ref{alg:comp} describes the evaluation of two patches.
The algorithm calculates two ratings, a message rating $r_m\in [0,1]$ and a diff
rating $r_d\in [0,1]$.
$r$ is the weighted arithmetic mean of $r_m$ and $r_d$, weighted by a heuristic
factor $w\in [0,1]$.
If the resulting rating $r < t_i$, the two commit hashes are classified as
dissimilar, if $t_i \leq r < t_a$, then manual evaluation is required, and if $r
\geq t_a$, the commits are classified as similar.
Given a commit hash, \textsc{GetCommit} returns the corresponding message and
diff.
\textsc{StripTags} removes all \emph{tags} (CC:, Signed-off-by:, Acked-by:,
\dots) as they are not relevant for comparing the content of commit messages.
Given the diff of a commit, \textsc{ChangedFiles} returns all touched files of the
diff.
\textsc{GetHunks} returns all hunks of the diff of a file while
\textsc{HunkByHeading} searches for the closest hunk which heading matches $x$
with a rating of at least $t_h$ given a section heading $x$ and the diff of a
file.
\textsc{Dist} takes either two strings or two lists of strings and returns a
rating between 0 and 1, where 0 denotes no commonalities and 1 denotes absolute
similarity.
Our implementation uses the Levenshtein distance, which is a well-known metric
of measuring the similarity of strings.
\begin{algorithm}[h]
\caption{Detection of similar patches}
\label{alg:comp}
\begin{algorithmic}[1]
\Function{comp}{$a, b, t_a, t_i, t_h, w$}
\If {not \Call{PreEval}{$a, b$}}
\State \Return False
\EndIf
\State $(\text{msg}_a, \text{diff}_a)\leftarrow$\Call{GetCommit}{$a$}
\State $(\text{msg}_b, \text{diff}_b)\leftarrow$\Call{GetCommit}{$b$}
\State $r_m \leftarrow $\Call{Dist}{\Call{StripTags}{$\text{msg}_a$}, \Call{StripTags}{$\text{msg}_b$}}
\State $r_d \leftarrow []$
\For {\textbf{each} $\text{file}\leftarrow$\Call{ChangedFiles}{$\text{diff}_a$}}
\State $\text{hunks}_a\leftarrow$\Call{GetHunks}{diff$_a$, file}
\State $\text{hunks}_b\leftarrow$\Call{GetHunks}{diff$_b$, file}
\State $r_f \leftarrow []$
\For {\textbf{each} $\text{lhunk}\leftarrow\text{hunks}_a$}
\State $\text{rhunk}\leftarrow$\Call{HunkByHeading}{$\text{hunks}_b, \text{lhunk}_{\text{head}}, t_h$}
\If {$\text{rhunk}$ is None}
\State \textbf{continue}
\EndIf
\State $r_f$.append(\Call{Dist}{$\text{lhunk}^+, \text{rhunk}^+$})
\State $r_f$.append(\Call{Dist}{$\text{lhunk}^-, \text{rhunk}^-$})
\EndFor
\State $r_d$.append(\Call{Mean}{$r_f$})
\EndFor
\State $r_d \leftarrow$\Call{Mean}{$r_d$}
\State $r\leftarrow w \cdot r_m + (1-w) \cdot r_d$
\If {$r\geq t_a$}
\State \Return True
\ElsIf {$r\geq t_i$}
\State \Return \Call{InteractiveReview}{$a, b$}
\EndIf
\State \Return False
\EndFunction
\end{algorithmic}
\end{algorithm}
\section{Conclusions}
We presented an approach and implementation for the quantitative analysis of
patch stacks and a semi-automatic method for identifying similar commits.
An evaluation and visualisation of the Preempt-RT{} patch stack was presented as case
study.
In future work, we will concentrate on deeper statistical analysis and
comparing the properties and soft\-ware-en\-gi\-nee\-ring implications of
patch stacks for a various projects.
We are also working on a measure to quantify the invasiveness of patches and
patch stacks, which will allow us to draw conclusions on the eventual
maintenance cost of such stacks.
\section{Discussion}
\begin{figure}
\centering
\resizebox{1.0\linewidth}{!}{
\input{img/PreemptRT-commitcount}
}
\caption{Preempt-RT{} patch stack: Evolution of the stack size since Linux kernel
version 3.0}
\label{fig:prtevo}
\end{figure}
After grouping all patches into equivalence classes and linking them to optional
commits of the base project, we can distinguish between two temporal conditions:
(1) Patches that first appeared on the patch stack and later appeared in the base
project (ports or \emph{forwardports}) and (2) patches that first appeared in the
base project and were ported back to older versions of the stack (backports).
Patches that are not linked to a commit of the base project are called
\emph{invariant}, as they only appear on the stack.
Across two releases of the patch stack, we observe a flow of patches:
(1) inflow -- new patches on the patch stack and backports.
(2) outflow -- patches that went upstream or patches that were dropped.
(3) invariant -- patches that remain on the stack.
In the follwing, we consider the evolution of the Preempt-RT{} patch stack as a case
study: First, we inspect the temporal evolution of patch stack
size, which is visualised in Figure~\ref{fig:prtevo}. Among all $554$
releases of the patch stack published since Jule 2011 (that in total consist of
almost $173\,000$ patches), we detected $1042$ different groups of patches.
$195$ of those groups were classified as backports, $153$ groups were classified
as forwardports.
Knowledge of the stack history allows us to determine the composition
of older patch stacks in terms of the direction of flow of constituents.
Retroactively, we can determine which patches of the stack went upstream at a
later point in time, and compute the amount of backported patches and invariant
patches.
Figure~\ref{fig:prtfuture} shows the composition of the latest releases of major
versions of the Preempt-RT{}\cite{prt} patch stack.
Green bars describe the amount of patches on the stack that eventually are
integrated into the upstream code base, red bars describe the amount of
backports, and the blue bars give the number of invariant patches.
Another covariate of interest is the duration a patch needs to go upstream
(i.e., the time between the first appearance on the patch stack and the
integration with the base project).
Figure~\ref{fig:prtups} shows the result of this analysis for the Preempt-RT{}
project.
Positive values on the $x$-axis describe forwardports, negative values describe
backports.
There is a prominent hot spot around zero days.
We interpret this spot to indicate close cooperation with the base
project: backporting of many patches only takes few days while the
author list of forward and backport patches overlaps.
\begin{figure}
\centering
\resizebox{1.0\linewidth}{!}{
\input{img/PreemptRT-upstream-analysis}
}
\caption{Preempt-RT{} patch stack: Distribution of integration times (in days)
for patches that are eventually integrated in mainline. Positive values
indicate forwardports, negative values indicate backports.}
\label{fig:prtups}
\end{figure}
\begin{figure}
\centering
\resizebox{1.0\linewidth}{!}{
\input{img/PreemptRT-l_sf_plot}
}
\caption{Preempt-RT{} patch stack: Comparing the composition of the last major
releases of the patch stacks}
\label{fig:prtfuture}
\end{figure}
\section{Introduction}
Special-purpose software, like industrial control, medical analysis, or other
domain-specific applications, is often composed of contributions from
general-purpose projects that provide basic building blocks.
Custom modifications implemented on top of them fulfill certain additional
requirements, while the development of \emph{mainline}, the primary branch of
the base project, proceeds independently.
Especially for software with high dependability requirements, it is crucial to
keep up to date with mainline: latest fixes must be applied and new general
features have to be introduced, as diverging software branches are hard to
maintain and lead to inflexible systems~\cite{mergeconflict}.
Parallel development often evolves in the form of \emph{patch stacks}:
feature-granular modifications of mainline releases.
Because of the dynamics exhibited by modern software projects, maintaining patch
stacks can become a significant issue in terms of effort and costs.
Our toolkit \texttt{PaStA}{}\footnote{\url{https://github.com/lfd/PaStA}}
(\textbf{Pa}tch \textbf{St}ack \textbf{A}nalysis) quantitatively analyses the
evolution of patch stacks by mining git~\cite{git} repositories and produces
data that can serve as input for statistical analysis.
It compares different releases of stacks and groups similar patches (patches
that lead to similar modifications) into equivalence classes.
This allows us to compare those classes against the base project to measure
integrability and influence of the patch stack on the base project.
Patches that remain on the external stack across releases are classified as
\emph{invariant} and are hypothesised to reflect the maintenance cost of the
whole stack.
A fine grained classification of different patch types that depends on the
actual modifications could function as a measure for the \emph{invasiveness} of
the stack.
In summary, we claim the following contributions:
\begin{itemize}
\item We provide an approach and tool for observing the evolution
of patch stacks.
\item We propose a language-independent
semi-automatic algorithm based on string distances that is suitable
for detecting similar patches on patch stacks.
\item We provide a case study on Preempt-RT{}~\cite{prt}, a realtime extension of
the Linux kernel that enjoys wide\-spread use in industrial appliances
for more than a decade, yet has not been integrated into standard Linux.
We measure its influence on mainline and visualise the development
dynamics of the stack.
\end{itemize}
|
1,116,691,497,759 | arxiv | \section{\label{sec:Introduction}Introduction}
The recent paper by \citet{Lentz2018a} (LQR) uncovered the role of particle exchange and inter-particle correlation in the dynamics of condensed fluids with non-local interactions on super-de Broglie lengths. The approach covered a wide range of interactions, including those with infinite-range such as the Coulombic potential of electrostatics and Newtonian gravity. The derived model of dynamics found that such Bose and Fermi systems exhibited extra-classical behavior in the form of exchange-correlation interactions, even above the de Broglie scale. The resulting equation of motion for the condensate is concise and highly tractable, making it an attractive model. For instance, the authors of LQR have used this model to describe cosmological structure formation of an axion dark matter candidate \cite{Lentz2018b,Lentz2018c}.
A pure condensed state is often too constraining of a description for many degenerate systems, however, particularly for states out of equilibrium. The process of forming or destroying a condensate and the interaction of multiple phases of a degenerate fluid are examples of mixed systems that are beyond the scope of LQR's derivation. This paper updates the previous derivation to allow for the occupation of a density of states, greatly enlarging the domain of utility. Exchange-correlation (XC) interactions are again seen to play a central and increasingly dynamical role in the presentation of interaction forces on the fluid's state. The resulting equations of motion for the fluid are found. Relations to the condensed and diffuse limits are then discussed. Intuitive and tangible examples of this model are also presented to demonstrate features of Bose and Fermi fluids in the presence of attractive, repulsive, and mixed interactions.
\section{\label{sec:Derivation}Derivation}
The many-body quantum mechanical system presented in LQR consists of $N$ identical Fermi or Bose particles, restricted to states that are symmetric over all non-spatial degrees such as spin. Further, the solution space is to be restricted to states that can be expressed as a product of spatial and non-spatial degrees, where the non-spatial degrees of freedom are in symmetric form. The Schr\"odinger equation expressed in the spatial basis then takes the form
\begin{equation}
i \hbar \partial_t \Psi \left( \vec{x}_1,...,\vec{x}_N;t \right) = H \Psi \left( \vec{x}_1,...,\vec{x}_N;t \right),
\end{equation}
where the Hamiltonian operator is represented by
\begin{equation}
H = - \sum_i^N \frac{\hbar^2 \nabla^2_i}{2 m} + \sum_{i < j}^N \phi \left(|\vec{x}_i-\vec{x}_j| \right),
\end{equation}
where the first term is the canonical non-relativistic kinetic-energy contribution from each particle and the second term represents the non-local inter-particle interactions, parameterized by the single smooth, real potential function $\phi$. Only non-local central interactions are considered here. The system of identical particles also exhibits (anti-)symmetry under particle exchange, written as
\begin{align}
\Psi \left( \vec{x}_1,...,\vec{x}_i,...,\vec{x}_j,...,\vec{x}_N;t \right) &= \mp \Psi \left( \vec{x}_1,...,\vec{x}_j,...,\vec{x}_i,...,\vec{x}_N;t \right) \nonumber \\
& \forall i,j
\end{align}
for fermions and bosons, respectively.
The mixed state of the many-body system can be tracked using the density matrix operator
\begin{equation}
\hat{\rho} = \sum_{\alpha \in A }p_{\alpha}(t) \ket{\alpha} \bra{\alpha},
\end{equation}
where $p_{\alpha}(t)$ is the probability of occupying in the $\ket{\alpha}$ state, and $A$ is the non-degenerate set of orthonormal vectors of the N-body Hilbert space. The evolution of the density operator is found via the operator form of Liouville's theorem
\begin{equation}
\dot{\hat{\rho}} = \frac{1}{i \hbar} \left[\hat{\rho},\hat{H}\right].
\end{equation}
The expected evolution of an operator $\hat{O}$ over the ensemble is then
\begin{equation}
\partial_t \braket{\hat{O}} = \frac{1}{i \hbar} \mathrm{Tr} \left( \hat{\rho} \left[\hat{O},\hat{H}\right]\right). \label{eqn:opevol}
\end{equation}
A many-body Wigner transform operator is constructed to describe the (quasi-)distribution of the system's states over the $N$ phase spaces $w_i = (\vec{x}_i,\vec{p}_i)$
\begin{align}
&\hat{f}^{(N)}(w_1,...,w_N,t) = \int d^3x'_1 \cdot \cdot \cdot d^3x'_N e^{i \sum_j^N \vec{p}_j \cdot \vec{x}'_j/\hbar} \times \nonumber \\
& \hat{\Psi}^{\dagger}(\vec{x}_1+\vec{x}'_1/2,...,\vec{x}_N+\vec{x}'_N/2,t)\times \nonumber \\
&\hat{\Psi}(\vec{x}_1-\vec{x}'_1/2,...,\vec{x}_N-\vec{x}'_N/2,t),
\end{align}
which will be used to track the motions of the fluid. The evolution of the Wigner operator's ensemble expectation $f^{(N)} = \braket{ \hat{f}^{(N)}}$ can be calculated using Eqn.~\ref{eqn:opevol} and is found to obey the many-body Liouville equation to lowest order in $\hbar/m$,
\begin{equation}
\partial_t f^{(N)} + \sum_i^N \frac{\vec{p}_i}{m} \vec{\nabla}_i f^{(N)} - \sum_{i \ne j}^N \vec{\nabla}_i \phi_{ij} \cdot \vec{\nabla}_{p_i} f^{(N)} = \mathcal{O}(\hbar/m), \label{eqn:DFEOM}
\end{equation}
where the use of the zeroth order terms puts us firmly in the super-de Broglie limit, also consistent with the Husimi distribution with an envelope such that $\sigma_x^2 \sigma_p^2$ is larger than $\hbar^2$ \citep{Husimi1940}. An additional advantage of operating in this limit is that the expected many-body Wigner function can be interpreted as a true distribution function (DF).
The simplification of the mixed N-body function is not as clear as for the condensate presented in LQR, as the Runge-Gross theorem \cite{Runge1984} no longer applies. We may still integrate out phase spaces $2,...,N$ of $f^{(N)}$ to find the motion of the single-body density
\begin{equation}
0=\partial_t f^{(1)} + \frac{\vec{p}_1}{m} \cdot \vec{\nabla}_1 f^{(1)} - (N-1)\int d^6w_2 \vec{\nabla} \phi_{12} \cdot \vec{\nabla}_{p_1} f^{(2)},
\end{equation}
though the two-body DF is no longer guaranteed to be functionally described in terms of the single-body DF.
The presence of correlations between bodies found in the many-body wave functions presented in LQR direct us to consider non-trivial correlations in the mixed system as well, though the behavior of the correlations is less clear. Determining the two-body DF equation of motion is done here by integrating out phase-spaces $3,...,N$ from Eqn.~\ref{eqn:DFEOM}
\begin{align}
0=\partial_t f^{(2)} +& \sum_{i=1}^2\Bigg[\frac{\vec{p}_i}{m} \cdot \vec{\nabla}_i f^{(2)} - \vec{\nabla}_i \phi_{12} \cdot \vec{\nabla}_{p_i} f^{(2)} \nonumber \\
&- (N-2)\int d^6w_3 \vec{\nabla}_i \phi_{i3} \cdot \vec{\nabla}_{p_i} f^{(3)}\Bigg],
\end{align}
which depends on the expected three-body DF $f^{(3)}$.
In this system it is possible to truncate this hierarchy of dependency by noting the exchange relations and structure of the two-body pairwise interactions permit the reduction of the three-body DF via the Kirkwood expression \citep{Pearl1988}
\begin{align}
& f^{(3)}(w_1,w_2,w_3,t) = \nonumber \\
&\frac{f^{(2)}(w_1,w_2,t) f^{(2)}(w_2,w_3,t) f^{(2)}(w_3,w_1,t)}{f^{(1)}(w_1,t) f^{(1)}(w_2,t) f^{(1)}(w_3,t)}.
\end{align}
One can also write the Kirkwood expression in terms of the dimensionless two-body correlation function $\tilde{g}(w_1,w_2,t) = f^{(2)}(w_1,w_2,t)/ f^{(1)}(w_1,t) f^{(1)}(w_2,t)$. The super-de Broglie dynamics of the mixed many-body system can therefore be reduced to the solutions of the single-body DF and the two-body correlation function
\begin{align}
&0=\partial_t f^{(1)}_1 + \frac{\vec{p}_1}{m} \cdot \vec{\nabla} f^{(1)} \nonumber \\
&- \frac{N-1}{N}\int d^6w_2 \vec{\nabla}_1 \Phi_{12} \cdot \vec{\nabla}_{p_1} \left(f^{(1)}_1 \tilde{g}_{12} f^{(1)}_2 \right) , \label{DF1EOM} \\
&0=\partial_t \tilde{g}_{12} + \sum_{i=1}^2\Bigg[ \frac{\vec{p}_i}{m} \cdot \vec{\nabla}_i \tilde{g}_{12} \nonumber \\
&- \frac{N-2}{N f^{(1)}_i}\int d^6w_3 \vec{\nabla}_i \Phi_{i3} \cdot \Bigg( \vec{\nabla}_{p_i} \left( \tilde g_{12}\tilde g_{13}\tilde g_{23} f^{(1)}_i f^{(1)}_3 \right) \nonumber \\
&- \tilde g_{12} \vec{\nabla}_{p_i}\left(\tilde{g}_{i3} f^{(1)}_i f^{(1)}_3 \right) \Bigg)\Bigg], \label{CorrEOM}
\end{align}
where $\Phi_{12} = N \phi_{12}$, and subscripts on correlation functions and single-body DFs indicate the phase spaces of their arguments. The trivial correlation limit ($\tilde g=1$) is seen to be a stationary point of the solution space, in the absence of external potentials. The condensed limit shown in LQR is also seen to be a potential solution of the generalized system, though the stability of any one such solution depends on the type of interaction and the correlation of other available states, condensed and not.
\section{\label{sec:Demonstrations} Examples}
This section presents several examples of the model described in Eqns.~\ref{DF1EOM}, \ref{CorrEOM}. Systems are chosen so that they are solvable either analytically or by using standard numerical ODE packages. Examples will also operate within the limits of small correlation perturbations $|\delta g| = |1- \tilde{g}| \ll 1$, and with sufficiently many particles that $(N-1)/N \to 1$.
\subsection{\label{sec:shells} Dynamics of Spherical Shells}
The first examples of this method are extensions of the example given in LQR, covering the collapse of cold spherical shells under self-gravitation, expansion under electrostatic repulsion, and oscillation in a bound state of a macroscopic van der Waals-like force. For simplicity, we again assume no tangential fluid motion, that the radial velocity dispersion is sufficiently small so as to leave the shell width unchanged over the simulated collapse, and that the two-body correlation function can be approximated as homogeneous over the shell. The ansatz for the one-body distribution is $f^{(1)}_{shell}(r,m v_r,t) = A \times \mathrm{exp} \left[ - (r-r_c)^2/ 2 \sigma_r^2 - (v_r - \dot{r}_c)^2/ 2 \sigma_v^2 \right]$, where $r_c$ is the shell center radius, $A$ is the DF normalization, $\sigma_r$ is the thickness of the shell, and $\sigma_v$ is the velocity dispersion of the shell. These assumptions reduce the complexity of the problem to the degrees of the shell's radius and the deviation of the correlation function from unity, $r_c(t)$, $\delta g(t)$.
The governing equation of the shell under an interaction force profile $- \partial_r \Phi = F $ may then be written to leading order in dispersion and correlation perturbation as
\begin{align}
\ddot{r}_c &= F(r_{soft}) \left(1 + \delta g \right), \\
\dot{\delta g} &= - 2 F(r_{soft}) \frac{\delta g}{\sigma_v},
\end{align}
where $r_{soft}^2 = \left(r^2_c + \sigma_r^2\right)$. The factor of two in the correlation equation of motion relates to the two phase spaces by which interaction enters the correlation equation of motion.
\subsubsection{\label{sec:Demonstration1} Collapse of Self-Gravitating Shell}
The collapsing spherical shell is subject to the Newtonian gravitational force
\begin{equation}
F(r) = -\frac{GM}{r^2},
\end{equation}
where $G$ is the Newton gravitational constant, and $M$ is the self-gravitating mass of the shell. A relevant example would be a cold dark matter candidate of a sufficiently small particle mass, such as the axion, that is expected to be extraordinarily degenerate at the presumed dark matter density and temperature \cite{Marsh2016,Guth2015,Sikivie2009}. Solutions to the collapse of an initially stationary sphere for a variety of perturbed correlations show that the rate of infall is amplified by correlation and that attractive infall amplifies the degree of correlation in a cycle of positive feedback, see Fig. \ref{sph_shell_coll}. The case of zero initial correlation (classical mean field theory) remains without correlation throughout, but the mean field is an unstable stationary point among attractive interactions. One can also see a similar behavior among fermions ($\delta g \le 0$), with collapse causing growing anti-correlations that slow the rate of collapse. Parameters chosen for Fig. \ref{sph_shell_coll} are $r(t=0)=1$, $\sigma_r=10^{-3}$, and $\sigma_v=10^{-1}$ in dynamical units. The growth of correlation under self-gravitation seems to contradict to the findings of \citet{Guth2015} that long-distance correlations are quenched by collapse.
\begin{figure*}[ht!]
\begin{center}
\includegraphics[width=\textwidth]{dyn_corr_sph_coll.pdf}
\caption{Demonstration of dynamical correlation on collapse of spherical shell. (Left) Radial trajectories of collapsing self-gravitating spherical shells of Bose fluid. (Right) Correlation perturbation evolution during collapse. Shells start from rest. The color gradient is logarithmic in initial correlation perturbations $\delta g_0 \in [10^{-50}, 10^{-10}]$, which cover a large range as the expected size of cosmological correlations at the start of collapse are as of yet unknown. The XC physics alters the infall of the shell, generally increasing the attraction and making the collapse more violent. The collapse also increases the size of the correlation, creating positive feedback. Near to collapse, the correlation exits the $\delta g \ll 1$ regime, compromising the reliability of the results at collapse.}
\label{sph_shell_coll}
\end{center}
\end{figure*}
\subsubsection{\label{sec:Demonstration2} Quenching of Repulsed Shells}
If the same spherical shell is subject to a long-range repulsive interaction, such as the Coulomb electrostatic force
\begin{equation}
F(r) = \frac{k C^2}{M r^2},
\end{equation}
where $k$ is the Coulomb constant and $C$ is the effective charge, the shell will not collapse but instead expand, with solutions shown in Fig.~\ref{sph_shell_repul}. The relevant example here would be a shell of cold ionic gas released from a trap. Correlations are seen to change the time of flight of the shell towards a particular radius. The dimensionless correlation perturbations also change their behavior, decaying towards zero for both Bose and Fermi systems. The classical mean field theory limit of $\tilde{g} = 1$ is now a stable stationary point. Such an example fulfills the intuition that (repulsive) interactions quench correlations.
\begin{figure*}[ht!]
\begin{center}
\includegraphics[width=\textwidth]{dyn_corr_sph_repul.pdf}
\caption{Demonstration of dynamical correlation on expansion of spherical shell under Coulombic repulsion. (Left) Radial trajectories of spherical shells of Bose and Fermi fluids. (Right) Correlation perturbation evolution during expansion. Shells start from rest. The color gradient is linear in initial correlation perturbations $\delta g_0 \in [-0.5,0.5]$, with Bose fluids having non-negative correlation perturbations and Fermi fluids having non-positive correlation perturbations. The XC physics alters the magnitude of the Coulombic force, altering the exit velocity of the shell. The repulsive force quenches the correlation perturbation for both fluid types.}
\label{sph_shell_repul}
\end{center}
\end{figure*}
\subsubsection{\label{sec:Demonstration3} Stable Bound States}
The final shell example is for a van der Waals-like force
\begin{equation}
F(r) = \frac{G M}{R} \left( \frac{-1}{r^3} + \frac{1.3 R}{r^4} \right),
\end{equation}
where $r=R$ is the initial radius of the shell. The initial condition of $r_c(t=0)=1$ places the shell in a classical bound state sufficiently far from the ground state at $r=1.3 R$. A relevant example may be a shell of neutral cold gas. The resultant oscillatory bound state is stable for small correlations, with correlations taking their extremal values as the shell passes through the interaction potential minimum, while at other times growing or shrinking in amplitude as the shell experiences an attractive or repulsive force respectively, as shown in Fig.~\ref{sph_shell_VdW}. Interestingly, the amplitude of the shell oscillation appears unchanged by the size of the seed correlation. Seed correlation appears to only change the period of the shell's oscillation.
\begin{figure*}[ht!]
\begin{center}
\includegraphics[width=\textwidth]{dyn_corr_sph_VdW.pdf}
\caption{Demonstration of dynamical correlation on the motion of a bound spherical shell subject to a Van der Waals force. (Left) Radial trajectories of spherical shells of Bose and Fermi fluids. (Right) Correlation perturbation evolution during oscillation. Shells start from rest. The color gradient is linear in initial correlation perturbations $\delta g_0 \in [-0.1,0.1]$, with Bose fluids having non-negative correlation perturbations and Fermi fluids having non-positive correlation perturbations. }
\label{sph_shell_VdW}
\end{center}
\end{figure*}
\subsection{\label{sec:bulk} Bulk Speed of Sound}
The impact of correlations on the bulk properties of Bose and Fermi fluids is also interesting. Consider perturbative sound waves propagating through a large bulk system subject to a finite-range repulsive force. For wavelengths much longer than the inter-particle spacing and the scattering length of the interaction, the effective potential becomes point-like, behaving in the kinetic regime as $\sim a \rho$, where $a$ is the interaction scale. The equations of motion for the density perturbations and correlation perturbations are then found to be
\begin{align}
\ddot{\delta}(\vec{x}_1,t) &= c_s^2 \nabla_1^2 \delta(\vec{x}_1,t) + \frac{c_s^2}{\rho} \nabla_1^2 \mu (\vec{x}_1,\vec{x}_1,t), \\
\ddot{\mu} (\vec{x}_1,\vec{x}_2,t) &= c_s^2 \nabla_1^2 \mu (\vec{x}_1,\vec{x}_2,t) + c_s^2 \nabla^2_2 \mu (\vec{x}_1,\vec{x}_2,t),
\end{align}
where $\delta$ is the spatial density perturbation, $\mu$ is a form of the spatial two-body density perturbation with units of density squared, $\rho$ is the mean bulk density, and $c_s$ is the base sound speed of the correlation-less fluid. Note that perturbations in the correlation function can propagate in two directions. The density perturbations can be considered as a propagating mode in the limit of small $\mu/\rho \delta$. The speed of sound of density perturbations is found to be
\begin{equation}
c'_s = c_s \sqrt{1+ \frac{\mu_0}{\delta_0 \rho}},
\end{equation}
where $\delta_0$ is initial amplitude of the density perturbations and $\mu_0$ gives the homogeneous correlation displacement and the wave's initial intitial correlation. The presence of correlation is seen to increase the speed of sound in Bose fluids relative to the mean field and decrease the speed of sound in Fermi fluids.
\section{\label{sec:Conclusions}Conclusions}
This paper presents a concise model of macroscopic dynamics for mixed systems of identical particles with non-local two-body interactions, expanding on the condensate dynamics presented in LQR. A density matrix approach was used to find the evolution of a mixed system of $N$ non-relativistic bodies. Those dynamics were transformed into a Wigner function, and its equation of motion was derived in the super-de Broglie limit. A closed system of Boltzmann-like field equations involving the single-body distribution function and the two-body correlation function was found to govern the many-body system. Deviations from the classical and standard mean-field descriptions were again found to be sourced by inter-body correlations, which now have their own dynamics. Correlations between particles are consistent with expected behavior in the condensed and diffuse limits.
Examples of the model were made using a thin cold spherical shell and sound waves on a homogeneous bulk, both operating with small correlations. Attractive forces such as Newtonian self-gravity were seen to grow correlations over the course of collapse, while repulsive forces such as the electrostatic Coulombic force quenched correlations as the shell expanded. A shell caught in the well of a macroscopic van der Waals-like interaction observed both of these behaviors as the shell oscillated about the potential minimum, growing and shrinking its correlations as it is subjected to attractive and repulsive forces, respectively; the net effect being an alteration in the period of oscillation. Growth or decay of correlations are seen to be quite rapid, owing to the multiple channels available to the two-body interactions in the correlation's governing equation. The speed of sound in bulk was seen to be impacted by the presence of correlations. Sound speed was seen to be decreased by XC for Fermi fluids and increased for Bose fluids. All four examples contained a stationary point at the mean field value of correlation $\tilde{g} = 1$, though the repulsive/attractive nature of the interaction force is capable of changing the stationary point from stable to unstable. The seeding of correlations does not appear possible with the stated model, though external interactions are expected to be the remedy to this. The authors' current application of the mixed degenerate system to Bose (axion-like) dark matter shows great promise for the genesis and growth of correlations that are first induced by gravitational interactions with the photon fluid and then build by self-gravitation during the radiation era of cosmology. There are many other potential applications of this technique. For example, researchers in condensed matter, nuclear astrophysics, cold atomic physics, and others may find the above description useful.
\section{\label{sec:Acknowledgements}Acknowledgements}
We would like to thank Bodo Schwabe, Sebastian Hoof, Mona Dentler, Viraf Mehta, and Fabian Heidrich-Meisner for their productive discussions in the refinement of this paper. We also gratefully acknowledge the support of the U.S. Department of Energy Office of High Energy Physics and the National Science Foundation. TQ was supported in part by the NSF grant AST-1514868. LR was supported in part by the DOE grant DE-SC0011665.
\bibliographystyle{apsrev4-1}
|
1,116,691,497,760 | arxiv | \section{Introduction}
\setcounter{equation}{0} Let $f$ be a transcendental entire
function and denote by $f^{n}, n \in \mathbb N$, the $n$th iterate of
$f$. The {\it Fatou set}, $F(f)$, is defined to be the set of
points, $z \in \mathbb C$, such that $(f^{n})_{n \in \mathbb N}$ forms a normal
family in some neighbourhood of $z$. The complement, $J(f)$, of
$F(f)$ is called the {\it Julia set} of $f$. An introduction to
the basic properties of these sets can be found in, for example,
[\ref{Berg}].
This paper concerns the {\it escaping set}
\[
I(f) = \{z: f^n(z) \to \infty \mbox{ as } n \to \infty \},
\]
which was first studied for a general transcendental entire
function $f$ by Eremenko [\ref{E}]. He proved that
\begin{equation}
I(f) \neq \emptyset,
\end{equation}
\begin{equation}
J(f) = \partial I(f),
\end{equation}
\begin{equation}
I(f) \cap J(f) \neq \emptyset,
\end{equation}
\begin{equation}
\overline{I(f)} \mbox{ has no bounded components.}
\end{equation}
In particular, note that $I(f)$ is unbounded.
Eremenko conjectured that it may be possible to replace
$\overline{I(f)}$ with $I(f)$ in (1.4). This problem still
remains open although it has been shown to be true for certain
classes of functions -- see, for example, [\ref{Bar}] and
[\ref{RRRS}]. However, it was shown in~[\ref{RRRS}] that
points in $I(f)$ cannot necessarily be connected to infinity by a
curve in $I(f)$, answering another question of Eremenko.
For many of the functions considered in [\ref{Bar}] and [\ref{RRRS}], $I(f)$ consists of an infinite family of unbounded curves. On the other hand, Eremenko's conjecture is also true when $I(f)$ is connected, since $I(f)$ is always unbounded. In fact, $I(f)$ can be connected in surprisingly simple situations. For example, consider the function defined by $f(z)=z+1+e^{-z}$. This function was studied by Fatou [\ref{F}, Example 1] who showed that $F(f)$ consists of a single completely invariant domain, $U$ say, which is in $I(f)$. (Such a set $U$ is now known as a {\em Baker domain}.) For this function, $I(f)$ is connected because $U \subset I(f) \subset \overline{U}=\mathbb C$.
In [\ref{RS}, Theorem 2] we proved the following result which shows that $I(f)$ is connected, and hence Eremenko's conjecture is true, in the more complicated situation where $F(f)$ has a multiply connected Fatou component. In the same paper, [\ref{RS}, Theorem~1], we showed that $I(f)$ always has at
least one unbounded component.
\newtheorem{1}{Theorem}
\begin{1}
Let $f$ be a transcendental entire function and suppose that
$F(f)$ has a multiply connected component $U$. Then $\overline{U}
\subset I(f)$ and $I(f)$ is connected.
\end{1}
{\it Remark.}
It is known [\ref{B84}, Theorem~3.1] that a multiply
connected component $U$ of $F(f)$ is a bounded wandering domain
and contains a Jordan curve $\gamma$ such that $f^n(\gamma) \to \infty$
and $f^n(\gamma)$ surrounds $0$ for sufficiently large $n \in
\mathbb N$.
In this paper we show that the escaping set is connected, and
hence Eremenko's conjecture is true, for other classes of
transcendental entire functions. In Section~2 we prove the
following generalisation of Theorem~1. This is the main result of
the paper. Note that $\widetilde{U}$ denotes the union of $U$ and
its bounded complementary components.
\newtheorem{2}[1]{Theorem}
\begin{2}
Let $f$ be a transcendental entire function and suppose that
there exists a bounded domain $U$ with $\partial U \subset I(f)$
and $U \cap J(f) \neq \emptyset$. Then
{\emph (a)} $\alpha_n = \partial \widetilde{f^n(U)}$ is in $I(f)$,
$\alpha_n \to \infty$ and $\alpha_n$ surrounds $0$ for
sufficiently large $n$;
{\emph (b)} $I(f)$ is connected.
\end{2}
Since $J(f) = \partial I(f)$, the hypothesis of Theorem~2 is
equivalent to the hypothesis that $\partial U \subset I(f)$ and
$U \cap I(f)^c \neq \emptyset$. We can thus describe this
hypothesis informally as there exists a `hole in $I(f)$'. It
follows from Theorem~2 that Eremenko's
conjecture is true whenever there is a hole in $I(f)$.
We will show in the final section of the paper that there are many functions which
satisfy the hypotheses of Theorem~2 but do not have any multiply
connected Fatou components (so Theorem~1 does not apply to them).
We proved Theorem~1 in [\ref{RS}] by considering the following subset of
$I(f)$, which was introduced by Bergweiler and Hinkkanen in
[\ref{BH}]:
\[
A(f) = \{z: \mbox{there exists } L \in \mathbb N \mbox{ such that }
|f^{n+L}(z)| > M(R,f^{n}), \mbox{ for } n \in \mathbb N\}.
\]
Here, for $r>0$,
\[
M(r,f) = \max_{|z|=r} |f(z)|
\]
and $R$ can be taken to be any value such that $R > \min_{z \in
J(f)}|z|$. We showed in~[\ref{RS}] that for any transcendental entire function $f$ the set
$A(f)$ is equal to the set
\[
B(f) = \{z: \mbox{ there exists } L \in \mathbb N \mbox{ such that
} f^{n+L}(z) \notin \widetilde{f^n(D)}, \mbox{ for } n \in \mathbb N \},
\]
where $D$ can be taken to be any open disc meeting $J(f)$, and also that $B(f)$ is connected under the hypotheses of Theorem~1; see [\ref{RS}, Theorem~2]. It follows from the `blowing up property' of the Julia set (see Lemma~2.1) that $B(f)$ is completely invariant.
In this paper, we introduce subsets of $B(f)$ of the form
\[
B_D(f) = \{ z: f^n(z)\notin \widetilde{f^n(D)}, \mbox{ for } n \in \mathbb N \},
\]
where $D$ is any open disc meeting $J(f)$. Note that for any transcendental entire function $f$ all the components of $B(f)$ and $B_D(f)$ are unbounded; see [\ref{RS}, Theorem~1 and its proof]. In Section~3 we prove results concerning the structure of such sets which lead to various sufficient conditions for $B_D(f)$, $B(f)$ and $I(f)$ to be connected. Our first theorem in that section is the following.
\newtheorem{3}[1]{Theorem}
\begin{3}
Let $f$ be a transcendental entire function, let $D$ be an open
disc meeting $J(f)$, and suppose that there exists a bounded domain
$U$ with $\partial U \subset B_D(f)$
and $U \cap J(f) \neq \emptyset$. Then
{\emph (a)} $\beta_n= f^n(\partial U) \to \infty$ and, for
sufficiently large $n$, $\beta_n$ is in $B_D(f)$ and $\beta_n$
surrounds $0$;
{\emph (b)} $B_D(f)$, $B(f)$ and $I(f)$ are all connected.
\end{3}
We then show that there are several conditions that are
equivalent to the hypotheses of Theorem~3. This enables us to
prove the following result.
\newtheorem{4}[1]{Theorem}
\begin{4}
Let $f$ be a transcendental entire function, let $D$ be an open
disc meeting $J(f)$, and suppose that there exist Jordan curves
$\gamma_n$ such that, for all $n \in \mathbb N$,
\[
\gamma_n \mbox{ surrounds } f^n(D)
\]
and
\[
f(\gamma_n) \mbox{ surrounds the bounded component of } \gamma_{n+1}^c.
\]
Then there exists a bounded domain $U$ with $\partial U \subset
B_D(f)$ and $U \cap J(f) \neq \emptyset$. Thus $B_D(f)$, $B(f)$
and $I(f)$ are all connected.
\end{4}
Note that the hypotheses of Theorem~3 (and hence Theorem~4)
imply that the hypotheses of Theorem~2 are satisfied; that is, there exists a hole in $I(f)$.
We are aware of two different classes of functions
that satisfy the hypotheses of Theorem~4. Firstly, by the remark
following Theorem~1, any entire function with a multiply connected
component of the Fatou set will satisfy these hypotheses.
Secondly, many functions of order less than 1/2 have been shown
to satisfy conditions which, as we explain in Section~4, are
stronger than the hypotheses of Theorem~4. This gives various
results, including the following.
\newtheorem{5}[1]{Corollary}
\begin{5}
Let $f$ be a transcendental entire function and suppose that
either
{\it (i)} there exist $\epsilon \in (0,1)$ and $R>0$ such that
\begin{equation}
\log \log M(r,f) < \frac{(\log r)^{1/2}}{(\log \log
r)^{\epsilon}}, \; \mbox{ for } r>R,
\end{equation}
or
{\it (ii)} the order of $f$ is $\rho < 1/2$ and
\begin{equation}
\frac{\log M(2r,f)}{\log M(r,f)} \to c\; \mbox{ as } r \to \infty,
\end{equation}
where $c$ is a finite constant that depends on $f$.
Then, for any open disc $D$ meeting $J(f)$, the hypotheses of Theorem 4 hold, so $B_D(f)$, $B(f)$ and $I(f)$ are all connected.
\end{5}
Recall that the order of a function $f$ is defined to be
\[
\rho = \overline{\lim_{r \to \infty}}\, \frac{\log \log M(r,f)}{\log
r}\,.
\]
The functions of order less than 1/2 in Corollary 5 were
originally studied in connection with a different question,
associated with Baker [\ref{B81}], namely, whether a function of
order at most 1/2, minimal type, can have any unbounded Fatou
components. It was shown in [\ref{S}, Theorem B and Theorem~C]
that the functions in Corollary 5 have no unbounded Fatou
components. Note that (1.5), which is not even satisfied by all
functions of order 0, is the best published growth condition on
$M(r,f)$ guaranteeing no unbounded Fatou components which does
not require some additional regularity condition such as (1.6).
See~[\ref{H}] for a survey article describing many other results
on this problem. Recently, we have shown that weaker conditions
than (1.5) and (1.6) are sufficient to ensure that a function of
order less than 1/2 has no unbounded Fatou components
(see~[\ref{SG}]), and Corollary~5 also holds under these weaker
conditions.
Finally, we give a criterion related to the escaping set which is
sufficient to ensure that a transcendental entire function has no unbounded Fatou
components.
\newtheorem{7}[1]{Theorem}
\begin{7}
Let $f$ be a transcendental entire function, let $D$ be an open
disc meeting $J(f)$, and suppose that there exists a bounded domain
$U$ with $\partial U \subset B_D(f)$ and $U \cap J(f) \neq
\emptyset$. Then $F(f)$ has no unbounded components.
\end{7}
Together with Theorem~4, this gives the following result.
\newtheorem{8}[1]{Theorem}
\begin{8}
Let $f$ be a transcendental entire function, let $D$ be an open
disc meeting $J(f)$, and suppose that there exist Jordan curves
$\gamma_n$ such that, for all $n \in \mathbb N$,
\[
\gamma_n \mbox{ surrounds } f^n(D)
\]
and
\[
f(\gamma_n) \mbox{ surrounds the bounded component of } \gamma_{n+1}^c.
\]
Then $F(f)$ has no unbounded components.
\end{8}
{\it Remark.} The similarities between Theorems~3 and~6, and between Theorems~4 and~7, arise indirectly from their common methods of proof. It is natural to ask whether there is any direct relationship between the condition that $I(f)$ (or $B(f)$) is connected and the absence of unbounded
components of $F(f)$. Note, however, that the Fatou example $f(z)=z+1+e^{-z}$ mentioned earlier has an unbounded Fatou component and $I(f)$ is connected.
\section{Proof of Theorem~2}
\setcounter{equation}{0}
We begin this section by stating some existing results that will
be used in the proof of Theorem~2. Firstly, we will use the
following well-known property of the Julia set; see, for example,
[\ref{Berg}, Section~2] for a proof. This is often called the
`blowing up property' of the Julia set.
\newtheorem{2.1}{Lemma}[section]
\begin{2.1}
Let $f$ be a transcendental entire function, let $K$ be a compact set with $K \cap E(f) = \emptyset$ and let
$U$ be an open neighbourhood of $z \in J(f)$. Then there exists
$N \in \mathbb N$ such that
\[
f^n(U) \supset K,\quad \mbox{for all } n \geq N.
\]
\end{2.1}
Here
\[
E(f) = \{z: O^-(z) \mbox{ is finite} \},
\]
where
\[
O^-(z) = \{w:f^n(w) = z, \mbox{ for some } n \in \mathbb N \}.
\]
The {\it exceptional set}, $E(f)$, contains at most one point.
We also use the following result; see [\ref{BH}] and [\ref{RS}].
\newtheorem{2.2}[2.1]{Lemma}
\begin{2.2}
Let $f$ be a transcendental entire function. The set $B(f)$ has the following properties:
{\it (a)} $J(f) = \partial B(f)$;
{\it (b)} $B(f)$ has no bounded components.
\end{2.2}
\begin{proof}[Proof of Theorem~2]
Let $f$ be a transcendental entire
function and suppose that there exists a bounded domain $U$ with
$\partial U \subset I(f)$ and $U \cap J(f) \neq \emptyset$. Then
\[
\alpha_n = \partial \widetilde{f^n(U)} \subset \partial f^n(U) \subset f^n(\partial U).
\]
So, since $I(f)$ is completely invariant, it follows from Lemma
2.1 that
\begin{equation}
\alpha_n \subset I(f), \; \alpha_n \to \infty \mbox{ and }
\alpha_n \mbox{ surrounds 0 for large enough } n.
\end{equation}
This proves part (a). We now show that $I(f)$ is connected.
We begin by showing that $B(f)$ belongs to one component of
$I(f)$. Suppose that $B_1$ and $B_2$ are two components of
$B(f)$. These are unbounded by Lemma~2.2(b) and so, by (2.1),
there exists $N \in \mathbb N$ such that
\[
B_i \cap \alpha_n \neq \emptyset, \; \mbox{ for } i = 1,2 \mbox{ and } n \geq N.
\]
Since $\alpha_n \subset I(f)$, for each $n \in \mathbb N$, it follows
that $B_1$ and $B_2$ belong to the same component of $I(f)$. Therefore, there exists a component $I_0$ of $I(f)$ such that
\begin{equation}
B(f) \subset I_0 \; \mbox{ and } \; \alpha_n \subset I_0, \;
\mbox{ for all } n \geq N.
\end{equation}
Now suppose that $z_0 \in J(f) \cap I(f)$. It follows from Lemma
2.2(a) and (2.2) that $z_0 \in \partial B(f) \subset
\overline{I}_0$. Thus $I_0 \cup \{z_0\}$ is connected. Since $z_0
\in I(f)$ and $I_0$ is a component of $I(f)$, it follows that $z_0
\in I_0$. Thus
\begin{equation}
J(f) \cap I(f) \subset I_0.
\end{equation}
To complete the proof we show that $F(f) \cap I(f) \subset I_0$.
Suppose that $V$ is a Fatou component in $I(f)$. (Note that a
Fatou component is either a subset of $I(f)$ or does not meet $I(f)$, by [\ref{Zip}, Theorem~3], for example.) We claim that there
exists a point in $\partial V \cap I(f)$. If $f^m(V)$ lies in a
multiply connected component $U$ of $F(f)$ for some $m \in \mathbb N$, then
it follows from Theorem~1 that $ f^m(\partial V) \subset \overline{U}\subset I(f)$. Thus $\partial V\subset I(f)$, by the complete invariance of $I(f)$.
Suppose on the other hand that $f^m(V)$ lies in a simply connected
component of $F(f)$ for all $m \in \mathbb N$. If $\partial V \subset
I(f)$, then the claim is proved. Otherwise, there exist $z_1 \in
\partial V$, $R>0$ and a sequence $(n_k)$ such that $|f^{n_k}(z_1)| < R$.
By (2.1) there exists $N \in \mathbb N$ such that
\begin{equation}
\alpha_N \mbox{ surrounds } \{z:|z| = R\} \; \mbox{ and } \;
B(0,R) \cap J(f) \neq \emptyset.
\end{equation}
Here we use the notation $B(w,r)=\{z:|z-w|<r\}$, where $w\in\mathbb C$ and $r>0$.
Since $V \subset I(f)$, there exists $k\in \mathbb N$ such that $f^{n_k}(V) \cap \alpha_N \neq \emptyset$. Since
$f^{n_k}(V)$ lies in a simply connected component of $F(f)$, there
are no curves in $f^{n_k}(V)$ surrounding points in $J(f)$. So, by
(2.4),
\[
\partial f^{n_k}(V) \cap \alpha_N \neq \emptyset.
\]
Thus $\partial f^{n_k}(V) \cap I(f) \neq \emptyset$. Since
$\partial f^{n_k}(V) \subset f^{n_k}(\partial V)$, we have
$\partial V \cap I(f) \neq \emptyset$ by the complete invariance
of $I(f)$.
So, whenever $V$ is a Fatou component in $I(f)$, we have
$\partial V \cap I(f)~\neq~\emptyset$. Since $\partial V \subset
J(f)$, it follows from (2.3) that $\partial V \cap I_0 \neq
\emptyset$. Now $V \cup (\partial V \cap I_0)$ is a connected
subset of $I(f)$ and so $V \subset I_0$. Hence $F(f) \cap I(f)
\subset I_0$. Together with (2.3), this shows that $I(f)$ is
connected. This completes the proof of Theorem~2.
\end{proof}
\section{Properties of the set $B_D(f)$}
\setcounter{equation}{0}
Let $f$ be a transcendental entire function and let $D$ be an
open disc meeting $J(f)$. Recall that
\[
B_D(f) = \{ z: f^n(z)\notin \widetilde{f^n(D)}, \mbox{ for } n \in \mathbb N \}.
\]
In what follows we often use the property that, if $G$ is a
bounded domain, then
\[
f(\widetilde{G}) \subset \widetilde{f(G)},
\]
which holds because if $\gamma$ is any simple closed curve in
$G$, then the image under $f$ of the inside of $\gamma$ lies
inside $f(\gamma)$ and hence in $\widetilde{f(G)}$.
\newtheorem{3.1}{Lemma}[section]
\begin{3.1}
Let $f$ be a transcendental entire function and let $D$ be an
open disc meeting $J(f)$. Then there exists $M \in \mathbb N$ such that
$f^m(B_D(f)) \subset B_D(f)$ for all $m \geq M$.
\end{3.1}
\begin{proof} Since $D \cap J(f) \neq \emptyset$, it follows from
the blowing up property, Lemma~2.1, that there exists $M \in {\bf
N}$ such that $\widetilde{f^m(D)} \supset D$ if $m \geq M$. Thus
\begin{equation}
\widetilde{f^{n+m}(D)} \supset \widetilde{f^n(\widetilde{f^m(D)})}
\supset \widetilde{f^n(D)},
\end{equation}
for each $n \in \mathbb N$, $m \geq M$. If $z \in B_D(f)$ then it
follows from (3.1) that, for each $n \in \mathbb N$,
\[
f^n(f^m(z)) = f^{n+m}(z) \in \mathbb C \setminus
\widetilde{f^{n+m}(D)} \subset \mathbb C \setminus \widetilde{f^n(D)},
\]
and hence $f^m(z)
\in B_D(f)$, for each $m \geq
M$. This completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem~3]
Suppose that there exists a bounded domain $U$ with $\partial U
\subset B_D(f)$ and $U \cap J(f) \neq \emptyset$, and put
\[
\beta_n = f^n(\partial U),\quad\text{for }n\ge 0.
\]
By Lemma~3.1, there exists $M\in\mathbb N$ such that $\beta_m\subset B_D(f)$ for $m\ge M$. In particular,
\[
\beta_{n+M}=f^{n+M}(\partial U)\subset\mathbb C \setminus \widetilde{f^n(D)},
\]
for each $n\in \mathbb N$. Since $\partial f^{n+M}(U) \subset \beta_{n+M}$, part~(a) of Theorem~3 now follows from the blowing up property of the Julia set, Lemma~2.1.
We now recall that the proof of Lemma~2.2(a) in [\ref{RS}, proof
of Theorem~1] shows that there are no bounded components of
$B_D(f)$. Together with part~(a), this is sufficient to show that
$B_D(f)$ is connected. Similarly, since $B_D(f) \subset B(f)$ and
$B(f)$ has no bounded components (see Lemma~2.2(b)), it follows
from part~(a) that $B(f)$ is connected. Finally, since $\partial U\subset B_D(f)
\subset I(f)$, it follows from Theorem~2 that $I(f)$ is connected.
\end{proof}
The next result states that, unlike the sets $B(f)$ and $I(f)$,
the set $B_D(f)$ is always closed.
\newtheorem{3.2}[3.1]{Lemma}
\begin{3.2}
Let $f$ be a transcendental entire function and let $D$ be an
open disc meeting $J(f)$. Then $B_D(f)$ is closed.
\end{3.2}
\begin{proof} Since $D$ is open, we have $\widetilde{f^n(D)}$ is
open and hence $f^{-n}(\widetilde{f^n(D)})$ is open, for each $n
\in \mathbb N$. Thus
\[
B_D(f)^c = \bigcup_{n \in \mathbb N}f^{-n}(\widetilde{f^n(D)})
\]
is open and hence $B_D(f)$ is closed.
\end{proof}
We now show that there are several properties of $B_D(f)$ that
are equivalent to the hypothesis on $B_D(f)$ given in Theorem~3.
These properties will be useful when proving that a given function satisfies
this hypothesis.
\newtheorem{3.3}[3.1]{Lemma}
\begin{3.3}
Let $f$ be a transcendental entire function and let $D$ be an open
disc meeting $J(f)$. The following are equivalent:
{\emph (a)} $B_D(f)^c$ has a bounded component;
{\emph (b)} there is a bounded domain $U$ with $\partial U
\subset B_D(f)$ and $U \cap J(f) \neq \emptyset$;
{\emph (c)} $\bigcup_{n \in \mathbb N}V_n$ is bounded, where $V_n$ is
the component of $f^{-n}(\widetilde{f^n(D)})$ that contains $D$.
\end{3.3}
\begin{proof} We begin by showing that (a) implies (b). So
suppose that $B_D(f)^c$ has a bounded component $U$. We know from
Lemma~3.2 that $U$ is open. Also $\partial U \subset B_D(f)$. Thus
\begin{equation}
\partial f^n(U) \subset f^n(\partial U) \subset \mathbb C
\setminus \widetilde{f^n(D)},
\end{equation}
for each $n \in \mathbb N$. Since $U \subset B_D(f)^c$, there exists $N
\in \mathbb N$ such that $f^N(U) \cap \widetilde{f^N(D)} \neq
\emptyset$. It follows from (3.2) that $f^N(U) \supset
\widetilde{f^N(D)}$ and so $f^N(U) \cap J(f) \neq \emptyset$. Thus
$U \cap J(f) \neq \emptyset$ and so (b) is true.
We now show that (b) implies (c). First note that if $V_n$ is the
component of $f^{-n}(\widetilde{f^n(D)})$ that contains $D$,
then $\bigcup_{n \in \mathbb N}V_n \subset B_D(f)^c$. Also
\begin{equation}
V_n \subset V_{n+1}, \mbox{ for each } n \in \mathbb N,
\end{equation}
since
\[
f^{n+1}(V_n) = f(f^n(V_n)) \subset f(\widetilde{f^n(D)}) \subset
\widetilde{f^{n+1}(D)}.
\]
Thus $\bigcup_{n \in \mathbb N}V_n$ is connected. If (b) is true then it
follows from Theorem~3(a) that there are no unbounded components
of $B_D(f)^c$ and so $\bigcup_{n \in \mathbb N}V_n$ is
bounded; that is, (c) is true.
Finally, we show that (c) implies (a). Put $V = \bigcup_{n \in
{\bf N}}V_n$ and suppose that (c) is true. Clearly $V \subset
B_D(f)^c$. We claim that $\partial V \subset B_D(f)$ and hence
(a) is true. To show this, we use proof by contradiction. So
suppose that $z_0 \in
\partial V$ and that $f^N(z_0) \in \widetilde{f^N(D)}$ for some
$N \in \mathbb N$. Since $\widetilde{f^N(D)}$ is open, there exists an
open neighbourhood $W$ of $z_0$ such that $f^N(W) \subset
\widetilde{f^N(D)}$ and hence
\begin{equation}
f^n(W) \subset \widetilde{f^n(D)}, \; \mbox{ for all } n \geq N.
\end{equation}
It follows from (3.3) that there exists $N_1 \geq N$ such that $W
\cap
\partial V_{N_1} \neq \emptyset$. Thus, by (3.4),
\[
f^{N_1}(\partial V_{N_1}) \cap \widetilde{f^{N_1}(D)} \neq
\emptyset.
\]
This, however, contradicts the fact that $V_{N_1}$ is a component of
$f^{-N_1}(\widetilde{f^{N_1}(V)})$ and so (a) is indeed true. This
completes the proof of Lemma~3.3.
\end{proof}
We are now in a position to prove Theorem~4.
\begin{proof}[Proof of Theorem~4]
Let $f$ be a transcendental entire function and
let $D$ be an open disc meeting $J(f)$. Suppose that there exist
Jordan curves $\gamma_n$ such that, for all $n \in \mathbb N$,
\begin{equation}
\gamma_n \mbox{ surrounds } f^n(D)
\end{equation}
and
\begin{equation}
f(\gamma_n) \mbox{ surrounds the bounded component of } \gamma_{n+1}^c.
\end{equation}
Now let $V_n$ be the component of $f^{-n}(\widetilde{f^n(D)})$
that contains $D$. We will show that $\bigcup_{n \in \mathbb N}V_n$ is
bounded. First note that, by the blowing up property, Lemma~2.1,
there exists $N \in \mathbb N$ such that
\begin{equation}
\widetilde{f^N(D)} \supset D.
\end{equation}
We claim that
\[
V_n \cap \gamma_N = \emptyset, \; \mbox{ for all } n \in \mathbb N.
\]
We prove this by contradiction. So, suppose that there exists $m
\in \mathbb N$ such that $V_m \cap \gamma_N \neq \emptyset$. Since $V_m
\supset D$, this implies that there exists a path $\sigma \subset
V_m$ joining a point in $D$ to a point in $\gamma_N$. By (3.7),
we have
\begin{equation}
f(D) \subset \widetilde{f^{N+1}(D)}
\end{equation}
and, by (3.5) and (3.6), $f(\gamma_N)$ surrounds the bounded
component of $\gamma_{N+1}^c$ and hence surrounds $f^{N+1}(D)$. So it
follows from (3.8) and (3.6) that $f(\sigma)$ joins a point in
$f(D)$ to a point in $\gamma_{N+1}$. Continuing this process
inductively, we find that $f^n(\sigma)$ joins a point in $f^n(D)$
to a point in $\gamma_{N+n}$. In particular, $f^m(\sigma)$ joins
a point in $f^m(D)$ to a point in $\gamma_{N+m}$. Since $\sigma
\subset V_m$ and $f^m(V_m) \subset \widetilde{f^m(D)}$, this
implies that
\begin{equation}
\widetilde{f^m(D)} \cap \gamma_{N+m} \neq \emptyset.
\end{equation}
By (3.7) we have $\widetilde{f^m(D)} \subset
\widetilde{f^{N+m}(D)}$ and so it follows from (3.9) that
\[
\widetilde{f^{N+m}(D)} \cap \gamma_{N+m} \neq \emptyset.
\]
This, however, contradicts (3.5) and so we must have
\begin{equation}
V_n \cap \gamma_N = \emptyset, \; \mbox{ for all } n \in \mathbb N.
\end{equation}
Since $V_n \supset D$, for all $n \in \mathbb N$, it follows from (3.5),
(3.7) and (3.10) that $V = \bigcup_{n \in \mathbb N} V_n$ lies in the
bounded component of $\gamma_N^c$ and is therefore bounded. So,
by Lemma~3.3, there exists a bounded domain $U$ with $\partial U
\subset B_D(f)$ and $U \cap J(f) \neq \emptyset$. Thus $B_D(f)$,
$B(f)$ and $I(f)$ are all connected, by Theorem~3.
\end{proof}
\section{Functions of small growth}
\setcounter{equation}{0}
As mentioned earlier, it is conjectured that if $f$ is a transcendental entire function
of order at most $1/2$, minimal type, then $F(f)$ has no unbounded components.
This question was first studied by Baker [\ref{B81}] and has
since been studied by many other authors; see~[\ref{H}]. Many of the papers in this area
use the following result [\ref{S}, Lemma~2.7], which is
a generalisation of a result by Baker [\ref{B81}, proof of
Theorem~2].
\newtheorem{4.1}{Lemma}[section]
\begin{4.1}
Let $f$ be a transcendental entire function and suppose that
there exist sequences $R_n, \rho_n \to \infty$ and $c(n) > 1$
such that
\begin{enumerate}
\item $R_{n+1} = M(R_n,f)$,
\item $R_n \leq \rho_n \leq R_n^{c(n)}$,
\item $m(\rho_n,f) > R_{n+1}^{c(n+1)}$.
\end{enumerate}
Then $F(f)$ contains no unbounded components.
\end{4.1}
Here, for $r>0$,
\[
m(r,f) = \min_{|z|=r}|f(z)|.
\]
We now show that if $f$ satisfies the hypotheses of Lemma~4.1,
then $f$ also satisfies the hypotheses of Theorem~4, so the sets $B_D(f)$,
$B(f)$ and $I(f)$ are all connected.
\newtheorem{4.2}[4.1]{Lemma}
\begin{4.2}
Let $f$ be a transcendental entire function and suppose that
there exist sequences $R_n, \rho_n \to \infty$ and $c(n) > 1$
such that
\begin{enumerate}
\item $R_{n+1} = M(R_n,f)$,
\item $R_n \leq \rho_n \leq R_n^{c(n)}$,
\item $m(\rho_n,f) > R_{n+1}^{c(n+1)}$.
\end{enumerate}
Then, for each open disc $D$ meeting $J(f)$, there exist Jordan
curves $\gamma_n$ such that, for all $n \in \mathbb N$,
\begin{equation}
\gamma_n \mbox{ surrounds } f^n(D)
\end{equation}
and
\begin{equation}
f(\gamma_n) \mbox{ surrounds the bounded component of } \gamma_{n+1}^c.
\end{equation}
Hence $B_D(f)$, $B(f)$ and $I(f)$ are all connected.
\end{4.2}
\begin{proof} Take $R_1$ such that $f(D) \subset B(0,R_1)$.
Then, if $R_{n+1} = M(R_n,f)$, we have $\widetilde{f^n(D)} \subset
B(0,R_n)$. If we put $\gamma_n = \{z:|z| = \rho_n\}$, then the
properties of the sequences $R_n$ and $\rho_n$ imply that (4.1)
and (4.2) are both satisfied. Thus $B_D(f)$, $B(f)$ and $I(f)$
are all connected, by Theorem~4.
\end{proof}
\begin{proof}[Proof of Corollary 5]
In [\ref{S}, Lemma 3.3, Lemma 4.2 and Lemma 4.5] we proved that functions satisfying (1.5) and functions of order $\rho < 1/2$ satisfying (1.6) must both satisfy the hypotheses of Lemma~4.1 (and so have no unbounded Fatou components); these proofs use estimates for $m(r,f)$ due to Baker [\ref{B58}] and Cartwright~[\ref{C}]. By Lemma~4.2, these functions also satisfy the hypotheses of
Theorem~4. This proves Corollary~5.
\end{proof}
\section{Unbounded Fatou components and the escaping set}
\setcounter{equation}{0}
In the previous section we stated that Lemma~4.1 forms the basis
of many proofs to show that certain functions of small growth have
no unbounded Fatou components. As we have seen, the
hypotheses of Lemma~4.1 are stronger than those of
Theorem~4 and hence stronger than those of Theorem~3. We now prove that the hypotheses of
Theorem~3 are in fact sufficient to ensure that a function has no
unbounded Fatou components -- that is, we prove Theorem~6. We use
the following results, the first of which is proved in
[\ref{Zip}, Theorem~3(b)], for example.
\newtheorem{5.1}{Lemma}[section]
\begin{5.1}
Let $f$ be a transcendental entire function, let $V$ be a Fatou component of $f$, and let $z_1, z_2 \in V$. If the Fatou components of $f$ which contain the forward orbits $f^n(z_i)$, $n=0,1,\ldots$, $i=1,2$, are all simply connected, then there
exists $A>0$ such that
\[
|f^n(z_2)| \leq A(|f^n(z_1)| + 1), \; \mbox{ for all } n \in \mathbb N.
\]
\end{5.1}
Next we need a result about the size of the image of a large disc under
an iterate of a transcendental entire function. A weaker result of this type was given in [\ref{RS}, Lemma~2.2].
\newtheorem{5.2}[5.1]{Lemma}
\begin{5.2}
Let $f$ be a transcendental entire function. Then there exists
$R(f)>0$ such that
\[
\widetilde{f^n(B(0,R))} \supset B(0,2M^n(R/2,f)), \; \mbox{ for
all } n \in \mathbb N, R>R(f).
\]
\end{5.2}
\begin{proof} We first recall some facts from Wiman-Valiron
theory. For a detailed account, see~[\ref{Ha}], for example.
Let $f(z)=\sum_{n=0}^{\infty} a_n z^n$ be a transcendental entire function. For $r>0$, let $N(r)$ denote the integer $n\ge 0$ such that $|a_n|r^n$ is maximal and let $z_r$ satisfy $|z_r|=r$ and $|f(z_r)|=M(r,f)$. The main result of Wiman-Valiron theory states that if $\tau>1/2$, then there is a set $E\subset (0,\infty)$ such that $\int_E 1/t\,dt<\infty$ and \[
f(z)\sim \left(\frac{z}{z_r}\right)^{N(r)} f(z_r),\quad \mbox{for }z\in B(z_r,r/N(r)^{\tau}),
\]
as $r\to\infty$, $r\notin E$. In [\ref{E}, proof of Theorem~1], this result was used to prove that, for $r$ large enough and $r\notin E$, we have
\begin{equation}
f(B(z_r,5r/N(r))\supset \{z:e^{-4}M(r,f)<|z|<e^4 M(r,f)\}.
\end{equation}
Now $N(r)\to\infty$ as $r\to\infty$. Thus, for $R$ sufficiently large, there exists $r\in (R/2,R)$ such that~(5.1) holds and $B(z_r,5r/N(r))\subset B(0,R)$. Therefore,
\[
\widetilde{f(B(0,R))} \supset B(0,2M(R/2,f)),
\]
for all sufficiently large $R$. The result then follows by induction.
\end{proof}
We are now in a position to prove Theorem~6.
\begin{proof}[Proof of Theorem~6]
Let $U$ be a bounded domain with $\partial U \subset B_D(f)$ and
$U \cap J(f) \neq \emptyset$, where $D$ is an open disc meeting
$J(f)$.
Suppose that $V$ is an unbounded component of the Fatou set.
Then $f$ has no multiply connected Fatou components, by the remark following Theorem~1. By Theorem~3(a), there exists $M \in \mathbb N$ such that
\[
V \cap \beta_n \neq \emptyset, \; \mbox{ for all } n \geq M,
\]
where $\beta_n = f^n(\partial U)$, for $n\ge 0$, and so $V \subset I(f)$. We now
take $R_0$ so large that
\begin{equation}
\beta_M \subset B(0,R_0)
\end{equation}
and $2M(R_0,f)> R(f)$, where $R(f)$ is as defined in Lemma~5.2.
Next we take $m \in \mathbb N$ such that
\begin{equation}
\widetilde{f^{m+M}(D)} \supset B(0,2M(R_0,f)).
\end{equation}
(This is possible by the blowing up property, Lemma~2.1.)
Finally, we take $z_1 \in V \cap \beta_M$ and $z_2 \in V \cap
\beta_{m+M}$. Then, by (5.2),
\[
|f^n(z_1)| \leq M(R_0,f^n), \; \mbox{ for all } n \in \mathbb N.
\]
Also, by (5.3) and Lemma~5.2, for all $n \in \mathbb N$,
\begin{eqnarray*}
\widetilde{f^{n+m+M}(D)} & \supset & \widetilde{f^n(B(0,2M(R_0,f))}
\supset B(0,2M^n(M(R_0,f),f))\\
& = & B(0,2M^{n+1}(R_0,f)) \supset B(0,2M(M(R_0,f^n),f)).
\end{eqnarray*}
Thus, since $z_2 \in \beta_{m+M}$ and $\beta_0=\partial U \subset B_D(f)$,
so $f^n(z_2) \notin \widetilde{f^{n+m+M}(D)}$, we have
\[
|f^n(z_2)| \geq 2M(M(R_0,f^n),f), \; \mbox{ for all } n \in \mathbb N.
\]
Hence
\[
\frac{|f^n(z_2)|}{|f^n(z_1)|} \geq
\frac{2M(M(R_0,f^n),f)}{M(R_0,f^n)} \to \infty \; \mbox{ as } n \to \infty,
\]
which contradicts Lemma~5.1. Therefore there are no
unbounded Fatou components.
\end{proof}
\section{Examples}
\setcounter{equation}{0}
Theorems~2,~3 and~4, and Corollary~5 give criteria on a transcendental
entire function for the sets $B_D(f)$, $B(f)$ and $I(f)$ to be connected, where
$D$ is any open disc meeting $J(f)$.
Here we point out that there are many functions which satisfy one of the
hypotheses of Corollary~5 (and hence satisfy the hypotheses of Theorems~2,~3
and~4) but which do not have multiply connected Fatou
components (so Theorem~1 does not apply to them).
Indeed, we show that there are such functions of all orders $\rho$,
$0\le\rho<1/2$. These are all new examples of
functions for which Eremenko's conjecture is true.
{\bf Example~1}\quad Bergweiler and Eremenko showed in [7] that
there are transcendental entire functions of arbitrarily small
growth for which the Julia set is the whole plane. In particular,
there exists such a function $f$ whose growth satisfies (1.5), and
hence has order 0. Thus, by Corollary~5(i), the sets $B_D(f)$, $B(f)$
and $I(f)$ are all connected. Clearly $f$ has no multiply connected Fatou components.
{\bf Example~2}\quad Baker [5] and Boyd [9] independently showed
that there are transcendental entire functions of arbitrarily
small growth for which every component of the Fatou set is simply
connected and every point in the Fatou set tends to~0 under
iteration. As before, there exists such a function $f$ whose
growth satisfies (1.5), and hence has order 0. Thus, by
Corollary~5(i), the sets $B_D(f)$, $B(f)$ and $I(f)$ are all connected. Clearly
$f$ has no multiply connected Fatou components.
{\bf Example~3}\quad For $0<\rho<1/2$, consider the infinite product
\[
f(z)=c\prod_{n=1}^{\infty}\left(1+\frac{z}{n^{1/\rho}}\right),
\]
where $c>0$. Then, for $r>0$,
\[
M(r)=M(r,f)=f(r)\quad \text{and}\quad \log \frac{M(r)}{c}=r\int_0^{\infty}\frac{n(t)}{t(r+t)}\,dt,
\]
where $n(r)$ is the number of zeros of~$f$ in $\{z:|z|\le r\}$;
see~[\ref{T}, page~271]. Since the work of Wiman~[\ref{W}], such
functions have been known to have very regular behaviour. For
example, we have $n(r)=[r^{\rho}]$ so we can deduce by contour
integration that
\[
\log \frac{M(r)}{c}\sim r^{\rho}\frac{\pi}{\sin \pi\rho}\;\text{ as }r\to\infty.
\]
In particular, $f$ has order $\rho$ and~(1.6) holds, so the sets $B_D(f)$, $B(f)$ and $I(f)$ are all connected by Corollary~5(ii).
Similarly,
\[
r\frac{M'(r)}{M(r)} = r\int_0^{\infty}\frac{n(t)}{(r+t)^2}\,dt \sim r^{\rho}\frac{\pi\rho}{\sin \pi\rho}\;
\text{ as }r\to\infty,
\]
so there exist $C>1$ and $r_0>0$ such that
\begin{equation}
\frac{rf'(r)}{f(r)}\ge C\frac{\log f(r)}{\log r}\,,\quad\text{for }r>r_0.
\end{equation}
Now, for $r_0$ large enough, the interval $[r_0,\infty)$ is
invariant under $f$. Thus $[r_0,\infty)\subset J(f)$ by~(6.1) and [\ref{Zip}, Theorem~4(a)]. Hence $f$ has no multiply connected
Fatou components, by the remark following Theorem~1. Note that if
$c>0$ is small enough, then $f$ has at least one attracting fixed
point so $F(f)\ne \emptyset$.
{\it Remarks.}\quad 1. Example~3 can be generalised somewhat. Let
\[
f(z)=c\prod_{n=1}^{\infty}\left(1+\frac{z}{r_n}\right),
\]
where $c>0$, $r_n>0$. If we assume that $n(r)\sim \lambda r^{\rho}$, where $\lambda>0$ and $0<\rho<1/2$, then it can be shown as above that~(1.6) and~(6.1) hold (see~[\ref{T}, page~271] for the calculation related to $\log M(r)$). Thus the sets $B_D(f)$, $B(f)$ and $I(f)$ are all connected, and $f$ has no multiply connected Fatou components.
An example of such a function in closed form is given by
\[
f(z)=\frac12\left(\cos z^{1/4} + \cosh z^{1/4}\right)=1+\frac{z}{4!}+\frac{z^2}{8!}+\cdots.
\]
In this case $\rho=1/4$, $c=1$ and $f(-r)=\cos(\frac1{\sqrt{2}}\, r^{1/4})\cosh(\frac1{\sqrt{2}}\, r^{1/4})$, for $r>0$, so the zeros of $f$ are at $-4\pi^4(n-\frac12)^4$, $n=1,2,\ldots$. A family of examples of this type, with orders $\rho=2^{-m},\;m=2,3,\ldots,$ is given in~[\ref{Mar}, page~279].
2. Baker~[\ref{B85}] constructed examples of entire functions of all possible orders $\rho\ge 0$ which have multiply connected Fatou components. His examples were constructed using infinite products, the zeros being chosen in a more irregular manner than those in Example~3.
3. In [\ref{MF}] we remarked that it was plausible
that a transcendental entire function with no multiply connected Fatou
components should have infinitely many unbounded components of
$B(f)$. The above examples show that this is not the case.
\section*{References}
\begin{enumerate}
\item\label{B58} I.N. Baker. Zusammensetzungen ganzer Funktionen.
{\it Math. Z., 69 (1958), 121-163.}
\item \label{B81} I.N. Baker. The iteration of polynomials and
transcendental entire functions. {\it J. Aust. Math. Soc., Ser. A,
30 (1981), 483--495.}
\item \label{B84} I.N. Baker. Wandering domains in the iteration
of entire functions. {\it Proc. Lond. Math. Soc., III. Ser., (3) 49 (1984),
563--576.}
\item \label{B85} I.N. Baker, Some entire functions with
multiply-connected wandering domains, {\it Ergodic Theory Dyn. Syst.,
5 (1985), 163--169.}
\item \label{B01} I.N. Baker. Dynamics of slowly growing entire
functions. {\it Bull. Aust. Math. Soc., 63 (2001), 367--377.}
\item \label{Bar} K.\ Bara\'nski.
Trees and hairs for some hyperbolic entire maps of finite order.
{\it Math.\ Z., 257 (2007), 33--59.}
\item \label{Berg} W. Bergweiler. Iteration of meromorphic functions.
{\it Bull. Am. Math. Soc., New Ser., 29 (1993), 151--188.}
\item \label{BE} W. Bergweiler and A.E. Eremenko. Entire functions
of slow growth whose Julia set coincides with the plane. {\it
Ergodic Theory Dyn. Syst., 20 (2000), 1577--1582.}
\item \label{BH} W. Bergweiler and A. Hinkkanen. On
semiconjugation of entire functions. {\it Math. Proc. Camb. Philos. Soc.,
126 (1999), 565--574.}
\item \label{Bo} D.A. Boyd. An entire function with slow growth
and simple dynamics. {\it Ergodic Theory Dyn. Syst., 22 (2002),
317--322.}
\item \label{C} M.L. Cartwright. {\it Integral functions.}
Cambridge Tracts in Mathematics and Mathematical Physics, No. 44,
Cambridge University Press, 1962.
\item \label{E} A.E. Eremenko. On the iteration of entire
functions. {\it Dynamical systems and ergodic theory,} Banach Cent. Publ.
23 (Polish Scientific Publishers, Warsaw, 1989)
339--345.
\item \label{F} P. Fatou. Sur l'it\'{e}ration des fonctions
transcendantes enti\`{e}res. {\it Acta Math. 47 (1926), 337-360.}
\item \label{Ha} W.K. Hayman. The local growth of power series: a
survey of the Wiman-Valiron method. {\it Can. Math. Bull. 17
(1974), 317--358.}
\item \label{H} A. Hinkkanen. Entire functions with bounded Fatou
components. To appear in {\it Transcendental dynamics and complex
analysis,} Cambridge University Press, 2008.
\item\label{Mar} A.I. Markushevich. {\it Theory of functions of a
complex variable, Volume II,} Prentice Hall, 1965.
\item \label{Zip} P.J. Rippon and G.M. Stallard. On sets where
iterates of a meromorphic function zip towards infinity. {\it
Bull. Lond. Math. Soc., 32 (2000), 528--536.}
\item \label{RS} P.J. Rippon and G.M. Stallard. On questions of
Fatou and Eremenko. {\it Proc. Am. Math. Soc., 133 (2005),
1119--1126.}
\item \label{MF} P.J. Rippon and G.M. Stallard. Escaping points of
meromorphic functions with a finite number of poles. {\it J. Anal.
Math., 96 (2005), 225-245.}
\item \label{SG} P.J. Rippon and G.M. Stallard. Functions of small
growth with no unbounded Fatou components. {\it Preprint.}
\item \label{RRRS} G.\ Rottenfu{\ss}er, J. R\"uckert, L. Rempe and D.\
Schleicher. Dynamic rays of bounded-type entire functions. {\it
Preprint.}
\item \label{S} G.M. Stallard. The iteration of entire functions
of small growth. {\it Math. Proc. Camb. Philos. Soc., 114 (1993),
43--55.}
\item \label{T} E.C. Titchmarsh. {\it The theory of functions}, Oxford University Press, 1939.
\item\label{W} A. Wiman. {\" U}ber die angen{\" a}herte Darstellung von ganzen Funktionen. {\it Ark. Mat., Astr. o. Fysik, 1 (1903), 105--111.}
\end{enumerate}
Department of Mathematics,\\
The Open University,\\
Walton Hall,\\
Milton Keynes MK7 6AA,\\
UK
E-mail: [email protected]; [email protected]
\end{document}
|
1,116,691,497,761 | arxiv | \section{The Friedmann-Robertson-Walker models}
\subsection{Geometry and topology}
Let us review and clarify the topological and geometrical aspects
of the Friedmann-Robertson-Walker (FRW) models of general
relativistic cosmology. Whilst doing so will contribute to the
overall intention of this paper to clarify, by means of precise
mathematical concepts, the notions of modern cosmology, there are
further philosophical motivations: firstly, to emphasise the
immense variety of possible topologies and geometries for our
universe, consistent with empirical (i.e. astronomical) data; and
secondly, to emphasise the great variety of possible other
universes.
The general interpretational doctrine adopted in this paper can be
referred to as `structuralism', in the sense advocated by Patrick
Suppes (1969), Joseph Sneed (1971), Frederick Suppe (1989), and
others\index{structuralism}. This doctrine asserts that, in
mathematical physics at least, the physical domain of a theory is
conceived to be an instance of a mathematical structure or
collection of mathematical structures. The natural extension of
this principle proposes that an entire physical universe is an
instance of a mathematical structure or collection of mathematical
structures.
Those expressions of structuralism which state that `the' physical
universe is an instance of a mathematical structure, tacitly
assume that our physical universe is the only physical universe.
If one removes this assumption, then structuralism can be taken as
the two-fold claim that (i) our physical universe is an instance
of a mathematical structure, and (ii), other physical universes,
if they exist, are either different instances of the same
mathematical structure, or instances of different mathematical
structures. If some aspects of our physical universe appear to be
contingent, that may indicate how other physical universes provide
different instances of the same mathematical structure possessed
by our universe. Alternatively, given that mathematical structures
are arranged in tree-like hierarchies, other physical universes
may be instances of mathematical structures which are sibling to
the structure possessed by our universe. In other words, the
mathematical structures possessed by other physical universes may
all share a common parent structure, from which they are derived
by virtue of satisfying additional conditions. This would enable
us to infer the mathematical structure of other physical universes
by first generalizing from the mathematical structure of our own,
and then classifying all the possible specializations of the
common, generic structure.
Hence, it is the aim of this paper not only to define the
mathematical structures used in modern cosmology to represent our
universe on large-scales, but to explore the variety of possible
instances of those structures, and to emphasise how those
mathematical structures are special cases of more general
mathematical structures. The intention is to establish the
mathematical structure possessed by our own universe, and to use
that to imply the nature of other universes.
\hfill \break
Geometrically, a FRW model is a $4$-dimensional Lorentzian
manifold $\mathcal{M}$ which can be expressed as a warped product,
(O'Neill 1983, Chapter 12; Heller 1992, Chapter 6):
$$
I \times_R \Sigma \,.
$$ $I$ is an open interval of the pseudo-Euclidean manifold
$\mathbb{R}^{1,1}$, and $\Sigma$ is a complete and connected
$3$-dimensional Riemannian manifold. The warping function $R$ is a
smooth, real-valued, non-negative function upon the open interval
$I$. It will otherwise be known as the scale factor.
If we deonote by $t$ the natural coordinate function upon $I$, and
if we denote the metric tensor on $\Sigma$ as $\gamma$, then the
Lorentzian metric $g$ on $\mathcal{M}$ can be written as
$$
g = -dt \otimes dt + R(t)^2 \gamma \,.
$$
One can consider the open interval $I$ to be the time axis of the
warped product cosmology. The $3$-dimensional manifold $\Sigma$
represents the spatial universe, and the scale factor $R(t)$
determines the time evolution of the spatial geometry.
In a conventional FRW model, the $3$-dimensional manifold $\Sigma$
is an isotropic and homogeneous Riemannian manifold. More
precisely, $\Sigma$ is globally isotropic. To explain the
significance of this, we shall review the notions of homogeneity
and isotropy.
A Riemannian manifold $(\Sigma,\gamma)$ is defined to be
homogeneous if the isometry group $I(\Sigma)$ acts transitively
upon $\Sigma$. For any pair of points $p,q \in \Sigma$ from a
homogeneous manifold, there will be an isometry $\phi$ such that
$\phi(p)=q$. If there is a unique isometry $\phi$ such that
$\phi(p)=q$ for each pair of points $p,q \in \Sigma$, then the
isometry group action is said to be simply transitive. If there is
sometimes, or always, more than one such isometry, then the
isometry group action is said to be multiply transitive.
In colloquial terms, one can say that the geometrical
characteristics at one point of a homogeneous Riemannian manifold,
match those at any other point.
To define isotropy, it is necessary to introduce the `isotropy
subgroup'. At each point $p \in \Sigma$ of a Riemannian manifold,
there is a subgroup $H_p \subset I(\Sigma)$ of the isometry group.
Referred to as the isotropy subgroup at $p$, $H_p$ is the set of
isometries under which $p$ remains fixed. Thus, $\psi \in H_p$ is
such that $\psi(p)=p$. The differential map $\psi_*$ of each $\psi
\in H_p$, bijectively maps the tangent space at $p$ onto itself.
By restricting the differential map $\psi_*$ of each $\psi \in
H_p$ to $T_p \Sigma$, the tangent space at $p$, one obtains a
linear representation of the isotropy subgroup $H_p$:
$$
j:H_p \rightarrow GL(T_p \Sigma)\,.
$$
We can refer to $j(H_p)$ as the linear isotropy subgroup at $p$.
Whilst $H_p$ is a group of transformations of $\Sigma$, $j(H_p)$
is a group of transformations of $T_p \Sigma$.
The Riemannian metric tensor field $\gamma$ upon the manifold
$\Sigma$, assigns a positive-definite inner product $\langle \; ,
\; \rangle_\gamma$ to each tangent vector space $T_p \Sigma$.
Hence, each tangent vector space can be considered to be an inner
product space
$$
(T_p \Sigma, \langle \; , \; \rangle _\gamma)\,.
$$
Whilst $H_p$ is a group of diffeomorphic isometries of the
Riemannian manifold $(\Sigma,\gamma)$, $j(H_p)$ is a group of
linear isometries of the inner product space $(T_p \Sigma,\langle
\; , \; \rangle_\gamma)$. For any pair of vectors $v,w \in T_p
\Sigma$, and for any $\psi \in j(H_p)$, this means that
$$
\langle \psi(v),\psi(w) \rangle = \langle v,w \rangle \,.
$$
We can therefore consider the representation $j$ to be an
orthogonal linear representation:
$$
j:H_p \rightarrow O(T_p \Sigma) \subset GL(T_p \Sigma) \,.
$$
We can now define a Riemannian manifold $(\Sigma,\gamma)$ to be
isotropic at a point $p$ if the linear isotropy group at $p$,
$j(H_p)$, acts transitively upon the unit sphere in the tangent
space $T_p \Sigma$.
This definition requires some elaboration. Firstly, the unit
sphere $S_p \Sigma \subset T_p \Sigma$ is defined as
$$
S_p \Sigma = \{v \in T_p \Sigma : \langle v,v \rangle_\gamma = 1
\} \,.
$$
The unit sphere represents all possible directions at the point
$p$ of the manifold $\Sigma$. Each vector $v \in S_p \Sigma$ can
be considered to point in a particular direction.
Now, the requirement that $j(H_p)$ acts transitively upon $S_p
\Sigma$, means that for any pair of points $v,w \in S_p \Sigma$ on
the unit sphere, there must be a linear isometry $\psi \in j(H_p)$
such that $\psi(v)=w$. If $j(H_p)$ acts transitively upon the unit
sphere $S_p \Sigma$, all directions at the point $p$ are
geometrically indistinguishable. If a Riemannian manifold
$(\Sigma,\gamma)$ is isotropic at a point $p$, then all directions
at the point $p$ are geometrically indistinguishable.
In the case of cosmological relevance, where $(\Sigma,\gamma)$ is
a $3$-dimensional Riemannian manifold which represents the spatial
universe, isotropy at a point $p$ means that all spatial
directions at $p$ are indistinguishable.
It is simple to show that $j(H_p)$ acts transitively upon the unit
sphere at a point $p$, if and only if it acts transitively upon a
sphere of any radius in $T_p \Sigma$. Hence, if $(\Sigma,\gamma)$
is isotropic at $p$, then $j(H_p)$ includes the so-called rotation
group $SO(T_p \Sigma) \cong SO(3)$. The orbits of the action are
the concentric family of $2$-dimensional spheres in $T_p \Sigma$,
plus the single point at the origin of the vector space.
If $j(H_p)$, the linear isotropy group at $p$, acts transitively
upon the unit sphere in $T_p \Sigma$, then each orbit of the
isotropy group action on $\Sigma$ consists of the points which lie
a fixed distance from $p$, and each such orbit is a homogeneous
surface in $\Sigma$, whose isometry group contains $SO(3)$.
An isotropic Riemannian manifold $(\Sigma,\gamma)$ is defined to
be a Riemannian manifold which is isotropic at every point $p \in
\Sigma$. To be precise, we have defined a globally isotropic
Riemannian manifold. We will subsequently introduce the notion of
local isotropy, which generalises the notion of global isotropy.
It is conventionally understood that when one speaks of isotropy,
one is speaking of global isotropy unless otherwise indicated. To
clarify the discussion, however, we will hereafter speak
explicitly of global isotropy.
From the perspective of the $4$-dimensional Lorentzian manifold
$\mathcal{M} = I \times_R \Sigma$, each point $p$ belongs to a
spacelike hypersurface $\Sigma_t = t \times \Sigma$ which is
isometric with $(\Sigma, R(t)^2\gamma)$. The hypersurface
$\Sigma_t$ is a $3$-dimensional Riemannian manifold of constant
sectional curvature. The tangent space $T_p\mathcal{M}$ contains
many $3$-dimensional spacelike subspaces, but only one,
$T_p\Sigma_t$, which is tangent to $\Sigma_t$, the hypersurface of
constant sectional curvature passing through $p$. The unit sphere
$S_p\Sigma_t$ in this subspace represents all the possible spatial
directions at $p$ in the hypersurface of constant sectional
curvature. Spatial isotropy means that the isotropy group at $p$
acts transitively upon this sphere. Whilst there is a spacelike
unit sphere in each $3$-dimensional spacelike subspace of
$T_p\mathcal{M}$, the spatial isotropy of the FRW models pertains
only to the transitivity of the isotropy group action upon
$S_p\Sigma_t$. However, there is also a null sphere at $p$
consisting of all the null lines in $T_p\mathcal{M}$. This sphere
represents all the possible light rays passing through $p$.
Letting $\partial_t$ denote the unit timelike vector tangent to
the one-dimensional submanifold $I$, the isotropy group action at
each $p$ maps $\partial_t$ to itself. Any vector in
$T_p\mathcal{M}$ can be decomposed as the sum of a multiple of
$\partial_t$ with a vector in the spacelike subspace
$T_p\Sigma_t$. Hence, the action of the isotropy group upon
$T_p\Sigma_t$ can be extended to an action upon the entire tangent
vector space $T_p\mathcal{M}$. In particular, the isotropy group
action can be extended to the null sphere. If the isotropy group
action is transitive upon the set of spatial directions
$S_p\Sigma_t$, then it will also be transitive upon the null
sphere at $p$.
In a conventional FRW model, the complete and connected
$3$-dimensional Riemannian manifold $(\Sigma,\gamma)$ is both
homogeneous and globally isotropic. In fact, any connected
$3$-dimensional globally isotropic Riemannian manifold must be
homogeneous. It is therefore redundant to add that a conventional
FRW model is spatially homogeneous.
Now, a complete, connected, globally isotropic $3$-dimensional
Riemannian manifold must be of constant sectional curvature $k$. A
complete, connected Riemannian manifold of constant sectional
curvature, of any dimension, is said to be a Riemannian space
form.
There exists a simply connected, $3$-dimensional Riemannian space
form for every possible value, $k$, of constant sectional
curvature.
\begin{theorem} A complete, simply connected, $3$-dimensional
Riemannian manifold of constant sectional curvature $k$, is
isometric to
\begin{itemize}
\item The sphere $S^3(r) \; \text{for} \; r=\surd(1/k) \; \text{if} \; k>0$
\item Euclidean space $\mathbb{R}^3 \; \text{if} \; k=0$
\item The hyperbolic space $H^3(r) \; \text{for} \; r= \surd(1/-k) \; \text{if} \; k<0$
\end{itemize}
where
$$S^3(r) = \{x \in \mathbb{R}^4 : \; \langle x,x \rangle = r^2\}$$ and
$$H^3(r) = \{x \in \mathbb{R}^{3,1} :\; \langle x,x \rangle = -r^2, \; x^0 >0\} \,.$$
\end{theorem}
$S^3(r)$, the sphere of radius $r$, is understood to have the
metric tensor induced upon it by the embedding of $S^3(r)$ in the
Euclidean space $\mathbb{R}^4$, and the hyperboloid $H^3(r)$ is
understood to have the metric tensor induced upon it by the
embedding of $H^3(r)$ in the pseudo-Euclidean space
$\mathbb{R}^{3,1}$.
Geometries which differ from each other by a scale factor are said
to be homothetic. Space forms are homothetic if and only if their
sectional curvature is of the same sign. There are, therefore, up
to homothety, only three $3$-dimensional simply connected
Riemannian space forms: $S^3$, the three-dimensional sphere;
$\mathbb{R}^3$, the three-dimensional Euclidean space; and $H^3$,
the three-dimensional hyperboloid.
Whilst it is true that every simply connected Riemannian space
form is globally isotropic, the converse is not true.
Real-projective three-space $\mathbb{R}\mathbb{P}^3$, equipped
with its canonical metric tensor, is also globally isotropic, but
is non-simply connected.
Up to homothety, there are four possible spatial geometries of a
conventional, globally isotropic FRW model: $S^3$, $\mathbb{R}^3$,
$H^3$, and $\mathbb{RP}^3$. Up to homothety, these are the only
complete and connected, globally isotropic $3$-dimensional
Riemannian manifolds, (Beem and Ehrlich 1981, p131).
$\mathbb{R}^3$ and $H^3$ are diffeomorphic, hence there are only
three possible spatial topologies of a globally isotropic FRW
model. Only $S^3$, $\mathbb{R}^3$, and $\mathbb{RP}^3$ can be
equipped with a globally isotropic, complete Riemannian metric
tensor.
A generalisation of the conventional FRW models can be obtained by
dropping the requirement of global isotropy, and substituting in
its place the condition that $(\Sigma,\gamma)$ must be a
Riemannian manifold of constant sectional curvature, a space form.
As already stated, every globally isotropic Riemannian
$3$-manifold is a space form. However, not every $3$-dimensional
Riemannian space form is globally isotropic. On the contrary,
there are \emph{many} $3$-dimensional Riemannian space forms which
are not globally isotropic.
One can obtain any $3$-dimensional Riemannian space form as a
quotient $\Sigma/\Gamma$ of a simply connected Riemannian space
form, where $\Gamma$ is a discrete, properly discontinuous,
fixed-point free subgroup of the isometry group $I(\Sigma)$,
(O'Neill 1983, p243 and Boothby 1986, p406, Theorem 6.5). Properly
discontinuous means that for any compact subset $C \subset
\Sigma$, the set $\{\phi \in \Gamma: \; \phi(C) \cap C \neq
\emptyset \}$ is finite. The quotient is guaranteed to be
Hausdorff if the action is properly discontinuous. $\Gamma$ acts
properly discontinuously if and only if $\Gamma$ is a discrete
group, hence there is some redundancy in the definition above.
The quotient $\Sigma/\Gamma$ is a Riemannian manifold if and only
if $\Gamma$ acts freely. The natural way of rendering the quotient
manifold $\Sigma/\Gamma$ a Riemannian manifold ensures that
$\Sigma$ is a Riemannian covering of $\Sigma/\Gamma$, (see O'Neill
1983, p191, for a general version of this where $\Sigma$ is a
semi-Riemannian manifold). The covering map $\eta:\Sigma
\rightarrow \Sigma/\Gamma$ is a local isometry, hence if $\Sigma$
is of constant sectional curvature $k$, then $\Sigma/\Gamma$ will
also be of constant sectional curvature $k$. If $\Sigma$ is simply
connected, then the fundamental group of the quotient manifold
$\Sigma/\Gamma$ will be isomorphic to $\Gamma$. i.e.
$\pi_1(\Sigma/\Gamma) = \Gamma$. Hence, for a non-trivial group
$\Gamma$, the quotient manifold will not be simply connected.
Every space form of constant sectional curvature $k > 0$ is a
quotient $S^3(r)/\Gamma$, every $k=0$ space form is a quotient
$\mathbb{R}^3/\Gamma$, and every $k < 0$ space form is a quotient
$H^3(r)/\Gamma$.
Let $N(\Gamma)$ denote the normalizer of $\Gamma$ in $I(\Sigma)$.
$N(\Gamma)$ is the largest subgroup of $I(\Sigma)$ which contains
$\Gamma$ as a normal subgroup. The isometry group of
$\Sigma/\Gamma$ is $N(\Gamma)/\Gamma$, (O'Neill 1983, p249).
Equivalently, the isometry group of the quotient manifold is the
`centralizer', or `commutant' $Z(\Gamma)$, the subgroup of
$I(\Sigma)$ consisting of elements which commute with all the
elements of $\Gamma$, (Ellis 1971, p11). In general, there is no
reason for $Z(\Gamma)$ to contain $\Gamma$ as a subgroup.
If $\Gamma$ is a discrete group acting freely on a manifold
$\mathcal{M}$, then there is a `fundamental cell' $C \subset
\mathcal{M}$, a closed subset, whose images under $\Gamma$
tessellate the space $\mathcal{M}$. Each orbit $\Gamma x$, for $x
\in \mathcal{M}$, contains either one interior point of $C$, or
two or more boundary points of $C$. A fundamental cell therefore
contains representatives of each orbit of $\Gamma$, and for almost
all orbits, the fundamental cell contains exactly one
representative. Given that a point of $\mathcal{M}/\Gamma$ is an
orbit of $\Gamma$, it follows that one can construct
$\mathcal{M}/\Gamma$ from $C$ by identifying boundary points in
the same orbit, (J.L.Friedman 1991, p543-545).
To say that the subsets $\{\phi(C): \phi \in \Gamma \}$,
tessellate the space $\mathcal{M}$, means that they provide a
covering of $\mathcal{M}$ by isometric closed subsets, no two of
which have common interior points.
In the case of a quotient $\Sigma/\Gamma$ of a simply connected
$3$-dimensional Riemannian space form $\Sigma$, by a discrete
group $\Gamma$ of properly discontinuous, freely acting
isometries, if $\Sigma/\Gamma$ is a compact $3$-manifold, then the
fundamental cell is a polyhedron, (J.L.Friedman 1991, p544). One
can construct such compact quotient manifolds by identifying the
faces of the polyhedron. The best-known example is the way in
which one can obtain the three-torus $\mathbb{T}^3$ by identifying
the opposite faces of a cube.
Although many space forms are not globally isotropic, they are
all, at the very least, locally isotropic. To define local
isotropy, it is necessary to use the concept of a local isometry.
One can define a local isometry of a Riemannian manifold
$(\Sigma,\gamma)$ to be a smooth map $\phi:\Sigma \rightarrow
\Sigma$, such that each differential map $\phi_{*p}:T_p \Sigma
\rightarrow T_{\phi(p)}\Sigma$ is a linear isometry. Equivalently,
the defining characteristic of a local isometry is that each $p
\in \Sigma$ has a neighbourhood $V$ which is mapped by $\phi$ onto
an isometric neighbourhood $\phi(V)$ of $\phi(p)$. Whilst a local
isometry $\phi:\Sigma \rightarrow \Sigma$ need not be a
diffeomorphism of $\Sigma$, it must at the very least be a local
diffeomorphism. It is also worth noting that every isometry must
be a local isometry.
At each point $p \in \Sigma$ of a Riemannian manifold, one can
consider the family of all local isometries of $(\Sigma,\gamma)$
which leave the point $p$ fixed. Each such local isometry maps a
neighbourhood $V$ of $p$ onto an isometric neighbourhood of the
same point $p$. This family of local isometries is the analogue of
the isotropy subgroup at $p$ of the global isometry group. The
differential map of each such local isometry, $\phi_{*p}: T_p
\Sigma \rightarrow T_p \Sigma$, is a linear isometry of the inner
product space $(T_p \Sigma,\langle \; , \; \rangle_\gamma)$.
One defines a Riemannian manifold $(\Sigma,\gamma)$ to be locally
isotropic at a point $p$ if the family of local isometries which
leave $p$ fixed, act transitively upon the unit sphere $S_p \Sigma
\subset T_p \Sigma$. If the local linear isotropy group at $p$
acts transitively upon $S_p \Sigma$, then the local isotropy group
at $p$ must contain $SO(3)$. Naturally, a locally isotropic
Riemannian manifold is defined to be a Riemannian manifold which
is locally isotropic at every point.
Outside the common neighbourhood $U$ of the local isometries in
the local isotropy group at $p$, the set of points which lie at a
fixed spatial distance from $p$, will not, in general, form a
homogeneous surface. If a locally isotropic space has been
obtained as the quotient of a discrete group action, then beyond
the neighbourhood $U$, the set of points which lie at a fixed
distance from $p$, will, in general, have a discrete isometry
group. Inside $U$, the set of points which lie at a fixed spatial
distance from $p$, will still form a homogeneous surface with an
isometry group that contains $SO(3)$.
Beyond the neighbourhood $U$, the orbits of the local isotropy
group of $p$ still coincide with sets of points that lie at a
fixed distance from $p$, but, to reiterate, these orbits are not
homogeneous surfaces. The local isotropy group of $p$, which
contains $SO(3)$, acts transitively upon these surfaces, but it
does not act as a group of isometries upon these surfaces.
Instead, only a discrete subgroup of $SO(3)$ provides the
isometries of these surfaces.
Beyond $U$, the orbits of the local isotropy group action have
preferred directions. If a locally isotropic space has been
obtained as the quotient of a discrete isometry group action, and
if that quotient action is defined by identifying the faces of a
polyhedron, then, beyond a certain distance from each point $p$,
the perpendiculars to the faces of the polyhedron define preferred
directions on the orbits of the local isotropy group action.
Projecting from the hypersurfaces $\Sigma_t$ of a warped product
$I \times_R \Sigma$ onto the $3$-dimensional locally isotropic
Riemannian manifold $\Sigma$, the past light cone $E^-(x)$ of an
arbitrary point $x = (t_0,p)$, passes through the orbits in
$\Sigma$ of the local isotropy group of $p \in \Sigma$ at
ever-greater distances from $p$ the further the light cone reaches
into the past\footnote{The past light cone $E^-(x)$ is the set of
points which can be connected to $x$ by a future directed null
curve.}. Hence, in a locally isotropic warped product, the past
light cone $E^-(x)$ will consist of homogeneous $2$-dimensional
surfaces close to $x$, but beyond a certain spatial distance,
looking beyond a certain time in the past, the constant time
sections of the light cone will consist of non-homogeneous
surfaces, which only have a discrete isometry group.
Observationally, this means that one would only see an isotropic
pattern of light sources up to a certain distance, or up to a
certain `look-back' time, away from the point of observation.
It is easy to see that every globally isotropic Riemannian
manifold must be locally isotropic. However, there are many
locally isotropic Riemannian manifolds which are not globally
isotropic. Whilst every Riemannian space form is locally
isotropic, only a simply connected space form is guaranteed to be
globally isotropic.
Present astronomical data indicates that the spatial universe is
locally isotropic about our location in space. Present data does
not reveal whether the spatial universe is globally isotropic
about our point in space. We have only received light from a
proper subset of the spatial universe because light from more
distant regions has not had time to reach us.
The Copernican Principle declares that the perspective which the
human race has upon the universe is highly typical. Combining this
philosophical principle with the astronomical evidence that the
spatial universe is locally isotropic about our point in space,
one infers that the spatial universe is locally isotropic about
every point in space. One infers that the spatial universe is
representable by a locally isotropic Riemannian manifold. Our
limited astronomical data means that it is unjustified to
stipulate that the spatial universe is globally isotropic.
Neither do our astronomical observations entail global
homogeneity; we only observe local homogeneity, and approximate
local homogeneity at that. One can define a Riemannian manifold
$(\Sigma,\gamma)$ to be locally homogeneous if and only if, for
any pair of points $p,q \in \Sigma$, there is a neighbourhood $V$
of $p$, which is isometric with a neighbourhood $U$ of $q$. There
will be a local isometry $\phi:\Sigma \rightarrow \Sigma$ such
that $\phi(V)=U$. Just as all of the space forms are locally
isotropic, so they are also locally homogeneous. A connected,
globally isotropic Riemannian manifold must be globally
homogeneous, and similarly, a connected locally isotropic
Riemannian manifold must be locally homogeneous, (Wolf 1967,
p381-382).
This generalisation of the conventional FRW models enlarges the
range of possible spatial geometries and topologies of our
universe. The topology of the spatial universe need not be
homeomorphic to either $\mathbb{R}^3$, $S^3$, or $\mathbb{RP}^3$.
Take the $3$-dimensional Euclidean space forms. These are the
complete, connected, flat $3$-dimensional Riemannian manifolds,
each of which is a quotient $\mathbb{R}^3/\Gamma$ of
$3$-dimensional Euclidean space by a discrete group $\Gamma$ of
properly discontinuous, fixed point free isometries.
There are actually $18$ non-homeomorphic $3$-dimensional manifolds
which can be equipped with a complete Riemannian metric tensor of
constant sectional curvature $k=0$. Of the $18$ there are $10$ of
compact topology, and $8$ of non-compact topology. The non-compact
$3$-dimensional Euclidean space forms include, (Wolf 1967,
p112-113):
\begin{enumerate}
\item $\mathbb{R}^1 \times \mathbb{R}^2 \equiv \mathbb{R}^3$
\item $\mathbb{R}^1 \times \text{(Cylinder)}$
\item $\mathbb{R}^1 \times \text{(Torus) i.e. } \mathbb{R}^1 \times \mathbb{T}^2$
\item $\mathbb{R}^1 \times \text{(Moebius band)}$
\item $\mathbb{R}^1 \times \text{(Klein bottle)}$
\end{enumerate}
The second factors in the five cases listed above, exhaust the
$2$-dimensional Euclidean space forms. Because the Moebius band
and the Klein bottle are non-orientable, cases 4. and 5. are
non-orientable $3$-manifolds. The first three cases are, however,
orientable.
Of the $8$ non-compact Euclidean space forms, $4$ are orientable,
and $4$ are non-orientable. Of the $10$ compact Euclidean space
forms, $6$ are orientable, and $4$ are non-orientable.
Notice that $\mathbb{R}^1 \times \text{(Cylinder)} \equiv
\mathbb{R}^1 \times (\mathbb{R}^1 \times S^1) \cong \mathbb{R}^2
\times S^1$. The open disc $B^2$ is homeomorphic with
$\mathbb{R}^2$, hence $\mathbb{R}^2 \times S^1 \cong B^2 \times
S^1$. The $3$-manifold $B^2 \times S^1$ is the interior of a solid
torus. The interior of a solid torus is a possible spatial
topology for a FRW universe. All our astronomical observations, in
conjunction with the Copernican principle, are consistent with the
spatial universe having the shape of a solid ring.
The compact $3$-dimensional Euclidean space forms include the
$3$-dimensional torus $\mathbb{T}^3$. The other nine compact flat
Riemannian manifolds can then be obtained from $\mathbb{T}^3$ as a
quotient $\mathbb{T}^3/\Gamma$, (Wolf 1967, p105). Of the ten
compact flat Riemannian $3$-manifolds, nine can be fibred over the
circle. In seven of these cases, the fibre is a $2$-torus, and in
the other two cases, the fibre is a Klein bottle, (Besse 1987,
p158, 6.20). Alternatively, one can treat two of the ten as circle
bundles over the Klein bottle, and one of these is the trivial
product bundle $S^1 \times (\text{Klein bottle})$, (Besse 1987,
p158, 6.19).
\hfill \break
In the case of the $3$-dimensional Riemannian space forms of
positive curvature $S^3/\Gamma$, the isometry group of $S^3$ is
$SO(4)$, and the quotient group $\Gamma$ must be a discrete
subgroup of $SO(4)$ which acts freely and discontinuously on
$S^3$. These subgroups come in three types (Rey and Luminet 2003,
p52-55; Wolf 1967, p83-87): $\mathbb{Z}_p$ the cyclic rotation
groups of order $p$, for $p \geq 2$; $D_m$ the dihedral groups of
order $2m$, for $m > 2$, the symmetry groups of the regular
$m$-sided polygons; and the symmetry groups of the regular
polyhedra, $T$, $O$ and $I$. There are actually five regular
polyhedra (the `Platonic solids'): the regular tetrahedron (4
faces), the regular hexahedron or cube (6 faces), the regular
octahedron (8 faces), the regular dodecahedron (12 faces), and the
regular icosahedron (20 faces). There are, however, only three
distinct symmetry groups, the tetrahedral group $T$, octahedral
group $O$, and icosahedral group $I$. The hexahedron has the
octahedral symmetry group $O$, and the dodecahedron has the
icosahedral symmetry group $I$. There are also double coverings of
the dihedral and polyhedral groups, denoted as
$D_m^*,T^*,O^*,I^*$.
The \emph{globally} homogeneous $3$-dimensional Riemannian space
forms of positive curvature $S^3/\Gamma$ can be listed as follows,
(Wolf 1967, p89, Corollary 2.7.2):
\begin{enumerate}
\item $S^3$
\item $\mathbb{RP}^3 \cong S^3/\mathbb{Z}_2$
\item $S^3/\mathbb{Z}_p \; \text{for} \; p > 2$
\item $S^3/D_m^* \; \text{for} \; m > 2$
\item $S^3/T^*$
\item $S^3/O^*$
\item $S^3/I^*$
\end{enumerate}
One can also take the quotient of $S^3$ with respect to groups
$\Gamma$ of the form $\mathbb{Z}_u \times D^*_v$, $\mathbb{Z}_u
\times T^*_v$, $\mathbb{Z}_u \times O^*$, or $\mathbb{Z}_u \times
I^*$, for certain values of $u$ and $v$, and where the $T^*_v$ are
subgroups of $T^*$ (Ellis 1971 p13). These spherical space forms
are merely locally homogeneous.
Note that, in contrast with the Euclidean case, there are an
infinite number of distinct spherical space forms because there is
no limit on $p$ or $m$.
The real projective space $\mathbb{RP}^3 \cong S^3/\mathbb{Z}_2$
is globally isotropic and orientable, but not simply connected.
The spaces $S^3/\mathbb{Z}_p$ are referred to as lens spaces,
while the Poincare manifold is homeomorphic with $S^3/I$.
All of the $3$-dimensional spherical Riemannian space forms are of
compact topology.
\hfill \break
In the case of the $3$-dimensional hyperbolic space forms, the
work of Thurston demonstrates that `most' compact and orientable
$3$-manifolds can be equipped with a complete Riemannian metric
tensor of constant negative sectional curvature. This means that
`most' compact, orientable $3$-manifolds can be obtained as a
quotient $H^3/\Gamma$ of hyperbolic $3$-space. The meaning of
`most' in this context involves Dehn surgery, (Besse 1987,
p159-160).
Every compact, orientable $3$-manifold can be obtained from $S^3$
by Dehn surgery along some link $L$. A link in a manifold is
defined to be a finite, disjoint union of simple closed curves $L
= J_1 \cup \cdots \cup J_n$. The first step of Dehn surgery along
a given link $L$, is to specify disjoint tubular neighbourhoods
$N_i$ of each component $J_i$. Each tubular neighbourhood $N_i$ is
homeomorphic with a solid torus $D^2 \times S^1$.
Having identified $n$ disjoint solid tori in $S^3$, some of which
may be knotted, one removes the interior $\text{Int}(N_i)$ of
each. That is, one takes the complement
$$
S^3 - (\text{Int}(N_1) \cup \cdots \cup \text{Int}(N_n)) \,.
$$The boundary surface $\partial N_i$ of the hole left by the
removal of $\text{Int}(N_i)$ is homeomorphic with a
$2$-dimensional torus $\mathbb{T}^2$. Thus, what remains is a
manifold bounded by $n$ disjoint $2$-dimensional tori.
Next, one takes $n$ copies of the solid torus $M_i$, and one sews
each solid torus back into $ S^3 - (\text{Int}(N_1) \cup \cdots
\cup \text{Int}(N_n))$. Each sewing instruction is specified by a
diffeomorphism
$$
\phi_i:\partial M_i \rightarrow \partial N_i \,.
$$ One defines a point $x \in \partial M_i$ on the boundary of the
solid torus $\partial M_i$ to be equivalent to the point
$\phi_i(x) \in \partial N_i$ on the boundary of the hole left by
the removal of $\text{Int}(N_i)$. The result is a new
$3$-manifold.
By varying the choice of link, and by varying the choice of sewing
instructions, one can obtain every compact, orientable
$3$-manifold. Furthermore, one can obtain every compact,
orientable $3$-manifold even if one limits the Dehn surgery to
hyperbolic links. A link $L$ in $S^3$ is defined to be a
hyperbolic link if $S^3 - L$ can be equipped with a complete
Riemannian metric tensor of constant negative sectional curvature.
Given a choice of link $L$, although it is true that some
collections of diffeomorphisms $\{\phi_i:\partial M_i \rightarrow
\partial N_i : \, i=1,...,n \}$ yield the same manifold, there are
still an uncountable infinity of distinct ways in which one can
sew the solid tori back in. Thurston has shown that, in the case
of a hyperbolic link, only a finite number of choices for the
sewing instructions yield a manifold which cannot support a
complete Riemannian metric tensor of constant negative sectional
curvature. It is in this sense that `most' compact, orientable
$3$-manifolds can be equipped with a complete Riemannian metric
tensor of constant negative sectional curvature. Given that all
the hyperbolic $3$-dimensional Riemannian space forms can be
obtained as quotients $H^3/\Gamma$, it follows that `most'
compact, orientable $3$-manifolds can be obtained as such a
quotient.
The cosmological corollary of Thurston's work is that there exists
a vast class of compact, orientable $3$-manifolds, which could
provide the topology of a $k < 0$ FRW universe. Note that there
are compact and non-compact quotients $H^3/\Gamma$.
The rigidity theorem for hyperbolic space-forms states that a
connected oriented $n$-dimensional manifold, compact or
non-compact, of dimension $n \geq 3$, supports at most one
Riemannian metric tensor of constant negative sectional curvature,
\textit{up to homothety}. Unfortunately, this last qualification
has been neglected in some places, and misunderstanding has
resulted amongst cosmologists. Given any $3$-dimensional
Riemannian space form $\Sigma$ of constant negative curvature $k$,
one can change the geometry by an arbitrary scale factor $f$ to
obtain a Riemannian manifold with the same topology, but with
constant negative curvature $k/f^2$. This, after all, is what the
time-dependent scale factor does with a hyperbolic universe in FRW
cosmology!\footnote{A scale factor $f$ used to define the
Riemannian geometry $(\Sigma,\gamma)$ should not be confused with
the dynamic scale factor $R(t)$ of a FRW universe.} Cornish and
Weeks falsely state that if a pair of $3$-dimensional hyperbolic
manifolds are homeomorphic, then they must be isometric, (Cornish
and Weeks 1998, p8). Rey and Luminet, (2003, p57-58), state that,
for $n \geq 3$, a connected oriented $n$-manifold can support at
most one hyperbolic metric, without adding the qualification, `up
to homothety'. They falsely state that if two hyperbolic manifolds
of dimension $n \geq 3$ have isomorphic fundamental groups, then
they must be isometric. There is no reason why the metric on a
manifold obtained as a quotient $H^3/\Gamma$ cannot be changed by
a scale factor from the metric it inherits in the quotient
construction. Alternatively, there is no reason why the canonical
metric on the universal cover $H^3$ cannot be changed by a scale
factor before the quotient is taken. This is equivalent to
introducing, as possible universal covers, the homothetic family
of hyperbolic 3-manifolds of radius $r$, $\{H^3(r): r \in
(0,\infty) \}$. The family of quotients $\{H^3(r)/\Gamma: r \in
(0,\infty) \}$ are also mutually homothetic. Each such quotient
$H^3(r)/\Gamma$ possesses the same fundamental group, namely
$\Gamma$, but the family of geometries $\{H^3(r)/\Gamma: r \in
(0,\infty) \}$ are merely homothetic, not isometric. Luminet and
Roukema (1999, p14) correctly state the rigidity theorem with the
vital qualification of a fixed scale factor on the universal
cover.
The rigidity theorem means that for a fixed $\Gamma$, (up to group
isomorphy), and for a fixed scale factor (`radius of curvature')
$r$ on the universal cover $H^3(r)$, there is a unique
$H^3(r)/\Gamma$ up to isometry. In the case of \emph{compact}
hyperbolic 3-manifolds, for fixed $\Gamma$ and $H^3(r)$, the
volume of the fundamental cell in $H^3(r)$, and therefore the
volume of the quotient $H^3(r)/\Gamma$, is unique. Hence, volumes
can be used to classify compact hyperbolic 3-manifolds as long as
one adds the vital qualification that the volumes are expressed in
units of the `curvature radius', i.e. in units of the scale
factor. If one fixes $\Gamma$, but permits $r$ to vary, then the
volume of the quotient can vary arbitrarily. It is the volume
expressed in curvature radius units which is unique, not the
absolute value of the volume. It has also been shown that the
volume of any compact hyperbolic 3-manifold, in curvature radius
units, is bounded from below by $V_{min} = 0.166 r^3$. This is
more a constraint on the relationship between volume and the scale
factor on the universal cover in the quotient construction, than
an absolute lower limit on volume.
These volume constraints, however, are significant because they
are absent in the case of the compact Euclidean 3-manifolds. Even
with the scale factor of the flat metric on $\mathbb{R}^3$ fixed,
and with $\Gamma$ fixed, one can choose fundamental cells of
arbitrary volume in $\mathbb{R}^3$, hence the volume of
$\mathbb{R}^3/\Gamma$ can be varied arbitrarily even when
expressed in units of curvature radius.
\hfill \break
To reiterate, a connected, locally isotropic Riemannian manifold
is only guaranteed to be locally homogeneous. The only
\emph{globally} homogeneous $3$-dimensional Riemannian space forms
are, (Wolf 1967, p88-89):
\begin{enumerate}
\item $k = 0$ \begin{enumerate}
\item $\mathbb{T}^3$
\item $\mathbb{R}^3$
\item $\mathbb{R}^2 \times S^1$
\item $\mathbb{R} \times \mathbb{T}^2$
\end{enumerate}
\item $k > 0$ \begin{enumerate}
\item $S^3$
\item $\mathbb{RP}^3 \cong S^3/\mathbb{Z}_2$
\item $S^3/\mathbb{Z}_p \; \text{for} \; p > 2$
\item $S^3/D_m^* \; \text{for} \; m > 2$
\item $S^3/T^*$
\item $S^3/O^*$
\item $S^3/I^*$
\end{enumerate}
\item $k < 0$ \begin{enumerate}
\item $H^3$ \end{enumerate}
\end{enumerate}
Clearly, there is no compact, globally homogeneous,
$3$-dimensional hyperbolic space-form because the only globally
homogeneous $3$-dimensional hyperbolic space-form is the
non-compact space $H^3$ (Wolf 1967, p90, Lemma 2.7.4 and p230,
Theorem 7.6.7). Also note that only four of the eighteen
$3$-dimensional Euclidean space-forms are globally homogeneous.
\subsection{The Friedmann equations}
Given a particular Lorentzian metric tensor field $g$, the
Einstein field equation determines the corresponding stress-energy
tensor field $T$. In coordinate-independent notation, the Einstein
field equation, without cosmological constant, can be expressed as
$$
T = 1/(8\pi G)(\text{Ric} - 1/2 \; \text{S} \, g) \,.
$$ $Ric$ denotes the Ricci tensor field determined by $g$,
and $\text{S}$ denotes the curvature scalar field. We have chosen
units here in which $c=1$. In the component terms used by
physicists,
$$
T_{\mu \nu} = 1/(8\pi G)(R_{\mu\nu} - 1/2 \; \text{S} \,
g_{\mu\nu}) \,.
$$
In the Friedmann-Robertson-Walker models, the warped product
metric, $$g = -dt \otimes dt + R(t)^2 \gamma \, ,$$ corresponds to
the stress-energy tensor of a perfect fluid:
$$
T = (\rho + p)dt \otimes dt + pg \,.
$$ $\rho$ and $p$ are both scalar fields on $\mathcal{M}$, constant on each
hypersurface $\Sigma_t = t \times \Sigma$, but time dependent.
$\rho$ is the energy density function, and $p$ is the pressure
function.
The scale factor $R(t)$, energy density $\rho(t)$, and pressure
$p(t)$ of a Friedmann-Robertson-Walker model satisfy the so-called
Friedmann equations, (O'Neill 1983, p346; Kolb and Turner 1990,
p49-50):
$$
\frac{8\pi G}{3}\rho(t) = \left( \frac{R'(t)}{R(t)} \right)^2 +
\frac{k}{R(t)^2} \, ,
$$
$$
-8\pi G p(t) = 2 \frac{R''(t)}{R(t)} + \left( \frac{R'(t)}{R(t)}
\right) ^2 + \frac{k}{R(t)^2} \,.
$$ $k$ is the constant sectional curvature of the $3$-dimensional
Riemannian space form $(\Sigma,\gamma)$; $R'(t) \equiv
\frac{dR(t)}{dt}$; and $R'(t)/R(t)$ is the Hubble parameter
$H(t)$. The sectional curvature of the hypersurface $\Sigma_t$ is
$k/R(t)^2$.
\subsection{The Hubble parameter, redshift, and horizons}
A Riemannian manifold $(\Sigma,\gamma)$ is equipped with a natural
metric space structure $(\Sigma,d)$. In other words, there exists
a non-negative real-valued function $d:\Sigma \times \Sigma
\rightarrow \mathbb{R}$ which is such that
\begin{enumerate}
\item $d(p,q) = d(q,p)$
\item $d(p,q) + d(q,r) \geq d(p,r)$
\item $d(p,q) = 0$ iff $p =q$
\end{enumerate}
The metric tensor $\gamma$ determines the Riemannian distance
$d(p,q)$ between any pair of points $p,q \in \Sigma$. The metric
tensor $\gamma$ defines the length of all curves in the manifold,
and the Riemannian distance is defined as the infimum of the
length of all the piecewise smooth curves between $p$ and $q$. In
the warped product space-time $I \times_R \Sigma$, the spatial
distance between $(t,p)$ and $(t,q)$ is $R(t)d(p,q)$. Hence, if
one projects onto $\Sigma$, one has a time-dependent distance
function on the points of space,
$$
d_t(p,q) = R(t)d(p,q) \,.
$$
Each hypersurface $\Sigma_t$ is a Riemannian manifold
$(\Sigma_t,R(t)^2\gamma)$, and $R(t)d(p,q)$ is the distance
between $(t,p)$ and $(t,q)$ due to the metric space structure
$(\Sigma_t,d_t)$.
The rate of change of the distance between a pair of points in
space is given by
$$\eqalign{
d/dt (d_t(p,q)) &= d/dt (R(t)d(p,q)) \cr &= R'(t)d(p,q) \cr &=
\frac{R'(t)}{R(t)}R(t)d(p,q) \cr &= H(t)R(t)d(p,q) \cr &=
H(t)d_t(p,q)\,. }
$$ The rate of change of distance between a pair of points is
proportional to the spatial separation of those points, and the
constant of proportionality is the Hubble parameter $H(t) =
R'(t)/R(t)$. Galaxies are embedded in space, and the distance
between galaxies increases as a result of the expansion of space,
not as a result of the galaxies moving through space. The rate of
change of the distance between ourselves and a galaxy is referred
to as the recessional velocity $v$ of the galaxy. Where $H_0$
denotes the current value of the Hubble parameter, the Hubble law
is simply $v = H_0d_0$. The recessional velocity corresponds to
the redshift in the spectrum of light received from the galaxy. If
$\lambda_o$ denotes the observed wavelength of light and
$\lambda_e$ denotes the emitted wavelength, the redshift $z$ is
defined as
$$
z = \frac{\lambda_o - \lambda_e}{\lambda_e} =
\frac{\lambda_o}{\lambda_e} - 1 \,.
$$ The distance between ourselves and a galaxy is inferred from a
knowledge of the absolute luminosity of `standard candles' in the
galaxy, and the observed apparent luminosity of those standard
candles.
Cosmology texts often introduce what they call `comoving' spatial
coordinates $(\theta,\phi,r)$. In these coordinates, galaxies
which are not subject to proper motion due to local
inhomogeneities in the distribution of matter, retain the same
spatial coordinates at all times. In effect, comoving spatial
coordinates are merely coordinates upon $\Sigma$ which are lifted
to $I \times \Sigma$ to provide spatial coordinates upon each
hypersurface $\Sigma_t$. The radial coordinate $r$ of a point $q
\in \Sigma$ is chosen to coincide with the Riemannian distance in
the metric space $(\Sigma,d)$ which separates the point at $r=0$
from the point $q$. Hence, assuming the point $p$ lies at the
origin of the comoving coordinate system, the distance between
$(t,p)$ and $(t,q)$ can be expressed in terms of the comoving
coordinate $r(q)$ as $R(t)r(q)$.
If light is emitted from a point $(t_e,p)$ of a warped product
space-time and received at a point $(t_0,q)$, then the integral,
$$
\int^{t_0}_{t_e}\frac{c}{R(t)} \, dt \, ,
$$ where $c$ is the speed of light, expresses the Riemannian
distance $d(p,q)$ in $\Sigma$ travelled by the light between the
point of emission and the point of reception. The present spatial
distance between the point of emission and the point of reception
is
$$
R(t_0)d(p,q) = R(t_0) \int^{t_0}_{t_e}\frac{c}{R(t)} \, dt \,.
$$
The distance which separated the point of emission from the point
of reception at the time the light was emitted is
$$
R(t_e)d(p,q) = R(t_e) \int^{t_0}_{t_e}\frac{c}{R(t)} \, dt \,.
$$
The following integral defines the maximum distance in
$(\Sigma,\gamma)$ from which one can receive light by the present
time $t_0$:
$$
d_{max}(t_0) = \int^{t_0}_{0}\frac{c}{R(t)} \, dt \,.
$$ From this, cosmologists define something called the `particle horizon',
$$
R(t_0) d_{max}(t_0) = R(t_0) \int^{t_0}_{0}\frac{c}{R(t)} \, dt
\,.
$$ We can only receive light from sources which are presently
separated from us by, at most, $R(t_0) d_{max}(t_0)$. In other
words, we can see the past states of luminous objects which are
presently separated from us by a distance of up to $R(t_0)
d_{max}(t_0)$, but we cannot see any further into the spatial
volume of the universe. $R(t_0) \int^{t_0}_{0} c/R(t) \, dt$ is
the present radius of the observable spatial universe.
\subsection{The critical density}
The critical density $\rho_c(t)$ in a FRW model is defined to be
$\rho_c(t) \equiv 3H(t)^2/8\pi G$, and the ratio of the density to
the critical density $\Omega(t) \equiv \rho(t)/\rho_c(t)$ is of
great observational significance. It follows from the Friedmann
equation for the density $\rho$ that one can infer the sign of the
spatial curvature $k$ from $\Omega$. Divide each side of the
Friedmann equation,
$$
\frac{8\pi G}{3}\rho(t) = \left( \frac{R'(t)}{R(t)} \right)^2 +
\frac{k}{R(t)^2} = H(t)^2 + \frac{k}{R(t)^2} \, ,
$$ by $H(t)^2$ to obtain
$$
\frac{8\pi G}{3}\frac{\rho(t)}{H(t)^2} = 1 + \frac{k}{H(t)^2
R(t)^2} \,.
$$ Now, given that $\Omega(t) = \rho(t)/\rho_c(t)$ it follows that
$$
\Omega(t) = \frac{8\pi G}{3}\frac{\rho(t)}{H(t)^2}\, ,
$$ and one obtains
$$
\Omega(t) - 1 = \frac{k}{H(t)^2 R(t)^2} \,.
$$ Assuming $H(t)^2 R(t)^2 \geq 0$ at the present time, the sign of $k$
must match the sign of $\Omega(t) - 1$ at the present time, (Kolb
and Turner 1990, p50).
If one can infer the current value of the Hubble parameter $H_0$
from observations, one can calculate the current value of the
critical density $\rho_c = 3H_0^2/8\pi G$. If one can also infer
the current average density of matter and energy $\rho_0$ from
observations, then one can calculate $\Omega_0 = \rho_0/\rho_c$.
If $\Omega_0 > 1$, then $k > 0$, if $\Omega_0 = 1$, then $k = 0$,
and if $\Omega_0 < 1$, then $k < 0$.
A FRW universe in which the observed density of matter and energy
is found to be greater than the critical density, must have
spatial curvature $k > 0$, and must be of compact spatial
topology. A FRW universe in which the observed density of matter
and energy is found to equal the critical density, could have the
topology of any one of the $18$ flat $3$-dimensional Riemannian
manifolds. A FRW universe in which the observed density of matter
and energy is found to be less than the critical density, must
have spatial curvature $k < 0$, and could have the topology of any
one of the vast family of 3-manifolds which can be equipped with a
metric tensor of constant negative sectional curvature.
Given a FRW universe, if $\Omega_0 > 1$ then the universe will
exist for a finite time, reaching a maximum diameter before
contracting to a future singularity; if $\Omega_0 = 1$, then the
universe will expand forever, but the expansion rate will converge
to zero, $R'(t) \rightarrow 0 \; \text{as} \; t \rightarrow
\infty$; and if $\Omega_0 < 1$ then the universe will expand
forever.
\subsection{Small universes}
It is commonly assumed in observational cosmology that the
observable spatial universe has the topology of a solid ball
$B^3$, and approximately Euclidean geometry. This assumption could
be derived from the further assumption that the spatial universe
is $\mathbb{R}^3$, with curvature $k=0$, but this would amount to
the selection of a very special geometry. In this context, Blau
and Guth point out, (1987, p532), that $k=0$ is a subset of
measure zero on the real line. As it stands, this is a slightly
glib comment. $k=0$ corresponds to all the flat space forms, not
just $\mathbb{R}^3$, and one requires a justification for placing
a measure on the set of space forms which is derived from a
measure on the set of their sectional curvature values.
It is widely believed that the solid ball topology and approximate
Euclidean geometry of our local spatial universe can be derived
from the assumption that the entire spatial universe is very much
larger than the observable spatial universe. However, this
assumption is not necessarily true, and, moreover, even if it is
true, it does not entail solid ball topology and approximate
Euclidean geometry for our local universe.
Suppose that the spatial universe is compact. A compact Riemannian
manifold $(\Sigma,\gamma)$ is a metric space of finite diameter.
(The diameter of a metric space is the supremum of the distances
which can separate pairs of points). If our universe is a FRW
universe in which the Riemannian $3$-manifold $(\Sigma,\gamma)$ is
a compact Riemannian manifold of sufficiently small diameter, then
the horizon distance $d_{max}(t_0) = \int^{t_0}_{0} c/R(t) \, dt$
at the present time $t_0 \sim 10^{10} yrs$ may have exceeded the
diameter of $(\Sigma,\gamma)$, or may be a sufficient fraction of
the diameter that it is invalid to assume the observable spatial
universe has the topology of a solid ball $B^3$. Thus, even if one
were to accept that the observable spatial universe has almost no
spatial curvature, it would not follow that the observable spatial
universe has the topology of a solid ball.
Given $diam(\Sigma_t,\gamma_t) = R(t) \, diam(\Sigma,\gamma)$,
$\int^{t_0}_{0} c/R(t) \, dt \geq diam(\Sigma,\gamma)$ if and only
if $R(t_0) \int^{t_0}_{0} c/R(t) \, dt \geq
diam(\Sigma_{t_0},\gamma_{t_0}) $. If $d_{max}(t_0) \geq
diam(\Sigma,\gamma)$, the horizon would have disappeared, and we
would actually be able to see the entire spatial universe at the
present time. No point of the spatial universe could be separated
from us by a distance greater than $diam(\Sigma,\gamma)$, so if
$d_{max}(t_0) \geq diam(\Sigma,\gamma)$, then we would have
already received light from all parts of the spatial universe.
Individual galaxies and clusters of galaxies could produce
multiple images upon our celestial sphere without the occurrence
of gravitational lensing. Light emitted from opposite sides of a
galaxy could form images in opposite directions upon our celestial
sphere. Light emitted in different directions from a galaxy might
travel different distances before reaching us, and would therefore
produce images of different brightness. Furthermore, the light
which travelled the shorter distance would provide an image of the
galaxy as it appeared at a more recent stage of its evolution.
Light emitted by a galaxy in one direction could circumnavigate
the universe on multiple occasions and produce multiple `ghost
images' upon our celestial sphere. If $\Sigma$ were
non-orientable, light which had circumnavigated the universe an
odd number of times would produce a mirror image from light which
had circumnavigated the universe an even number of times.
One can define a compact FRW universe to be `small' if the size of
the horizon exceeds the size of the spatial universe,
$d_{max}(t_0) \geq diam(\Sigma,\gamma)$. In such universes, the
global spatial topology and geometry can have locally observable
consequences. Different compact spatial topologies and geometries
would produce different patterns of multiple and ghost images upon
the celestial sphere. In addition, a small compact universe would
produce patterns of paired circles in the Cosmic Microwave
Background Radiation (CMBR), (Cornish, Spergel and Starkman 1998).
Luminet and Roukema (1999, p15) point out that those compact
hyperbolic 3-manifolds with the smallest sizes, in curvature
radius units, are the most interesting in terms of cosmological
observational effects. This requires some explanation and
clarification. The size of a compact hyperbolic spatial universe
is not constrained by the topology of the compact hyperbolic
3-manifold. The size of a compact hyperbolic spatial universe can
vary arbitrarily. For a fixed compact hyperbolic 3-manifold, its
size must indeed be a fixed multiple of its `curvature radius',
(and powers thereof), but its curvature radius can vary
arbitrarily as a function of time. Moreover, there are two factors
to the curvature radius of the spatial universe: there is the
curvature radius $r$ used to define the scale factor on the
universal cover of the quotient construction that obtains the
spatial geometry $(\Sigma,\gamma)$; and there is the
time-dependent scale factor $R(t)$. Let $(\Sigma,\gamma) =
H^3(r)/\Gamma$. Then
$$
Vol \; (\Sigma,\gamma) = (Vol \; H^3/\Gamma)r^3
$$ and
$$\eqalign{
Vol \; (\Sigma_t,\gamma_t) &= (Vol \; (\Sigma,\gamma)) R(t)^3 \cr
&= (Vol \; H^3/\Gamma)r^3 R(t)^3 \, .}
$$ Hence, the size of a compact hyperbolic spatial universe is not a
fixed multiple of the time-dependent scale factor $R(t)$ and its
powers. Having fixed a compact hyperbolic topology for the spatial
universe, and having fixed a profile for the time-dependent scale
factor $R(t)$, one can independently vary the radius of curvature
$r$ in the universal cover $H^3(r)$ used to obtain that compact
hyperbolic topology. By so doing, one can arbitrarily vary the
volume of the spatial universe at the present time, $Vol \;
(\Sigma_{t_0},\gamma_{t_0})$, without changing either the topology
of the spatial universe or the profile of the time-dependent scale
factor $R(t)$. Whatever the compact hyperbolic topology chosen for
the spatial universe, one can judiciously choose a value of $r$
which enables the present spatial universe to fit inside the
present horizon. Conversely, whatever the compact hyperbolic
topology chosen for the spatial universe, one can always choose a
value of $r$ which makes the present spatial universe much larger
than the present horizon.
Suppose that one selects a compact hyperbolic topology which is
such that if one chooses $r=1$, then the current size of the
spatial universe will be much larger than the current size of the
particle horizon,
$$diam(\Sigma_{t_0},\gamma_{t_0})\gg R(t_0)\int^{t_0}_{0} \frac{c}{R(t)} \,
dt.$$ To remedy this situation, one can choose a very small value
for $r$ so that $1/r \gg 1$. Without changing the spatial
topology, this reduces the current size of the spatial universe to
$$
diam(\Sigma'_{t_0},\gamma'_{t_0}) = r \cdot
diam(\Sigma_{t_0},\gamma_{t_0}) \ll
diam(\Sigma_{t_0},\gamma_{t_0}) \,.
$$ Thus, a judicious choice of $r$ enables the present spatial universe
to fit inside the particle horizon.
However, varying $r$ does vary the current spatial curvature
$k(t_0) = k/R(t_0)^2$, which is constrained by observation. If one
begins with $r=1$ and $k=-1$, then re-setting $r \ll 1$ entails
changing to $k \ll -1$, and a change to a much more negative value
of $k(t_0) = k/R(t_0)^2$.
In most cosmology texts, the spatial curvature $k$ is set to $+1$,
$-1$ or $0$. Assuming $k \neq 0$, if one chooses a space form
$(\Sigma,\gamma)$ in which $|k| \neq 1$, i.e. if one chooses a
radius of curvature $r \neq 1$ on the universal cover of the space
form, then to re-set $k$ without changing the physical model, one
must re-set the time-dependent scale factor $R(t)$ so that it
incorporates the scale factor $r$. One re-sets the scale factor to
$$
R(t)_{new} = \frac{R(t)_{old}}{\sqrt{|k|}} = R(t)_{old}\, r \, .
$$ Note that $1/r = \sqrt{|k|}$.
Re-setting $k$ and $R(t)$ enables one to express the
time-dependence of the spatial curvature as
$$
k(t) = \frac{\pm 1}{R(t)_{new}^2} = \frac{\pm 1}{R(t)^2_{old}\cdot
r^2 }= \frac{k}{R(t)^2_{old}} \, .
$$
Assuming that $k \neq 0$, if observations suggest that the current
spatial curvature $k(t_0) = \pm 1/R(t_0)^2_{new}$ is very close to
zero, then it entails that $R(t_0)_{new}$ is very large. Hence,
unless the size of the compact hyperbolic manifold,
$diam(\Sigma,\gamma)$, is very small in curvature radius units,
then the current size of the spatial universe,
$diam(\Sigma_{t_0},\gamma_{t_0})$, will be much greater than the
current size of the particle horizon. Hence, those compact
hyperbolic 3-manifolds with the smallest sizes, in curvature
radius units, are the most interesting in terms of cosmological
observational effects.
\subsection{Inflation}
Cosmologists have postulated that the early universe underwent a
period of exponential, acceleratory expansion called `inflation'.
If inflation did take place, it means that the horizon distance
$d_{max}(t_0) = \int^{t_0}_{0} c/R(t) \, dt$ is much smaller than
it would have been otherwise. In the case of a compact spatial
universe $(\Sigma,\gamma)$, inflation makes the size of the
observable universe a smaller fraction of the size of the entire
universe than would otherwise have been the case. Given that
$R(t_0) \int^{t_0}_{0} c/R(t) \, dt$ is the present radius of the
observable spatial universe, and given that $R(t_0)\,
diam(\Sigma,\gamma)$ is the diameter of the present spatial
universe in the case of compact $(\Sigma,\gamma)$, the ratio
$$
\frac{R(t_0) \int^{t_0}_{0} c/R(t) \, dt}{R(t_0)\,
diam(\Sigma,\gamma)} = \frac{\int^{t_0}_{0} c/R(t) \,
dt}{diam(\Sigma,\gamma)}
$$ gives the present size of the observable spatial universe as a fraction
of the present size of the entire spatial universe. Clearly,
inflation reduces the value of the integral $\int^{t_0}_{0} c/R(t)
\, dt$, and therefore makes the size of the observable universe a
smaller fraction of the size of the entire universe than would
otherwise have been the case.
The advocates of inflation assert that in a universe which has
undergone inflation, the observable spatial universe must be very
much smaller than the entire spatial universe. This does not
follow from the last proposition, and is not necessarily the case.
If the present horizon distance $d_{max}(t_0) = \int^{t_0}_{0}
c/R(t) \, dt$ is reduced, it merely entails a decrease in the
diameter of the compact $3$-manifolds whose global topology and
geometry could have observable consequences at the present time.
Inflation does not entail that the observable spatial universe has
the topology of a solid ball.
It is widely asserted that if the early universe underwent a
period of inflationary expansion, which drove the time-dependent
scale factor to very high values, then the spatial curvature
$k/R(t_0)^2$ of the present universe must be very close to zero
even if the spatial universe is spherical or hyperbolic. Given
that
$$\Omega(t) = 1 + \frac{k}{H(t)^2 R(t)^2}\, ,$$ it is also
asserted that inflation produces a universe in which $\Omega$ is
very close to unity. These assertions rest upon the tacit
assumption that the sectional curvature $k$ of the Riemannian
manifold $(\Sigma,\gamma)$ in the warped product is very small. No
matter how large the time-dependent scale factor is, either as a
result of inflationary expansion or deceleratory FRW expansion,
the absolute value of $k$ can be chosen to be sufficiently large
that it cancels out the size of $R(t_0)^2$. With a judicious
choice of $k \gg 1$, a spherical universe which is 14 billion
years old, and which underwent inflation, could have spatial
curvature $k/R(t_0)^2 \approx 1$. Similarly, with a judicious
choice of $k \ll -1$, a hyperbolic universe which is 14 billion
years old, and which underwent inflation, could have spatial
curvature $k/R(t_0)^2 \approx -1$. (Whilst the issue is the
proximity of $k/R(t_0)^2$ to zero, I have chosen $| k/R(t_0)^2 |
\approx 1$ to represent a significant amount of spatial
curvature).
The time profile of the spatial curvature, $k(t) = k/R(t)^2 =
Sgn(k)/(R(t) \cdot r)^2 $, is determined by two independent
inputs: the sectional curvature $k$ of $(\Sigma,\gamma)$, and the
profile of the time-dependent scale factor, $R(t)$. The sectional
curvature $k$ is empirically meaningful: it is the value of $k(t)$
at the time that $R(t)=1$. If the profile of the time-dependent
scale factor, $R(t)$, is not fixed, then $k$ is not the value of
$k(t)$ at a fixed time in this family of models. The time at which
$R(t) =1$ varies from one model to another, depending upon the
functional expression chosen for $R(t)$. If $R(t)$ grows faster,
as it clearly does when $R(t)$ has the form of an exponential
function, then $R(t) =1$ at a much earlier time, and $k$ is the
value of $k(t)$ at a much earlier time. Having fixed a choice of
$R(t)$, one can independently vary $k$ to obtain an empirically
distinct family of models that share a time-dependent scale factor
with the same profile. Each such model has a different $k(t)$ time
profile. No matter how steep the chosen profile for $R(t)$, one
can find a family of models in which $k$ is sufficiently large
(or, equivalently, in which $r$ is sufficiently small), that the
present value of spatial curvature, $k(t_0) = k/R(t_0)^2 = \pm 1/
(R(t_0) \cdot r)^2$, remains significant.
For a spherical universe, whatever value inflation drives
$R(t_0)^2$ to, there is a value of sectional curvature
$0<k'<\infty$, such that for any $k \in [k',\infty)$, $k/R(t_0)^2
\geq 1$. There is only a finite range of values $(0,k')$ for which
$k/R(t_0)^2 < 1$, but an infinite range for which $k/R(t_0)^2 \geq
1$. Similarly, for a hyperbolic universe, there is a value of
sectional curvature $0>k''>-\infty$, such that for any $k \in
[k'',-\infty)$, $k/R(t_0)^2 \leq -1$. There is only a finite range
of values $(0,k'')$ for which $k/R(t_0)^2 > -1$, but an infinite
range for which $k/R(t_0)^2 \leq -1$. Whatever value inflation
drives $R(t_0)^2$ to, there is only a finite range of sectional
curvature values consistent with negligible spatial curvature in
the present day, but an infinite range which would produce a
significant amount of curvature in the present day.
Thus, inflation does not entail that a universe with an age of the
order $10^{10}yrs$ must have spatial curvature very close to zero.
This is true irrespective of whether the spatial universe is
compact or non-compact. Hence, contrary to the opinion held by
most cosmologists, the fact that our observable spatial universe
has a spatial curvature very close to zero, cannot be explained by
postulating a period of inflationary expansion alone. Even if the
entire spatial universe is very much larger than the observable
spatial universe, it does not entail that the observable spatial
universe must be approximately Euclidean. To explain the observed
`flatness' of our local spatial universe without preselecting
$\mathbb{R}^3$, one must either conjoin the postulate of inflation
with the postulate that $| k | \approx 1$, or the inflation-driven
growth in $R(t)$ must be commensurate with the magnitude of $k$.
Inflation is often presented as a solution to the flatness
`problem' and the horizon `problem', the latter of which will be
dealt with in the more extensive treatment of inflation contained
in part $\textrm{II}$ of this paper. The argument above is to the
effect that inflation does not solve the flatness `problem'.
\subsection{Dark energy}
Observations in the last decade using Type $\textrm{I}a$
supernovae as `standard candles' appear to indicate that the
expansion of our universe is accelerating. Type $\textrm{I}a$
supernovae at redshifts of $z \approx 0.5$ appear to be fainter
than would otherwise be the case. A universe which is currently
accelerating can be given an age estimate which is consistent with
the ages of the oldest stars in globular clusters. Under the
assumption of deceleratory expansion, the current value of the
Hubble parameter tended to yield an estimated age for the universe
which was less than the ages of the oldest stars in the universe.
The acceleratory expansion can be explained within general
relativity by an additional component to the energy density and
pressure in the Friedmann equations. This additional component,
referred to as `dark energy', must be such that it produces a
repulsive gravitational effect. The structure of a Lorentzian
manifold itself is not sufficient to guarantee that gravity will
be attractive. Given a timelike vector $Z \in T_p \mathcal{M}$,
one has a tidal force operator $F_Z: Z^\bot \rightarrow Z^\bot$
defined on each $W \in Z^\bot$ by
$$F_Z(W) = R(\,\cdot \,,Z,W,Z)\, ,$$ where $R$ is the Riemann curvature tensor.
$F_Z(W)$ is physically interpreted as the tidal force due to
gravity, acting on a particle in spatial direction $W$, for an
instantaneous observer $Z$. Hence, $\langle F_Z(W), W \rangle \leq
0$ means that gravity is attractive in direction $W$. Now, the
trace of the tidal force operator,
$$
\text{tr}\; F_Z = \sum_{i=1}^{3} \langle F_Z(e_i), e_i \rangle =
-Ric(Z,Z)\, ,
$$ where $\{e_1,e_2,e_3\}$ is an orthonormal basis of the subspace
$Z^\bot$, gives the sum of the tidal force over three orthogonal
directions in $Z^\bot$. Hence, the `timelike convergence
condition' that $Ric(Z,Z) \geq 0$, for all timelike vectors $Z$,
stipulates that the net effect of gravity is attractive.
A component in the Friedmann equations will have a repulsive
gravitational effect if it possesses a negative pressure $p <
-\frac{1}{3}\rho$. Thus, in terms of an `equation of state', which
expresses pressure as a function of energy density $p = f(\rho)$,
dark energy is such that $p = w\rho$ with $w< - \frac{1}{3}$.
A non-zero and positive cosmological constant provides a special
case of dark energy. In the general case of a non-zero
cosmological constant $\Lambda$, the Einstein field equations
become
$$
8\pi G \, T_{\mu \nu} - \Lambda g_{\mu\nu} = R_{\mu\nu} - 1/2 \;
\text{S} \, g_{\mu\nu}\, ,
$$ and the Friedmann equations become, (Heller 1992,
p101):
$$
\frac{8\pi G}{3}\rho(t) = \left( \frac{R'(t)}{R(t)} \right)^2 +
\frac{k}{R(t)^2} - \frac{1}{3}\Lambda \, ,
$$
$$
-8\pi G p(t) = 2 \frac{R''(t)}{R(t)} + \left( \frac{R'(t)}{R(t)}
\right) ^2 + \frac{k}{R(t)^2} - \Lambda \,.
$$
The presence of a non-zero cosmological constant is equivalent to
an additional component of the energy density and pressure in the
Friedmann equations without cosmological constant. Let $\rho_m$
denote the energy density due to matter alone. With the
cosmological constant, the Friedmann equation for energy density
can be written as
$$
\frac{8\pi G}{3}\rho_m(t) + \frac{1}{3}\Lambda = \left(
\frac{R'(t)}{R(t)} \right)^2 + \frac{k}{R(t)^2} \,.
$$ Setting $\rho_\Lambda = \Lambda/8\pi G$
one can further re-write this equation as
$$
\frac{8\pi G}{3}(\rho_m(t) + \rho_\Lambda) = \left(
\frac{R'(t)}{R(t)} \right)^2 + \frac{k}{R(t)^2} \,.
$$ Hence, the presence of a non-zero cosmological constant corresponds to
an additional, \emph{time-independent} component $\rho_\Lambda$ to
the energy density. One re-defines $\Omega$ as
$$
\Omega = \frac{\rho_m + \rho_\Lambda}{\rho_c} = \frac{\rho_m +
\rho_\Lambda}{3H^2/8\pi G} = \frac{\rho_m}{3H^2/8\pi
G}+\frac{\rho_\Lambda}{3H^2/8\pi G} \equiv \Omega_m +
\Omega_\Lambda \,.
$$
With the cosmological constant, the Friedmann equation for
pressure can be written as
$$
-8\pi G p(t) + \Lambda = 2 \frac{R''(t)}{R(t)} + \left(
\frac{R'(t)}{R(t)} \right) ^2 + \frac{k}{R(t)^2} \,.
$$ Setting $p_\Lambda = -\Lambda/8\pi G$, so $\Lambda = - p_\Lambda 8\pi G$,
one can further re-write this equation as
$$
-8\pi G (p(t) - p_\Lambda) = 2 \frac{R''(t)}{R(t)} + \left(
\frac{R'(t)}{R(t)} \right) ^2 + \frac{k}{R(t)^2} \,.
$$
A positive cosmological constant behaves like an repulsive
component to gravity because $p_\Lambda = -\rho_\Lambda$. In terms
of an equation of state $p_\Lambda = w\rho_\Lambda$, a
cosmological constant has $w=-1$.
In the case of a non-zero cosmological constant, the dark energy
can be interpreted as a property of space-time, rather than a
property of some exotic field in space-time. If the additional
component to the energy density in the Friedmann equations without
cosmological constant represents some exotic field, then this
additional energy density can be time-dependent and can possess a
time-dependent equation of state. In contrast, a non-zero
cosmological constant can only correspond to a constant energy
density and a constant equation of state. The latest astronomical
evidence, (Adam G.Riess \textit{et al} 2004), indicates that the
dark energy is a non-zero cosmological constant.
If the cosmological constant is non-zero, one can no longer infer
the sign of the spatial curvature and the long-term dynamical
behaviour of a FRW universe from $\Omega_m$, but because the
Friedmann equation is unchanged when the cosmological constant is
incorporated into the total $\rho$, they can still be inferred
from $\Omega$, with $\Omega = \Omega_m + \Omega_\Lambda$ in this
case.
The current observational evidence leads cosmologists to believe
that $\Omega$ is approximately unity, with $\Omega_m \approx 0.3$
and $\Omega_\Lambda \approx 0.7$. It might, however, be noted that
circa 1967, observations appeared to indicate a surplus number of
quasars at redshift $z=2$, (Heller 1992, p102). These observations
were explained by postulating a Lemaitre model, a FRW model with a
positive cosmological constant, in which the expansion of the
universe is punctuated by an almost static period, a type of
plateau in the scale factor when it is displayed as a function of
time. The quasar observations transpired to be a selection effect,
and the community of cosmologists reverted to their belief that
$\Lambda = 0$. Perhaps in a similar vein, it has been suggested
that the dimming of Type $\textrm{I}a$ supernovae at redshifts of
$z \approx 0.5$ could be due to screening from `grey dust', or due
to intrinsically fainter Type $\textrm{I}a$ supernovae at
redshifts of $z \approx 0.5$.
\section{Spatially homogeneous cosmologies}
As a continuation to the rationale of the opening section, the
philosophical purpose of this section is to explain and emphasise
the immense variety of spatially homogeneous cosmological models
which are consistent with astronomical observation, or which serve
to highlight the variety of possible universes similar to our own.
This section will also clarify the Bianchi classification, and the
relationship between the spatially homogeneous models and the FRW
models.
\hfill \break
The spatially homogeneous class of cosmological models are usually
presented as a generalisation of the Friedmann-Robertson-Walker
cosmological models. The generalisation is said to be obtained by
dropping the requirement of spatial isotropy, but retaining the
requirement of spatial homogeneity. The FRW models are considered
to be special cases of the class of spatially homogeneous models.
The topology of a typical spatially homogeneous cosmological model
is a product $I \times \Sigma$ of an open interval $I \subset
\mathbb{R}^1$ with a connected $3$-dimensional manifold $\Sigma$.
The $4$-dimensional manifold $\mathcal{M} = I \times \Sigma$ is
ascribed a Lorentzian metric tensor which induces a homogeneous
Riemannian metric $\gamma_t$ on each hypersurface $\Sigma_t = t
\times \Sigma$. Thus, each pair $(\Sigma_t,\gamma_t)$ is a
homogeneous $3$-dimensional Riemannian manifold. A spatially
homogeneous cosmological model is a Lorentzian manifold
$\mathcal{M}$ in which the orbits of the isometry group
$I(\mathcal{M})$ consist of such a one-parameter family of
spacelike hypersurfaces.
As with the FRW models, there is a spatial topology $\Sigma$
associated with each spatially homogeneous cosmological model.
However, the spatial geometry of a spatially homogeneous model can
vary in a more complex manner than the single scale factor
variation of a FRW model. In other words, there is no need for a
spatially homogeneous model to be a warped product. The class of
cosmological models obtained by taking warped products $I \times
_R \Sigma$ in which $\Sigma$ is a (globally) homogeneous
$3$-dimensional Riemannian manifold, only constitutes a proper
subset of the entire class of spatially homogeneous cosmological
models.
The time variation of the spatial geometry in a spatially
homogeneous cosmology is, in general, expressed by a matrix of
scale factors, rather than a single scale factor. Whilst a warped
product geometry can be expressed as $-dt \otimes dt + R(t)^2
\gamma$, in a general spatially homogeneous cosmology, each
component of spatial geometry can be subject to time variation,
hence the metric can be expressed as $-dt \otimes dt +
\gamma_{ab}(t)\omega^a(t) \otimes \omega^b(t)$, where
$\omega^a(t), a=1,2,3$ are one-forms on $\Sigma_t$ invariant under
the action of the isometry group $I(\Sigma_t)$.
Now consider a connected $3$-dimensional homogeneous Riemannian
manifold $(\Sigma,\gamma)$. Associated with $(\Sigma,\gamma)$ are
the isometry group $I(\Sigma)$ and the isotropy subgroup $H$. The
isometry group can be of dimension 6,4, or 3. Moreover, it is true
that
$$
\text{dim}\; \Sigma = \text{dim}\; I(\Sigma) - \text{dim} \; H \,.
$$ Thus, when $I(\Sigma)$ is of dimension 6, $H$ will be of
dimension 3; when $I(\Sigma)$ is of dimension 4, $H$ will be of
dimension 1; and when $I(\Sigma)$ is of dimension 3, the isotropy
group $H$ will be trivial.
In any dimension, it can be shown that every homogeneous
Riemannian manifold $(\Sigma, \gamma)$ is diffeomorphic to some
Lie group. In particular, every homogeneous $3$-dimensional
Riemannian manifold $(\Sigma, \gamma)$ is diffeomorphic to some
$3$-dimensional Lie group. The $3$-dimensional Riemannian manifold
$\Sigma$ is diffeomorphic with the quotient Lie group
$I(\Sigma)/H$, the quotient of the isometry group by the isotropy
subgroup.
If one has a homogeneous $3$-dimensional Riemannian manifold
$(\Sigma,\gamma)$ which has a 3-dimensional isometry group
$I(\Sigma)$, then $I(\Sigma)/H \cong I(\Sigma)$, and the
Riemannian manifold is diffeomorphic with its own isometry group.
In the event that a homogeneous $3$-dimensional Riemannian
manifold $(\Sigma,\gamma)$ has an isometry group $I(\Sigma)$ of
dimension 4 or 6, the quotient $I(\Sigma)/H$ will be distinct from
$I(\Sigma)$, and the Riemannian manifold will not be diffeomorphic
with its own isometry group.
By the definition of homogeneity, the isometry group $I(\Sigma)$
of a homogeneous Riemannian manifold $(\Sigma,\gamma)$ must act
transitively. However, when the isometry group $I(\Sigma)$ is of
dimension 3, the action is simply transitive, and when $I(\Sigma)$
is of dimension 4 or 6, the action is multiply transitive.
Not only is every homogeneous Riemannian manifold $(\Sigma,
\gamma)$ diffeomorphic to some Lie group, but conversely, any Lie
group can be equipped with a metric which renders it a homogeneous
Riemannian manifold. Thus, the topologies of all the
$3$-dimensional Lie groups equal the possible topologies for a
$3$-dimensional homogeneous Riemannian manifold. A list of all the
$3$-dimensional Lie groups will exhaust the possible topologies
for a $3$-dimensional homogeneous Riemannian manifold. However,
this list of topologies is repetitious; although every
$3$-dimensional Lie group will provide the topology for a
$3$-dimensional homogeneous Riemannian manifold, two distinct Lie
groups can possess the same topology.
To obtain a classification of all the connected $3$-dimensional
Lie groups, the first step is to obtain a classification of all
the simply connected $3$-dimensional Lie groups. Simply connected
Lie groups are in a one-to-one correspondence with Lie algebras,
and there is a classification of the isomorphism classes of
$3$-dimensional Lie algebras called the Bianchi classification.
Hence, the Bianchi classification provides a classification of the
simply connected $3$-dimensional Lie groups. The Bianchi
classification of all the $3$-dimensional Lie algebras only
provides a coarse-grained classification of the connected
$3$-dimensional Lie groups because many Lie groups can possess the
same Lie algebra. However, all the Lie groups which share the same
Lie algebra will be `locally isomorphic', and will have a common
simply connected, universal covering Lie group. Each connected
$3$-dimensional Lie group $G$ is obtained from its universal cover
$\widetilde{G}$ as the quotient $\widetilde{G}/N$ of its universal
cover with respect to a discrete, normal subgroup $N$. If
$\widetilde{G}$ has Lie algebra $\mathfrak{g}$, then the quotient
$\widetilde{G}/N$ will also have Lie algebra $\mathfrak{g}$. A
discrete normal subgroup of a connected Lie group is contained in
the centre of the Lie group, hence $N$ is a central, discrete,
normal subgroup.
Once the simply connected $3$-dimensional Lie groups have been
classified, the second step is to classify all the discrete normal
subgroups of each simply connected $3$-dimensional Lie group, up
to conjugacy. Step two yields a family of Lie groups
$\widetilde{G}/N_i,\widetilde{G}/N_j,...$ which share the same
Bianchi type, but which are distinct, possibly non-diffeomorphic
Lie groups. These two steps together provide a classification of
all the connected $3$-dimensional Lie groups.
To reiterate, a list of all the connected $3$-dimensional Lie
groups provides an exhaustive, but repetitious list of all the
possible homogeneous spatial topologies. Note that a list which
exhausts all the possible homogeneous spatial topologies, does not
provide a list of all the possible homogeneous spatial geometries.
A $3$-dimensional Lie group can support more than one homogeneous
metric.
\hfill \break
Let us turn, then, to the Bianchi classification of the
isomorphism classes of $3$-dimensional Lie algebras. Given a
choice of basis $\{e_1,e_2,e_3\}$ for a $3$-dimensional Lie
algebra, the structure constants $C_{ij}^k$ are defined to be such
that $[e_i,e_j] = C_{ij}^k e_k$. The Bianchi classification is
based upon the fact that Lie algebras can be characterised in
terms of their structure constants $C_{ij}^k$, and the fact that
for a $3$-dimensional Lie algebra, the structure constants can be
expressed as
$$
C_{ij}^k = \epsilon_{ijl}B^{lk} + \delta^k_j a_i - \delta^k_i a_j
\, ,
$$ where $B$ is a symmetric $3 \times 3$ matrix, and $\textbf{a}$ is a $1
\times 3$ column vector, (Dubrovin \textit{et al} 1992, Part I,
\S24.5, p230). The Jacobi identity which constrains the structure
constants of a Lie algebra entails that
$$
B^{ij}a_j = 0 \,.
$$
Although the structure constants of a Lie algebra are
basis-dependent, the classification of $3$-dimensional Lie
algebras is basis-independent. Hence, the classification uses the
fact that one can choose a basis in which $B$ is a diagonal matrix
with $B^{ii} = \pm 1, 0$ for $i =1,2,3$, and $\textbf{a} =
(a,0,0)$. In this basis, the structure constants are such that
$$\eqalign{
[e_1,e_2] &= a e_2 + B^{33}e_3 \cr [e_2,e_3] &= B^{11}e_1 \cr
[e_3,e_1] &= B^{22}e_2 - a e_3 \,.}
$$
With this choice of basis, it also follows that $B^{11}a = 0$,
hence either $B^{11}$ or $a$ is zero. The Bianchi types, denoted
by Roman numerals, are duly defined in Table \ref{Bianchi}.
\begin{table}[h]
\caption{Bianchi classification of 3-dimensional Lie algebras}
\label{Bianchi}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\noalign{\hrule}
Type & a & $B^{11}$ & $B^{22}$ & $B^{33}$ \\ \hline
$\textrm{I}$ & 0 & 0 & 0 & 0 \\
$\textrm{II}$ & 0 & 1 & 0 & 0 \\
$\textrm{VI}_0$ & 0 & 1 & -1 & 0 \\
$\textrm{VII}_0$ & 0 & 1 & 1 & 0 \\
$\textrm{VIII}$ & 0 & 1 & 1 & -1 \\
$\textrm{IX}$ & 0 & 1 & 1 & 1 \\
$\textrm{V}$ & 1 & 0 & 0 & 0 \\
$\textrm{IV}$ & 1 & 0 & 0 & 1 \\
$\textrm{VI}_h \; (h < 0) \; a \neq 1$ & $\sqrt{-h}$ & 0 & 1 & -1 \\
$\textrm{III}$ & 1 & 0 & 1 & -1 \\
$\textrm{VII}_h \; (h > 0)$ & $\sqrt{h}$ & 0 & 1 & 1 \\ \hline
\end{tabular}
\end{center}
\end{table}
Note that $\textrm{III} = \textrm{VI}_{-1}$ if we remove the
restriction that $a \neq 1$ for type $\textrm{VI}_h$.
\hfill \break
Before proceeding further, some salient definitions concerning Lie
algebras are required. Given a Lie algebra $\mathfrak{g}$, one can
inductively define the \emph{lower central series} of subalgebras
$\mathscr{D}_k\mathfrak{g}$ by
$$
\mathscr{D}_1\mathfrak{g} = [\mathfrak{g},\mathfrak{g}], \; \;
\mathscr{D}_k\mathfrak{g} = [\mathfrak{g},
\mathscr{D}_{k-1}\mathfrak{g}] \,.
$$ A Lie algebra is defined to be nilpotent if $\mathscr{D}_k\mathfrak{g}=
0$ for some $k$.
Secondly, one can inductively define the \emph{derived series} of
subalgebras $\mathscr{D}^k\mathfrak{g}$ by
$$
\mathscr{D}^1\mathfrak{g} = [\mathfrak{g},\mathfrak{g}], \; \;
\mathscr{D}^k\mathfrak{g} = [\mathscr{D}^{k-1}\mathfrak{g},
\mathscr{D}^{k-1}\mathfrak{g}] \,.
$$ A Lie algebra is defined to be solvable if $\mathscr{D}^k\mathfrak{g}=
0$ for some $k$.
An ideal in a Lie algebra is a Lie subalgebra $\mathfrak{h}
\subset \mathfrak{g}$ which is such that $[X , Y ] \in
\mathfrak{h}$ for all $X \in \mathfrak{h}$, $Y \in \mathfrak{g}$.
A Lie algebra can be defined to be semi-simple if it has no
nonzero solvable ideals. (Fulton and Harris 1991, p122-123).
Under the Bianchi classification, there are six `Type A' Lie
algebras: $\textrm{I}$, $\textrm{II}$, $\textrm{VI}_0$,
$\textrm{VII}_0$, $\textrm{VIII}$, and $\textrm{IX}$. These are
the Lie algebras of the six unimodular $3$-dimensional connected
Lie groups. As Lie algebras, they are trace-free. All the other
Lie algebras are `Type B'.
Bianchi types $\textrm{VIII}$ and $\textrm{IX}$ are the only
semi-simple real $3$-dimensional Lie algebras. All the other
Bianchi types are solvable Lie algebras.\footnote{The ensuing
discussion of simple, solvable, and nilpotent Lie algebras was
motivated by a private communication from Karl H.Hofmann} In
particular, Bianchi types $\textrm{VIII}$ and $\textrm{IX}$ are
both simple Lie algebras. Type $\textrm{VIII}$ is
$\mathfrak{sl}(2,\mathbb{R}) \cong \mathfrak{so}(2,1)$, and type
$\textrm{IX}$ is $\mathfrak{so}(3) \cong \mathfrak{su}(2)$.
Bianchi types $\textrm{I}$ and $\textrm{II}$ are the only
nilpotent solvable real $3$-dimensional Lie algebras. The Bianchi
type $\textrm{I}$ is the abelian Lie algebra $\mathbb{R}^3$, and
any abelian Lie algebra is automatically nilpotent and solvable.
The Bianchi type $\textrm{II}$ is the $3$-dimensional Heisenberg
Lie algebra, a $2$-step nilpotent Lie algebra,
$$
[\mathfrak{g},\mathfrak{g}] \neq 0, \; \;
[\mathfrak{g},[\mathfrak{g},\mathfrak{g}]] = 0 \, ,
$$ with a $1$-dimensional centre.
The other seven classes of Lie algebra all contain non-nilpotent
solvable Lie algebras. Of the `Type A' Lie algebras,
$\textrm{VI}_0$ and $\textrm{VII}_0$ are the non-nilpotent
solvable ones. Type $\textrm{VI}_0$ is the Lie algebra of
$E(1,1)$, the group of motions of the Euclidean plane equipped
with a Minkowski metric. Type $\textrm{VII}_0$ is the Lie algebra
of $E(2)$, the group of motions of the Euclidean plane equipped
with a spacelike metric.
Within the `Type B' Lie algebras, Bianchi types $\textrm{VI}_h$
and $\textrm{VII}_h$ provide one-parameter families of Lie
algebras for $0 < h < \infty$, for which the $\textrm{VI}_0$ and
$\textrm{VII}_0$ Lie algebras are limiting cases as $h \rightarrow
0$.
The Type B Bianchi algebra $\textrm{III}$ is such that
$\textrm{III} = \textrm{VI}_{-1}$ if we remove the restriction
that $a \neq 1$ for type $\textrm{VI}_h$. The Bianchi type
$\textrm{III}$ algebra brings us to the Levi-Malcev decomposition.
The sum of all the solvable ideals in a Lie algebra $\mathfrak{g}$
is a maximal solvable ideal called the radical $\mathfrak{r}$. The
quotient $\mathfrak{g}/\mathfrak{r}$ is a semi-simple Lie algebra.
There exist mutually conjugate subalgebras of $\mathfrak{g}$,
called Levi subalgebras $\mathfrak{l}$, which are maximal
semi-simple subalgebras, and which map isomorphically onto
$\mathfrak{g}/\mathfrak{r}$. The Levi-Malcev decomposition states
that for any Lie algebra $\mathfrak{g}$, there is a Levi
subalgebra $\mathfrak{l}$ such that $\mathfrak{g} =\mathfrak{r}
\oplus \mathfrak{l}$.
Now, a semi-simple Lie algebra has no non-zero solvable ideals,
hence a semi-simple algebra has no radical. A real $3$-dimensional
semi-simple Lie algebra therefore has a trivial Levi
decomposition, coinciding with its own Levi subalgebra. On the
other hand, a solvable Lie algebra is coincident with its own
radical, so it too has a trivial Levi decomposition. A real
$3$-dimensional Lie algebra with a non-trivial Levi decomposition
would have to be the sum of one $2$-dimensional Lie algebra and a
$1$-dimensional Lie algebra. Now, there is only one real
$1$-dimensional Lie algebra, the abelian Lie algebra $\mathbb{R}$.
Being abelian, it must be solvable and not semi-simple, hence it
could only provide the radical $\mathfrak{r}$ in the Levi
decomposition of a real $3$-dimensional Lie algebra. The Levi
subalgebra $\mathfrak{l}$ in such a decomposition would therefore
have to be a real $2$-dimensional semi-simple Lie algebra. In
fact, a real semi-simple Lie algebra is at least
$3$-dimensional,\footnote{Private communication with Karl
H.Hofmann} hence no real $3$-dimensional Lie algebra possesses a
non-trivial Levi decomposition. To reiterate, every real
$3$-dimensional Lie algebra is either semi-simple or solvable.
There is a unique non-abelian real $2$-dimensional Lie algebra,
$V^2$, but it is not semi-simple. The type $\textrm{III}$ Bianchi
algebra is the direct sum $\mathbb{R} \oplus V^2$, but the type
$\textrm{III}$ algebra is solvable, and this is not a Levi
decomposition.
\hfill \break
The spatially homogeneous cosmological models in which each pair
$(\Sigma_t,\gamma_t)$ has a $3$-dimensional isometry group are
referred to as Bianchi cosmological models. The case in which
$I(\Sigma_t)$ is $3$-dimensional is obviously the case in which
the isotropy group at each point is trivial. In this case, the
Riemannian manifold $\Sigma_t$ is diffeomorphic to the isometry
group $I(\Sigma_t)$. Moreover, in this case, the Bianchi
classification of $3$-dimensional Lie algebras can contribute to
the classification of the homogeneous spatial geometries because
the Lie algebra of Killing vector fields on $\Sigma_t$ is
isomorphic with the Lie algebra of the isometry group
$I(\Sigma_t)$. Those homogeneous $3$-dimensional Riemannian
manifolds which have $3$-dimensional isometry groups can be
classified according to Bianchi types \textrm{I} - \textrm{IX}. To
be clear, the Lie algebra of Killing vector fields on a
homogeneous $3$-dimensional Riemannian manifold $\Sigma_t$ is
always isomorphic with the Lie algebra of the isometry group
$I(\Sigma_t)$, but the Bianchi classification only provides a
classification of the Lie algebras of Killing vector fields in the
case in which $I(\Sigma_t)$ is $3$-dimensional.
Groups with the same Lie algebra are not, in general, isomorphic
Lie groups, hence $3$-dimensional homogeneous geometries of the
same Bianchi type do not, in general, have the same
$3$-dimensional isometry groups, and are not, in general,
isometric geometries. Distinct geometries can share the same Lie
algebra of Killing vector fields.
The spatially homogeneous models in which each pair
$(\Sigma_t,\gamma_t)$ has a 4-dimensional isometry group are
called rotationally symmetric in contrast with the spherical
symmetry of the FRW models. Whilst the isotropy group at each
point of an FRW model contains the $3$-dimensional group $SO(3)$,
the isotropy group at each point of a rotationally symmetric model
is the $1$-dimensional group $SO(2)$. This $1$-dimensional group
acts transitively upon the set of directions within a
$2$-dimensional plane of the tangent space. The rotationally
symmetric models are often referred to as Kantowski-Sachs models.
In fact, the latter term should be reserved for models with a
$4$-dimensional isometry group $I(\Sigma_t)$ in which there is no
$3$-dimensional isometry subgroup which acts simply transitively
upon $\Sigma_t$.
Finally, those spatially homogeneous models in which each pair
$(\Sigma_t,\gamma_t)$ has a $6$-dimensional isometry group, and in
which the time variation of the spatial geometry is given by a
single scale factor, are Friedmann-Robertson-Walker models. It is
true that the conventional FRW models, which are spatially
globally isotropic, are special cases of the class of spatially
homogeneous cosmological models. A globally isotropic
$3$-dimensional Riemannian manifold does indeed have a
6-dimensional isometry group. However, the generalized class of
FRW models takes one outside the class of spatially homogeneous
models. A locally isotropic $3$-dimensional Riemannian manifold
need not be homogeneous. It is therefore incorrect to consider the
entire class of FRW models as a subclass of the spatially
homogeneous cosmological models. In many of the generalized FRW
models, the spatial geometry has an isometry group of dimension
lower than 6, or no Lie group of isometries at all.
Although the 6-dimensional isometry group $I(\Sigma_t)$ of each
hypersurface $\Sigma_t$ in a spatially homogeneous
Friedmann-Robertson-Walker model is not diffeomorphic with the
Riemannian manifold $\Sigma_t$, it does contain $3$-dimensional
Lie subgroups which act simply transitively upon $\Sigma_t$, and
the Lie algebras of these subgroups do fall under the Bianchi
classification. The isometry group of the $\mathbb{R}^3$ FRW model
contains the Bianchi type $\textrm{I}$ group of translations on
$\mathbb{R}^3$ as a simply transitive subgroup. The isotropy group
at each point is a $3$-dimensional subgroup of Bianchi type
$\textrm{VII}_0$. In the case of the $H^3$ FRW model, the isometry
group contains a simply transitive $3$-dimensional subgroup of
Bianchi type $\textrm{V}$, whilst the isotropy group is a
$3$-dimensional subgroup of type $\textrm{VII}_h$. In the case of
the $S^3$ FRW model, the isometry group contains a simply
transitive $3$-dimensional subgroup of Bianchi type $\textrm{IX}$,
whilst the isotropy group is a $3$-dimensional subgroup also of
type $\textrm{IX}$. (Rey and Luminet 2003, p43-44).
In contrast, the spatially homogeneous Kasner cosmology has a
$3$-dimensional isometry group of Bianchi type $\textrm{I}$, and
no isotropy group, whilst the spatially homogeneous Mixmaster
cosmology has a $3$-dimensional isometry group of Bianchi type
$\textrm{IX}$, and no isotropy group.
Note that in space-times which can be sliced up into a
one-parameter family of homogeneous spacelike hypersurfaces
$(\Sigma_t,\gamma_t)$, each bearing a specific Bianchi type, there
is no guarantee that the Bianchi type of each homogeneous
hypersurface will be the same; the Bianchi type can change with
time, (Rainer and Schmidt 1995).
A $3$-dimensional geometry whose isometry group admits a simply
transitive $3$-dimensional subgroup, must be homeomorphic with
that $3$-dimensional group. All the simply connected
$3$-dimensional Lie groups in the Bianchi classification are
either diffeomorphic to $\mathbb{R}^3$, or diffeomorphic to $S^3$
in the case of the $\textrm{IX}$ type. Hence, any globally
homogeneous $3$-dimensional geometry which falls under the Bianchi
classification, has a topology which is either covered by
$\mathbb{R}^3$ or $S^3$.
In the case of a homogeneous $3$-dimensional Riemannian manifold
$\Sigma$, which has a $3$-dimensional isometry group $I(\Sigma)$
to which it is diffeomorphic, a quotient $\Sigma/\Gamma$ with
respect to a discrete normal subgroup of $I(\Sigma)$ is
diffeomorphic to the quotient Lie group $I(\Sigma)/\Gamma$. Given
that the isometry group of such a quotient is $N(\Gamma)/\Gamma$,
and given that $\Gamma$ is a normal subgroup, $N(\Gamma) =
I(\Sigma)$, and it follows that the isometry group of the quotient
is $I(\Sigma)/\Gamma$. If the manifold is diffeomorphic with its
isometry group, then the quotient manifold is diffeomorphic with
the isometry group of the quotient. Given that the isometry group
of the quotient can also be expressed as the centralizer
$Z(\Gamma)$, it follows that the quotient is a homogeneous
Riemannian manifold if and only if the centralizer $Z(\Gamma)$
acts transitively on $\Sigma$, (Ellis 1971, p11). Given that a
discrete normal subgroup of a connected Lie group must be central,
the centralizer $Z(\Gamma)$ will, in this instance, contain
$\Gamma$ as a subgroup.
\hfill \break
Assuming the Copernican principle is true, the observed local
isotropy of our universe can be used to exclude a number of
$3$-manifolds which would otherwise be candidates for the spatial
topology. The reasoning here follows from two key facts:
\begin{enumerate}
\item A locally isotropic $3$-dimensional Riemannian manifold must
be of constant sectional curvature (Wolf 1967, p381-382)
\item A $3$-dimensional manifold can only possess a Riemannian
metric tensor of constant sectional curvature if its universal
covering manifold is diffeomorphic to either $\mathbb{R}^3$ or
$S^3$.
\end{enumerate}
A simply connected manifold is its own universal cover, hence a
simply connected $3$-manifold which is not diffeomorphic to either
$\mathbb{R}^3$ or $S^3$ will not be able to support a Riemannian
metric tensor of constant sectional curvature, and will therefore
not be able to represent a locally isotropic spatial universe.
To find such a manifold, we note that for a connected product
manifold $M \times N$, the fundamental group $\pi_1(M \times N)$
is such that
$$
\pi_1(M \times N) \cong \pi_1(M) \times \pi_1(N) \,.
$$ Hence, if $M$ and $N$ are both simply connected, then $M \times
N$ will be simply connected. $S^2$ and $\mathbb{R}^1$ are both
simply connected, hence the hypercylinder $S^2 \times
\mathbb{R}^1$ is also simply connected.
$S^2 \times \mathbb{R}^1$ is not diffeomorphic to either
$\mathbb{R}^3$ or $S^3$, hence the hypercylinder $S^2 \times
\mathbb{R}^1$ cannot support a Riemannian metric tensor of
constant sectional curvature, and cannot therefore represent a
locally isotropic spatial universe.
Including $S^2 \times \mathbb{R}^1$ itself, there are seven
$3$-manifolds which have $S^2 \times \mathbb{R}^1$ as their
universal covering, (Scott 1983, p457-459). Each such manifold has
a universal covering which is neither $\mathbb{R}^3$ nor $S^3$,
hence each such manifold cannot support a Riemannian metric tensor
of constant sectional curvature, and cannot therefore represent a
locally isotropic spatial universe. Of these seven manifolds,
three are non-compact and four are compact. The non-compact cases
consist of $S^2 \times \mathbb{R}^1$ itself, the trivial line
bundle $\mathbb{RP}^2 \times \mathbb{R}^1$, and a non-trivial line
bundle over $\mathbb{RP}^2$. The compact cases consist of
$\mathbb{RP}^2 \times S^1$, the connected sum $\mathbb{RP}^3 \#
\mathbb{RP}^3$, and a pair of line bundles over $S^2$, one of
which is the trivial bundle $S^2 \times S^1$.
\hfill \break
Thurston has identified eight globally homogeneous, simply
connected $3$-dimensional Riemannian manifolds which admit a
compact quotient, (Rey and Luminet 2003, p39-42). $\mathbb{R}^3$,
$S^3$, and $H^3$ provide three of these, but the remaining five
are non-isotropic. These five geometries and their quotients are
neither globally nor locally isotropic. Moreover, the quotients of
these five geometries are only guaranteed to be locally
homogeneous. The hypercylinder $S^2 \times \mathbb{R}^1$ is
obviously one of these five geometries. The others are $H^2 \times
\mathbb{R}^1$, $\widetilde{SL2\mathbb{R}}$, the universal covering
of the $3$-dimensional group $SL(2,\mathbb{R})$, $Nil$, the
$3$-dimensional Lie group of $3 \times 3$ Heisenberg matrices, and
$Sol$, a Lie group consisting of $\mathbb{R}^3$ equipped with a
non-standard group product.
The quotients of $H^2 \times \mathbb{R}^1$ include all the
products of $T_g$ with either $S^1$ or $\mathbb{R}^1$, where the
$T_g$ are the compact, orientable surfaces of genus $g > 1$,
equipped with metrics of constant negative curvature, and
constructed from the $2$-sphere by attaching $g$ handles.
$Sol$ has a disconnected isometry group with eight components, the
identity component of which is $Sol$ itself, (Koike \textit{et al}
1994, p12). The other four geometries possess a $4$-dimensional
isometry group. $S^2 \times \mathbb{R}^1$, $H^2 \times
\mathbb{R}^1$, $\widetilde{SL2\mathbb{R}}$ and $Nil$ are therefore
rotationally symmetric models. However, only the hypercylinder
$S^2 \times \mathbb{R}^1$ provides a Kantowski-Sachs model. Whilst
the isometry group of $S^2 \times \mathbb{R}^1$ has no
$3$-dimensional subgroups which act simply transitively upon $S^2
\times \mathbb{R}^1$, the isometry group of $H^2 \times
\mathbb{R}^1$ has a Bianchi type $\textrm{III} = \textrm{VI}_{-1}$
subgroup, the isometry group of $\widetilde{SL2\mathbb{R}}$ has a
Bianchi type $\textrm{VIII}$ subgroup, and the isometry group of
$Nil$ has a Bianchi type $\textrm{II}$ subgroup, each of which
acts simply transitively. The isometry group of $Sol$ is a Bianchi
type $\textrm{VI}_0$ group. (Rey and Luminet 2003, p45).
There are only three distinct topologies amongst the eight
Thurston geometries. $\mathbb{R}^3$, $H^3$, $H^2 \times
\mathbb{R}^1$, $\widetilde{SL2\mathbb{R}}$, $Nil$, and $Sol$ are
all homeomorphic to $\mathbb{R}^3$. $S^3$ and $S^2 \times R^1$
provide the other two topologies. (Koike \textit{et al} 1994,
p19).
Note that not all of the globally homogeneous, simply connected
$3$-dimensional Lie groups from the Bianchi classification admit a
compact quotient. For example, Bianchi type $\textrm{IV}$ and the
one-parameter family in Bianchi type $\textrm{VI}_h$ do not admit
a compact quotient, and therefore do not provide a Thurston
geometry.
\hfill \break
Assuming the Copernican principle is true, the observed local
isotropy of our universe can be used to exclude any $3$-manifold
which is not a prime manifold. A prime manifold is a manifold
which has no non-trivial connected sum decomposition. Primeness is
a necessary condition for a $3$-manifold to accept a metric of
constant sectional curvature, hence any non-trivial connected sum
of prime manifolds can be excluded as a candidate for the spatial
topology of our universe. Note that any compact $3$-manifold can
be decomposed as a finite connected sum of prime $3$-manifolds,
and any compact orientable $3$-manifold can be decomposed as a
\emph{unique} finite connected sum of primes. Although $M \# S^3
\cong M$ for any $3$-manifold $M$, the connected sum construction
provides a method of obtaining a plentiful family of compact
orientable $3$-manifolds which are inconsistent with the
conjunction of the Copernican principle and our observation of
local isotropy. This should be balanced with Thurston's assertion
that `most' compact orientable $3$-manifolds accept a metric of
constant negative curvature.
Note also that whilst primeness is a necessary condition for a
$3$-manifold to accept a metric of constant sectional curvature,
it is not a sufficient condition. $S^2 \times S^1$ is a prime
manifold, but it cannot accept a metric of constant sectional
curvature, as noted above from the fact that its universal cover
is $S^2 \times \mathbb{R}^1$. To reiterate, only a $3$-manifold
with either $\mathbb{R}^3$ or $S^3$ as universal cover can accept
a metric of constant sectional curvature.
\section{The epistemology of cosmology}
To elucidate the nature and scope of astronomical and cosmological
knowledge, the philosophical purpose of this section is to
precisely clarify, using the concept of the celestial sphere, the
relationship between general relativity and astronomical
observation and measurement. \textit{En route}, the nature of
colour in astronomical observation is clarified, and an
iconoclastic scenario suggested by Arp \textit{et al} (1990) is
used as a case study of the relationship between astronomical
observation and cosmological theory. The nature of the Cosmic
Microwave Background Radiation (CMBR), and its variations, is
clarified, together with a definition and explanation of the
angular power spectrum. The paper concludes with some comments on
the overall status of the FRW models.
\hfill \break
The mathematical formalism of general relativity can be connected
to empirical observation and measurement by means of the concept
of the celestial sphere. One can associate a celestial sphere with
each point of each timelike curve in a Lorentzian manifold
$(\mathcal{M},g)$. In general relativity, the history of an
idealised observer is represented by a timelike curve $\gamma:I
\rightarrow \mathcal{M}$ in a Lorentzian manifold
$(\mathcal{M},g)$, which is such that the tangent to the curve at
each point is a future-pointing, timelike unit vector, (Sachs and
Wu 1977, p41). Hence, one can associate a celestial sphere with
each moment in the history of an idealised observer. At each
moment $\tau$ in the proper time of an observer, there is a
corresponding point $p=\gamma(\tau)$ in the manifold. The tangent
to the curve $\gamma$ at $p$, denoted as $Z$, determines a direct
sum decomposition of the tangent space $T_p \mathcal{M}$:
$$
\mathbb{R}Z \oplus Z^\perp \,.
$$ $\mathbb{R}Z$, the span of $Z$, is the local time axis, and
$Z^\perp$, the set of vectors orthogonal to $Z$, represents the
local rest space of the observer. $Z^\perp$ is isometric with
$\mathbb{R}^3$, and the observer's celestial sphere is the sphere
of unit radius in $Z^\perp$. One can consider the pair $(p,Z)$ as
an instantaneous observer, (Sachs and Wu 1977, p43). Each
instantaneous observer has a private celestial sphere.
Recall now that a light ray/photon is represented by a null
geodesic, and the tangent vector at each point of a null geodesic
is the energy-momentum of the photon. The observation of an
incoming light ray/photon by an instantaneous observer $(p,Z)$,
will be determined by the energy-momentum tangent vector $Y$ of
the null geodesic at $p$.
Given a vector space equipped with an indefinite inner product,
$g(\; ,\; )$, and given an orthonormal basis $\{e_1,...,e_n\}$
such that $\epsilon_i = g(e_i,e_i)$, any vector $v$ in the space
can be expressed as
$$
v = \epsilon_1 g(v,e_1)e_1 + \cdots + \epsilon_n g(v,e_n)e_n \,.
$$ Given that $g(Z,Z) = -1$, the direct sum decomposition determined
by $Z$ enables one to express an arbitrary vector $Y \in T_p
\mathcal{M}$ as
$$
Y = -g(Y,Z)Z + P \, ,
$$ where $P$ is a spacelike vector in the local rest space
$Z^\perp$. There is a unit spacelike vector $B$ such that $P =
bB$, for some real number $b$. Letting $e=-g(Y,Z)$, it follows
that an arbitrary vector $Y$ can be expressed as
$$
Y = eZ + bB \,.
$$
In the case of interest here, where $Y$ is the energy-momentum
tangent vector of a null geodesic at $p$, the null condition means
that $\langle Y,Y \rangle = 0$. This entails that
$$\eqalign{
\langle eZ+bB,eZ+bB\rangle &= \langle eZ,eZ \rangle + \langle
bB,bB \rangle \cr &= - e^2 + b^2 \cr &= 0 \, ,}$$ which is
satisfied if and only if $e=b$. Hence, for a null vector $Y$,
$$
Y = eZ +eB \,.
$$ Letting $U = -B$, we have
$$
Y = e(Z-U) = -g(Y,Z)Z + g(Y,Z)U \, ,
$$ with $P=-eU$. $U$ is a unit spacelike vector in the celestial
sphere, pointing in the spacelike direction from which the photon
with the null vector $Y$ emanates.
The instantaneous observer $(p,Z)$ will detect the photon of light
to be of energy $e = -g(Y,Z) \in (0,\infty)$, and to come from the
spatial direction $U \in Z^\perp$, where $Y = e(Z-U)$, (Sachs and
Wu 1977, p46 and p130). The measured frequency of the light will
simply be $\nu=e/h$, and the wavelength will be $\lambda=c/\nu$,
or simply $\lambda=\nu^{-1}$ if `geometric units' are used, in
which $c=1$.
Observers in a different state of motion at the same point $p$ in
space, will be represented by different timelike vectors at $p$.
Two distinct timelike vectors $V,W \in T_p \mathcal{M}$ will
determine different direct sum decompostions of $T_p \mathcal{M}$.
As a consequence, observers in a different state of motion will
have different local rest spaces, $V^\perp$ and $W^\perp$, and
will have different celestial spheres. This results in the
aberration of light: different observers will disagree about the
position of a light source, (Sachs and Wu 1977, p46). Moreover,
different observers at a point $p$ will measure the same photon of
light to have different energy. $(p,V)$ would measure $e =
-g(Y,V)$, and $(p,W)$ would measure $e = -g(Y,W)$.
In the simplest of cases, the colour of an object perceived by an
observer is determined by the energy, within the visible spectrum,
at which most of the photons are emitted or reflected from that
object. Hence, the colour of light detected from some source will
be dependent upon one's motion with respect to the source. Let us
agree to define an intrinsic property of an object to be a
property which the object possesses independently of its
relationships to other objects. Let us also agree to define an
extrinsic property of an object to be a property which the object
possesses depending upon its relationships with other objects. The
colour of an object, determined by the energy of the light it
emits or reflects, is not an intrinsic property of an object. The
colour of an object varies depending upon the relationship between
that object and an observer, hence the colour of an object is an
extrinsic property. The perception of colour has a number of
additional subtleties associated with it, which we will detail at
a later juncture.
We are ultimately interested in cosmology, so we shall consider
here only the way in which the formalism of general relativity is
linked with astronomical observations. Given an instantaneous
observer $(p,Z)$, one can associate with it a celestial sphere
$\mathscr{S}_Z$ and a direction-energy space $\mathscr{S}_Z \times
(0,\infty)$, (Sachs and Wu 1977, p141). Recall that each photon,
corresponding to a forward-pointing null vector $Y \in
T_p\mathcal{M}$, has an energy $e = -g(Y,Z)$ and a spatial
direction $U \in \mathscr{S}_Z$, hence the notion of a
direction-energy space. $\mathscr{S}_Z \times (0,\infty)$ is
diffeomorphic to the forward light cone $\mathscr{V}_0^+$, which
in turn is diffeomorphic to $Z^\perp - \textbf{0} \cong
\mathbb{R}^3 - \textbf{0}$, (Sachs and Wu 1977, p147). Hence, one
can introduce spherical coordinates $(e,\theta,\phi)$ in which the
radial coordinate corresponds to the energy $e$. In these
coordinates, the Euclidean metric tensor on $Z^\perp \cong
\mathbb{R}^3$ can be expressed as
$$
g = e^2(d\theta^2 + \sin^2 \theta d\phi^2) \,.
$$ In these coordinates, the determinant of the metric is
$\det g = e^4 \sin^2 \theta$. The natural metric volume element of
a Riemannian metric $g$ in a coordinate system $(x_1,...,x_n)$ is
defined to be
$$
\Omega = (\sqrt{| \det g | }) \, dx^1 \wedge \cdots \wedge dx^n \,
, $$ hence in the case above, the natural metric volume element is
$$\eqalign{
\Omega &= e^2 \sin \theta \; de \wedge d\theta \wedge d\phi \cr &=
de \wedge e d\theta \wedge e \sin \theta d\phi \cr &= de \wedge
e^2(d\theta \wedge \sin \theta d\phi) \cr &= de \wedge e^2\omega
\, ,}
$$ where $\omega$ is the standard metric volume element on the
$2$-sphere.
Sachs and Wu introduce a photon distribution function $N_Z$ on the
direction-energy space of an instantaneous observer, (1977, p142),
$$N_Z:\mathscr{S}_Z \times (0,\infty) \rightarrow [0,\infty) \,.$$
Given that $de \wedge e^2\omega = e^2de \wedge \omega$, for a
range of energies $[a,b] \subset (0,\infty)$ and a compact subset
of the celestial sphere $\mathscr{K} \subset \mathscr{S}_Z$,
$$
\int_{\mathscr{K} \times [a,b]}N_Z \, \Omega = \int_\mathscr{K}
\omega \int_a^b e^2N_Z \, de \,.
$$
Sachs and Wu define this integral to be the number of photons per
unit spatial volume in the energy range $[a,b]$ emanating from the
compact region $\mathscr{K}$ of the observer's celestial sphere,
(p142). They interpret $e^2N_Z$ as the number of photons per unit
spatial volume per unit solid angle per unit energy interval. As
they subsequently explain, (p147-148), because photons travel at
unit speed in the `geometric units' employed, they travel a unit
distance in unit time. Hence, the number of photons which occupy a
unit spatial volume is equal to the number of photons which pass
through a unit area perpendicular to their direction of motion in
unit time. Therefore $e^2N_Z$ can also be interpreted as the
number of photons which pass through a unit perpendicular area per
unit time per unit solid angle upon the celestial sphere per unit
energy interval. In terms of astronomical observations, the unit
area is the unit area of some photon collection device, such as
the surface of a radio telescope, or the mirrored surface of an
optical telescope.
Making the independent variables explicit, $e^2N_Z$ is a function
$e^2N_Z(a,t,\theta,\phi,e)$, where $a$ denotes a point on the
surface on the photon collection device, $t$ denotes time,
$\theta$ and $\phi$ are coordinates upon the celestial sphere of
the instantaneous observer, and $e$ is the energy. A different
function $e^3N_Z(a,t,\theta,\phi,e)$ specifies the amount of
energy passing through a unit perpendicular area per unit time per
unit solid angle upon the celestial sphere per unit energy
interval. When $e$ is replaced with the frequency of the
radiation, $\nu = e/h$, the function $e^3N_Z(a,t,\theta,\phi,\nu)$
specifies the amount of energy passing through a unit
perpendicular area per unit time per unit solid angle upon the
celestial sphere per unit frequency interval. In the astronomy
literature, this function is referred to as the \emph{specific
intensity} of radiation. Its dimensions are Watts ($W$) per square
metre ($m^{-2}$) per Hertz ($Hz^{-1}$) per steradian
($sterad^{-1}$). The specific intensity is often denoted as
$I_\nu$ to emphasise that it is a function of the frequency $\nu$
of radiation. In this event, $I$ is often reserved to denote the
integral of the specific intensity over all frequencies
$$
I = \int_0^\infty I_\nu \, d\nu \,.
$$ The resulting function $I(a,t,\theta,\phi)$ specifies the amount
of energy passing through a unit perpendicular area per unit time
per unit solid angle upon the celestial sphere, over all
frequencies.
Suppose that a light source such as a star, a nebula or a galaxy
corresponds to a compact region $\mathscr{K}$ upon the celestial
sphere of an observer. The \emph{flux density} $F$ of the light
source is obtained by integrating the intensity $I$ over the
region $\mathscr{K}$. To be precise, one integrates $I \cos \alpha
$, where $\alpha$ is the angle between each point in $\mathscr{K}$
and the perpendicular to the surface area of the measuring device,
(Karttunen \textit{et al} 2003, p81). In the case of a light
source which subtends a small solid angle upon the celestial
sphere, and a measuring instrument pointed directly at the light
source, $\cos \alpha \approx 1$. One can deal with either a
frequency-dependent flux
$$F_\nu = \int_{\mathscr{K}}I_\nu \cos \alpha \; \omega \, ,$$ or the total flux $$F=
\int_\mathscr{K} \omega \int_0^\infty I_\nu \cos \alpha \; d\nu
\,.
$$ The dimensions of $F_\nu$ are $W \, m^{-2} \, Hz^{-1}$, whilst
the dimensions of $F$ are $W \, m^{-2}$.
The flux density $F(r)$ observed from a light source at a distance
$r$ is another name for the \emph{apparent luminosity} $l(r)$ of
the light source at distance $r$. Assuming space is approximately
Euclidean on the length scales involved, and assuming that the
light is emitted isotropically from the source, the \emph{absolute
luminosity} $L$ of the light source is defined to be $L = 4\pi r^2
F(r)$. The absolute luminosity is simply the power output of the
light source, the amount of energy emitted per unit time, in all
directions. That power is spread out over spheres of increasing
surface area $4\pi r^2$ at increasing distances $r$, hence the
flux decreases as a function of distance $F(r) = L/4\pi r^2$.
The brightness of an object, either in astronomy, or in perception
with the naked eye, corresponds not to the specific intensity of
the light received from that object, but to the flux density of
the light. Assuming that an object and observer are not in
relative motion and that the space between the object and observer
is static, then the specific intensity of the light received from
the object is independent of the distance separating the observer
from the object, whilst the flux density is inversely proportional
to the square of the distance, (Karttunen \textit{et al} 2003,
p89). If an object and observer are either in relative motion, or
the space between them is dynamic, then the flux density will also
depend upon the redshift/blueshift.
Sachs and Wu suggest (p142) that the brightness of a rose
corresponds to the specific intensity $e^3N_Z$. The specific
intensity is independent of distance because it measures the flux
density per unit solid angle. At greater distances, a unit solid
angle collects photons emitted from a larger fraction of the
surface area of the object, but due to the greater distance, the
unit solid angle collects a smaller fraction of the photons
emitted from the surface area under its purview. These effects
cancel. The brightness of an object to the naked eye decreases
with distance, hence specific intensity does not correspond to the
naked eye perception of brightness.
The brightness of an object to the naked eye corresponds not to
the total flux density of the object, but to the flux integrated
over the visible range of frequencies:
$$\eqalign{
F_{[a,b]} &= \int_\mathscr{K} \omega \int_a^b I_\nu \cos \alpha \;
d\nu \cr &= \int_a^b F_\nu \, d\nu \,.} $$
If the angle subtended by a luminous object remains constant, but
the intensity of the light it radiates increases, then the flux
density $\equiv$ brightness of the object will increase. Hence,
although the brightness of an object should not be conflated with
the intensity of the light radiated by the object, it is
legitimate to explain an increase in the brightness of an object
as being the result of an increase in the intensity of the light
it emits.
Picking up an issue alluded to above, the colour of an object
perceived by an observer is determined by the intensity of the
light emitted or reflected from that object, over the range of
visible wavelengths, in the reference frame of that observer. The
visible spectrum contains those colours which can be identified in
a rainbow, or in the light refracted from a prism. These `spectral
colours' each correspond to a particular wavelength or range of
wavelengths. If the intensity of light over the visible spectrum
is peaked at a certain wavelength in an observer's reference
frame, then that observer perceives the corresponding colour.
However, the human perceptual system introduces colours and
structures amongst the set of colours, which do not exist in the
visible spectrum itself, (Clark 1998). For a start, whilst the
visible spectrum has the topology of a closed interval $[0,1]$ of
the real line, and a consequent linear ordering, the set of
colours perceived by humans has the topology of the circle $S^1$,
and, obviously, no such linear ordering relationship. The visible
spectrum ranges from the blue end at 400nm to the red end at
700nm. A type of purple, called magenta, exists between blue and
red in the set of humanly perceived colours, and completes the
circle.
Magenta can be defined as a mixture of red and blue, and this
introduces the second difference between the visible spectrum and
the set of humanly perceived colours. Let us adopt the common
nomenclature, and refer to the latter as the set of `hues'. One
can mix hues that do correspond to spectral colours, to produce
new hues which don't correspond to spectral colours. Such hues
correspond to an intensity curve which has multiple peaks over the
visible spectrum. Different combinations of hue can produce the
same mixed hue; these hue combinations are called `metamers'. This
means that different intensity curves over the visible spectrum,
with different combinations of wavelength peaks, can produce the
same perception of colour. There is a many-one correspondence
between intensity curves and perceived colours.
In general, three parameters are used to characterise the space of
colours in the human perceptual system. The exact parameters used
depend upon whether one is dealing with reflected light from a
surface, emitted light from a source, or the light which falls
upon a photographic emulsion after passing through an aperture.
With this qualification, the three parameters are hue, saturation,
and lightness, the latter sometimes being thought of as the shade
of a colour. Shade is the relative amount of lightness or darkness
of a colour. For a particular hue, you get a lighter shade by
mixing it with white, and a darker shade by mixing it with black.
Lightness measures the overall intensity of a colour; lighter
shades are therefore brighter. The saturation of a colour measures
the ratio of the intensity at the dominant wavelengths to the
intensity at other wavelengths. If the dominant wavelengths of a
hue are highly peaked, then that hue has high saturation. If the
peaks are quite small compared to the intensity at other
wavelengths, then the hue tends towards an achromatic grey, and is
said to have low saturation. Pastel colours are low saturation
hues. For achromatic light, the lightness scale ranges from white
to black through all the various intervening greys. No light at
all at visible wavelengths produces the perception of black. Equal
combinations of light at different wavelengths within the visible
spectrum produce achromatic light, and each hue has a complement,
such that when that hue is combined with its complement, the
result is achromatic light.
One can treat hue, saturation, and lightness as cylindrical polar
coordinates upon the space of colours in the human perceptual
system. The circle of hues has the angular coordinate, saturation
provides a radial coordinate in the plane, and lightness provides
the `vertical' coordinate. Note, however, that for darker shades,
the saturation range is more restricted, so one is dealing with
something more akin to a cone than a cylinder. Note also that
there are other coordinatizations in use, such as the `colour
sphere'.
At a rather high level of idealisation, Sachs and Wu (1977, p142)
suggest that one can regard all astronomers who have ever lived as
a single instantaneous observer $(p,Z)$. I will slightly relax
this idealisation, and suggest instead that one can associate a
single celestial sphere with the human race. Whilst each
individual has a private celestial sphere, at another level of
idealisation there is a celestial sphere which is common to all
humans upon the Earth. Gazing skywards on a clear night, the stars
appear to be speckled across the inner surface of an inverted
bowl. This is one hemisphere of our common celestial sphere. The
history of the human race can be represented as a timelike curve,
and as Sachs and Wu suggest (1977, p131), one can use parallel
transport to identify the celestial spheres associated with the
points of a timelike curve. Thence, (changing notation slightly),
all the possible astronomical observations made by the human race
could be encoded as a time-dependent function of only three
variables $I_t(\theta,\phi,\nu)$. The function $I_t$ specifies the
intensity at time $t$ of electromagnetic radiation at any
frequency $\nu$ over the entire celestial sphere. The time
variation of this function provides all the raw astronomical data
that a species located upon a single planet could ever have. In
terms of using the raw data upon our celestial sphere to make
cosmological inferences, it should be noted that only $1\%$ of the
light which intersects our celestial sphere comes from beyond our
galaxy, (Disney 2000, p4).
The conventional coordinates upon a sphere are such that $\theta
\in [0,\pi]$ and $\phi \in [0,2\pi)$. Astronomers use a variety of
slightly different, but closely related celestial coordinates. For
example, the \emph{equatorial system} (Nicolson 1977, p42-43) uses
the intersection of the plane of the Earth's equator with the
celestial sphere to determine a great circle on the celestial
sphere called the celestial equator. Right ascension $\alpha \in
[0,2\pi)$ then provides a coordinate upon the celestial equator,
starting at the vernal equinox and running Eastward. Declination
$\delta \in [-\frac{1}{2}\pi,\frac{1}{2}\pi]$ then specifies the
angular distance North or South of the celestial
equator.\footnote{The vernal equinox is the point of intersection
of the ecliptic and the celestial equator at which the Sun moves
from the Southern celestial hemisphere into the Northern celestial
hemisphere (Nicolson 1977, p234). The ecliptic is the great circle
which the Sun traces upon the celestial sphere due to the Earth's
annual orbit around the Sun (1977, p73). It can also be thought of
as the intersection of the Earth's orbital plane with the
celestial sphere. Because the Earth's axis, and therefore its
equator, are inclined at approximately $23\frac{1}{2}\deg$ to the
orbital plane, the celestial equator is inclined at the same angle
to the ecliptic. The ecliptic intersects the celestial equator at
two points, the vernal equinox and the autumnal equinox.}
The timelike vector $Z$ that specifies which local rest space, and
thence which celestial sphere, is selected for the human race,
will be determined by taking the vector sum of the motion of the
Local Group of galaxies relative to the microwave background
radiation, the motion of the Milky Way within the Local Group, the
motion of the Sun within the Milky Way, and the motion of the
Earth around the Sun.
General relativity enables us to interpret the complete array of
astronomical images upon the celestial sphere, as the projection
onto the celestial sphere of all the light sources contained
within our past light cone $E^-(x)$. The past light cone $E^-(x)$
of our point in space-time $x \in \mathcal{M}$, is a
$3$-dimensional null hypersurface whose universal covering is a
manifold of topology $S^2 \times \mathbb{R}^1$. One can use the
right ascension and declination coordinates $(\alpha,\delta)$ upon
the $S^2$ factor, and in a simple type of expanding universe, one
can use redshift $z$ as the $\mathbb{R}^1$-coordinate.
To interpret the raw data $I_t(\theta,\phi,\nu)$ upon the
celestial sphere it is necessary to use theories of light emission
and absorption processes. These theories enable us to interpret
the raw data in terms of the electromagnetic spectra of chemical
elements and compounds, and in terms of the statistical mechanics
and thermodynamics of the matter which either emits the radiation,
or absorbs some parts of it.
The best example of this is provided by the cosmic microwave
background radiation (CMBR). This radiation has a spectrum which
is very close to that of `Planckian' blackbody radiation, often
called thermal radiation. Blackbody radiation at temperature $T$
is radiation whose specific intensity is given by, (Sachs and Wu,
p144-145),
$$
I_\nu = e^3N_Z = e^3(2h^{-3}[exp(h\nu/kT)-1]^{-1}) \, ,
$$ where $k$ is the Boltzmann constant.
It is known both from theory, and from Earth-bound experiment and
observation, that only radiation which is in a state of
equilibrium with matter can have a blackbody spectrum. The
radiation is said to be `thermalised' by its interaction with
matter. It is only when there is no net transfer of energy between
the radiation and the matter, that the radiation will be
blackbody. Deep inside a star, where the gas is opaque, the
radiation will be blackbody radiation. Similarly, the radiation
inside the evacuated cavity of an opaque-walled box, whose walls
are maintained at a constant temperature, will be blackbody. The
opacity is necessary because it is the interaction between the
matter and the radiation which makes the radiation blackbody.
Now, as Layzer puts it ``the present day Universe is just as
transparent to the [microwave] background radiation as it is to
ordinary light. We are not living in the equivalent of an opaque
box or inside an opaque gas. This means that the background [i.e.
the CMBR] could not have acquired its distinctive blackbody
characteristics under present conditions. The background radiation
must be a relic of an earlier period of cosmic history, when the
Universe was far denser and more opaque," (Layzer, 1990, p147).
Although the reasoning here is correct, Arp \textit{et al} (1990)
challenged the empirical claim that the present universe is
effectively transparent to radiation at all wavelengths. It is
commonly believed that radiation emitted from stars is able to
propagate freely through space, with only negligible absorption
and scattering by interstellar/intergalactic gas and dust, and
planets. The matter which does absorb radiation is distributed in
a clumpy, discrete manner across the sky, yet the CMBR is
continuum radiation across the entire celestial sphere. Thus, it
is reasoned, the CMBR could not have been produced in the present
universe.
Arp \textit{et al} argued that the CMBR we observe, was emitted
recently and locally. They suggested that there is some form of
intergalactic material, ``with the property of being strongly
absorptive of microwaves, yet of being almost translucent in both
the visible and longer radio wave regions of the spectrum," (1990,
p809). They suggested that our present universe is opaque in the
microwave, and that starlight is absorbed and scattered by this
intergalactic material to produce an isotropic blackbody microwave
spectrum across our celestial sphere. Although stars are discrete
sources of light, because the hypothetical intergalactic material
is distributed uniformly, it could produce a continuum of
radiation across the celestial sphere.
When a photon of starlight is absorbed by interstellar gas, the
gas re-radiates the energy that is absorbed, but it does so by
emitting a sequence of lower-energy photons, and it emits the
photons in random directions. This characteristic might be able to
explain the isotropy of the CMBR. The intergalactic material might
be re-radiating starlight equally in all directions.
If the present universe were opaque in the microwave, it would no
longer follow that the CMBR must be a relic of an earlier period
of cosmic history. One of the primary pillars providing empirical
support for FRW cosmology would have crumbled.
Arp \textit{et al} suggested that metallic filaments, particularly
iron filaments, blasted into intergalactic space by supernovae,
would provide the requisite microwave opacity. Arp \textit{et al}
concluded quite splendidly ``The commonsense inference from the
planckian nature of the spectrum of the microwave background and
from the smoothness [i.e.uniformity] of the background is that, so
far as microwaves are concerned, we are living in a fog and that
the fog is relatively local. A man who falls asleep on the top of
a mountain and who wakes in a fog does not think he is looking at
the origin of the Universe. He thinks he is in a fog," (1990,
p810).
At the risk of sounding churlish, the case of a man who falls
asleep atop a mountain is not relevantly analogous to the
astronomical predicament of the human race. If we had made
observations of distant objects in the microwave for some years,
without any impediment, but after a period of academic sleep, we
then returned to find an isotropic obscuration in the microwave,
we would indeed be justified in thinking that a microwave fog had
developed. The position of the human race is that we have found a
microwave fog from the time that we began looking.
It is well-known that Alpher and Herman predicted in 1948, using
FRW cosmology, that the present universe should be permeated by a
residue of electromagnetic radiation from the early universe. This
radiation was detected by Penzias and Wilson in 1965. Rhook and
Zangari point out that ``because the existence of a background of
microwave radiation was predicted as a consequence of the big
bang, its account, unlike that of rivals, was granted immunity
against accusations of being ad hoc. Competing theories were then
forced into constructing post hoc explanations for the radiation
which did not carry the force of being prior predictions, and
which themselves lay open to charges of being ad hoc," (1994,
p230).
According to a FRW model of our universe there was no net transfer
of energy between the radiative component of the energy density
and the matter component of the energy density, until the universe
was $10^4-10^5$ yrs old. At that time, the `epoch of last
scattering', the universe had expanded to the point that the
equilibrium reactions between the photons and the plasma of matter
could no longer be maintained, and the universe became transparent
to all but a negligible fraction of the radiation. Blackbody
radiation was emitted throughout space, and the FRW models
represent this radiation to cool as the universe expands, until it
reaches microwave frequencies in the present era. The FRW models
therefore predict the continuum, blackbody, microwave radiation
that we observe today.
The verification of FRW cosmology by the detection of the CMBR is
the hypothetico-deductive method at its finest. The physical
processes responsible for the CMBR cannot be deduced from the
empirical characteristics of the CMBR, as the work of Arp
\textit{et al} demonstrates. Instead, one hypothesizes the FRW
models, one deduces the empirical predictions, and one compares
and verifies the predictions with the astronomical data. The mere
possibility that there could be an alternative explanation for the
CMBR, is not a decisive argument against FRW cosmology.
\hfill \break
The CMBR observed by the COBE and WMAP satellites, and a variety
of Earth-bound/balloon-borne measuring devices, possesses an
approximately blackbody spectrum across the entire celestial
sphere, for all values of $\theta$ and $\phi$. However, the
temperature of the blackbody spectrum varies as a function of
$\theta$ and $\phi$. The CMBR has a blackbody spectrum in all
directions, but there are different blackbody curves in different
directions. The temperature $T$ of the CMBR is a real-valued
function $T(\theta,\phi)$ upon the celestial sphere. Let $\langle
T \rangle$ denote the mean temperature, averaged over the entire
celestial sphere. The function
$$
\delta T(\theta,\phi) = \frac{T(\theta,\phi)- \langle T
\rangle}{\langle T \rangle} \equiv \frac{\Delta T}{\langle T
\rangle}(\theta,\phi)
$$ expresses the temperature deviations (or `fluctuations') as a
fraction of the mean temperature (Coles and Lucchin 1995, p92).
This temperature fluctuation function is itself a real-valued
function upon the celestial sphere, and one can decompose it into
an infinite linear combination of the spherical harmonic functions
upon the sphere, (Coles and Lucchin 1995, p366),
$$
\delta T(\theta,\phi) = \sum_{l = 0}^{\infty} \sum_{m = -l}^{m=+l}
c_{lm} Y^m_l(\theta,\phi) \,.
$$ Note that on a specific celestial sphere, the coefficients
$c_{lm}$ which define the function $\delta T(\theta,\phi)$ are not
functions of $(\theta,\phi)$ themselves. $\delta T(\theta,\phi)$
is a function of $(\theta,\phi)$ because the $Y^m_l(\theta,\phi)$
are functions of $(\theta,\phi)$. The coefficients $c_{lm}$ only
vary across the statistical ensemble of all possible celestial
spheres within our universe.
The spherical harmonics $\{Y^m_l(\theta,\phi): \, l \in
\mathbb{N}, \, m \in (-l,-l+1,...,+l)\}$ form an orthonormal basis
of the Hilbert space $L^2(S^2)$ of square-integrable functions
upon the sphere. They can be defined as
$$
Y^m_l(\theta,\phi) = N^m_l P^{| m |}_l(\cos \theta)e^{im\phi} \, ,
$$ with $ N^m_l$ a normalization constant, and $P^{| m
|}_l(u)$ a Legendre function. Any square-integrable function
$f(\theta,\phi)$ on $S^2$ can then be expressed as a linear
combination
$$
f(\theta,\phi) = \sum_{l = 0}^{\infty} \sum_{m = -l}^{m=+l} c_{lm}
Y^m_l(\theta,\phi) \, ,
$$ with the spherical harmonic coefficients $c_{lm}$ given by
$$
c_{lm} = \langle Y^m_l,f \rangle = \int_{S^2}
\overline{Y}^m_l(\theta,\phi)f(\theta,\phi) \, d\Omega \,.
$$ Note that the angular brackets here denote the inner product on
the space of functions on $S^2$, not to be confused with the use
of angular brackets to denote a mean value.
Physicists tend to refer to the terms in a spherical harmonic
decomposition as `modes'. The term corresponding to $l=0$ is
referred to as the monopole term, $l=1$ terms are called dipole
terms, $l=2$ terms are quadrupole terms, etc. A dipole anisotropy
in the temperature of the CMBR is a periodic variation which
completes $1$ cycle around the sky; it has one `hot' pole and one
`cold' pole. A quadrupole anisotropy is a periodic variation in
the temperature of the CMBR which completes $2$ cycles around the
sky. Mode $l$ anisotropies complete $l$ cycles around the sky.
Higher $l$ modes correspond to temperature fluctuations on smaller
angular scales. For higher $l$ modes, the angular scale
$\vartheta$ of the fluctuation is $\vartheta \approx 60\deg/l$,
(Coles and Lucchin 1995, p367). After subtracting the effects of
the Earth's diurnal rotation, its orbit around the Sun, the motion
of the Sun within the Milky Way galaxy, and the motion of the
Milky Way within the Local Group, we observe from the Earth a
dipole anisotropy in the CMBR upon the celestial sphere. This is a
dipole anisotropy upon our own private celestial sphere due to the
proper motion of the Local Group of galaxies at approx. $600km
s^{-1}$. This dipole anisotropy in the temperature of the CMBR can
be expressed as (Coles and Lucchin 1995, p93)
$$
T(\vartheta) = \langle T \rangle + \Delta T_{dipole} \cos
\vartheta \,.
$$
It is only when one calculates the effect of the proper motion of
the Local Group, and one `subtracts' that effect from the observed
CMBR, that one obtains radiation which is uniform across the
celestial sphere, to at least one part in $10,000$, $\Delta
T/\langle T \rangle < 10^{-4}$, on any angular scale. After
compensating for the effect of our proper motion, the average
temperature of the CMBR is approximately $2.7K$.
The COBE satellite discovered in 1992 that superimposed upon the
dipole temperature anisotropy, there are very small scale
variations in the temperature of the microwave blackbody spectrum
across the entire celestial sphere.
Because radiation was in equilibrium with matter just before they
decoupled, the variations in the CMBR indicate variations in the
density of matter at the time of decoupling. These variations are
believed to be the origins of what have today become galaxies. In
a FRW model, the subsequent formation of galaxies has a negligible
effect upon the CMBR. Thus, the variations in the CMBR are thought
to indicate inhomogeneity at the so-called `epoch of last
scattering'.
Of deep observational significance at the present time is the CMBR
\emph{angular power spectrum}. To clarify precisely what this is,
it will be necessary to carefully distinguish between two
different mathematical expressions. To obtain the first
expression, begin by noting that whilst the mean value of the
temperature fluctuations is zero, $\langle \delta T \rangle = 0$,
the variance, the mean value of the square of the fluctuations,
$\langle (\delta T)^2 \rangle$, is non-zero.
Consider $| \delta T |^2(\theta,\phi)$. Given the expansion of
$\delta T$ in the spherical harmonics, it follows that
$$
| \delta T |^2(\theta,\phi) = \sum_{l = 0}^{\infty} \sum_{m =
-l}^{m=+l} \sum_{l' = 0}^{\infty} \sum_{m' = -l'}^{m'=+l'}
c_{lm}^* c_{l'm'}
\overline{Y}^m_l(\theta,\phi)Y^{m'}_{l'}(\theta,\phi) \,.
$$ $\overline{Y}^m_l(\theta,\phi)$ and $Y^{m'}_{l'}(\theta,\phi)$
don't vary over the ensemble of all celestial spheres, so if
$\langle | \delta T |^2 \rangle (\theta,\phi)$ is taken to be the
mean value of $| \delta T |^2(\theta,\phi)$ over the ensemble, it
can be expressed as
$$
\langle | \delta T |^2 \rangle (\theta,\phi) = \sum_{l =
0}^{\infty} \sum_{m = -l}^{m=+l} \sum_{l' = 0}^{\infty} \sum_{m' =
-l'}^{m'=+l'} \langle c_{lm}^* c_{l'm'} \rangle
\overline{Y}^m_l(\theta,\phi)Y^{m'}_{l'}(\theta,\phi) \,.
$$ Now, given that $\langle c_{lm}^* c_{l'm'} \rangle = \langle | c_{lm} |^2
\rangle \delta_{l l'}\delta_{m m'}$, this expression reduces to
$$
\langle | \delta T |^2 \rangle (\theta,\phi) = \sum_{l =
0}^{\infty} \sum_{m = -l}^{m=+l} \langle | c_{lm} |^2 \rangle |
Y^{m}_{l}(\theta,\phi) | ^2 \,.
$$ Noting that $\delta T$ is real-valued, this means
$$
\langle (\delta T) ^2 \rangle (\theta,\phi) = \sum_{l =
0}^{\infty} \sum_{m = -l}^{m=+l} \langle | c_{lm} |^2 \rangle |
Y^{m}_{l}(\theta,\phi) | ^2 \,.
$$
This expression is clearly dependent on $(\theta,\phi)$. A second
approach yields an expression with no such dependence:
The function $\delta T(\theta,\phi)$ is a vector in the Hilbert
space of functions $L^2(S^2)$. This space of functions, as a
Hilbert space, is equipped with an inner product $\langle \; , \;
\rangle$, and a norm $\| \; \|$. (Again, the angular brackets of
the inner product here should not be confused with the angular
brackets that define a mean value). The norm defines the length of
a vector in the vector space of functions. Consider the square of
the norm $\| \delta T \|^2$ of the function $\delta
T(\theta,\phi)$:
$$\eqalign{
\| \delta T \|^2 &= \langle \delta T, \delta T \rangle \cr &=
\left \langle \sum_{l = 0}^{\infty} \sum_{m = -l}^{m=+l} c_{lm}
Y^m_l(\theta,\phi), \sum_{l = 0}^{\infty} \sum_{m = -l}^{m=+l}
c_{lm} Y^m_l(\theta,\phi) \right \rangle \cr &= \sum_{l =
0}^{\infty} \sum_{m = -l}^{m=+l} | c_{lm} |^2 \,.}
$$ This follows because $\langle Y^m_l(\theta,\phi),
Y^{m'}_{l'}(\theta,\phi)\rangle = \delta_{l l'}\delta_{m m'}$
Using angular brackets to denote the mean once again, $\langle \|
\delta T \|^2 \rangle$ denotes the mean of the squared length of
the function vector, taken over all possible celestial spheres.
From the last expression, it follows that
$$
\langle \| \delta T \|^2 \rangle = \sum_{l = 0}^{\infty} \sum_{m =
-l}^{m=+l} \langle | c_{lm} |^2 \rangle \,.
$$ This is the sum of the mean of the square modulus value of all the
coefficients from the spherical harmonic expansion of $\delta T$.
The mean $\langle | c_{lm} |^2 \rangle$ is the mean of $| c_{lm}
|^2$ taken over the ensemble of celestial spheres. $| c_{lm} |^2$
is fixed for each celestial sphere.
By an ergodic hypothesis, for large $l$ this average is
approximated by an average taken over all the modes with the same
$l$ on our private celestial sphere
$$
\sum_{m = -l}^{m=+l} \langle | c_{lm} |^2 \rangle =
\frac{1}{(2l+1)}\sum_{m=-l}^{m=+1} | c_{lm} |^2 \,.
$$
The angular power spectrum is
$$
C_l = \frac{1}{(2l+1)}\sum_{m=-l}^{m=+1} | c_{lm} |^2 \,.
$$ Hence
$$
\langle \| \delta T \|^2 \rangle = A + \sum_{l = l_b}^{\infty}C_l
\,.
$$ $A$ is the contribution from the small $l$ spherical harmonics, and
$l_b$ is the lower bound at which the ergodic hypothesis becomes
valid. For large $l$, $C_l$ is the contribution to the mean of the
squared length of the temperature fluctuation function vector from
the mode $l$ spherical harmonics.
The value of $l$ for the highest peak in the CMBR power spectrum
corresponds to hot and cold spots of a specific angular size on
the celestial sphere, (Tegmark 2002, p2). The exact angular size
of these spots can be used to determine if the curvature of space
is positive, negative or zero. If the peak in the CMBR power
spectrum corresponds to spots which subtend a specific value close
to $0.5 \deg$, then space is flat, (2002, p2). If space has
positive curvature, then the angles of a triangle add up to more
than $180 \deg$, and the size of the CMBR spots would be greater
than $0.5 \deg$. If space has negative curvature, then the angles
of a triangle add up to less than $180 \deg$, and the size of the
CMBR spots would be less than $0.5 \deg$. The current data on the
CMBR indicates that the spot size is very close to $0.5 \deg$, but
cannot determine the exact value. Thus, the current data merely
confirms the long-held belief that the curvature of space is very
close to zero.
Tegmark falsely states that ``many of the most mathematically
elegant models, negatively curved yet compact spaces, have been
abandoned after the recent evidence for spatial flatness," (2002,
p3). Unless Tegmark means that the evidence indicates a $k=0$
universe, (which it doesn't), this remark might betray the
misunderstanding of the rigidity theorem for hyperbolic manifolds
alluded to before. Negative values of spatial curvature very close
to zero exclude the possibility of a compact hyperbolic universe
which is sufficiently small for the topology to be detectable, but
it does not exclude the possibility that the spatial universe does
have a compact hyperbolic topology.
The CMBR power spectrum can also be used to determine whether our
spatial universe is a \emph{small} compact universe. Whilst a
compact universe of volume much greater than the Hubble volume
would leave no imprint upon the CMBR, a small compact universe
would affect the CMBR power spectrum on large angular scales, and
could leave paired circles in the CMBR at antipodal positions on
the celestial sphere. No paired circles have been discovered, but
the WMAP satellite has revealed anomalies in the CMBR power
spectrum on large angular scales. The quadrupole $l=2$ mode was
found to be about $1/7$ the strength predicted for an infinite
flat universe, while the octopole $l=3$ mode was $72\%$ of the
strength predicted for such a non-compact $k=0$ universe, (Luminet
\textit{et al} 2003, p3).
Tegmark states that ``the interim conclusion about the overall
shape of space is thus `back to basics': although mathematicians
have discovered a wealth of complicated manifolds to choose from
and both positive and negative curvature would have been allowed
\emph{a priori}, all available data so far is consistent with the
simplest possible space, the infinite flat Euclidean space that we
learned about in high school," (2002, p3). As emphasised above, it
is also the case that all the data remains consistent with
positive or negative curvature, and with multiply connected
topology as well as simply connected topology. No such `back to
basics' conclusion can be drawn.
\hfill \break
The present universe only approximates a FRW model on length
scales greater than $100$Mpc. On smaller length scales, the
universe exhibits large inhomogeneities and anisotropies. The
distribution of matter is characterised by walls, filaments and
voids up to $100$Mpc, with large peculiar velocities relative to
the rest frame defined by the CMBR.
Whilst the CMBR indicates that the matter in the universe was
spatially homogeneous to a high degree when the universe was
$10^{4}-10^{5}$yrs old, the distribution of galaxies is an
indicator of the distribution of matter in the present era, when
the universe is $\sim 10^{10}$ yrs old. Given perturbations from
exact homogeneity which were sufficiently large relative to the
speed of expansion when the universe was $10^{4}-10^{5}$yrs in
age, one would expect the degree of homogeneity to decrease with
the passage of time. Small initial inhomogeneities result in some
regions which are denser than the average. A positive feedback
process then ensues. The regions of greater than average density
gravitationally attract matter from the surroundings, thus
increasing the excess density of matter. As the excess density of
matter increases, a greater force is exerted on the surrounding
matter, thus continuing to increase the agglomeration of matter.
Gravity magnifies small initial inhomogeneities. Hence, the FRW
models become increasingly inaccurate as the universe gets older.
The length scale on which the universe can be idealised as being
homogeneous, grows as a function of time, hence the length scale
on which a FRW model is valid, grows as a function of time. Not
only do the FRW models constitute a first approximation, but they
constitute an increasingly inaccurate first approximation.
|
1,116,691,497,762 | arxiv | \section{Omitted Details for Section~\ref{sec:motivatinguni}}\label{sect:recurreldetails}
\lstset{language=prog}
\lstset{tabsize=3}
\newsavebox{\progsearch}
\begin{lrbox}{\progsearch}
\begin{lstlisting}[mathescape]
$\mathsf{randsearch}(ar,i,j,d)$ {
1: if ($i=j$ and $ar[i]\ne d$)
2: return $-1$;
3: else if ($i=j$ and $ar[i]=d$)
4: return $i$;
5: else
6: $k\leftarrow \mathrm{uniform}(i,j)$;
7: if ($ar[k]=d$)
8: return $k$;
9: else if ($ar[k]<d$ and $k<j$)
10: return $\mathsf{randsearch}(ar, k+1, j,d)$;
11: else if ($ar[k]>d$ and $i<k$)
12: return $\mathsf{randsearch}(ar,1,k-1,d)$;
13: else
14: return $-1$;
end if
end if
}
\end{lstlisting}
\end{lrbox}
\begin{figure}
\centering
\usebox{\progsearch}
\caption{Sherwood's {\sc Randomized-Search}}
\label{fig:randsearch}
\end{figure}
\noindent{\bf Example~\ref{ex:randsearch}.} [{\sc Randomized-Search}]
Consider the Sherwood's {\sc Randomized-Search\ } algorithm (cf.~\cite[Chapter~9]{McConnellbook}) depicted in Fig.~\ref{fig:randsearch}.
The algorithm checks whether an integer value $d$ is present within the index range $[i,j]$ ($0\le i\le j$)
in an integer array $ar$ which is sorted in increasing order and is without duplicate entries.
The algorithm outputs either the index for $d$ in $ar$ or $-1$ meaning that $d$ is not present in
the index range $[i,j]$ of $ar$.
The description of the pseudo-code is as follows.
The first four lines deal with the base case when there is only one index in the index range.
The remaining lines deal with the recursive case:
in line 6, an index $k$ is uniformly sampled from $\{i,i+1,\dots,j\}$;
line 7--8 check whether $k$ is the output;
line 9--12 perform the recursive calls depending on whether $ar[k]<d$ or not;
finally, line 13--14 handle the case when $d<ar[i]$ or $d>ar[j]$.
Let $T:\Nset\rightarrow\Nset$ be the function such that for any $n\in\Nset$, we have
$T(n)$ is the supremum of the expected execution times upon all inputs $(ar,i,j)$ with $j-i+1=n$.
We derive a recurrence relation for $T$ as follows.
Let $n\in\Nset$ and $(ar,i,j), d$ be any input such that $n=j-i+1$.
We clarify two cases below:
\begin{enumerate}
\item there exists an $i\le k^*< j$ such that $ar[k^*]\le d < ar[k^*+1]$, where $ar[j+1]$ is interpreted $\infty$ here;
\item $ar[j]\le d$ or $d< ar[i]$.
\end{enumerate}
In both cases, we have $T(1)=1$.
In Case 1, we deduce from the pseudo-code in Fig.~\ref{fig:randsearch} that
\begin{displaymath}
T(n)\le 6+\frac{1}{n}\cdot \max\limits_{1\le \ell^*< n} \left(\displaystyle\sum_{\ell=1}^{\ell^*} T(n-\ell)+ \displaystyle\sum_{\ell=\ell^*+1}^{n} T(\ell-1)\right)
\end{displaymath}
for all $n\ge 2$, where the maximum ranges over all $\ell^*:=k^*-i+1$'s.
In Case 2, similarly we deduce that
\begin{displaymath}
T(n)\le 6+\frac{1}{n}\cdot\max\left\{\displaystyle\sum_{\ell=1}^{n-1} T(n-\ell), ~~\displaystyle\sum_{\ell=2}^{n} T(\ell-1)\right\}
\end{displaymath}
Thus a preliminary version $G'$ of the recurrence relation is $\mathrm{T}(1)=1$ and
\[
\mathrm{T}(n)=6+\frac{1}{n}\cdot\max\limits_{1\le \ell^*< n} \left(\displaystyle\sum_{\ell=1}^{\ell^*} \mathrm{T}(n-\ell)+ \displaystyle\sum_{\ell=\ell^*+1}^{n} \mathrm{T}(\ell-1)\right) \\
\]
for all $n\ge 2$.
Let $T':\Nset\rightarrow\Rset$ be the unique solution to $G'$.
Then from the fact that $T'(2)\ge T'(1)$, by induction $T'$ is monotonically increasing.
Thus the maximum
\[
\max\limits_{1\le \ell^*<n} \left(\displaystyle\sum_{\ell=1}^{\ell^*} T'(n-\ell)+ \displaystyle\sum_{\ell=\ell^*+1}^{n} T'(\ell-1)\right)
\]
is attained at $\ell^*=\left\lfloor\frac{n}{2}\right\rfloor$ for all $n\ge 2$.
Then $G'$ is transformed into our final recurrence relation as follows:
\[
\begin{cases}
\mathrm{T}(\mathfrak{n})=6+\frac{1}{\mathfrak{n}}\cdot\left( \displaystyle\sum_{\mathfrak{j}=\left\lceil\frac{\mathfrak{n}}{2}\right\rceil}^{\mathfrak{n}-1}\mathrm{T}(\mathfrak{j})+ \displaystyle\sum_{\mathfrak{j}=\left\lfloor\frac{\mathfrak{n}}{2}\right\rfloor}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j})\right) \\
\mathrm{T}(1)=1
\end{cases}.
\]
We note that the worst-case complexity for this algorithm is $\Theta(n)$.\qed
\lstset{language=prog}
\lstset{tabsize=3}
\newsavebox{\progsort}
\begin{lrbox}{\progsort}
\begin{lstlisting}[mathescape]
$\mathsf{quicksort}(ar,i,j)$ {
1: if ($i<j$)
2: $k\leftarrow \mathrm{uniform}(i,j)$;
3: $m\leftarrow \mathsf{pivot}(ar,i,j,ar[k])$;
4: if ($i\le m-1$)
5: $\mathsf{quicksort}(ar,i,m-1)$;
end if
6: if ($m+1\le j$)
7: $\mathsf{quicksort}(ar,m+1,j)$;
end if
end if
}
\end{lstlisting}
\end{lrbox}
\begin{figure}
\centering
\usebox{\progsort}
\caption{Randomized {\sc Quick-Sort}}
\label{fig:quicksort}
\end{figure}
\noindent{\em Example \ref{ex:quicksort}.}[{\sc Quick-Sort}]
Consider the {\sc Quick-Sort} algorithm~\cite[Chapter~7]{DBLP:books/daglib/0023376} depicted in Fig.~\ref{fig:quicksort}, where every input $(ar,i,j)$ is assumed to satisfy that $0\le i\le j$ and $ar$ is an array of integers which does not contain duplicate numbers.
The description of the pseudo-code is as follows:
first, line 2 samples an integer uniformly from $\{i,\dots, j\}$;
then, line 3 calls a subroutine $\mathsf{pivot}$ which (i) rearranges $ar$ such that integers in $ar$ which are less than $ar[k]$ come first,
then $ar[k]$, and finally integers in $ar$ greater than $ar[k]$, and (ii) outputs the new index $m$ of $ar[k]$ in $ar$; and
finally, lines 4--7 handle recursive calls to sub-arrays.
From the pseudo-code, the following recurrence relation is easily obtained:
\[
\mathrm{T}(\mathfrak{n})=2\cdot\mathfrak{n}+ 2\cdot (\sum_{\mathfrak{j}=1}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j}))/{\mathfrak{n}}
\]
where $\mathrm{T}(\mathfrak{n})$ represents the maximal expected execution time where
$\mathfrak{n}$ is the array length and the execution time of {\em pivoting} is represented
by $2\cdot \mathfrak{n}$.
We note that the worst-case complexity for this algorithm is $\Theta(n^2)$.\qed
\lstset{language=prog}
\lstset{tabsize=3}
\newsavebox{\progdiameter}
\begin{lrbox}{\progdiameter}
\begin{lstlisting}[mathescape]
$\mathsf{diameter}(S)$ {
1: if ($|S|=1$)
2: return 0;
else
3: $p\leftarrow \mathrm{uniform}(S)$;
4: $d\leftarrow \max_{p'\in S} \mbox{\sl dist}(p,p')$
5: $U\leftarrow \bigcap_{p'\in S}\{p''\in\Rset^3 \mid \mbox{\sl dist}(p'',p')\le d\}$
6: $S'\leftarrow S\backslash U$
7: if ($S'=\emptyset$)
8: return $d$
9: else
10: return $\mathsf{diameter}(S')$
end if
end if
}
\end{lstlisting}
\end{lrbox}
\lstset{language=prog}
\lstset{tabsize=3}
\newsavebox{\progselect}
\begin{lrbox}{\progselect}
\begin{lstlisting}[mathescape]
$\mathsf{quickselect}(ar,i,j,d)$ {
1: if ($i=j$) return $a[i]$;
2: else
3: $k\leftarrow \mathrm{uniform}(i,j)$;
4: $m\leftarrow \mathsf{pivot}(ar,i,j,ar[k])$;
5: if ($m-i+1=d$)
6: return $ar[m]$;
7: else if ($m-i+1<d$)
8: return $\mathsf{quickselect}(ar,m+1,j,d)$;
9: else if ($m-i+1>d$)
10: return $\mathsf{quickselect}(ar,i,m-1,d)$;
end if
end if
}
\end{lstlisting}
\end{lrbox}
\begin{figure}
\begin{minipage}{0.55\textwidth}
\centering
\usebox{\progselect}
\caption{Randomized {\sc Quick-Select}}
\label{fig:quickselect}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\centering
\usebox{\progdiameter}
\caption{{\sc Diameter-Computation}}
\label{fig:diameter}
\end{minipage}
\end{figure}
\noindent{\bf Example~\ref{ex:quickselect}.} [{\sc Quick-Select}]
Consider the {\sc Quick-Select} algorithm (cf.~\cite[Chapter~9]{DBLP:books/daglib/0023376}) depicted in Fig.~\ref{fig:quickselect} which upon any input $(ar,i,j)$ and $d$ such that
$0\le i\le j$, $1\le d\le j-i+1$ and $ar$ contains no duplicate integers, finds the $d$-th largest integer in $ar$.
Note that for an array of size $n$, and $d=n/2$, we have the {\sc Median-Find} algorithm.
The description of the pseudo-code is as follows: line 1 handles the base case;
line 3 starts the recursive case by sampling $k$ uniformly from $\{i,\dots,j\}$;
line 4 rearranges $ar$ and returns an $m$ in the same way as $\mathsf{pivot}$ in {\sc Quick-Sort}
(cf. Example~\ref{ex:quicksort});
line 5 handles the case when $ar[k]$ happens to be the $d$-th largest integer in $ar$;
and finally, line 7--10 handle the recursive calls.
Let $T:\Nset\rightarrow\Nset$ be the function such that for any $n\in\Nset$, we have $T(n)$ is the supremum of the
expected execution times upon all inputs $(ar,i,j)$ with $j-i+1=n$.
By an analysis on where the $d$-th largest integer lies in $ar$ which is similar to the analysis on $d$ in Example~\ref{ex:randsearch},
a preliminary recurrence relation is obtained such that $\mathrm{T}(1)=1$ and
\[
\mathrm{T}(n)=4+2\cdot n+\frac{1}{n}\cdot\max\limits_{1\le \ell^*\le n} \left(\displaystyle\sum_{\ell=1}^{\ell^*-1} \mathrm{T}(n-\ell)+ \displaystyle\sum_{\ell=\ell^*+1}^{n} \mathrm{T}(\ell-1)\right). \\
\]
By similar monotone argument in Example~\ref{ex:randsearch}, the maximum of the right-hand-side expression above is attained at $\ell^*=\left\lfloor\frac{n+1}{2}\right\rfloor$ for all $n\ge 2$.
By the fact that $\left\lfloor\frac{n+1}{2}\right\rfloor=\left\lceil \frac{n}{2}\right\rceil$ for all $n\ge 2$, the following recurrence relation is obtained:
\[
\begin{cases}
\mathrm{T}(\mathfrak{n})=4+2\cdot\mathfrak{n}+
\frac{1}{\mathfrak{n}}\cdot \left(\displaystyle\sum_{\mathfrak{j}=\left\lfloor \frac{n}{2}\right\rfloor+1}^{n-1} \mathrm{T}(\mathfrak{j})+ \displaystyle\sum_{\mathfrak{j}=\left\lceil \frac{n}{2}\right\rceil}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j})\right)\\
\mathrm{T}(1)=1
\end{cases}
\]
To fit our univariate recurrence expression, we use over-approximation, and the final recurrence relation for this example is
\[
\begin{cases}
\mathrm{T}(\mathfrak{n})\!=\!4+2\cdot\mathfrak{n}+
\frac{1}{\mathfrak{n}}\cdot \left(\displaystyle\sum_{\mathfrak{j}=\left\lfloor \frac{n}{2}\right\rfloor}^{n-1} \mathrm{T}(\mathfrak{j})+ \displaystyle\sum_{\mathfrak{j}=\left\lceil \frac{n}{2}\right\rceil}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j})\right)\\
\mathrm{T}(1)=1
\end{cases}.
\]
We note that the worst-case complexity for this algorithm is $\Theta(n^2)$.\qed
\noindent{\em Example~\ref{ex:diameter}.}[{\sc Diameter-Computation}]
Consider the {\sc Diameter-Computation} algorithm (cf.~\cite[Chapter 9]{DBLP:books/cu/MotwaniR95}) to compute the diameter of an input finite set $S$ of three-dimensional points.
A pseudo-code to implement this is depicted in Fig.~\ref{fig:diameter}.
The description of the pseudo-code is as follows:
line 1--2 handle the base case;
line 3 samples a point $p$ uniformly from $S$;
line 4 calculates the maximum distance in $S$ from $p$;
line 5 calculates the intersection of all balls centered at points in $S$ with uniform radius $d$;
line 6 calculates the set of points outside $U$;
lines 7--8 handle the situation $S'=\emptyset$ which implies that $d$ is the diameter;
lines 9--10 handle the recursive call to $S'$.
Due to uniform choice of $p$ at line 3, the size of $S'$ is uniformly in $[0,|S|-1]$;
it then follows a pivoting (similar to that in Example~\ref{ex:quickselect} and Example~\ref{ex:quicksort}) by line $5$
w.r.t the linear order over $\{\max_{p'\in S}{\mbox{\sl dist}(p,p')}\mid p\in S\}$.
Lines 5--6 can be done in $\mathcal{O}(|S|\cdot \log{|S|})$ time for Euclidean distance,
and $\mathcal{O}(|S|)$ time for $L_1$ metric~\cite{DBLP:books/cu/MotwaniR95}.
Depending on Eucledian or $L_1$ metric we obtain two different recurrence relations.
For Eucledian we have the following relation:
\[
\mathrm{T}(\mathfrak{n})=2+\mathfrak{n}+ 2\cdot \mathfrak{n}\cdot\ln{\mathfrak{n}} + (\sum_{\mathfrak{j}=1}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j}))/{\mathfrak{n}} ;
\]
with the execution time for lines 5--6 being taken to be $2\cdot \mathfrak{n}\cdot\ln{\mathfrak{n}}$, and
and for $L_1$ metric we have the following relation:
\[
\mathrm{T}(\mathfrak{n})=2+\mathfrak{n}+ 2\cdot \mathfrak{n} + (\sum_{\mathfrak{j}=1}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j}))/{\mathfrak{n}} \\
\]
with the execution time for lines 5--6 being taken to be
$2\cdot \mathfrak{n}$.
We note that the worst-case complexity for this algorithm is as follows:
for Euclidean metric it is $\Theta(n^2 \cdot \log n)$ and for the $L_1$ metric
it is $\Theta(n^2)$.\qed
\lstset{language=prog}
\lstset{tabsize=3}
\newsavebox{\progsortselect}
\begin{lrbox}{\progsortselect}
\begin{lstlisting}[mathescape]
$\mathsf{sortbyselect}(ar,i,j)$ {
1: if ($i<j$)
2: $m\leftarrow \mathsf{quickselect}(ar,i,j,\lfloor\frac{j-i+1}{2}\rfloor)$;
3: if ($i< m-1$)
4: $\mathsf{sortbyselect}(ar,i,m-1)$;
end if
5: if ($m+1<j$)
6: $\mathsf{sortbyselect}(ar,m+1,j)$;
end if
end if
}
\end{lstlisting}
\end{lrbox}
\begin{figure}
\centering
\usebox{\progsortselect}
\caption{Sorting with {\sc Quick-Select}}
\label{fig:sortselect}
\end{figure}
\noindent{\em Example \ref{ex:sortselect}.}[Sorting with {\sc Quick-Select}]
Consider a sorting algorithm depicted in Fig.~\ref{fig:sortselect}
which selects the median through the {\sc Quick-Select} algorithm.
The recurrence relation is directly obtained as follows:
\[
\mathrm{T}(\mathfrak{n})=4+ T^*(\mathfrak{n})+\mathrm{T}\left(\lfloor{\mathfrak{n}}/{2}\rfloor\right)+\mathrm{T}\left(\lceil{\mathfrak{n}}/{2}\rceil\right)
\]
where $T^*(\centerdot)$ is an upper bound on the expected running time of {\sc Quick-Select}
(cf. Example~\ref{ex:quickselect}).
We note that the worst-case complexity for this algorithm is $\Theta(n^2)$.\qed
\section{Omitted Details for Section~\ref{sec:motivatingbi}}\label{app:motivatingbi}
\noindent{\bf Example~\ref{ex:coupon}.}[{\sc Coupon-Collector}]
Consider the {\sc Coupon-Collector} problem~\cite[Chapter~3]{DBLP:books/cu/MotwaniR95} with $n$ different types of coupons ($n\in\Nset$).
The randomized process proceeds in rounds: at each round, a coupon is collected uniformly at random from the coupon types
(i.e., each coupon type is collected with probability $\frac{1}{n}$); and
the rounds continue until all the $n$ types of coupons are collected.
We model the rounds as a recurrence relation with two variables
$\mathfrak{n},\mathfrak{m}$, where $\mathfrak{n}$ represents the total number
of coupon types and $\mathfrak{m}$ represents the remaining number of uncollected
coupon types.
The recurrence relation is as follows:
\[
\mathrm{T}(\mathfrak{n},1)=\mathfrak{n}\cdot 1; \qquad
\mathrm{T}(\mathfrak{n},\mathfrak{m})=\mathfrak{n}/{\mathfrak{m}}+ \mathrm{T}(\mathfrak{n},\mathfrak{m}-1)
\]
where $\mathrm{T}(\mathfrak{n},\mathfrak{m})$ is the expected number of rounds, $\frac{\mathfrak{n}}{\mathfrak{m}}$
represents the expected number of rounds to collect a new (i.e., not-yet-collected) coupon type
when there are still $\mathfrak{m}$ type of coupons to be collected, and $\mathfrak{n}$ (for $\mathrm{T}(\mathfrak{n},1)$)
represents the expected number of rounds to collect a new coupon type when there is only one new coupon type to be collected.
We note that the worst-case complexity for this process is $\infty$.\qed
\noindent{\bf Example~\ref{ex:channel}.}[{\sc Channel-Conflict Resolution}]
We consider two network scenarios in which $n$ clients are trying to get
access to a network channel.
This problem is also called the {\sc Resource-Contention Resolution}~\cite[Chapter~13]{Kleinbergbook}.
In this problem, if more than one client tries to access the channel,
then no client can access it, and if exactly one client requests access to
the channel, then the request is granted.
While centralized deterministic algorithms exist (such as Round-Robin) for
the problem, to be implemented in a distributed or concurrent setting,
randomized algorithms are necessary.
\smallskip\noindent{\em Distributed setting.}
In the distributed setting, the clients do not share any information.
In this scenario, in each round, every client requests an access to the
channel with probability $\frac{1}{n}$.
We are interested in the expected number of rounds until every client gets
at least one access to the channel.
At each round, let $m$ be the number of clients who have not got any access.
Then the probability that a new client (from the $m$ clients) gets the access is $m\cdot \frac{1}{n}\cdot (1-\frac{1}{n})^{n-1}$.
Thus, the expected rounds that a new client gets the access is $\frac{n}{m}\cdot \frac{1}{(1-\frac{1}{n})^{n-1}}$.
Since the sequence $\left\{(1-\frac{1}{n})^{n-1}\right\}_{n\in\Nset}$ converges decreasingly to $\frac{1}{e}$ when
$n\rightarrow\infty$, this expected time is no greater than
$e\cdot\frac{n}{m}$.
Then for this scenario, we obtain an over-approximating recurrence relation
\[
\mathrm{T}(\mathfrak{n},1)=\mathfrak{n}\cdot 1; \qquad
\mathrm{T}(\mathfrak{n},\mathfrak{m})=(\mathfrak{n}\cdot{e})/{\mathfrak{m}}+ \mathrm{T}(\mathfrak{n},\mathfrak{m}-1)
\]
for the expected rounds until which every client gets at least one access to the channel.
Note that in this setting no client has any information about any other client.
\smallskip\noindent{\em Concurrent setting.}
In the concurrent setting, the clients share one variable, which is the number
of clients which has not yet been granted access.
Also in this scenario, once a client gets an access the client does not
request for access again.
Moreover, the shared variable represents the number of clients $m$
that have not yet got access.
In this case, in reach round a client that has not access to the channel yet,
requests access to the channel with probability $\frac{1}{m}$.
Then the probability that a new client gets the access becomes
$m\cdot \frac{1}{m}\cdot (1-\frac{1}{m})^{m-1}$.
It follows that the expected time that a new client gets the access becomes
$\frac{1}{(1-\frac{1}{m})^{m-1}}$
which is smaller than $e$.
Then for this scenario, we obtain an over-approximating recurrence relation
\[
\mathrm{T}(\mathfrak{n},1)=1\cdot 1; \qquad
\mathrm{T}(\mathfrak{n},\mathfrak{m})=1\cdot e+ \mathrm{T}(\mathfrak{n},\mathfrak{m}-1)
\]
We also note that the worst-case complexity for both is $\infty$.\qed
\section{Details for Overapproximations}\label{app:overapprox}
To prove results for overapproximations for recurrence expressions, we need the following well-known theorem.
\begin{theorem}[Taylor's Theorem (with Lagrange's Remainder){~\cite[Chapter 6]{BasicCalculus}}]
For any function $f:[a,b]\rightarrow \Rset$ ($a,b\in\Rset$ and $a<b$), if $f$ is ($k+1$)-order differentiable, then for all $x\in [a,b]$, there exists a $\xi\in (a,x)$ such that
\[
f(x)=\left(\sum_{j=0}^k \frac{f^{(j)}(a)}{j!}\cdot (x-a)^j\right)+\frac{f^{(k+1)}(\xi)}{(k+1)!}\cdot (x-a)^{k+1}~~.
\]
\end{theorem}
We also recall that
\[
\sum_{j=1}^{\infty}\frac{1}{j^2}=\frac{\pi^2}{6}\mbox{ and }\sum_{j=1}^{\infty}\frac{1}{j^3}=\alpha
\]
where $\alpha$ is the Ap\'{e}ry's constant which lies in $[1.2020,1.2021]$.
Moreover, we have the following result using integral-by-part technique
and Newton-Leibniz Formula.
\begin{lemma}\label{lemm:basiccalculus}
For all $a,b\in (0,\infty)$ such that $a<b$, the following assertions hold:
\begin{eqnarray*}
\displaystyle
(1)\ \int_a^b \frac{1}{x}\,\mathrm{d}x = \ln{x}{\Big|}_{a}^b \enspace ; \displaystyle
~~~(2)\ \int_a^b \ln{x}\,\mathrm{d}x = \left(x\cdot\ln{x}-x\right){\Big|}_{a}^b \enspace\\[1ex]
\displaystyle
(3)\ \int_a^b x\cdot\ln{x}\,\mathrm{d}x = \left(\frac{1}{2}\cdot x^2\cdot\ln{x}-\frac{1}{4}\cdot x^2\right){\Big|}_{a}^b~~ \enspace.
\end{eqnarray*}
\end{lemma}
Furthermore, we need the following simple lemmas.
The following lemma provides a tight approximation for floored expressions, the proof of which is a simple case distinction between even and odd cases.
\begin{lemma}\label{lemm:flooroverapprox}
For all natural numbers $n$, we have
$\frac{n-1}{2}\le \left\lfloor\frac{n}{2}\right\rfloor\le \frac{n}{2}\le \left\lceil\frac{n}{2}\right\rceil \le \frac{n+1}{2}$~~.
\end{lemma}
The following lemma handles over-approximation of simple summations.
\begin{lemma}\label{lemm:sumoverapprox}
For any natural number $n\ge 2$ and real number $c$, one has that
$\frac{\sum_{j=1}^{n-1} c}{n}\le c\mbox{ and }\frac{\left( \sum_{j=\left\lceil\frac{\mathfrak{n}}{2}\right\rceil}^{n-1} c+ \sum_{j=\left\lfloor\frac{\mathfrak{n}}{2}\right\rfloor}^{n-1} c\right)}{n}\le c$~~.
\end{lemma}
Then we prove the following two propositions.
\textbf{Proposition~\ref{prop:lnflooroverapprox}.}
For any natural number $n\ge 2$, we have
\[
(1)\ \ln{n}-\ln{2}-\frac{1}{n-1}\le \ln{\left\lfloor \frac{n}{2}\right\rfloor}\le \ln{n}-\ln{2} \enspace ;
\]
\[
(2)\ \ln{n}-\ln{2}\le \ln{\left\lceil \frac{n}{2}\right\rceil}\le \ln{n}-\ln{2}+\frac{1}{n}~~.
\]
\begin{proof}
Let $n\ge 2$ be a natural number. The first argument comes from the facts that
\[
\ln{\left\lfloor \frac{n}{2}\right\rfloor}\le \ln{\frac{n}{2}}=\ln{n}-\ln{2}
\]
and
\begin{align*}
\ln{\left\lfloor \frac{n}{2}\right\rfloor} & \ge \ln{\frac{n-1}{2}} \\
& = \ln{\frac{n}{2}}-\left(\ln{\frac{n}{2}}-\ln{\frac{n-1}{2}}\right) \\
& = \ln{n}-\ln{2}-\frac{1}{2}\cdot \frac{1}{\xi_n}~~\left(\xi_n\in\left(\frac{n-1}{2},\frac{n}{2}\right)\right)\\
& \ge \ln{n}-\ln{2}-\frac{1}{n-1}
\end{align*}
where we use the fact that
\[
\left\lfloor \frac{n}{2}\right\rfloor\ge \frac{n-1}{2}
\]
and $\xi_n$ is obtained from Taylor's Theorem.
The second argument comes from the facts that
\[
\ln{\left\lceil \frac{n}{2}\right\rceil}\ge \ln{\frac{n}{2}}=\ln{n}-\ln{2}
\]
and
\begin{align*}
\ln{\left\lceil \frac{n}{2}\right\rceil} & \le \ln{\frac{n+1}{2}}\\
& =\ln{\frac{n}{2}}+\left(\ln{\frac{n+1}{2}}-\ln{\frac{n}{2}} \right)\\
& =\ln{n} -\ln{2} + \left(\ln{\frac{n+1}{2}}-\ln{\frac{n}{2}} \right)\\
& =\ln{n}-\ln{2}+\frac{1}{2}\cdot \frac{1}{\xi'_n}\quad \left(\xi'_n\in \left(\frac{n}{2}, \frac{n+1}{2}\right)\right)\\
& \le \ln{n}-\ln{2}+\frac{1}{n}
\end{align*}
where the first inequality is due to the fact that
\[
\left\lceil \frac{n}{2}\right\rceil\le \frac{n+1}{2}
\]
and $\xi'_n$ is obtained from Taylor's Theorem.\qed
\end{proof}
\textbf{Proposition~\ref{prop:nminusoneoverapprox}.}
For any natural number $n\ge 2$, we have
\[
\ln{n}-\frac{1}{n-1}\le\ln{(n-1)}\le \ln{n}-\frac{1}{n} .
\]
\begin{proof}
The lemma follows directly from the fact that
\[
\ln{n}-\ln{(n-1)}=\frac{1}{\xi}
\]
for some $\xi\in (n-1,n)$, which can be obtained through Taylor's Theorem.\qed
\end{proof}
\noindent\textbf{Proposition~\ref{prop:integralapproximation}.}
For any natural number $n\geq 2$, we have:
\begin{equation}\label{eq:reciprocalapprox}
\int_1^n \frac{1}{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} \frac{1}{j}\in \left[-0.7552,-\frac{1}{6}\right]
\end{equation}
\begin{equation}\label{eq:logarithmapprox}
\int_1^n \ln{x}\,\mathrm{d}x-\left(\sum_{j=1}^{n-1} \ln{j}\right) - \frac{1}{2}\cdot \int_1^n \frac{1}{x}\,\mathrm{d}x\in \left[-\frac{1}{12}, 0.2701\right]
\end{equation}
\begin{equation}\label{eq:xlogxapprox}
\int_1^n x\cdot \ln{x}\,\mathrm{d}x-\left(\sum_{j=1}^{n-1} j\cdot\ln{j}\right)-\frac{1}{2}\cdot\int_1^n \ln{x}\,\mathrm{d}x+\frac{1}{12}\cdot \int_1^n \frac{1}{x}\,\mathrm{d}x-\frac{n-1}{2}\in \left[-\frac{19}{72},0.1575\right].
\end{equation}
\begin{proof}
Let $n$ be a natural number such that $n\ge 2$.
We first estimate the difference
\[
\int_1^n \frac{1}{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} \frac{1}{j}~~.
\]
To this end, we deduce the following equalities:
\begin{align*}
&\int_1^n \frac{1}{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} \frac{1}{j} \\
= &\sum_{j=1}^{n-1}\int_j^{j+1} \left[\frac{1}{x}-\frac{1}{j}\right]\,\mathrm{d}x \\
= &\sum_{j=1}^{n-1}\int_0^{1} \left[\frac{1}{j+x}-\frac{1}{j}\right]\,\mathrm{d}x \\
= & \sum_{j=1}^{n-1}\int_0^{1} \left[-\frac{1}{j^2}\cdot x +\frac{1}{\xi^3_{j,x}}\cdot x^2\right]\,\mathrm{d}x\\
= & -\frac{1}{2}\cdot \left(\sum_{j=1}^{n-1}\frac{1}{j^2}\right)+ \sum_{j=1}^{n-1}\int_0^{1}\frac{1}{\xi^3_{j,x}}\cdot x^2\,\mathrm{d}x~~,\\
\end{align*}
where $\xi_{j,x}$ is a real number in $(j, j+x)$ obtained from Taylor's Theorem with Lagrange's Remainder.
The first and fourth equalities come from the linear property of Riemann Integral;
the second one follows from the variable substitution $x'=x-j$;
the third one follows from Taylor's Theorem.
Using the fact that $\xi_{j,x}\in (j, j+1)$, one obtains that
\begin{equation}\label{eq:reciprocalupperapprox}
\int_1^n \frac{1}{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} \frac{1}{j} \le -\frac{1}{2}\cdot \left(\sum_{j=1}^{n-1}\frac{1}{j^2}\right) + \frac{1}{3}\cdot \left(\sum_{j=1}^{n-1}\frac{1}{j^3}\right)
\end{equation}
and
\begin{equation}\label{eq:reciprocallowerapprox}
\int_1^n \frac{1}{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} \frac{1}{j} \ge -\frac{1}{2}\cdot \left(\sum_{j=1}^{n-1}\frac{1}{j^2}\right) + \frac{1}{3}\cdot \left(\sum_{j=2}^{n}\frac{1}{j^3}\right).
\end{equation}
Then (\ref{eq:reciprocalapprox}) follows from the facts that
\begin{align*}
\int_1^n \frac{1}{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} \frac{1}{j} & \le \sum_{j=1}^{n-1}\left(-\frac{1}{2\cdot j^2} + \frac{1}{3\cdot j^3}\right)\\
& \le -\frac{1}{2\cdot 1^2} + \frac{1}{3\cdot 1^3}\\
& =-\frac{1}{6}
\end{align*}
and
\begin{align*}
\int_1^n \frac{1}{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} \frac{1}{j} & \ge \sum_{j=1}^{n-1}\left(-\frac{1}{2\cdot j^2} + \frac{1}{3\cdot j^3}\right)-\frac{1}{3}+\frac{1}{3\cdot n^3}\\
& \ge -\frac{\pi^2}{12}+\frac{\alpha}{3}-\frac{1}{3}\\
& \ge -0.7552
\end{align*}
where in both situations we use the fact that $2\cdot j^2\le 3\cdot j^3$ for all $j\in\Nset$.
Then we consider the difference
\[
\int_1^n \ln{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} \ln{j}~.
\]
First, we derive that
\begin{align*}
&\int_1^n \ln{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} \ln{j} \\
= &\sum_{j=1}^{n-1}\int_j^{j+1} \left[\ln{x}-\ln{j}\right]\,\mathrm{d}x \\
= &\sum_{j=1}^{n-1}\int_0^{1} \left[\ln{(j+x)}-\ln{j}\right]\,\mathrm{d}x \\
= & \sum_{j=1}^{n-1}\int_0^{1} \left[\frac{1}{j}\cdot x -\frac{1}{2\cdot \xi^2_{j,x}}\cdot x^2\right]\,\mathrm{d}x\\
= & \frac{1}{2}\cdot \left(\sum_{j=1}^{n-1}\frac{1}{j}\right)- \sum_{j=1}^{n-1}\int_0^{1}\frac{1}{2\cdot \xi^2_{j,x}}\cdot x^2\,\mathrm{d}x \\
\end{align*}
where $\xi_{j,x}$ is a real number in $(j, j+1)$ obtained from Taylor's Theorem.
Using the fact that $\xi_{j,x}\in (j, j+1)$, one can obtain that
\begin{align}\label{eq:logarithmupperapprox}
& \int_1^n \ln{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} \ln{j} \\
\le & \frac{1}{2}\cdot \left(\sum_{j=1}^{n-1}\frac{1}{j}\right)-\frac{1}{6}\cdot\sum_{j=2}^{n}\frac{1}{j^2}\nonumber \\
\le & \frac{1}{2}\cdot \int_1^n \frac{1}{x}\,\mathrm{d}x + \frac{1}{4}\cdot \left(\sum_{j=1}^{n-1}\frac{1}{j^2}\right) - \frac{1}{6}\cdot \left(\sum_{j=2}^{n}\frac{1}{j^3}\right)\nonumber\\
&\qquad{}-\frac{1}{6}\cdot\sum_{j=2}^{n}\frac{1}{j^2} \nonumber \\
= & \frac{1}{2}\cdot \int_1^n \frac{1}{x}\,\mathrm{d}x + \sum_{j=1}^{n-1}\left(\frac{1}{12\cdot j^2}-\frac{1}{6\cdot j^3}\right)\nonumber\\
& \qquad{}+\frac{1}{3}-\frac{1}{6\cdot n^3}-\frac{1}{6\cdot n^2} \nonumber
\end{align}
where the second inequality follows from Inequality~(\ref{eq:reciprocallowerapprox}), and
\begin{align}\label{eq:logarithmlowerapprox}
& \int_1^n \ln{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} \ln{j} \\
\ge & \frac{1}{2}\cdot \left(\sum_{j=1}^{n-1}\frac{1}{j}\right)-\frac{1}{6}\cdot\sum_{j=1}^{n-1}\frac{1}{j^2}\nonumber\\
\ge & \frac{1}{2}\cdot\int_1^n \frac{1}{x}\,\mathrm{d}x+\frac{1}{12}\cdot \left(\sum_{j=1}^{n-1}\frac{1}{j^2}\right) - \frac{1}{6}\cdot \left(\sum_{j=1}^{n-1}\frac{1}{j^3}\right)\nonumber\\
= & \frac{1}{2}\cdot\int_1^n \frac{1}{x}\,\mathrm{d}x + \sum_{j=1}^{n-1}\left(\frac{1}{12\cdot j^2}-\frac{1}{6\cdot j^3}\right)\nonumber
\end{align}
where the second inequality follows from Inequality~(\ref{eq:reciprocalupperapprox}).
Then from Inequality~(\ref{eq:logarithmupperapprox}) and Inequality~(\ref{eq:logarithmlowerapprox}), one has that
\begin{align*}
& \int_1^n \ln{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} \ln{j} \\
\le & \frac{1}{2}\cdot\int_1^n \frac{1}{x}\,\mathrm{d}x+\sum_{j=1}^{\infty}\left(\frac{1}{12\cdot j^2}-\frac{1}{6\cdot j^3}\right)+\frac{1}{3} \\
\le & \frac{1}{2}\cdot \int_1^n \frac{1}{x}\,\mathrm{d}x+\frac{\pi^2}{72}-\frac{\alpha}{6}+\frac{1}{3} \\
\le & \frac{1}{2}\cdot \int_1^n \frac{1}{x}\,\mathrm{d}x+0.2701
\end{align*}
and
\begin{align*}
& \int_1^n \ln{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} \ln{j} \\
\ge & \frac{1}{2}\cdot\int_1^n \frac{1}{x}\,\mathrm{d}x+\left(\frac{1}{12}-\frac{1}{6}\right)\\
\ge & \frac{1}{2}\cdot\int_1^n \frac{1}{x}\,\mathrm{d}x -\frac{1}{12}
\end{align*}
where in both situations we use the fact that $12\cdot j^2\le 6\cdot j^3$ for all $j\ge 2$. The inequalities above directly imply the inequalities in (\ref{eq:logarithmapprox}).
Finally, we consider the difference
\[
\int_1^n x\cdot \ln{x}\,\mathrm{d}x-\sum_{j=m}^{n-1} j\cdot\ln{j}~~.
\]
Following similar approaches, we derive that for all natural numbers $n\ge 2$,
\begin{align*}
&\int_1^n x\cdot \ln{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} j\cdot\ln{j} \\
= &\sum_{j=1}^{n-1}\int_j^{j+1} \left[x\cdot \ln{x}-j\cdot\ln{j}\right]\,\mathrm{d}x \\
= &\sum_{j=1}^{n-1}\int_0^{1} \left[(j+x)\cdot \ln{(j+x)}-j\cdot\ln{j}\right]\,\mathrm{d}x \\
= & \sum_{j=1}^{n-1}\int_0^{1} \left[(\ln{j}+1)\cdot x +\frac{1}{2\cdot \xi_{j,x}}\cdot x^2\right]\,\mathrm{d}x\\
= & \frac{1}{2}\cdot \left(\sum_{j=1}^{n-1}\ln{j}\right)+\frac{n-1}{2}+ \sum_{j=1}^{n-1}\int_0^{1}\frac{1}{2\cdot \xi_{j,x}}\cdot x^2\,\mathrm{d}x \\
\end{align*}
where $\xi_{j,x}\in (j, j+1)$. Thus, one obtains that
\begin{align}\label{eq:xlogxupperapprox}
& \int_1^n x\cdot \ln{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} j\cdot\ln{j} \le \\
& \qquad\frac{1}{2}\cdot \left(\sum_{j=1}^{n-1}\ln{j}\right)+\frac{n-1}{2}+ \frac{1}{6}\cdot\sum_{j=1}^{n-1}\frac{1}{j}\nonumber
\end{align}
and
\begin{align}\label{eq:xlogxlowerapprox}
&\int_1^n x\cdot \ln{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} j\cdot\ln{j}\ge \\
&\qquad\frac{1}{2}\cdot \left(\sum_{j=1}^{n-1}\ln{j}\right)+\frac{n-1}{2}+\frac{1}{6}\cdot\sum_{j=1}^{n-1}\frac{1}{j}-\frac{1}{6}+\frac{1}{6\cdot n}.\nonumber
\end{align}
By plugging Inequalities in~(\ref{eq:reciprocallowerapprox}) and~(\ref{eq:logarithmlowerapprox}) into Inequality~(\ref{eq:xlogxupperapprox}), one obtains that
\begin{align*}
& \int_1^n x\cdot \ln{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} j\cdot\ln{j} \\
\le & \frac{1}{2}\cdot\left[\int_1^n \ln{x}\,\mathrm{d}x-\frac{1}{2}\cdot\int_1^n \frac{1}{x}\,\mathrm{d}x - \sum_{j=1}^{n-1}\left(\frac{1}{12\cdot j^2}-\frac{1}{6\cdot j^3}\right)\right] \\
& {}+\frac{n-1}{2}\\
& {}+\frac{1}{6}\cdot\left[\int_1^n \frac{1}{x}\,\mathrm{d}x +\frac{1}{2}\cdot \left(\sum_{j=1}^{n-1}\frac{1}{j^2}\right) - \frac{1}{3}\cdot \left(\sum_{j=2}^{n}\frac{1}{j^3}\right)\right] \\
\le & \frac{1}{2}\cdot\int_1^n \ln{x}\,\mathrm{d}x-\frac{1}{12}\cdot \int_1^n \frac{1}{x}\,\mathrm{d}x+\frac{n-1}{2}\\
& {}+\sum_{j=1}^{n-1}\left(\frac{1}{24\cdot j^2}+\frac{1}{36\cdot j^3}\right)+\frac{1}{18}-\frac{1}{18\cdot n^3} \\
\le & \frac{1}{2}\cdot\int_1^n \ln{x}\,\mathrm{d}x-\frac{1}{12}\cdot \int_1^n \frac{1}{x}\,\mathrm{d}x+\frac{n-1}{2}+\frac{\pi^2}{144}+\frac{\alpha}{36}+\frac{1}{18} \\
\le & \frac{1}{2}\cdot\int_1^n \ln{x}\,\mathrm{d}x-\frac{1}{12}\cdot \int_1^n \frac{1}{x}\,\mathrm{d}x+\frac{n-1}{2} + 0.1575
\end{align*}
for all natural numbers $n\ge 2$.
Similarly, by plugging Inequalities in (\ref{eq:reciprocalupperapprox}) and (\ref{eq:logarithmupperapprox}) into Inequality~(\ref{eq:xlogxlowerapprox}), one obtains
\begin{align*}
& \int_1^n x\cdot \ln{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} j\cdot\ln{j} \\
\ge & \frac{1}{2}\cdot\left[\int_1^n \ln{x}\,\mathrm{d}x-\frac{1}{2}\cdot\int_1^n \frac{1}{x}\,\mathrm{d}x - \sum_{j=1}^{n-1}\left(\frac{1}{12\cdot j^2}-\frac{1}{6\cdot j^3}\right)-\frac{1}{3}\right] \\
& {}+\frac{n-1}{2}\\
& {}+\frac{1}{6}\cdot\left[\int_1^n \frac{1}{x}\,\mathrm{d}x +\frac{1}{2}\cdot \left(\sum_{j=1}^{n-1}\frac{1}{j^2}\right) - \frac{1}{3}\cdot \left(\sum_{j=1}^{n-1}\frac{1}{j^3}\right)\right]-\frac{1}{6} \\
\ge & \frac{1}{2}\cdot\int_1^n \ln{x}\,\mathrm{d}x-\frac{1}{12}\cdot \int_1^n \frac{1}{x}\,\mathrm{d}x+\frac{n-1}{2}\\
& {}+\sum_{j=1}^{n-1}\left(\frac{1}{24\cdot j^2}+\frac{1}{36\cdot j^3}\right)-\frac{1}{3} \\
\ge & \frac{1}{2}\cdot\int_1^n \ln{x}\,\mathrm{d}x-\frac{1}{12}\cdot \int_1^n \frac{1}{x}\,\mathrm{d}x+\frac{n-1}{2}+\frac{1}{24}+\frac{1}{36}-\frac{1}{3}\\
= & \frac{1}{2}\cdot\int_1^n \ln{x}\,\mathrm{d}x-\frac{1}{12}\cdot \int_1^n \frac{1}{x}\,\mathrm{d}x+\frac{n-1}{2} -\frac{19}{72}
\end{align*}
Then the inequalities in (\ref{eq:xlogxapprox}) are clarified.\qed
\end{proof}
\noindent{\em Example~\ref{ex:overapprox}.}
Consider the summation
\[
\displaystyle\sum_{j=\left\lceil\frac{n}{2}\right\rceil}^{n-1}\ln{j}+ \displaystyle\sum_{j=\left\lfloor\frac{n}{2}\right\rfloor}^{n-1} \ln{j}\quad (n\ge 4).
\]
By Proposition~\ref{prop:integralapproximation}, we can over-approximate it as
\[
2\cdot\left(\Gamma_{\ln{\mathfrak{n}}}\left(n\right)+\frac{1}{12}\right)
-\left(\Gamma_{\ln{\mathfrak{n}}}\left(\left\lceil\frac{n}{2}\right\rceil\right)+\Gamma_{\ln{\mathfrak{n}}}\left(\left\lfloor\frac{n}{2}\right\rfloor\right)-0.5402\right)
\]
which is equal to
\begin{align*}
& 2\cdot n\cdot\ln{n}-2\cdot n-\ln{n}-\left\lceil\frac{n}{2}\right\rceil\cdot\ln{\left\lceil\frac{n}{2}\right\rceil}-\left\lfloor\frac{n}{2}\right\rfloor\cdot\ln{\left\lfloor\frac{n}{2}\right\rfloor}\\
&{}+\left\lceil\frac{n}{2}\right\rceil+\left\lfloor\frac{n}{2}\right\rfloor+\frac{\ln{\left\lfloor\frac{n}{2}\right\rfloor}}{2}+\frac{\ln{\left\lceil\frac{n}{2}\right\rceil}}{2}+\frac{1}{6}+0.5402.
\end{align*}
Then using Proposition~\ref{prop:lnflooroverapprox}, we can further obtain the following over-approximation
\begin{align*}
& 2\cdot n\cdot\ln{n}-2\cdot n-\ln{n}+0.7069-\frac{n}{2}\cdot\left(\ln{n}-\ln{2}\right)-\frac{n-1}{2}\cdot\left(\ln{n}-\ln{2}-\frac{1}{n-1}\right)\\
&{}+\frac{n+1}{2}+\frac{n}{2}+\frac{\ln{n}-\ln{2}}{2}+\frac{\ln{n}-\ln{2}+\frac{1}{n}}{2}
\end{align*}
which is roughly
$n\cdot\ln{n}-(1-\ln{2})\cdot n+\frac{1}{2}\cdot\ln{n}+0.6672+\frac{1}{2\cdot n}$.\qed
\section{Proofs for Sect.~\ref{sect:unisynth}}\label{app:unisynth}
\noindent\textbf{Lemma~\ref{lemm:unitrans}.}
Let $\mathfrak{f}\in\{\ln{\mathfrak{n}},\mathfrak{n},\mathfrak{n}\cdot\ln{\mathfrak{n}}\}$
and $c$ be a constant.
For all univariate recurrence expressions $\mathfrak{g}$, there exists pseudo-polynomials
$p$ and $q$ such that coefficients (i.e., $a_i,b_i$'s in~(\ref{eq:pseudopoly}))
of $q$ are all non-negative, $C_q>0$ and the following assertion holds:
for all $d>0$ and for all $n\ge 2$, with $h=d\cdot \mathsf{Subst}({\mathfrak{f}})+c$,
the inequality $\mathsf{OvAp}(\mathfrak{g}, h)(n)\le h(n)$ is equivalent to
$d\cdot p(n)\ge q(n)$.
\begin{proof}
From Definition~\ref{def:unioverapprox}, $n\mapsto n\cdot(n-1)\cdot\mathsf{OvAp}(\mathfrak{g}, h)(n)$ is a pseudo-polynomial.
Simple rearrangement of terms in inequality $\mathsf{OvAp}(\mathfrak{g}, h)(n)\le h(n)$ gives the desired
pseudo-polynomials.
Moreover, the fact that all coefficients in $\mathfrak{g}$ (from~(\ref{eq:unirecurrel})) are positive,
is used to derive that all coefficients of $q$ are non-negative and $C_q>0$.\qed
\end{proof}
\noindent\textbf{Proposition~\ref{prop:unisufflarge}.}
Let $p,q$ be pseudo-polynomials such that $C_q>0$ and
all coefficients of $q$ are non-negative.
Then there exists a real number $d>0$ such that
$d\cdot p(n)\ge q(n)$
for sufficiently large $n$ iff $\mathrm{deg}(p)\ge \mathrm{deg}(q)$ and $C_p>0$.
\begin{proof} We present the two directions of the proof.
(``\emph{If}'':) Suppose that $\mathrm{deg}(p)\ge \mathrm{deg}(q)$ and $C_p>0$.
Then the result follows directly from the facts that
(i) $\frac{q(n)}{p(n)}>0$ for sufficiently large $n$ and (ii) $\lim\limits_{n\rightarrow\infty}\frac{q(n)}{p(n)}$ exists and is non-negative.
(``\emph{Only-if}'':) Let $d$ be a positive real number such that $d\cdot p(n)\ge q(n)$ for sufficiently large $n$. Then $C_p>0$, or otherwise $d\cdot p(n)$ is either constantly zero or negative for sufficiently large $n$.
Moreover, $\mathrm{deg}(p)\ge \mathrm{deg}(q)$, since otherwise $\lim\limits_{n\rightarrow\infty}\frac{q(n)}{p(n)}=\infty$.\qed
\end{proof}
\noindent\textbf{Proposition~\ref{prop:unisufflargeN}.}
Consider two univariate pseudo-polynomials $p,q$ such that $\mathrm{deg}(p)\ge \mathrm{deg}(q)$, all coefficients of $q$ are non-negative and $C_p,C_q>0$.
Then given any $\epsilon\in (0,1)$,
\[
\frac{q(n)}{p(n)}\le \frac{\mathbf{1}_{\mathrm{deg}(p)=\mathrm{deg}(q)}\cdot \frac{C_q}{C_p}+\epsilon}{1-\epsilon}
\]
for all $n\ge N_{\epsilon,p,q}$ (for $N_{\epsilon,p,q}$ of Definition~\ref{def:unisuffN}).
\begin{proof}
Let $p,q$ be given in Definition~\ref{def:unisuffN}.
Fix an arbitrary $\epsilon\in (0,1)$ and let $N_{\epsilon,p,q}$ be given in Definition~\ref{def:unisuffN}.
Then for all $n\ge N_{\epsilon,p,q}$, (i) both $p(n),q(n)$ are positive and (ii)
\begin{align*}
\frac{q(n)}{\overline{p}(n)} &\le \sum_{i=1}^{k'} a'_i\cdot \frac{N^{i}\cdot \ln{N}}{\overline{p}(N)}+\sum_{i=1}^{\ell'} b'_i\cdot \frac{N^{i}}{\overline{p}(N)}\\
&\le \mathbf{1}_{\mathrm{deg}(p)=\mathrm{deg}(q)}\cdot\frac{C_q}{C_p}+\epsilon
\end{align*}
and
\begin{align*}
\frac{p(n)}{\overline{p}(n)} &\ge 1-\left[-1+\sum_{i=1}^{k} |a_i|\cdot \frac{N^{i}\cdot \ln{N}}{\overline{p}(N)}+\sum_{i=1}^{\ell} |b_i|\cdot \frac{N^{i}}{\overline{p}(N)}\right]\\
&\ge 1-\epsilon.
\end{align*}
It follows that for all $n\ge N_{\epsilon,p,q}$,
\[
\frac{q(n)}{p(n)}\le \frac{\mathbf{1}_{\mathrm{deg}(p)=\mathrm{deg}(q)}\cdot \frac{C_q}{C_p}+\epsilon}{1-\epsilon}~~.
\]
The desired result follows.\qed
\end{proof}
\noindent\textbf{Theorem~\ref{thm:soundnessunidec}.}[Soundness for $\mbox{\sl UniDec}$]
If $\mbox{\sl UniDec}$ outputs ``$\mbox{\sl yes}$'', then there exists a univariate guess-and-check function in form~(\ref{eq:uniguess})
for the inputs $G$ and $\mathfrak{f}$.
The algorithm is a linear-time algorithm in the size of the input recurrence relation.
\begin{proof}
From Definition~\ref{def:uniguess} and the special form (\ref{eq:uniguess}) for univariate guess-and-check functions, a function in form (\ref{eq:uniguess}) which satisfies the inductive argument of Definition~\ref{def:uniguess} can be modified to satisfy also the base condition of Definition~\ref{def:uniguess} by simply raising $d$ to a sufficiently large amount.
Then the correctness of the algorithm follows from Theorem~\ref{thm:uniguess} and the sufficiency of Proposition~\ref{prop:unisufflarge}.
Furthermore, the algorithm runs in linear time since the transformation from the inequality $\mathsf{OvAp}(\mathfrak{g}, h)(n)\le h(n)$ into $d\cdot p(n)\ge q(n)$ (cf. Lemma~\ref{lemm:unitrans}) takes linear time in the size of the input recurrence relation.
\qed
\end{proof}
\smallskip
\noindent\textbf{Theorem~\ref{thm:soundnessunisynth}.}[Soundness for $\mbox{\sl UniSynth}$]
If the algorithm $\mbox{\sl UniSynth}$ outputs a real number $d$, then $d\cdot \mathsf{Subst}(\mathfrak{f})+c$ is a univariate guess-and-check function for $G$.
\begin{proof}
Directly from the construction of the algorithm, Theorem~\ref{thm:uniguess}, Proposition~\ref{prop:unisufflarge} and Proposition~\ref{prop:unisufflargeN}.\qed
\end{proof}
\section{Detailed Experimental Results}\label{app:experiments}
The detailed experimental results are given in Table~\ref{tbl:detailedexperiments}.
We use $\checkmark$ to represent $\mbox{\sl yes}$ and $\times$ for $\mbox{\sl fail}$.
In addition to Table~\ref{tab:experiments}, we include values for $N_{\epsilon,p,q}$ in Definition~\ref{def:unisuffN}.
For the separable bivariate examples, recall that $n$ does not change, and in these examples, the reduction
to the univariate case is the function of $m$.
\begin{table*}
\centering
\begin{tabular}{ |c|c|c|c|c|c|c| }
\hline
\multirow{2}{*}{\sc Program} & \multirow{2}{*}{\sc $\mathfrak{f}$} & \multirow{2}{*}{\sc UniDec } & \multicolumn{4}{|c|}{\sc UniSynth(\checkmark)}\\
\cline{4-7}
& & & $\epsilon$ & $N_{\epsilon,p,q}$ & $d$ & $d_{100}$ \\
\hline
\hline
\multirow{4}{*}{\sc R.-Sear.} & \multirow{4}{*}{\sc $\ln \mathfrak{n}$}
& \multirow{4}{*}{\sc \checkmark} & $0.5$ & $13$ & $40.107$ & \multirow{4}{*}{\sc $15.129$ } \\
\cline{4-6}
&&& $0.3$ & $25$ & $28.363$ & \\
\cline{4-6}
&&& $0.1$ & $97$ & $21.838$ & \\
\cline{4-6}
&&& $0.01$ & $1398$ & $19.762$ &\\
\hline
\hline
\multirow{6}{*}{\sc Q.-Sort} & $\ln \mathfrak{n}$ & $\times$ & \multirow{2}{*}{\sc -} & \multirow{2}{*}{\sc -} & \multirow{2}{*}{\sc -} &\multirow{2}{*}{\sc -} \\
\cline{2-3}
& $\mathfrak{n}$ & $\times$ & & & & \\
\cline{2-7}
& \multirow{4}{*}{\sc $\mathfrak{n}\ln \mathfrak{n}$}
& \multirow{4}{*}{\sc \checkmark} & $0.5$ & $10$ & $9.001$ & \multirow{4}{*}{\sc $3.172$ } \\
\cline{4-6}
&&& $0.3$ & $21$ & $6.143$ & \\
\cline{4-6}
&&& $0.1$ & $91$ & $4.556$ & \\
\cline{4-6}
&&& $0.01$ & $1458$ & $4.051$ &\\
\hline
\hline
\multirow{5}{*}{\sc Q.-Select} & $\ln \mathfrak{n}$ & $\times$ & - & - & - & - \\
\cline{2-7}
& \multirow{4}{*}{\sc $\mathfrak{n}$}
& \multirow{4}{*}{\sc \checkmark} & $0.5$ & $33$ & $17.001$ & \multirow{4}{*}{\sc $7.909$ } \\
\cline{4-6}
&&& $0.3$ & $54$ & $11.851$ & \\
\cline{4-6}
&&& $0.1$ & $160$ & $9.001$ & \\
\cline{4-6}
&&& $0.01$ & $1600$ & $8.091$ &\\
\hline
\hline
\multirow{6}{*}{\sc Diam. A} & $\ln \mathfrak{n}$ & $\times$ & \multirow{2}{*}{\sc -} & \multirow{2}{*}{\sc -} & \multirow{2}{*}{\sc -} & \multirow{2}{*}{\sc -} \\
\cline{2-3}
& $\mathfrak{n}$ & $\times$ & & & & \\
\cline{2-7}
& \multirow{4}{*}{\sc $\mathfrak{n}\ln \mathfrak{n}$}
& \multirow{4}{*}{\sc \checkmark} & $0.5$ & $3$ & $9.001$ & \multirow{4}{*}{\sc $4.525$ } \\
\cline{4-6}
&&& $0.3$ & $3$ & $6.143$ & \\
\cline{4-6}
&&& $0.1$ & $4$ & $4.556$ & \\
\cline{4-6}
&&& $0.01$ & $4$ & $4.525$ &\\
\hline
\hline
\multirow{5}{*}{\sc Diam. B} & $\ln \mathfrak{n}$ & $\times$ & - & - & - & - \\
\cline{2-7}
& \multirow{4}{*}{\sc $\mathfrak{n}$}
& \multirow{4}{*}{\sc \checkmark} & $0.5$ & $9$ & $13.001$ & \multirow{4}{*}{\sc $5.918$ } \\
\cline{4-6}
&&& $0.3$ & $14$ & $9.001$ & \\
\cline{4-6}
&&& $0.1$ & $40$ & $6.778$ & \\
\cline{4-6}
&&& $0.01$ & $400$ & $6.071$ &\\
\hline
\hline
\multirow{6}{*}{\sc Sort-Sel.} & $\ln \mathfrak{n}$ & $\times$ & \multirow{2}{*}{\sc -} & \multirow{2}{*}{\sc -} & \multirow{2}{*}{\sc -} & \multirow{2}{*}{\sc -} \\
\cline{2-3}
& $\mathfrak{n}$ & $\times$ & & & & \\
\cline{2-7}
& \multirow{4}{*}{\sc $\mathfrak{n}\ln \mathfrak{n}$}
& \multirow{4}{*}{\sc \checkmark} & $0.5$ & $18$ & $50.052$ & \multirow{4}{*}{\sc $16.000$} \\
\cline{4-6}
&&& $0.3$ & $29$ & $24.852$ & \\
\cline{4-6}
&&& $0.1$ & $87$ & $17.313$ & \\
\cline{4-6}
&&& $0.01$ & $866$ & $16.000$ & \\
\hline
\hline
\multirow{4}{*}{\sc Coupon} & \multirow{4}{*}{\sc $\mathfrak{n}\cdot\ln \mathfrak{m}$}
& \multirow{4}{*}{\sc \checkmark} & $0.5$ & $2$ & $3.001$ & \multirow{4}{*}{\sc $0.910$ } \\
\cline{4-6}
&&& $0.3$ & $2$ & $1.858$ & \\
\cline{4-6}
&&& $0.1$ & $2$ & $1.223$ & \\
\cline{4-6}
&&& $0.01$ & $2$ & $1.021$ &\\
\hline
\hline
\multirow{4}{*}{\sc Res. A} & \multirow{4}{*}{\sc $\mathfrak{n}\cdot\ln \mathfrak{m}$}
& \multirow{4}{*}{\sc \checkmark} & $0.5$ & $2$ & $6.437$ & \multirow{4}{*}{\sc $2.472$ } \\
\cline{4-6}
&&& $0.3$ & $2$ & $4.312$ & \\
\cline{4-6}
&&& $0.1$ & $2$ & $3.132$ & \\
\cline{4-6}
&&& $0.01$ & $2$ & $2.756$ &\\
\hline
\hline
\multirow{5}{*}{\sc Res. B} & $\ln \mathfrak{m}$ & $\times$ & - & - & - & -\\
\cline{2-7}
&\multirow{4}{*}{\sc $\mathfrak{m}$}
& \multirow{4}{*}{\sc \checkmark} & $0.5$ & $2$ & $6.437$ & \multirow{4}{*}{\sc $2.691$ } \\
\cline{4-6}
&&& $0.3$ & $2$ & $4.312$ & \\
\cline{4-6}
&&& $0.1$ & $2$ & $3.132$ & \\
\cline{4-6}
&&& $0.01$ & $2$ & $2.756$ &\\
\hline
\end{tabular}
\caption{Detailed experimental results where all running times (averaged over $5$ runs) are less than $0.02$ seconds
(between $0.01$ and $0.02$ seconds).}
\label{tbl:detailedexperiments}
\end{table*}
\section{Expected-Runtime Analysis}\label{sect:expruntime}
\vspace{-1em}
We focus on synthesizing logarithmic, linear, and almost-linear asymptotic bounds for recurrence relations.
Our goal is to decide and synthesize asymptotic bounds in the simple form:
$d\cdot \mathfrak{f}+\mathfrak{g}, \mathfrak{f}\in\{\ln{\mathfrak{n}},\mathfrak{n},\mathfrak{n}\cdot\ln{\mathfrak{n}}\}$ .
Informally, $\mathfrak{f}$ is the major term for time complexity, $d$ is the coefficient of $\mathfrak{f}$ to be synthesized,
and $\mathfrak{g}$ is the time complexity for the base case specified in (\ref{eq:unirecurrel}) or (\ref{eq:birecurrel}).
\smallskip\noindent{\bf Univariate Case:} The algorithmic problem in univariate case is as follows:
\begin{compactitem}
\item {\em Input:} a univariate recurrence relation $G$ taking the form (\ref{eq:unirecurrel}) and an
expression $\mathfrak{f}\in\{\ln{\mathfrak{n}},\mathfrak{n},\mathfrak{n}\cdot\ln{\mathfrak{n}}\}$.
\item {\em Output: Decision problem.} Output ``$\mbox{\sl yes}$'' if $T_G \in \mathcal{O}(\mathsf{Subst}(\mathfrak{f}))$, and ``$\mbox{\sl fail}$'' otherwise.
\item {\em Output: Quantitative problem.}
A positive real number $d$ such that
\begin{equation}\label{eq:uniguess}
T_G(n) \leq d\cdot \mathsf{Subst}(\mathfrak{f})(n)+c
\end{equation}
for all $n \geq 1$,
or ``$\mbox{\sl fail}$'' otherwise, where $c$ is from (\ref{eq:unirecurrel}).
\end{compactitem}
\begin{remark}
First note that while in the problem description we consider the form $\mathfrak{f}$
part of input for simplicity, since there are only three possibilites we can simply
enumerate them, and thus have only the recurrence relation as input.
Second, in the algorithmic problem above, w.l.o.g, we consider that every
$\mathfrak{e}$ in (\ref{eq:unirecurrel}) or (\ref{eq:birecurrel}) involves at least
one $\mathrm{T}(\centerdot)$-term and one non-$\mathrm{T}(\centerdot)$-term;
this is natural since (i) for algorithms with recursion at least one $\mathrm{T}(\centerdot)$-term
should be present for the recursive call and at least one non-$\mathrm{T}(\centerdot)$-term for non-recursive base step. \qed
\end{remark}
\smallskip\noindent{\bf Bivariate Case:} The bivariate-case problem is an extension of the univariate one,
and hence the problem definitions are similar, and we present them succinctly below.
\begin{compactitem}
\item {\em Input:} a bivariate recurrence relation $G$ taking the form (\ref{eq:birecurrel}) and an expression
$\mathfrak{f}$ (similar to the univariate case).
\item {\em Output: Decision problem.} Output ``$\mbox{\sl yes}$'' if $T_G \in \mathcal{O}(\mathsf{Subst}(\mathfrak{f}))$, and ``$\mbox{\sl fail}$'' otherwise;
\item {\em Output: Quantitative problem.}
A positive real number $d$ such that
$T_G(n,m) \leq d\cdot \mathsf{Subst}(\mathfrak{f})(n,m) +c\cdot\mathsf{Subst}(\mathfrak{h})(n)$
for all $n,m \geq 1$,
or ``$\mbox{\sl fail}$'' otherwise, where $c,\mathfrak{h}$ are from (\ref{eq:birecurrel}).
Note that in the expression above the term $\mathfrak{b}$ does not appear as it can be captured with
$\mathfrak{f}$ itself.
\end{compactitem}
Recall that in the above algorithmic problems obtaining the finite behaviour of
the recurrence relations is easy (through evaluation of the recurrences using
dynamic programming), and the interesting aspect is to decide the asymptotic
infinite behaviour.
\section{Related Work}
\vspace{-1em}
Automated program analysis is a very important problem with a long tradition~\cite{DBLP:journals/cacm/Wegbreit75}.
The following works consider various approaches for automated worst-case
bounds~\cite{DBLP:conf/aplas/HoffmannH10,DBLP:conf/esop/HoffmannH10,DBLP:conf/popl/HofmannJ03,DBLP:conf/esop/HofmannJ06,DBLP:conf/csl/HofmannR09,DBLP:conf/fm/JostLHSH09,DBLP:conf/popl/JostHLH10,Hoffman1,DBLP:conf/icfp/AvanziniLM15,DBLP:conf/se/SinnZV16} for amortized analysis,
and the SPEED project~\cite{SPEED1,SPEED2,DBLP:conf/cav/GulavaniG08}
for non-linear bounds using abstract interpretation.
All these works focus on the worst-case analysis, and do not consider
expected-runtime analysis.
Our main contribution is automated analysis of recurrence relations.
Approaches for recurrence relations have also been considered in the
literature.
Wegbreit~\cite{DBLP:journals/cacm/Wegbreit75} considered solving recurrence relations through either simple difference equations or generating functions.
Zimmermann and Zimmermann~\cite{Zimmermann1989} considered solving recurrence relations by transforming them into difference equations.
Grobauer~\cite{DBLP:conf/icfp/Grobauer01} considered generating recurrence relations from DML
for the worst-case analysis.
Flajolet~\emph{et al.}~\cite{DBLP:journals/dam/FlajoletGT92} considered allocation problems.
Flajolet~\emph{et al.}~\cite{DBLP:journals/tcs/FlajoletSZ91} considered solving recurrence relations
for randomization of combinatorial structures (such as trees) through generating functions.
The COSTA project~\cite{DBLP:journals/entcs/AlbertAGGPRRZ09,DBLP:conf/sas/AlbertAGP08,DBLP:conf/esop/AlbertAGPZ07}
transforms Java bytecode into recurrence relations and solves them through ranking functions.
Moreover, The PURRS tool~\cite{BagnaraPZZ05} addresses finite linear recurrences (with bounded summation), and some restricted linear infinite recurrence relations (with unbounded summation).
Our approach is quite different because we consider analyzing recurrence relations
arising from randomized algorithms and expected-runtime analysis through over-approximation of unbounded summations through integrals, whereas
previous approaches either consider recurrence relations for worst-case bounds or combinatorial structures, or use generating functions or difference equations to solve the recurrence relations.
For intraprocedural analysis ranking functions have been widely studied~\cite{BG05,DBLP:conf/cav/BradleyMS05,DBLP:conf/tacas/ColonS01,DBLP:conf/vmcai/PodelskiR04,DBLP:conf/pods/SohnG91,DBLP:conf/vmcai/Cousot05,DBLP:journals/fcsc/YangZZX10,DBLP:journals/jossac/ShenWYZ13},
which have then been extended to non-recursive probabilistic programs as
ranking supermartingales~\cite{SriramCAV,HolgerPOPL,DBLP:conf/popl/ChatterjeeFNH16,DBLP:conf/cav/ChatterjeeFG16,ChatterjeeNZ2017,DBLP:journals/corr/ChatterjeeF17}.
Such approaches are related to almost-sure termination, and not deriving
optimal asymptotic expected-runtime bounds (such as $\mathcal{O}(\log n)$, $\mathcal{O}(n \log n)$).
Proof rules have also been considered for recursive (probabilistic)
programs in~\cite{DBLP:journals/fac/Hesselink94,JonesPhdThesis,DBLP:conf/lics/OlmedoKKM16},
but these methods cannot be automated and require manual proofs.
\vspace{-1.5em}
\section{Conclusion}
\vspace{-1em}
In this work we considered efficient algorithms for automated analysis of randomized
recurrences for logarithmic, linear, and almost-linear bounds.
Our work gives rise to a number of interesting questions.
First, an interesting theoretical direction of future work would be to consider
more general randomized recurrence relations (such as with more than two variables,
or interaction between the variables).
While the above problem is of theoretical interest, most interesting examples are
already captured in our class of randomized recurrence relations as mentioned above.
Another interesting practical direction would be automated techniques
to derive recurrence relations from randomized recursive programs.
\vspace{-1.5em}
\subsubsection*{Acknowledgements}
We thank all reviewers for valuable comments.
The research is partially supported by Vienna Science and Technology Fund (WWTF) ICT15-003,
Austrian Science Fund (FWF) NFN Grant No. S11407-N23 (RiSE/SHiNE), ERC Start grant (279307: Graph Games), the Natural Science Foundation of China (NSFC) under Grant No. 61532019 and the CDZ project CAP (GZ 1023).
\section{Experimental Results}\label{sect:experiments}
\vspace{-1em}
We consider the classical examples illustrated in Section~\ref{sec:motivatinguni}
and Section~\ref{sec:motivatingbi}.
In Table~\ref{tab:experiments} for experimental results we consider the following recurrence relations $G$:
{\sc R.-Sear.} corresponds to the recurrence relation (\ref{eq:relrandsearch}) for Example~\ref{ex:randsearch};
{\sc Q.-Sort} corresponds to the recurrence relation (\ref{eq:relquicksort}) for Example~\ref{ex:quicksort};
{\sc Q.-Select} corresponds to the recurrence relation (\ref{eq:relquickselect}) for Example~\ref{ex:quickselect};
{\sc Diam. A} (resp. {\sc Diam. B}) corresponds to the recurrence relation (\ref{eq:reldiametera}) (resp. the recurrence relation (\ref{eq:reldiameterb})) for Example~\ref{ex:diameter};
{\sc Sort-Sel.} corresponds to recurrence relation (\ref{eq:relsortselect}) for Example~\ref{ex:sortselect}, where we use the result from setting $\epsilon=0.01$ in {\sc Q.-Select};
{\sc Coupon} corresponds to the recurrence relation (\ref{eq:relcoupon}) for Example~\ref{ex:coupon};
{\sc Res. A} (resp. {\sc Res. B}) corresponds to the recurrence relation (\ref{eq:relresourcea}) (resp. the recurrence relation (\ref{eq:relresourceb})) for Example~\ref{ex:channel}.
In the table, $\mathfrak{f}$ specifies the input asymptotic bound, $\epsilon$ and Dec is the input which specifies either we use algorithm $\mbox{\sl UniDec}$ or the synthesis algorithm $\mbox{\sl UniSynth}$ with the given $\epsilon$ value, and $d$ gives the value synthesized w.r.t the given $\epsilon$ ($\checkmark$ for $\mbox{\sl yes}$).
We describe $d_{100}$ below.
We need approximation for constants such as $e$ and $\ln{2}$, and use the interval $[2.7182,2.7183]$
(resp., $[0.6931, 0.6932]$) for tight approximation of $e$ (resp., $\ln{2}$).
\smallskip\noindent{\em The value $d_{100}$.}
For our synthesis algorithm we obtain the value $d$. The optimal value of the associated constant with the asymptotic bound, denoted $d^*$, is defined as
follows.
For $z\geq 2$, let
$d_{z}:=\max\left\{\frac{T_G(n)-c}{\mathsf{Subst}(\mathfrak{f})(n)}\mid 2\le n\le z\right\}$
($c$ is from (\ref{eq:unirecurrel})).
Then the sequence $d_z$ is increasing in $z$, and its limit is the optimal constant, i.e., $d^* =\lim_{z \to \infty} d_z$.
We consider $d_{100}$ as a lower bound on $d^*$ to compare against the value of $d$ we synthesize.
In other words, $d_{100}$ is the minimal value such that (\ref{eq:uniguess}) holds for $1\le n\le 100$, whereas
for $d^*$ it must hold for all $n$, and hence $d^* \geq d_{100}$.
Our experimental results show that the $d$ values we synthesize for $\epsilon=0.01$ is quite close
to the optimal value.
We performed our experiments on Intel(R) Core(TM) i7-4510U CPU, 2.00GHz, 8GB RAM.
All numbers in Table~\ref{tab:experiments} are over-approximated up to $10^{-3}$, and the running time of all experiments are less than $0.02$ seconds.
From Table~\ref{tab:experiments}, we can see that optimal $d$ are effectively over-approximated.
For example, for {\sc Quick-Sort} (Eq.~(\ref{eq:relquicksort})) (i.e, {\sc Q.-Sort} in the table), our algorithm detects $d=4.051$ and the optimal one lies somewhere in $[3.172, 4.051]$.
The experimental results show that we obtain the results extremely efficiently (less than $1/50$-th of a second).
For further details see Table~\ref{tbl:detailedexperiments} in Appendix~\ref{app:experiments}.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
Recur. Rel. & $\mathfrak{f}$ & $\epsilon,\mbox{\sl Dec} $ & $d$ & $d_{100}$ &
Recur. Rel. & $\mathfrak{f}$ & $\epsilon,\mbox{\sl Dec} $ & $d$ & $d_{100}$ \\
\hline
\multirow{5}{*}{{\sc R.-Sear.}} & \multirow{5}{*}{$\ln{\mathfrak{n}}$} & $\mbox{\sl UniDec}$ & $\mbox{\sl \checkmark}$ & \multirow{5}{*}{$15.129$}
& \multirow{5}{*}{{\sc Sort-Sel.}} & \multirow{5}{*}{$\mathfrak{n}\cdot\ln{\mathfrak{n}}$} & $\mbox{\sl UniDec}$ & $\mbox{\sl \checkmark}$ & \multirow{5}{*}{$16.000$} \\
\cline{3-4}\cline{8-9} & & $0.5$ & $40.107$ & & & & $0.5$ & $50.052$ & \\
\cline{3-4}\cline{8-9} & & $0.3$ & $28.363$ & & & & $0.3$ & $24.852$ & \\
\cline{3-4}\cline{8-9} & & $0.1$ & $21.838$ & & & & $0.1$ & $17.313$ & \\
\cline{3-4}\cline{8-9} & & $0.01$ & $19.762$ & & & & $0.01$ & $16.000$ & \\
\hline
\multirow{5}{*}{{\sc Q.-Sort}} & \multirow{5}{*}{$\mathfrak{n}\cdot\ln{\mathfrak{n}}$} & $\mbox{\sl UniDec}$ & $\mbox{\sl \checkmark}$ & \multirow{5}{*}{$3.172$} & \multirow{5}{*}{{\sc Coupon}} & \multirow{5}{*}{$\mathfrak{n}\cdot\ln{\mathfrak{m}}$} & $\mbox{\sl UniDec}$ & $\mbox{\sl \checkmark}$ & \multirow{5}{*}{$0.910$} \\
\cline{3-4}\cline{8-9} & & $0.5$ & $9.001$ & & & & $0.5$ & $3.001$ & \\
\cline{3-4}\cline{8-9} & & $0.3$ & $6.143$ & & & & $0.3$ & $1.858$ & \\
\cline{3-4}\cline{8-9} & & $0.1$ & $4.556$ & & & & $0.1$ & $1.223$ & \\
\cline{3-4}\cline{8-9} & & $0.01$ & $4.051$ & & & & $0.01$ & $1.021$ & \\
\hline
\multirow{5}{*}{{\sc Q.-Select}} & \multirow{5}{*}{$\mathfrak{n}$}
& $\mbox{\sl UniDec}$ & $\mbox{\sl \checkmark}$ & \multirow{5}{*}{$7.909$} & \multirow{5}{*}{{\sc Res. A}} & \multirow{5}{*}{$\mathfrak{n}\cdot\ln{\mathfrak{m}}$} & $\mbox{\sl UniDec}$ & $\mbox{\sl \checkmark}$ & \multirow{5}{*}{$2.472$} \\
\cline{3-4}\cline{8-9} & & $0.5$ & $17.001$ & & & & $0.5$ & $6.437$ & \\
\cline{3-4}\cline{8-9} & & $0.3$ & $11.851$ & & & & $0.3$ & $4.312$ & \\
\cline{3-4}\cline{8-9} & & $0.1$ & $9.001$ & & & & $0.1$ & $3.132$ &\\
\cline{3-4}\cline{8-9} & & $0.01$ & $8.091$ & & & & $0.01$ & $2.756$ &\\
\hline
\multirow{5}{*}{{\sc Diam. A}} & \multirow{5}{*}{$\mathfrak{n}\cdot\ln{\mathfrak{n}}$}
& $\mbox{\sl UniDec}$ & $\mbox{\sl \checkmark}$ & \multirow{5}{*}{$4.525$} &
\multirow{5}{*}{{\sc Res. B}} & \multirow{5}{*}{$\mathfrak{m}$}
& $\mbox{\sl UniDec}$ & $\mbox{\sl \checkmark}$ & \multirow{5}{*}{$2.691$} \\
\cline{3-4}\cline{8-9} & & $0.5$ & $9.001$ & & & & $0.5$ & $6.437$ & \\
\cline{3-4}\cline{8-9} & & $0.3$ & $6.143$ & & & & $0.3$ & $4.312$ & \\
\cline{3-4}\cline{8-9} & & $0.1$ & $4.556$ & & & & $0.1$ & $3.132$ & \\
\cline{3-4}\cline{8-9} & & $0.01$ & $4.525$ & & & & $0.01$ & $2.756$ & \\
\hline
\multirow{5}{*}{{\sc Diam. B}} & \multirow{5}{*}{$\mathfrak{n}$} & $\mbox{\sl UniDec}$ & $\mbox{\sl \checkmark}$ & \multirow{5}{*}{$5.918$} &
\multirow{5}{*}{{-}} & \multirow{5}{*}{-} & - & - & \multirow{5}{*}{-} \\
\cline{3-4}\cline{8-9} & & $0.5$ & $13.001$ & & & & - & - &\\
\cline{3-4}\cline{8-9} & & $0.3$ & $9.001$ & & & & - & - &\\
\cline{3-4}\cline{8-9} & & $0.1$ & $6.778$ & & & & - & - &\\
\cline{3-4}\cline{8-9} & & $0.01$ & $6.071$ & & & & - & - & \\
\hline
\end{tabular}
\caption{Experimental results where all running times (averaged over $5$ runs) are less than $0.02$ seconds, between
$0.01$ and $0.02$ in all cases.}
\label{tab:experiments}
\end{table}
\subsection{Discussion}
\vspace{-0.5em}
We discuss some aspects of the recurrence relations we consider and possible generalizations.
\begin{compactenum}
\item {\em Recurrence relations.}
We note that the recurrence relations we consider capture various classical randomized programs,
ranging for sorting ({\sc Quick-Sort}), searching ({\sc Randomized-Search}), {\sc Resource-Allocation},
{\sc Coupon-Collector}, which are basic in several analysis.
Thus the recurrence relations we consider indeed captures a diverse range of widely used programs
such as sort, search, etc.
\item {\em Generalization of recurrence relations.}
Our setting of recurrence relations can also be easily generalized.
For example, consider a restricted non-uniform sampling that has one uniform probability value for
the first half of the array, and a different uniform value for the second half.
Then we can truncate the summation (for expected-time complexity) into two halves and apply our methods.
\item {\em Generalization of bounds.}
In this work we focus on almost-linear bounds.
Our approach could be extended to upper bounds such as $\mathcal{O}\left(n^k\right)$ or
$\mathcal{O}\left(n^k\cdot \log n\right)$ ($k\ge 2$) since the definite integral over these
functions can be obtained in closed form through integral-by-part technique, and
can be approximated efficiently as in Proposition~\ref{prop:integralapproximation}.
\end{compactenum}
While our techniques can be generalized to other recurrence relations and bounds,
in this work we focus on the recurrence relations and bounds for two reasons:
(a)~most of the classical examples does not require such generalizations; and
(b)~the simpler classes allow clear exposition of the core ideas.
\section{Guess-and-Check Functions}\label{sect:mfunc}
We follow the standard guess-and-check technique to solve simple recurrence relations.
Below we first fix a univariate recurrence relation $G$ taking the form (\ref{eq:unirecurrel}).
\smallskip
\begin{definition}[Univariate Guess-and-Check Functions]\label{def:uniguess}
Let $G$ be a univariate recurrence relation taking the form (\ref{eq:unirecurrel}).
A function $h:\Nset\rightarrow\Rset$ is a \emph{guess-and-check} function for $G$ if there exists a natural number $N\in\Nset$ such that
\begin{compactitem}
\item {\em (Base Condition)} $T_G(n)\le h(n)$ for all $1\le n\le N$, and
\item {\em (Inductive Argument)} $\mathsf{Subst}(\mathfrak{e},h) (n)\le h(n)$ for all $n> N$.
\end{compactitem}
\end{definition}
By an easy induction on $n$ (starting from the $N$ specified in Definition~\ref{def:uniguess})
we obtain the following result.
\medskip
\begin{theorem}[Guess-and-Check, Univariate Case]\label{thm:uniguess}
If a function $h:\Nset\rightarrow\Rset$ is a \emph{guess-and-check} function for a univariate recurrence relation $G$ taking the form (\ref{eq:unirecurrel}), then $T_G(n)\le h(n)$ for all $n\in\Nset$.
\end{theorem}
We do not present explicitly present the definition for guess-and-check functions in the bivariate case,
since we will present a reduction of the analysis of separable bivariate recurrence relations to that
of the univariate ones (cf. Section~\ref{sect:bisynth}).
\begin{comment}
\begin{definition}[Bivariate Guess-and-Check Functions]\label{def:biguess}
Let $G$ be a bivariate recurrence relation taking the form (\ref{eq:birecurrel}).
A function $h:\Nset\times\Nset\rightarrow\Rset$ is a \emph{guess-and-check} function for $G$ if there exists a natural number $N\in\Nset$ such that
\begin{compactitem}
\item {\em (Base Condition)} $T_G(n,m)\le h(n,m)$ for all $n\in\Nset$ and $1\le m\le N$, and
\item {\em (Inductive Argument)} $\mathsf{Subst}(\mathfrak{e}, h)(n,m)\le h(n,m)$ for all $n\in\Nset$ and $m> N$.
\end{compactitem}
\end{definition}
\smallskip
\begin{theorem}[Guess-and-Check, Bivariate Case]
If a function $h:\Nset\times\Nset\rightarrow\Rset$ is a \emph{guess-and-check} function for a bivariate recurrence relation $G$ taking the form (\ref{eq:birecurrel}),
then $T_G(n,m)\le h(n,m)$ for all $n,m\in\Nset$.
\end{theorem}
\begin{proof}
By an easy induction on $m$ (starting from the $N$ specified in Definition~\ref{def:biguess}).
\end{proof}
\end{comment}
\section{Introduction}
\vspace{-1em}
\noindent{\em Static analysis for quantitative bounds.}
Static analysis of programs aims to reason about programs without
running them.
The most basic properties for static analysis are qualitative properties,
such as safety, termination, liveness, that for every trace of a program
gives a Yes or No answer (such as assertion violation or not, termination
or not).
However, recent interest in analysis of resource-constrained systems,
such as embedded systems, as well as for performance analysis,
quantitative performance characteristics are necessary.
For example, the qualitative problem of termination asks whether a given
program always terminates, whereas the quantitative problem asks to
obtain precise bounds on the number of steps, and is thus a more
challenging problem.
Hence the problem of automatically reasoning about resource bounds (such
as time complexity bounds) of programs is both of significant
theoretical as well as practical interest.
\smallskip\noindent{\em Worst-case bounds.}
The worst-case analysis of programs is the fundamental problem in computer
science, which is the basis of algorithms and complexity theory.
However, manual proofs of worst-case analysis can be tedious and also require
non-trivial mathematical ingenuity, e.g., the book {\em The Art of Computer Programming}
by Knuth presents a wide range of involved techniques to derive such precise
bounds~\cite{DBLP:books/aw/Knuth73a,DBLP:books/aw/Knuth73}.
There has been a considerable research effort for automated analysis of such
worst-case bounds for programs, see~\cite{SPEED1,SPEED2,Hoffman1,Hoffman2}
for excellent expositions on the significance of deriving precise worst-case
bounds and the automated methods to derive them.
For the worst-case analysis there are several techniques, such as
worst-case execution time analysis~\cite{DBLP:journals/tecs/WilhelmEEHTWBFHMMPPSS08}, resource analysis using abstract
interpretation and type systems~\cite{SPEED2,DBLP:journals/entcs/AlbertAGGPRRZ09,DBLP:conf/popl/JostHLH10,Hoffman1,Hoffman2}, ranking functions~\cite{BG05,DBLP:conf/cav/BradleyMS05,DBLP:conf/tacas/ColonS01,DBLP:conf/vmcai/PodelskiR04,DBLP:conf/pods/SohnG91,DBLP:conf/vmcai/Cousot05,DBLP:journals/fcsc/YangZZX10,DBLP:journals/jossac/ShenWYZ13},
as well as recurrence relations~\cite{DBLP:conf/icfp/Grobauer01,DBLP:journals/entcs/AlbertAGGPRRZ09,DBLP:conf/sas/AlbertAGP08,DBLP:conf/esop/AlbertAGPZ07}.
\smallskip\noindent{\em Expected-runtime bounds.}
While several works have focused on deriving worst-case bounds for programs,
quite surprisingly little work has been done to derive precise bounds for
expected-runtime analysis, with the exception of~\cite{DBLP:journals/tcs/FlajoletSZ91},
which focuses on randomization in combinatorial structures (such as trees).
This is despite the fact that expected-runtime analysis is an equally important
pillar of theoretical computer science, both in terms of theoretical and
practical significance.
For example, while for real-time systems with hard constraints worst-case
analysis is necessary, for real-time systems with soft constraints the more
relevant information is the expected-runtime analysis.
Below we highlight three key significance of expected-runtime analysis.
\begin{compactenum}
\item {\em Simplicity and desired properties:}
The first key aspect is {\em simplicity}: often much simpler
algorithms (thus simple and efficient implementations) exist
for expected-runtime complexity as compared to worst-case complexity.
A classic example is the {\sc Selection} problem that given a set of $n$
numbers and $0\leq k \leq n$, asks to find the $k$-th largest number
(eg, for median $k=n/2$).
The classical linear-time algorithm for the problem (see~\cite[Chapter~9]{DBLP:books/daglib/0023376})
is quite involved, and its worst-case analysis to obtain linear time bound is
rather complex.
In contrast, a much simpler algorithm exists (namely, {\sc Quick-Select})
that has linear expected-runtime complexity.
Moreover, randomized algorithms with expected-runtime complexity enjoy
many desired properties, which deterministic algorithms do not have.
A basic example is {\sc Channel-Conflict Resolution}
(see Example~\ref{ex:channel}, Section~\ref{sec:motivatingbi})
where the simple randomized algorithm can be implemented in a distributed or
concurrent setting, whereas deterministic algorithms are quite cumbersome.
\item {\em Efficiency in practice:}
Since worst-case analysis concerns with corner cases that rarely arise,
many algorithms and implementations have much better expected-runtime complexity,
and they perform extremely well in practice.
A classic example is the {\sc Quick-Sort} algorithm, that has quadratic worst-case
complexity, but almost linear expected-runtime complexity, and is one of the most
efficient sorting algorithms in practice.
\item {\em Worst-case analysis ineffective:}
In several important cases
the worst-case analysis is completely ineffective.
For example, consider one of the textbook stochastic process, namely the
{\sc Coupon-Collector} problem, where there are $n$ types of coupons to be collected,
and in each round, a coupon type among the $n$ types is obtained uniformly at random.
The process stops when all types are collected.
The {\sc Coupon-Collector} process is one of the basic and classical stochastic
processes, with numerous applications in network routing, load balancing,
etc (see~\cite[Chapter 3]{DBLP:books/cu/MotwaniR95} for applications of
{\sc Coupon-Collector} problems).
For the worst-case analysis, the process might not terminate (worst-case bound infinite),
but the expected-runtime analysis shows that the expected termination
time is $\mathcal{O}(n \cdot \log n)$.
\end{compactenum}
\smallskip\noindent{\em Challenges.}
The expected-runtime analysis brings several new challenges as compared to
the worst-case analysis.
First, for the worst-case complexity bounds, the most classical characterization
for analysis of recurrences is the {\em Master Theorem} (cf.~\cite[Chapter~1]{DBLP:books/daglib/0023376}) and Akra-Bazzi's Theorem~\cite{DBLP:journals/coap/AkraB98}.
However, the expected-runtime analysis problems give rise to recurrences that
are not characterized by these theorems since our recurrences normally involve an unbounded summation resulting from a randomized selection of integers from $1$ to $n$ where $n$ is unbounded.
Second, techniques like ranking functions (linear or polynomial ranking
functions) cannot derive efficient bounds such as $\mathcal{O}(\log n)$
or $\mathcal{O}(n \cdot \log n)$.
While expected-runtime analysis has been considered for combinatorial structures
using generating function~\cite{DBLP:journals/tcs/FlajoletSZ91},
we are not aware of any automated technique to handle recurrences arising from
randomized algorithms.
\smallskip\noindent{\em Analysis problem.}
We consider the algorithmic analysis problem of recurrences
arising naturally for randomized recursive programs.
Specifically we consider the following:
\begin{compactitem}
\item We consider two classes of recurrences:
(a)~{\em univariate} class with one variable (which represents the array
length, or the number of input elements, as required in problems such as
{\sc Quick-Select, Quick-Sort} etc); and
(b)~{\em separable bivariate} class with two variables (where the two independent
variables represent the total number of elements and total number of successful cases,
respectively, as required in problems such as {\sc Coupon-Collector, Channel-Conflict Resolution}).
The above two classes capture a large class of expected-runtime analysis problems,
including all the classical ones mentioned above.
Moreover, the main purpose of expected-runtime analysis is to obtain efficient bounds.
Hence we focus on the case of logarithmic, linear, and almost-linear bounds
(i.e., bounds of form $\mathcal{O}(\log n)$, $\mathcal{O}(n)$ and
$\mathcal{O}(n \cdot \log n)$, respectively, where $n$ is the size of the
input).
Moreover, for randomized algorithms, quadratic bounds or higher are rare.
\end{compactitem}
Thus the main problem we consider is to automatically derive such efficient bounds
for randomized univariate and separable bivariate recurrence relations.
\smallskip\noindent{\em Our contributions.}
Our main contribution is a sound approach for analysis of recurrences for expected-runtime analysis.
The input to our problem is a recurrence relation and the output is either
logarithmic, linear, or almost-linear as the asymptotic bound, or fail.
The details of our contributions are as follows:
\begin{compactenum}
\item {\em Efficient algorithm.}
We first present a linear-time algorithm for the univariate case, which is
based on simple comparison of leading terms of pseudo-polynomials.
Second, we present a simple reduction for separable bivariate
recurrence analysis to the univariate case.
Our efficient (linear-time) algorithm can soundly infer logarithmic,
linear, and almost-linear bounds for recurrences of
one or two variables.
\item {\em Analysis of classical algorithms.}
We show that for several classical algorithms, such as
{\sc Randomized-Search, Quick-Select, Quick-Sort, Coupon-Collector, Channel-Conflict Resolution}
(see Section~\ref{sec:motivatinguni} and Section~\ref{sec:motivatingbi}
for examples), our sound approach can obtain the asymptotically optimal
expected-runtime bounds for the recurrences.
In all the cases above, either the worst-case bounds (i)~do not exist
(e.g., {\sc Coupon-Collector}), or
(ii)~are quadratic when the expected-runtime bounds are linear or almost-linear
(e.g., {\sc Quick-Select, Quick-Sort});
or (iii)~are linear when the expected-runtime bounds are logarithmic
(e.g., {\sc Randomized-Search}).
Thus in cases where the worst-case bounds are either not applicable,
or grossly overestimate the expected-runtime bounds, our technique is both
efficient (linear-time) and can infer the optimal bounds.
\item {\em Implementation.}
Finally, we have implemented our approach, and we present experimental results on the
classical examples to show that we can efficiently achieve the automated expected-runtime
analysis of randomized recurrence relations.
\end{compactenum}
\noindent{\em Novelty and technical contribution.}
The key novelty of our approach is an automated method to analyze
recurrences arising from randomized recursive programs, which are
not covered by Master theorem.
Our approach is based on a guess-and-check technique.
We show that by over-approximating terms in a recurrence relation through
integral and Taylor's expansion, we can soundly infer logarithmic, linear and
almost-linear bounds using simple comparison between leading terms of
pseudo-polynomials.
\section{Recurrence Relations}\label{sec:recurrel}
\vspace{-1em}
We present our mini specification language for recurrence
relations for expected-runtime analysis.
The language is designed to capture running time of recursive randomized
algorithms which involve (i)~only one function call whose expected-runtime
complexity is to be determined, (ii)~at most two integer parameters,
and (iii)~involve randomized-selection or divide-and-conquer techniques.
We present our language separately for the univariate and bivariate cases.
In the sequel, we denote by $\Nset$, $\Nset_0$, $\Zset$, and $\Rset$ the
sets of all positive integers, non-negative integers, integers, and real
numbers, respectively.
\vspace{-1.5em}
\subsection{Univariate Randomized Recurrences}\label{sect:univariate}
\vspace{-1em}
Below we define the notion of univariate randomized recurrence relations.
First, we introduce the notion of univariate recurrence expressions.
Since we only consider single recursive function call, we use `$\mathrm{T}$'
to represent the (only) function call.
We also use `$\mathfrak{n}$' to represent the only parameter in the function
declaration.
\smallskip\noindent{\bf Univariate recurrence expressions.}
The syntax of \emph{univariate recurrence expressions} $\mathfrak{e}$ is
generated by the following grammar:
\begin{align*}
\mathfrak{e} & ::= c\mid \mathfrak{n}\mid \ln{\mathfrak{n}} \mid \mathfrak{n}\cdot \ln{\mathfrak{n}}\mid \frac{1}{\mathfrak{n}}\mid \mathrm{T}\left(\mathfrak{n}-1\right) \mid \mathrm{T}\left(\left\lfloor\frac{\mathfrak{n}}{2}\right\rfloor\right) \mid \mathrm{T}\left(\left\lceil\frac{\mathfrak{n}}{2}\right\rceil\right)\\
&\mid \frac{\sum_{\mathfrak{j}=1}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j})}{\mathfrak{n}}\mid \frac{1}{\mathfrak{n}}\cdot\left( \textstyle\sum_{\mathfrak{j}=\left\lceil\mathfrak{n}/2\right\rceil}^{\mathfrak{n}-1}\mathrm{T}(\mathfrak{j})+ \textstyle\sum_{\mathfrak{j}=\left\lfloor\mathfrak{n}/{2}\right\rfloor}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j})\right)\mid c\cdot \mathfrak{e}\mid \mathfrak{e}+\mathfrak{e}
\end{align*}
where $c\in [1,\infty)$ and $\ln(\centerdot)$ represents the natural logarithm function with base $e$.
Informally, $\mathrm{T}(\mathfrak{n})$ is the (expected) running time of a recursive randomized program which involves only one recursive routine indicated by $\mathrm{T}$ and only one parameter indicated by $\mathfrak{n}$.
Then each $\mathrm{T}(\centerdot)$-term in the grammar has a direct algorithmic meaning:
\begin{compactitem}
\item $\mathrm{T}\left(\mathfrak{n}-1\right)$ may mean a recursion to a sub-array with length decremented by one;
\item
$\mathrm{T}\left(\left\lfloor\frac{\mathfrak{n}}{2}\right\rfloor\right)$ and $\mathrm{T}\left(\left\lceil\frac{\mathfrak{n}}{2}\right\rceil\right)$
may mean a recursion related to a divide-and-conquer technique;
\item finally,
$\frac{\sum_{\mathfrak{j}=1}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j})}{\mathfrak{n}}\mbox{ and }\frac{1}{\mathfrak{n}}\cdot\left( \sum_{\mathfrak{j}=\left\lceil\frac{n}{2}\right\rceil}^{\mathfrak{n}-1}\mathrm{T}(\mathfrak{j})+ \sum_{\mathfrak{j}=\left\lfloor\frac{\mathfrak{n}}{2}\right\rfloor}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j})\right)$
may mean a recursion related to a randomized selection of an array index.
\end{compactitem}
\smallskip\noindent{\em Substitution.}
Consider a function $h:\Nset\rightarrow\Rset$ and univariate recurrence expression ${\mathfrak{e}}$.
The {\em substitution function}, denoted by $\mathsf{Subst}({\mathfrak{e}},h)$, is the function from $\Nset$ into $\Rset$ such that the value
for $n$ is obtained by evaluation through substituting $h$ for $\mathrm{T}$ and $n$ for $\mathfrak{n}$ in ${\mathfrak{e}}$, respectively.
Moreover, if $\mathfrak{e}$ does not involve the appearance of `$\mathrm{T}$', then we use the abbreviation
$\mathsf{Subst}({\mathfrak{e}})$ i.e., omit $h$.
For example, (i)~if ${\mathfrak{e}}= \mathfrak{n} + \mathrm{T}(\mathfrak{n}-1)$, and $h: n \mapsto n\cdot \log n$,
then $\mathsf{Subst}({\mathfrak{e}},h)$ is the function $n \mapsto n+ (n-1)\cdot \log (n-1)$,
and (ii)~if ${\mathfrak{e}}= 2\cdot \mathfrak{n}$, then
$\mathsf{Subst}({\mathfrak{e}})$ is
$n \mapsto 2n$.
\smallskip\noindent{\bf Univariate recurrence relation.}
A {\em univariate recurrence relation} $G=(\mathsf{eq}_1,\mathsf{eq}_2)$ is a pair of equalities as follows:
\begin{equation}\label{eq:unirecurrel}
\mathsf{eq}_1: \ \mathrm{T}(\mathfrak{n})=\mathfrak{e}; \qquad \qquad \mathsf{eq}_2: \ \mathrm{T}(1)=c
\end{equation}
where $c\in (0,\infty)$ and $\mathfrak{e}$ is a univariate recurrence expression.
For a univariate recurrence relation $G$ the {\em evaluation sequence} $\mathsf{Eval}(G)$ is as follows:
$\mathsf{Eval}(G)(1)=c$, and for $n \geq 2$, given $\mathsf{Eval}(G)(i)$ for $1\leq i < n$, for the
value $\mathsf{Eval}(G)(n)$ we evaluate the expression $\mathsf{Subst}(\mathfrak{e},\mathsf{Eval}(G))$,
since in $\mathfrak{e}$ the parameter $\mathfrak{n}$ always decreases and is thus
well-defined.
\smallskip\noindent{\em Finite vs infinite solution.}
Note that the above description gives a computational procedure to compute
$\mathsf{Eval}(G)$ for any finite $n$, in linear time in $n$ through dynamic programming.
The interesting question is to algorithmically analyze the infinite behavior.
A function $T_G:\Nset\rightarrow\Rset$ is called a solution to $G$
if $T_G(n)=\mathsf{Eval}(G)(n)$ for all $n \geq 1$.
The function $T_G$ is unique and explicitly defined as follows:
(1)~\emph{Base Step.} $T_G(1):=c$; and (2)~\emph{Recursive Step.} $T_G(n):=\mathsf{Subst}(\mathfrak{e},T_G)(n)$ for all $n\ge 2$.
The interesting algorithmic question is to reason about the asymptotic infinite behaviour of $T_G$.
\vspace{-1.5em}
\subsection{Motivating Classical Examples}\label{sec:motivatinguni}
\vspace{-1em}
In this section we present several classical examples of randomized
programs whose recurrence relations belong to the
class of univariate recurrence relations described in Section~\ref{sect:univariate}.
We put details of pseudocode and how to derive the recurrence relations in this section in Appendix~\ref{sect:recurreldetails}.
Moreover in all cases the base step is $\mathrm{T}(1)=1$, hence we discuss the recursive case.
\begin{example}[{\sc Randomized-Search}]\label{ex:randsearch}
Consider the Sherwood's {\sc Randomized-Search\ } algorithm (cf.~\cite[Chapter~9]{McConnellbook}).
The algorithm checks whether an integer value $d$ is present within the index range $[i,j]$ ($0\le i\le j$)
in an integer array $ar$ which is sorted in increasing order and is without duplicate entries.
The algorithm outputs either the index for $d$ in $ar$ or $-1$ meaning that $d$ is not present in
the index range $[i,j]$ of $ar$.
The recurrence relation for this example is as follows:
\begin{equation}\label{eq:relrandsearch}
\textstyle\mathrm{T}(\mathfrak{n})=6+\frac{1}{\mathfrak{n}}\cdot\big( \sum_{\mathfrak{j}=\left\lceil\mathfrak{n}/{2}\right\rceil}^{\mathfrak{n}-1}\mathrm{T}(\mathfrak{j})+ \sum_{\mathfrak{j}=\left\lfloor\mathfrak{n}/{2}\right\rfloor}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j})\big)
\end{equation}
We note that the worst-case complexity for this algorithm is $\Theta(n)$.\qed
\end{example}
\begin{example}[{\sc Quick-Sort}]\label{ex:quicksort}
Consider the {\sc Quick-Sort} algorithm~\cite[Chapter~7]{DBLP:books/daglib/0023376}.
The recurrence relation for this example is:
\begin{equation}\label{eq:relquicksort}
\textstyle\mathrm{T}(\mathfrak{n})=2\cdot\mathfrak{n}+ 2\cdot (\sum_{\mathfrak{j}=1}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j}))/{\mathfrak{n}}
\end{equation}
where $\mathrm{T}(\mathfrak{n})$ represents the maximal expected execution time where
$\mathfrak{n}$ is the array length and the execution time of {\em pivoting} is represented
by $2\cdot \mathfrak{n}$.
We note that the worst-case complexity for this algorithm is $\Theta(n^2)$.\qed
\end{example}
\begin{example}[{\sc Quick-Select}]\label{ex:quickselect}
Consider the {\sc Quick-Select} algorithm (cf.~\cite[Chapter~9]{DBLP:books/daglib/0023376}).
The recurrence relation for this example is
\begin{equation}\label{eq:relquickselect}
\textstyle\mathrm{T}(\mathfrak{n})\!=\!4+2\cdot\mathfrak{n}+
\frac{1}{\mathfrak{n}}\cdot \left(\sum_{\mathfrak{j}=\left\lfloor \mathfrak{n}/2\right\rfloor}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j})+ \sum_{\mathfrak{j}=\left\lceil \mathfrak{n}/2\right\rceil}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j})\right)
\end{equation}
We note that the worst-case complexity for this algorithm is $\Theta(n^2)$.\qed
\end{example}
\begin{example}[{\sc Diameter-Computation}]\label{ex:diameter}
Consider the {\sc Diameter-Computation} algorithm (cf.~\cite[Chapter 9]{DBLP:books/cu/MotwaniR95}) to compute the diameter of an input finite set $S$ of three-dimensional points.
Depending on Eucledian or $L_1$ metric we obtain two different recurrence relations.
For Eucledian we have the following relation:
\begin{equation}\label{eq:reldiametera}
\textstyle\mathrm{T}(\mathfrak{n})=2+\mathfrak{n}+ 2\cdot \mathfrak{n}\cdot\ln{\mathfrak{n}} + (\sum_{\mathfrak{j}=1}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j}))/{\mathfrak{n}} ;
\end{equation}
and for $L_1$ metric we have the following relation:
\begin{equation}\label{eq:reldiameterb}
\textstyle\mathrm{T}(\mathfrak{n})=2+\mathfrak{n}+ 2\cdot \mathfrak{n} + (\sum_{\mathfrak{j}=1}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j}))/{\mathfrak{n}} \\
\end{equation}
We note that the worst-case complexity for this algorithm is as follows:
for Euclidean metric it is $\Theta(n^2 \cdot \log n)$ and for the $L_1$ metric
it is $\Theta(n^2)$.\qed
\end{example}
\begin{example}[Sorting with {\sc Quick-Select}]\label{ex:sortselect}
Consider a sorting algorithm
which selects the median through the {\sc Quick-Select} algorithm.
The recurrence relation is directly obtained as follows:
\begin{equation}\label{eq:relsortselect}
\textstyle\mathrm{T}(\mathfrak{n})=4+ T^*(\mathfrak{n})+\mathrm{T}\left(\lfloor{\mathfrak{n}}/{2}\rfloor\right)+\mathrm{T}\left(\lceil{\mathfrak{n}}/{2}\rceil\right)
\end{equation}
where $T^*(\centerdot)$ is an upper bound on the expected running time of {\sc Quick-Select}
(cf. Example~\ref{ex:quickselect}).
We note that the worst-case complexity for this algorithm is $\Theta(n^2)$.\qed
\end{example}
\vspace{-1.5em}
\subsection{Separable Bivariate Randomized Recurrences}\label{sec:bivariate}
\vspace{-1em}
We consider a generalization of the univariate recurrence relations to a
class of bivariate recurrence relations called \emph{separable bivariate recurrence relations}.
Similar to the univariate situation, we use `$\mathrm{T}$' to represent the (only) function call and `$\mathfrak{n}$', `$\mathfrak{m}$' to represent namely the two integer parameters.
\smallskip\noindent{\bf Separable Bivariate Recurrence Expressions.}
The syntax of \emph{separable bivariate recurrence expressions} is illustrated by $\mathfrak{e},\mathfrak{h}$ and $\mathfrak{b}$ as follows:
\begin{align*}
\mathfrak{e} & ::= \mathrm{T}\left(\mathfrak{n}, \mathfrak{m}-1\right) \mid \mathrm{T}\left(\mathfrak{n},\left\lfloor{\mathfrak{m}}/{2}\right\rfloor\right) \mid \mathrm{T}\left(\mathfrak{n},\left\lceil{\mathfrak{m}}/{2}\right\rceil\right)\\
& \mid \frac{\sum_{\mathfrak{j}=1}^{\mathfrak{m}-1} \mathrm{T}(\mathfrak{n},\mathfrak{j})}{\mathfrak{m}}
\mid \frac{1}{\mathfrak{m}}\cdot\left( \textstyle\sum_{\mathfrak{j}=\left\lceil {\mathfrak{m}}/{2}\right\rceil}^{\mathfrak{m}-1}\mathrm{T}(\mathfrak{n},\mathfrak{j})+
\textstyle\sum_{\mathfrak{j}=\left\lfloor {\mathfrak{m}}/{2}\right\rfloor}^{\mathfrak{m}-1} \mathrm{T}(\mathfrak{n},\mathfrak{j})\right)\mid c\cdot \mathfrak{e}\mid \mathfrak{e}+\mathfrak{e}
\\
\mathfrak{h} & ::= c\mid \ln{\mathfrak{n}}\mid \mathfrak{n}\mid \mathfrak{n}\cdot\ln{\mathfrak{n}}\mid c\cdot\mathfrak{h}\mid \mathfrak{h}+\mathfrak{h}\quad\mathfrak{b} ::= c\mid \frac{1}{\mathfrak{m}} \mid \ln{\mathfrak{m}}\mid \mathfrak{m}\mid \mathfrak{m}\cdot\ln{\mathfrak{m}}\mid c\cdot \mathfrak{b}\mid \mathfrak{b}+\mathfrak{b}
\end{align*}
The differences are that (i) we have two independent parameters $\mathfrak{n},\mathfrak{m}$, (ii) $\mathfrak{e}$ now represents an expression composed of only $\mathrm{T}$-terms, and (iii) $\mathfrak{h}$ (resp. $\mathfrak{b}$) represents arithmetic expressions for $\mathfrak{n}$ (resp. for $\mathfrak{m}$).
This class of separable bivariate recurrence expressions (often for brevity bivariate recurrence expressions)
stresses a dominant role on $\mathfrak{m}$ and a minor role on $\mathfrak{n}$, and is intended to model randomized algorithms where some parameter (to be represented by $\mathfrak{n}$) does not change value.
\smallskip\noindent{\em Substitution.}
The notion of substitution is similar to the univariate case.
Consider a function $h:\Nset\times\Nset\rightarrow\Rset$, and a bivariate recurrence expression ${\mathfrak{e}}$.
The {\em substitution function}, denoted by $\mathsf{Subst}({\mathfrak{e}},h)$, is the function from $\Nset\times\Nset$ into $\Rset$
such that $\mathsf{Subst}({\mathfrak{e}},h)(n,m)$ is the real number evaluated through substituting $h,n,m$ for
$\mathrm{T},\mathfrak{n},\mathfrak{m}$, respectively.
The substitution for $\mathfrak{h},\mathfrak{b}$ is defined in a similar way, with the difference that they both induce a univariate function.
\smallskip\noindent{\bf Bivariate recurrence relations.}
We consider {\em bivariate recurrence relations} $G=(\mathsf{eq}_1,\mathsf{eq}_2)$, which consists of
two equalities of the following form:
\begin{equation}\label{eq:birecurrel}
\mathsf{eq}_1: \ \mathrm{T}(\mathfrak{n},\mathfrak{m})=\mathfrak{e}+\mathfrak{h}\cdot\mathfrak{b}; \quad \qquad \mathsf{eq}_2: \ \mathrm{T}(\mathfrak{n},1)=\mathfrak{h}\cdot c
\end{equation}
where $c\in(0,\infty)$ and $\mathfrak{e},\mathfrak{h},\mathfrak{b}$ are from the grammar above.
\smallskip\noindent{\em Solution to bivariate recurrence relations.}
The evaluation of bivariate recurrence relation is similar to the univariate
case.
Similar to the univariate case,
the unique solution $T_G:\Nset\times\Nset\rightarrow\Rset$ to a recurrence relation $G$ taking the form (\ref{eq:birecurrel}) is a function defined recursively as follows:
(1)~\emph{Base Step.} $T_G(n,1):=\mathsf{Subst}({\mathfrak{h}})(n)\cdot c$ for all $n\in\Nset$; and
(2)~\emph{Recursive Step.} $T_G(n,m):=\mathsf{Subst}({\mathfrak{e}},T_G)(n,m)+\mathsf{Subst}(\mathfrak{h})(n)\cdot\mathsf{Subst}(\mathfrak{b})(m)$
for all $n\in\Nset$ and $m\ge 2$.
Again the interesting algorithmic question is to reason about the infinite behaviour of $T_G$.
\vspace{-1.5em}
\subsection{Motivating Classical Examples}\label{sec:motivatingbi}
\vspace{-0.5em}
In this section we present two classical examples of randomized algorithms
where the randomized recurrence relations are bivariate.
We put the detailed illustration for this two examples in Appendix~\ref{app:motivatingbi}.
\begin{example}[{\sc Coupon-Collector}]\label{ex:coupon}
Consider the {\sc Coupon-Collector} problem~\cite[Chapter~3]{DBLP:books/cu/MotwaniR95} with $n$ different types of coupons ($n\in\Nset$).
The randomized process proceeds in rounds: at each round, a coupon is collected uniformly at random from the coupon types
the rounds continue until all the $n$ types of coupons are collected.
We model the rounds as a recurrence relation with two variables
$\mathfrak{n},\mathfrak{m}$, where $\mathfrak{n}$ represents the total number
of coupon types and $\mathfrak{m}$ represents the remaining number of uncollected
coupon types.
The recurrence relation is as follows:
\begin{equation}\label{eq:relcoupon}
\mathrm{T}(\mathfrak{n},1)=\mathfrak{n}\cdot 1; \qquad
\mathrm{T}(\mathfrak{n},\mathfrak{m})=\mathfrak{n}/{\mathfrak{m}}+ \mathrm{T}(\mathfrak{n},\mathfrak{m}-1)
\end{equation}
where $\mathrm{T}(\mathfrak{n},\mathfrak{m})$ is the expected number of rounds.
We note that the worst-case complexity for this process is $\infty$.\qed
\end{example}
\begin{example}[{\sc Channel-Conflict Resolution}]\label{ex:channel}
We consider two network scenarios in which $n$ clients are trying to get
access to a network channel.
This problem is also called the {\sc Resource-Contention Resolution}~\cite[Chapter~13]{Kleinbergbook}.
In this problem, if more than one client tries to access the channel,
then no client can access it, and if exactly one client requests access to
the channel, then the request is granted.
In the distributed setting, the clients do not share any information.
In this scenario, in each round, every client requests an access to the
channel with probability $\frac{1}{n}$.
Then for this scenario, we obtain an over-approximating recurrence relation
\begin{equation}\label{eq:relresourcea}
\mathrm{T}(\mathfrak{n},1)=\mathfrak{n}\cdot 1; \qquad
\mathrm{T}(\mathfrak{n},\mathfrak{m})=(\mathfrak{n}\cdot{e})/{\mathfrak{m}}+ \mathrm{T}(\mathfrak{n},\mathfrak{m}-1)
\end{equation}
for the expected rounds until which every client gets at least one access to the channel.
In the concurrent setting, the clients share one variable, which is the number
of clients which has not yet been granted access.
Also in this scenario, once a client gets an access the client does not
request for access again.
For this scenario, we obtain an over-approximating recurrence relation
\begin{equation}\label{eq:relresourceb}
\mathrm{T}(\mathfrak{n},1)=1\cdot 1; \qquad
\mathrm{T}(\mathfrak{n},\mathfrak{m})=1\cdot e+ \mathrm{T}(\mathfrak{n},\mathfrak{m}-1)
\end{equation}
We also note that the worst-case complexity for both the scenarios is $\infty$.\qed
\end{example}
\section{The Synthesis Algorithm}\label{sect:synalg}
\vspace{-1em}
In this section, we present our algorithms to synthesize
asymptotic bounds for randomized recurrence relations.
\smallskip\noindent{\em Main ideas.}
The main idea is as follows.
Consider as input a recurrence relation taking the form (\ref{eq:unirecurrel}) and an univariate recurrence expression
$\mathfrak{f}\in\{\ln{\mathfrak{n}}, \mathfrak{n},\mathfrak{n}\cdot\ln{\mathfrak{n}}\}$ which specifies the desired asymptotic bound.
We first define the standard notion of a guess-and-check function which provides a sound approach for asymptotic bound.
Based on the guess-and-check function, our algorithm executes the following steps for the univariate case.
\begin{compactenum}
\item First, the algorithm sets up a scalar variable $d$ and then constructs the template $h$ to be $n\mapsto d\cdot \mathsf{Subst}(\mathfrak{f})(n)+c$
for a univariate guess-and-check function.
\item Second, the algorithm computes an over-approximation $\mathsf{OvAp}(\mathfrak{e}, h)$ of $\mathsf{Subst}(\mathfrak{e}, h)$
such that the over-approximation $\mathsf{OvAp}(\mathfrak{e}, h)$ will involve terms from $\mathfrak{n}^k,\ln^\ell{\mathfrak{n}}$ (for $k,\ell\in\Nset_0$) only.
Note that $k,\ell$ may be greater than $1$, so the above expressions are not necessarily linear (they can be quadratic or cubic for example).
\item Finally, the algorithm synthesizes a value for $d$
such that $\mathsf{OvAp}(\mathfrak{e},h)(n)\le h(n)$ for all $n\ge 2$ through truncation
of $[2,\infty)\cap\Nset$ into a finite range and a limit behaviour analysis (towards $\infty$).
\end{compactenum}
Our algorithm for bivariate cases is a reduction to the univariate case.
\smallskip\noindent{\bf Guess-and-Check functions.}
We follow the standard guess-and-check technique to solve simple recurrence relations.
Below we first fix a univariate recurrence relation $G$ taking the form (\ref{eq:unirecurrel}).
By an easy induction on $n$ (starting from the $N$ specified in Definition~\ref{def:uniguess})
we obtain Theorem~\ref{thm:uniguess}.
\begin{definition}[Univariate Guess-and-Check Functions]\label{def:uniguess}
Let $G$ be a univariate recurrence relation taking the form (\ref{eq:unirecurrel}).
A function $h:\Nset\rightarrow\Rset$ is a \emph{guess-and-check} function for $G$ if there exists a natural number $N\in\Nset$ such that:
(1) {\em (Base Condition)} $T_G(n)\le h(n)$ for all $1\le n\le N$, and
(2) {\em (Inductive Argument)} $\mathsf{Subst}(\mathfrak{e},h) (n)\le h(n)$ for all $n> N$.
\end{definition}
\begin{theorem}[Guess-and-Check, Univariate Case]\label{thm:uniguess}
If a function $h:\Nset\rightarrow\Rset$ is a \emph{guess-and-check} function for a univariate recurrence relation $G$ taking the form (\ref{eq:unirecurrel}), then $T_G(n)\le h(n)$ for all $n\in\Nset$.
\end{theorem}
We do not explicitly present the definition for guess-and-check functions in the bivariate case,
since we will present a reduction of the analysis of separable bivariate recurrence relations to that
of the univariate ones (cf. Section~\ref{sect:bisynth}).
\smallskip\noindent{\bf Overapproximations for Recurrence Expressions.}
We now develop tight overapproximations for logarithmic terms.
In principle, we use Taylor's Theorem to approximate logarithmic terms such as $\ln{(n-1)},\ln{\lfloor\frac{n}{2}\rfloor}$,
and integral to approximate summations of logarithmic terms.
All the results below are technical and depends on basic calculus
(the detailed proofs are in the Appendix~\ref{app:overapprox}).
\begin{proposition}\label{prop:lnflooroverapprox}
For all natural number $n\ge 2$:
\[
(1)\ \ln{n}-\ln{2}-\frac{1}{n-1}\le \ln{\left\lfloor \frac{n}{2}\right\rfloor}\le \ln{n}-\ln{2};
(2)\ \ln{n}-\ln{2}\le \ln{\left\lceil \frac{n}{2}\right\rceil}\le \ln{n}-\ln{2}+\frac{1}{n}~~.
\]
\end{proposition}
\begin{proposition}\label{prop:nminusoneoverapprox}
For all natural number $n\ge 2$:
$\ln{n}-\frac{1}{n-1}\le\ln{(n-1)}\le \ln{n}-\frac{1}{n}$~~.
\end{proposition}
\begin{proposition}\label{prop:integralapproximation}
For all natural number $n\geq 2$:
\begin{compactitem}
\item $\int_1^n \frac{1}{x}\,\mathrm{d}x-\sum_{j=1}^{n-1} \frac{1}{j}\in \left[-0.7552,-\frac{1}{6}\right]$;
\item $\int_1^n \ln{x}\,\mathrm{d}x-\left(\sum_{j=1}^{n-1} \ln{j}\right) - \frac{1}{2}\cdot \int_1^n \frac{1}{x}\,\mathrm{d}x\in \left[-\frac{1}{12}, 0.2701\right]$;
\item $\int_1^n x\cdot \ln{x}\,\mathrm{d}x-\left(\sum_{j=1}^{n-1} j\cdot\ln{j}\right)-\frac{1}{2}\cdot\int_1^n \ln{x}\,\mathrm{d}x+\frac{1}{12}\cdot \int_1^n \frac{1}{x}\,\mathrm{d}x-\frac{n-1}{2}\in \left[-\frac{19}{72},0.1575\right]$.
\end{compactitem}
\end{proposition}
Note that Proposition~\ref{prop:integralapproximation} is non-trivial since it approximates summation of reciprocal and logarithmic terms up to a constant deviation.
For example, one may approximate $\sum_{j=1}^{n-1} \ln{j}$ directly by $\int_1^n \ln{x}\,\mathrm{d}x$, but this approximation deviates up to a logarithmic term from Proposition~\ref{prop:integralapproximation}.
From Proposition~\ref{prop:integralapproximation}, we establish a tight approximation for summation of logarithmic or reciprocal terms.
\begin{example}\label{ex:overapprox}
Consider the summation $\sum_{j=\left\lceil\frac{n}{2}\right\rceil}^{n-1}\ln{j}+ \sum_{j=\left\lfloor\frac{n}{2}\right\rfloor}^{n-1} \ln{j}\quad (n\ge 4)$.
By Proposition~\ref{prop:integralapproximation}, we can over-approximate it as
\[
2\cdot\left(\Gamma_{\ln{\mathfrak{n}}}\left(n\right)+\frac{1}{12}\right)
-\left(\Gamma_{\ln{\mathfrak{n}}}\left(\left\lceil\frac{n}{2}\right\rceil\right)+\Gamma_{\ln{\mathfrak{n}}}\left(\left\lfloor\frac{n}{2}\right\rfloor\right)-0.5402\right)
\]
where $\Gamma_{\ln{\mathfrak{n}}}(n) :=
\int_1^n\ln{x}\,\mathrm{d}x-\frac{1}{2}\cdot\int_1^n\frac{1}{x}\,\mathrm{d}x
=
n\cdot\ln{n}-n-\frac{\ln{n}}{2}+1$.
By using Proposition~\ref{prop:lnflooroverapprox}, the above expression is roughly
$n\cdot\ln{n}-(1-\ln{2})\cdot n+\frac{1}{2}\cdot\ln{n}+0.6672+\frac{1}{2\cdot n}$
(for details see Appendix~\ref{app:overapprox}).\qed
\end{example}
\begin{remark}\label{rmk:exthigerdegree}
Although we do approximation for terms related to only almost-linear bounds, Proposition~\ref{prop:integralapproximation} can be extended to logarithmic bounds with higher degree (e.g., $n^3\ln n$) since integration of such bounds can be obtained in closed forms.\qed
\end{remark}
\vspace{-1.5em}
\subsection{Algorithm for Univariate Recurrence Relations}\label{sect:unisynth}
\vspace{-1em}
We present our algorithm to synthesize a guess-and-check function in form~(\ref{eq:uniguess})
for univariate recurrence relations.
We present our algorithm in two steps. First, we present the decision version,
and then we present the quantitative version that synthesizes the associated constant.
The two key aspects are over-approximation and use of pseudo-polynomials,
and we start with over-approximation.
We relegate some technical details to Appendix~\ref{app:unisynth}.
\begin{definition}[Overapproximation]\label{def:unioverapprox}
Let $\mathfrak{f}\in\{\ln{\mathfrak{n}},\mathfrak{n},\mathfrak{n}\cdot\ln{\mathfrak{n}}\}$.
Consider a univariate recurrence expression $\mathfrak{g}$, constants $d$ and $c$,
and the function $h= d \cdot \mathsf{Subst}(\mathfrak{f}) + c$.
We define the {\em over-approximation function}, denoted $\mathsf{OvAp}(\mathfrak{g},h)$,
recursively as follows.
\begin{itemize}
\item {\em Base Step A.} If $\mathfrak{g}$ is one of the following:
$c', \mathfrak{n}, \ln{\mathfrak{n}}, \mathfrak{n}\cdot \ln{\mathfrak{n}},\frac{1}{\mathfrak{n}}$,
then $\mathsf{OvAp}(\mathfrak{g},h):=\mathsf{Subst}({\mathfrak{g}})$.
\item {\em Base Step B.} If $\mathfrak{g}$ is a single term which involves $\mathrm{T}$,
then we define $\mathsf{OvAp}(\mathfrak{g},h)$
from over-approximations
Proposition~\ref{prop:lnflooroverapprox}--~\ref{prop:integralapproximation}.
In details, $\mathsf{OvAp}(\mathfrak{g},h)$
is obtained from $\mathsf{Subst}(\mathfrak{g},h)$ by first over-approximating any summation through Proposition~\ref{prop:integralapproximation}
(i.e., through those $\Gamma_{(\centerdot)}$ functions defined below Proposition~\ref{prop:integralapproximation}), then over-approximating any
$\ln{(\mathfrak{n}-1)}, \left\lfloor\frac{\mathfrak{n}}{2}\right\rfloor, \left\lceil \frac{\mathfrak{n}}{2}\right\rceil, \ln{\left\lfloor\frac{\mathfrak{n}}{2}\right\rfloor}, \ln{\left\lceil \frac{\mathfrak{n}}{2}\right\rceil}$
by Proposition~\ref{prop:lnflooroverapprox} and Proposition~\ref{prop:nminusoneoverapprox}.
The details of the important over-approximations are illustrated explicitly
in Table~\ref{tbl:unioverapprox}.
\item {\em Recursive Step.} We have two cases:
(a)~If $\mathfrak{g}$ is $\mathfrak{g}_1+\mathfrak{g}_2$,
then $\mathsf{OvAp}(\mathfrak{g},h)$ is $\mathsf{OvAp}(\mathfrak{g}_1,h)+\mathsf{OvAp}(\mathfrak{g}_2,h)$.
(b)~If $\mathfrak{g}$ is $c'\cdot\mathfrak{g}'$, then
$\mathsf{OvAp}(\mathfrak{g},h)$ is $c'\cdot\mathsf{OvAp}(\mathfrak{g}',h)$.
\end{itemize}
\end{definition}
\begin{table}
\caption{Illustration for Definition~\ref{def:unioverapprox} where the notations are given in the top-left corner.}
\label{tbl:unioverapprox}
\centering
\scalebox{0.82}{
\begin{tabular}{|c|c|c|c|}
\hline
Notation & Expression & $\mathfrak{f}$, $\mathrm{T}$-term & Over-approximation \\
\hline
$\mathfrak{e}_1$ & $\mathrm{T}(\mathfrak{n}-1)$ & $\ln{\mathfrak{n}}$, $\mathfrak{e}_1$ &
$\ln{\mathfrak{n}}-\frac{1}{\mathfrak{n}}$\\
\hline
$\mathfrak{e}_2$ & $\mathrm{T}\left(\left\lfloor\frac{\mathfrak{n}}{2}\right\rfloor\right)$ & $\ln{\mathfrak{n}}$, $\mathfrak{e}_2$ & $\ln{\mathfrak{n}}-\ln{2}$
\\
\hline
$\mathfrak{e}_3$ & $\mathrm{T}\left(\left\lceil\frac{\mathfrak{n}}{2}\right\rceil\right)$ & $\ln{\mathfrak{n}}$, $\mathfrak{e}_3$ & $\ln{\mathfrak{n}}-\ln{2}+\frac{1}{\mathfrak{n}}$ \\
\hline
$\mathfrak{e}_4$ & $\frac{1}{\mathfrak{n}}\cdot \sum_{\mathfrak{j}=1}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j})$ & $\ln{\mathfrak{n}}$, $\mathfrak{e}_4$ & $\ln{\mathfrak{n}}-1-\frac{\ln{\mathfrak{n}}}{2\cdot\mathfrak{n}} +\frac{13}{12}\cdot\frac{1}{\mathfrak{n}}$\\
\hline
$\mathfrak{e}_5$ & $\frac{1}{\mathfrak{n}}\cdot\left(\sum_{\mathfrak{j}=\left\lceil\frac{\mathfrak{n}}{2}\right\rceil}^{\mathfrak{n}-1}\mathrm{T}(\mathfrak{j})+ \sum_{\mathfrak{j}=\left\lfloor\frac{\mathfrak{n}}{2}\right\rfloor}^{\mathfrak{n}-1} \mathrm{T}(\mathfrak{j})\right)$ &
$\ln{\mathfrak{n}}$, $\mathfrak{e}_5$ & $\ln{\mathfrak{n}}-(1-\ln{2})+\frac{\ln{\mathfrak{n}}}{2\cdot \mathfrak{n}}+\frac{0.6672}{\mathfrak{n}}+\frac{1}{2\cdot \mathfrak{n}^2} $ \\
\hline
\hline
$\mathfrak{f}$, $\mathrm{T}$-term & Over-approximation & $\mathfrak{f}$, $\mathrm{T}$-term & Over-approximation \\
\hline
$\mathfrak{n}$, $\mathfrak{e}_1$ & $\mathfrak{n}-1$ & $\mathfrak{n}\cdot\ln{\mathfrak{n}}$, $\mathfrak{e}_1$ & $\mathfrak{n}\cdot\ln{\mathfrak{n}}-\ln{\mathfrak{n}}-1+\frac{1}{\mathfrak{n}}$ \\
\hline
$\mathfrak{n}$, $\mathfrak{e}_2$ & $\frac{\mathfrak{n}}{2}$ & $\mathfrak{n}\cdot\ln{\mathfrak{n}}$, $\mathfrak{e}_2$ & $\frac{1}{2}\cdot\mathfrak{n}\cdot \ln{\mathfrak{n}}-\frac{\ln{2}}{2}\cdot\mathfrak{n}$ \\
\hline
$\mathfrak{n}$, $\mathfrak{e}_3$ & $\frac{\mathfrak{n}+1}{2}$ & $\mathfrak{n}\cdot\ln{\mathfrak{n}}$, $\mathfrak{e}_3$ & $\frac{\mathfrak{n}\cdot \ln{\mathfrak{n}}}{2}-\frac{\ln{2}}{2}\cdot \mathfrak{n}+\frac{1-\ln{2}}{2}+\frac{\ln{\mathfrak{n}}}{2}+\frac{1}{2\cdot\mathfrak{n}}$ \\
\hline
$\mathfrak{n}$, $\mathfrak{e}_4$ & $\frac{\mathfrak{n}-1}{2}$ & $\mathfrak{n}\cdot\ln{\mathfrak{n}}$, $\mathfrak{e}_4$ & $\frac{\mathfrak{n}\cdot\ln{\mathfrak{n}}}{2}-\frac{\mathfrak{n}}{4}-\frac{\ln{\mathfrak{n}}}{2}+\frac{\ln{\mathfrak{n}}}{12\cdot\mathfrak{n}}+\frac{0.5139}{\mathfrak{n}}$ \\
\hline
\multirow{2}{*}{$\mathfrak{n}$, $\mathfrak{e}_5$} & \multirow{2}{*}{$\frac{3}{4}\cdot \mathfrak{n}-\frac{1}{4\cdot \mathfrak{n}}$} & \multirow{2}{*}{$\mathfrak{n}\cdot\ln{\mathfrak{n}}$, $\mathfrak{e}_5$} & $\frac{3}{4}\cdot \mathfrak{n}\cdot \ln{\mathfrak{n}}-0.2017\cdot \mathfrak{n}-\frac{1}{2}\cdot \ln{\mathfrak{n}}$ \\
& & & $-0.2698+\frac{\ln{\mathfrak{n}}}{8\cdot\mathfrak{n}}+\frac{1.6369}{\mathfrak{n}}+\frac{1}{2\cdot\mathfrak{n}\cdot(\mathfrak{n}-1)}+\frac{1}{4\cdot \mathfrak{n}^2}$ \\
\hline
\end{tabular}
}
\vspace{-1em}
\end{table}
\begin{example}\label{ex:reloverapprox}
Consider the recurrence relation for Sherwood's {\sc Randomized-Search} (cf.~(\ref{eq:relrandsearch})).
Choose $\mathfrak{f}=\ln{\mathfrak{n}}$ and then the template $h$ becomes
$n\mapsto d\cdot \ln{n}+1$.
From Example~\ref{ex:overapprox}, we have that the over-approximation for
$6+\frac{1}{\mathfrak{n}}\cdot\left( \sum_{\mathfrak{j}=\left\lceil\frac{\mathfrak{n}}{2}\right\rceil}^{\mathfrak{n}-1}\mathrm{T}(\mathfrak{j})+ \sum_{\mathfrak{j}=\left\lfloor\frac{\mathfrak{\mathfrak{n}}}{2}\right\rfloor}^{\mathfrak{\mathfrak{n}}-1} \mathrm{T}(\mathfrak{j})\right)$
when $n\ge 4$ is
$7+ d\cdot \left[\ln{n}-(1-\ln{2})+\frac{\ln{n}}{2\cdot n}+\frac{0.6672}{n}+\frac{1}{2\cdot n^2}\right]$
(the second summand comes from an over-approximation of
$\frac{1}{\mathfrak{n}}\cdot\left( \sum_{\mathfrak{j}=\left\lceil\frac{\mathfrak{n}}{2}\right\rceil}^{\mathfrak{n}-1}d\cdot \ln{\mathfrak{j}}+ \sum_{\mathfrak{j}=\left\lfloor\frac{\mathfrak{\mathfrak{n}}}{2}\right\rfloor}^{\mathfrak{\mathfrak{n}}-1} d\cdot \ln{\mathfrak{j}}\right)$).\qed
\end{example}
\begin{remark}
Since integrations of the form $\int x^k\ln^l x\,\mathrm{d}x$ can be calculated in closed forms (cf. Remark~\ref{rmk:exthigerdegree}), Table~\ref{tbl:unioverapprox} can be extended to logarithmic expressions with higher order, e.g., $\mathfrak{n}^2\ln \mathfrak{n}$.\qed
\end{remark}
\smallskip\noindent{\em Pseudo-polynomials.}
Our next step is to define the notion of (univariate) pseudo-polynomials
which extends normal polynomials with logarithm.
This notion is crucial to handle inductive arguments in the definition
of (univariate) guess-and-check functions.
\begin{definition}[Univariate Pseudo-polynomials]
A univariate pseudo-polynomial (w.r.t logarithm) is a function $p:\Nset\rightarrow\Rset$ such that there exist non-negative integers $k,\ell \in \Nset_0$ and real numbers $a_i,b_i$'s such that for all $n\in\Nset$,
\vspace{-0.5em}
\begin{equation}\label{eq:pseudopoly}
\textstyle p(n)=\sum_{i=0}^{k} a_i\cdot n^{i}\cdot \ln{n}+\sum_{i=0}^{\ell} b_i\cdot n^{i}.
\end{equation}
\vspace{-1em}
\end{definition}
W.l.o.g, we consider that in the form (\ref{eq:pseudopoly}), it holds that
(i) $a^2_k+b^2_\ell\ne 0$,
(ii) either $a_k\ne 0$ or $k=0$, and (iii) similarly either $b_\ell\ne 0$ or $\ell=0$.
\smallskip\noindent{\em Degree of pseudo-polynomials.}
Given a univariate pseudo-polynomial $p$ in the form (\ref{eq:pseudopoly}), we define the \emph{degree} $\mathrm{deg}(p)$ of $p$ by:
$\mathrm{deg}(p)= k+\frac{1}{2}$ if $k\ge \ell\mbox{ and }a_k\ne 0$ and $\ell$ otherwise.
Intuitively, if the term with highest degree involves logarithm, then we increase the degree by $1/2$,
else it is the power of the highest degree term.
\smallskip\noindent{\em Leading term $\overline{p}$.}
The \emph{leading term} $\overline{p}$ of a pseudo-polynomial $p$ in the form (\ref{eq:pseudopoly}) is a function
$\overline{p}:\Nset\rightarrow\Rset$ defined as follows:
$\overline{p}(n)=a_{k}\cdot n^{k}\cdot \ln{n} \mbox{ if }k\ge \ell\mbox{ and }a_k\ne 0$; and
$b_{\ell}\cdot n^{\ell} \mbox{ otherwise }$; for all $n\in\Nset$.
Furthermore, we define $C_p$ to be the (only) coefficient of $\overline{p}$.
With the notion of pseudo-polynomials, the inductive argument of guess-and-check functions can be soundly transformed into an inequality between pseudo-polynomials.
\begin{lemma}\label{lemm:unitrans}
Let $\mathfrak{f}\in\{\ln{\mathfrak{n}},\mathfrak{n},\mathfrak{n}\cdot\ln{\mathfrak{n}}\}$
and $c$ be a constant.
For all univariate recurrence expressions $\mathfrak{g}$, there exists pseudo-polynomials
$p$ and $q$ such that coefficients (i.e., $a_i,b_i$'s in~(\ref{eq:pseudopoly}))
of $q$ are all non-negative, $C_q>0$ and the following assertion holds:
for all $d>0$ and for all $n\ge 2$, with $h=d\cdot \mathsf{Subst}({\mathfrak{f}})+c$,
the inequality $\mathsf{OvAp}(\mathfrak{g}, h)(n)\le h(n)$ is equivalent to $d\cdot p(n)\ge q(n)$.
\end{lemma}
\begin{remark}\label{rem:unitrans}
In the above lemma, though we only refer to existence of pseudo-polynomials $p$ and $q$,
they can actually be computed in linear time, because $p$ and $q$ are obtained by simple rearrangements
of terms from $\mathsf{OvAp}(\mathfrak{g}, h)$ and $h$, respectively.
\end{remark}
\begin{example}\label{ex:inequality}
Let us continue with Sherwood's {\sc Randomized-Search}.
Again choose $h=d\cdot\ln{\mathfrak{n}}+1$.
From Example~\ref{ex:reloverapprox}, we obtain that for every $n\ge 4$, the inequality
\begin{align*}
d\cdot\ln{n}+1\ge 7+ d\cdot \left[\ln{n}-(1-\ln{2})+\frac{\ln{n}}{2\cdot n}+\frac{0.6672}{n}+\frac{1}{2\cdot n^2}\right]
\end{align*}
resulting from over-approximation and the inductive argument of guess-and-check functions
is equivalent to
$d\cdot\left[(1-\ln{2})\cdot n^2-\frac{n\cdot\ln{n}}{2}-0.6672\cdot n-\frac{1}{2}\right]\ge 6\cdot n^2$.\qed
\end{example}
As is indicated in Definition~\ref{def:uniguess},
our aim is to check whether $ \mathsf{OvAp}(\mathfrak{g}, h)(n)\le h(n)$
holds for sufficiently large $n$.
The following proposition provides a sufficient and necessary condition for checking whether
$d\cdot p(n)\ge q(n)$ holds for sufficiently large $n$.
\begin{proposition}\label{prop:unisufflarge}
Let $p,q$ be pseudo-polynomials such that $C_q>0$ and
all coefficients of $q$ are non-negative.
Then there exists a real number $d>0$ such that
$d\cdot p(n)\ge q(n)$
for sufficiently large $n$ iff $\mathrm{deg}(p)\ge \mathrm{deg}(q)$ and $C_p>0$.
\end{proposition}
Note that by Definition~\ref{def:uniguess} and the special form (\ref{eq:uniguess}) for univariate guess-and-check functions, a function in form (\ref{eq:uniguess}) needs only
to satisfy the inductive argument in order to be a univariate guess-and-check function: once a value for $d$ is synthesized for a sufficiently large $N$, one can scale the value so that the base condition
is also satisfied.
Thus from the sufficiency of Proposition~\ref{prop:unisufflarge}, our decision algorithm that checks the existence of some guess-and-check function in form (\ref{eq:uniguess}) is presented below.
Below we fix an input univariate recurrence relation $G$ taking the form (\ref{eq:unirecurrel}) and an input expression $\mathfrak{f}\in\{\ln{\mathfrak{n}},\mathfrak{n},\mathfrak{n}\cdot\ln{\mathfrak{n}}\}$.
\textbf{Algorithm} $\mbox{\sl UniDec}$: Our algorithm, namely $\mbox{\sl UniDec}$, for the decision problem of
the univariate case, has the following steps.
\begin{compactenum}
\item {\em Template.} The algorithm establishes a scalar variable $d$ and sets up the template
$d\cdot \mathfrak{f}+c$ for a univariate guess-and-check function.
\item {\em Over-approximation.}
Let $h$ denote $d \cdot \mathsf{Subst}(\mathfrak{f}) +c$. The algorithm calculates the over-approximation function $\mathsf{OvAp}(\mathfrak{e},h)$,
where $\mathfrak{e}$ is from (\ref{eq:unirecurrel}).
\item {\em Transformation.} The algorithm transforms the inequality
$\mathsf{OvAp}(\mathfrak{e},h)(n) \le h(n) ~~(n\in\Nset)$
for inductive argument of guess-and-check functions through Lemma~\ref{lemm:unitrans} equivalently into
$d\cdot p(n)\ge q(n)~~(n\in\Nset)$,
where $p,q$ are pseudo-polynomials obtained in linear-time through rearrangement of terms from
$\mathsf{OvAp}(\mathfrak{e},h)$ and $h$ (see Remark~\ref{rem:unitrans}).
\item {\em Coefficient Checking.} The algorithm examines cases on $C_p$. If $C_p> 0$ and $\mathrm{deg}(p)
\ge \mathrm{deg}(q)$,
then algorithm outputs ``$\mbox{\sl yes}$'' meaning that ``there exists a univariate guess-and-check function'';
otherwise, the algorithm outputs ``$\mbox{\sl fail}$''.
\end{compactenum}
\begin{theorem}[Soundness for $\mbox{\sl UniDec}$]\label{thm:soundnessunidec}
If $\mbox{\sl UniDec}$ outputs ``$\mbox{\sl yes}$'', then there exists a univariate guess-and-check function in form~(\ref{eq:uniguess})
for the inputs $G$ and $\mathfrak{f}$.
The algorithm is a linear-time algorithm in the size of the input recurrence relation.
\end{theorem}
\begin{example}
Consider Sherwood's {\sc Randomized-Search} recurrence relation (cf.~(\ref{eq:relrandsearch})) and $\mathfrak{f}=\ln{\mathfrak{n}}$ as the input.
As illustrated in Example~\ref{ex:reloverapprox} and Example~\ref{ex:inequality}, the algorithm
asserts that the asymptotic behaviour is $\mathcal{O}(\ln{n})$.\qed
\end{example}
\begin{remark}
From the tightness of our over-approximation (up to only constant deviation) and
the sufficiency and necessity of Proposition~\ref{prop:unisufflarge},
the $\mbox{\sl UniDec}$ algorithm can handle a large class of univariate recurrence relations.
Moreover, the algorithm is quite simple and efficient (linear-time).
However, we do not know whether our approach is complete. We suspect that there is certain intricate recurrence relations that will make our approach fail.
\end{remark}
\noindent{\bf Analysis of examples of Section~\ref{sec:motivatinguni}.}
Our algorithm can decide the following optimal bounds for the examples of Section~\ref{sec:motivatinguni}.
\begin{compactenum}
\item For Example~\ref{ex:randsearch} we obtain an $\mathcal{O}(\log n)$ bound (recall worst-case bound is $\Theta(n)$).
\item For Example~\ref{ex:quicksort} we obtain an $\mathcal{O}(n\cdot\log n)$ bound (recall worst-case bound is $\Theta(n^2)$).
\item For Example~\ref{ex:quickselect} we obtain an $\mathcal{O}(n)$ bound (recall worst-case bound is $\Theta(n^2)$).
\item For Example~\ref{ex:diameter} we obtain an $\mathcal{O}(n\cdot\log n)$ (resp. $\mathcal{O}(n)$) bound for Euclidean metric
(resp. for $L_1$ metric), whereas the worst-case bound is $\Theta(n^2\cdot\log n)$ (resp. $\Theta(n^2)$).
\item For Example~\ref{ex:sortselect} we obtain an $\mathcal{O}(n\cdot\log n)$ bound (recall worst-case bound is $\Theta(n^2)$).
\end{compactenum}
In all cases above, our algorithm decides the asymptotically optimal bounds for the expected-runtime
analysis, whereas the worst-case analysis grossly over-estimate the expected-runtime bounds.
\smallskip\noindent{\bf Quantitative bounds.}
Above we have already established that our linear-time decision algorithm can establish
the asymptotically optimal bounds for the recurrence relations of several classical algorithms.
We now take the next step to obtain even explicit quantitative bounds, i.e., to synthesize
the associated constants with the asymptotic complexity.
To tackle these situations, we derive a following proposition which gives explicitly a
threshold for ``sufficiently large numbers''.
We first explicitly constructs a threshold for ``sufficiently large numbers''.
Then we show in Proposition~\ref{prop:unisufflargeN}
that $N_{\epsilon,p,q}$ is indeed what we need.
\begin{definition}[Threshold $N_{\epsilon,p,q}$ for Sufficiently Large Numbers]\label{def:unisuffN}
Let $p,q$ be two univariate pseudo-polynomials
$p(n)=\sum_{i=0}^{k} a_i\cdot n^{i}\cdot \ln{n}+\sum_{i=0}^{\ell} b_i\cdot n^{i}$~,
$q(n)=\sum_{i=0}^{k'} a'_i\cdot n^{i}\cdot \ln{n}+\sum_{i=0}^{\ell'} b'_i\cdot n^{i}$
such that $\mathrm{deg}(p)\ge \mathrm{deg}(q)$ and $C_p,C_q>0$. Then given any $\epsilon\in (0,1)$,
the number $N_{\epsilon,p,q}$ is defined as the smallest natural number such that both $x,y$ (defined below) is smaller than $\epsilon$:
\begin{compactitem}
\item $x=-1+\sum_{i=0}^{k} |a_i|\cdot \frac{N^{i}\cdot \ln{N}}{\overline{p}(N)}+\sum_{i=0}^{\ell} |b_i|\cdot \frac{N^{i}}{\overline{p}(N)}$ ;
\item $y=-\mathbf{1}_{\mathrm{deg}(p)=\mathrm{deg}(q)}\cdot\frac{C_q}{C_p}+\sum_{i=0}^{k'} |a'_i|\cdot \frac{N^{i}\cdot \ln{N}}{\overline{p}(N)}+\sum_{i=0}^{\ell'} |b'_i|\cdot \frac{N^{i}}{\overline{p}(N)}$ .
\end{compactitem}
where $\mathbf{1}_{\mathrm{deg}(p)=\mathrm{deg}(q)}$ equals $1$ when ${\mathrm{deg}(p)=\mathrm{deg}(q)}$ and $0$ otherwise.
\end{definition}
\begin{proposition}\label{prop:unisufflargeN}
Consider two univariate pseudo-polynomials $p,q$ such that $\mathrm{deg}(p)\ge \mathrm{deg}(q)$, all coefficients of $q$ are non-negative and $C_p,C_q>0$.
Then given any $\epsilon\in (0,1)$,
$\frac{q(n)}{p(n)}\le \frac{\mathbf{1}_{\mathrm{deg}(p)=\mathrm{deg}(q)}\cdot \frac{C_q}{C_p}+\epsilon}{1-\epsilon}$
for all $n\ge N_{\epsilon,p,q}$ (for $N_{\epsilon,p,q}$ of Definition~\ref{def:unisuffN}).
\end{proposition}
With Proposition~\ref{prop:unisufflargeN}, we describe our algorithm $\mbox{\sl UniSynth}$ which outputs explicitly a value for $d$ (in (\ref{eq:uniguess})) if $\mbox{\sl UniDec}$ outputs yes.
Below we fix an input univariate recurrence relation $G$ taking the form (\ref{eq:unirecurrel}) and an input expression $\mathfrak{f}\in\{\ln{\mathfrak{n}},\mathfrak{n},\mathfrak{n}\cdot\ln{\mathfrak{n}}\}$.
Moreover, the algorithm takes $\epsilon>0$ as another input, which is basically a parameter to choose the threshold for finite behaviour.
For example, smaller $\epsilon$ leads to large threshold, and vice-versa.
Thus we provide a flexible algorithm as the threshold can be varied with the choice of $\epsilon$.
\textbf{Algorithm} $\mbox{\sl UniSynth}$: Our algorithm for the quantitative problem has the following steps:
\begin{compactenum}
\item {\em Calling $\mbox{\sl UniDec}$.} The algorithm calls $\mbox{\sl UniDec}$, and if it
returns ``$\mbox{\sl fail}$'', then return ``$\mbox{\sl fail}$'', otherwise execute the following steps.
Obtain the following inequality $d\cdot p(n)\ge q(n)~~(n\in\Nset)$
from the transformation step of $\mbox{\sl UniDec}$.
\item {\em Variable Solving.}
The algorithm calculates $N_{\epsilon, p,q}$ for a given $\epsilon\in(0,1)$ by e.g. repeatedly increasing $n$ (see Definition~\ref{def:unisuffN})
and outputs the value of $d$ as the least number such that
the following two conditions hold:
(i)~for all $2\le n< N_{\epsilon, p,q}$, we have $\mathsf{Eval}(G)(n)\le d\cdot \mathsf{Subst}({\mathfrak{f}})(n)+c$
(recall $\mathsf{Eval}(G)(n)$ can be computed in linear time), and
(ii)~we have $d\ge \frac{\mathbf{1}_{\mathrm{deg}(p)=\mathrm{deg}(q)}\cdot \frac{C_q}{C_p}+\epsilon}{1-\epsilon}$.
\end{compactenum}
\begin{theorem}[Soundness for $\mbox{\sl UniSynth}$]\label{thm:soundnessunisynth}
If the algorithm $\mbox{\sl UniSynth}$ outputs a real number $d$, then $d\cdot \mathsf{Subst}(\mathfrak{f})+c$ is a univariate guess-and-check function for $G$.
\end{theorem}
\begin{example}
Consider the recurrence relation for Sherwood's {\sc Randomized-Search} (cf.~(\ref{eq:relrandsearch})) and $\mathfrak{f}=\ln{\mathfrak{n}}$.
Consider that $\epsilon:=0.9$.
From Example~\ref{ex:reloverapprox} and Example~\ref{ex:inequality}, the algorithm establishes the inequality
$d\ge \frac{ 6}{(1-\ln{2})-\frac{\ln{n}}{2\cdot n}-\frac{0.6672}{n}-\frac{1}{2\cdot n^2}}$
and finds that $N_{0.9,p,q}=6$.
Then the algorithm finds $d=204.5335$ through the followings:
(a) $\mathsf{Eval}(G)(2)=7\le d\cdot \ln{2}+1$;
(b) $\mathsf{Eval}(G)(3)=11\le d\cdot \ln{3}+1$;
(c) $\mathsf{Eval}(G)(4)=15\le d\cdot \ln{4}+1$;
(d) $\mathsf{Eval}(G)(5)=17.8\le d\cdot \ln{5}+1$;
(e) $d\ge \frac{\frac{6}{1-\ln{2}}+0.9}{1-0.9}$.
Thus, by Theorem~\ref{thm:uniguess}, the expected running time of the algorithm
has an upper bound $204.5335\cdot \ln{n}+1$.
Later in Section~\ref{sect:experiments}, we show that one can obtain a much better $d=19.762$ through our algorithms by choosing $\epsilon:=0.01$, which is quite good since the optimal value lies in $[15.129, 19.762]$ (cf. the first item {\sc R.-Sear.} in Table~\ref{tab:experiments}).\qed
\end{example}
\vspace{-1.5em}
\subsection{Algorithm for Bivariate Recurrence Relations}\label{sect:bisynth}
\vspace{-1em}
In this part, we present our results for the separable bivariate recurrence relations.
The key idea is to use separability to reduce the problem to univariate recurrence relations.
There are two key steps which we describe below.
\smallskip\noindent{\em Step~1.}
The first step is to reduce a separable bivariate recurrence relation to a univariate one.
\begin{definition}[From $G$ to $\Uni{G}$]
Let $G$ be a separable bivariate recurrence relation taking the form~(\ref{eq:birecurrel}).
The univariate recurrence relation $\Uni{G}$ from $G$ is defined by eliminating any occurrence of $\mathfrak{n}$ and replacing any occurrence of $\mathfrak{h}$ with $1$.
\end{definition}
Informally, $\Uni{G}$ is obtained from $G$ by simply eliminating the roles of $\mathfrak{h},\mathfrak{n}$.
The following example illustrates the situation for {\sc Coupon-Collector} example.
\begin{example}
Consider $G$ to be the recurrence relation (\ref{eq:relcoupon}) for {\sc Coupon-Collector} example.
Then $\Uni{G}$ is as follows:
$\mathrm{T}(\mathfrak{n})=\frac{1}{\mathfrak{n}}+ \mathrm{T}(\mathfrak{n}-1)$ and $\mathrm{T}(1)=1$.
\qed
\end{example}
\smallskip\noindent{\em Step~2.}
The second step is to establish the relationship between $T_G$ and $T_{\Uni{G}}$, which is handled by the following proposition,
whose proof is an easy induction on $m$.
\begin{proposition}\label{prop:reduction}
For any separable bivariate recurrence relation $G$ taking the form (\ref{eq:birecurrel}), the solution $T_G$ is equal to $(n,m)\mapsto \mathsf{Subst}(\mathfrak{h})(n) \cdot T_{\Uni{G}}(m)$.
\end{proposition}
\smallskip {\em Description of the Algorithm.}
With Proposition~\ref{prop:reduction}, the algorithm for separable bivariate recurrence relations is straightforward:
simply compute $\Uni{G}$ for $G$ and then call the algorithms for univariate case presented in Section~\ref{sect:unisynth}.
\smallskip\noindent{\bf Analysis of examples in Section~\ref{sec:motivatingbi}.}
Our algorithm can decide the following optimal bounds for the examples of Section~\ref{sec:motivatingbi}.
\begin{compactenum}
\item For Example~\ref{ex:coupon} we obtain an $\mathcal{O}(n\cdot \log m)$ bound, whereas the worst-case bound is $\infty$.
\item For Example~\ref{ex:channel} we obtain an $\mathcal{O}(n\cdot\log m)$ bound for distributed setting and $\mathcal{O}(m)$ bound for concurrent setting, whereas the worst-case bounds are both $\infty$.
\end{compactenum}
Note that for all our examples, $m \leq n$, and thus we obtain $\mathcal{O}(n\cdot \log n)$ and $\mathcal{O}(n)$ upper bounds for expected-runtime
analysis, which are the asymptotically optimal bounds.
In all cases above, the worst-case analysis is completely ineffective as the worst-case bounds are infinite.
Moreover, consider Example~\ref{ex:channel}, where the optimal number of rounds is $n$ (i.e., one process every round,
which centralized Round-Robin schemes can achieve).
The randomized algorithm, with one shared variable, is a decentralized algorithm that achieves $O(n)$ expected
number of rounds (i.e., the optimal asymptotic expected-runtime complexity).
|
1,116,691,497,763 | arxiv | \section{Introduction}
Non-collider experiments and astronomical observations have given us our first hints of physics beyond the Standard Model, via the discoveries of neutrino oscillations, dark energy, and dark matter. The implication of these discoveries for fundamental physics is still unknown. The energy scale of the new degrees of freedom giving rise to neutrino oscillations could be as high as $10^{16}$ GeV, as in Grand Unified theories, or as low as $0.05$ eV, as in Dirac neutrino mass models. Even more mysterious is the nature of dark energy and dark matter, and the associated energy scale or scales. If the new physics is light, it must be very weakly coupled to the Standard Model, or it would already have been discovered. Neutrino oscillation measurements offer an unmatched portal into any new nonstandard sectors containing light fermions, because neutrinos can mix with neutral spin 1/2 particles, and because oscillations over long baselines are extraordinarily sensitive to extremely tiny effects.
The long baseline experiment MINOS, which uses similar near and far detectors to reduce systematic errors, has observed the disappearance of both muon neutrinos and muon antineutrinos in the far detector \cite{Michael:2006rx,Gogos:2007zz,Adamson:2008zt,Evans:2008zzc}. The antineutrino data comes from a 7\% antineutrino contamination of the beam and is severely statistics limited \cite{Adamson:2009ju}. Nonetheless, it is interesting to note that
the antineutrino disappearance rate is larger than the rate expected from neutrino disappearance by almost 2 sigma \cite{Evans:2008zzc}. Currently MINOS is running in antineutrino mode, offering a unique opportunity to precisely measure the parameters governing the long baseline oscillations of muon antineutrinos. It is the purpose of this paper to offer a framework for searching for new physics in the antineutrino data. Most recent analyses \cite{Evans:2008zzc,Dighe:2008bu,Koranga:2009ep,Arias:2009fk,Barenboim:2009ts,Diaz:2009qk,Arias:2009fk} of anomalies in antineutrino data have focussed on CPT violation in the neutrino mass matrix
\cite{Murayama:2000hm,Barenboim:2001ac,Skadhauge:2001kk,Bilenky:2001ka,Jacobson:2003wc,Kostelecky:2003cr}. However there is no theoretical motivation for CPT violation, and CPT violation requires Lorentz violation which is complicated to incorporate in a complete theory that is consistent with other data. In contrast, in this paper we propose a simple, renormalizable, Lorentz invariant field theory, which is consistent with other experiments and which allows a significant difference between muon neutrino and antineutrino disappearance in the MINOS experiment.
Many other papers have analyzed the consistency of neutrino oscillation data with sterile neutrinos \cite{Barger:1990bg,Goswami:1995yq,GonzalezGarcia:1998dp,Maltoni:2004ei,GonzalezGarcia:2007ib,Maltoni:2007zf} and new forces \cite{Valle:1987gv}. Our model differs from those considered previously in three ways. First, we consider a relatively long range but weakly coupled new force for which the size of the matter effect can be considerably larger than the usual MSW effect, in a theory which is nevertheless consistent with precision electroweak constraints. Second, we consider smaller neutrino mass squared differences that primarily affect long baseline experiments for muon neutrinos. Thirdly, many previous analyses were concerned with the effects of sterile neutrinos on electron neutrino oscillations, while we are not attempting to address any anomalies involving electron neutrinos and are considering a model where the sterile neutrino has no electron neutrino component.
\section{Review of the standard picture of neutrino oscillations}
A standard picture of 3 flavor neutrino oscillations has been successful in explaining phenomena observed by many diverse long baseline experiments \cite{Maki:1962mu,GonzalezGarcia:2010er}.
The $e$, $\mu$, and $\tau$ flavor eigenstates are related to the mass eigenstates via a unitary transformation, parameterized by three angles, neglecting a CP-violating phase:
\begin{equation}\label{stdosc}
\left( \begin{array}{c}
\nu_e \\
\nu_{\mu} \\
\nu_{\tau}
\end{array} \right) = \left( \begin{array}{c c c}
1 & 0 & 0 \\
0 & c_{23} & s_{23} \\
0 & -s_{23} & c_{23}
\end{array} \right)\left( \begin{array}{c c c}
c_{13} & 0 & s_{13} \\
0 & 1 & 0 \\
-s_{13} & 0 & c_{13}
\end{array} \right)\left( \begin{array}{c c c}
c_{12} & s_{12} & 0 \\
-s_{12} & c_{12} & 0 \\
0 & 0 & 1
\end{array} \right)\left( \begin{array}{c}
\nu_1 \\
\nu_2 \\
\nu_3
\end{array} \right)\equiv U_{LNS}\left( \begin{array}{c}
\nu_1 \\
\nu_2 \\
\nu_3
\end{array} \right)\ ,
\end{equation}
where $c_{ij} = \cos\theta_{ij}$, $s_{ij} = \sin\theta_{ij}$. A good fit to long baseline neutrino oscillation data may be found \cite{Amsler:2008zzb} for angles
\begin{equation}\label{stdangles}
\begin{split}
\sin^2 2 \theta_{12} &= 0.87\pm0.03,\\
\sin^2 2\theta_{13} &< 0.19 \textrm{ (90\% C.L.}),\\
\sin^2 2\theta_{23} &= 1.0^{+0}_{-0.08} ,
\end{split}
\end{equation}
and neutrino mass squared differences:
\begin{equation}\label{stdmasses}
\begin{split}
\Delta m_{12}^2 &= (7.6\pm0.2)\cdot10^{-5}\textrm{ eV}^2/c^4 ,\\
\big|\Delta m_{23}^2\big| &= (2.4\pm0.1)\cdot10^{-3}\textrm{ eV}^2/c^4 .
\end{split}
\end{equation}
Note that in this picture, for $L/E < 1000$ km/GeV, the smaller mass squared difference $\Delta m_{12}^2$ gives oscillation probabilities which are always less than a percent, and which may be neglected compared with experimental uncertainties. Therefore for $L/E < 1000$ km/GeV, the observed oscillations are only sensitive to $\Delta m_{23}^2$ and, due to the small size of $\theta_{13}$, are primarily muon flavor into tau flavor.
In matter, the oscillation parameters of neutrinos are modified due to the weak potential, a phenomena known as the MSW effect \cite{Wolfenstein:1977ue,Mikheev:1986gs,Langacker:1986jv}. In the standard picture, only oscillations involving $\nu_e$ from a charged current weak potential are affected, as the $\nu_\mu$ and $\nu_\tau $ neutrinos only experience a flavor universal, neutral current weak potential in matter.
The MINOS experiment studies the oscillations of both muon neutrinos and muon antineutrinos at a distance of 735 km, over an energy range from 1 to 50 GeV. The survival probability as a function of energy for muon neutrinos, $P(\nu_{\mu}\to\nu_{\mu})$, is well parameterized by Eqs.~\ref{stdangles} and \ref{stdmasses}. The survival probability for muon antineutrinos, $P(\bar{\nu}_{\mu}\to\bar{\nu}_{\mu})$ is only marginally consistent with the standard picture, and is not well characterized by the oscillation parameters in Eqs.~\ref{stdangles} and \ref{stdmasses}. The marginal agreement of the neutrino and antineutrino oscillation data at MINOS has raised interest, particularly because in vacuum, if CPT is a good symmetry, then the two oscillation probabilities are equal:
\begin{equation}
\textrm{CPT} \quad \Rightarrow \quad P(\nu_{\mu}\to\nu_{\mu}) = P(\bar{\nu}_{\mu}\to\bar{\nu}_{\mu})\, ,
\end{equation}
and in the standard oscillation picture, the presence of matter does not significantly change this relation at the MINOS baseline.
\section{Sterile neutrinos, a new force, and MINOS }
The MINOS antineutrino disappearance data has led several authors to investigate the possibility that CPT violation may have been observed at MINOS. However, because the neutrinos are passing through matter, alternative explanations include a nonstandard interaction between the neutrinos and the matter which distinguishes between neutrinos and antineutrinos, a new kind of neutrino with a different weak charge, or both.
In this work we discuss a model that can generate a difference between muon neutrino and muon antineutrino disappearance probabilities in matter at the MINOS energies and baseline. The basic framework is simple. We extend the Standard Model to allow the active neutrinos to mix with a sterile neutrino that has no Standard Model weak interaction. Because a sterile neutrino has a different weak charge than the active neutrinos, in matter muon neutrinos and muon antineutrinos will have different oscillation parameters, although this effect is small and only marginally detectable at the MINOS baseline.
In Ref.~\cite{Nelson:2007yq}, two of us
proposed a new $U(1)$ gauge force, coupled to the difference between baryon and lepton numbers ($B-L$), which can enhance this matter effect, in order to account for the anomalous results of the MiniBooNE and LSND experiments. We showed that provided the new boson was very weakly coupled ($g \mathop{}_{\textstyle \sim}^{\textstyle <} 10^{-5}$), such a boson of mass greater than $\approx0.2 $~keV could escape all laboratory constraints. If such a weakly coupled boson is lighter than a few MeV, it can provide a potential in the earth's crust which is much larger than the standard MSW potential. Furthermore, in Ref.~\cite{Nelson:2008tn} we showed that a light vector boson could have a chameleon nature, with very different effective mass in extreme environments, allowing it to escape astrophysical and cosmological constraints.
Due to the very different energies and baselines of LSND, MiniBooNE, and MINOS, a minimal model which gives an observable anomalous neutrino/antineutrino oscillation difference at MINOS would not necessarily significantly affect LSND and MiniBooNE.
The best fit points that we will find for the model from the MINOS neutrino and antineutrino oscillation data have a contribution to the potential from the $B-L$ interaction which is approximately 4 orders of magnitude smaller than the one required to reconcile the shorter baseline MiniBooNE and LSND experiments. This allows the $B-L$ coupling $g$ to be lower by 2 orders of magnitude for the same $B-L$ vector boson mass. Consequently, the constraints on the $B-L$ force discussed in Ref.~\cite{Nelson:2007yq} do not limit the model in describing neutrino/antineutrino oscillation anomalies at MINOS, and in general at long baseline oscillation experiments.
As in the usual seesaw model of neutrino masses \cite{GellMann:1976pg,Minkowski:1977sc,Weinberg:1979sa,Mohapatra:1979ia,Bilenky:1980cx,Yanagida:1980xy,Lazarides:1980nt,Weinberg:1980bf,Schechter:1980gr,Cheng:1980qt}, we consider two types of neutrinos -- neutrinos $\nu_i$ with lepton number $L=1$ which carry Standard Model weak interactions and neutrinos $N_i$ with $L=-1$ which are sterile under the Standard Model. In general the number of sterile neutrinos is not fixed by terrestrial experiment. We allow for both lepton number conserving and violating mass terms, with both types of mass terms required to give active-sterile mixing. The allowed lepton number conserving mass terms, which arise from a Yukawa coupling to the Standard Model Higgs, are
\begin{equation}
m_{ij}\nu_iN_j + \textrm{ h.c.}\ .
\end{equation}
We also have lepton number violating Majorana masses for the sterile neutrinos
\begin{equation}
M_{ij}N_iN_j + \textrm{ h.c.}
\end{equation}
The Majorana terms are gauge invariant under the Standard Model. With gauged $B-L$, they may arise from an expectation value of a $B-L$ charged scalar.
Unlike in the usual seesaw model, we assume that at least one of the sterile neutrinos is very light. For simplicity of analysis we will assume any other sterile neutrinos, such as would be required to cancel a gauge $B-L$ anomaly, are heavier and more weakly mixed and will integrate them out. We thus consider an oscillation picture involving 4 light neutrinos-- three active and one sterile.
Neutrinos are produced and detected as one of the three active neutrino flavors, which propagate as a superposition of all 4 mass eigenstates. A 4-by-4 unitary matrix $U$ transforms between the flavor and mass eigenstate bases. In general, four flavor neutrino oscillations are affected by 6 mixing angles and three $CP$ violating phases. We make some assumptions to reduce the number of free parameters and simplify the analysis. We assume the neutrino masses to be hierarchical, with the heaviest mass eigenstate mostly sterile. We assume the sterile neutrino is a mixture of the 2 heaviest states only. We can then write the mixing matrix as \begin{equation} U=U_aU_s,\end{equation} where
$U_a$ is a block diagonal matrix
\begin{equation} U_a=\left( \begin{array}{c c c c}
& & & \\
& U_{LNS} & &\\
& & &1
\end{array} \right)\end{equation}
representing mixing between the three active neutrinos and
\begin{equation} U_s=\left( \begin{array}{c c c c}
1& 0&0&0 \\
0 & 1&0 &0\\
0 & 0 &\cos\theta_{34} &\sin\theta_{34} \\
0&0&-\sin\theta_{34} &\cos\theta_{34}
\end{array} \right)\end{equation}
describes mixing between active and sterile neutrinos. In the flavor basis, the Hamiltonian is
\begin{equation}
\mathcal{H}_f = \frac{1}{2E}U_aU_s \mathcal{M}_m^2 U_s^T U_a^T + \mathcal{V}_f .
\end{equation}
$\mathcal{M}_m$ is the mass matrix in the mass basis and $\mathcal{V}_f$ is the potential matrix in the flavor basis,
\begin{equation}
\mathcal{V}_f = {\rm diag}(V_{cc} - V_{nc} - V_{B-L},\,-V_{nc} - V_{B-L},\,-V_{nc} - V_{B-L},\, V_{B-L}) .
\end{equation}
The potential $V_{cc}$ arises from the weak charged current, $V_{nc}$ from the weak neutral current, and $V_{B-L}$ from a new $B-L$ gauge interaction. The values are:
\begin{equation}
V_{cc}=\pm\sqrt2 G_F N_e, \textrm{ } V_{nc} = \pm \frac{G_F }{\sqrt2}N_n, \textrm{ and } V_{B-L} = \frac{\tilde{g}^2}{m_V^2}N_n\, ,
\end{equation}
where $N_e$ is the electron density, $N_n$ is the neutron density, $\tilde{g}$ is the $B-L$ coupling constant, and $m_V$ is the mass of the $B-L$ vector boson. With a reasonable matter density of $\rho \approx 2.7$ g/cm$^3$, the neutral current potential has magnitude
\begin{equation}
V_{nc} \approx 5.0\cdot10^{-5}\textrm{ neV}.
\end{equation}
We will define the parameter $V \equiv \frac12 V_{nc} + V_{B-L}$, which will arise in the oscillation probability formula. The factor of $1/2$ arises because the neutral current potential only affects active neutrinos. $V$ in matter is positive for neutrinos and negative for antineutrinos.
For simplicity, and because the MINOS range of $L/E$ is not large enough to be sensitive to the smaller mass squared differences, we neglect the 2 smaller masses and take
\begin{equation}
\mathcal{M}_m = {\rm diag}(0,0,m,M) .
\end{equation}
With this restricted mixing, $U_{12}$ commutes with $U_s$ and $\mathcal{M}_m$. Therefore, we can neglect $U_{12}$ in $U_a$, since it does not contribute to oscillations. For simplicity, we also neglect the small angle $\theta_{13}$. Then the electron neutrino does not significantly participate in oscillations at the MINOS baseline, meaning $V_{cc}$ is irrelevant and $U_a$ commutes with $\mathcal{V}_f$. We can further reduce parameters by assuming the mass $m$ results entirely from mixing with the light sterile neutrino via the seesaw mechanism, receiving no significant contribution from the heavier neutrinos we have integrated out. The active-sterile mixing angle then satisfies
\begin{equation}
\tan \left(2\theta_{34}\right) = \frac{2\sqrt{Mm}}{M-m} .
\end{equation}
We can now write the Hamiltonian in an interaction basis where .
\begin{equation}
H_f = U_aU_s' H_i U_s'{}^TU_a^T ,
\end{equation}
with
\begin{equation}
U_s'H_i U_s'{}^T = \frac{1}{2E}U_s\mathcal{M}_m^2U_s^T + \mathcal{V}_f .
\end{equation}
The diagonalized Hamiltonian is
\begin{equation}
H_i = \frac{1}{2E}{\rm diag}\left(0,0,\tilde{m}^2,\tilde{M}^2\right) ,
\end{equation}
and the mixing matrix between the sterile neutrino and a linear combination of $\nu_\mu$ and $\nu_\tau$ is
\begin{equation}
U_s' = \left( \begin{array}{c c c c}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & \cos\theta_s & \sin\theta_s \\
0 & 0 & -\sin\theta_s & \cos\theta_s
\end{array} \right) .
\end{equation}
In terms of the other parameters,
\begin{equation}\label{mangle}
\begin{split}
\tan\left( 2\theta_s\right) = \frac{\sin\left( 2\theta_{34}\right)}{\cos\left(2\theta_{34}\right) + \alpha} ,
\end{split}
\end{equation}
where
\begin{equation}
\alpha \equiv \frac{4EV}{M^2 - m^2} .
\end{equation}
Note that this mixing angle is larger in the antineutrino sector, where $\alpha$ is negative, and is maximal for
\begin{equation}\cos(2\theta_{34}) = \frac{-4EV}{M^2 - m^2} \quad \Rightarrow \quad |4EV| = (M-m)^2 \ . \end{equation}
This effect can give substantial conversion of muon antineutrinos into sterile fermions for a range of energies at the MINOS baseline.
The eigenvalues of the Hamiltonian are
\begin{equation}\label{eigen}
\begin{split}
\tilde{m}^2 &= m^2 + \frac{M^2 - m^2}{2}\left[1 + \alpha - \sqrt{1 + 2\alpha\cos(2\theta_{34}) + \alpha^2}\right] ,\\
\tilde{M}^2 &= m^2 + \frac{M^2 - m^2}{2}\left[1 + \alpha + \sqrt{1 + 2\alpha\cos(2\theta_{34}) + \alpha^2}\right] .
\end{split}
\end{equation}
The unitary matrix relating the interaction and flavor bases is
\begin{equation}
U_aU_s' = \left( \begin{array}{c c c c}
1 & 0 & 0 & 0\\
0 & c_{23} & s_{23}c_s & s_{23}s_s \\
0 & -s_{23} & c_{23}c_s & c_{23}s_s \\
0 & 0 & -s_s & c_s
\end{array} \right) ,
\end{equation}
with $c_s = \cos\theta_s$, $s_s = \sin\theta_s$. Taking $\theta_{23} = \pi/4$ gives the simple mixing matrix
\begin{equation}
U_aU_s' = \left( \begin{array}{c c c c}
\vspace{0.05in}
1 & 0 & 0 & 0 \\
\vspace{0.05in}
0 & \frac{1}{\sqrt2} & \frac{1}{\sqrt2}c_s & \frac{1}{\sqrt2}s_s \\
\vspace{0.05in}
0 & -\frac{1}{\sqrt2} & \frac{1}{\sqrt2}c_s & \frac{1}{\sqrt2}s_s \\
0 & 0 & -s_s & c_s
\end{array} \right) .
\end{equation}
The muon neutrino survival probability is then
\begin{equation}
P(\nu_\mu\to\nu_\mu) = 1 - \cos^2\theta_s \sin^2\left(\frac{\tilde{m}^2 L}{4E}\right) - \sin^2\theta_s \sin^2\left(\frac{\tilde{M}^2 L}{4E}\right) - \cos^2\theta_s\sin^2\theta_s \sin^2\left(\frac{(\tilde{M}^2 - \tilde{m}^2)L}{4E}\right) .
\end{equation}
The same formula applies to antineutrinos, although the values of $\tilde m, \tilde M$ and $\theta_s$ are different because of the opposite sign of $\alpha$ in Eqs.~\ref{mangle} and \ref{eigen}.
\section{Comparison of Neutrino Oscillation Model with MINOS Data}
Armed with the muon neutrino survival probability, we can constrain the parameter space of the model using the results of the MINOS experiment. We will perform a combined fit of the model parameters $m$, $V$, and $M$ using the MINOS $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ survival data. We briefly discuss implications for the K2K \cite{Ahn:2002up} and SuperK \cite{Fukuda:1998mi} experiments, and the constraints from the CDHS results \cite{Dydak:1983zq}.
We perform a $\chi^2$ fit to the MINOS $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ data, simultaneously varying the three model parameters $m$, $M$, and $V$. The parameters are restricted to the ranges
\begin{align}
10^{-4} \textrm{ eV } < \,\,&m < 1 \textrm{ eV } \\
0.1 \textrm{ eV } < \,\,&M < 10 \textrm{ eV } \\
0.25\cdot10^{-4} \textrm{ neV } < \,\,&V < 10 \textrm{ neV }
\end{align}
and each parameter is sampled from a logarithmic distribution (so that, for example, the same number of points are sampled for $M$ in the ranges $(0.1, 1)$ eV and $(1, 10)$ eV). We use as our $\chi^2$ function
\begin{equation}
\chi^2 = 2\left(N_{\tiny{\textrm{exp}}} - N_{\tiny{\textrm{obs}}} + N_{\tiny{\textrm{obs}}}\ln\left(\frac{N_{\tiny{\textrm{obs}}}}{N_{\tiny{\textrm{exp}}}}\right)\right)
\end{equation}
which is more suitable for samples with small statistics \cite{Amsler:2008zzb}. To narrow down the range of parameter space, we use an approximation the $\chi^2$ value to estimate which regions of parameter space will provide good fits to the data. A sampling of approximately $2.5\cdot10^5$ points was sufficient to fill out the parameter space and identify the 68.3\%, 95.5\%, and 99.7\% confidence level contours. A more detailed sampling was used to identify the best fit point.
The MINOS data is divided into 24 total bins, 17 in the neutrino data and 7 in the antineutrino data. Fitting the three parameters $m$, $M$, and $V$ yields a best fit point
\begin{equation}
m = 0.0394 \textrm{ eV, } \quad M = 0.157 \textrm{ eV, } \quad V = 2.01\cdot10^{-4} \textrm{ neV.}
\end{equation}
The best fit point has a total $\chi^2 = 24.8 = 1.24$/dof. Note that the value of the potential at the best fit point is about 8 times the neutral current contribution to the potential $V$, which is $0.25\cdot10^{-4}$ neV. The mostly-active mass $m$ at the best fit point is close to the corresponding value of $(\Delta m_{\textrm{atm}}^2)^{1/2}$ in the standard picture, 0.049 eV.
\begin{figure}[htbp]
\subfloat[\large{$M$ vs. $m$}] {\label{mM} \includegraphics[width = .32\textwidth] {mMcontour.pdf}}
\subfloat[\large{$V$ vs. $m$}] {\label{mV} \includegraphics[width = .32\textwidth] {mlightVcontour.pdf}}
\subfloat[\large{$V$ vs. $M$}] {\label{MV} \includegraphics[width = .32\textwidth] {MVcontour.pdf}}
\caption{Regions within $68\%$ (blue), $95.5\%$ (red), and $99.7\%$ (green) confidence limits, along with the best fit point (star), for pairs of parameters. The neutral current contribution to $V$, which is the lower limit, is the dashed line.}
\label{ParamSpace}
\end{figure}
In Fig.~\ref{ParamSpace}, we show the regions within the $68.3\%$, $95.5\%$, and $99.7\%$ confidence limits of the best fit point for pairs of parameters. In each plot, we have projected out the other parameter, taking the minimum $\chi^2$ over the parameter not displayed. The figures clearly indicate there are two regions of parameter space that are favored by the model. Both regions have $\tilde{m}^2$ near $\Delta m_{\textrm{atm}}^2$ for energies near the oscillation dip for both neutrinos and anti-neutrinos.
The two regions of parameter space fitting the MINOS data have very different characteristics. One region has a nearly constant value of $m$, approximately the best fit value of $m$, and prefers larger values of $M$ and smaller mixing with the sterile neutrino. This region contains the natural extrapolation of the model to the standard picture, where $m^2 = \Delta m_{\textrm{atm}}^2$ and $M\to\infty$, which removes mixing between active and sterile neutrinos.
The second favored region has larger values of $V$ and smaller values of $m$. In the second region $m$ and $M$ are inversely related: as $m$ increases in this region, $M$ decreases. The allowed parameters produce an oscillation maximum for neutrinos at the MINOS baseline at about the same energy as the standard picture, while antineutrinos have a nonstandard looking pattern. In the second region the oscillation pattern for neutrinos can differ significantly from the standard picture in low density material and for neutrino oscillations at different baselines.
Two other experiments with evidence for muon neutrino disappearance are K2K and SuperK. K2K uses a muon neutrino beam with baseline of 250 km and energy range from 0.5 GeV to 3 GeV. SuperK detects neutrinos produced by cosmic rays in the atmosphere and infers a baseline distance from their direction, with upward going neutrinos traveling much farther than downward going neutrinos. SuperK cannot distinguish neutrino events from antineutrinos, however most of their events are from neutrinos due to the larger neutrino cross section. K2K does not have any antineutrino data. SuperK data prefers oscillations of muon neutrinos into tau neutrinos over sterile states \cite{Fukuda:2000np}. In our model muon neutrinos at SuperK should mostly oscillate into $\tau$ neutrinos, since the mixing angle of neutrinos with sterile neutrinos is small at all energies and matter densities. However, due to the enhanced mixing angle of antineutrinos with sterile neutrinos in matter, for some parameters a significant fraction of the upward going muon antineutrinos at SuperK could convert into sterile states.
We do not attempt an analysis of constraints from the K2K data, because the statistics are very limited, or of the SuperK data, because such an analysis requires access to details of the data. However we note that
the first region of our fit looks very much like the standard picture at shorter baseline experiments such as K2K, where the effects of the potential are small, and in low density material, such as experienced by downward going neutrinos at SuperK, since in the first region the potential is small, mixing with the sterile state is small, and muon neutrino oscillations are dominated by $m$, which is near the standard atmospheric value.
In the second region however the potential is larger and the neutrino oscillation pattern looks nonstandard at all distances other than the MINOS baseline. This region could conceivably be constrained by an analysis of SuperK data. For instance, the muon neutrino survival probability for downward going neutrinos at SuperK will be higher than the standard picture because matter effects are small in the atmosphere and because the vacuum mass squared difference between the neutrinos with large mixing angle is smaller. A SuperK analysis was done for mass varying neutrinos \cite{us,Kaplan:2004dq} whose mass was different in the earth and in the atmosphere, and this analysis did produce some constraints \cite{Abe:2008zza}. In particular, a model with no muon neutrino oscillations in air was excluded at a 3.5 $\sigma$ level. Thus a similar analysis should be able to exclude the regions of parameter space with very small $m$.
All of our fit points are consistent with short baseline tests of muon neutrino disappearance. The strongest such constraints come from CDHS \cite{Dydak:1983zq} which is sensitive to mass squared differences larger than 0.3 eV$^2$, and has the strongest mixing angle constraints for muon neutrino mixing for a mass squared difference of 2.5 eV$^2$. For this model the we find the CDHS constraints are weaker than those of MINOS, due to the fact that the sterile mixing angle is quite small for larger mass squared differences. For comparison, we give the the effective mixing angle $\theta_{\rm eff} $ for 2 neutrino interpretations of muon neutrino disappearance at short baseline:
\begin{equation}
\sin^2(2\theta_{\rm eff})\approx\sin^2\theta_s(1+\cos^2\theta_s) \ .
\end{equation}
For the case where $\alpha$ is negligible, the fit points always have $\sin^2(2\theta_{\rm eff})$ smaller than $0.13$ for effective mass squared differences larger than $0.3$ eV$^2$. For effective mass squared differences larger than 2 eV$^2$, $\sin^2(2\theta_{\rm eff})< 0.05$, compared with the strongest CDHS constraint of $\sin^2(2\theta_{\rm eff})< 0.053$. Nonnegligible $\alpha$ makes the effective neutrino mixing angle smaller. Thus the CDHS results do not provide any additional constraint.
We also remove the $B-L$ potential from the model, so that only the neutral current potential is present, and perform a two parameter fit to $m$ and $M$. This fit finds a minimum $\chi^2 = 28.1 = 1.34$/dof, and lies within the $68\%$ confidence level contour of the best fit point for the three parameter fit. The best fit values are
\begin{equation}
V_{B-L} = 0:\, m = 0.0420 \textrm{ eV, } \,\, M = 0.309 \textrm{ eV.}
\end{equation}
and in good agreement with the MINOS neutrino and antineutrino data.
\begin{figure}[htbp]
\subfloat[\large{neutrinos}] {\label{nu} \includegraphics[width = .48\textwidth] {Nu_ModelSpectrum.pdf}}
\subfloat[\large{antineutrinos}] {\label{nubar} \includegraphics[width = .48\textwidth] {NuBar_ModelSpectrum.pdf}}
\caption{Muon flavor survival probability for the various models (solid red is the standard picture, dashed blue is the model with $V_{B-L}$, and dotted purple is the model without) against the data as a function of energy.}
\label{ModelSpectra}
\end{figure}
In Fig.~\ref{ModelSpectra}, we plot the predicted muon flavor survival probabilities as a function of energy at MINOS for the model at the best fit points with and without $V_{B-L}$ against the data. For reference, we also include the survival probability for the standard picture.
An analysis of the MINOS neutrino data has shown that oscillation into $\nu_\tau$ is preferred over $\nu_s$. The model is consistent with this observation, as neutrinos oscillate primarily into $\nu_\tau$. Antineutrino have larger oscillations into the sterile flavor than do neutrinos, and for the best fit parameters oscillations of muon antineutrinos into sterile neutrinos dominate oscillations in to $\bar{\nu}_\tau$ at higher energies. In Fig.~\ref{MuTauSterile}, we plot the $\tau$ and sterile flavor conversion rates for our model at the best fit point for both neutrinos and antineutrinos at MINOS. In the same figure we also show the same data at the best fit point of the model with $V_{B-L}$ set to zero.
\begin{figure}[htbp]
\subfloat[\large{neutrinos, best fit $V_{B-L}$ }] {\label{nuV} \includegraphics[width = .48\textwidth] {Nu_MuTauSterileOsc.pdf}}
\subfloat[\large{antineutrinos, best fit $V_{B-L}$ }] {\label{nubarV} \includegraphics[width = .48\textwidth] {NuBar_MuTauSterileOsc.pdf}}
\subfloat[\large{neutrinos, no $V_{B-L}$}] {\label{nuNoV} \includegraphics[width = .48\textwidth] {Nu_MuTauSterileOsc_NoV.pdf}}
\subfloat[\large{antineutrinos, no $V_{B-L}$}] {\label{nubarNoV} \includegraphics[width = .48\textwidth] {NuBar_MuTauSterileOsc_NoV.pdf}}
\caption{Predicted event rates with data as a function of neutrino energy, for the best fit point. The solid blue curve shows the $\mu$ flavor survival rate, while the dashed red curve and dotted purple curve show the rates for conversion into $\tau$ and sterile flavors respectively.}
\label{MuTauSterile}
\end{figure}
Note that the rate of conversion of active neutrinos into sterile is predicted to be small, due to the small vacuum active-sterile mixing angle $\theta_{34}$. However the rate for active antineutrinos oscillations into sterile neutrinos is enhanced in matter.
The MINOS experiment has also searched for oscillations into sterile neutrinos via measurements of the neutral current rate at the far detector \cite{Koskinen:2009zz,Sousa:2009wp,Adamson:2010wi}. They report a ratio $R$ of observed to expected neutral current events. The effect of $\nu_e$ appearance, which can mimic a neutral current, is treated as an uncertainty in the analysis. The result is
\begin{eqnarray}}% can be used as {equation} or {eqnarray
R&=0.99\pm0 .09({\rm stat})\pm 0.07({\rm syst}) (-0.08) \ \ \ &E<3\ {\rm GeV}\cr
R&=1.09\pm0.12({\rm stat})\pm0.10 ({\rm syst})(-0.13)\ \ \ &3<E<120\ {\rm GeV}\cr
\end{eqnarray}
where $\nu_e$ appearance is assumed to be negligible, and the last number in parentheses shows the effect on $R$ of assuming the maximally-allowed $\nu_e$ appearance. These results are obtained from a neutrino run, with only a 7\% antineutrino component to the beam.
We have also computed $R$ for the best fit point, with and without a $B-L$ potential, in our model. We have neglected the possibility of sterile neutrino scattering via the $B-L$ gauge boson and mimicking a neutral current, because we are assuming the $B-L$ coupling constant is small enough so that this is negligible. We find:
\begin{eqnarray}}% can be used as {equation} or {eqnarray
R&=0.88 \ \ \ &E<3,\ {\rm GeV},\ {\rm neutrinos,\ best \ fit \ }V_{B-L}\cr
R&=0.83\ \ \ &E<4,\ {\rm GeV},\ {\rm antineutrinos,\ best \ fit \ }V_{B-L}\cr
R&=0.89\ \ \ &3<E<50, \ {\rm GeV},\ {\rm neutrinos,\ best \ fit \ }V_{B-L}\cr
R&=0.71\ \ \ &4<E<50, \ {\rm GeV},\ {\rm antineutrinos,\ best \ fit \ }V_{B-L}\cr
R&=0.90 \ \ \ &E<3,\ {\rm GeV},\ {\rm neutrinos,\ no \ }V_{B-L}\cr
R&=0.90\ \ \ &E<4,\ {\rm GeV},\ {\rm antineutrinos,\ no \ }V_{B-L}\cr
R&=0.90\ \ \ &3<E<50, \ {\rm GeV},\ {\rm neutrinos,\ no\ }V_{B-L}\cr
R&=0.89\ \ \ &4<E<50, \ {\rm GeV},\ {\rm antineutrinos,\ no \ }V_{B-L}.\end{eqnarray}
The antineutrino data is split up at 4 GeV energy instead of 3 because the lowest bin for the antineutrinos goes up to 4 GeV.
Note that while the neutrino neutral current rate is in agreement with the reported MINOS data, there is an interesting possibility that the antineutrino run at MINOS could detect a significant depletion in the neutral current rate, particularly at higher energies.
\section{Conclusions}
There is no compelling evidence for an antineutrino oscillation anomaly at the MINOS experiment. The data is statistics limited, and the current discrepancy is not statistically significant. However, this discrepancy offers the opportunity to discuss models that produce interesting experimental signatures at neutrino oscillation experiments. A neutrino/antineutrino oscillation anomaly can be explained by CPT violation, but it can also be explained by more conservative, well-motivated, and theoretically sound models.
As we have shown in this work, the addition of a sterile neutrino can generate a difference in the neutrino and antineutrino oscillation probabilities due purely to the weak neutral current interactions, and the addition of a $B-L$ gauge interaction will coherently add to the potential and enhance the oscillation probability difference. The model we have proposed can fit well the reported MINOS neutrino and antineutrino data, and matter effects are a nice framework to explore neutrino oscillations outside of the standard picture.
As long baseline neutrino experiments enter an era of greatly increased baseline, intensity, and diversity, it is important to keep in mind that the history of neutrino experiment is full of surprises. Long baseline neutrino oscillation experiments using antineutrinos offer a unique and sensitive window to possible exotic sectors and forces, and it is important to keep a variety of possibilities in mind when analyzing the data.
\medskip
\noindent {\bf Acknowledgments}
This work was partially supported by the DOE under contract DE-FGO3-96-ER40956. The work of Netta Engelhardt was partially supported by the NSF under the University of Washington REU program. We thank Banibrata Mukhopadhyay, Tommy Ohlsson, and Tony Mann for correspondence.
|
1,116,691,497,764 | arxiv | \section{Introduction}
\label{intro}
The starting point of this article on Hopf algebras is a well-known result due to A. Grothendieck, as outlined in \cite{Sweedler}, saying that the category of finite-dimensional, commutative and cocommutative Hopf $K$-algebras over a field $K$ is abelian. This result was extended by M. Takeuchi to the category of commutative and cocommutative Hopf $K$-algebras, not necessarily finite-dimensional \cite{Takeuchi}.
The category $\mathbf{Hopf}_{K,coc}$ of cocommutative Hopf $K$-algebras is not additive, thus it can not be abelian. In the present article we investigate some of its fundamental exactness properties, showing that it is a homological category (Section \ref{homological}), and that it is Barr-exact (Section \ref{exact}), leading to the conclusion that the category $\mathbf{Hopf}_{K,coc}$ is semi-abelian \cite{Janelidze2002367} when the field $K$ is of characteristic zero (Theorem \ref{semi-abelian}). This result establishes a new link between the theory of Hopf algebras and the more recent one of semi-abelian categories, both of which can be viewed as wide generalizations of group theory.
Since a category $\mathbf{C}$ is abelian if and only if both $\mathbf{C}$ and its dual $\mathbf{C}^{op}$ are semi-abelian, this observation can be seen as a ``non-commutative'' version of Takeuchi's theorem mentioned above. The fact that the category $\mathbf{Hopf}_{K,coc}$ is semi-abelian was independently obtained by Clemens Berger and Stephen Lack. A recent article of Christine Vespa and Marc Wambst shows that the abelian core of $\mathbf{Hopf}_{K,coc}$ is the category of commutative and cocommutative Hopf $K$-algebras \cite{Vespa}.
In the present work we also prove the existence of a non-abelian torsion theory $(\mathbf{T},\mathbf{F})$ in $\mathbf{Hopf}_{K,coc}$ (Theorem \ref{hereditary torsion}), where the torsion subcategory $\mathbf{T}$ is the category of \emph{primitive Hopf K-algebras}, which is isomorphic to the category of Lie $K$-algebras, and the torsion-free subcategory
$\mathbf{F}$ is the category of \emph{group Hopf K-algebras}, which is isomorphic to the category of groups.
The categories of groups and of Lie $K$-algebras are two typical examples of semi-abelian categories: this shows again that the theories of cocommutative Hopf algebras and of semi-abelian categories are strongly intertwined.
The category $\mathbf{Hopf}_{K,coc}$ inherits
some fundamental exactness properties from groups and Lie algebras thanks to the well-known canonical decomposition of a cocommutative Hopf algebra into a semi-direct product of a group Hopf algebra and a primitive Hopf algebra (a result associated with the names Cartier-Gabriel-Kostant-Milnor-Moore).
The present work opens the way to some new applications of categorical Galois theory \cite{Janelidze} in the category of cocommutative Hopf $K$-algebras, since the reflection from this category to the torsion-free subcategory of group Hopf algebras enjoys all the properties needed for this kind of investigations, as we briefly explain in Section \ref{torsion}. We conclude the article by observing that the semi-abelian category $\mathbf{Hopf}_{K,coc}$ is also an action representable category \cite{BJK} (Corollary \ref{representable object actions}).
\section{Preliminaries}
\subsection{Semi-abelian categories.\\}
Semi-abelian categories \cite{Janelidze2002367} are finitely complete, pointed, exact in the sense of M. Barr \cite{Barr}, protomodular in the sense of D. Bourn \cite{protomodularity}, with finite coproducts.
These categories have been introduced to capture some typical algebraic properties valid for non-abelian algebraic structures such as groups, Lie algebras, rings, crossed modules, varieties of $\Omega$-groups in the sense of P. Higgins \cite{Higgins01071956} and compact groups. As already mentioned in the introduction, every abelian category is in particular semi-abelian.
Although protomodularity is a property that can be expressed in any category with finite limits, in the pointed context, i.e. when there is a zero object $0$ in $\mathbf{C}$, protomodularity amounts to the fact that the following formulation of the \emph{Split Short Five Lemma} holds:
given a commutative diagram
\[\xymatrixcolsep{5pc}\xymatrix{
0 \ar[r]^-{} & K \ar[r]^-{k} \ar[d]_-{\kappa} & A \ar@<0.5ex>[r]^-{f} \ar[d]_-{\alpha} & B \ar[d]^-{\beta} \ar@<0.5ex>[l]^-{s} \\
0 \ar[r]^-{} & K' \ar[r]_-{k'} & A' \ar@<0.5ex>[r]^-{f'} & B' \ar@<0.5ex>[l]^-{s'} }\]
\begin{flushleft}
where $k=ker(f)$, $k'=ker(f')$, $f\circ s = 1_{B}$, and $f'\circ s' = 1_{B'}$ (i.e. $f$ and $f'$ are split epimorphisms with sections $s$ and $s'$), if both $\kappa$ and $\beta$ are isomorphisms, then so is $\alpha$.
\end{flushleft}
Any protomodular category $\mathbf{C}$ is a \textit{Mal'tsev} category \cite{CLP}: this means that every (internal) reflexive relation $\mathbf{C}$ is an (internal) equivalence relation. Recall that a reflexive relation on an object $X$ is a diagram of the form
\begin{equation}\label{reflexive}
\xymatrix{
R \ar@<1ex>[r]^-{p_{1}} \ar@<-1ex>[r]_-{p_{2}} & X \ar[l]|-{\delta},
}
\end{equation}
where $p_1$ and $p_2$ are jointly monic, and $p_{1}\circ \delta = 1_{X} = p_{2}\circ \delta$; such a reflexive relation $R$ is an equivalence relation when, moreover, there exist $\sigma: R \longrightarrow R$ and $\tau : R \times_{X} R \longrightarrow R$ as in the following diagram
\[\xymatrixcolsep{6pc}
\xymatrix{
R \times_{X} R \ar[r]^-{\tau} & R \ar@(ur,ul)_-{\sigma} \ar@<1ex>[r]^-{p_{1}} \ar@<-1ex>[r]_-{p_{2}} & X \ar[l]|-{\delta}
}\]
such that $p_{1}\circ \sigma = p_{2}$ and $p_{2}\circ \sigma = p_{1}$ (symmetry), and $p_{1} \circ \tau = p_{1}\circ \pi_{1}$ and $p_{2} \circ \tau = p_{2}\circ \pi_{2}$ (transitivity), where $\pi_{1}$ and $\pi_{2}$ are the projections in the following pullback:
\[\xymatrix{
R \times_{X} R \ar[r]^-{\pi_{2}} \ar[d]_-{\pi_{1}} \ar@{}[rd]|<<<{\copy\pullbackbox} & R \ar[d]^-{p_{1}}\\
R \ar[r]_-{p_{2}} & X
}\]
In the present article, by a \textit{regular} category is meant a finitely complete category where every morphism can be factorized as a regular epimorphism followed by a monomorphism, and where regular epimorphisms are pullback stable. A regular category $\mathbf C$ is said to be \textit{Barr-exact} if, moreover, every equivalence relation is effective, i.e. every equivalence relation is the kernel pair of a morphism in $\mathbf C$. A category which is pointed, protomodular and regular is said to be \textit{homological} \cite{BorceuxBourn}. In this context several basic diagram lemmas of homological algebra hold true (such as the snake lemma, the $3$-by-$3$-Lemma, etc.).
We end these preliminaries with the following diagram indicating some implications between the different contexts recalled above:
\[\xymatrixrowsep{1pc}\xymatrix{
& \text{semi-abelian} \ar[ld] \ar[rd] & \\
\text{homological} \ar[d] \ar[rrd] & & \text{Barr-exact} \ar[d]\\
\text{protomodular} \ar[d] & & \text{regular} \\
\text{Mal'tsev} & &
}\]
\subsection{The category $\mathbf{Hopf}_{K,coc}$ of cocommutative Hopf $K$-algebras.\\}
The category we study in this article is the category of Hopf $K$-algebras over a field $K$, denoted by $\mathbf{Hopf}_{K}$. The objects in $\mathbf{Hopf}_{K}$ are Hopf $K$-algebras, i.e. sextuples $(H,M,u,\Delta,\epsilon,S)$ where $(H,M,u)$ is a $K$-algebra and $(H,\Delta,\epsilon)$ is a $K$-coalgebra, such that these two structures are compatible, i.e. maps $M$ and $u$ are $K$-coalgebras morphisms, making $(H,M,u,\Delta,\epsilon)$ a $K$-bialgebra. The linear map $S$ is called the \textit{antipode}, and makes the following diagram commute:
\[\xymatrix{
H \ar[rrd]_-{\epsilon} \ar[r]^-{\Delta} & H\otimes H \ar@<0.5ex>[rr]^-{S \otimes id} \ar@<-0.5ex>[rr]_-{id \otimes S} & & H\otimes H \ar[r]^-{M} & H \\
& &K \ar[rru]_-{u}& &
}\]
\vspace*{0.2cm}
Morphisms in $\mathbf{Hopf}_{K}$ are exactly morphisms of $K$-bialgebras (i.e. morphisms that are both morphisms of $K$-algebras and $K$-coalgebras), as morphisms of $K$-bialgebras always preserve antipodes.
To denote the comultiplication map of a Hopf algebra $H$, we will use the Sweedler notation: $\forall h\in H$, $\Delta(h) = h_{1} \otimes h_{2}$ by omitting the summation sign. A Hopf algebra $H$ is said to be \textit{cocommutative} if its comultiplication map $\Delta$ makes the following diagram commute, where $\sigma$ is the linear map such that $\sigma(x\otimes y)= y\otimes x$, $\forall x,y \in H$
\[\xymatrix{
& H \otimes H \ar[rd]^-{\sigma}& \\
H\ar[rr]_-{\Delta} \ar[ru]^-{\Delta} & & H \otimes H
}\]
The category of cocommutative Hopf $K$-algebras will be denoted by $\mathbf{Hopf}_{K,coc}$.
In the category $\mathbf{Hopf}_{K,coc}$ there are two full subcategories which will be of importance for our work: the category $\mathbf{GrpHopf}_{K}$ of group Hopf algebras, and the category $\mathbf{PrimHopf}_{K}$ of primitive Hopf algebras, whose definitions we are now going to recall.
\begin{enumerate}
\item
The \textit{group Hopf algebra on a group G}, denoted by $K[G]$, is the free vector space on $G$ over the field $K$, i.e.\ $K[G] = \{\sum_{g\in G} \alpha_{g}g$, where $(\alpha_{g})_{g\in G}$ is a family of scalars with only a finite number being non zero$\}$ and $\{g\mid g\in G\}$ is a basis of $K[G]$. The group Hopf algebra $K[G]$ can be equipped with a structure of cocommutative Hopf algebra, by taking the multiplication induced by the group law, and comultiplication $\Delta: K[G] \longrightarrow K[G] \otimes K[G]$, counit $\epsilon:K[G]\to K$ and antipode $S:K[G]\to K[G]$ the linear maps defined on the base elements respectively by $\Delta(g) = g\otimes g$, $\epsilon(g)=1$ and $S(g)= g^{-1}$, $\forall g \in G$.
This assignment defines a functor $K[-] \colon \mathbf{Grp} \rightarrow \mathbf{Hopf}_{K}$ from the category of groups to the category of Hopf algebras, which has a right adjoint $\mathcal G \colon \mathbf{Hopf}_{K} \rightarrow \mathbf{Grp} $, that associates to any Hopf algebra $H$ its group of grouplike elements ${\mathcal G}(H)=\{x\in H~|~\Delta(x)=x\otimes x, \epsilon(x)=1\}$. We can now consider the category $\mathbf{GrpHopf}_{K}$, which is the full subcategory of $\mathbf{Hopf}_{K}$ whose objects are group Hopf algebras, that is Hopf algebras generated as algebra by grouplike elements. The functor $K[-] \colon \mathbf{Grp} \rightarrow \mathbf{GrpHopf}_{K}$ is an isomorphism of categories.
Indeed, it is clear that $K[-] \circ {\mathcal G}$ is the identity functor on the category of group Hopf algebras, and let us recall why the functor ${\mathcal G} \circ K[-]$ is the identity functor on the category of groups. If $G$ is a group and $K[G]$ the group Hopf K-algebra on $G$, one clearly has that $G \subseteq \mathcal{G} \big(K[G]\big)$ (in fact, this inclusion is the unit of the adjunction described above). Conversely, let $x= \sum_{g\in G} \alpha_{g}g\in K[G]$ be a group-like element. We have $\Delta(x) = \sum_{g\in G} \alpha_{g}g\otimes g$ but also $\Delta(x) = x\otimes x=\sum_{g,h\in G}\alpha_g\alpha_h g\otimes h$. Since $\{g\otimes h ~|~g,h\in G\}$ forms a basis of $K[G]\otimes K[G]$, we have $\forall g,h \in G$: $\alpha_g^2=\alpha_g$, and $\alpha_g\alpha_h=0$ if $g\neq h$. It follows that all $\alpha_g$'s should be zero except one that should be $1_K$, thus $x$ is in $G$.
\item The \textit{universal enveloping algebra of a Lie algebra L}, denoted by $U(L)$, is defined by the quotient $U(L)=T(L)/I$, where $T(L)$ is the tensor algebra on the vector space underlying $L$, and $I$ is the two-sided ideal of $T(L)$ generated by the elements of the form $x\otimes x' - x'\otimes x - [x,x']$, $\forall x,x' \in L$. Remark that the elements of $L$ generate $U(L)$ as an algebra. The universal enveloping algebra $U(L)$ can be equipped with a structure of cocommutative (and non commutative) Hopf algebra, by taking the concatenation as multiplication, and comultiplication $\Delta: U(L) \longrightarrow U(L) \otimes U(L)$, counit $\epsilon:U(L)\to K$ and antipode $S:U(L)\to U(L)$ the algebra maps defined on the generators by $\Delta(x) = x\otimes 1 + 1\otimes x$, $\epsilon(x)=0$ and $S(x)=-x$,
$\forall x \in L$.
Recall that for any Hopf algebra $H$ an element $x\in H$ is called a \textit{primitive element} if $\Delta(x) = x\otimes 1 + 1\otimes x$ (and consequently, $\epsilon(x)=0$).
The above constructions lead to a pair of adjoint functors, where the functor $U:\mathbf{LieAlg}_{K}\to \mathbf{Hopf}_{K}$ is a left adjoint to $P:\mathbf{Hopf}_{K}\to \mathbf{LieAlg}_{K}$.
We can now consider the category $\mathbf{PrimHopf}_{K}$, which is the full subcategory of $\mathbf{Hopf}_{K}$
whose objects are primitive Hopf algebras, that is Hopf algebras generated as algebra by primitive elements. In the case where $K$ is of characteristic $0$, the category $\mathbf{PrimHopf}_{K}$ is known to be isomorphic to the category $\mathbf{LieAlg}_{K}$ of Lie $K$-algebras \cite[Theorem 5.18]{Milnor}.
\end{enumerate}
\vspace*{0.3cm}
\begin{remark}
\textsf{
As can be seen from the formula for comultiplication, both group Hopf algebras and primitive Hopf algebras are cocommutative. Therefore the categories $\mathbf{GrpHopf}_{K}$ and $\mathbf{PrimHopf}_{K}$ are also full subcategories of $\mathbf{Hopf}_{K,coc}$.
The functors $U, P, K[-], \mathcal{G}$ and their adjunctions are represented in the following diagram:
\[\xymatrixcolsep{6pc}
\xymatrix{
\mathbf{Grp} \ar@<1ex>[r]^-{K[-]}_-{\perp} & \mathbf{Hopf}_{K,coc} \ar@<1ex>[r]^-{P}_-{\top} \ar@<1ex>[l]^-{\mathcal G} & \mathbf{LieAlg}_{K} \ar@<1ex>[l]^-{U}
}\]
}
\end{remark}
\section{The category of cocommutative Hopf algebras over a field of characteristic zero is homological}
\label{homological}
The category $\mathbf{Hopf}_{K,coc}$ is certainly pointed, with the zero object $K$, that will be denoted by $0$, from now on. $\mathbf{Hopf}_{K,coc}$ is complete and cocomplete, since it is locally presentable \cite{Porst1}. We will now establish its protomodularity and regularity.
\subsection{Protomodularity of the category $\mathbf{Hopf}_{K}$}\label{protomodular}
Let us consider the following commutative diagram of short exact sequences in the category $\mathbf{Hopf}_{K,coc}$:
\begin{equation}\label{diagram}
\xymatrixcolsep{5pc} \xymatrix{
0 \ar[r]^-{} & A \ar[r]^-{} \ar[d]_-{id_{A}} & C_{1} \ar[r]^-{} \ar[d]^-{\theta} & B \ar[d]^-{id_{B}} \ar[r]^-{} & 0 \\
0 \ar[r]^-{} & A \ar[r]_-{} & C_{2} \ar[r]_-{} & B \ar[r]^-{} & 0 }
\end{equation}
Thanks to the explicit descriptions of equalizers and coequalizers given in \cite{Andru1} one can easily prove that the kernel and the cokernel of $\theta$ are the zero object. This proves that $\theta$ is a monomorphism and an epimorphism of Hopf $K$-algebras. Since monomorphisms are injections and epimorphisms are surjections in the category $\mathbf{Hopf}_{K,coc}$ \cite{Chirva1,Nasta1}, this shows that $\theta$ is an isomorphism of Hopf $K$-algebras and so the category $\mathbf{Hopf}_{K,coc}$ is protomodular.
The category $\mathbf{Hopf}_{K,coc}$ is actually even \emph{strongly protomodular} \cite{strong}, since it has finite limits and it can be viewed as the category of internal groups in the category of cocommutative $K$-coalgebras (see \cite{Street}, for instance).
\subsection{Semi-direct products of cocommutative Hopf algebras}
Let $B$ be a cocommutative Hopf algebra. A {\em $B$-module Hopf algebra} is a Hopf algebra $A$ that is at the same time a left $B$-module with action $\rho :B\otimes A\to A, \rho(b\otimes a)=b\cdot a$ such that $\rho$ is a morphism of bialgebras. The semi-direct product (also known as {\em smash product}) of $B$ and $A$, denoted by $A \rtimes B$, is the Hopf algebra whose underlying vector space is the tensor product $A\otimes B$ and with the following structure: the unit is $u_{A\rtimes B}=u_A\otimes u_B$ and multiplication given by
$$(a\otimes b)(a'\otimes b')=a(b_1\cdot a')\otimes b_2b',$$
for all $a,a'\in A$ and $b,b'\in B$. The coalgebra structure is given by the tensor product coalgebra, i.e. $\Delta_{A\rtimes B}= (id_A \otimes \sigma \otimes id_{B})(\Delta_{A} \otimes \Delta_{B})$ and $\epsilon_{A\rtimes B}=\epsilon_A\otimes \epsilon_B$. The antipode is given by
$S_{A\rtimes B}(a\otimes b) = S_B(b_1)\cdot S_A(a)\otimes S_B(b_2)$.\\
The following Lemma is a reformulation of Theorem $4.1$ in \cite{Molnar}:
\begin{lemma}\label{Molnar}
Every split short exact sequence in $\mathbf{Hopf}_{K,coc}$
\[\xymatrixcolsep{5pc}\xymatrix{
0 \ar[r]^-{} & A \ar[r]^-{k} & H \ar@<0.5ex>[r]^-{p} & B \ar@<0.5ex>[l]^-{s} \ar[r] & 0 }\]
is canonically isomorphic to the semi-direct product exact sequence
\[\xymatrixcolsep{5pc}\xymatrix{
0 \ar[r]^-{} & A \ar[r]^-{i_1} & A \rtimes B \ar@<0.5ex>[r]^-{p_2} & B \ar@<0.5ex>[l]^-{i_2} \ar[r] & 0 }\]
where $i_1=id_A\otimes u_B$, $i_2=u_A\otimes id_B$ and $p_2=\epsilon_A\otimes id_B$.
\end{lemma}
\begin{proof}
The arrow $ h \colon A \rtimes B \rightarrow H$ in the diagram below is given by $h(a\otimes b)= k(a) s(b)$ for all $a\otimes b \in A \rtimes B$.
\[\xymatrixcolsep{5pc}\xymatrix{
0 \ar[r]^-{} & A \ar[r]^-{i_1} & A \rtimes B \ar@<0.5ex>[r]^-{p_2} \ar[d]^h & B \ar@<0.5ex>[l]^-{i_2} \ar[r] & 0 \\
0 \ar[r]^-{} & A \ar@{=}[u] \ar[r]^-{k} & H \ar@<0.5ex>[r]^-{p} & B \ar@<0.5ex>[l]^-{s} \ar[r]\ar@{=}[u] & 0 }
\]
This is a morphism of split short exact sequences, and therefore $h$ is an isomorphism by protomodularity of $\mathbf{Hopf}_{K,coc}$.
\end{proof}
We use this lemma to reformulate the well-known structure theorem for cocommutative Hopf algebras over an algebraically closed field of characteristic zero (see for instance \cite{Sweedler}, page 279 in combination with Lemma 8.0.1(c)) in terms of split exact sequences.
\begin{theorem}[Cartier-Gabriel-Moore-Milnor-Kostant]\label{MM} Every cocommutative Hopf K-algebra $H$, over an algebraically closed field $K$ of characteristic $0$, is isomorphic to the semi-direct product $$H \cong U(L_{H})\rtimes K[G_{H}]$$ of the universal enveloping algebra of a Lie algebra $U(L_{H})$ with the group Hopf algebra $K[G_{H}]$, where $L_{H}$ and $G_{H}$ are given respectively by the space of primitive elements and the group of group-like elements of $H$. Consequently, with notations of Lemma \ref{Molnar}, for each $H$ there exists a canonical split exact sequence of cocommutative Hopf algebras of the following form
\[\xymatrix{
0 \ar[r]^-{} & U(L_{H}) \ar[r]^-{i_{H}} & H \ar@<0.5ex>[r]^-{p_{H}} & K[G_{H}] \ar@<0.5ex>[l]^-{s_{H}} \ar[r]^-{} & 0 }\]
where $i_{H} = h \circ i_{1}$, $s_{H} = h \circ i_{2}$ and $p_{H} = p_{2} \circ h^{-1}$.
\end{theorem}
\begin{remark}\label{morphism}
Every morphism $f: H_{1} \longrightarrow H_{2}$ of cocommutative Hopf $K$-algebras gives rise to a morphism of split exact sequences of following form
\[\xymatrixcolsep{5pc}\xymatrix{
0 \ar[r]^-{} & U(L_{H_{1}}) \ar[r]_-{i_{H_1}} \ar[d]_-{f_{1}:=U(P(f))} & H_1 \ar[r]_-{p_{H_1}} \ar[d]_-{f} & K[G_{H_{1}}] \ar[d]^-{f_{2}:=K[\mathcal G(f)]} \ar[r]^-{} \ar@/_/[l]_-{s_{H_1}}& 0 \\
0 \ar[r]^-{} & U(L_{H_{2}}) \ar[r]_-{i_{H_2}} & H_2 \ar[r]_-{p_{H_2}} & K[G_{H_{2}}] \ar[r]^-{} \ar@/_/[l]_-{s_{H_2}} & 0 }\]
\end{remark}
\subsection{Regularity of the category $\mathbf{Hopf}_{K,coc}$}
\subsubsection{The regular epimorphism/monomorphism factorization in $\mathbf{Hopf}_{K,coc}$}
Let $f: A \longrightarrow B$ be a morphism of cocommutative Hopf $K$-algebras. By the protomodularity of $\mathbf{Hopf}_{K,coc}$, it is well known that regular epimorphisms are the same as cokernels, i.e. normal epimorphisms. Thus, to construct the regular epimorphism/monomorphism factorization of the morphism $f$, we consider the kernel $i:Hker(f)\to A$ of $f$ and the cokernel $p:A\to HCoker (i)$ of $i$, both computed in the category of $\mathbf{Hopf}_{K,coc}$ :
\[\xymatrixcolsep{8pc}\xymatrixrowsep{1pc}\xymatrix{
Hker(f) \ar@{^{(}->}[r]^-{i}& A \ar[r]^-{f} \ar@{->>}[d]_-{p} & B \\
& HCoker (i) \ar@{>->}[ur]_-{m} &
}\]
The existence of this factorization $m$ such that $m\circ p=f$ follows from the universal property of the cokernel $p$ of $i$. It remains to prove that $m$ is a monomorphism, which is equivalent in $\mathbf{Hopf}_{K,coc}$ to showing that $m$ is an injection. In the category $\mathbf{Hopf}_{K,coc}$ the above factorization is obtained as in the category of vector spaces since
$HCoker(i)=\dfrac{A}{A\big(Hker(f)\big)^{+} A}$
(by \cite{Andru1}), and $ker(f) = A\big(Hker(f)\big)^{+} A,$ (by \cite{shudo1988,Newman19751}).
Note that any epimorphism of cocommutative Hopf $K$-algebras is then a normal epimorphism, and the following classes of epimorphisms coincide in $\mathbf{Hopf}_{K,coc}$:
\hspace*{2cm} normal epis = regular epis = epis = surjective morphisms.
\subsubsection{Pullback stability of regular epimorphisms in the category $\mathbf{Hopf}_{K,coc}$}
To prove the pullback stability of regular epimorphisms in the category $\mathbf{Hopf}_{K,coc}$, the approach we follow is to apply the pullback stability of regular epimorphisms in the two full subcategories $\mathbf{GrpHopf}_{K}$ and $\mathbf{PrimHopf}_{K}$ of $\mathbf{Hopf}_{K,coc}$, which are both semi-abelian, and closed under pullbacks and quotients in $\mathbf{Hopf}_{K,coc}$. From the regularity of these two categories and the decomposition Theorem \ref{MM} we shall deduce the regularity of $\mathbf{Hopf}_{K,coc}$.
\begin{lemma}\label{Subcategories}
The two full subcategories $\mathbf{GrpHopf}_{K}$ and $\mathbf{PrimHopf}_{K}$, of $\mathbf{Hopf}_{K,coc}$, are semi-abelian categories. Both these categories are closed under quotients and pullbacks in $\mathbf{Hopf}_{K,coc}$.
\end{lemma}
\begin{proof}
As recalled in Preliminaries 2.2, the two full subcategories $\mathbf{GrpHopf}_{K}$ and $\mathbf{PrimHopf}_{K}$ are isomorphic to the category $\mathbf{Grp}$ of groups and to the category $\mathbf{LieAlg}_{K}$ of Lie $K$-algebras, respectively. The fact that $\mathbf{Grp}$ and $\mathbf{LieAlg}_{K}$ are semi-abelian is well-known (see \cite{BorceuxBourn}, for instance).
The categories $\mathbf{GrpHopf}_{K}$ and $\mathbf{PrimHopf}_{K}$ are closed under quotients since morphisms of Hopf $K$-algebras preserve both group-like and primitive elements. These categories are closed under products in the category $\mathbf{Hopf}_{K,coc}$, as explained in \cite{Milnor}. To see that $\mathbf{PrimHopf}_{K}$ is closed under subobjects in $\mathbf{Hopf}_{K,coc}$, let us consider a monomorphism $m: A \rightarrowtail U(L)$ in $\mathbf{Hopf}_{K,coc}$ with codomain a primitive Hopf algebra. The morphism $m$, being a Hopf algebra morphism, preserves group-like elements and the group of group-like elements is trivial in the primitive Hopf algebra $U(L)$. Since $m$ is injective, we see that the group of group-like elements of $A$ is trivial as well. By applying Theorem \ref{MM}, we conclude that $A$ has to be a primitive Hopf algebra as well. Similar arguments show that $\mathbf{GrpHopf}_{K}$ is closed under subobjects in $\mathbf{Hopf}_{K,coc}$. It follows that the subcategories $\mathbf{GrpHopf}_{K}$ and $\mathbf{PrimHopf}_{K}$ are closed under pullbacks in $\mathbf{Hopf}_{K,coc}$.
\end{proof}
\begin{remark}
In the following we shall assume that $K$ is an algebraically closed field. It can be checked that this is not a restriction: indeed, given a field $K$ and $\phi:K\to\overline K$ an embedding of $K$ in an algebraic closure $\overline{K}$, one has the adjunction
\[\xymatrixcolsep{6pc}
\xymatrix{
\mathbf{Hopf}_{\overline{K},coc} \ar@/_/[r]_-{R_\phi}^-{\perp} &\mathbf{Hopf}_{K,coc} \ar@/_/[l]_-{L_\phi}
}\]
where $R_\phi$ is the ``restriction of scalars functor'' and $L_\phi=- \otimes_K \overline K$ its left adjoint, the ``extension of scalars'' functor. Being a left adjoint, $L_\phi$ preserves regular epimorphisms and moreover $L_\phi$ reflects regular epimorphisms and preserves finite limits.
Accordingly, knowing that $\mathbf{Hopf}_{\overline{K},coc}$ is regular (respectively, exact), one can deduce from this that $\mathbf{Hopf}_{{K},coc}$ is regular (resp. exact) as well.
\end{remark}
The following result concerning split short exact sequences in $\mathbf{Hopf}_{K,coc}$ will be useful in the proof of the regularity of this category:
\begin{lemma}\label{surjective}
Given the following commutative diagram of split short exact sequences in $\mathbf{Hopf}_{K,coc}$:
\[\xymatrixcolsep{5pc}\xymatrix{
0 \ar[r]^-{} & A_1 \ar[r]_-{i_{H_1}} \ar[d]_-{{h}_{A}} & H_1 \ar[r]_-{p_{H_1}} \ar[d]_-{h} & B_1 \ar[d]^-{{h}_B} \ar[r]^-{} \ar@/_/[l]_-{s_{H_1}}& 0 \\
0 \ar[r]^-{} & A_2 \ar[r]_-{i_{H_2}} & H_2 \ar[r]_-{p_{H_2}} & B_2 \ar[r]^-{} \ar@/_/[l]_-{s_{H_2}} & 0 }\]
We have that $h$ is surjective if and only if both $h_A$ and $h_B$ are surjective.
\end{lemma}
\begin{proof}
We apply Lemma \ref{Molnar} to the exact sequences in the statement of the Lemma, we obtain the following commutative diagram which is canonically isomorphic to the previous one:
\[\xymatrixcolsep{5pc}\xymatrix{
0 \ar[r]^-{} & A_1 \ar[r]_-{i_{1}} \ar[d]_-{{h}_{A}} & A_1 \rtimes B_1 \ar[r]_-{p_{1}} \ar[d]_-{h_A \otimes h_B} \ar@/_/@{.>}[l]_-{\xi_{1}} & B_1 \ar[d]^-{{h}_B} \ar[r]^-{} \ar@/_/[l]_-{s_{1}}& 0 \\
0 \ar[r]^-{} & A_2 \ar[r]_-{i_{2}} & A_2 \rtimes B_2 \ar[r]_-{p_{2}} \ar@/_/@{.>}[l]_-{\xi_{2}} & B_2 \ar[r]^-{} \ar@/_/[l]_-{s_{2}} & 0 }\]
Hence, the morphism $h$ is surjective if and only if $h_A \otimes h_B \colon A_1 \rtimes B_1 \to A_2 \rtimes B_2$ is surjective.
If $h_A$ and $h_B$ are surjective, then $h_A \otimes h_B$ is surjective by considering this morphism on its underlying vector space. For the converse implication, if $ h_A \otimes h_B $ is surjective, let us note that
for each semi-direct product $A_i\rtimes B_i$, the underlying coalgebra is exactly the categorical product of the coalgebras $A_i$ and $B_i$; we denote $\xi_i=id_{A_i}\otimes \epsilon_{B_i}$ for the coalgebra-projection of $A_i\rtimes B_i$ onto $A_i$ (which is not a Hopf algebra morphism).
It is clear that $h_A \circ \xi_1 = \xi_2 \circ (h_A \otimes h_B)$, as coalgebra morphisms. Since $\xi_{2}$ is a split epimorphism and $h_A \otimes h_B$ is surjective, we conclude that $h_A$ is surjective. It is clear that $h_{B}$ is surjective whenever $h$ is.
\end{proof}
\begin{theorem}
Consider the following pullback $(P,\pi_{A},\pi_{B})$ in the category $\mathbf{Hopf}_{K,coc}$:
\begin{equation}\label{pullback}
\xymatrix{
P \ar[r]^-{\pi_{B}} \ar[d]_-{\pi_{A}} \ar@{}[rd]|<<<{\copy\pullbackbox} & B \ar[d]^-{g}\\
A \ar[r]_-{f} & C
}
\end{equation}
\vspace*{0.2cm}
if $f$ is a regular epimorphism then $\pi_{B}$ is a regular epimorphism.
\end{theorem}
\begin{proof}
Thanks to Lemma \ref{Subcategories}, regular epimorphisms are pullback stable whenever the Hopf algebras $A$, $B$ and $C$ in diagram \eqref{pullback} belong to $\mathbf{GrpHopf}_{K}$, or to $\mathbf{PrimHopf}_{K}$, respectively.
Let us now consider $A$, $B$ and $C$ cocommutative Hopf $K$-algebras over a field $K$ of characteristic zero. By Theorem \ref{MM}, we have: $A \cong U(L_{A}) \rtimes K[G_{A}]$, $B \cong U(L_{B}) \rtimes K[G_{B}]$ and $C \cong U(L_{C}) \rtimes K[G_{C}]$.
Thanks to Remark \ref{morphism}, we can build the following commutative diagram.
\[\xymatrixcolsep{5pc}\xymatrixrowsep{0.7pc}
\hspace{-1cm} \xymatrix{
P_{1} \ar[rrr]^-{{\pi}_{B_1}} \ar[ddd]_-{\pi_{A_1}} \ar[rd]_-{i_{P}} \ar@{}[rd]|<<<<<{\copy\pullbackbox} &&& U(L_B) \ar[rd]^-{i_{B}} \ar[ddd]^-{g_{1}} \\
& P \ar[rrr]^-{\pi_{B}} \ar[ddd]_-{\pi_{A}} \ar[rd]_-{p_{P}} \ar@{}[rd]|<<<<<{\copy\pullbackbox} &&& B \ar[rd]_-{p_{B}} \ar[ddd]^-{g} \\
&& P_{2} \ar[rrr]^-{{\pi}_{B_2}} \ar@/_/[lu]_(.3){s_{P}} \ar[ddd]_-{\pi_{A_2}} \ar@{}[rd]|<<<<<{\copy\pullbackbox} &&& K[G_B] \ar@/_/[lu]_{s_{B}} \ar[ddd]^-{g_{2}} \\
U(L_A) \ar[rrr]_-{f_{1}} \ar[rd]_-{i_{A}} &&& U(L_C) \ar[rd]^-{i_{C}} \\
& A \ar[rrr]_-{f} \ar[rd]_-{p_{A}} &&& C \ar[rd]_-{p_{C}} \\
&& K[G_A] \ar@/_/[lu]_(.3){s_{A}} \ar[rrr]_-{f_{2}} &&& K[G_C] \ar@/_/[lu]_{s_{C}} \\
}\]
where $(P_{1},\pi_{A_1},\pi_{B_1})$ is the pullback of $f_{1}$ and $g_{1}$, $(P_{2},\pi_{A_2},\pi_{B_2})$ is the pullback of $f_{2}$ and $g_{2}$.
When $f$ is surjective, the surjectivity of $f_{1}$ and $f_{2}$ follows both from Lemma \ref{surjective} applied to the lower part of the diagram. The front and back squares of the diagram are in $\mathbf{GrpHopf}_{K}$ and in $\mathbf{PrimHopf}_{K}$, respectively, thus both ${\pi}_{B_1}$ and ${\pi}_{B_2}$ are surjective (by Lemma \ref{Subcategories}). Applying Lemma \ref{surjective} again (in converse direction), we obtain that $\pi_B$ is also surjective.
\end{proof}
\section{A torsion theory in the category $\mathbf{Hopf}_{K,coc}$}\label{torsion}
In the non-abelian context of homological categories it is natural to define and study a general notion of torsion theory, that extends the one introduced by S.E. Dickson in the frame of abelian categories \cite{Dickson}. This study was first initiated in \cite{Bourn200618}, and further developed in \cite{Duckerts20121837} \cite{EveraertGran}, also in relationship with semi-abelian homology theory.
Let us recall the definition of a torsion theory in the homological context:
\begin{definition}
\textsf{
In a homological category $\mathbf{C}$, a \textit{torsion theory} is given by a pair ($\mathbf{T}$,$\mathbf{F}$) of full and replete (i.e. isomorphism closed) subcategories of $\mathbf{C}$ such that:
\begin{enumerate}[i.]
\item For any object $X$ in $\mathbf{C}$, there exists a short exact sequence:
\[\xymatrix{
0 \ar[r]^-{} & T \ar[r]^-{t_{X}} & X \ar[r]^-{\eta_{X}} & F \ar[r]^-{} & 0 }\]
where $0$ is the zero object in $\mathbf{C}$, $T \in \mathbf{T}$ and $F \in \mathbf{F}$.
\item The only morphism $f: T\longrightarrow F$ from $T \in \mathbf{T}$ to $F \in \mathbf{F}$ is the zero morphism.
\end{enumerate}
}
\end{definition}
When ($\mathbf{T}$,$\mathbf{F}$) is a torsion theory, $\mathbf{T}$ is called the \textit{torsion} subcategory of $\mathbf{C}$, and $\mathbf{F}$ its \textit{torsion-free} subcategory.
Among the many examples in the homological context, let us just mention the following ones:
\begin{example}
\textsf{
\begin{enumerate}
\item
Every torsion theory in an abelian category $\mathbf{C}$. For instance: the pair ($\mathbf{Ab}_{t}$, $\mathbf{Ab}_{tf}$) in the category $\mathbf{Ab}$ of abelian groups, where $\mathbf{Ab}_{t}$ and $\mathbf{Ab}_{tf}$ denote the full and replete subcategories of the category of abelian groups whose objects are torsion and torsion-free abelian groups, respectively.
\item
The pair ($\mathbf{NilCRng}$, $\mathbf{RedCRng}$) in the category $\mathbf{CRng}$ of commutative rings, where $\mathbf{NilCRng}$ and $\mathbf{RedCRng}$ denote the full subcategories of nilpotent commutative rings, and of reduced commutative rings (i.e. commutative rings with no nontrivial nilpotent elements), respectively.
\item
The pair ($\mathbf{Grp(Ind)}$, $\mathbf{Grp(Haus)}$) in the category $\mathbf{Grp(Top)}$ of topological groups, where $\mathbf{Grp(Ind)}$ and $\mathbf{Grp(Haus)}$ denote the full subcategories of indiscrete groups and of Hausdorff groups, respectively.
\end{enumerate}
}
\end{example}
From now on, in the homological category of cocommutative Hopf $K$-algebras, $\mathbf{T}$ will always denote the (full) subcategory $\mathbf{PrimHopf}_{K}$ of primitive Hopf algebras, and $\mathbf{F}$ the (full) subcategory $\mathbf{GrpHopf}_{K}$ of group Hopf algebras.
\begin{theorem}\label{hereditary torsion}
The pair ($\mathbf{PrimHopf}_{K}$,$\mathbf{GrpHopf}_{K}$) is a hereditary torsion theory in $\mathbf{Hopf}_{K,coc}$.
\end{theorem}
\begin{proof}
Thanks to Theorem \ref{MM}, we know that we can associate the following short exact sequence with any cocommutative Hopf $K$-algebra $H$:
\[\xymatrix{
0 \ar[r]^-{} & U(L_{H}) \ar[r]^-{i_{H}} & H \ar[r]^-{p_{H}} & K[G_{H}] \ar[r]^-{} & 0 }\]
Any morphism $f \colon U(L) \rightarrow K[G]$ from a primitive Hopf algebra $U(L) \in \mathbf{PrimHopf}_{K}$ to a group Hopf algebra $K[G] \in \mathbf{GrpHopf}_{K}$ is the zero morphism in $\mathbf{Hopf}_{K,coc}$. Indeed, any primitive Hopf algebra $U(L)$ is generated by its primitive elements, which are preserved by any morphism of Hopf algebras. Since a group Hopf algebra $K[G]$ does not contain any non-zero primitive element, it follows that $f$ is the zero morphism. It follows that ($\mathbf{PrimHopf}_{K}$,$\mathbf{GrpHopf}_{K}$) is a torsion theory, which is actually hereditary thanks to Lemma \ref{Subcategories}.
\end{proof}
As it follows from the results in \cite{Bourn200618} the reflector $I$ in the adjunction
\[\xymatrixcolsep{6pc}
\xymatrix{
\mathbf{F} \ar@/_/[r]_-{H}^-{\perp} &\mathbf{Hopf}_{K,coc} \ar@/_/[l]_-{I}
}\]
is semi-left-exact in the sense of Cassidy-Hebert-Kelly \cite{JAZ:4910656}, i.e. it preserves all pullbacks of the form
\begin{equation*}
\xymatrix{
P \ar[r]^-{p_{2}} \ar[d]_-{p_{1}} \ar@{}[rd]|<<<{\copy\pullbackbox} & Y \ar[d]^-{\eta_{Y}}\\
H(X) \ar[r]_-{H(f)} & HI(Y)
}
\end{equation*}
where $\eta_{Y}$ is the $Y$-component of the unit of the adjunction and $f$ lies in the subcategory $\mathbf{F}$.
The adjunction is then \emph{admissible} in the sense of categorical Galois theory \cite{Janelidze}: this opens the way to further investigations in the direction of semi-abelian homology \cite{Duckerts20121837}. The fact that the torsion theory is hereditary and $\mathbf{Hopf}_{K,coc} $ a homological category implies that the corresponding \emph{Galois coverings} are precisely those regular epimorphisms $f \colon A \rightarrow B$ in $\mathbf{Hopf}_{K,coc} $ with the property that the kernel $Hker(f)$ is in $\mathbf{F}$ (by applying Theorem $4.5$ in \cite{GranRossi}). This fact is crucial to describe generalized Hopf formulae for homology, as explained in \cite{EveraertGran}.
\section{The category of cocommutative Hopf algebras over a field of characteristic zero is semi-abelian}
\label{exact}
In order to prove that $\mathbf{Hopf}_{K,coc}$ is semi-abelian, it remains to show that equivalence relations are effective.
For this, we shall show that any equivalence relation $R$ as in diagram \eqref{reflexive} in $\mathbf{Hopf}_{K,coc}$ is the kernel pair of its coequalizer $q \colon X \rightarrow \overline{X}$. We first apply Theorem \ref{MM} to the equivalence relation $R$, obtaining the following commutative diagram, where the morphisms $q_{1}$, $q_{2}$ and $q$ are the coequalizers of $p_{11}$ and $p_{21}$, $p_{12}$ and $p_{22}$, $p_{1}$ and $p_{2}$, respectively, and ($Eq(q)$,$\pi_{1}$,$\pi_{2}$) is the kernel pair of $q$.
\[\xymatrixcolsep{4.5pc} \xymatrix{
0 \ar[r]^-{} & U(L_{R}) \ar@/_/[rrd]_<<<{i'_{R}} \ar[r]^-{i_{R}} \ar@<2ex>[dd]^-{p_{21}} \ar@<-2ex>[dd]_-{p_{11}} & R \ar@<2ex>[dd]^-{p_{2}} \ar@<-2ex>[dd]_-{p_{1}} \ar[rr]_-{p_{R}} \ar[rd]^-{\theta}& & K[G_{R}] \ar@<2ex>[dd]^-{p_{22}} \ar@<-2ex>[dd]_-{p_{12}} \ar[r]^-{} & 0 \\
& & & Eq(q) \ar@<0.3ex>[ld]^-{\pi_{2}} \ar@<-0.3ex>[ld]_-{\pi_{1}} \ar@/_/[ru]^-{p'_{R}} & & \\
0 \ar[r]^-{} & U(L_{X}) \ar[r]^-{i_{X}} \ar[d]_-{q_{1}} \ar[uu]|-{\Delta_{X_{1}}} & X \ar[rr]_-{p_{X}} \ar[uu]|-{\Delta_{X}} \ar[d]_-{q}& & K[G_{X}] \ar[d]^-{q_{2}} \ar[uu]|-{\Delta_{X_{2}}} \ar[r]^-{} & 0 \\
& \overline{U(L_{X})} \ar[r]_-{\overline{i_{X}}} & \overline{X} \ar[rr]_-{\overline{p_{X}}}& &\overline{K[G_{X}]} &
}\]
Thanks to Lemma \ref{Subcategories}, the left column is exact, i.e. $U(L_{R}) = {Eq}(q_1)$. On the other hand, let us explain why the right column is also exact. First observe that $(K[G_{R}],p_{12}, p_{22})$ is a reflexive relation on $K[G_{X}]$, while the category $\mathbf{GrpHopf}_{K}$ of group Hopf algebras is exact Mal'tsev and closed under pullbacks and quotients in $\mathbf{Hopf}_{K,coc}$. It follows that $K[G_{R}]$ is the kernel pair of $q_2$ in $\mathbf{Hopf}_{K,coc}$.
The universal property of ($Eq(q)$,$\pi_{1}$,$\pi_{2}$) gives the unique arrow $\theta \colon R \rightarrow Eq(q)$ with $\pi_1 \circ \theta = p_1$ and $\pi_2 \circ \theta = p_2$. One can check that the arrow $\overline{i_{X}}$ is a monomorphism, by using the fact that the coequalizers of equivalence relations in $\mathbf{Hopf}_{K,coc}$ are computed as in $\mathbf{Coalg}_{K,coc}$. Consequently, the lower row in the diagram above is an exact sequence as is then the lower row in the following diagram.
\[\xymatrixcolsep{5pc}\xymatrix{
0 \ar[r]^-{} & U(L_{R}) \ar[r]^-{i_{R}} \ar[d]_-{id_{U(L_{R})}} & R \ar[r]^-{p_{R}} \ar[d]^-{\theta} & K[G_{R}] \ar[d]^-{id_{K[G_{R}]}} \ar[r]^-{} & 0 \\
0 \ar[r]^-{} & U(L_{R}) \ar[r]_-{i'_{R}} & Eq(q) \ar[r]_-{p'_{R}} & K[G_{R}] \ar[r]^-{} & 0 }\]
By applying the Split Short Five Lemma to the above commutative diagram, it follows that the morphism $\theta$ is an isomorphism, and the equivalence relation $R$ is effective.
One accordingly has the following:
\begin{theorem}\label{semi-abelian}
For any field $K$ of characteristic $0$, the category $\mathbf{Hopf}_{K,coc}$ of cocommutative Hopf $K$-algebras is semi-abelian.
\end{theorem}
This result has the following interesting
\begin{corollary}\label{representable object actions}
For any field $K$ of characteristic $0$, the category $\mathbf{Hopf}_{K,coc}$ is action representable \\ (i.e. it has representable object actions in the sense of \cite{BJK}).
\end{corollary}
\begin{proof}
This follows from Theorem 4.4 in \cite{BJK}, the exactness of $\mathbf{Hopf}_{K,coc}$ (Theorem \ref{semi-abelian}), and the fact that $\mathbf{Hopf}_{K,coc}$ can be viewed as the category of internal groups in the cartesian closed category of cocommutative $K$-coalgebras (see 2.3 in \cite{Grunenfelder}).
\end{proof}
\section*{Acknowledgement}
The authors are very grateful to Clemens Berger and Stephen Lack for many useful suggestions on the subject of the article.
\section*{}
\begin{flushleft}
\begin{small}
\textbf{Funding} : The second author is funded by a grant from the National Fund for Scientific Research of Belgium.
\end{small}
\end{flushleft}
|
1,116,691,497,765 | arxiv | \section{Introduction}
A relatively new field of study, {\it ergodic optimization} has displayed under a new point of view several distinct problems in dynamical systems, and enjoyed the benefits of allying techniques from optimization theory and ergodic theory to address them. Its usual setup is a dynamical system $f:X\to X$, where $X$ is a topological space, and a potential function $\phi:X\to\mathbb{R}$, and the prototypical problem in the field is to determine, among all $f$ invariant Borel probability measures $\mathcal{M}_{inv}(f)$, if there exists measures that maximize the functional $P_{\phi}:\mathcal{M}_{inv}(f)\to\mathbb{R},\, P_\phi(\mu)=\int \phi d\mu$ and to further characterize these maximizing measures in term of their support.
Several problems can be put under this context, like finding Lyapunov exponents, action minimizing solutions to Lagrangian systems and the zero temperature limits of Gibbs equilibrium states in thermodynamical formalism. Some of the first ideas of the field appeared in the early work \cite{CG} and a very good introduction to the subject is \cite{jenkinsondcds}, where the fundamental results of the theory are displayed alongside the main lines of research.
One of these research lines seeks to determine, when $X$ is compact and $\phi$ is continuous, what are the typical support of the maximizing measures (note that the existence of at least one maximizing measure is assured in this case by the compactness of the set of invariant probability measures in the weak-* topology). This is inspired by the classical conjecture of Ma\~n\'e that, generically, the measures that minimize the Lagrangian action in Lagrangian flows are supported in periodic orbits.
There are some different conceptual approaches to this question. First, one may be interested in a specific dynamical property (as, for instance, Lyapunov exponents or rotation numbers) and so the potential is determined by the choice of the dynamical system. Examples of this are \cite{jenkinsonmorris, garibaldinonlin, CDI, CLT}. Another approach, followed for instance in \cite{bouschwalters, Lopesasterisque, Morrisdcds2000}, involves fixing a dynamical system $f$, usually with some specific dynamical condition like hiperbolicity or expansiveness, and varying the potential in a suitable space.
In this work we follow yet a different line, searching to understand how the maximizing measures behave when the potential is fixed, but the dynamics are allowed to change in a given space. In \cite{ATfundamenta} it is shown that, if $M$ is a compact Riemannian manifold of dimension $n\ge 2$, then for any continuous $\phi:M\to \mathbb{R}$ there exists a dense set of homeomorphisms of $M$ with a maximizing measure supported on aperiodic orbit, but in \cite{ATDCDS} it is shown that this set is meager. And in \cite{ATnonlin} it is shown that for a dense set of endomorphisms of the circle, there exists a $\phi$ maximizing measure supported on a periodic orbit. In this note we extend this last result, showing that
\begin{thm}\label{maintheorem}
Let $M$ be a compact Riemannian manifold and $\phi_0:M\to\mathbb{R}$ continuous. Then there exists a dense subset $\mathcal{A}$ of End($M$) such that, for every $f\in\mathcal{A}$ there exists a $\phi_0$ maximizing measure supported on a periodic orbit.
\end{thm}
Where End($M$) is the set of continuous surjections of $M$ endowed with the $\mathcal{C}^0$ metric, $d(f,g)=\sup_{x\in M}d(f(x),g(x))$.
The strategy of the proof, similar to the one used in \cite{ATnonlin}, is to make a series of local perturbations in order to obtain a periodic source with large $\phi_0$ average while controlling the Birkhoff averages of the return map to the perturbation support. The proof of \cite{ATnonlin} relied on the local ordered structure of the domain, particularly in the definition of the support of the perturbations and in controlling the Birkhoff averages, two key points that were not adaptable to higher dimensions. In here we dealt with these difficulties by supporting perturbations in convex sets and analyzing the maximal Birkhoff sums on homothetic copies of the perturbation support, and by controlling the radial rate of escape from the periodic source.
The paper is organized as follows: In the next section we present some preliminary lemmas and notations, and in section 3 prove the theorem. Since the argument is perturbative, for a given endomorphism we analyze several possibilities, each dealt with in a different subsection, and show for each possibility how to construct a
perturbed endomorphism close to $f$ with the desired property.
\section{Preliminaries}
We start with some notations and preliminary results. Let $M$ be a compact Riemannian manifold and End($M$) the set of endomorphisms of $M$, its continuous surjections. We endow End($M$) with its usual topology of uniform convergence and define the metric $d(f,g)= \sup_{x\in M}(d(f(x), g(x)),\, f,g \in \hbox{End(}M)$.
Given $f\in\hbox{End(}M)$ we denote by $\mathcal M_{inv}(f)$ the set of $f$ invariant Borel probability measures, which is non-empty, convex and also compact in the weak-* topology. The subset of ergodic measures of $\mathcal{M}_{inv}(f)$ is denoted by $\mathcal M_{erg}(f)$.
Given $\phi:M\to\mathbb{R}$ continuous and $f\in \hbox{End(}M)$, we define $P_\phi:\mathcal{M}_{inv}(f)\to\mathbb{R}, \, P_{\phi}(\mu)=\int \phi d\mu$. As the functional $P_\phi$ is affine and $\mathcal M_{inv}(f)$ is a convex compact set, $P_\phi$ must have a maximum point at an extremal point of $\mathcal M_{inv}(f)$. Since the extremal points of $\mathcal M_{inv}$ are precisely the ergodic measures, there exists some $\mu_{\max} \in \mathcal M_{erg}(f)$ that maximizes $P_\phi$. We denote $S_nf(x):=\displaystyle \sum_{i=0}^{n-1}\phi(f^i(x))$ to the $n$ Birkhoff sum of $\phi$.
The following lemma is a direct consequence of Atkinson's Lemma (see \cite{atkinson})
\begin{lemma}\label{atik}
Let $\phi:M\rightarrow \mathbb{R}$ be a continuous function, $f\in $End($M)$ and $\mu \in \mathcal M_{erg}(f)$, such that,
$$\int \phi(x)d\mu(x)=0.$$
Then for $\mu$-almost all $x \in M$, there exist $n_k\to\infty$ such that,
$$f^{n_k}(x)\rightarrow x \: \mbox{ and } \: S_{n_k}f (x) \rightarrow 0.$$
\end{lemma}
We begin with the following simple result
\begin{lemma}
There exists a dense subset of End($M$) such that, for all $x\in M$ and all $f$ in this subset, the set $ \{y\in M: f(y)=x \}$ is finite.
\end{lemma}
\begin{proof}
Let $f_0$ be an endomorphism, and let $\varepsilon>0$. We will find some $f$ with the stated property $\varepsilon$ close to $f_0$. First, let $\delta>0$ be such that, for all $x_1,x_2\in M$, if $d(x_1,x_2)<\delta$, then $d(f(x_1),f(x_2))<\varepsilon/2K$, where $K$ is the ratio of the radii of the circunscribed and inscribed spheres in the $n$ dimensional regular simplex.
Since every $n$-dimensional differential manifold admits a triangulation and $M$ is compact, we can assume that $M$ has a triangulation $\Im_1$ with finitely many triangles, such that each simplex has diameter less then $\delta$ and let $\Im_2$ be a subtriangulation of $\Im_1$ such that, for each $\Delta_i\in\Im_1$ there exists some $\tilde\Delta_{j_i}\in\Im_2$ which is contained in the interior of $\Delta_i$.
Now we define $f:M\rightarrow M$, in a way that $f$ is a linear bijection in each triangle of $\Im_2$ and such that, in local coordinates, $f(\tilde\Delta_{j_i})$ is a simplex that contains $f_0(\Delta_i)$ and is contained in a sphere or radius $\varepsilon/2$. It should be immediate that $f$ is a continuous surjection, since $M=\bigcup_{i\in I}f_0(\Delta_i)\subset \bigcup_{j\in J}f(\tilde\Delta_{j_i})\subset f(M)$. Moreover if $x\in \tilde\Delta_j$ for some $\Delta \in \Im$ then $d(f_0(x),f(x))<\varepsilon/2$, this implies $d(f_0,f)<\varepsilon.$
As $f$ is linear in each simplex of $\Im_2$, the set $\{y\in \tilde\Delta_j \cap M: f(y)=x \}$ is either empty or unitary, and therefore $\{y\in M: f_2(y)=x \}$ has cardinality smaller than or equal to the number of simplexes in $\Im_2$
\end{proof}
The structure of proof of Theorem \ref{maintheorem} is the following. Let $\phi_0:M\to\mathbb{R}$ be fixed. We start with an endomorphism $f$ which we assume that, for every $x\in M$, the pre-image of $x$ is finite and we construct successive small perturbations to produce an endomorphism $\widetilde f$ which is $\varepsilon$ close to $f$ and such that $\widetilde f$ has a $\phi_0$ maximizing measure supported on a periodic orbit.
Let $\mu_{\max}\in \mathcal M_{erg}(f)$ be a $\phi_0$ maximizing measure and let $\phi= \phi_0-\int{\phi d\mu_{\max}}$, so that $\int{\phi d\mu_{\max}}=0$, and we remark that, for any endomorphism $g$, $\mu$ is a $\phi_0$ maximizing measure if and only if it is a $\phi$ maximizing measure.
\begin{lemma}\label{at}
For all $x\in \supp(\mu_{\max})$ and $\varepsilon>0$ there exist $y\in B_{\varepsilon}(x)$ and $n>0$ with $f^n(y)\in B_{\varepsilon}(x)$ and $S_nf(y)\ge0$.
\end{lemma}
\begin{proof}
Let $x\in\supp(\mu_{\max})$. By \ref{atik}, since $\mu_{max}(B_{\varepsilon}(x))>0$, there exist $x_1\in \supp(\mu_{\max})$ and $n_k$, such that $f^{n_k}(x_1)\rightarrow x_1$ as $k\rightarrow \infty$, and such that $S_nf(x_1)\to 0$. Let $k_1>0$ be such that $f^{n_{k_1}}(x_1)\in B_{\varepsilon}(x)$ and let $a_1=S_{n_{k_1}}f(x_1)$. If $a_1\ge 0$ we set $y=x_1, n=n_{k_1}$ and we are done. If $a_1<0$, let $n_{k_2}>n_{k_1}$ be such that $f^{n_{k_2}}(x_1)\in B_{\varepsilon}(x)$ and such that $S_{n_{k_2}}f(x_1)>a_1$. Then, as
$S_{n_{k_2}}f(x_1)= S_{n_{k_1}}f(x_1)+ S_{(n_{k_2}-n_{k_1})}f( f^{n_{k_1}}(x_1))$, we set $y = f^{n_{k_1}}(x_1)$ and $n = n_{k_2}-n_{k_1}$ and we are done.
\end{proof}
The next proposition is a consequence of the $\mathcal M_f$ compactness.
\begin{proposition}\label{jproposition}
For every constant $a>0$, there exist a positive integer $m_0=m_0(a)$ such that, for all $m\geq m_0$ and $x\in M$
$$\frac{1}{m}S_mf(x)\leq \frac{a}{2}. $$
\end{proposition}
\begin{proof}
This follows from $$\limsup_{n\rightarrow \infty} \max_{x\in M}\frac{1}{n}S_nf(x)=\sup_{\mu \in \mathcal M_{inv}(f)} \int{\phi d\mu}=0, $$ proposition 2.1 of \cite{jenkinsondcds}
\end{proof}
\section{Contruction of the perturbed endomorphism}
Fix $x\in \supp(\mu)$ and let $\varepsilon>0$. There are two possibilities,
\begin{itemize}
\item[I] {For all $y\in B_{\varepsilon}(x)$ and all $n>0$, if $f^n(y)\in B_{\varepsilon}(x)$ then $S_nf(y)\le 0$}
\item[II]{There exists $x_0\in B_{\varepsilon}(x)$ and $n_0>0$ such that $\widetilde f^{n_0}(x_0)\in B_{\varepsilon}(x)$ and $S_{n_0}f(x_0)>0$}.
\end{itemize}
\subsection{Case I}
Let us show first how to construct $\widetilde f$ in the case I :
Denote, for simplicity, $B=B_{\varepsilon}(x)$.
We assume that for all $y\in B$ and all $n>0$, if $f^n(y)\in B$ then $S_nf(y)\le 0$. From Lemma \ref{at} there exists $x_0\in B$ and $n_0>0$ such that $S_{n_0}f(x_0)\ge 0$, and so $S_{n_0}f(x_0)= 0$. Let $0<n_1\le n_0$ be the first return of $x_0$ to $B$. Note that, as
$$0= S_{n_0}f(x_0)= S_{n_1}f(x_0)+S_{n_0-n_1}f(f^{n_1}(x_0))\le S_{n_1}f(x_0),$$ where the inequality comes from assuming that we are in case I, then $S_{n_1}f(x_0)\ge 0$ and, again from the assumption, $S_{n_1}f(x_0)=0$.
Let $T:M\to M$ be a homeomorphism such that $T(f^{n_1}(x_0))=x_0$, and such that $T$ is the identity outside of $B$, let $\widetilde f= T\circ f$. Note that $x_0$ is a $n_1$ periodic point for $\widetilde f$. Let $\mu_1$ be the measure uniformly distributed on the points of the $\widetilde f$ orbit of $x_0$.
\begin{lemma}\label{casoI}
$\mu_1$ is a $\phi$ maximizing measure for $\widetilde f$
\end{lemma}
\begin{proof}
Clearly $\int \phi d\mu_1= \frac{1}{n_1}S_{n_1}\widetilde f(x_0)= \frac{1}{n_1}S_{n_1} f(x_0)=0$. Furthermore, if $z\in M$ is such that there exists $n_z$ such that, if $n>n_z$ then $\widetilde f^{n}(z)\notin B$, then $\limsup_{n\to\infty}\frac{1}{n}S_n\widetilde f(z)= \limsup_{n\to\infty}\frac{1}{n}S_n\widetilde f(\widetilde f^{n_z}(z))= \limsup_{n\to\infty}\frac{1}{n}S_n f(\widetilde f^{n_z}(z))\le 0$ where the last inequality comes from the fact that $P_\phi(\mu)\le 0$ for all $\mu\in \mathcal{M}_{inv}(f)$.
On the other hand, if the return times of $z$ to $B$ are $0\le t_0<t_1<t_2.....$ with $t_k\to\infty$, then
$$\frac{1}{t_k}S_{t_k} \widetilde f(z)= \frac{1}{k}\left(S_{t_0}\widetilde f(z) +\sum_{j=1}^{k} S_{t_j-t_{j-1}}\widetilde f( \widetilde f^{t_{j-1}}(z))\right)\le \frac{1}{k}S_{t_0}\widetilde f(z)\to 0$$
so that $\lim_{n\to\infty} \frac{1}{n} S_n\widetilde f(z)\le \int \phi d\mu_1$ for all $z\in M$ and we have the result
\end{proof}
\subsection{Case II }
Assume now we are in case II, and let $a_0= \frac{1}{n_0}S_{n_0}f(x_0)>0$.
Denote by $B_{\varepsilon}[z]$ the closed ball with center $z$ and radius $\varepsilon$.
Let $m_0=m_0(a_0)>n_0>0$ be the integer from Proposition (\ref{jproposition}), and for each $k\in \{1,2,\ldots,m_0\}$ consider the compact sets
$$K_k=B_{\varepsilon}[x_0]\cap f^{-k}(B_{\varepsilon}[x_0]).$$
For each $k$, let $c_k=\sup_{z\in K_k}\frac{1}{k}S_kf(z)$ and let $\overline{c}=\sup\{c_1, ... ,c_{m_0}\}$. Note that, by the choice of $x_0,\, c_{n_0}\ge a_0.$ Furthermore, by Proposition (\ref{jproposition}), if $n>m_0$ then for all $z \in M,\,\frac{1}{n}S_nf(z)\le \frac{a_0}{2}.$ This, and the choice of $\overline c$ implies that, for each $z\in B_{\varepsilon}[x_0]$ and $n>0$ such that $f^n(z)\in B_{\varepsilon}[x_0]$, we have $\frac{1}{n}S_nf(z)\le \overline c$.
We consider 2 distinct possibilities:
\begin{itemize}
\item[(a)] {There exists $q\in B_{\varepsilon}[x_0],\, n_q>0$ such that $\frac{1}{n_q}S_{n_q}f(q)=\overline c$ and $f^{n_q}(q)$ lies in the {\it open} ball $B_{\varepsilon}(x_0)$}
\item[(b)] {For all $z\in B_{\varepsilon}[x_0], n>0$ if $\frac{1}{n}S_nf(z)=\overline c$ and $f^n(z)\in B_{\varepsilon}[x_0]$, then $f^n(z)\in \partial B_{\varepsilon}[x_0]$}
\end{itemize}
\subsubsection{Case $(a)$}
If $(a)$ happens then we can define $\widetilde{f}=T\circ f$, where $T$ is the identity outside of $B_{\varepsilon}[x_0]$ and $T$ is an endomorphism of $B_{\varepsilon}[x_0]$ satisfying $T(f^{n_q}(q))=q$.
The next lemma show us that the $\widetilde{f}$ invariant measure supported on the periodic orbit of $q$ is a $\phi$-maximizing measure.
\begin{lemma}\label{somamenor}
For any $z\in M, \liminf_{n\to\infty}\frac{1}{n}S_n\widetilde{f}(z)\le \frac{1}{n_q}S_{n_q}\widetilde{f}(q)=\overline c$
\end{lemma}
\begin{proof}
Let $z \in M$ and first assume that $z$ is such that there exists some $\overline n$ such that $\widetilde{f}^i(z)\notin B_{\varepsilon}[x_0]$ whenever $i\ge \overline{n}$, then
$$\liminf_{n\to\infty}\frac{1}{n}S_n\widetilde{f}(z)= \liminf_{n\to\infty}\frac{1}{n}S_n\widetilde{f}( \widetilde f^{\overline{n}}(z))=\liminf_{n\to\infty}\frac{1}{n}S_nf( \widetilde f^{\overline{n}}(z))\le 0$$
where the second equality follows from the fact that $f(y)=\widetilde{f}(y)$ whenever $y\notin B_{\varepsilon}[x_0]$, and the inequality follows since the maximal $\phi$ average for $f$ is 0, and from $\sup_{z\in M}\limsup_{n\to\infty}\frac{1}{n}S_nf( \widetilde f^{\overline{n}}(z))\le \sup_{\mu\in\mathcal{M}_{inv}(f)} \int \phi d\mu$. As $\frac{1}{n_q}S_{n_q}f(q)>0$, we are done in this case.
Now assume that there exists an increasing sequence of times $N_k\to\infty, k\ge 1$ such that $\widetilde{f}^{i}(z)$ belongs to $B_{\varepsilon}[x_0]$ if and only if $i=N_k$ for some integer $k$. Then it holds that
$$\frac{1}{N_{k+1}-N_k}\widetilde{f}(\widetilde{f}^{N_k}(z))=\frac{1}{N_{k+1}-N_k}f(\widetilde{f}^{N_k}(z))$$ and
\begin{eqnarray}
\frac{1}{N_k} S_{N_k}\widetilde{f}(z)&=& \displaystyle \frac{1}{N_k} \left( \sum_{i=0}^{N_1-1}\phi \circ \widetilde{f}^i(z)+\ldots+\sum_{i=N_{k-1}}^{N_k-1}\phi \circ \widetilde{f}^i(z) \right) \nonumber \\
&=& \displaystyle \frac{1}{N_k}\left(\frac{N_1-N_0}{N_1-N_0}S_{N_1-N_0}\widetilde{f}(z)+\ldots+ \frac{N_k-N_{k-1}}{N_k-N_{k-1}}S_{N_k-N_{k-1}}\widetilde{f}\left( \widetilde{f}^{N_{k-1}}(z)\right) \right) \nonumber \\
&=& \displaystyle \frac{1}{N_k}\left(\sum_{i=1}^{k}(N_i-N_{i-1})\frac{1}{N_i-N_{i-1}}S_{N_i-N_{i-1}}f(\widetilde{f}^{N_{i-1}}(z))\right) \label{continha1} \\
&\leq & \frac{1}{N_k}\sum_{i=1}^{k}(N_i-N_{i-1})\overline c= \overline c. \nonumber
\end{eqnarray}
Where the inequality (\ref{continha1}) follows from $S_kf(z)=S_k\widetilde{f}(z)$, as $\widetilde{f}^i(z) \notin B_{\varepsilon}[x_0]$, $0\leq i \leq k-1$.
\end{proof}
The previous lemma shows that, if $z$ is a typical point of an $\widetilde{f}$ ergodic invariant measure $\mu$, then $\lim_{n}\frac{1}{n}S_n\widetilde{f}(z)=\int\phi d\mu\le \frac{1}{n_q}S_{n_q}\widetilde{f}(q)$ and we are done.
\subsubsection{Case $(b)$}
There exists some $z_1\in B_{\varepsilon}[x_0]$ and $n_{z_1}>0$ such that $f^{n_{z_1}}(z_1)\in\partial B_{\varepsilon}[x_0]$, and such that $\frac{1}{n_{z_1}}S_{n_{z_1}}f(z_1)=\overline c$. Let us call $q_0= f^{n_{z_1}}(z_1)$.
Since each point in $M$ has finitely many preimages, the set $P=(\bigcup_{i=1}^{m_0}f^{-i}(q_0))\cap B_{\varepsilon}[x_0]$ is finite, as is
$$\tilde P=\{ z\in P\mid \exists n_z>0 \hbox{ such that} f^{n_z}(z)=q_0 \hbox{ and } \frac{1}{n_z}S_{n_z}f(z)=\overline c\}.$$
Let $q\in \tilde P$ be a point which is closest to $q_0$ and let $n_q$ be such that $f^{n_q}(q)=q_0$ and $\frac{1}{n_q}S_{n_q}(q)=\overline c$.
Finally, let $E$ be some closed convex set contained in $B_{\varepsilon}[x_0]$ such that, if $d(z_1,z_2)\ge d(q,q_0)$ and $z_1,z_2\in E$, then $\{z_1,z_2\}=\{q,q_0\}$.
\begin{proposition}\label{cconexa}
There exist $\delta >0$ such that, if $z$ is not in the connected component of $$f^{-j}\left(B_{\delta}[q_0]\right)\bigcap \left(B_{\delta}[q_0]\cup E \right)$$ that contains $q$, then $$\frac{1}{j}S_jf(z)<\overline c$$ for all $j\in \mathbb{N}^*$.
\end{proposition}
\begin{proof}
By the choice of $E,\,\tilde P\cap E=\{q\}$ and so for any $z\not=q$ in $P\cap E$ and $n_z$ such that $f^{n_z}(z)=q_0,\, \frac{1}{n_z}S_{n_z}f(z)$ is strictly smaller than $\overline c$. Thus, by the continuity of $f$ and $\phi$, there exist $\delta_1(z)>0$ such that if $d(z,y)<\delta_1(z)$ we have $\frac{1}{i}S_if(y)<\frac{1}{n_q}S_{n_q}f(q)$. Moreover, for each $\delta_1(z)$ there exist $\delta_2(z)$ such that the connected component of $f^{-i}(B_{\delta_2(z)}(q_0))$ which contain $z$ is contained in $B_{\delta_1(z)}(z)$. Finally there exists $\delta_3>0$ such that, if $f^{-i}(B_{\delta_3}(q_0))$ intersects $E$ then there is some point of $P$ in this component. By taking $\delta=\min_{x\in P}\{\delta_2(x),\delta_3\}$ we are done.
\end{proof}
Denote the set $E\cup B_{\delta}[q_0]$ by $I$, we will construct a new endomorphism $\widetilde{f}=T\circ f$, where $T|_{M\setminus I}(z)=z$ , and such that there exist a $\widetilde{f}$-periodic point in $I$ whose average is strictly positive.
Let
$$D=I\cap \left( \bigcup_{i=1}^{\infty} f^{-i}(I) \right).$$
Over $D$ we define the following functions:
$$N_{ret}(x)=\inf\{j\in \mathbb{N}^*:f^j(x)\in I \}$$
$$f_2(x)=f^{N_{ret}(x)}(x)$$
$$ \psi(x)=\frac{1}{N_{ret}(x)} \sum_{i-0}^{N_{ret}(x)-1}\phi(f^i(x))$$
By the Proposition \ref{cconexa}, let $W_0$ be the connected component of $f_2^{-1}(B_{\delta}[q_0])$ which contains $q$, if $z\in D$ and $\psi(z)>\psi(q)$, then $z\in W_0$. Denote by $z_{\max}$ the point in $\bar{W_0}$ that maximizes $\psi(z)$. Choose $\alpha \in W_0$ sufficiently close to $z_{\max}$ such that the inequality
\begin{equation}
4 m_0 [\psi(z_{\max})-\psi(\alpha)]\leq |\psi(\alpha)-\psi(q)|
\label{alfa}
\end{equation}
is true, and such that $f_2(\alpha)\in int(B_{\delta}[q_0])$.
Now we consider $L$ to be the line segment joining $\alpha$ and $f_2(\alpha)$, $T_1:M\rightarrow M$ an homeomorphism mapping $f_2(\alpha)$ to $\alpha$, that is, $T_1(f_2(\alpha))=\alpha$ and such that $T_1$ is the identity outside $V(L)$, where
$$V(L)=\{ z\in M: d(z,L)< \delta_3 \},$$ and $\delta_3>0$, chosen such that $V(L)$ is contained in the interior of $I$.
In figure (\ref{T2}), the shadow part is the neighborhood of the line segment $L$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.4]{fronteira1.png}
\caption{Neighborhood of $L$.}
\label{T2}
\end{figure}
We define now $\widetilde f$ by the composition $\widetilde f = T_1\circ f$. Note that $\alpha$ is a $n_q$ periodic point for $\widetilde{f}$ and that the $\phi$ average over the orbit of $\alpha$ is $\psi(\alpha)\ge\frac{1}{n_q}S_{n_q}f(q)>0$. Yet the dynamics defined by $\widetilde{f}$ may have some new invariant measures whose $\phi$ average is strictly larger than $\psi(\alpha)$.
Still, it should be clear that, as in the proof of Lemma \ref{somamenor} if $z$ is such that the $\widetilde{f}$ orbit of $z$ returns to $D$ finitely many times, then $\limsup_{n\to\infty}\frac{1}{n}S_nf(z)\le 0$, and if $z\in D$ returns infinitely-many times by $\widetilde{f}$ to the set $I$, but its orbit does not intersect $W_0$ (or just intersects it finitely many times), then if $n_1,n_2,\ldots$ are the return times to $D$, we have, by \ref{cconexa}:
\begin{equation}\label{naoemw0}
\lim_{k\rightarrow \infty}\frac{1}{n_k}\sum_{i=0}^{n_k-1}\phi(\widetilde{f}^i(x))\leq \psi(q)=\overline c.
\end{equation}
Now we define $W_{\alpha}=W_0\bigcap f_2^{-1}\left( V(L) \right )$. If $z\in W_0 \setminus W_{\alpha}$ then $f_2(z)\in B_{\delta}[q_0]$ and we remark that $ B_{\delta}[q_0]$ is disjoint from $W_0$. So, if there is some future time $n_1>n_q$ such that $\widetilde{f}^{n_1}(z) \in W_0$, we can write $n_1=n_q+k$ with $f^{n_q}(z)=f_2(z)\in B_{\delta}[q_0]$ and $f^{n_q+k}(z) \in W_{0}$. The following estimate will be useful
\begin{eqnarray}\label{naoemwalpha}
\displaystyle \frac{1}{n_{1}}\sum_{i=0}^{n_{1}-1}\phi(\widetilde{f}^i(z))-\psi(\alpha)=&\displaystyle \frac{n_q\psi(z)+k\psi(f_2(z))}{n_z+k}-\psi(\alpha) \nonumber \\
\leq &\displaystyle \frac{n_q \psi(z_{\max})+k\psi(q)}{n_q+k}-\psi(\alpha) \nonumber \\ \leq & \displaystyle\frac{n_q \psi(z_{\max})+\psi(q)}{n_z+1}-\psi(\alpha) \nonumber \label{eqq}\\ \leq & \displaystyle\frac{n_q (\psi(z_{\max})-\psi(\alpha))-(\psi(\alpha)-\psi(q))}{n_q+1}\leq 0,
\end{eqnarray}
where the last inequality follows from (\ref{alfa}).
\subsection{The last pertubation}
In order to finish the demonstration of the Theorem (\ref{maintheorem}) we need to control the averages of those elements which have infinitely many returns on $W_{\alpha}$. In this section we construct a new pertubation $T_2$ such that $\alpha$ will be a source for the new endomorphism $T_2 \circ \widetilde{f}$, and $W_{\alpha}$ is contained in its basin of repulsion.
Let $\tilde{D}=I\cap \left( \bigcup_{i=1}^{\infty} \widetilde{f}^{-i}(I) \right )$ be the set of points who return to $I$ by the function $\widetilde{f}$. Over this set we define the following functions:
\begin{eqnarray}
\tilde{N}_{ret}(x)&=& \inf \{j\in \mathbb{N}^*: \widetilde{f}^j(x)\in I \} \nonumber \\
\widetilde{f}_2(x)&=& \widetilde{f}^{\tilde{N}_{ret}(x)}(x) \nonumber \\
\tilde{\psi}(x)&=& \frac{1}{\tilde{N}_{ret}(x)}S_{\tilde{N}_{ret}(x)}\widetilde{f}(x) \nonumber
\end{eqnarray}
The following propositions are immediate from the definitions:
\begin{itemize}
\item[a)] $\tilde{D}=D$;
\item[b)] $\tilde{N}_{ret}(x)=N_{ret}(x)$ for all $x \in \tilde{D}$;
\item[c)] $\tilde{\psi}(x)=\psi(x)$ for all $x \in \tilde{D}$;
\item[d)] $\widetilde{f}_2(x)=T_1 \circ f_2(x)$ and if $x\notin f_2^{-1}\left(V(L)\right)$, then $\widetilde{f}_2(x)=f_2(x)$.
\end{itemize}
\
Let $\psi_{\max}:I\rightarrow \mathbb{R}$ be the following function:
$$\psi_{\max}(z)=\max_{y\in B_{d(\alpha,z)}[\alpha]}\psi(y).$$
Note that $\psi_{\max}(\alpha)=\psi(\alpha)$, $\psi(z)\leq \psi_{\max}(z)$, for all $z\in I$ and if $d(\alpha,z_1)<d(\alpha,z_2)$ then $\psi_{\max}(z_1)\leq \psi_{\max}(z_2).$
Define the function $P:\mathbb{R} \rightarrow \mathbb{R}$:
$$P(s)=\sup_{z\in B_s(\alpha)} \psi(z)-\psi(\alpha). $$
This function is continuous non decreasing with $P(0)=0$, moreover, given $z$ with $d(z,\alpha)=s$, then $P(s)=\psi_{\max}(z)-\psi(\alpha)$.
Let $R_1, R_2,\, 0<R_1<R_2$ be such that $W_{\alpha}\subset B_{R_1}(\alpha) \subset B_{R_2}(\alpha)$ and $B_{R_2}(\alpha) \subset E$ as shown in figure (\ref{T1}):
\begin{figure}[h]
\centering
\includegraphics[scale=0.7]{fronteira2.png}
\caption{Perturbation region}
\label{T1}
\end{figure}
The perturbation $T_2$ differs from the identity only at $B_{R_2}(\alpha)$. Define the set $A_s=\{x\in B_{R_1}(\alpha):s\leq d(\alpha,z) \leq R_1 \}$ for all $0\leq s \leq R_1$ and we define the function
$$Q(s)=\inf_{z\in A_s}d(\widetilde{f}_{2}(z),\alpha).$$
$Q(s)$ is a non-decreasing continuous function. Since for all $z\in M,\,\widetilde{f}_2^{-1}(z)$ is a finite set, $\alpha$ has only one pre-image by $\widetilde{f}_2$ in $B_{R_1}(\alpha)$. This implies $Q(0)=0$, and if $s>0$ then $Q(s)>0$.
Given $s_0=R_1$, we define two sequences $s_i$ and $r_i$ by:
$$P(s_i)=\frac{1}{2^i}P(R_1)\; \mbox{ and }\; r_i=\min \{ Q(s_i), s_i \}.$$
Now we define the function $\lambda(z)$ by the rule:
If $r_{k+1}< d(z,\alpha)< r_{k}<R_1$ then $$s_k\leq \lambda(z) d(z,\alpha) \leq s_{k-1} .$$
One such function can be $\displaystyle \lambda(z)=\frac{s_k+(s_{k-1}-s_k)\frac{d(x,\alpha)-r_{k+1}}{r_k-r_{k+1}}}{d(z,\alpha)}$.
If $d(z,\alpha)>R_1$ then $\lambda(z)=1.$
Finally consider $\hat{f}=T_2\circ \widetilde{f}$, where $T_2$ on $B_{R_2}(\alpha)$ is the identity, and $T_2$ on $B_{R1}(\alpha)$ is defined by
$$T_2(z)=\lambda(z)(z-\alpha)+\alpha.$$
Note the function $T_2$ is a continuous function and a radial expansion with variable speed $\lambda(z)$.
Define $\hat{f}_2(z)=T_2\circ \widetilde{f}_2(z)$.
\begin{lemma} \label{lemaseq}
If $z\in W_{\alpha}$ with $d(\hat{f}_2(z),\alpha)<s_k$ then
$$d(z,\alpha) \leq s_{k+1}.$$
\end{lemma}
\begin{proof}
If $d(z,\alpha)>s_{k+1}$ then $d(\widetilde{f}_2(z), \alpha)>r_{k+1}$ this implies $$d(\hat{f}_2(z),\alpha)=d(\lambda(\widetilde{f}_2(z))(\widetilde{f}_2(z)-\alpha)+\alpha,\alpha)=\lambda(\widetilde{f}_2(z))d(\widetilde{f}_2(z),\alpha)\geq s_k ,$$ we can conclude, if $d(\hat{f}_2(z),\alpha)<s_k$, then $d(z,\alpha) \leq s_{k+1}.$
\end{proof}
\begin{proposition}\label{proposicaox}
If $z, \hat{f}_2(z), \ldots, \hat{f}_2^N(z) \in W_{\alpha}$, then $d(z,\alpha) \leq s_N$.
\end{proposition}
\begin{proof}
Let us prove the proposition by induction over $N$. For $N=0$ we have $d(z,\alpha)\leq R_1$ therefore $d(z,\alpha) \leq s_0$, the induction hypothesis is that we assume true the assertion for $k=N-1$.
For $k=N$, if $y=\hat{f}_2(z)$ and as $y, \hat{f}_2(y), \ldots, \hat{f}_2^{N-1}(y) \in W_{\alpha}$ then by the induction hypothesis:
$$d(y,\alpha)\leq s_{N-1} \Rightarrow d(\hat{f}_2(z),\alpha)\leq s_{N-1} \mbox{ by lemma (\ref{lemaseq}) } d(z,\alpha) \leq s_N.$$
\end{proof}
\begin{lemma} \label{lemadesig}
If $z, \hat{f}_2(z), \ldots, \hat{f}_2^k(z) \in W_{\alpha}$, then $N_{ret}(\hat{f}_2^i(z))=N_{ret}(q)$ for all \linebreak $i=1,\ldots,k$ and
$$\frac{1}{kN_{ret}(q)}S_{kN_{ret}(q)}\hat{f}(z)\leq \psi(\alpha)+\frac{1}{k}(\psi(x_{\max})-\psi(\alpha))$$
\end{lemma}
\begin{proof}
If $z, \hat{f}_2(z), \ldots, \hat{f}_2^k(z) \in W_{\alpha}$, then $N_{ret}(\hat{f}_2^i(z))=N_{ret}(q)$ for all $i=1,\ldots,k$.
Note that
\begin{eqnarray}
\frac{1}{kN_{ret}(q)}S_{kN_{ret}(q)}\hat{f}(z)&=& \frac{1}{kN_{ret}(q)}\sum_{i=0}^{k-1}N_{ret}(q) \psi (\hat{f}_2^i(z)) \nonumber \\
& \leq & \frac{1}{k}\sum_{i=0}^{k-1}\psi_{\max} (\hat{f}_2^i(z)) \mbox{ and as } P(d(z,\alpha))=\psi_{\max}(z)-\psi(\alpha) \nonumber \\
& \leq & \psi(\alpha)+\frac{1}{k}\sum_{i=0}^{k-1}P(d(\hat{f}_2^i(z),\alpha)). \nonumber
\end{eqnarray}
By the proposition (\ref{proposicaox}) $d(\hat{f}_2^i(z),\alpha)\leq s_{k-i}$, therefore
$$P(d(\hat{f}_2^i(z),\alpha))\leq P(s_{k-i})=\frac{1}{2^{k-i}}P(R_1),$$
moreover, $P(R_1)\leq \psi(z_{max})-\psi(\alpha)$. This way we can conclude:
\begin{eqnarray}
\frac{1}{k} \sum_{i=0}^{k-1}P(d(\hat{f}_2^i(z),\alpha)) &\leq & \frac{1}{k}\left( \frac{1}{2^k}+\frac{1}{2^{k-1}}+\ldots +\frac{1}{2}\right) P(R_1) \nonumber \\
& \leq & \frac{1}{k}P(R_1) \leq \frac{1}{k} \left( \psi(x_{\max})-\psi(\alpha) \right), \nonumber
\end{eqnarray}
and we are done.
\end{proof}
As the $\phi$ integral over the measure equidistributed over the $\hat f$ orbit of $\alpha$ is $\psi(\alpha)$, the final step in the proof of Theorem \ref{maintheorem} is
\begin{proposition}
For all $z\in M,\, \lim_{n\to\infty}\frac{1}{n}S_n\hat f(z)\le \psi(\alpha)$.
\end{proposition}
\begin{proof}
First, note that, if the $\hat f$ orbit of $z$ does visit $W_0$ infinitely many times, then \ref{naoemw0} and the same argument applied in lemmas \ref{casoI} and \ref{somamenor} show the
result.
Second, if the $\hat f$ orbit of $z$ visits $W_0$ infinitely many times, but only visits $W_{\alpha}$ a finite number of times, then using (\ref{naoemwalpha}) and again using the reasoning in lemmas \ref{casoI} and \ref{somamenor} we have the result.
Now assume $z\in W_{\alpha}$ returns to $W_{\alpha}$ infinitely many times, and let $n_j$ the sequence of times such that $\hat{f}^{n_j}(z)=(T_2\circ \widetilde{f})^{n_j}(x)\in I$, where $n_0=0$ and $n_{i+1}=n_i+N_{ret}(\hat f ^{n_i}(x))$. Consider the following subsequences of $(n_k)_{k\in \mathbb{N}}$:
\begin{itemize}
\item $a_j$, where $a_1=0$ and $a_{i+1}$ is the smallest integer larger than $a_i$ such that $\hat f ^{n_{a_{i+1}}}(x)$ is in $W_{\alpha}$, but $\hat f ^{n_{a_{i+1}-1}}(x)$ is not.
\item $b_j$, where $b_i$ is the smallest integer larger than $a_i$ such that $\hat f ^{n_{b_i -1}}(x)$ is in $W_{\alpha}$, but $\hat f ^{n_{b_i}}(x)$ is not.
\end{itemize}
We have that:
\begin{eqnarray}\label{equacao12}
\frac{1}{n_{a_k}} S_{n_{a_k}}\hat f(x)=\displaystyle \frac{1}{n_{a_k}} \displaystyle \sum_{l=0}^{a_k-1}N_{ret}(\hat f^{n_l}(x))\psi(\hat f^{n_l}(x))=\\ = \frac{1}{n_{a_k}} \displaystyle \sum_{j=1}^{k-1} \left( \displaystyle \sum_{l=a_j}^{b_j-1} N_{ret}(\hat f^{n_l}(x))\psi(\hat f^{n_l}(x)) + \displaystyle \sum_{l=b_j}^{a_{j+1}-1} N_{ret}(\hat f^{n_l}(x))\psi(\hat f^{n_l}(x)) \right)\nonumber,
\end{eqnarray}
furthermore if $a_j\leq i\leq b_j-1$ then $\hat{f}^{n_i}(x)\in W_{\alpha}$ and if $b_j\leq i\leq a_{j+1}-1$ then $\hat{f}^{n_i}(x)\notin W_{\alpha}$, for all $x\in W_{\alpha}$, this way:
\begin{equation}
\displaystyle \sum_{l=b_j}^{a_{j+1}-1} N_{ret}(\hat f^{n_l}(x))\psi(\hat f^{n_l}(x))\leq (n_{a_{j+1}}-n_{b_j})\psi(q),\label{final1}
\end{equation}
For the other term in expression (\ref{equacao12}) we have:
\begin{eqnarray}
& &\displaystyle \sum_{l=a_j}^{b_j-1} N_{ret}(\hat f^{n_l}(x))(\psi(\hat f^{n_l}(x))-\psi(\alpha))= \displaystyle \sum_{l=a_j}^{b_j-1} N_{ret}(q)\psi(\hat{f}^{n_l}(x))-(b_j-a_j)N_{ret}(q)\psi(\alpha) \nonumber \\
&\leq & (b_j-a_j) N_{ret}(q)\left( \psi(\alpha)+\frac{1}{b_j-a_j}[\psi(x_{\max})-\psi(\alpha) ] \right)-(b_j-a_j)N_{ret}(q)\psi(\alpha)\nonumber \\
&= & N_{ret}(q)(\psi(x_{\max})-\psi(\alpha))\leq \frac{\psi(\alpha)-\psi(q)}{2} \nonumber
\end{eqnarray}
which holds by lemma (\ref{lemadesig}) and by (\ref{alfa}) since $N_{ret}(q)< m_0$. Then
\begin{equation}
\displaystyle \sum_{l=a_j}^{b_j-1} N_{ret}(\hat f^{n_l}(x))\psi(\hat f^{n_l}(x)) \leq \frac{\psi(\alpha)-\psi(q)}{2}+(n_{b_j}-n_{a_j})\psi(\alpha). \label{final2}
\end{equation} Replacing (\ref{final1}) and (\ref{final2}) in (\ref{equacao12}) we have:
\begin{equation}
\frac{1}{n_{a_k}}\displaystyle \sum_{j=1}^{k-1}\left[ \psi(\alpha)\left(n_{b_j}-n_{a_j}-\frac{1}{2} \right)+\psi(q) \left( n_{a_{j+1}}-n_{b_j}+\frac{1}{2} \right) \right]
\end{equation}
by the manner $\alpha$ was chosen we have $\psi(\alpha)\geq \psi(q)$, so
\begin{eqnarray}
\frac{1}{n_{a_k}}\displaystyle \sum_{j=1}^{k-1}\left[ \psi(\alpha)\left(n_{b_j}-n_{a_j}-\frac{1}{2} \right)+\psi(q) \left( n_{a_{j+1}}-n_{b_j}+\frac{1}{2} \right) \right] \nonumber \\
\leq \frac{1}{n_{a_k}}\displaystyle \sum_{j=1}^{k-1} \psi(\alpha)(n_{a_{j+1}}-n_{a_j})=\psi(\alpha)-\frac{n_{a_1}}{n_{a_k}}\psi(\alpha)\rightarrow \psi(\alpha).\nonumber
\end{eqnarray}
concluding the proof of the proposition and the theorem
\end{proof}
\bibliographystyle{plain}
|
1,116,691,497,766 | arxiv | \section{Introduction}
\label{intro}
Spin manipulation and control is at the core of
spintronics, a technology that uses the spin of the electrons, rather than
their charge, to transfer and/or process information.\cite{awsc,fabian}
The Rashba spin-orbit (SO) coupling arising in materials lacking
structural inversion symmetry\cite{rashba} plays a leading role in this field
because its strength can be tuned by an applied electric field and
by specific material engineering methods. The SO induced lifted spin
degeneracy may then be used in spin filtering devices and spin transistors.
Whether the main effect of SO coupling is limited to the spin splitting
or it is accompanied by substantial modifications in other electronic properties,
which could be detrimental for the spin propagation,
is of course crucial for the functioning of spin-based devices.
In this respect, an important issue calls into play the role of the SO
interaction on the coupling of electrons to the lattice vibrations (phonons).
In particular, a sensible problem is whether the polaron, that is the quasiparticle
composed by the electron and its phonon cloud, is strengthened or weakened by
the Rashba SO interaction.
In previous works, an enhancement of the polaronic character has been obtained
for a two-dimensional (2D) electron gas with linear Rashba coupling for both
short-range (Holstein model Ref.[\onlinecite{holstein}]) and long-range
(Fr\"ohlich model Ref.[\onlinecite{froelich}]) electron-phonon (el-ph)
interactions.\cite{cgf2007a,cgf2007b,grima2008a} On the contrary, a recent calculation on the 2D
tight-binding Holstein-Rashba model on the square lattice has shown that a large
el-ph interaction gets effectively suppressed by the Rashba SO coupling.\cite{berciu}
At present therefore the role of the Rashba SO coupling on the polaron properties
is not clear, and different models and approximations appear to give quite contradicting
results.
In this article the tight-binding Holstein-Rashba model for one electron coupled to adiabatic
phonons is considered and the corresponding non-linear Schr\"odinger equation for the
polaron wave function is solved numerically.
It is shown that, for el-ph couplings such that the electron is delocalized in the zero SO limit,
the Rashba term creates a large polaron state, with polaron localization length inversely
proportional to the SO strength. Furthermore, the small polaron regime appearing at large el-ph
couplings and zero SO gets weakened (or even suppressed) for sufficiently strong SO couplings.
Hence, the Holstein-Rashba polaron is strengthened or weakened by the SO interaction depending on
whether the el-ph coupling is respectively weak or strong, thereby reconciling the different trends
reported in Refs.[\onlinecite{cgf2007a,berciu}] into one single picture.
\begin{figure*}[t]
\protect
\includegraphics[scale=1.78,clip=true]{fig1}
\caption{(Color online) Total energy difference $\Delta E=E-E_0$ for the adiabatic Holstein-Rashba model
as a function of the
el-ph coupling $\lambda$ and for different values of the SO Rashba interaction $\gamma$. $E_0$ is the
ground state energy for $\lambda=0$. Different symbols refer to different solutions of the nonlinear Schr\"odinger
equation \eqref{nonlin}, and the ground state is given by the solution with lower $\Delta E$ values.
Insets: corresponding electron density probability at $\mathbf{R}=\mathbf{0}$.} \label{fig1}
\end{figure*}
\section{model}
\label{model}
By presenting the spinor operator $\Psi^\dagger_\mathbf{R}=(c^\dagger_{\mathbf{R}\uparrow},c^\dagger_{\mathbf{R}\downarrow})$,
where $c^\dagger_{\mathbf{R}\alpha}$ creates an electron with spin $\alpha=\uparrow,\downarrow$
on site $\mathbf{R}$, the tight-binding Holstein-Rashba Hamiltonian on the square lattice can
be written as $H=H_0+H_{ph}+H_{el-ph}$, where \cite{sheng}
\begin{align}
\label{h0}
H_0=&-t\sum_\mathbf{R}\left(\Psi^\dagger_\mathbf{R}\Psi_{\mathbf{R}+\hat{\mathbf{x}}}
+\Psi^\dagger_\mathbf{R}\Psi_{\mathbf{R}+\hat{\mathbf{y}}}\right) \nonumber \\
&-i\frac{\gamma}{2}\sum_\mathbf{R}\left(\Psi^\dagger_\mathbf{R}\sigma_y\Psi_{\mathbf{R}+\hat{\mathbf{x}}}
-\Psi^\dagger_\mathbf{R}\sigma_x\Psi_{\mathbf{R}+\hat{\mathbf{y}}}\right)+{\rm H.c.},
\end{align}
is the lattice Hamiltonian for a free electron with transfer integral $t$ and SO coupling $\gamma$.
$\sigma_x$ and $\sigma_y$ are Pauli matrices. The lattice constant is taken to be unity,
and $\hat{\mathbf{x}}$ and $\hat{\mathbf{y}}$ are unit vectors along the $x$ and $y$ directions, respectively.
The Hamiltonian \eqref{h0} is easily diagonalized in momentum space, and the resulting electron
dispersion is composed of two branches: $E_\mathbf{k}^\pm = -2t[\cos(k_x)+\cos(k_y)]
\pm \gamma \sqrt{\sin(k_x)^2+\sin(k_y)^2}$. The lowest branch, $E_\mathbf{k}^-$, has a four-fold
degenerate minimum $E_0=-4t\sqrt{1+\gamma^2/(8t^2)}$ for momenta $\mathbf{k}=(\pm k_0,\pm k_0)$ with
$k_0=\arctan[\gamma/(\sqrt{8}t)]$.\cite{berciu}
The Hamiltonian for Einstein phonons with mass $M$ and frequency $\omega_0$ is given by:
\begin{equation}
\label{hph}
H_{ph}=\sum_\mathbf{R}\left(\frac{P_\mathbf{R}^2}{2M}+\frac{1}{2}M\omega_0^2 X_\mathbf{R}^2\right),
\end{equation}
where $P_\mathbf{R}$ and $X_\mathbf{R}$ are impulse and displacement phonon operators. Finally, the
el-ph Hamiltonian contribution is
\begin{equation}
\label{helph}
H_{el-ph}=\sqrt{2M\omega_0}\,g\sum_\mathbf{R}\Psi^\dagger_\mathbf{R}\Psi_\mathbf{R}X_\mathbf{R},
\end{equation}
where $g$ is the el-ph interaction matrix element.
The (quasi-) 2D materials and heterostructures which display non-zero
Rashba couplings (semiconductor quantum wells, surface states of metals and semimetals)
are wide electron bandwidth systems with $t$ of the order of $1$ eV, while the typical phonon energy
scale is of the order of few to tens meV.\cite{note1} These systems are expected therefore to be
well within the adiabatic regime $\omega_0/t\ll 1$. In the following, however, only the strict
adiabatic limit $\omega_0/t=0$ is considered, which simplifies considerably the problem and, as
shown below, permits to identify the critical parameters governing the electron localization transitions.
The adiabatic limit $\omega_0/t=0$ is obtained formally from Eqs. \eqref{hph} and \eqref{helph} by
setting $M\rightarrow\infty$ and keeping $K=M\omega_0^2$ finite.
Since for $M\rightarrow\infty$ the phonon kinetic energy is zero, the ground state in the adiabatic limit
is obtained by finding the displacement configuration $X_\mathbf{R}^0$ which
minimizes the total energy $E=\langle H \rangle$, where the brackets mean the expectation value
with respect to the electron wave function and the lattice displacement. Hence, since by Hellmann-Feynman theorem $X_\mathbf{R}^0=\sqrt{2M\omega_0}\,g
\langle\psi\vert\Psi^\dagger_\mathbf{R}\Psi_\mathbf{R}\vert\psi\rangle/K$, the ground state energy becomes
\begin{equation}
\label{Hmin}
E_{GS}=\langle\psi\vert H_0\vert\psi\rangle-
E_P\sum_\mathbf{R}\langle\psi\vert\Psi^\dagger_\mathbf{R}\Psi_\mathbf{R}\vert\psi\rangle^2,
\end{equation}
where $E_P=g^2/\omega_0$ is independent of $M$, and $\vert\psi\rangle=\sum_{\mathbf{R},\alpha}\phi_{\mathbf{R}\alpha}c^\dagger_{\mathbf{R}\alpha}\vert 0\rangle$.
The ground state electron wave function $\phi_{\mathbf{R}\alpha}$ can be found from Eq.\eqref{Hmin} by applying the variational
principle, leading to the following non-linear Schr\"odinger equation:
\begin{align}
\label{nonlin}
\varepsilon\Phi_\mathbf{R}=&-t\sum_{n=\pm}(\Phi_{\mathbf{R}+n\hat{\mathbf{x}}}
+\Phi_{\mathbf{R}+n\hat{\mathbf{y}}})-2E_P\mid\Phi_\mathbf{R}\mid^2\Phi_\mathbf{R}\nonumber \\
&-i\frac{\gamma}{2}\sum_{n=\pm}n(\sigma_y \Phi_{\mathbf{R}+n\hat{\mathbf{x}}}
-\sigma_x \Phi_{\mathbf{R}+n\hat{\mathbf{y}}}),
\end{align}
where $\Phi_\mathbf{R}=(\phi^*_{\mathbf{R}\uparrow},\phi^*_{\mathbf{R}\downarrow})^+$ and
$\varepsilon=E_{GS}+E_P\sum_\mathbf{R}\vert\Phi_\mathbf{R}\vert^4$.
Finally, the ground state energy $E_{GS}$ is obtained by solving
Eq.\eqref{nonlin} iteratively, with $\sum_{\mathbf{R},\alpha}\vert\phi_{\mathbf{R}\alpha}\vert^2 = 1$,
and by inserting the resulting wave function into Eq.\eqref{Hmin}.
\section{Results}
\label{results}
Solutions of \eqref{nonlin} for lattices of $N=101\times 101$ sites are plotted in Fig.~\ref{fig1} as a
function of the el-ph coupling constant $\lambda=E_P/(4t)=g^2/(4t\omega_0)$ and for four different
values of $\gamma$. For $\gamma=0$, Fig.~\ref{fig1}(a), we recover the well-known behavior of the adiabatic
Holstein model in two-dimensions:\cite{2D}
a delocalized solution with $E_{\rm GS}=E_0=-4t$ (filled circles)
extending to the whole range of $\lambda$ values considered, and a localized one (filled squares) having
energy lower than $E_0$ for $\lambda\geq \lambda^*=0.835$.
The delocalized/localized nature of the solutions is illustrated in the
inset of Fig.~\ref{fig1}(a) where the electron density probability $\vert\Phi_\mathbf{R}\vert^2=
\sum_\alpha\vert\phi_{\mathbf{R}\alpha}\vert^2$ is plotted for $\mathbf{R}=\mathbf{0}$.
The solution having lower energy for $\lambda\geq \lambda^*$ is a small polaron state, with
more than $90$\% of its wave function localized at the origin.
Let us now consider the $\gamma>0$ case.
As shown in Figs.~\ref{fig1}(b)-(c), a nonzero Rashba term gives rise to a new feature absent for $\gamma=0$.
Namely, besides the two solutions already discussed for the $\gamma=0$ case,
a third solution appears (filled triangles), which has lower energy than the
delocalized and small polaron states in a region of intermediate values of $\lambda$.
It is thus possible to identify
a second critical coupling, $\lambda^{**}$, such that for $\lambda^{**}\leq \lambda\leq\lambda^*$ the ground state
is given by this third solution. Furthermore, the transition to the small polaron state (identified by $\lambda^*$)
gets shifted to larger el-ph couplings as $\gamma/t$ increases, thereby confirming the results
of Ref.[\onlinecite{berciu}] obtained by a different method and for $\omega_0/t\neq 0$.
A map of the behavior of $\lambda^*$ and $\lambda^{**}$ as $\gamma$ is varied
is reported in the $\gamma/t$-$\lambda$ phase diagram of Fig.~\ref{fig2}, where
the filled circles are the calculated values of $\lambda^{**}$, while the filled squares mark the onset
of the small polaron regime ($\lambda^*$).\cite{note2} The resulting diagram is therefore composed of three
separate regions: a delocalized electron with $E_{GS}=E_0$ for $\gamma/t>\lambda^{**}$ (white region),
a small polaron state for large el-ph couplings ($\lambda>\lambda^*$), and a new ground state
in the region comprised between the $\lambda^{**}$ and $\lambda^*$ lines.
\begin{figure}[t]
\protect
\includegraphics[scale=0.85,clip=true]{fig2}
\caption{(Color online) Phase diagram of the 2D adiabatic Holstein-Rashba model.
The $\lambda^{**}$ and $\lambda^{*}$ transition lines are the phase boundaries separating
the different states of the polaron. The dashed curve has been obtained from the
maximum of $d^2 E/\lambda^2$ and identifies a smooth crossover from large to small
polaron for large $\gamma/t$ values. The solid line is the variational result of Eq.\eqref{var7}.
The graded gray (violet) scale refers to the polaron density probability at $\mathbf{R}=\mathbf{0}$.}\label{fig2}
\end{figure}
As it can be inferred from the insets of Fig.~\ref{fig1}
and from the gray (violet) scale of Fig.~\ref{fig2}, in this region the density probability
at $\mathbf{R}=\mathbf{0}$, $\vert\Phi_\mathbf{0}\vert^2$, is lower than the small polaron solution,
but substantially larger than zero as long as $\gamma\neq 0$, and increasing with $\gamma/t$.
The region between the $\lambda^{**}$ and $\lambda^*$ lines identifies therefore
a large polaron state created by the SO interaction, with a localized wave function which
may extent over several lattice sites.
The large polaron nature of this solution is substantiated in
Fig.~\ref{fig3}, where the polaron localization radius $R_P$, extracted from a fit of
$\vert\Phi_\mathbf{R}\vert^2$ to $\exp(-\vert\mathbf{R}\vert/R_P)$ (see inset),
is plotted as a function of $\gamma/t$ for $\lambda=0.4$, $0.6$, and $0.8$.
Although a numerical evaluation of $R_P$ for $\gamma/t\rightarrow 0$ is hampered by the finite size
of the lattice, $R_P$ turns out to be approximately
proportional to $t/\gamma$, suggesting therefore that the large polaron evolves continuously
towards a delocalized electron as $\gamma/t\rightarrow 0$.
\begin{figure}[t]
\protect
\includegraphics[scale=0.8,clip=true]{fig3}
\caption{(Color online) Polaron radius $R_P$ of the large polaron state as a function of $\gamma/t$
and for different el-ph couplings $\lambda$.
Inset: Density probability (symbols) of the large polaron for $\lambda=0.4$ and $\gamma/t=0.25$, $0.5$, and $1.0$
as a function of distance $R=\vert\mathbf{R}\vert$ along the $(1,0)$ direction.
The solid lines are fits to $\exp(-\vert\mathbf{R}\vert/R_P)$.} \label{fig3}
\end{figure}
Further insight on the large polaron state, and in particular on its behavior as $\gamma/t\rightarrow 0$,
can be gained by a simple variational calculation in the continuum. In fact, as long as $R_P$ is
much larger than the lattice constant ($R_P\gg 1$) then an upper bound for $E_{GS}$ can be obtained from
a minimization of the energy functional
\begin{align}
\label{var1}
\mathcal{E}[\Phi(\mathbf{r})]=&\int\! d\mathbf{r}\,\Phi^\dagger(\mathbf{r})\!
\left[t\hat{p}^2+\gamma(\sigma_y\hat{p}_x-
\sigma_x\hat{p}_y)\right]\!\Phi(\mathbf{r})\nonumber\\
&-E_P\int d\mathbf{r}\vert\Phi(\mathbf{r})\vert^4,
\end{align}
where $\hat{p}_q=-i\partial/\partial q$ is the electron momentum operator ($q=x,y$ and
$\hbar=1$) and $\hat{p}^2=\hat{p}_x^2+\hat{p}_y^2$. In the above expression,
$\Phi(\mathbf{r})$ is a suitable ansatz for the ground state spinor, which is assumed to vary
slowly over distances comparable to the lattice spacing. In writing Eq.\eqref{var1},
only the lowest order terms in the lattice constant have been retained,
which amounts to consider a parabolic band with a Rashba coupling linear in the momentum operators.
One can then use for $\Phi(\mathbf{r})$ an ansatz which has been already introduced in studying
the effects of a linear Rashba term on the 2D
Fr\"ohlich polaron and the 2D hydrogen atom:\cite{grima2008a,grima2008b}
\begin{equation}
\label{var2}
\Phi({\bf r})=A\exp(-ar)\left[
\begin{array}{l}
J_0(b r)\\
J_1(b r)\,e^{i\varphi}\!
\end{array}\right].
\end{equation}
Here, $r=\vert\mathbf{r}\vert$ and $\varphi$ is the azimuthal angle, $A$ is a normalization
constant, $J_0$ and $J_1$ are Bessel functions, and $a$ and $b$
are variational parameters.
By using \eqref{var2} and the properties of the Bessel functions, equation \eqref{var1}
reduces to
\begin{align}
\label{var3}
\mathcal{E}&=t(a^2+b^2)-\gamma b-\frac{E_P}{2\pi}\frac{\int_0^\infty\!dr\, r e^{-4ar}F(br)^2}
{\left[\int_0^\infty\!dr\, r e^{-2ar}F(br)\right]^2} \nonumber \\
&\simeq t(a^2+b^2)-\gamma b -\frac{2E_P a^2}{\pi}\ln\left(\frac{b}{\sqrt{e}a}\right),
\end{align}
where $F(br)=J_0(br)^2+J_1(br)^2$. The second equality stems from assuming $a\ll b$,
which is the relevant limit of the large polaron regime. Minimization of $\mathcal{E}$ with
respect to $a$ and $b$ leads to two possible solutions: $b=\gamma/(2t)$ and $a=0$, which corresponds
to a delocalized electron with $\mathcal{E}_{\rm min}=E_0=-4t-\gamma^2/(4t)$, and
\begin{equation}
\label{var4}
a=b\exp\left(-1-\frac{\pi}{8\lambda}\right),
\,\,b=\frac{\gamma/(2t)}{1-4\lambda\exp\left[-2-\pi/(4\lambda)\right]/\pi},
\end{equation}
which represents the large polaron solution with
\begin{equation}
\label{var5}
\mathcal{E}_{\rm min}-E_0=
-\frac{\lambda}{\pi}\frac{\gamma^2}{t}\exp\left(-2-\frac{\pi}{4\lambda}\right),
\end{equation}
for $\lambda$ small. Since Eq.\eqref{var5} is an upper bound for $\Delta E_{\rm GS}=E_{\rm GS}-E_0$,
then that the large polaron state has energy always lower than the delocalized electron.
Furthermore, by realizing that the variational parameter $a$ represents the polaron radius through $a=1/(2R_P)$,
it turns out from Eq.\eqref{var4} that $R_P$ scales as $t/\gamma$, in agreement therefore with the
results of Fig.~\ref{fig3}.
The finding that a large polaron is formed for $\gamma/t\neq 0$ is in accord with the observation
of Ref.[\onlinecite{cgf2007a}] that perturbation theory breaks down in the adiabatic limit for
any finite $\lambda$. This breakdown basically stems from the one-dimensional-like divergence of
the density of states (DOS) of a parabolic band with linear Rashba coupling.\cite{cgf2007a,cgf2007b}
Although the variational result presented above correctly predicts the appearance of the large polaron state
as soon as $\gamma/t\neq 0$, it fails nevertheless in describing the $\lambda^{**}$
transition line of Fig.~\ref{fig2} separating the large polaron state from the delocalized solution.
This is because the lowest order expansion in the lattice constant of Eq.~\eqref{var1} neglects
higher order powers of the momentum operator arising from the lattice Rashba term, which
shift the van Hove divergence of the DOS from $E_0$ to higher energies,\cite{berciu}
thereby making the perturbation theory non-singular.
To investigate this point within the variational method, it suffices to
expand the discrete Hamiltonian up to the third order in the lattice constant. This corresponds
to add to the energy functional \eqref{var1} the following contribution
\begin{equation}
\label{var6}
\mathcal{E}'[\Phi(\mathbf{R})]=\frac{\gamma}{6}\int\! d\mathbf{r}\,\Phi^\dagger(\mathbf{r})\!
\left(\sigma_x\hat{p}_y^3-\sigma_y\hat{p}_x^3\right)\!\Phi(\mathbf{r}),
\end{equation}
which, by using again the ansatz \eqref{var2} and for $a\ll b$, leads to the third order correction term
$\mathcal{E}'=(\gamma/8)(b^3+3a^2b-\pi a^3)$ to Eq.\eqref{var3}. It is then easy
to shown that $[\mathcal{E}+\mathcal{E}']_{\rm min}-E_0$ is negative
[with $E_0=-4t-\gamma^2/(4t)+\gamma^4/(128 t^3)$] as long as $\gamma/t< \lambda^{**}_{\rm var}$, where
for $\lambda$ small
\begin{equation}
\label{var7}
\lambda^{**}_{\rm var}=8\sqrt{\frac{2\lambda}{\pi}}\exp\!\left(-1-\frac{\pi}{8\lambda}\right).
\end{equation}
Although Eq.~\eqref{var7} provides only a lower bound for
$\lambda^{**}$ (solid line in Fig.~\ref{fig2}), it shows nevertheless that, as $\gamma/t$ is enhanced for
fixed $\lambda$, the transition from the large polaron to the delocalized electron state
originates from higher order of the SO interaction than the linear Rashba coupling.
\section{Discussion and conclusions}
\label{concl}
Let us discuss now the significance of the results reported above for materials of interest and
possible consequences for spintronics applications. First of all, it is important to identify the
region in the phase diagram of Fig.~\ref{fig2} where realistic values of $\gamma/t$ and $\lambda$
are expected to fall. This is easily done by realizing that the largest Rashba SO coupling to date
is that found in the surface stats of Bi/Ag(111) surface alloys\cite{ast} for which
$\gamma/t\approx 1.4$ can be estimated. Other 2D systems and heterostructures have lower or much lower
$\gamma/t$ values. Concerning the coupling to the phonons, a survey\cite{kroger} on the el-ph interaction
at metal surfaces evidences that $\lambda$ is usually lower than $0.6$-$0.7$ (see also Ref.[\onlinecite{hofmann}]),
at least for the surface states with large SO splittings (\emph{i.e.} Ag, Cu, Bi). It is therefore
a rather conserving assumption to confine to $\gamma/t\lesssim 1$ and $\lambda \lesssim 1$ the region
of interest for the microscopic parameters which, as shown in Fig.~\ref{fig2}, is substantially
dominated by the SO induced large polaron state.
Hence, upon tuning of the Rashba SO coupling, a delocalized electron at $\gamma/t=0$ can in principle
be changed into a self-trapped large polaron state for $\gamma/t>0$, with obvious consequences on the
spin propagation in the system. In passing, it is worth noticing
that the small polaron regime instead is affected rather weakly by the SO interaction for $\gamma/t \lesssim 1$,
while its weakening gets pronounced only for unrealistically large values of
$\gamma/t$ (see also Fig.~\ref{fig1}).
Before concluding, it is important to discuss a last important point. Although the adiabatic limit
employed here allows for a clear identification of the $\lambda^*$ and $\lambda^{**}$ transition lines,
the energy gain associated to the large polaron formation becomes very small in the weak coupling and
small SO limits [see Eq.~\eqref{var5}]. In this regime, the inclusion of quantum fluctuations which
arise as soon as $\omega_0/t\neq 0$ may wash out completely any signature (like e.g. an anomalous
enhancement of the electron effective mass $m^*$) of the large polaron state,
even for $\omega_0/t$ small, while they should remain visible for larger $\lambda$ and $\gamma/t$
values. For a more complete description of the SO effects on the Holstein-Rashba polaron, it is
therefore necessary to extend the study to the non-adiabatic regime $\omega_0/t\neq 0$, by keeping
however in mind that, as discussed above, relevant materials have $\omega_0/t \ll 1$.
In summary, the complete phase diagram of the 2D adiabatic Holstein el-ph Hamiltonian in the
presence of Rashba SO coupling has been calculated.
It has been shown that a self-trapped large polaron state is created by the SO
interaction in a wide region of the phase diagram, and that its localization radius can be modulated by the
SO coupling. This result implies that, for realistic values of the microscopic parameters, the appearance
of a self-trapped large polaron state is a potentially detrimental factor for spin transport.
\acknowledgements
The author thanks E. Cappelluti, S. Ciuchi, and F. Marsiglio for valuable comments.
|
1,116,691,497,767 | arxiv | \section{Experimental Details}
Both polycrystalline samples were prepared by a solid-state reaction of La$_2$O$_3$, BaCO$_3$, and MnO$_2$\@.
After repeating the sintering process in an Ar flow, the obtained ceramics was annealed in flowing O$_2$, which leads to LaBaMn$_2$O$_6$\@.
The use of pure Ar gas is to avoid the formation of the A-site disorder.
La$_{0.5}$Ba$_{0.5}$MnO$_3$ was obtained by a solid state reaction in 1\% O$_2$/Ar gas, followed by an annealing in O$_2$ gas.
The details are described elsewhere \cite{nakajima04}.
Both samples were characterized crystallographically with an X-ray diffractometer, and magnetically with a SQUID magnetometer.
The obtained data were found to be in accordance with those in the literature \cite{nakajima03}.
NMR was performed with a phase-coherent spectrometer by using a spin-echo technique.
\begin{figure}[t]
\includegraphics[scale=0.27]{fig1a.eps}%
\includegraphics[scale=0.27]{fig1b.eps}%
\caption{Field-swept $^{139}$La-NMR spectra of LaBaMn$_2$O$_6$ at 1.6 K with (a) $f_1$ = 6.016 MHz and (b) $f_2$ = 37.600 MHz\@.}
\end{figure}
We can easily distinguish between magnetic resonances of nuclei in the FM and AFI(CE) regions.
First, the resonance in the FM region is characterized by a very strong NMR signal and a small applied rf field required for the maximum echo signal due to the coupling between nuclear and electronic spins.
The second distinction between these resonances arises from the different local field at La and Mn sites due to Mn spin and charge order.
The La-NMR signal from the AFI(CE) region is not observable in zero field, because the transferred field from the nearest Mn neighbors cancels at La site in this spin structure \cite{kapusta00, note:dipole}.
On Mn site, previous works in manganese perovskites \cite{papavassilou00} have shown that zero-field antiferromagnetic resonances at the localized Mn$^{4+}$ and Mn$^{3+}$ are around 300-320 MHz and 400-420 MHz, respectively, whereas the signal from FM region is located at an intermediate frequency due to the fast electron transfer between Mn$^{3+}$ and Mn$^{4+}$ ions.
\begin{figure}[t]
\includegraphics[scale=0.43]{fig2.eps}%
\caption{$^{55}$Mn-NMR spectra of (a) LaBaMn$_2$O$_6$ under $H$ = 0, 0.7, and 1.35 T and of (b) La$_{0.5}$Ba$_{0.5}$MnO$_3$\ in zero field at 4.2 K\@.}
\end{figure}
Figure 1 shows the field-swept $^{139}$La-NMR spectra of LaBaMn$_2$O$_6$ at the fixed frequencies of (a) $f_1 =$ 6.016 MHz and (b) $f_2 =$ 37.600 MHz at 1.6 K\@.
The signal around 1.0 T in Fig.~1 (a) is attributed to the resonance from the antiferromagnetic region, because this resonance occurs close to $f_1/^{139}\gamma$ = 1.0 T, namely the hyperfine field cancels at La site in this environment.
On the other hand, the signal around 1.0 T in Fig.~1 (b) is associated with the resonance from the ferromagnetic region with the hyperfine field of 5.2 T via positive transferred couplings.
These assignments are assured by the enhancement observed in the spin-echo intensity only for the signal in Fig.~1 (b).
Thus, these $^{139}$La-NMR spectra give evidence for the coexistence of ferromagnetic and antiferromagnetic regions at the ground state.
$^{55}$Mn-NMR spectra show this coexistence as well.
In Fig.~2 are shown the $^{55}$Mn-NMR spectra of LaBaMn$_2$O$_6$ at $H$ = 0, 0.7 and 1.35 T, and (b) that of La$_{0.5}$Ba$_{0.5}$MnO$_3$ in zero field at 4.2 K\@.
The peak at 370 MHz for LaBaMn$_2$O$_6$ in zero field may be attributed to some region with the disordered structure, because the spectrum for La$_{0.5}$Ba$_{0.5}$MnO$_3$ has a maximum at the same frequency with a ferromagnetic enhancement.
The degree $S$ of A-site order in LaBaMn$_2$O$_6$ is roughly estimated to be about $S$ = 92 \% from the spin-echo intensity, which is comparable to $S = 96\pm2$ \% for PrBaMn$_2$O$_6$ estimated by the X-ray and neutron diffractions \cite{nakajima04, note:degree}.
Here, we use the fact that the FM region occupies about 50 \% of the whole volume of LaBaMn$_2$O$_6$ at 4.2 K as estimated later.
It is also assumed that the enhancement factors are comparable between the resonance for La$_{0.5}$Ba$_{0.5}$MnO$_3$ and that for FM region in LaBaMn$_2$O$_6$\@.
The majority of the signal is associated with the intrinsic LaBaMn$_2$O$_6$ and the origin of their peaks may be distinguished by their shifts in an external field.
The largest peak at 380 MHz (indicated by a solid arrow) is associated with the FM region, ensured by the observation that an external field $H$ shifts this resonance frequency by $-^{55}\gamma H$ and depresses the ferromagnetic enhancement by removing the domain wall that has the largest contribution to the enhancement.
Comparison with the NMR spectra of the related manganites \cite{papavassilou00} indicates that the peaks at 323 MHz and 395 MHz (indicated by dotted arrows) are ascribed to the nuclei of localized Mn$^{4+}$ and Mn$^{3+}$ ions in the AFI(CE) state, respectively.
This is assured by the fact that these peaks are unshifted by an external field.
Thus, both $^{139}$La- and $^{55}$Mn-NMR spectra provide evidence for the coexistence of antiferromagnetic and ferromagnetic regions at the ground state in spite of the ordering of La and Ba, and these regions are assigned to the insulating charge ordered and the metallic states, respectively.
However, the origin of the peak at 349 MHz is unclear at present.
Note that similar additional lines are also observed in the AFI(CE) phase of the A-site ordered YBaMn$_2$O$_6$ \cite{ohno05}.
It may be, therefore, related to the layer-type crystallographical and/or spin structures with a fourfold periodicity along the $c$ axis \cite{kageyama03}.
\begin{figure}[t]
\includegraphics[scale=0.43]{fig3.eps}%
\caption{$^{55}$Mn-NMR spectra of LaBaMn$_2$O$_6$ in zero field at several temperatures.}
\end{figure}
Next, we demonstrate the development of the coexisting state as temperature varies.
Figure 3 shows the temperature dependence of the $^{55}$Mn-NMR spectrum for LaBaMn$_2$O$_6$ in zero field.
All spectra are measured at a given temperature after warming from the lowest temperature, otherwise a hysteretic behavior may be observed in the NMR spectra.
Although the line for FM region is observed in the whole measured temperature range, the lines for AFI(CE) region are undetected above 175 K, which is consistent with the appearance of AFI(CE) phase below 200 K as shown later.
The peak frequencies for the FM and AFI(CE) regions, which is proportional to the time-averaged Mn local moments $\mu_{\rm Mn}$, decrease monotonically with increasing temperature.
\begin{figure}[t]
\includegraphics[scale=0.43]{fig4.eps}%
\caption{Temperature dependences of (a) $H_{\rm int}^{\rm F}$ (closed circles) and $H_{\rm int}^{\rm AF}$ (open squares), (b) $I_{\rm B}/(H_{\rm int}^{\rm F})^2$ normalized by the value at 220 K, and (c) $1/T_2$'s measured at the peak from the FM region of LaBaMn$_2$O$_6$ (closed circles) and that of La$_{0.5}$Ba$_{0.5}$MnO$_3$ (open circles).
}
\end{figure}
In Fig.~4 (a) is shown the temperature dependence of the internal field estimated from the peak frequency, where closed circles and open squares correspond to the lines from nuclei in the FM region and those of Mn$^{4+}$ in the AFI(CE) region, respectively.
The internal field $H_{\rm int}^{\rm F}$ at Mn site in the FM region decreases monotonically from 36.2 T at 4.2 K with increasing temperature and seems to approach zero at $T_{\rm C} \sim$ 330 K\@.
This temperature dependence seems to be a characteristic of a second-order phase transition, as contrasted with the case in La$_{0.5}$Ca$_{0.5}$MnO$_3$ and Pr$_{0.7}$Ba$_{0.3}$MnO$_3$ exhibiting a first-order-like transition from FM to paramagnetic metal \cite{allodi98,savosta97}.
Note that any anomaly in $H_{\rm int}^{\rm F}$ is not observed around 200 K below which the AFI(CE) region appears inside the FM phase.
This is presumably because the typical size of FM region is mesoscale rather than nanoscale as suggested later and, hence $H_{\rm int}^{\rm F}$ dominated by the on-site core polarization is hardly affected by the spin rearrangement in the region where the FM phase transforms to the AFI(CE) one.
On the other hand, the internal field $H_{\rm int}^{\rm AF}$ in the AFI(CE) region, being still high at 150 K, indicates that the antiferromagnetic spin structure arises from the ferromagnetic one through a first-order-like transition.
In connection with this fact, the peak frequency 349 MHz at 4.2 K becomes close to that for the FM region with increasing temperature, which may be responsible for the apparent broadening of the NMR line from the FM region above 77 K\@.
To gain further insight into the phase separation, we compare our results with those obtained by the neutron diffraction experiments \cite{nakajima03}.
While $H_{\rm int}^{\rm F}$ is proportional to $\mu_{\rm Mn}$ in the FM region, the magnetic Bragg neutron-reflection intensity $I_{\rm B}$ has a relation $I_{\rm B} \propto (V_{\rm FM}/V_0) \mu_{\rm Mn}^2$, where $V_{\rm FM}$ and $V_0$ are the volumes of the FM region and the whole sample, respectively.
The value of $I_{\rm B}/(H_{\rm int}^{\rm F})^2$ is, therefore, proportional to the FM volume fraction, $V_{\rm FM}/V_0$\@.
Figure 4 (b) shows the temperature dependence of $I_{\rm B}/(H_{\rm int}^{\rm F})^2$ normalized by the value at 220 K \cite{note:enhance}.
The FM volume fraction decreases below 200 K and remains about 50\% below 100 K, which suggests a formation of the AFI(CE) phase inside of the FM phase.
The estimated FM volume fraction is somewhat smaller than 70 \% reported from the magnetization \cite{nakajima03}.
The bulk ferromagnet La$_{0.5}$Ba$_{0.5}$MnO$_3$ contained in LaBaMn$_2$O$_6$ may be responsible for this discrepancy.
Finally, the temperature dependence of Mn spin-spin relaxation rate $1/T_2$ is shown in Fig.~4 (c), where closed and open circles are the data measured at the peaks from the nuclei in the FM region of LaBaMn$_2$O$_6$ and those of La$_{0.5}$Ba$_{0.5}$MnO$_3$, respectively.
$1/T_2$'s are determined by a fit with a relaxation $I(\tau)=I(0) \exp(-2\tau/T_2)$, where $I(\tau)$ is a spin-echo intensity at a pulse interval $\tau$ in the spin-echo measurements.
They show almost the same temperature dependence, that is, an exponential rise $1/T_2 \propto \exp(aT)$ in a wide temperature range above 60 K\@.
Such a temperature variation in $1/T_2$, which is rather different from the case in a conventional ferromagnetic metal, has been reported for a number of double-exchange FM manganites \cite{savosta99} including La$_{0.5}$Ba$_{0.5}$MnO$_3$\@.
This result indicates that the nature of FM region coexisting with the AFI(CE) one in LaBaMn$_2$O$_6$ is microscopically identical to the bulk FM phase of La$_{0.5}$Ba$_{0.5}$MnO$_3$, irrespective of the order/disorder at the A-site of perovskite.
In addition, this suggests mesoscopic rather than nanoscopic nature of FM region in LaBaMn$_2$O$_6$\@.
The small deviation from the exponential rise for both samples below 60 K may be ascribed to an additional relaxation process, such as the Suhl-Nakamura interaction in which the nuclear spins interact via electronic spin waves \cite{savosta99,suhl58,nakamura58}.
No indication of critical behavior around 200 K in $1/T_2$ for LaBaMn$_2$O$_6$ may be related to the first-order-like formation of the AFI(CE) phase inside the FM phase.
In conclusion, it is demonstrated by the NMR spectra that the FM and AFI(CE) regions coexist at the ground state of LaBaMn$_2$O$_6$ in spite of the A-site order, where the AFI(CE) region occupies about half volume of the whole sample.
This exotic coexisting state appears below 200 K via a first-order-like formation of the AFI(CE) state inside the FM one.
The temperature dependence of $1/T_2$ indicates that the FM region coexisting with the AFI(CE) one in LaBaMn$_2$O$_6$ is microscopically identical to the bulk FM phase of La$_{0.5}$Ba$_{0.5}$MnO$_3$, irrespective of the order/disorder at the A-site of perovskite.
This suggests mesoscopic rather than nanoscopic nature of FM region in LaBaMn$_2$O$_6$\@.
This work was in part supported by Grant-in-Aid for Scientific Research (Nos.\ 16540324 and 17740231) from MEXT of Japan and by Yazaki Memorial Foundation for Science and Technology\@.
|
1,116,691,497,768 | arxiv | \section{Introduction}
We use the shorthand ``koosh" to describe the hyperconical, spherically symmetric Kerr-Schild geometry with the line element:
\begin{equation}
ds^2=-\kappa^2 dt^2+\frac{1}{\kappa^2}dr^2+ r^2 d\theta^2+r^2 \sin^2\theta \, d\phi^2.
\label{uniaxialmet}
\end{equation}
This metric describes a hypercone in four dimensions in that the circumference of a circle of proper radius $r^*$ is $2\pi \kappa r^*$. Its Riemann tensor has only one independent nonzero element $R^{\theta \phi}_{~~\theta \phi}=(1-\kappa^2)/r^2$.
Since spherically symmetric Kerr-Schild metrics may be written in the form $-g_{tt}=1/g_{rr}=1-2m(r)/r$, where $m(r)$ is a ``mass" function, we can identify that for a koosh
\begin{equation}
m=\lambda r,
\label{uniaxialmass}
\end{equation}
such that $\kappa=\sqrt{1-2 \lambda}$.
Demanding that the $t$ coordinate remains timelike requires
$ 2\lambda<1$.
The only nonzero energy momentum tensor components are
\begin{align}
T^t_{~t}=T^r_{~r}=-\frac{\lambda}{4\pi r^2}
\end{align}
This means that the eigenvalue structure is Segre Type [(11)(1,1)] with
\begin{align}
\Lambda_0=\Lambda_1,\quad \Lambda_2=\Lambda_3=0
\label{stringeos}
\end{align}
where $\Lambda_0$ is the eigenvalue associated with a timelike eigenvector and $\Lambda_{1,2,3}$ are the other eigenvalues. This can be though of as a ``stringy" equation of state, in that it applies to solutions with vacuum cosmic strings \cite{PhysRevD.23.852,PhysRevD.31.3288,Letelier:1979ej,gursesnambu}.
This solution seems to have been initially discovered by Lettelier as a cloud of radially alligned strings \cite{Letelier:1979ej}, arranged like the filaments on a koosh ball toy and giving rise to our name. It was independently examined as a model for a ``global monopole"\cite{PhysRevLett.63.341}. Several later papers have also examined the koosh or similar systems as stringy systems or monopoles arising from various theories \cite{Guendelman:1991qb,Delice:2003zp,2019arXiv191008166B}.
One intriguing recent development is the consideration of the $\lambda\rightarrow1/2^-$ limit. If such a system is cut off at a finite radius by a shell, it acts as an interior to the Schwarzschild black hole, and further have the correct mass scaling characteristics without changing the interior density profile. This is a ``quasiblack hole" configuration \cite{Lemos:2020hsz}. The quasiblack hole does have the problem that it is a singular configuration, but it is noteworthy that the hyperconical geometry of metric Eq.~\eqref{uniaxialmet} only applies to global monopoles at sufficient radius from the center, at extremely small radii the global monopole described in \cite{PhysRevLett.63.341} is de Sitter like and nonsingular. Replacing the interior of a quasiblack hole with a very extreme global monopole would lead to a system with some properties like certain gravastar \cite{MazurMottola:2001,MazurMottola:2004,MazurMottola:2015,Visser:2003ge} or dark energy star models \cite{ChapHohlLaughSant:2001} (in that the compact object is bounded by some kind of thin shell at or near the horizon) and other properties like Bardeen \cite{Bardeen:1968} or similar (e.g.\cite{Dymnikova:1992,AyonBeato:1998ub,Hayward:2005gi}) type nonsigular black holes (in that the interior is Kerr-Schild and the pressure everywhere follows $p_r=-\rho$, there is a de Sitter center, and the solution decreases in density as one moves outward).
\iffalse
\textbf{needs work, speculation... possibly move to conclusions?} Pure vacuum energy/de Sitter regions are popular for... A pure vacuum energy interior can not carry angular momentum density \cite{2022PhRvD.105b4002B}, and it is also impossible for a pure vacuum energy region to undergo a change in density if the object acretes additional matter \cite{Beltracchi:2018ait}. However, at least in spherical symmetry, the anisotropic TOV gives that items with $p_r=-\rho$ are immune from their own gravity making which is an interesting and (desireable?) property for alternative models of compact objects
However, if quantum mechanical effects can lead to shell like stresses on horizons as proposed in \cite{MottolaVaulin:2006}, hybrid objects like cuttoff global monopole quasiblackholes may be worth further consideration.
\textbf{end needs work}
\fi
There have also been various examinations of rotating solutions generated from the Newman-Janis algorithm \cite{Newman:1965tw,Newman:1965my,Gurses:1975vu} which have string cloud behavior in their static versions \cite{Sakti:2019iku}. While the Newman-Janis algorithm preserves Segre type [(11)(1,1)], the stringy equation of state is not preserved in passing to rotation under the Newman-Janis algorithm \cite{Beltracchi2021a}.
In this paper we modify the Hartle formalism \cite{Hartle:1967he,Hartle:1968si} to produce a perturbative model for a slowly rotating koosh which preserves the stringy equation of state. Originally, the Hartle formalism involved perfect fluid (Segre type [(111),1]) equations of state. Effects from first order in rotation for anisotropic systems had been previously considered for anisotropic neutron star models \cite{2015CQGra..32n5008S} and anisotropic continuous pressure gravastar models \cite{2008PhRvD..78h4011C}. Very recently, a treatment more similar to Hartle's involving the second order in rotation deformation terms for a particular anisotropic Segre type [(11)1,1] neutron star model was presented \cite{2021EPJC...81..698P}. The situation with a koosh is extremely anisotropic in that one of the distinct eigenvalues is always zero, and the Segre type is different than what has been considered previously.
\section{Axisymetric spacetimes and the Hartle Formalism }
\label{Sec:Eineqs}
One convenient notation of the general axisymetric metric in coordinates $(t,r,\theta,\phi)$ comes from \cite{Chandrasekhar:1984}
\begin{align}
ds^2 = -e^{2\nu} dt^2 + e^{2\psi} \big(d \phi - \omega dt\big)^2 + e^{2 \alpha} dr^2 + e^{2 \beta} d\theta^2
\label{axisymstat}
\end{align}
where the five functions $\nu, \psi, \alpha, \beta, \omega$ are functions of $r$ and $\theta$. The functions $\nu,\psi,\omega$ can be isolated as scalar functions because of the existence of the time and axial Killing vectors
\begin{align}
K_{(t)}^\mu=(1,0,0,0),\\
K_{(\phi)}^\mu=(0,0,0,1).
\end{align}
Adopting the nomenclature from \cite{2022PhRvD.105b4001B}, we define additional vectors $l$ and $N$, which in these coordinates are
\begin{align}
l^\mu=(1,0,0,\omega),\\
N_\mu=(-1,-e^{\alpha-\nu},0,0),
\end{align}
such that
\begin{align}
l=K_{(t)}+\omega K_{(\phi)},\quad N\cdot N=0,~N\cdot l=-1.
\end{align}
With these auxiliary vectors, we can define two physically relevant scalar quantities, being the surface gravity parameter
\begin{align}
S_G=N_\mu l_\nu(\nabla^\mu l^\nu)=\frac{1}{2}e^{-\alpha-\nu}\frac{\partial}{\partial r}e^{2\nu}
\end{align}
and angular momentum density parameter
\begin{align}
\mathcal{J}=-N_\mu l_\nu(\nabla^\mu K_{(\phi)}^\nu)=-\frac{1}{2}e^{2\psi-\alpha-\nu}\frac{\partial \omega}{\partial r}.
\end{align}
These scalars are related to the Komar mass and angular momentum(see \cite{Komar:1959} for the introduction of the concepts, and \cite{2022PhRvD.105b4001B} for information about the particular formulation), which can be defined as surface integrals at a given radius, or as the sum of a surface integral at a smaller radius and a volume integral of components of the energy momentum tensor between the smaller and given radii
\begin{align}
M_K(r) &= \frac{1}{4 \pi G}\int_{\partial V_+}\! (S_G + \omega \mathcal{J})\, dA
=\int_V \sqrt{-g}\ \Big(\! \!-T^t_{\ t} + T^r_{\ r} + T^\theta_{\ \theta} + T^\phi_{\ \phi} \Big)\, dr\,d\theta\,d\phi\, +\, \frac{1}{4 \pi G} \int_{\partial V_-} \! (S_G + \omega \mathcal{J})\, dA,
\label{KomarMass}\\
J_K(r) &= \frac{1}{8 \pi } \int_{\partial V_+}\! \! \mathcal{J} \, dA = \int_V \sqrt{-g}\ T^t_{\ \phi} \,dr\,d\theta\,d\phi +\frac{1}{8 \pi } \int_{\partial V_-}\! \!\mathcal{J} \, dA.
\label{KomarAngmom}
\end{align}
For his perturbative framework, Hartle expanded the line element \eqref{axisymstat} to second order in the angular momentum as~\cite{Hartle:1967he}
\begin{align}
ds^2 = & - e^{2\nu_0(r)} \Big[ 1+2 h_0(r) + 2 h_2(r)\, P_2(\cos\theta) \Big] dt^2 \nonumber\\ &
+ \frac{r}{r-2m(r)} \left\{1+\frac{2}{r-2m(r)} \Big[ m_0(r) + m_2(r)\, P_2(\cos\theta) \Big] \right\} dr^2 \nonumber\\&
+ r^2 \Big[ 1+2 k_2(r)\, P_2(\cos\theta) \Big] \Big[ d\theta^2 + \sin^2\!\theta\,\big(d\phi - \omega(r) dt \big)^2 \Big].
\label{kingmet}
\end{align}
The function $\omega(r)$ is the first-order contribution that gives rise to inertial frame dragging. Here $P_l(\cos\theta)$ is the Legendre polynomial of order $l$, $m(r)$ and $\nu_0(r)$ are the metric functions of the nonrotating solution, and $h_l(r)$, $m_l(r)$, $k_l(r)$ are the monopole ($l=0$) and quadrupole ($l=2$) contributions of second order in rotation respectively. The choice $k_0(r)=0$ is part of Hartle's choice of gauge.
The Hartle metric \eqref{kingmet} is equivalent to second order to general metric \eqref{axisymstat} with the identifications (see eg \cite{Chandrasekhar:1974})
\begin{subequations}
\begin{align}
&e^\nu = e^{\nu_0(r)} \big[ 1 + h_0(r) + h_2(r) \,P_2(\cos \theta)\big]\\
&e^\psi = r \sin\theta\, \big[1 + k_2(r)\, P_2(\cos \theta)\big]\\
&e^\alpha = \sqrt{\frac{r}{r-2 m(r)}} \left\{1 + \frac{m_0(r)+m_2(r)P_2(\cos \theta)}{r- 2 m(r)}\right\} \\
&e^\beta = r \big[1 + k_2(r) \, P_2(\cos \theta)\big]\\
&\omega=\omega(r).
\end{align}
\end{subequations}
\subsection{Hartle's energy momentum tensor}
Originally, Hartle's metric was paired with a perfect fluid energy momentum tensor. We describe its construction here for completeness, but since we are interested in a system with anisotropic pressures we use a different method to construct and examine the energy momentum tensor which is described in the following section. With a background metric of the form
\begin{align}
ds^2 = - e^{2\nu_0(r)} \, dt^2 + \frac{dr^2}{1-\frac{2m(r)}{r}} + r^2 \, d\theta^2 + r^2 \, \sin^2\theta \, d\phi^2 .
\label{sphmetintro}
\end{align} and the unperturbed energy momentum tensor $T^\mu_{~\nu}=diag(-\rho,p,p,p)$. Einstein's equations give the following relationships
\begin{align}
\frac{\partial m}{\partial r} & = 4\pi r^2 \rho ,
\label{eq:Einstein-rhointro}
\\
\frac{\partial\nu_0}{\partial r} & = \frac{(m+4\pi r^3 p)}{r^2 \left(1-\frac{2m}{r}\right)} ,
\label{eq:Einstein-printro}
\\
- \frac{\partial p}{\partial r}&=\frac{ \left( m+4\pi r^3 p \right) \left( \rho+p \right)}{r^2 \left( 1 - \frac{2 m}{r} \right) }.
\label{Eq:Forceintro}
\end{align}
Given an equation of state and appropriate boundary conditions, one may in theory solve this system for the unperturbed metric functions. In the notation of Hartle~\cite{Hartle:1967he}, the perturbed energy momentum tensor is
\begin{align}
T^{\mu\nu} = ({\matheul E} + {\matheul P}) u^\mu u^\nu + {\matheul P} g^{\mu\nu},
\label{Tfluid}
\end{align}
where ${\matheul E}$ and ${\matheul P}$ are the energy density and pressure in the comoving frame of the rotating fluid, and $u^\mu$ is its four-velocity
\begin{equation}
u^t=\frac{1}{\sqrt{-g_{tt}-2\Omega g_{t\phi}-\Omega^2 g_{\phi \phi}}},\qquad u^\phi=\Omega u^t,\qquad u^r=u^\theta=0.
\end{equation}
To order $\Omega^2$,
\begin{align}
& {\matheul E} = \rho(r) + {\matheul E}_0(r) + {\matheul E}_2(r) P_2(\cos\theta),
\\
& {\matheul P} = p(r) + {\matheul P}_0(r) + {\matheul P}_2(r) P_2(\cos\theta) ,
\end{align}
where ${\matheul E}_0(r)$, ${\matheul E}_2(r)$, ${\matheul P}_0(r)$, ${\matheul P}_2(r)$ are monopole and quadrupole perturbation functions of order $\Omega^2$, where in the case of these perfect fluid systems the rotation parameter $\Omega$ has a simple interpretation as a uniform angular velocity ($\Omega$ loses such a simple interpretation for vacuum energy type solutions, such as the pure vacuum Hartle-Thorne solution \cite{Hartle:1968si} and de Sitter like solutions\cite{UchiYoshida:2014,Uchikata:2015,Pani:2015,UchiPani:2016,2022PhRvD.105b4002B} because the four velocity drops out the energy momentum tensor \ref{Tfluid}) .
Note that in \cite{Hartle:1968si}, they define fractional changes
\begin{align}
& {\matheul P} = p(r) + (\rho+p)(\delta p_0(r) + \delta p_2(r) P_2(\cos\theta)),\\
& {\matheul E} = \rho(r) + \frac{d\rho}{dp}(\rho+p)(\delta p_0(r) + \delta p_2(r) P_2(\cos\theta)),
\end{align}
which are commonly used in other works. With Eq.~\eqref{Tfluid}, the Einstein tensor for Eq.~\eqref{kingmet}, appropriate boundary conditions, and the equation of state, one may solve for the perturbation functions.
\section{koosh}
The stringy equation of state $\Lambda_0=\Lambda_1$, $\Lambda_2=\Lambda_3=0$ is radically different than a perfect fluid equation of state $\Lambda_3=\Lambda_2=\Lambda_1=f(\Lambda_0)$. However, we find that if we examine the Einstein tensor order by order we can identify Hartle perturbation metrics which correspond to rotating kooshes and satisfy the stringy equation of state.
Keeping terms which are zeroth or first order in rotation, we find the metric is specified by
\begin{equation}
ds^2=-\kappa^2 dt^2+\frac{1}{\kappa^2}dr^2+ r^2 d\theta^2+r^2 \sin^2\theta d\phi^2-2 r^2\sin^2\theta\omega(r).
\label{kooshW1met}
\end{equation}
and the energy-momentum tensor has nonzero components
\begin{align}
T^t_{~t}=T^r_{~r}=-\frac{\lambda}{4\pi r^2},\\
T^t_{~\phi}=-\frac{r \sin ^2(\theta ) \left(4 \omega '(r)+r \omega ''(r)\right)}{16 \pi }\\
T^\phi_{~t}=-\frac{ \left(2 (2 \lambda -1) r \left(4 \omega '(r)+r \omega
''(r)\right)+8 \lambda \omega (r)\right)}{32 \pi r^2}
\end{align}
This suggests two special\footnote{As we will see in the next subsection, preservation of the equation of state is not enough to fully specify the frame dragging. However, because these examples for frame dragging lead to terms going to zero, the other equations simplify and exact solutions for the other functions can be found.} frame dragging solutions
\begin{align}
T^t_{~\phi}=0 \rightarrow4 \omega '(r)+r \omega ''(r)=0\rightarrow \omega(r)=W_1+\frac{W_2}{r^3}\label{vacuumdrag1} \\
T^\phi_{~t}=0\rightarrow \left(4 \omega '(r)+r \omega
''(r)\right)=\frac{-4 \lambda \omega (r)}{(2 \lambda -1) r}\rightarrow \omega(r)=\frac{W_a r^{-\sqrt{\frac{9-2 \lambda }{4-8 \lambda }}}+W_b
r^{\sqrt{\frac{9-2 \lambda }{4-8 \lambda }}}}{r^{3/2}}\label{alternatedrag1}
\end{align}
Notice that Eq. \eqref{vacuumdrag1} gives the same dragging as the vacuum Hartle-Thorne \cite{Hartle:1968si} and vacuum energy de-Sitter type solutions \cite{UchiYoshida:2014,Uchikata:2015,Pani:2015,UchiPani:2016,2022PhRvD.105b4002B}, we will therefore call it ``vacuum dragging". Interestingly, there is no Komar angular momentum Eq.~\eqref{KomarAngmom} associated with this frame dragging for the volume term of the Koosh as and we have $ T^t_{~\phi}=0$. Note that in the case of the Hartle-Thorne solution and the vacuum energy de-Sitter type solutions the $W_2$ term is associated with an angular momentum concentrated inside the region of interest (such as a rotating star in the standard Hartle-Thorne picture \cite{Hartle:1968si} or a delta function in \cite{2022PhRvD.105b4002B}). The $W_1$ term can be associated with angular momentum concentrated outside the region of interest, specifically arising from a rotating eternal shell for the vacuum energy de-Sitter type solutions considered in \cite{UchiYoshida:2014,Uchikata:2015,Pani:2015,UchiPani:2016,2022PhRvD.105b4002B}.
\subsection{preservation of the equations of state}
The full second order energy momentum tensor is
\begin{align}
T^t_{~t}=&\frac{-\lambda}{4\pi r^2}+\Bigg[\frac{r \left(2 \omega \left(r \omega ''+4 \omega '\right)+r \left(\omega
'\right)^2\right)}{48 \pi }-\frac{ m_0'}{4 \pi r^2}+P_2(\cos\theta)\Bigg(\frac{\kappa ^2
\left(r k_2''+3 k_2'\right)}{4 \pi r}-\frac{ 2k_2+ m_2'}{4 \pi r^2}-\frac{3 m_2 }{4 \pi \kappa ^2
r^3}\nonumber\\&-\frac{r \left(2 \omega \left(r \omega ''+4 \omega '\right)+r \left(\omega
'\right)^2\right)}{48 \pi }\Bigg)\Bigg]\\
T^t_{~\phi}=&-\{\frac{r \sin ^2(\theta ) \left(r \omega ''+4 \omega '\right)}{16 \pi }\}\\
T^\phi_{~t}=&-\{\frac{\left(4 \lambda \omega -\kappa^2 r \left(r \omega ''+4
\omega '\right)\right)}{16 \pi r^2}\}\\
T^\phi_{~\phi}=&\Bigg[\frac{\kappa ^2 r^2 \left(r h_0''+h_0'\right)-r m_0'+m_0}{8 \pi
r^3}+\frac{h_2 \kappa ^2 r+m_2}{8 \pi \kappa ^2 r^3}-\frac{r \left(2
\omega \left(r \omega ''+4 \omega '\right)+3 r \left(\omega
'\right)^2\right)}{48 \pi }+\frac{P_2(\cos\theta)}{8 \pi r^3} \Bigg(\nonumber\\& r \left( \kappa ^2 r \left(r
\left(h_2''+k_2''\right)+h_2'+2 k_2'\right)-4 h_2- m_2'+\frac{r^3 \omega '}{6}
\left(3 r \omega '+8 \omega \right)+ \frac{r^4 \omega \omega ''}{3}\right)+
m_2-\frac{4 m_2}{\kappa^2}\Bigg)\Bigg]\\
T^r_{~r}=&\frac{-\lambda}{4\pi r^2}+\Bigg[\frac{ \kappa ^2 r^2 h_0'- m_0}{4 \pi r^3}+\frac{r^2 \left(\omega '\right)^2}{48 \pi }-\frac{P_2(\cos\theta)}{4\pi r^3}\Bigg(3 h_2 r+2 k_2 r+ m_2+\frac{r^5
\left(\omega '\right)^2}{12}- \kappa ^2 r^2 (h_2'+k_2')\Bigg)\Bigg]\\
T^\theta_{~\theta}=&\Bigg[\frac{ \kappa ^2 r^2 (h_0'+ r h_0'')- r m_0'+ m_0}{8
\pi r^3}-\frac{ h_2 \kappa ^2 r+ m_2}{8
\pi \kappa ^2 r^3}-\frac{ r^2 \left(\omega '\right)^2}{48
\pi }+\frac{P_2(\cos\theta)}{8 \pi r^3}\Bigg(\nonumber\\& r \left( \kappa ^2 r^2 h_2''+ \kappa ^2 r h_2'+ \kappa ^2 r^2
k_2''+2 \kappa ^2 r k_2'- m_2'+\frac{r^4 \left(\omega '\right)^2}{6}\right)-2 h_2
r+ \left(1-\frac{2}{\kappa^2}\right) m_2\Bigg)\Bigg]\\
T^r_{~\theta}=&r^2\kappa^2 T^\theta_{~r}=\Bigg[\frac{3 \sin (2 \theta ) \left(\kappa ^2 r^2
\left(h_2'+k_2'\right)-r h_2 \kappa ^2 -m_2\right)}{16 \pi r^2}\Bigg
\end{align}
Here terms in big square brackets are second order, terms in curly brackets are first order, and we use the shorthand $\kappa$ from earlier.
For a stationary axisymmetric metric of the form Eq.~\eqref{axisymstat} the eigenvalues of the energy momentum tensor follow a pattern due to its block diagonal structure, and can be written as
\begin{align}
\Lambda_0=\frac{1}{2}\Big(T^t_{~t}+T^\phi_{~\phi}-\sqrt{(T^t_{~t}-T^\phi_{~\phi})^2-4T^\phi_{~t}T^t_{~\phi}}\Big)\\
\Lambda_3=\frac{1}{2}\Big(T^t_{~t}+T^\phi_{~\phi}+\sqrt{(T^t_{~t}-T^\phi_{~\phi})^2-4T^\phi_{~t}T^t_{~\phi}}\Big)\\
\Lambda_1=\frac{1}{2}\Big(T^r_{~r}+T^\theta_{~\theta}-\sqrt{(T^r_{~r}-T^\theta_{~\theta})^2-4T^\theta_{~r}T^r_{~\theta}}\Big)\\
\Lambda_2=\frac{1}{2}\Big(T^r_{~r}+T^\theta_{~\theta}+\sqrt{(T^r_{~r}-T^\theta_{~\theta})^2-4T^\theta_{~r}T^r_{~\theta}}\Big)
\end{align}
When expanded to second order, the eigenvalues of this energy-momentum tensor are
\begin{align}
\Lambda_0=&T^t_t-\frac{r \sin^2(\theta)(4\omega'+r\omega'')(4 \lambda \omega-\kappa^2 r (r \omega ''+4\omega '))}{64\pi\lambda}\\
\Lambda_1=&T^r_{~r}\\
\Lambda_2=&T^\theta_{~\theta}\\
\Lambda_3=&T^\phi_{~\phi}+\frac{r \sin^2(\theta)(4\omega'+r\omega'')(4 \lambda \omega-\kappa^2 r (r \omega ''+4\omega '))}{64\pi\lambda}
\end{align}
To this second order, we have $\Lambda_1=T^r_{~r}$ and $\Lambda_2=T^\theta_{~\theta}$ because the effects from the $T^r_{~\theta}T^\theta_{~r}$ cross term are pushed to higher order.
One expression for eigenvectors in $(t,r,\theta,\phi)$ coordinates, correct to second order, is
\begin{align}
x_0^\mu=\left(1,0,0,\omega -\frac{\kappa ^2 r \left(r \omega ''+4 \omega
'\right)}{4 \lambda }\right)\\
x_1^\mu=\left(0,1,\frac{3 \sin (2 \theta ) \left(m_2-\kappa ^2 r \left(r
\left(h_2'+k_2'\right)-h_2\right)\right)}{4 \kappa ^2 \lambda
r^2},0\right)\\
x_2^\mu=\left(0,-\frac{3 \sin (2 \theta ) \left(m_2-\kappa ^2 r(
r\left(h_2'+k_2'\right)-h_2 )\right)}{4 \lambda
},1,0\right)\\
x_3^\mu=\left(-\frac{r^3 \sin ^2(\theta ) \left(r \omega ''+4 \omega '\right)}{4
\lambda },0,0,1\right)
\end{align}
Note that the $x_0^\mu,x_3^\mu$ have the zeroth order term and a first order term while the $x_1^\mu,x_2^\mu$ have the zeroth order term and a second order term. Notice that in the vacuum dragging Eq.~\eqref{vacuumdrag1} case $x_3^\mu$ is purely along the $\phi$ direction and in the alternate frame dragging case $x_0^\mu$ is purely along the $t$ direction.
Now that we have expressions for the components and eigenvalues in the energy momentum tensor for arbitrary perturbation functions, we can now use a form of the stringy equation of state \ref{stringeos}, being $\Lambda_2=0,\Lambda_3=0,\Lambda_0-\Lambda_1=0$, to derive differential equations which the perturbation functions should follow. Notice that each of these equations will separate into a monopole term and a quadrupole term.
Since $\Lambda_2$ and $\Lambda_3$ are seperately zero, if we take their difference we also obtain $\Lambda_2-\Lambda_3=0$, which leads to the algebraic condition allowing for the elimination of $h_2$
\begin{align}
h_2= \frac{- m_2}{\kappa ^2 r}-\frac{ r^4 \left(4 (7 \lambda -4) \left(\omega
'\right)^2-\kappa ^2 r^2 \left(\omega ''\right)^2-8 \kappa
^2 r \omega ' \omega ''\right)}{24 \lambda }
\label{h2rule}.
\end{align}
Next, the equation of state \eqref{stringeos} requires the condition $\Lambda_0-\Lambda_1-2\Lambda_2=0$, using the quadrupole part of this condition and Eq.~\eqref{h2rule} gives a differential equation
\begin{align}
m_2=&-\frac{\kappa ^2 r^2}{72 \lambda } \Big(-4 \kappa ^2 r^4 \omega ' \left((31 \lambda -16) r
\omega ^{(3)}-\kappa ^2
r^2 \omega ^{(4)}+2 (65 \lambda -33) \omega
''\right)+\nonumber\\&\kappa ^4 r^5 \left(r^2 (\omega ^{(3)})^2+r
\omega '' \left(26 \omega ^{(3)}+r \omega
^{(4)}\right)+83 \left(\omega ''\right)^2\right)-12
\lambda m_2''+16 (5 \lambda -2) (7 \lambda -4) r^3 \left(\omega
'\right)^2\Big), \label{m2rule}
\end{align}
where $(3),(4)$ specify 3rd and 4th order derivatives with $r$, which specifies $m_2$. Finally, using the quadrupole part of $\Lambda_0-\Lambda_1=0$ gives a differential equation for $k_2$
\begin{align}
12 \kappa ^4 \lambda r^2 \left(r k_2''+2 k_2'\right) =&72 \lambda m_2-\kappa ^2 r^5 \Big(-2 (\lambda (56 \lambda -47)+8)
\left(\omega '\right)^2+2 \kappa ^2 (7 \lambda -3) r^2 \left(\omega
''\right)^2-\nonumber\\&\kappa ^4 r^2 \omega^{(3)}\left(r \omega
''+4 \omega '\right)-4 (\lambda (34 \lambda -31)+7) r \omega ' \omega
''\Big) \label{k2rule}
\end{align}
For the monopole functions, we can use the remaining monopole term from $\Lambda_0-\Lambda_1=0$ to obtain an expression for $h_0'$, being
\begin{align}
h_0'=\frac{ m_0- r m_0'}{ \kappa ^2 r^2}+\frac{ r^3 \left(r \omega ''+4
\omega '\right)^2}{24 \lambda }
\label{h0rule}
\end{align}
If we take the derivative of this we can replace both $h_0'$ and $h_0''$ in the other monopole equation (with the condition Eq.~\eqref{h2rule}, $\Lambda_2=0$ and $\Lambda_3=0$ have identical forms), we obtain a final differential equation for $m_0$
\begin{align}
m_0''=-\frac{r^3}{24 \lambda } \left(8 (13 \lambda -6) \left(\omega '\right)^2-\kappa ^2 r^2
\omega '' \left(2 r \omega^{(3)}+13 \omega ''\right)-8 \kappa ^2 r
\omega ' \left(r \omega^{(3)}+8 \omega ''\right)\right).
\label{m0rule}
\end{align}
Importantly, the stringy equation of state can now be satisfied (to second order) regardless of what $\omega$ is, provided the second order functions obey the above rules. In theory, a system with any given function for $\omega$ could be used with Eqs. (\ref{h2rule},\ref{m2rule},\ref{k2rule},\ref{h0rule},\ref{m0rule}) to find second order functions such that the equation of state is preserved.
Presumably $\omega$ could be determined by some ansatz about the form of the energy momentum tensor beyond the equation of state, such as the uniform angular velocity assumption in the standard Hartle framework.
\section{example frame draggings}
Despite the fact that the preservation of the equation of state is not sufficient to specify the frame dragging, we have isolated two cases which have interesting behavior, being the vacuum frame dragging case Eq.~\eqref{vacuumdrag1} and the alternate case Eq.~\eqref{alternatedrag1}. In both of these cases we find exact solutions to the differential equations for the perturbation functions.
\subsection{vacuum frame dragging}
For our first example, consider that the frame dragging is of the ``vacuum" like form Eq.~\eqref{vacuumdrag1}. Based on the behavior of the vacuum like frame dragging in other systems, we might expect this solution to apply when the Komar angular momentum density is concentrated inside (for $W_2$) or outside (for $W_1$) the region of spacetime in which the background metric \eqref{uniaxialmet} applies.
In the vacuum frame dragging case, the Eq.~\eqref{m0rule} simply becomes
\begin{align}
m_0''=-\frac{3 W_2^2}{r^5}
\end{align}
which may be trivially integrated to obtain
\begin{align}
m_0=-\frac{W_2^2}{4r^3}+z_1 r+z_2\label{m0vac}
\end{align}
where $z_n$ are integration constants for the monopole functions. With $m_0$, the Eq.~\eqref{h0rule} likewise simplifies and may be trivially intergrated, giving
\begin{align}
h_0'=\frac{1}{\kappa^2}\Big(\frac{z_2}{r^2}-\frac{W_2^2}{r^5}\Big)\\
h_0=\frac{1}{\kappa^2}\Big(-\frac{z_2}{r}+\frac{W_2^2}{4r^4}\Big)+z_3.
\end{align}
This fully specifies the monopole functions.
The quadrupole functions are slightly more complicated. Eq.~\eqref{m2rule} leads to
\begin{align}
m_2=\frac{1}{6} \kappa ^2 r^2 m_2''+\frac{2 \kappa ^2 (3 \lambda -1)
W_2^2}{r^3},\\
m_2=q_1 r^{\frac{1}{2} \left(1-S\right)}+q_2 r^{\frac{1}{2} \left(1+S\right)}+\frac{2 \kappa ^2 (3 \lambda -1) W_2^2}{(4 \lambda
-1) r^3}\label{m2vac}
\end{align}
where we introduce the shorthand $S=\sqrt{\frac{24}{\kappa^2}+1}$ and the integration constants for the quadrupole functions $q_n$. We use Eqs.~\eqref{h2rule} to obtain the expression for $h_2$
\begin{align}
h_2=\frac{3W_2^2}{2r^4}-\frac{m_2}{\kappa^2 r},
\end{align}
where $m_2$ takes the form from Eq.~\eqref{m2vac}. Finally, we obtain
\begin{align}
12 \kappa ^4 \lambda r^2 \left(2 k_2'+r k_2''\right)=\frac{18 \lambda \left(2 \lambda (8 \lambda -5)+\frac{1}{4 \lambda
-1}\right) W_2^2}{r^3}-72 \lambda \left(q_1
r^{\frac{1-S}{2}}+q_2 r^{\frac{S+1}{2}}\right),\\
k_2=\frac{q_1 r^{-\frac{S+1}{2}}}{\kappa ^2}+\frac{q_2
r^{\frac{S-1}{2}}}{\kappa ^2}+\frac{q_3}{r}+q_4+\frac{(4 (3-8
\lambda ) \lambda +1)W_2^2}{8 \left(1-4\lambda\right)\kappa^2 r^4}
\end{align}
The nonzero eigenvalue becomes
\begin{align}
\Lambda_0=&\Lambda_1=\frac{-\lambda}{4\pi r^2}+\Bigg[-\frac{z_1}{4\pi r^2}+\nonumber\\&\frac{P_2(\cos \theta)}{2 \pi r^2} \left(\frac{q_3 (-3+2 \lambda )}{2
r}-q_4+\frac{\left(q_1 r^{-\frac{1}{2} (1+S)}+q_2
r^{\frac{S-1}{2}}\right) \lambda }{\kappa ^2}+\frac{W_2^2 \lambda
(-1+4 (3-4 \lambda ) \lambda )}{4 r^4 \left(1-4\lambda\right)\kappa^2}\right)\Bigg]\label{vaceig}
\end{align}
Notice how $W_1,~z_2,~z_3$ do not show up in the energy momentum tensor eigenvalue. The entire energy-momentum tensor simplifies considerably, becoming
\begin{align}
T^{t}_{~t}=T^{r}_{~r}=\Lambda_0=\Lambda_1\\
r^2\kappa^2T^{\theta}_{~r}= T^{r}_{~\theta}=\Bigg[\frac{-3\sin(2\theta)(W_2^2\lambda+q_3 r^3\kappa^2)}{16\pi r^5}\Bigg]\\
T^{\phi}_{~t}=\frac{-\lambda(W_2+r^3 W_1)}{4\pi r^5},
\end{align}
with all other components vanishing to this order.
The $S_G$ and $\mathcal{J}$ scalars are
\begin{align}
S_G=&\Bigg[\frac{ \log (\kappa )}{\kappa } \left(\frac{z_2}{ r^2}-\frac{W_2^2}{
r^5}+P_2(\cos\theta) \left(\frac{1}{2} q_1 r^{-\frac{3+S}{2} }
(1+S)-\frac{1}{2} q_2 r^{\frac{S-3}{2}} (S-1)+\frac{2W_2^2
\kappa ^2}{r^5 (1-4 \lambda )}\right)\right)\Bigg]\label{SG1}\\
\mathcal{J}=&\Big\{\frac{-3 W_2 \kappa \sin^2\theta}{2r^2}\Big\}\label{J1}
\end{align}
It is noteworthy that the surface gravity parameter $S_G$ of the unperturbed koosh is zero because the $e^{2\nu}$ goes to a constant, so its derivative is zero (the Komar mass of the unperturbed koosh is also zero, as for the unperturbed koosh $S_G,\omega$ and $\mathcal{J}$ being integrated in Eq.~\eqref{KomarMass} are all separately zero). For the system with vacuum frame dragging, the $S_G$ is second order in rotation and $\mathcal{J}$ is first order in rotation.
\subsubsection{divergences of perturbation functions in vacuum frame dragging}
If the integration constants $W_2,~z_2,~q_1,~q_3$ are nonzero, then there will be a divergence of the metric perturbation functions as $r\rightarrow0$. There will likewise be divergences as $r\rightarrow\infty$ in the quadrupole sector if $q_2$ is nonzero. There is a $r\rightarrow\infty$ divergence in the function $m_0$ if the integration constant $z_1$ is present , but it is noteworthy that $m_0$ enters into the metric \eqref{kingmet} in the combination $m_0/(r-2m)=m_0/(\kappa^2r)$, which has no divergences associated with $z_1$. Note however that for a given system any of these terms might be present if the background metric \eqref{uniaxialmet} is only valid over a certain domain, as is the case with global monopoles (which differs at small $r$) or something that is cut off by a shell (which would differ at large $r$). The integration constants $W_1,z_3,q_4$ do not cause any divergences in the metric functions. Strictly speaking, there is a divergence in the nonzero eigenvalue of the energy momentum tensor Eq.~\eqref{vaceig} associated with $q_4$ as $r\rightarrow0$, but it has the same $1/r^2$ divergence as the background $\lambda/4\pi r^2$ term, so the $q_4$ term can still be considered small with respect to the background term. A similar formally divergent but small compared to the background term in Eq.~\eqref{vaceig} comes from $z_1$. The $q_1,q_3,W_2$ terms in Eq.~\eqref{vaceig} all diverge faster than the background term as $r\rightarrow0$, and the $q_2$ term will diverge as $r\rightarrow\infty$.
Independently of the divergences at large or small radii, there are particular values of $\lambda$ in which there are divergences. It is not surprising, given the pathological nature of the background line element \eqref{uniaxialmet} in the $\lambda\rightarrow1/2,\kappa^2\rightarrow0$ quasiblack hole limit, that there are a multitude of divergences in the perturbation functions, associated with $W_2,z_2,q_1,q_2$. However, divergences due to these terms also show up in the eigenvalue Eq.~\eqref{vaceig} and the surface gravity parameter Eq.~\eqref{SG1}, which are scalars, showing that divergences caused by these integration constants in the quasiblack hole limit are not simply a coordinate artifact. Elimination of $W_2$ means that the frame dragging inside the quasiblack hole would go to a constant, which based on the behavior in the de Sitter type solutions might indicate the angular momentum is localized to the shell.
A more surprising feature is the fact that for nonzero $W_2$, the $m_2$ function (and hence $h_2$) will diverge at all radii when $\lambda=1/4$, in which the background metric exhibits no special behavior. This divergence at $\lambda=1/4$ also manifests in the $W_2$ term of Eq.~\eqref{vaceig}, and in the surface gravity parameter Eq.~\eqref{SG1}, indicating it has coordinate independent significance. A summary of divergences in scalar quantities under different conditions and the associated integration constants is given in Table \ref{divtable1}.
\begin{table}[]
\begin{center}
\begin{tabular}{ |c|c|c|c|c| }
~~~&$r=0$& $r=\infty$ & $\lambda=1/2$ & $\lambda=1/4$\\
\hline
$\Lambda$&$W_2,q_1,q_3$;$z_1^*,q_4^*$ &$q_2$ &$W_2,q_1,q_2$ &$W_2$ \\
\hline
$S_G$& $W_2,z_2,q_1$& $q_2$&$W_2,z_2,q_1,q_2$& $W_2$\\
\hline
$\mathcal{J}$&$W_2$ & & & \\
\hline
$\omega$& $W_2$& & & \\
\end{tabular}
\caption{Table showing integration constants which cause divergences in scalar quantities given certain conditions. An asterisk indicates that, while a divergence is formally present, it diverges in the same manner as the background and remains subdominant, $z_1$ and $q_4$ are associated with divergences of this type. $W_1$ and $z_3$ do not lead to divergences in any of these quantities. \label{divtable1}}
\end{center}
\end{table}
\subsection{Other frame dragging}
Recall that there is a second special frame dragging with regard to the off diagonal first order in rotation components, being Eq.~\eqref{alternatedrag1}. To simplify the notation, we introduce the shorthand
\begin{align}
Z=\sqrt{\frac{9-2 \lambda }{1-2 \lambda }}
\end{align}
such that
\begin{align}
\omega=\frac{W_a r^{-Z/2}+W_b
r^{Z/2}}{r^{3/2}}.
\end{align}
With this in hand, one may solve the monopole equations Eqs.~(\ref{m0rule},\ref{h0rule}) and obtain
\begin{align}
m_0=\frac{1}{24} W_b^2 (Z-3) r^Z-\frac{1}{24} W_a^2 (Z+3) r^{-Z}+r
z_a+z_b\label{m0alt}\\
h_0= \frac{W_a^2 (Z+3) r^{-Z-1}}{6 \kappa ^2 (Z+1)}+\frac{W_b^2 (Z-3)
r^{Z-1}}{6 \kappa ^2 (Z-1)}-\frac{8 \lambda W_a W_b+6 \kappa
^2 z_b}{6 \kappa ^4 r}+z_c
\end{align}
The solutions to the quadrupole equations (\ref{m2rule},\ref{h2rule},\ref{k2rule}) are
\begin{align}
m_2=&q_a r^{\frac{S+1}{2}}+q_b r^{\frac{1-S}{2}}+\frac{1}{6}
W_a^2 r^{-Z} (-2 \lambda (Z+1)+Z+9)+\frac{1}{6} W_b^2 r^Z (2
\lambda (Z-1)-Z+9)\label{m2alt}\\
h_2=& \frac{\frac{1}{12} W_a^2 r^{-Z-1} (2 \lambda -2 \lambda
Z+Z-9)+\frac{1}{12} W_b^2 r^{Z-1} (2 \lambda +(2 \lambda -1)
Z-9)- q_a r^{\frac{S-1}{2}}-q_b r^{\frac{-S-1}{2}
}}{\kappa ^2}\\
k_2= &\frac{q_a r^{\frac{S}{2}-\frac{1}{2}} (-2 \lambda +(2 \lambda -1)
S+25)}{\kappa ^4 (S-1) S}-\frac{q_b r^{\frac{1}{2} (-S-1)} (2
\lambda +(2 \lambda -1) S-25)}{\kappa ^4 S
(S+1)}-\frac{q_c}{r}+q_d+\nonumber\\&\frac{W_a^2 r^{-Z-1} \left(16
\lambda ^2 (Z+1)-10 \lambda (9+7Z)-9( 5Z-9)\right)}{12 \kappa ^2 (2
\lambda -9) (Z+1)}+\frac{W_b^2 r^{Z-1} \left(16 \lambda ^2
(Z-1)+10\lambda (9-7 Z)-9 (5 Z+9)\right)}{12 \kappa ^2 (2 \lambda -9)
(Z-1)}\nonumber\\&-\frac{4 \lambda W_a W_b}{3 \kappa ^4 r}-\frac{4
\lambda W_a W_b \log (r)}{3 \kappa ^4 r}
\end{align}
where $S=\sqrt{\frac{24}{\kappa^2}+1}$ as before. Notice that the $q_a$ and $q_b$ terms in Eq. (\ref{m2vac}) and the $q_1,~q_2$ terms in (\ref{m2alt}) are analogous, as are the $z_1,~z_2$ and $z_a,~z_b$ terms in Eqs. (\ref{m0vac}) and (\ref{m0alt}). This is because they are associated with the homogeneous, or $\omega=0$, equations for $m_0$ and $m_2$, which become
\begin{align}
\text{Eq.(\ref{m0rule})}\rightarrow m_0''=0\\
\text{Eq.(\ref{m2rule})}\rightarrow m_2=\frac{\kappa^2 r^2}{6}m_2''
\end{align}
Because the terms are associated with $q_a,~q_b$ and $z_a,z_b$ terms are associated with the homogeneous $m_n$ equations analogous terms can show up in the $m_n$ functions for all frame draggings.
The nonzero eigenvalue of the energy-momentum tensor becomes
\begin{align}
&\Lambda_0=\Lambda_1=\frac{-\lambda}{4\pi r^2}+\Bigg[\frac{W_a W_b \lambda }{6 \pi r^3
\kappa ^2}-\frac{z_a}{4 \pi r^2}+\frac{\lambda \left(r^{-3+Z} W_b^2+r^{-3-Z}
W_a^2\right)}{12 \pi \kappa ^2}+P_2(\cos \theta)\Bigg( \frac{q_c (3-2 \lambda )}{4 \pi
r^3}-\frac{q_d}{2 \pi r^2}\nonumber\\&+\frac{W_a W_b \lambda (5-2 \lambda +(6-4 \lambda ) \log
(r))}{6 \pi r^3 \kappa ^4}+\frac{\lambda}{2\pi
\kappa ^2}\Big(q_a r^{(S-5)/2}+q_b r^{-(S+5)/2}\Big)+\nonumber\\&\frac{16\lambda(5-2\lambda)}{\kappa^2} \left(\frac{r^{-3-Z}
W_a^2}{96\pi(1+Z)}-\frac{r^{-3+Z} W_b^2}{96\pi(Z-1)}\right)+\frac{16 Z \lambda(3+4\lambda(\lambda-5))}{9+4\lambda(\lambda-5)}
\left(\frac{r^{-3-Z} W_a^2}{96\pi(1+Z)}+\frac{r^{-3+Z}
W_b^2}{96\pi(Z-1)}\right)\Bigg)\Bigg]
\label{altereig}
\end{align}
The energy momentum tensor follows the structure
\begin{align}
& T^{t}_{~t}=T^{r}_{~r}=\Lambda_0=\Lambda_1,\\
& r^2\kappa^2T^{\theta}_{~r}= T^{r}_{~\theta}=\frac{\sin(2\theta)}{8\pi r^2}\Bigg[\frac{3 q_c \kappa ^2}{2}+\frac{\left(3 \left(S-25+(S-1) \left(S \kappa
^2-2 \lambda \right)\right)\right) \left(q_a
r^{\frac{1+S}{2}}+q_b r^{\frac{1-S}{2}}\right)}{4 S \kappa
^2}+\nonumber\\&\frac{2 W_a W_b \lambda \log (r)}{\kappa ^2}-\frac{3\lambda}{4}
\left(r^{-Z} W_a^2+r^Z
W_b^2\right)+\frac{Z}{4} \left(\frac{2\lambda^2-13\lambda}{9-2 \lambda }\right) \left(r^{-Z} W_a^2-r^Z
W_b^2\right)\Bigg],\\
& T^{t}_{~\phi}=-\frac{\left(\lambda \sin ^2(\theta )\right) \left(W_a
r^{-\frac{3}{2}-\frac{Z}{2}}+W_b
r^{-\frac{3}{2}+\frac{Z}{2}}\right)}{4 \pi \kappa ^2},
\end{align}
with all other components vanishing to this order. Notice that in this case as well as in the vacuum dragging case, we had $T^\phi_{~\phi}=0$ and $T^t_{~t}=\Lambda_0$. This is because we either had one or the other of $T^\phi_{~t}$ or $T^t_{~\phi}$ as zero for the particular frame dragging function. For other frame dragging functions for which the product $T^\phi_{~t}T^t_{~\phi}\ne0$, we no longer require $T^t_{~t}=\Lambda_0$ and $T^\phi_{~\phi}=0$ to satisfy the general expressions.
The $S_G$ and $\mathcal{J}$ scalars become
\begin{align}
S_G=&\frac{\log(\kappa)}{\kappa}\Bigg[\frac{1}{6} r^{-2+Z} W_b^2 (Z-3)-\frac{1}{6} r^{-2-Z} W_a^2
(3+Z)+\frac{z_b}{r^2}+\frac{4 \lambda W_a W_b}{3 r^2
\kappa ^2}+P_2(\cos\theta) \Big(\frac{1}{2} q_a r^{-\frac{3}{2}+\frac{S}{2}}
(1-S)\nonumber\\&+\frac{1}{2} q_b r^{-\frac{3}{2}-\frac{S}{2}}
(1+S)+\frac{1}{12} \left((Z-1) \kappa ^2-8\right) \left(r^{-2-Z}
W_a^2 (-1-Z)+r^{-2+Z} W_b^2 (1+Z)\right)\Big)\Bigg]\label{altersg}\\
\mathcal{J}=&\frac{1}{4} \kappa \sin ^2(\theta ) \left(W_b (Z-3)
r^{\frac{Z-1}{2}}-W_a (Z+3) r^{\frac{1}{2} (-Z-1)}\right)
\end{align}
\subsubsection{divergences in the alternate frame dragging}
Because the functions in the alternate frame dragging involve more notational shorthand to write in a reasonable amount of space, it is helpful to review how the different shorthand parameters are related to $\lambda$ before examining what divergences may exist. Plots of the shorthand paramaters are depicted in Figure \ref{params}, where we see that $5\leq S\le\infty$ and $3\leq Z\le\infty$.
\begin{figure}
\centering
\includegraphics{shorthandplot1.pdf}
\caption{Graph showing the behavior of the shorthand items with respect to the parameter $\lambda$. Over the interval $0\le\lambda<1/2$ that we are interested in, $\kappa^2$ goes from 1 to 0, $Z$ goes from 3 to $\infty$, and $S$ goes from 5 to $\infty$}
\label{params}
\end{figure}
There are many terms in the perturbation functions and energy-momentum tensor components which have powers of $\kappa$ in denominators, which would of course lead to divergences in the quasiblackhole limit. Notice that within the eigenvalue Eq.~\eqref{altereig} and within the surface gravity parameter Eq.~\eqref{altersg}, both $W_a$ and $W_b$ have to be zero in order for these scalars to not diverge at all radii in the quasiblackhole limit, which implies no frame dragging could be present, or that the alternate frame dragging is not really compatible with a quasiblackhole configuration. The other terms in denominators i.e. $Z-1,Z+1,S,S-1,S+1,S^2-1,2\lambda-9$ do not have a zero over the $0\le\lambda<1/2$ interval, so there are no intermediate values of $\lambda$ causing divergences like $\lambda=1/4$ did for the vacuum dragging case.
There are however terms which diverge either as $r\rightarrow0$ or $r\rightarrow\infty$. Because $Z\ge3$, the presence of nonzero $W_a$ will cause divergences as $r\rightarrow0$ in the frame dragging, all second order functions, and all nonzero energy-momentum tensor components for general $\lambda$. The $W_b$ terms are typically associated with divergences as $r\rightarrow\infty$. However, for $\lambda=0,Z=3$, the $W_b$ term in frame dragging goes to a constant, and the $W_b$ terms in the monopole perturbation functions and energy-momentum tensor components/ eigenvalues vanish, but the quadrupole perturbation functions still have divergent $W_b$ terms as $r\rightarrow\infty$. Among the monopole integration constants, $z_a$ causes a divergence in $m_0$ as $r\rightarrow\infty$ (note the caveat that the combination in the line element $m_0/(r-2m)=m_0/(\kappa^2r)$ remains finite since $z_a$ and $z_1$ are analogous terms from the homogeneous solution) and a formally divergent but small compared to the background term in Eq.~\eqref{altereig}, $z_b$ causes a divergence in $h_0$ as $r\rightarrow0$, and $z_c$ does not cause any divergences. Among the quadrupole constants, $q_a$ is associated with divergences as $r\rightarrow\infty$, $q_b$ and $q_c$ are associated with divergences as $r\rightarrow0$, and $q_d$ causes another formally divergent but small compared to the background term in the eigenvalue Eq.~\eqref{altereig}. We give a table summarizing divergences for the alternate frame dragging in Table \ref{divtable2}. It is important to reiterate that one may not care about divergent terms if they occur outside the domain where the background solution would be valid, such as $r\rightarrow \infty$ for objects cut off by a shell or $r \rightarrow 0$ for global monopole like objects with a smoothed core.
\begin{table}[]
\begin{center}
\begin{tabular}{ |c|c|c|c| }
~~~&$r=0$& $r=\infty$ & $\lambda=1/2$ \\
\hline
$\Lambda$&$W_a,W_a\times W_b,q_b,q_c$;$z_a^*,q_d^*$ &$W_b,q_a$ & $W_a,W_b,W_a\times W_b,q_a,q_b$ \\
\hline
$S_G$&$W_a,W_a\times W_b,z_b,q_b$ &$W_b,q_a$ &$W_a,W_b,W_a\times W_b,z_b,q_a,q_b$\\
\hline
$\mathcal{J}$&$W_a$ & $W_b$&\\
\hline
$\omega$& $W_a$& $W_b$ & \\
\end{tabular}
\caption{Table showing integration constants which cause divergences in scalar quantities given certain conditions. An asterisk indicates that, while a divergence is formally present, it diverges in the same manner as the background and remains subdominant. Unlike the previous case, both frame dragging $W$ constants are associated with divergences, and we also have divergent terms proportional to the product $W_a\times W_b$. Additionally, there is no intermediate value of $\lambda$ associated with divergences in these quantities. Corresponding behaviour with respect to conditions for divergences of the integration constants from the homogeneous sector of the equations ($z_a,z_b,q_a,q_b$ versus $z_1,z_2,q_1,q_2$) is evident in comparing the results with Table \ref{divtable1}. The integration constant $z_c$ has no associated divergences in these quantities. \label{divtable2}}
\end{center}
\end{table}
\section{Conclusion}
Within general relativity, rotating axisymmetric systems are of considerable interest. In this paper we show that a modified Hartle formalism (using the same form of perturbed metric but different conditions on the energy-momentum tensor) is capable of producing rotating solutions in a perturbative framework which preserve heavily anisotropic equations of state far different from the original application of perfect fluids. This is accomplished by deriving differential equations for the second order perturbation functions and presenting closed form solutions to a system which could be interpreted as describing a region of some global monopole or string cloud with rotation, although additional information beyond the equation of state is required to specify what frame dragging will be appropriate in a given physical situation. Examining specific physical situations and attempting to determine the appropriate frame dragging function (and by extension the other functions) is a possible avenue for future work.
One other possible extension of this work is examination of slowly rotating nonlinear electrodynamics monopoles. In the static case, Bardeen type nonsingular black holes can also arise from nonlinear electrodynamics theories (see e.g. \cite{AyonBeato:1998ub,AyonBeato:2000zs,Balart:2014cga} ) because of their [(11)(1,1)] Segre type. It is also known that rotating versions of these black holes and nonlinear electrodynamics monopoles can be generated by the Newman-Janis algorithm (for instance \cite{2016EPJC...76..273A,Bambi:2013ufa}), but these rotating versions typically no longer follow the equation of state associated with the underlying nonlinear electrodynamics theory supposed to generate the static version \cite{Lombardo_2004,Beltracchi2021a}. However, the koosh is also a Segre type [(11)(1,1)] system, can describe a monopole in a particular nonlinear electrodynamics theory \cite{2019arXiv191008166B}, and a modified version of the Hartle formalism was able to give rotating solutions which preserved its equation of state. This gives hope that, at least within the slowly rotating nearly spherical limit, rotating solutions for Bardeen type black holes or other nonlinear electrodynamics systems which satisfy the underlying equation of state may be found in a similar manner.
\centerline{\bf Acknowledgement}
I would like to thank Paolo Gondolo and Emil Mottola for suggestions on this manuscript.
|
1,116,691,497,769 | arxiv | \section{Introduction}
In differential geometry, the Ricci flow is an intrinsic geometric flow.
It is a process of deformation of the Riemannian metric, independent of any embedding or immersion.
The Ricci flow is given by an evolution equation for the metric and the associated Ricci tensor on a Riemannian manifold depending on a variable called time.
Ricci flows, formally analogous to the diffusion of heat, are among the important tools of theoretical physics.
In 1982, R.\,S. Hamilton \cite{Ham82} introduced the concept of Ricci flow and proved its
existence. This idea is contributed to the proof of Thurston's Geometrization Conjecture and consequently the Poincar\'e Conjecture.
Ricci soliton is a special solution of the Ricci flow equation and it
moves only by a one-parameter family of diffeomorphism and scaling.
Ricci solitons represent a natural generalization of Einstein metrics on a Riemannian
manifold, being generalized fixed points of Hamilton's Ricci flow, considered as a dynamical system.
In 2008, R. Sharma \cite{Shar} initiated the study of Ricci solitons in contact Riemannian geometry, in particular on K-contact manifolds. After that
Ricci solitons have been studied on different kinds of almost contact metric manifolds, e.g.
$\al$-Sasakian \cite{IngBag}, trans-Sasakian \cite{BagIng13}, Kenmotsu
\cite{BagIngAsh13}, \cite{NagPre}, nearly Kenmotsu \cite{AyaYil}, $f$-Kenmotsu \cite{GalCra}, etc.
In paracontact geometry, Ricci solitons are first investigated
by G. Calvaruso and D. Perrone \cite{CalPer}.
As a generalization of a Ricci soliton, the notion of $\eta$-Ricci soliton has been introduced by J.\,T. Cho and M. Kimura \cite{ChoKim}.
In the context of
paracontact geometry $\eta$-Ricci solitons have been investigated in
\cite{Bla15}, \cite{Bla16}, \cite{BlaCra}, \cite{PraHad}.
Although this topic was first studied in Riemannian geometry,
in recent years Ricci solitons and their generalizations
have also been studied in pseudo-Riemannian manifolds,
mostly with Lorentzian metrics
(\cite{BagIng12}, \cite{Bla16}, \cite{BlaPer}, \cite{BlaPerAceErd}, \cite{Mat}).
Usually, an almost contact manifold is equipped with
a metric called compatible, which can be Riemannian or pseudo-Riemannian.
Then, the contact endomorphism
acts as an isometry with respect to the
metric on the contact distribution.
The alternative possibility is when the contact endomorphism
acts as an anti-isometry.
In this case, the metric is necessarily pseudo-Riemannian and it is known as a B-metric.
Other important characteristic of almost contact B-metric structure which differs
it from the metric one is that the associated $(0,2)$-tensor of the B-metric is also a
B-metric.
Thus, it is obtained an almost contact B-metric manifold.
In another point of view, it is an almost contact complex Riemannian manifold.
The differential geometry of almost contact B-metric manifolds has been studied since 1993 \cite{GaMiGr}, \cite{ManGri93}.
In the present paper, our goal is to introduce and investigate a generalization of the Ricci soliton
compatible with the almost contact B-metric structure, and its potential to be the Reeb vector field.
We are studying these objects on some important kinds of manifolds under consideration: Einstein-like,
Sasaki-like and having a torse-forming Reeb vector field.
In relation with the proved assertions, we comment explicit examples in dimension 3 and 5 of Lie groups
with the structure studied.
\section{Almost Contact B-Metric Manifolds}
Let us consider an \emph{almost contact B-metric manifold} denoted by $\M$. This means that $M$
is a $(2n+1)$-dimensional differentiable manifold with an almost
contact structure $(\f,\xi,\eta)$, where $\f$ is an endomorphism
of the tangent bundle $TM$, $\xi$ is a Reeb vector field and $\eta$ is its dual contact 1-form. Moreover,
$M$ is equipped with a pseu\-do-Rie\-mannian
metric $g$ of signature $(n+1,n)$, such that the following
algebraic relations are satisfied: \cite{GaMiGr}
\begin{equation}\label{strM}
\begin{array}{c}
\f\xi = 0,\qquad \f^2 = -\I + \eta \otimes \xi,\qquad
\eta\circ\f=0,\qquad \eta(\xi)=1,\\
g(\f x, \f y) = - g(x,y) + \eta(x)\eta(y),
\end{array}
\end{equation}
where $\I
$ is the identity transformation on $\Gamma(TM)$. In the latter equality and further, $x$, $y$, $z$, $w$ will stand for arbitrary elements of $\Gamma(TM)$ or vectors in the tangent space $T_pM$ of $M$ at an arbitrary
point $p$ in $M$.
Some immediate consequences of \eqref{strM} are the following equations
\begin{equation}\label{strM2}
\begin{array}{ll}
g(\f x, y) = g(x,\f y),\qquad &g(x, \xi) = \eta(x),
\\
g(\xi, \xi) = 1,\qquad &\eta(\n_x \xi) = 0,
\end{array}
\end{equation}
where $\n$ is the Levi-Civita connection of $g$.
The associated metric $\g$ of $g$ on $M$ is defined by
\(\g(x,y)=g(x,\f y)+\eta(x)\eta(y)\). The manifold
$(M,\f,\xi,\eta,\g)$ is also an almost contact B-metric manifold.
The B-metric $\g$ is also of signature $(n+1,n)$.
The Levi-Civita connection of $\g$ is denoted by
$\nn$.
A classification of almost contact B-metric manifolds, which contains eleven basic classes $\F_1$, $\F_2$, $\dots$, $\F_{11}$, is given in
\cite{GaMiGr}. This classification is made with respect
to the tensor $F$ of type (0,3) defined by
\begin{equation}\label{F=nfi}
F(x,y,z)=g\bigl( \left( \nabla_x \f \right)y,z\bigr).
\end{equation}
The following identities are valid:
\begin{equation}\label{F-prop}
\begin{array}{l}
F(x,y,z)=F(x,z,y)\\
\phantom{F(x,y,z)}
=F(x,\f y,\f z)+\eta(y)F(x,\xi,z)
+\eta(z)F(x,y,\xi),\\
F(x,\f y, \xi)=(\n_x\eta)y=g(\n_x\xi,y).
\end{array}
\end{equation}
The special class $\F_0$,
determined by the condition $F=0$, is the intersection of the basic classes and it is known as the
class of the \emph{cosymplectic B-metric manifolds}.
Let $\left\{e_i;\xi\right\}$ $(i=1,2,\dots,2n)$ be a basis of
$T_pM$ and let $\left(g^{ij}\right)$ be the inverse matrix of the
matrix $\left(g_{ij}\right)$ of $g$. Then the following 1-forms
are associated with $F$:
\begin{equation}\label{t}
\theta(z)=g^{ij}F(e_i,e_j,z),\quad
\theta^*(z)=g^{ij}F(e_i,\f e_j,z), \quad \omega(z)=F(\xi,\xi,z).
\end{equation}
These 1-forms are known also as the Lee forms of the considered manifold. Obviously, the identities
$\om(\xi)=0$ and $\ta^*\circ\f=-\ta\circ\f^2$ are always valid.
\subsection{Sasaki-like almost contact B-metric manifolds}
In \cite{IvMaMa45}, it is defined the class of \emph{Sasaki-like spaces} in the set of almost
contact B-metric manifolds (also known as almost contact complex Riemannian manifolds) by the condition its complex cone to be a K\"ahler-Norden manifold.
Then, these Sasaki-like spaces are determined by the condition
\begin{equation}\label{defSl}
\begin{array}{l}
\left(\nabla_x\f\right)y=-g(x,y)\xi-\eta(y)x+2\eta(x)\eta(y)\xi,
\end{array}
\end{equation}
which may be rewritten as $\left(\nabla_x\f\right)y=g(\f x,\f y)\xi+\eta(y)\f^2 x$.
\begin{remark}\label{rem:Sl}
Obviously, the considered manifolds of Sasaki-like type form a subclass of the basic class $\F_4$
of the classification in \cite{GaMiGr} and they
have Lee forms of the form $\ta=2n\,\eta$, $\ta^*=\om=0$.
\end{remark}
Moreover,
the following identities are valid for them \cite{IvMaMa45}
\begin{equation}\label{curSl}
\begin{array}{ll}
\n_x \xi=-\f x, \qquad &\left(\n_x \eta \right)(y)=-g(x,\f y),\\
R(x,y)\xi=\eta(y)x-\eta(x)y, \qquad &R(\xi,y)\xi=\f^2y, \\[0pt]
\rho(x,\xi)=2n\, \eta(x),\qquad &\rho(\xi,\xi)=2n,
\end{array}
\end{equation}
where $R$ and $\rho$ stand for the curvature tensor and the Ricci tensor.
As a consequence we have
\begin{equation*}
R(\xi,y)z=g(y,z)\xi-\eta(z)y, \qquad \eta\bigl(R(x,y)z\bigr)=\eta(x)g(y,z)-\eta(y)g(x,z);
\end{equation*}
\begin{equation*}
\begin{array}{lll}
\nabla_{\xi}\xi=0, \qquad &\nabla_{\xi}\f=0, \qquad &\n_{\xi} \eta=0,\\
\D\eta=0,\qquad & R(\f x,\f y)\xi=0, \qquad & \rho(\f x,\xi)=0.
\end{array}
\end{equation*}
Obviously, the sectional curvature of an arbitrary $\xi$-section $\al=\Span\{x,\xi\}$ is $k(x,\xi)=1$.
\subsubsection{Example of a Sasaki-like manifold}\label{ex1}
As Example 2 in \cite{IvMaMa45}, it is considered
the Lie group $G$ of
dimension $5$
with a basis of left-invariant vector fields $\{e_0,\dots, e_{4}\}$
and the corresponding Lie algebra is
defined by the commutators
\begin{equation}\label{comEx1}
\begin{array}{ll}
[e_0,e_1] = p e_2 + e_3 + q e_4,\quad &[e_0,e_2] = - p e_1 -
q e_3 + e_4,\\[0pt]
[e_0,e_3] = - e_1 - q e_2 + p e_4,\quad &[e_0,e_4] = q e_1
- e_2 - p e_3,\qquad p,q\in\R.
\end{array}
\end{equation}
Then, $G$ is equipped with an almost contact B-metric structure by
\begin{equation}\label{strEx1}
\begin{array}{l}
g(e_0,e_0)=g(e_1,e_1)=g(e_2,e_2)=-g(e_{3},e_{3})=-g(e_{4},e_{4})=1,
\\[0pt]
g(e_i,e_j)=0,\quad
i,j\in\{0,1,\dots,4\},\; i\neq j,
\\[0pt]
\xi=e_0, \quad \f e_1=e_{3},\quad \f e_2=e_{4},\quad \f e_3=-e_{1},\quad \f e_4=-e_{2}.
\end{array}
\end{equation}
The components of the
Levi-Civita connection are the following
\begin{equation}\label{nijEx1}
\begin{array}{c}
\begin{array}{ll}
\n_{e_0} e_1 = p e_2 + q e_4,\qquad &
\n_{e_0} e_2 = - p e_1 - q e_3, \\
\n_{e_0} e_3 = - q e_2 + p e_4,\qquad &
\n_{e_0} e_4 = q e_1 - p e_3,
\end{array}
\\
\begin{array}{c
\n_{e_1}e_0 = - e_3,\quad
\n_{e_2} e_0 = - e_4,\quad
\n_{e_3} e_0 = e_1,\quad
\n_{e_4}e_0 =e_2,
\end{array}
\\
\begin{array}{c
\n_{e_1}e_3 = \n_{e_2} e_4 = \n_{e_3} e_1 = \n_{e_4}e_2 = - e_0.
\end{array}
\end{array}
\end{equation}
It is verified that the constructed almost contact B-metric manifold
$(G,\f,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}g)$ is Sasaki-like.
\subsection{Einstein-like almost contact B-metric manifolds}
Let us introduce the following
\begin{definition}
An almost contact B-metric manifold $\M$ is said to be
\emph{Einstein-like} if its Ricci tensor $\rho$ satisfies
\begin{equation}\label{defEl}
\begin{array}{l}
\rho(x,y)=a\,g(x,y) +b\,\g(x,y) +c\,\eta(x)\eta(y)
\end{array}
\end{equation}
for some triplet of constants $(a,b,c)$.
\end{definition}
In particular, when $b=0$ and $b=c=0$, the manifold is called an \emph{$\eta$-Einstein manifold} and an \emph{Einstein manifold}, respectively. Other particular cases of Einstein-like types of the considered manifolds are given in \cite{ManNak15} and \cite{HMMek}.
As a consequence of \eqref{defEl} we obtain that the corresponding scalar curvature is the following constant
\begin{equation}\label{tauEl}
\begin{array}{l}
\tau=(2n+1)a +b +c.
\end{array}
\end{equation}
\begin{remark}
The Reeb vector field $\xi$ of an Einstein-like $\M$ is an eigenvector
of the Ricci operator $Q=\rho^{\sharp}$, \ie $g(Qx,y)=\rho(x,y)$,
with the corresponding eigenvalue $a+b+c$.
\end{remark}
Then, applying \eqref{strM} and \eqref{strM2} into \eqref{defEl}, we obtain the truthfulness of the following
\begin{proposition}
The Ricci tensor $\rho$ of an Einstein-like manifold $\M$ has the following properties:
\begin{equation}\label{roEl1}
\begin{array}{l}
\begin{array}{l}
\rho(\f x,\f y)=-\rho(x,y)+(a+b+c)\eta(x)\eta(y),
\end{array}\\
\begin{array}{ll}
\rho(\f x,y)=\rho(x,\f y), \qquad & \rho(\f x,\xi)=0,
\\
\rho(x,\xi)=(a+b+c)\eta(x), \qquad & \rho(\xi,\xi)=a+b+c,
\end{array}
\end{array}
\end{equation}
\begin{equation}\label{roEl2}
\begin{array}{l}
\left(\n_x\rho\right)(y,z)=b\,g\bigl(\left(\n_x\f\right)y,z\bigr) \\
\phantom{\left(\n_x\rho\right)(y,z)=}
+(b+c)\left\{g(\n_x \xi,y)\eta(z) +g(\n_x \xi,z)\eta(y)\right\}.
\end{array}
\end{equation}
\end{proposition}
\begin{proposition}
Let $\M$, $\dim M=2n+1$, have a scalar curvature $\tau$ and let the manifold be Sasaki-like and Einstein-like with a triplet of constants $(a,b,c)$.
Then we have
\begin{equation}\label{tauElSl2}
a+b+c=2n,\qquad \tau=2n(a+1),
\end{equation}
\begin{equation}\label{roElSl}
\begin{array}{l}
\left(\n_x\rho\right)(y,z)=-(b+c)\left\{g(x,\f y)\eta(z) +g(x,\f z)\eta(y)\right\}\\
\phantom{\left(\n_x\rho\right)(y,z)=}{}
+b\left\{g(\f x,\f y)\eta(z) +g(\f x,\f z)\eta(y)\right\}.
\end{array}
\end{equation}
\begin{equation}\label{roElSl3}
\begin{array}{c}
a=2n+\frac{1}{2n}(\Div^*{\rho})(\xi),\qquad
b=-\frac{1}{2n}(\Div{\rho})(\xi),\\
c=\frac{1}{2n}\bigl\{(\Div{\rho})(\xi)-(\Div^*{\rho})(\xi)\bigr\},
\end{array}
\end{equation}
where $\Div$ and $\Div^*$ denote the divergence with respect to $g$ and $\g$, respectively.
\end{proposition}
\begin{proof}
Bearing in mind the last equalities of \eqref{curSl} and \eqref{roEl1}, as well as \eqref{tauEl}, we obtain \eqref{tauElSl2}.
The expression \eqref{roElSl} follows from \eqref{defSl} and \eqref{roEl2}.
Taking the traces over $x$ and $y$ in \eqref{roElSl} with respect to both metrics and using $\g^{ij}=-\f^i_kg^{ik}+\xi^i\xi^j$,
we get \eqref{roElSl3}.
\end{proof}
\begin{corollary}\label{cor:ElSl}
Let $\M$ be Sasaki-like and Einstein-like. Then we have:
\begin{enumerate}
\item[(i)] It is scalar-flat if and only if $a=-1$.
\item[(ii)] It is Ricci-symmetric if and only if it is an Einstein manifold.
\item[(iii)] Its Ricci tensor is $\eta$-parallel and parallel along $\xi$.
\item[(iv)] It is $\eta$-Einstein if and only if $\Div Q\in\ker\eta$.
\item[(v)] It is Einstein if and only if $\Div{Q}, \Div^*{Q}\in\ker\eta$.
\end{enumerate}
\end{corollary}
\begin{proof}
The equivalence in (i) is obvious from \eqref{tauElSl2},
and (ii) is easily deduced from \eqref{roElSl}.
The assertion (iii) is valid since \eqref{roElSl} yields $\left(\n\rho\right)|_{\ker\eta}=0$ and $\n_\xi\rho=0$.
The statement in (iv) is true because of the value of $b$ in \eqref{roElSl3} and the identities
$\left(\n_x \rho\right)(y,z)=g\left(\left(\n_x Q\right)y,z\right)$
and
$(\Div\rho)(z)=g(\Div Q,z)$. Similarly, we establish the truthfulness of (v).
\end{proof}
Bearing in mind \eqref{tauElSl2}, we obtain immediately that the scalar curvature
of any Einstein Sasaki-like manifold $\M$ is $\tau=2n(2n+1)$.
\subsubsection{Example of an Einstein-like manifold}\label{ex1-E}
We consider the example of almost contact B-metric manifold $(G,\f,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}g)$ from \S\,\ref{ex1}.
Taking into account \eqref{comEx1}, \eqref{strEx1} and \eqref{nijEx1},
we compute the components of the curvature tensor $R_{ijkl}=R(e_i,e_j,e_k,e_l)$
and those of the Ricci tensor $\rho_{ij}=\rho(e_i,e_j)$.
The non-zero of them are determined by the following equalities and the property $R_{ijkl}=-R_{jikl}=-R_{ijlk}$:
\begin{equation}\label{Rex1}
\begin{array}{l}
R_{0110}=R_{0220}=-R_{0330}=-R_{0440}=1,\\
R_{1234}=R_{1432}=R_{2341}=R_{3412}=1,\\
R_{1331}=R_{2442}=1,\qquad
\rho_{00}=4.
\end{array}
\end{equation}
Bearing in mind \eqref{defEl}, we establish that $(G,\f,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}g)$
is $\eta$-Einstein, because we obtain $\rho=4\eta\otimes\eta$ and the constants are
\begin{equation}\label{abcS}
(a,b,c)=(0,0,4).
\end{equation}
\subsection{Almost contact B-metric manifolds with a torse-forming Reeb vector field}\label{sec:tf}
Let us recall, a vector field $\xi$ is called \emph{torse-forming} if $\n_x \xi=f\,x+\al(x)\xi$, where $f$ is a smooth function and $\al$ is an 1-form on the manifold. Taking into account the last equality in \eqref{strM2}, the 1-form $\al$ on an almost contact B-metric manifold $\M$ is determined by $\al=-f\,\eta$ and then we have the following mutually equivalent equalities
\begin{equation}\label{tf}
\begin{array}{l}
\n_x \xi=-f\,\f^2x,\qquad \left(\n_x \eta \right)(y)=-f\,g(\f x,\f y).
\end{array}
\end{equation}
If $f = 0$, the vector field $\xi$ in \eqref{tf} is
a parallel vector field.
This case is trivial and we omit it in our considerations, \ie we assume that $f$ is not identically zero.
Conditions \eqref{tf} imply $\ta^*(\xi)=2n\,f$ and $\ta(\xi)=\om=0$, using \eqref{F-prop} and \eqref{t}.
Then, taking into account the components of $F$ in the basic classes $\F_i$, $i\in\{1,\dots,11\}$, given in \cite{HM1},
we deduce
\begin{remark}\label{rem:tf}
Manifolds $\M$ with torse-forming $\xi$ belong to
the class $\F_1\oplus\F_2\oplus\F_3\oplus\F_5\oplus\F_6\oplus\F_{10}$.
Among the basic classes, only $\F_5$ can contain such manifolds.
\end{remark}
Let us note that manifolds of the class $\F_5$ are counterparts of $\bt$-Kenmotsu manifolds in the case of almost contact metric manifolds.
If $\xi$ is torse-forming for $\M\in\F_5$, then it is valid the following
\begin{equation}\label{tfF5}
\begin{array}{l}
\left(\n_x \f \right)y=-f\{g(x,\f y)\xi+\eta(y)\f x\}.
\end{array}
\end{equation}
Taking into account \remref{rem:Sl} and \remref{rem:tf}, we obtain the following
\begin{proposition}
The Reeb vector field $\xi$ of any Sasaki-like manifold $\M$ is not a torse-forming vector field.
\end{proposition}
From \eqref{roEl2} and \eqref{tf}, similarly to (ii) from \corref{cor:ElSl}, we get
\begin{corollary} Let $\M$ with torse-forming $\xi$ be Einstein-like.
Then, it is Ricci-symmetric if and only if it is an Einstein manifold.
\end{corollary}
\subsubsection{Example of an almost contact B-metric manifold with a torse-forming Reeb vector field}\label{ex2}
In \cite{HMMek}, it is given a 3-dimensional $\F_5$-manifold on a Lie group $L$ with
a basis $\{e_0, e_1, e_2\}$ of left invariant vector fields.
The corresponding Lie algebra and the almost contact B-metric structure are determined as follows:
\begin{equation}\label{strL}
\begin{array}{c}
[e_0,e_1]=p e_1, \qquad [e_0,e_2]=p e_2, \qquad
[e_1,e_2]=0,\qquad p\in\R,\\
\begin{array}{l}
\f e_0=0,\qquad \f e_1=e_{2},\qquad \f e_{2}=- e_1,\qquad \xi=
e_0,\\
\eta(e_0)=1,\qquad \eta(e_1)=\eta(e_{2})=0,
\end{array}\\
\begin{array}{l}
g(e_0,e_0)=g(e_1,e_1)=-g(e_{2},e_{2})=1, \\
g(e_0,e_1)=g(e_0,e_2)=g(e_1,e_2)=0.
\end{array}
\end{array}
\end{equation}
The obtained manifold $(L,\f,\xi,\eta,g)$ is characterized as an $\F_5$-manifold by the following non-zero components
\begin{equation}\label{FijkF5}
\begin{array}{c}
\n_{e_1}e_1=-\n_{e_2}e_2=p e_0,\qquad
\n_{e_1}e_0=-p e_1,\qquad
\n_{e_2}e_0=-p e_2\\
F_{102}=F_{120}=F_{201}=F_{210}=\frac{1}{2}\ta^*_0=-p.
\end{array}
\end{equation}
Also, there are computed the components $R_{ijkl}$ of $R$
and $\rho_{ij}$ of $\rho$.
The non-zero of them are determined by the following equalities and the property $R_{ijkl}=-R_{jikl}=-R_{ijlk}$:
\begin{equation*}\label{res}
\begin{array}{l}
-R_{0101}=R_{0202}=R_{1212}=\frac12\rho_{00}=\frac12\rho_{11}
=-\frac12\rho_{22}=-p^2.
\end{array}
\end{equation*}
Moreover, the values of
$\tau$
and the sectional curvatures $k_{ij}=k(e_i,e_j)$ are
\begin{equation*}\label{curvF5}
\begin{array}{l}
\frac16 \tau=k_{12}=k_{01}=k_{02}=-p^2.
\end{array}
\end{equation*}
It is shown that the constructed manifold is Einstein because $\rho=-2p^2 g$ holds, \ie \eqref{defEl} is valid for constants
\begin{equation}\label{abcF5}
(a,b,c)=(-2p^2,0,0).
\end{equation}
According to the results for $\n\xi$ in \eqref{FijkF5}, we make sure that $\xi$ is a torse-forming vector field with constant
\begin{equation}\label{fF5}
f=-p.
\end{equation}
\section{Ricci-like solitons on almost contact B-metric manifolds}\label{sect-1}
A pseudo-Riemannian manifold $(M,g)$ is called a \emph{Ricci soliton} if it admits a smooth non-zero vector field $v$
on $M$ such that \cite{Ham82}
\begin{equation*}\label{Rs}
\begin{array}{l}
\frac12 \mathcal{L}_v g + \rho + \lm\, g =0,
\end{array}
\end{equation*}
where $\mathcal{L}$ denotes the Lie derivative,
$\rho$ stands for the Ricci tensor field and $\lm$ is a constant.
A Ricci soliton is called \emph{shrinking}, \emph{steady} or \emph{expanding} according to whether $\lm$ is negative, zero or positive, respectively \cite{ChoLuNi}.
In the trivial case when $v$ is a Killing vector field,
the Ricci soliton is an Einstein manifold.
A generalization of the Ricci soliton on an almost contact metric manifold $(M,\f,\xi,\eta,g)$ for example is
the \emph{$\eta$-Ricci soliton} defined
by \cite{ChoKim}
\begin{equation*}\label{eRs}
\begin{array}{l}
\frac12 \mathcal{L}_v g + \rho+ \lm\, g + \nu\, \eta\otimes\eta=0,
\end{array}
\end{equation*}
where $\nu$ is also a constant. Obviously, an $\eta$-Ricci soliton with $\nu=0$ is a Ricci soliton.
Further in the present paper, $(M,\f,\xi,\eta,g)$ stands for an almost contact B-metric manifold.
Due to the presence of two associated metrics $g$ and $\g$ on $M$,
we introduce a generalization of the Ricci soliton and the $\eta$-Ricci soliton
as follows.
\begin{definition}
An almost contact B-metric manifold $\M$ is called a
\emph{Ricci-like soliton} with potential vector field $\xi$ if its Ricci tensor $\rho$ satisfies the following condition for a triplet of constants $(\lm,\mu,\nu)$
\begin{equation}\label{defRl}
\begin{array}{l}
\frac12 \mathcal{L}_{\xi} g + \rho + \lm\, g + \mu\, \g + \nu\, \eta\otimes \eta =0.
\end{array}
\end{equation}
\end{definition}
For the Lie derivative of $g$ along $\xi$ we have the following expression
\begin{equation}\label{Lg=nxi}
\left(\mathcal{L}_{\xi} g \right)(x,y)=g(\n_x \xi,y)+g(x,\n_y \xi)
\end{equation}
and then \eqref{defRl} implies for the scalar curvature the following
\[
\tau=-\Div\xi-(2n+1)\lm -\mu-\nu.
\]
\begin{proposition}\label{prop-RlEl}
Let $\M$ with a Ricci tensor $\rho$ be Einstein-like with constants $(a,b,c)$ and let the manifold admit a Ricci-like soliton with potential $\xi$ and constants $(\lm,\mu,\nu)$.
Then the following properties are valid:
\begin{enumerate}
\item[(i)] $a+b+c+\lm+\mu+\nu=0$;
\item[(ii)] $\xi$ is geodesic, \ie $\n_{\xi}\xi=0$;
\item[(iii)] $\left(\n_{\xi} \f\right)\xi=0$, $\n_{\xi} \eta=0$, $\om=0$;
\item[(iv)] $\left(\n_{\xi}\rho\right)(y,z)=b\,g\bigl(\left(\n_{\xi}\f\right)y,z\bigr)$.
\end{enumerate}
\end{proposition}
\begin{proof}
Using \eqref{defEl}, \eqref{defRl} and \eqref{Lg=nxi}, we obtain
\begin{equation}\label{ElRl}
\begin{array}{l}
g(\n_x \xi,y)+g(\n_y \xi,x)=-2\{(a+\lm)g(x,y)+(b+\mu)g(x,\f y)\\
\phantom{g(\n_x \xi,y)+g(\n_y \xi,x)=-2\{}
+(b+c+\mu+\nu)\eta(x)\eta(y)\}.
\end{array}
\end{equation}
Substituting $y$ for $\xi$ in \eqref{ElRl}, we obtain (i) and immediately after that (ii).
Bearing in mind \eqref{F=nfi}, \eqref{F-prop} and \eqref{t}, the property (ii) is equivalent to the vanishing of
$\left(\n_{\xi} \f\right)\xi$, $\n_{\xi} \eta$ and $\om$, i.e. (iii) is valid.
On the other hand, \eqref{roEl2} and $\n_{\xi} \xi=0$ yields (iv).
\end{proof}
Taking into account the characterization of the basic classes $\F_i$, $i\in\{1,\A\dots,\A11\}$ of $\M$ by the components of $F$, given in \cite{HM1}, and the assertions (iii) and (iv) of the latter proposition, we deduce the following
\begin{corollary}
Let $\M$ satisfy the hypothesis of \propref{prop-RlEl}. Then, we have:
\begin{enumerate}
\item[(i)] The manifold does not belong to $\F_{11}$ or to its direct sum with other basic classes.
\item[(ii)] The Ricci tensor is parallel along $\xi$ if and only if the manifold does not belong to $\F_{10}$ or to its direct sum with other basic classes.
\item[(iii)] The Ricci tensor is parallel along $\xi$ if and only if the manifold is $\eta$-Einstein.
\end{enumerate}
\end{corollary}
\subsection{Ricci-like solitons on Sasaki-like almost contact B-metric manifolds}\label{sec:RlSl}
\begin{theorem}\label{thm:RlSl}
Let $\M$ be a $(2n+1)$-dimensional Sasaki-like manifold and let $a$, $b$, $c$, $\lm$, $\mu$, $\nu$ be constants that satisfy the following equalities:
\begin{equation}\label{SlElRl-const}
a+\lm=0,\qquad b+\mu-1=0,\qquad c+\nu+1=0.
\end{equation}
Then, the manifold admits
a Ricci-like soliton with potential $\xi$ and constants $(\lm,\A\mu,\A\nu)$, where $\lm+\mu+\nu=-2n$,
if and only if
it is Einstein-like with constants $(a,b,c)$, where $a+b+c=2n$.
In particular, we get:
\begin{enumerate}
\item[(i)] The manifold admits an $\eta$-Ricci soliton with potential $\xi$ and constants $(\lm,-2n-\lm)$ if and only if
the manifold is Einstein-like with constants $(-\lm,1,\lm+2n-1)$.
\item[(ii)] The manifold admits a shrinking Ricci soliton with potential $\xi$ and constant $-2n$ if and only if
the manifold is Einstein-like with constants $(2n,1,-1)$.
\item[(iii)] The manifold is $\eta$-Einstein with constants $(a,2n-a)$ if and only if
it admits a Ricci-like soliton with potential $\xi$ and constants $(-a,1,a-2n-1)$.
\item[(iv)] The manifold is Einstein with constant $2n$ if and only if
it admits a Ricci-like soliton with potential $\xi$ and constants $(-2n,1,-1)$.
\end{enumerate}
\end{theorem}
\begin{proof}
Taking into account the expression of $\n_x \xi$ from \eqref{curSl} and applying it in \eqref{Lg=nxi} and \eqref{defRl},
we obtain
that condition \eqref{defRl}
is equivalent to
\begin{equation}\label{SlRl-rho}
\rho=-\lm g+(1-\mu)\g-(1+\nu)\eta\otimes\eta,
\end{equation}
which coincides with \eqref{defEl} under conditions \eqref{SlElRl-const}.
The equality $a+b+c=2n$ is known from \eqref{tauElSl2}, whereas
$\lm+\mu+\nu=-2n$ is a consequence of \eqref{SlRl-rho} for $\rho(x,\xi)$ and the corresponding formula from \eqref{curSl}.
Therefore, the main assertion is valid.
The assertions (i), (ii), (iii) and (iv) are corollaries of the main assertion for the cases
$\mu=0$, $\mu=\nu=0$, $b=0$ and $b=c=0$, respectively.
\end{proof}
\subsubsection{Example of an Ricci-like soliton on the manifold}\label{ex1-R}
Let us return to the example of Sasaki-like almost contact B-metric manifold $(G,\f,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}g)$ from \S\,\ref{ex1}, which is also $\eta$-Einstein, according to \S\,\ref{ex1-E}.
Now, let us check whether it admits a Ricci-like soliton with potential $\xi$.
Using \eqref{strEx1}, \eqref{nijEx1} and \eqref{Lg=nxi},
we compute the components $\left(\LL_\xi g\right)_{ij}=\left(\LL_\xi g\right)(e_i,e_j)$ and
the non-zero of them are the following
\begin{equation}\label{Lex1}
\begin{array}{l}
\left(\LL_\xi g\right)_{13}=\left(\LL_\xi g\right)_{24}=\left(\LL_\xi g\right)_{31}=\left(\LL_\xi g\right)_{42}=2.
\end{array}
\end{equation}
After that, taking into account
\eqref{strEx1}, \eqref{Rex1} and \eqref{Lex1}, we obtain that \eqref{defRl} is satisfied for
\begin{equation}\label{lmnS}
(\lm,\mu,\nu)=(0,1,-5)
\end{equation}
and $(G,\f,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}g)$
admits a Ricci-like soliton with potential $\xi$.
\begin{remark}
The constructed 5-dimensional $\eta$-Einstein Sasaki-like manifold $(G,\f,\allowbreak{}\xi,\eta,g)$ admitting a Ricci-like soliton with potential $\xi$
has relevant constants, determined by \eqref{abcS} and \eqref{lmnS}, that satisfy \eqref{SlElRl-const} and therefore this example
supports the case (iii) of \thmref{thm:RlSl} and the rest results in \S\,\ref{sec:RlSl}.
\end{remark}
\subsection{Ricci-like solitons with torse-forming potential on almost contact B-metric manifolds}\label{sec:Rltf}
\begin{theorem}\label{thm:Rltf}
Let $\xi$ on $\M$, $\dim M=2n+1$, be torse-forming with function $f$.
The manifold admits
a Ricci-like soliton with potential $\xi$ and constants $(\lm,\mu,\nu)$
if and only if
the manifold is Einstein-like with constants $(a,b,c)$
provided that
$f$ is a constant and the following equalities are satisfied:
\begin{equation}\label{tfElRl-const}
a+\lm+f=0,\qquad b+\mu=0,\qquad c+\nu-f=0.
\end{equation}
In particular, we have:
\begin{enumerate}
\item[(i)] The manifold admits an $\eta$-Ricci soliton with potential $\xi$ and constants $(\lm,\nu)$ if and only if
the manifold is $\eta$-Einstein with constants $(a,c)$, where the equality $a+c+\lm+\nu=0$ is valid.
\item[(ii)] The manifold admits a Ricci soliton with potential $\xi$ and constant $\lm$ if and only if
the manifold is $\eta$-Einstein with constants $(-\lm-f,f)$.
\item[(iii)] The manifold is Einstein with constant $a$ if and only if
it admits an $\eta$-Ricci with potential $\xi$ and constants $(-a-f,f)$.
\end{enumerate}
\end{theorem}
\begin{proof}
If we have a Ricci-like soliton with torse-forming potential $\xi$ on $\M$,
then the first equality of \eqref{tf}, \eqref{defRl} and \eqref{Lg=nxi} yield the form of its Ricci tensor as follows
\begin{equation*}\label{tfRl-rho}
\rho=-(\lm+f) g-\mu\g-(\nu-f)\eta\otimes\eta.
\end{equation*}
According to \eqref{defEl}, the latter equality shows that the manifold is Einstein-like
if and only if $f$ is a constant. In this case
\eqref{tfElRl-const} are valid.
Vice versa, let $\M$ have torse-forming $\xi$ with function $f$ and let the manifold be Einstein-like with constants $(a,b,c)$.
Then, applying \eqref{tf} and \eqref{defEl} in \eqref{Lg=nxi} and \eqref{defRl} for constants $(\lm,\mu,\nu)$,
we establish that condition \eqref{defRl} is satisfied if and only if \eqref{tfElRl-const} holds. Therefore, $f$ is a constant.
The particular cases (i), (ii) and (iii) follow from the main assertion for
$b=\mu=0$, $b=\mu=\nu=0$ and $b=c=\mu=0$, respectively.
\end{proof}
\begin{proposition}\label{prop-RlEltf}
Let $\M$ be Einstein-like with constants $(a,b,c)$ and let it admit a Ricci-like soliton with constants $(\lm,\mu,\nu)$ and a potential $\xi$ which is torse-forming with constant $f$.
Then $f$ is determined by
\begin{equation}\label{tf1}
f=\varepsilon\sqrt{-\frac{a+b+c}{2n}}=\varepsilon\sqrt{\frac{\lm+\mu+\nu}{2n}}, \qquad \varepsilon=\pm 1.
\end{equation}
\end{proposition}
\begin{proof}
The expression of $\n\xi$ in \eqref{tf} for contant $f$ implies
\begin{equation}\label{Rxif}
R(x,y)\xi=f^2\{\eta(x)y-\eta(y)x\},\qquad \rho(x,\xi)=-2nf^2\eta(x).
\end{equation}
Comparing the second equality of \eqref{Rxif}
to the fourth one in \eqref{roEl1}, we get the equation
$-2n f^2=a+b+c$. The latter equality and (i) of \propref{prop-RlEl} imply \eqref{tf1}.
\end{proof}
The first equality of \eqref{Rxif} implies the following
\begin{remark}
The $\xi$-sections $\al=\Span\{x,\xi\}$ of $\M$, where $\xi$ is torse-forming with constant $f$,
have a constant negative sectional curvature $k(\al)=-f^2$.
\end{remark}
The following corollary follows from the expression $\tau=(2n+1)a$ for an Einstein manifold, \propref{prop-RlEltf} and assertion (iii) of \propref{thm:Rltf}.
\begin{corollary}\label{cor:E}
The manifold $\M$, $\dim M=2n+1$, is Einstein with a negative scalar curvature $\tau$ if and only if
the manifold admits an $\eta$-Ricci soliton with potential $\xi$
and constants $(\lm=-\frac{\tau}{2n+1}-f, \nu=f)$,
where $f$ is the constant for the torse-forming vector field $\xi$
and $f=\varepsilon\sqrt{\frac{-\tau}{2n(2n+1)}}$, $\varepsilon=\pm 1$.
\end{corollary}
For the obtained $\eta$-Ricci soliton in \corref{cor:E}, it is clear that the sign of $\nu$ depends on $\varepsilon$.
Then, we differ the following cases:
1) If $\tau<-1-\frac{1}{2n}$, then the both values of $\lm$ are positive, whereas $\nu$ is either positive or negative.
2) If $\tau=-1-\frac{1}{2n}$, then $(\lm,\nu)$ is either $\left(\frac{1}{n},-\frac{1}{2n}\right)$
or $\left(0,\frac{1}{2n}\right)$.
3) If $-1-\frac{1}{2n}<\tau<0$, then one value of $\lm$ is positive and the other is negative,
with the sign of $\nu$ opposite that of $\lm$.
Since any $\F_5$-manifold has a torse-forming vector field $\xi$, we conclude the following
\begin{proposition}\label{prop-RlEltfRs}
Let $\M\in\F_5\setminus\F_0$ be Einstein-like with constants $(a,b,c)$ and let it admit a Ricci-like soliton with constants $(\lm,\mu,\nu)$ and a potential $\xi$. Then we have:
\begin{enumerate}
\item[(i)] It is Ricci-symmetric if and only if it is an Einstein manifold.
\item[(ii)] Its Ricci tensor is $\eta$-parallel and parallel along $\xi$.
\end{enumerate}
\end{proposition}
\begin{proof}
By virtue of \eqref{roEl2}, \eqref{tf}, \eqref{tfF5} and the first equality of \eqref{tfElRl-const}, we get
\begin{equation*}\label{roElF5}
\begin{array}{l}
\left(\n_x\rho\right)(y,z)=(a+\lm)\bigl\{(b+c)\left\{g(\f x, \f y)\eta(z) +g(\f x, \f z)\eta(y)\right\} \\
\phantom{\left(\n_x\rho\right)(y,z)=(a+\lm)\bigl\{(b}
+b\left\{g(x,\f y)\eta(z) +g(x,\f z)\eta(y)\right\}\bigr\}.
\end{array}
\end{equation*}
Using the latter expression, in a similar way as in (ii) and (iii) of \corref{cor:ElSl}, we establish the truthfulness of the assertions.
\end{proof}
\subsubsection{Example of a Ricci-like soliton on an almost contact B-metric manifold with torse-forming $\xi$}\label{ex2-R}
Let us return to the $\F_5$-manifold $(L,\f,\xi,\eta,g)$ from \S\,\ref{ex2}, for which $\xi$ is a torse-forming vector field
with constant
$f=-p$.
Using \eqref{FijkF5} and \eqref{Lg=nxi}, we compute the components $\left(\LL_\xi g\right)_{ij}$ and
the non-zero of them are the following
\begin{equation}\label{Lex2}
\begin{array}{l}
\left(\LL_\xi g\right)_{11}=-\left(\LL_\xi g\right)_{22}=-2p.
\end{array}
\end{equation}
Taking into account
\eqref{strL} and \eqref{Lex2}, we obtain that \eqref{defRl} is satisfied for constants
\begin{equation}\label{lmnF5}
(\lm,\mu,\nu)=(p+2p^2,0,-p)
\end{equation}
and $(L,\f,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}g)$
admits a Ricci-like soliton with torse-forming potential $\xi$.
\begin{remark}
The constructed Einstein manifold $(L,\f,\xi,\eta,g)$ admitting an $\eta$-Ricci soliton with torse-forming potential $\xi$
has relevant constants, determined by \eqref{abcF5}, \eqref{fF5} and \eqref{lmnF5}, that satisfy \eqref{tfElRl-const} and therefore this example
supports the case (iii) of \thmref{thm:Rltf} and the rest results in \S\,\ref{sec:Rltf}.
\end{remark}
\subsection*{Acknowledgment}
The author was supported by projects MU19-FMI-020 and FP19-FMI-002 of the Scientific Research Fund,
University of Plovdiv Paisii Hilendarski, Bulgaria.
|
1,116,691,497,770 | arxiv | \section{Introduction}
\vspace{-0.8 em}
Human intelligence has long been the goal of artificial intelligence. Humans can effectively solve multi-task learning, lifelong learning, transfer learning, few-shot learning, generalization, exploration, prediction, and decision. AI algorithms, on the other hand, still struggle with these problems\cite{legg2007universal}. For example, supervised learning based on neural networks typically requires large training datasets, and suffers from catastrophic forgetting\cite{kirkpatrick2017overcoming,rusu2016progressive,goodfellow2013empirical}; most reinforcement learning algorithms\cite{parisotto2015actor,schaul2015universal,rusu2015policy} specialize in a single task through huge amounts of trial and error, and are difficult to generalize to new tasks.
So how is human intelligence formed? This remains an open question, and we hypothesize that there are at least three key elements. 1) \emph{Intrinsic motivation}. Humans are intrinsically motivated to truly understand the world. This understanding is crucial to intelligence because it applies to and is required by all tasks that humans will be faced with. 2) \emph{Unified network}. Humans use a single unified network – the brain – to tackle all tasks. This allows knowledge to be accumulated and shared across tasks, and thus increase the level of intelligence over time. 3) \emph{Limited complexity}. Humans face several constraints on complexity. They have limited time and data to learn each task; they have limited space to encode and pass on knowledge; the brain has limited energy to operate. These constraints force humans to develop intelligence and find efficient solutions.
Interestingly, we find that abstraction learning is a way to combine these elements, and thus key to achieving human-like intelligence. Here, we define abstractions as the set of concepts and laws that the world is built upon. By this definition, learning abstractions is equivalent to discovering a universal model of the world and using it to interpret observations. This satisfies intrinsic motivation, and has two implications. First, because only a single world model is yielded by abstraction learning, it can be maintained in a unified network. This model can then be shared and improved across all tasks, facilitating knowledge transfer and generalization. Second, because abstraction learning enables interpretation of complex observations as concise concepts and laws, task models can be greatly simplified. Consequently, fewer samples are required to learn each task, and less energy is needed to perform each task.
While there is biological evidence suggesting the existence of abstractions in the brain (e.g. the grandmother cell\cite{gross2002genealogy,clark2000theory,knorkski1967integrative}), abstraction learning remains largely unexplored in AI research. One of the reasons why abstraction learning is harder than other machine learning tasks such as classification is because it is difficult to establish a clear objective, or target for training\cite{bengio2013representation}. Two broadly related areas are probabilistic graphical models and neural networks. Probabilistic graphical models\cite{roweis1998algorithms,olshausen1996emergence,smolensky1986information} use abstractions to define random variables and graph structures. These abstractions serve to decompose the joint probability distribution and simplify computation, but they are given by human experts rather than learned by the model itself. Neural networks, on the other hand, aim to learn effective representations for specific tasks. Taking auto-encoder\cite{yann1987modeles,bourlard1988auto,hinton1994autoencoders} as an example, one can learn a set of representation for data by unsupervised learning, typically aiming for dimensionality reduction. The process of closely matching the output with the original data enforces the auto-encoder to learn the abstraction. Through visualization\cite{zeiler2014visualizing,simonyan2013deep,springenberg2014striving}, these artificial neurons can be qualitatively regarded as representing abstract concepts. In lifelong learning, knowledge shared between tasks was considered as abstractions in recent works, such as using a regularizer that prevents the parameters from drastic changes in their values\cite{kirkpatrick2017overcoming} or blocking any changes to the old task parameters\cite{rusu2016progressive}. However, these abstractions are not directly learned, but emerge as a by-product of minimizing task losses. Therefore, the quality of these abstractions cannot be guaranteed, depriving neural networks of the benefits that real abstractions would offer.
In this paper, we aim to learn abstractions directly. This raises at least three challenges. First, while abstractions may be straightforward to express in natural language, they are hard to specify in the language of neurons and synapses. Second, because abstractions are like hidden variables that do not appear in the training data, there is no simple objective function that distinguishes between right and wrong abstractions. Third, even if we find such an objective function, which would probably be non-differentiable, how it should influence the network structure remains unclear. To overcome these challenges, we propose a novel framework named ONE that formulates abstraction learning as Optimization via Network Evolution. ONE incorporates three levels of innovations:
\quad $\bullet $ \textbf{Partition structure prior, structure prior, with pre-allocated abstraction neurons.} These abstraction neurons accumulate abstractions, and separate the network into a task-agnostic cognition part and task-specific decision parts. The cognition part generates abstractions, and the decision parts select abstractions. Task losses do not modify abstractions, thus improving reusability and avoiding catastrophic forgetting.
\quad $\bullet $ \textbf{Constrained optimization formulation, integrating three abstraction properties.} 1) Variety. Abstractions should cover the various concepts and laws behind observations. 2) Simplicity. Each observation must be described succinctly by only a few abstractions, thus simplifying task-specific decision making. 3) Effectiveness. Each abstraction should be effective for multiple tasks, enhancing task performance and knowledge sharing.
\quad $\bullet $ \textbf{Network evolution algorithm, producing and improving abstraction structures.} The constrained optimization problem for abstraction learning involves optimizing the network structure, which is beyond the capabilities of standard gradient-based methods. To solve this problem, we introduce connection growth to search through the structure space, local competition to improve search efficiency, and use the objective function to guide search direction.
Through extensive experiments on the MNIST dataset, we demonstrate that ONE successfully converges and learns abstractions that accomplish different tasks. Importantly, ONE shows elementary human-like intelligence in three aspects. First, ONE performs tasks based on simple abstractions, and thus activate only a small proportion of the whole network for each task. This leads to better generalization and less energy consumption. Second, when faced with new tasks, ONE is able to generate new abstractions. These abstractions accumulate, so as ONE learns more tasks, it makes more use of existing abstractions and generates fewer new ones. This facilitates task transfer and boosts learning speed. Third, ONE does not forget, and can thus learn continually.
\section{Partition structure prior}
\vspace{-0.8 em}
To our knowledge, although abstraction has been proposed in a large number of literature\cite{hinton1986learning,bengio2012deep,krizhevsky2012imagenet,collobert2011natural}, the current research remains basically at the level of giving general concept and description of abstractions, which we refer to as virtual abstractions. We consider that the process human beings realizing basic intellectual activities can be divided into the cognitive process for concepts and laws and the decision making process related to tasks. Because of the lack of physical meanings of virtual abstractions, it is difficult to draw a clear line between the parts of cognition and decision-making, which makes the channels of knowledge sharing among tasks obscure and the decision-making process of a single task complicated. Therefore, in this section, we propose a specific form of entity abstraction and a partition structure based on the abstraction locked layer (see Figure \ref{fig:Partition structure of ONE model}).
Moreover, the structures (abstractions) activated by limited amounts of tasks cannot reach the level of understanding the whole world, abstractions needs to be continuously generated, accumulated, and reused by different tasks. In other words, each task makes full use of the existing abstractions, and only generates necessary abstractions that are conducive to the new task. This promotes the accumulation and reuse of abstractions and avoids forgetting knowledge acquired in the past.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{1_pe.pdf}
\caption[Partition structure of ONE model]{Partition structure of ONE model: ONE performs structure search in the cognitive part and parameters learning in the decision part, with an abstraction locked layer in the middle of them. In the cognitive part, the mechanisms of growth, extinction, and competition promote the production of excellent structures (abstractions), where the dashed line represents death, the number of pulses denotes the firing intensity of neurons and the dark units are winners. The mechanisms of each layer is applied to all layers of the cognitive part. }
\label{fig:Partition structure of ONE model}
\vspace{-1.75 em}
\end{figure}
The whole network can be seen as a feedforward neural network with a fixed number of neurons, which has been divided into the cognition and decision parts by an abstraction locked layer (see Figure \ref{fig:Partition structure of ONE model}). The abstraction layer separates neurons into two types: task-agnostic and task-specific. Each neuron in the abstraction layer is the output of a subnetwork composed of task-agnostic neurons. These subnetworks can overlap and take any form, in addition, they should be sparse and hierarchical. The task-specific neurons are organized into non-overlapping groups. The neurons within each group form fully connected layers and are intended for a specific task. Each group connects sparsely with the abstraction layer, selecting the relevant abstractions for each task. Inspired by biology\cite{dayan2001theoretical}, there are three mechanisms that promote the production of excellent abstractions: 1) \textbf{Growth:} This a way of exploring the structure space. The more extensively we explore, the better structure we expect to find. 2) \textbf{Competition:} It promotes the variety and simplicity of abstractions and prunes unpromising structures on the fly. 3) \textbf{Extinction:} It aims to optimize the abstractions for task effectiveness.
Formally, suppose there are $K$ tasks in total. Each task is associated with a dataset $\mathcal{D}^{\left(k\right)}$ and a loss function $\mathcal{L}^{\left(k\right)}$, and requires to learn a mapping $f^{\left(k\right)}:\mathcal{X}^{\left(k\right)}\to \mathcal{Y}^{\left(k\right)}, k=1,2,\dots,K$. All input spaces have the same dimension, $\mathcal{X}^{\left(k\right)}\subseteq\mathbb{R}^{n}$, but the output spaces can be arbitrary, for example, discrete for classification and continuous for regression.
Let $L$ denote the number of layers formed by task-agnostic neurons, with the input layer being the first layer and the abstraction layer being the $L$-th layer. Let $I_{\left(l\right)}$ denote the set of neurons in the $l$-th layer, and $N_{\left(l\right)}$ the size of $I_{\left(l\right)}$. Each task-agnostic neuron represents a mapping $\Phi_{i}:\mathcal{X}\to\mathbb{R}$, where $\mathcal{X}=\bigcup\nolimits_{k=1}^{K}\mathcal{X}^{\left(k\right)}, i\in\bigcup\nolimits_{l=1}^{L}I_{\left(l\right)}$. Let $\Phi_{\left(l\right)}=\left[\Phi_{i}\right]_{i\in I_{\left(l\right)}}^{T}$ denote the concatenation of mappings of each layer, then $\Phi_{\left(l\right)}:\mathcal{X}\to\mathbb{R}^{N_{\left(l\right)}}$ can be defined recursively:
\vspace{-0.5 em}
\begin{align
\Phi_{\left(1\right)}\left(x\right)&=x,\\
\Phi_{\left(l\right)}\left(x\right)&=\sigma_{\left(l\right)}\left(W_{\left(l-1\right)}\Phi_{\left(l-1\right)}\left(x\right)+b_{\left(l\right)}\right),\ l=2,\ 3,\ \dots,\ L,
\end{align}
\vspace{0.2 em}
where $W_{\left(l-1\right)}\in \mathbb{R}^{N_{\left(l\right)}\times N_{\left(l-1\right)}}$ is the weight matrix, $b_{\left(l\right)}\in \mathbb{R}^{N_{\left(l\right)}}$is the bias vector, and $\sigma_{\left(l\right)}: \mathbb{R}^{N_{\left(l\right)}}\to \mathbb{R}_{+}^{N_{\left(l\right)}}$ is the activation function. Note that $\Phi_{\left(l\right)}\left(x\right)$ is non-negative for $l\geq2$, and that the neuron $i$ is included in the feedforward process and thus "active" only if $\Phi_{i}\left(x\right)>0$. This means the network operates in a distributed manner, activating a portion of subnetworks and abstraction neurons for a given input. In other words, the network distributes its capacity over the input space.
In essence, ONE instantiates a special decomposition of $f^{\left(k\right)}$, which is $f^{\left(k\right)}=g^{\left(k\right)}\circ\Phi_{\left(L\right)}$, where $g^{\left(k\right)}$ represents the fully-connected layers in each task-specific group. If $\Phi_{\left(L\right)}$ represents really good abstractions, this decomposition can be highly efficient, because each $g^{\left(k\right)}$ will be quite simple and only need a few parameters. This means the majority of computation for each task is shared, and a new task merely adds a small increment to the whole network.
There are at least four differences between ONE and other neural networks (NNs): 1) \textbf{Partition structure:} NNs are mostly designed to implement specific tasks, making the entire network serve the tasks. 2) \textbf{Local parameter updating:} NNs’ parameters are fully updated during back propagation, which is one of reasons that causes knowledge acquired in old tasks being forgotten in the process of implementing new tasks. In pre-trained models\cite{hinton2006reducing,mesnil2011unsupervised}, parameters are updated in a similar way to ours, but the knowledge is only a small part of all knowledge, which cannot be further accumulated in later learning. 3) \textbf{Abstraction locked layer:} The traditional NNs do not point out the specific form of abstractions, but only stay on the concept and description level. 4) \textbf{Dynamic network based on growth, competition and extinction:} The works of \citet{zhou2012online,philipp2017nonparametric,cortes2016adanet} explored dynamic neural networks, while none of them considered multi-task setting. The network\cite{xiao2014error} grows and branches only on the topmost layer, while ONE can change structures at any layer in the cognitive part.
\section{Constraint optimization formulation}
\vspace{-0.8 em}
The structure mentioned above is just a framework of abstraction generation, which satisfies reusability but not guarantee other properties of abstractions. So, we formulate the development of abstractions into a constrained optimization problem, where the constraints ensure the variety and simplicity of abstractions, and the objective function measures the effectiveness of abstractions.
\textbf{Variety constraint.} ONE is intended for multiple tasks that can be radically different, so it must have the ability to generate and maintain various abstractions required to accomplish these tasks. Even if they are not required, a variety of abstractions would offer a wider perspective and probably lead to better solutions.
In ONE, each abstraction is represented by an abstraction neuron, which in turn corresponds to a subnetwork. Therefore, a variety of abstractions require a variety of subnetworks. The variety also provides an opportunity to search for specialized structures that express abstractions best.
Because the network capacity is fixed, in order to realize variety, we need to control the size of subnetworks and make sure that they do not overlap much. Accordingly, we introduce two constraints as followed.
\vspace{0 em}
\begin{align}
&A1:\quad \sum\nolimits_{j\in I_{\left(l\right)}}\mathbb{I}\left\{\Phi_{j}\left(x\right)>0\right\}\leq {V1}_{\left(l\right)},\ l=2,3,\ \dots,\ L-1,\ \forall x\in\mathcal{X},\\
&A2:\quad \sum\nolimits_{h\in I_{\left(l+1\right)}}\mathbb{I}\left\{W_{j\to h}\neq 0\right\}\leq {V2}_{\left(l\right)}^{\ j},\ \forall j\in I_{\left(l\right)},\ l=2,\ 3,\ \dots,\ L-1,\ \forall x\in\mathcal{X}.
\end{align}
\vspace{0 em}
Here, $\mathbb{I}\left\{\cdot\right\}$ is the indicator function. The first constraint limits the number of active neurons at each layer for any given input, thus controls the size of the activated subnetworks. The second constraint limits the number of outgoing connections of each neuron, thus reduces the overlap among subnetworks.
\textbf{Simplicity constraint.} Once we develop good abstractions, we shall be able to grasp these concepts and principles, and use them to guide our actions and decisions. This greatly simplifies the tasks we encounter.
We introduce simplicity into ONE from two aspects. First, we require that only a few abstraction neurons are activated for any given input. This not only fosters simple explanation of the input, but also limits the size of task-specific subnetworks, enforcing simple decision making. Second, we expect abstractions to capture the commonality among entities. For clarity, we consider classification tasks only. We impose an upper bound on the total number of abstraction neurons that can be activated for all inputs belonging to the same category.
More concretely, let $\mathcal{C}^{\left(k\right)}$ denote the set of possible categories in the $k$-th classification task, and $\mathcal{X}^{c}$ denote the input space for each category $c\in \mathcal{C}^{\left(k\right)}$. We encourage simplicity by adding the following two constraints.
\vspace{0 em}
\begin{align}
&A3:\quad \sum\nolimits_{i\in I_{\left(L\right)}}\mathbb{I}\left\{\Phi_{i}\left(x\right)>0\right\}\leq S1,\ \forall x\in \mathcal{X},\\
&A4:\quad \sum\nolimits_{i\in I_{\left(L\right)}}\mathbb{I}\left\{ \sum\limits_{x\in\mathcal{X}^{c}}\mathbb{I}\left\{\Phi_{i}\left(x\right)>0\right\}>0 \right\}\leq S2,\ \forall c\in\mathcal{C}^{\left(k\right)},\ k=1,\ 2,\ \dots,\ K.
\end{align}
\vspace{0 em}
\textbf{Effectiveness constraint.} Although abstractions are task-agnostic, some of them are particularly relevant to certain tasks while others are not. To boost task performance, we propose a matching process that selects highly effective abstraction neurons for each task. These selected neurons are then included in separate fully connected layers to adapt to task-specific details.
Thus, the network achieves effectiveness for tasks through two stages of optimization. First, it optimizes the selection variables $e_{i}^{\left(k\right)}$ for each abstraction neuron $i \in I_{\left(L\right)}$ and each task $k$. If neuron $i$ is selected for task $k$, $e_{i}^{\left(k\right)}=1$, and otherwise $e_{i}^{\left(k\right)}=0$. Second, the network optimizes the weights of the fully connected layers $W_{\left(\geq L\right)}^{\left(k\right)}$ to reduce task losses. For neurons with $e_{i}^{\left(k\right)}=0$, their connection weights to the $k$-th task-specific group $W_{i}^{\left(k\right)}$ are constrained to be zero.
We now define the objective function for optimizing $e_{i}^{\left(k\right)}$. For classification tasks, it is reasonable to assume that an abstraction neuron can promote task effectiveness if it has some preliminary ability to distinguish different categories. We evaluate the ability through the distribution of the neuron’s activations over inputs of all categories for any given task. If the neuron is activated by inputs of only a few categories, we expect that it has such ability. More formally, we optimize $e_{i}^{\left(k\right)}$ to select the $E$ neurons whose activation distributions have the lowest entropy.
\vspace{0 em}
\begin{equation}
\min\limits_{e_{i}^{\left(k\right)}}\sum\limits_{i\in I_{\left(L\right)}}e_{i}^{\left(k\right)}\cdot H_{i}^{\left(k\right)}
\end{equation}
\begin{equation}
\begin{aligned}
subject\ to \\
&A5:\quad \sum\nolimits_{i\in I_{\left(L\right)}}e_{i}^{\left(k\right)}=E,\ k=1,\ 2,\ \dots,\ K,\\
&A6:\quad W_i^{(k)}=0,\ if\ e_i^{(k)}=0,\ \forall i\in I_{(l)},\ k=1,\ 2,\ ...,\ K.
\end{aligned}
\end{equation}
\vspace{0 em}
Here
\vspace{0 em}
\begin{align}
&H_{i}^{\left(k\right)}=-\sum\nolimits_{c\in\mathcal{C}^{\left(k\right)}}P_{i}^{c}\cdot\log P_{i}^{c},\\
&P_{i}^{c}=\frac{\sum\nolimits_{x\in\mathcal{X}^{c}}\mathbb{I}\left\{\Phi_{i}\left(x\right)>0\right\}}{\sum\nolimits_{c\in\mathcal{C}^{\left(k\right)}}\sum\nolimits_{x\in\mathcal{X}^{c}}\mathbb{I}\left\{\Phi_{i}\left(x\right)>0\right\}}.
\end{align}
\vspace{0 em}
\textbf{Overall optimization problem.} We now present the overall optimization problem that the network needs to solve. It integrates both the requirements for abstractions and the performance on each task.
\vspace{-0.5 em}
\begin{equation}
\min\limits_{W,e_{i}^{\left(k\right)}}\sum\limits_{k=1}^{K}\left(\mathcal{L}^{\left(k\right)}+\sum\limits_{i\in I_{\left(L\right)}}e_{i}^{\left(k\right)}\cdot H_{i}^{\left(k\right)}\right),
\end{equation}
\begin{equation}
\boxed{
subject\ to\ A1,\ A2,\ A3,\ A4,\ A5,\ A6.}
\end{equation}
Note that we do not include $\Phi_{i}$ in optimization variables, because they depend implicitly on the weights $W_{\left(<L\right)}$ between task-agnostic neurons. Note also that because $H_{i}^{\left(k\right)}$ depends on $\Phi_{i}$, the term $\sum\nolimits_{i\in I_{\left(L\right)}}e_{i}^{\left(k\right)}\cdot H_{i}^{\left(k\right)}$ is optimized jointly on $W$ and $e_{i}^{\left(k\right)}$. This implies that the differentiation ability of abstractions can also be improved.
\section{Network evolution algorithm}
\vspace{-0.8 em}
We observe that this optimization problem presents two major difficulties. First, the objective function includes discrete variables $e_{i}^{\left(k\right)}$. Also, the constraints depend on discrete values $\mathbb{I}\left\{\Phi_{j}\left(x\right)>0\right\}$ and $\mathbb{I}\left\{W_{j\to m}\neq 0\right\}$. Therefore, this difficulty of discreteness results from optimizing structure along with parameters, which is essential because the best structure for abstraction is unknown. The second difficulty comes from the nature of learning. In particular, tasks are not carried out simultaneously, but divided into batches and implemented sequentially. Consequently, abstractions for all tasks cannot be acquired at a time, but need to accumulate over time. This means the structure of the network keeps changing.
Standard gradient-based methods only work for continuous parameters and fixed structures, and thus are not suitable for our problem. Moreover, there is the same problem of optimizing discrete variables and dynamic structures in nature, which gets solved though natural evolution. Therefore, we propose an evolution-inspired optimization framework to search for good abstractions. More concretely, we identify three key elements of natural evolution: production, competition, and selection. Production generates a vast number of individuals, which then go through local competition. Nature selects the fittest individuals from the winners, and use them to guide a new round of production.
We incorporate these three components into our optimization framework to perform efficient parallel search of the structure space for good abstractions. In particular, we use production to grow new structures and maintain the variety abstractions. This corresponds to drawing a large sample from the structure space. These structures compete locally and only a small proportion of them get activated. This promotes the simplicity of abstractions, and improves search efficiency. The activated structures are then evaluated for their effectiveness and selected by each task. Only good structures survive, and they will affect the direction of production afterwards. We now describe each of these components in detail.
\textbf{Production.} In our framework, production refers to growing connections among task-agnostic neurons. This is necessary for two reasons. First, production is a way of exploring the structure space. The more extensively we explore, the better structure we expect to find. Second, production enables the network to better adapt to new tasks which may require new structures (see Figure \ref{fig:Production}).
We integrate production into the feedforward pass of the network, which means connections are created layer by layer. We design three production principles that help enhance the efficiency, sufficiency, and diversity of exploration. 1) Efficiency. In each feedforward pass, only the activated neurons can grow connections to the next layer. In this way, newly created structures are more likely to be activated and evaluated. Also, this helps capture regularities in the input data. 2) Sufficiency. To ensure that sufficient structures are explored, production is triggered if there are not enough activations in the next layer. 3) Diversity. The created structures should also be diverse. This is required by a variety of abstractions. We promote diversity by reducing the production probability of neurons which already have a large number of outgoing connections.
\begin{figure}
\begin{minipage}[t]{0.45\textwidth}
\centering
\includegraphics[width=\linewidth]{2_pe.pdf}
\caption[Production]{Production. The dark units repre
-sent activated neurons, while the light ones
do not. The dashed units denotes the newly
activated neurons.}
\label{fig:Production}
\end{minipage}
\quad
\begin{minipage}[t]{0.45\textwidth}
\centering
\includegraphics[width=0.84\linewidth]{3_pe.pdf}
\caption[Competion]{Competition. The dark units denote winners, while the light ones denote losers. The dashed units indicate the dying neurons.}
\label{fig:Competition}
\end{minipage}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[height=0.75in,width=1in]{4.pdf}
\subcaption{Death}
\label{fig:Selection Death}
\end{subfigure}\quad
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[height=0.75in,width=1in]{5.pdf}
\subcaption{Growth}
\label{fig:Selection Death}
\end{subfigure}
\caption[Selection]{(a) The third abstraction is not good enough and it dies along with the structure it is connected to. (b) The third one is new abstraction produced by a new structure.}
\label{fig:Selection}
\end{figure}
\textbf{Competition.} We introduce competition among task-agnostic neurons within each layer. Specifically, for a given input, the neurons with the highest output values win the competition. Only the winners propagate their outputs to the next layer; other neurons are de-activated. This is similar in spirit to winner-take-all circuits which are common modules in the brain.
We believe that competition brings two benefits. First, it promotes the variety and simplicity of abstractions, because it controls the size of activated subnetworks and the number of activated abstractions. Second, it prunes unpromising structures on the fly – structures that contain deactivated neurons will not be evaluated and selected for the given task. This is essential to maintain search efficiency. Because the structure space is enormous and task-specific evaluation is costly, we can only afford to evaluate a very small proportion of all possible structures.
Since the network capacity is fixed, we need to give priority to promising structures and delete others if necessary. Therefore, we keep track of the promising structures by maintain a reward variable $r_{j}$ for each neuron $j$. Specifically, $r_{j}$ is incremented each time neuron $j$ wins a competition, and decremented when it loses.When $r_j$ is less than a threshold, the corresponding neuron dies (see Figure \ref{fig:Competition}).
\textbf{Selection.} Selection aims to optimize the abstractions for task effectiveness. Because the objective function is non-differentiable, we take an iterative approach instead of standard gradient-based methods. At each iteration, we select the most effective abstractions from those that satisfy the variety and simplicity constraints. We preserve these abstractions and their corresponding structures, and eliminate others to allow new abstractions to be produced (see Figure \ref{fig:Selection}). In order to make production more efficient, we guide the direction of production by giving advantage to components of effective structures in the production process.
Evaluating task effectiveness is costly, because computing the entropy of activation distributions requires a certain amount of input data. Consequently, compared to production and competition, selection operates on a larger time scale, and leads to significant modifications to the network structure.
The detailed algorithm for these three components is in the appendix.
\section{Experiments}
In all experiments of this section, we verify the feasibility of the model on the issues of single task and multiple consecutive tasks respectively. The prediction accuracy and effectiveness of the model are the main concerns, andwe verify effectiveness in terms of the forgotten rate of knowledge and the sharing rate of abstractions. In order to evaluate the performance of ONE more clearly, no special strategies, such as augmenting data with transformations, dropout or noise, were used. We did not fine-tune any parameters such as parameters initialization in the abstraction layer, either. Thus, our objective is to evaluate the feasibility of the model rather than achieving the absolute best testing scores. Moreover, it is well known that one of the drawbacks of evolution algorithms is that the speed of convergence is slow, so we only validate the feasibility on MNIST dataset. In the future, we will verify our ideas on more datasets.
The MNIST dataset consists of 60K training images and 10K test images of handwritten digits from 0 to 9, where the size of each image is $28\times 28$. In the following experiments, we used all available training data to train the model. Furthermore, we divided the whole classification task into 5 tasks, each of which is a binary classification task on each class.
Our model introduces several hyper-parameters , which we refer to as extinction and growth parameters, in the process of generating abstractions. In terms of growth parameters, parameter $a_l$ controls the minimum number of neurons that each training sample activates on layer $l$, parameter $b_l$ indicates that each neuron in layer $l$ can be connected to $b_l$ neurons in layer $l-1$ at most, parameter $c_l$ controls the number of neurons that win in layer $l$ by directly enforcing a winner-take-all sparsity constraint and coordinates with $a_l$ to produce good neurons. In terms of extinction parameters, parameter $d$ and parameter $e$ jointly indicate that when the activation number of a neuron is greater than $d$ and the activation number of synapses connected with the neuron is less than $e$, then the synaptic dies. And if a neuron does not have any outgoing connection, the neuron dies. If growth parameters are set too large, they will promote the exploration of the structure, so as to find the optimal structure, but it is easy to cause the redundancy of the structure. On the contrary, if growth parameters are set too small, the structure exploration is slow and the number of shared neurons are reduced, which is not conducive to the generation of abstractions. What’s more, too large extinction parameters will not be conducive to the accumulation of abstractions, while too small parameters lead to abstraction redundancy. Instead of using the 2D structure of the images, we turn each image into a vector of pixels, where the pixel values were rescaled to $[0,\ 1]$. According to the general configuration of CNNs, we set the parameters as follows: the configuration of layers in the cognitive part: $\left\{784,\ 500,\ 500,\ 500\right\}$, the configuration of layers in the decision part: $\left\{200,\ 50,\ 2\right\}$, $a=\left\{200,\ 160,\ 100\right\}$, $b=\left\{50,\ 50,\ 50\right\}$, $c=\left\{100,\ 80,\ 60\right\}$, $d=3000$, $e=300$.
The second column of Table \ref{table:Test accuracy} shows the experimental results in five tasks. ONE obtains pretty good test scores, which proves the feasibility of the model. The reason why the recognition rates for task 2 and 5 are not as good as those for the rest 3 tasks may be that the difference between the digits is small.
\begin{figure*}[t]
\begin{minipage}{\linewidth}
\centering
\begin{minipage}[c]{0.56\linewidth}
\captionsetup{type=figure}
\centering
\subcaptionbox{Abstraction generation\label{fig:Abstraction generation}}
[0.49\linewidth] {\includegraphics[height=0.95in]{6.pdf}}
\subcaptionbox{Abstraction sharing\label{fig:Abstraction sharing}}
[0.49\linewidth] {\includegraphics[height=0.95in]{7.pdf}}
\caption{(a) shows the proportion of abstractions generated by the previous tasks in the total abstractions used by the current task. (b) shows the total number of abstractions as tasks increase.}
\label{fig:Abstraction generation and sharing}
\end{minipage}
\quad
\begin{minipage}[c]{0.4\linewidth}
\centering
\captionof{table}{Test accuracy on MNIST without data augmentation.}
\label{table:Test accuracy}
\resizebox{1.0\linewidth}{!}
{\begin{tabular}{@{}ccccc@{}}
\toprule[1.0pt]
& Subtask & Single Task & Consecutive tasks\\
\midrule[0.5pt]
& 1 & 99.81\% & 99.81\% \\
& 2 & 97.06\% & 97.01\% \\
& 3 & 98.83\% & 98.67\% \\
& 4 & 99.04\% & 99.45\% \\
& 5 & 96.72\% & 97.23\% \\
\bottomrule[1.0pt]
\end{tabular}}
\end{minipage}
\end{minipage}
\end{figure*}
In incremental learning, tasks come in sequence. All parameters in the cognitive part are shared, including structure, weight and bias, while the parameters in the decision part for each task are not shared. The third column of Table \ref{table:Test accuracy} shows the test accuracy of each task in incremental learning. Our model, ONE, performs almost the same as these batch models in single-task experiments, and even outperforms them in the last two tasks. Figure \ref{fig:Abstraction sharing} shows that ONE performs each task based on part of the total abstraction (500), and thus activates only a small proportion of the whole network for each task. To go further, we count the abstraction sharing rate between current tasks and all previous tasks (see figure \ref{fig:Abstraction generation}). The result shows that fewer new abstractions are produced as tasks increase, which means that abstractions produced by ONE are highly reusable. This reduces energy consumption and boosts learning speed. And the reason why the test accuracy increases in incremental learning is that some abstractions produced in other tasks are useful in current task, while they cannot be generated by current data. Moreover, because of the accumulation of the abstractions and the non-overlap of the decision part for different tasks, ONE does not forget, and thus can learn continually. We believe that abstractions play an extremely important role in lifelong learning, and the advantages of ONE will be more prominent as the number of tasks increases.
\section{Conclusion}
We proposed ONE, a completely new framework for learning abstractions and achieving human-like intelligence. The partition structure provides the foundation for abstraction accumulation across tasks, enabling knowledge transfer and lifelong learning. The constrained optimization formulation directly specifies the properties of abstractions, turning abstraction learning into a well-defined problem. The network evolution algorithm effectively searches through the structure space, ensuring the quality of abstractions.
While human beings may represent and learn abstractions in a different way, we have demonstrated that the abstractions learned by ONE can offer similar benefits as those learned by human beings. In the future, we plan to extend the experiments to larger datasets and perform cross-domain abstraction learning. Another interesting direction is to learn abstractions of tasks. This would allow sharing among task-specific parts, and also improve interpretability of decision making.
\bibliographystyle{plainnat}
|
1,116,691,497,771 | arxiv | \section{Introduction}
\subsection{} \label{sec:lrpair} Let $\kk$ be a field of characteristic zero, and
$A = \oplus_{n \geq 0} A_n$ be
an $\mathbb{N}$-graded Poisson algebra of degree $-1$, that is, a graded, commutative,
associative, unital $\kk$-algebra, whose multiplication is of degree $0$ (\emph{i.e.},
if $ a \in A_p, \: b \in A_q$, then $ a \cdot b \in A_{p+q}$),
together with a $\kk$-linear Lie algebra bracket of degree $-1$ (that is,
if $a \in A_p, \: b \in A_q$, we have $\left\{ a, b \right\} \in A_{p+q-1}$).
Both operations are subject to the Leibniz identity
\begin{equation} \label{eq:leibniz} \left\{ a, b \cdot c \right\} = c \cdot \left\{ a,b \right\} + b \cdot
\left\{ a, c \right\}.
\end{equation}
Then the pair $\mathscr{O} = A_0$, $\mathscr{E} = A_1$ is a \emph{Lie--Rinehart pair} (also known as a \emph{Lie
algebroid} in the geometric context). More precisely,
\begin{enumerate}
\item $\mathscr{O}$ is a commutative associative unital $\kk$-algebra.
\item $\mathscr{E}$ is an $\mathscr{O}$-module.
\item $\mathscr{E}$ is a $\kk$-linear Lie algebra with bracket $[-,-] = \{-,-\}|_\mathscr{E}$.
\item $\mathscr{E}$ acts on $\mathscr{O}$ by derivations, \emph{i.e.} there exists a $\kk$-linear map of Lie
algebras $\pi: \mathscr{E} \rightarrow \operatorname{Der}_\kk \mathscr{O}$, given by $\pi(a)(f) = \{a,f\}$ for
$a \in \mathscr{E}$ and $f \in \mathscr{O}$.
\end{enumerate}
These data are compatible in the following way:
\[ [a, f\cdot b] = f \cdot [a,b]+ \pi(a)(f) \cdot b, \qquad f \in \mathscr{O}, \: a,b
\in \mathscr{E}. \]
Conversely, given a Lie--Rinehart pair one canonically constructs a Poisson
algebra of degree $-1$ on the space $\operatorname{Sym}^*_\mathscr{O} \mathscr{E}$, by extending the Lie
bracket from $A_1 = \mathscr{E}$ using the Leibniz rule
\eqref{eq:leibniz}. As a commutative algebra, the symmetric algebra is freely
generated over $\mathscr{O}$ by its degree $1$ part given by $\mathscr{E}$. If we start with a graded
Poisson algebra of degree $1$ as above, and construct the corresponding
Lie--Rinehart pair, we obtain a canonical surjection
$ \operatorname{Sym}^*_\mathscr{O} \mathscr{E} \twoheadrightarrow A$,
which is the identity in degrees $0$ and $1$.
This provides a one--to--one correspondence between Lie--Rinehart pairs over $\kk$ and
graded Poisson algebras of degree $-1$, freely generated in degrees $0$ and $1$.
\subsection{} \label{sec:1.2}
Recall that a \emph{graded Poisson vertex algebra (PVA)} $A$ of degree $-1$ is a graded, commutative, associative, unital algebra
$ A = \oplus_{n \geq 0} A_n$,
together with a derivation of degree $1$, $\partial: A_\bullet \rightarrow
A_{\bullet+1}$, and a \emph{Lie conformal bracket} of degree $-1$, which is defined as an
operation
\[ A \otimes A \longrightarrow A[\lambda], \qquad a \otimes b \longmapsto
\{a_\lambda b\} = \sum_{j \geq
0} a_{(j)}b \frac{\lambda^j}{j!}, \]
making $A$ into a $\kk$-Lie conformal algebra \cite{kac:vertex} such that
\[ a \in A_p, \: b \in A_q \Longrightarrow a_{(j)}b \in A_{p+q-j-1}. \]
These operations are related by the Leibniz rule
\[ \left\{ a_\lambda b \cdot c\right\}= b \cdot \left\{ a_\lambda c \right\} + c
\cdot \left\{ a_\lambda b \right\}. \]
One of the main features of Poisson vertex algebras is that they are the quasi-classical limits of
Borcherds' vertex algebras.
Every vertex algebra $V$ comes equipped with a canonical decreasing filtration
known as the \emph{Li filtration} \cite{li05}:
\[V = F_{0} V \supset F_{1} V \supset \dots \supset F_p V \supset F_{p+1} V
\supset \cdots \]
The associated graded algebra $A = \mathrm{gr}_F V = \oplus_{p \geq 0} F_{p}V / F_{p+1} V$
carries a structure of Poisson vertex algebra. This is a commutative
associative graded algebra (the normally ordered product of $V$ descends to the
multiplication of $A$), with a unit (given by the image of the vacuum vector
$\mathbb{1} \in V$) and a derivation of degree $1$ (given by the translation
operator of $V$). In addition, the $\lambda$-bracket of $V$ descends to a Lie
conformal structure on $A$ satisfying the Leibniz rule with respect to the
multiplication.
The canonical example of a Poisson vertex algebra is given by the ring of
functions of the
\emph{arc space} of a Poisson manifold. If $P$ is a Poisson algebra, we put
$J_0 = P$, and let $J$ be the unital associative commutative algebra with a
derivation of degree 1, freely generated (as a differential algebra) by $J_0$.
For example, we will have $J_1 = \Omega^1 J_0$, the module of K\"ahler
differentials of $J_0$, and $\partial: J_0 \rightarrow J_1$ will be given by the
universal derivation. Then the Poisson bracket of $J_0$ can be uniquely extended
to a $\lambda$-bracket on $J$ by sesquilinearity and the Leibniz rule
\cite{arakawa}.
In the particular case where $P = \kk[x_1,\dots,x_n]$ is a polynomial algebra, we have that
$J = \kk[x_1^{(p)}, \dots, x_n^{(p)}]_{p \geq 0}$ is a differential
polynomial algebra. The derivation is defined by $\partial x_i^{(p)} =
x_i^{(p+1)}$ and the degrees of the generators are given by $\deg x_i^{(p)} =
p$.
It was noted in \cite{arakawa} that the quasi-classical limit of any vertex algebra is a quotient of the free
algebra $J$ described in the previous paragraph. Indeed,
given a vertex algebra $V$, one considers its associated graded algebra with respect to the Li
filtration, $A = \mathrm{gr}_F V$. Then $P := A_0$ is a Poisson algebra (known as
Zhu's $C_2$-quotient) and by the universal property, it follows that $A$ is a
quotient of the arc algebra $J$ of $P$ described above.
\subsection{}\label{sec:1.2-bis}
A \emph{chiral} or \emph{vertex} version of \S\ref{sec:lrpair} is
given as follows.
Let $A = \oplus_{n \geq 0} A_n$ be a graded Poisson vertex algebra of degree $-1$.
It was observed by Ekstrand and Zabzine \cite{EZ11}, following works of Bressler
\cite{bressler1}, Roytenberg \cite{Roy09} and the second author \cite{Hel09},
that the pair $\mathscr{O} = A_0$, $\mathscr{E} = A_1$ is a \emph{Courant--Dorfman algebra}\footnote{Note that in \cite{EZ11}, Ekstrand and Zabzine used the term `weak Courant--Dorfman algebra', rather than `Courant--Dorfman algebra'. However, as in the present article we will mainly follow \cite{Roy09}, hereafter we will omit the adjective `weak'.},
which consists of the following data:
\begin{enumerate}
\item
a commutative $\kk$-algebra $\mathscr{O}$.
\item
an $\mathscr{O}$-module $\mathscr{E}$.
\item
a symmetric bilinear form $\langle-,-\rangle\colon \mathscr{E}\otimes \mathscr{E}\to \mathscr{O}$, given
by $\langle a, b\rangle = a_{(1)}b$.
\item
an $\mathscr{E}$-valued derivation of $\mathscr{O}$: $\partial\colon \mathscr{O} \to \mathscr{E}$.
\item
a $\kk$-bilinear operation called the \emph{Dorfman bracket} $[-,-]\colon
\mathscr{E}\otimes \mathscr{E}\to \mathscr{E}$, given by $[a,b] = a_{(0)}b$.
\end{enumerate}
These data are required to satisfy the following compatibility conditions:
\begin{equation}
\begin{aligned}
[a, f \cdot b]&= f [a,b]+\langle a,\partial f\rangle b,
\\
\langle a,\partial \langle b,c\rangle\rangle&= \langle [a,b],c\rangle+\langle
a,[b,c]\rangle,
\\
\partial\langle a,b\rangle&=[a,b]+[b,a],
\\
[a,[b,c]]&=[[a,b],c]+[b,[a,c]],
\\
0&=[\partial f,a],
\\
0&=\langle\partial f,\partial g\rangle,
\end{aligned}
\label{eq:intro-axiomas-Couarnt-Dorfman}
\end{equation}
for all $a,b,c \in \mathscr{E}$ and $f,g\in \mathscr{O}$. If $\langle-,- \rangle$ is
non-degenerate, then the last two conditions and the first one are redundant and one
obtains the definition of \emph{Courant algebroids} of \cite{Roy00}.
Conversely, given a Courant--Dorfman algebra $(\mathscr{O}, \mathscr{E}, \partial)$, one canonically constructs a
graded Poisson vertex algebra $A$, freely generated in degrees $0$ and $1$,
such that $A_0 = \mathscr{O}$ and $A_1 = \mathscr{E}$. This gives a one--to--one correspondence between
graded Poisson vertex algebras of degree $-1$, freely generated in degrees $0$
and $1$, and Courant--Dorfman algebras.
\subsection{} There are different ways to produce noncommutative generalizations
or \emph{deformations} of the correspondences unveiled in
\S\ref{sec:lrpair}--\ref{sec:1.2-bis}. Given a Lie--Rinehart pair $(\mathscr{O},\mathscr{E})$ one can
construct its universal enveloping algebra $U(\mathscr{E})$ \cite{bs}, which is a filtered
associative algebra generalizing the algebra of differential operators in the
case $\mathscr{E} = \operatorname{Der}_{\kk} \mathscr{O}$, and the universal enveloping algebra of a Lie algebra in
the case $\mathscr{O} = \kk$. Its associated graded algebra coincides with $\operatorname{Sym}^*_\mathscr{O} \mathscr{E}$ as a
Poisson algebra.
The chiral rendition is more complicated. Given a
Courant--Dorfman algebra $(\mathscr{O}, \mathscr{E}, \partial)$
one aims at constructing a filtered vertex algebra, whose
associated graded Poisson vertex algebra recovers the given Courant--Dorfman
algebra in degrees $0$ and $1$. As explained in \cite{gerbes2} and \cite{bd}, there is an obstruction to constructing this algebra.
\subsection{}
\label{sec:main-results}
Another noncommutative generalization of the correspondence in \S\ref{sec:lrpair} was
given by Van den Bergh \cite{VdB08,VdB08a} by constructing a noncommutative
generalization of a Poisson algebra using his notion of
\emph{double Poisson algebra}. It consists of an associative $\kk$-algebra $A$ and a linear map
\[
\lr{-,-}\colon A\otimes A\longrightarrow A\otimes A,
\]
satisfying suitable skewsymmetry and Leibniz rules, as well as the (double) Jacobi identity; see \eqref{eq:double-bracket} and \eqref{double-Jacobi-Poisson}. Remarkably, if $a\in A$, the Leibniz rule \eqref{eq:double-bracket.b} can be rephrased by saying that $\lr{a,-}\colon A\to A\otimes A$ is a \emph{double derivation} (cf. \S\ref{sec:nc-diff-forms-double-derivations}) rather than a plain derivation.
Besides giving a powerful notion of noncommutative Poisson algebras with
significant applications in geometric representation theory, Van den Bergh's
double Poisson algebras have been used in \cite{ORS,Artha} to develop an
interesting theory of noncommutative Hamiltonian Ordinary Differential Equations
(ODEs), which extends the well-known connection in classical mechanics between
(classical) Poisson geometry and (commutative) Hamiltonian ODEs. Furthermore, in
classical field theory, the study of Hamiltonian Partial Differential Equations
(PDEs) is realized by Poisson vertex algebras.
In their groundbreaking article \cite{DSKV15}, De Sole,
Kac and Valeri started the study of integrable PDEs in noncommutative variables by introducing the fundamental notion of \emph{double Poisson vertex algebras}, which are a cross between Van den Bergh's double Poisson algebras and familiar Poisson vertex algebras, as discussed in \S\ref{sec:1.2}.
They rely on the notion of a
\emph{double $\lambda$-bracket} that consists of a bilinear map
\[
\lr{-{}_\lambda-}\colon\mathcal{V}\otimes\mathcal{V}\longrightarrow (\mathcal{V}\otimes \mathcal{V})[\lambda]
\]
satisfying appropriate sesquilinearity conditions, Leibniz rules, skewsymmetry and the Jacobi identity (see Definitions \ref{def:double-Poisson-vertex-alg} and \ref{DPVA}), resembling the corresponding axioms in the definition of double Poisson algebras. Note that in \cite{CW}, Casati and Wang introduced a \emph{multiplicative} analogue of double Poisson vertex algebras with applications in differential-difference Hamiltonian equations (see also \cite{CM} for a multiplicative perspective on the theory of integrable systems in noncommutative variables).
In the light of the discussion depicted in \S\ref{sec:1.2-bis}, it is natural to ask whether the double Poisson vertex algebras as introduced in \cite{DSKV15} can be put in correspondence with suitable noncommutative generalizations of Courant--Dorfman algebras.
The main goal of this article is to show that the answer is in the affirmative.
Firstly, in \cite{ACF17} (see also \cite{Fer16}), \'Alvarez-C\'onsul and the first author were able to unveil noncommutative analogues of the prominent Courant algebroids, called \emph{double Courant algebroids}. They were obtained by developing a differential-graded symplectic (Batalin--Vilkovisky) version of the noncommutative geometry based on double derivations as in \cite{CBEG07,VdB08,VdB08a} modeled around the notion of \emph{bisymplectic $\mathbb{N}Q$-algebras}: noncommutative versions of symplectic $\mathbb{N}Q$-manifolds originally introduced in \cite{ACF15}.
Hence, following ideas of \cite{Roy00,Sev00}, from a bisymplectic $\mathbb{N}Q$-algebra of weight 2 the axioms characterizing double Courant algebroids were established. Based on them, in this article we succeed in introducing the notion of \emph{double Courant--Dorfman algebras} as a 5-tuple $(A,E,\ii{-,-},\partial,\cc{-,-})$, where $E$ is a bimodule over an associative $\kk$-algebra $A$, a derivation $\partial\colon A\to E$ and a \emph{pairing} $\ii{-,-}\colon E\otimes E\to A\otimes A$ in the sense of \cite{VdB08a} --- see \S\ref{sec:pairings}.
The main ingredient is the
\emph{double Courant--Dorfman bracket} on the tensor algebra $T_A E$: a bilinear map
\[ \cc{-,-}\colon T_AE\otimes T_AE\longrightarrow T_AE\otimes T_AE, \]
of degree -1. It is required that these data satisfy natural compatibility conditions reminiscent of \eqref{eq:intro-axiomas-Couarnt-Dorfman}; see Definitions \ref{def:double-Courant-Dorfman-bracket} and \ref{CD-Definition}.
We could write that double Poisson algebras and double Poisson vertex algebras were noncommutative generalizations of familiar Poisson algebras and Poisson vertex algebras, respectively, since, as shown in \eqref{eq:KR-double-Poisson-algebras} and Proposition \ref{prop:KR-double-Poisson-vertex-algebras}, they satisfy the paradigm in noncommutative geometry: the \emph{Kontsevich--Rosenberg principle}. As explained in \S\ref{sec:KR-DPVA-SECTION}, it states that a noncommutative algebro-geometric structure on an associative algebra should naturally induce the corresponding standard algebro-geometric structure on its representation varieties. One of the main results of this article is Theorem \ref{thm:KR-double-Courant-Dorfman} where we prove that our double Courant--Dorfman algebras, as introduced in Definition \ref{CD-Definition}, do satisfy the Kontsevich--Rosenberg principle, inducing the familiar Courant--Dorfman algebras on representation varieties.
As in \cite[\S7]{VdB08} and \cite[\S3.7]{DSKV15}, to prove this theorem, we take advantage of the explicit description based on indices of the coordinate ring of the representation variety and bimodules over it --- see \eqref{eq:KR-relations-defining-AV} and \eqref{eq:KR-relaciones-bimodules}.
Once we ascertain that double Courant--Dorfman algebras are the objects that we were looking for, we are in position to prove Theorem \ref{theorem-CD-DPVA}, which is the main result of this article.
It states that there exists a one--to--one correspondence between
double Courant--Dorfman algebras and graded double Poisson vertex algebras of
degree $-1$, freely generated in degrees $0$ and $1$, providing a noncommutative version of the
correspondence between Poisson vertex algebras and Courant--Dorfman algebras presented in \S\ref{sec:1.2-bis}.
The key point of the proof is the following identity that relates double Poisson $\lambda$-brackets and the data characterizing double Courant--Dorfman algebras:
\[
\lr{e{}_\lambda f}=\cc{e,f}+\ii{e,f}\lambda,
\]
for all $e,f\in E$.
\subsection{} As shown in \cite{vaintrob}, there exists yet another geometric interpretation for the notion of a
Lie algebroid $(\mathscr{O}, \mathscr{E})$ in terms of supermanifolds: the exterior algebra $\wedge_\mathscr{O} \mathscr{E}$ is the algebra
of functions of a \emph{symplectic supermanifold} of weight $1$. Similarly,
associated to a Courant algebroid, Roytenberg \cite{Roy00} constructs a symplectic
$\mathbb{N}Q$-manifold of weight 2.
Furthermore, as pointed out in \S\ref{sec:main-results},
\'Alvarez-C\'onsul and the first author \cite{ACF17} defined double Courant algebroids using bisymplectic $\mathbb{N}Q$-algebras of weight 2, which are
noncommutative counterparts of the symplectic $\mathbb{N}Q$-manifolds of weight $2$ of Roytenberg.
Hence, this article completes the chart below, by showing the
relation between double Courant--Dorfman algebras and double Courant algebroids, noncommutative symplectic
$\mathbb{N}Q$-geometries and double Poisson vertex algebras:
\begin{center}
\begin{tabular}[center]{|c|c|c|}
\hline
\textbf{Geometry} & \textbf{Graded geometry} & \textbf{Algebra} \\
\hline
\hline
\begin{tabular}{@{}c@{}}Lie algebroids/ \\ Lie--Rinehart {pairs}\end{tabular}
& \begin{tabular}{@{}c@{}}Symplectic supermanifold \\ of weight 1\end{tabular}
& Poisson
\\
\hline
\begin{tabular}{@{}c@{}}Courant algebroids/ \\ Courant--Dorfman algebras\end{tabular}
& \begin{tabular}{@{}c@{}}Symplectic $\mathbb{N}Q$-manifolds \\ of weight 2\end{tabular}
& PVA
\\
\hline
\begin{tabular}{@{}c@{}}Double Courant algebroids/ \\ double Courant--Dorfman algebras\end{tabular}
& \begin{tabular}{@{}c@{}}Bisymplectic $\mathbb{N}Q$-algebras \\ of weight 2\end{tabular}
& double PVA
\\
\hline
\end{tabular}
\end{center}
\subsection{}
The organization of this article is as follows. After establishing
basic notational conventions in Section
\ref{sec:notation}, we collect in
Section \ref{sub:basics} the basic notions of noncommutative geometry needed in the rest of the article.
In Section \ref{sec:DPVA} we recall the notion of
double Poisson vertex algebras following \cite{DSKV15}.
In Section \ref{sec:CD-algebras} we recall the definitions of Courant--Dorfman algebras
following \cite{Roy09} and we define their noncommutative counterparts: double Courant--Dorfman algebras, which are one of the key objects of our work.
In Section
\ref{sec:equivalence} we prove our main result, Theorem \ref{theorem-CD-DPVA}, establishing an equivalence
between double Courant--Dorfman algebras and double Poisson vertex algebras, freely
generated in degrees $0$ and $1$. Finally, Section \ref{sec:KR-double-CD} is devoted to establish the Kontsevich--Rosenberg principle for double
Courant--Dorfman algebras.
In Appendix \ref{app:some-useful} we collect two some technical results that we need to prove our main results, which may be of independent interest.\\
\noindent\textbf{Acknowledgments.}
The first author is supported by the Alexander von Humboldt Stiftung in the framework of an Alexander von Humboldt professorship endowed by the German Federal Ministry of Education and Research.
During the initial stage of this work, he was supported by IMPA and CAPES through their postdoctorate of excellence fellowships at UFRJ. He deeply acknowledges their support and excellent working conditions.
The second author is partially supported by CNPq grant number 305688/2019-7.
The authors wish to thank Henrique Bursztyn, Alejandro Cabrera and Mario Garc\'ia-Fern\'andez for useful conversations. Special thanks are due to Daniele Valeri for his kind and enlightening emails, and to Luis \'Alvarez-C\'onsul for his interest in this project and encouragement, as well as for drawing our attention to the crucial article \cite{DSKV15}.
\section{Notation and Conventions} \label{sec:notation}
Throughout this article, all associative and commutative algebras will
be unital and finitely generated over a base field $\kk$ of
characteristic 0. In fact, a $\kk$-algebra $A$ will mean a unital and finitely generated associative (not necessarily commutative) $\kk$-algebra. Unless otherwise stated, all unadorned notation will mean over the base field $\kk$; in particular, $\otimes\mathrel{\mathop:}=} %{\coloneqq} %{:= \otimes_\kk$ and $\operatorname{Hom}(-,-):=\operatorname{Hom}_\kk(-,-)$. The opposite algebra and the enveloping algebra of an
associative algebra $A$ will be denoted $A^{\op}$ and $A^{\e}\mathrel{\mathop:}=} %{\coloneqq} %{:=
A\otimes A^{\op}$, respectively. We will identify the category of $A$-bimodules and the category of (left) $A^{\e}$-modules.
Given two $A$-bimodules $E$ and $F$, their tensor product $E\otimes F$ has two commuting $A$-bimodule structures, called the \emph{outer} and the \emph{inner} $A$-bimodule structures, denoted $(E\otimes F)_{\out}$ and $(E\otimes F)_{\inn}$, respectively, given by
\begin{align*}
a'(e\otimes f) a''&= (a'e)\otimes (fa'')\quad \text{ on } (E\otimes F)_\out,
\\
a'*(e\otimes f) *a''&= (ea'')\otimes (a'f) \quad \text{ on } (E\otimes F)_\inn,
\end{align*}
for all $a',a''\in A$, and $e\otimes f\in E\otimes F$. In particular, the underlying $A$-bimodule of $A$ determines two $A$-bimodules with the same underlying vector space $A\otimes A$: $(A\otimes A)_{\out}$ and $(A\otimes A)_{\inn}$. Unless explicitly stated, we endow the tensor product $E\otimes F$ (and in particular $A\otimes A_{\out}$) with the outer $A$-bimodule structure.
We will systematically use Sweedler's notation, consisting of dropping the summation sign: if $v\in E\otimes F$, we write $v=v^{\prime}\otimes v^{\prime\prime}\in E\otimes F$, where $v^{\prime}\in E$ and $v^{\prime\prime}\in F$.
Furthermore, following \cite[\S 1.3]{DSKV15}, given $n\in\mathbb{N}_{\geq 2}$, for every $i=0,\dots ,n-1$, we define the $i$-th left and right $A$-module structures of $A^{\otimes n}$ by
\begin{equation}
\begin{aligned}
a^\prime*_i(a_1\otimes\cdots \otimes a_n)&=a_1\otimes\cdots\otimes a_i\otimes a^\prime a_{i+1}\otimes\cdots\otimes a_n,
\\
(a_1\otimes\cdots\otimes a_n)*_i a^{\prime\prime}&=a_1\otimes\cdots\otimes a_{n-i}a^{\prime\prime} \otimes\cdots\otimes a_n,
\end{aligned}
\label{square-op}
\end{equation}
for all $a^{\prime},a^{\prime\prime},a_1,\dots,a_n\in A$.
The index denotes the number of ``jumps''. In particular, the outer bimodule structure can be written as $a^{\prime}*_0(a_1\otimes a_2) *_0 a^{\prime\prime}$, whilst the inner bimodule structure as $a^{\prime}*_1(a_1\otimes a_2)*_1 a^{\prime\prime}$. Moreover, following \cite[\S 1.3]{DSKV15}, let $E_i$ be an $A$-bimodule for $1\leq i\leq 3$, and given $e_i\in E_i$, we define
\begin{equation}
\begin{aligned}
e_1\otimes_1 (e_2\otimes e_3) &\mathrel{\mathop:}=} %{\coloneqq} %{:= e_2\otimes e_1\otimes e_3\in E_2\otimes E_1\otimes E_3,
\\
(e_1\otimes e_2)\otimes_1 e_3&\mathrel{\mathop:}=} %{\coloneqq} %{:= e_1\otimes e_3\otimes e_2\in E_1\otimes E_3\otimes E_2.
\end{aligned}
\label{jumping-notation}
\end{equation}
We define the \emph{bidual} of the $A$-bimodule $E$ as the $A$-bimodule
\begin{equation}
M^\vee:=\operatorname{Hom}_{A^\e}(E,(A\otimes A)_\out),
\label{eq:bidual-def}
\end{equation}
where the $A$-bimodule structure is induced by the one in $(A\otimes A)_\inn$ (\emph{i.e.} $(a\phi b)e=a*(\phi e)*b=(\phi^{\prime}e)b\otimes a(\phi^{\prime\prime}e)$, for all $\phi\in E^\vee$, $e\in E$, $a,b\in A$).
Let $n\in\mathbb{N}$ and $V_1,\dots,V_n$ be vector spaces. We will denote by $\mathbb{S}_n$ the group of permutations of $n$ elements $\{1,\dots,n\}$, which acts on $V_1\otimes\cdots\otimes V_n$ in the usual way.
If $s\in\mathbb{S}_n$, we have a linear map
\[
\sigma_s\colon V_1\otimes\cdots\otimes V_n\longrightarrow V_{s^{-1}(1)}\otimes\cdots\otimes V_{s^{-1}(n)},
\]
given by the formula
\[
\sigma_s(v)=v_{s^{-1}(1)}\otimes\cdots\otimes v_{s^{-1}(n)},
\]
for all $v=v_1\otimes\cdots\otimes v_n$, with $v_i\in V_i$.
If $s=(1\dots n)$ is the cyclic permutation (which sends 1 to 2), we use the notation
\[
(v_1\otimes\cdots\otimes v_{n-1}\otimes v_n)^{\sigma}=v_n\otimes v_1\otimes\cdots\otimes v_{n-1}.
\]
\section{Basics on noncommutative geometry}
\label{sub:basics}
In this section, following \cite{CBEG07, VdB08, VdB08a}, we will introduce the basics on noncommutative geometry that we shall need below. One of the main features of our approach to noncommutative geometry is that the role of vector fields is played by \emph{double derivations} that we will define in \S\ref{sec:nc-diff-forms-double-derivations}. Since double Poisson vertex algebras in \S\ref{sec:DPVA} will be based on them, in \S\ref{sec:double-Poisson}, we introduce \emph{double Poisson algebras} \cite{VdB08}; furthermore, they are noncommutative analogues of Poisson algebras --- see \S\ref{sec:KR-double-CD}.
Finally, we define (symmetric) pairings in \S\ref{sec:pairings}, which will be extensively used in \S\ref{sec:CD-algebras}--\ref{sec:KR-double-CD}.
\subsection{Noncommutative differential forms and double derivations}
\label{sec:nc-diff-forms-double-derivations}
Let $A$ be a unital associative $\kk$-algebra (the unit is denoted by $1$), and $E$ be an $A$-bimodule.
A \emph{derivation} of $A$ into $E$ is an
additive map $\theta\colon A\to E$ satisfying $\theta(1)=0$, and the Leibniz rule
$\theta(ab) =(\theta a)b+a(\theta b)$, for all $a,b\in A$.
The vector space of derivations $\theta\colon A\to E$ is denoted $\operatorname{Der}(A,E)$. If $E=A$ (regarded as an $A$-bimodule over itself), we simply write $\operatorname{Der} A$.
We define the $A$-bimodule of \emph{noncommutative differential 1-forms} $\Omega^1_{{\operatorname{nc}}} A$ as the kernel of the multiplication of $A$ (regarded as an $A$-subbimodule of $A\otimes A$).
It carries the \emph{universal derivation} $\du\,$, defined as the derivation $\du\,\colon A\to \Omega^1_{{\operatorname{nc}}} A$, $a\mapsto \du a\mathrel{\mathop:}=} %{\coloneqq} %{:= 1\otimes a-a\otimes 1$. Cuntz--Quillen \cite{CQ95} proved that the pair $(\Omega^1_{{\operatorname{nc}}}A,\du\,)$ satisfies the following universal property: for any derivation $\theta\colon A\to E$, there exists a unique $A$-bimodule morphism $i_\theta\colon\Omega^1_{{\operatorname{nc}}}A\to E$ such that $\theta=i_\theta\circ\du\,$.
It is convenient to rephrase this universal property by saying that there
exists a canonical isomorphism of $A$-bimodules
\begin{equation}\label{sub:ncdiffforms.corep}
\operatorname{Der}(A,E)\lra{\cong}\operatorname{Hom}_{A^\e}(\Omega^1_{{\operatorname{nc}}}A,E),\quad \theta\longmapsto i_\theta,
\end{equation}
whose inverse map is given by
$i_\theta\mapsto\theta=i_\theta\circ\du\,$.
Taking the bidual \eqref{eq:bidual-def} of the $A$-bimodule $\Omega^1_{{\operatorname{nc}}}A$, we obtain
\begin{equation}
(\Omega^1_{{\operatorname{nc}}}A)^\vee\mathrel{\mathop:}=} %{\coloneqq} %{:=\operatorname{Hom}_{A^\e}\big(\Omega^1_{{\operatorname{nc}}}A,(A\otimes A)_\out\big)\simeq \operatorname{Der}\big(A, (A\otimes A)_{\out}\big),
\label{double canonical isomorphism}
\end{equation}
where the last isomorphism comes from \eqref{sub:ncdiffforms.corep} with $E=(A\otimes A)_{\out}$.
Hence, we define the $A$-bimodule
$\D A\mathrel{\mathop:}=} %{\coloneqq} %{:=\operatorname{Der} (A,(A\otimes A)_{\out})$, whose $A$-bimodule structure is induced by the inner $A$-bimodule structure (that is, $(a\Delta b)(c)=a*\Delta(c)*b=\Delta^\prime(c)b\otimes a\Delta^{\prime\prime}(c)$, for all $a,b,c\in A$, $\Delta\in\D A$). The elements $\Delta\colon A\to A\otimes A$ of $\D A$ are called \emph{double derivations}.
\subsection{Double Poisson algebras}
\label{sec:double-Poisson}
A \emph{double bracket} \cite{VdB08} on $A$ is a linear map
\[
\lr{-,-}\colon A\otimes A\longrightarrow A\otimes A,\quad a\otimes b \longmapsto \lr{a,b}^\prime\otimes\lr{a,b}^{\prime\prime},
\]
satisfying
\begin{subequations}
\label{eq:double-bracket}
\begin{align}
\lr{a,b}&=\lr{b,a}^\sigma,\label{eq:double-bracket.a}
\\
\lr{a,bc}&=b\lr{a,c}+\lr{a,b}c, \label{eq:double-bracket.b}
\end{align}
\end{subequations}
for all $a,b,c\in A$.
Note that \eqref{eq:double-bracket.b} means that $\lr{a,-}$ is a double derivation in its second entry (with respect to the outer structure on $A\otimes A$). Also, \eqref{eq:double-bracket.a} and \eqref{eq:double-bracket.b} imply that $\lr{ab,c}=a*\lr{b,c}+\lr{a,c}*b$; in other words, a double bracket is a double derivation in its first entry for the inner bimodule structure on $A\otimes A$.
To define an analogue of the Jacobi identity in this setting, we require a suitable extension of the double bracket $\lr{-,-}$. Let $a\in A$, $b=b_1\otimes\cdots\otimes b_n \in A^{\otimes n}$, then we define
\begin{equation}
\begin{aligned}
\lr{a,b}_L&\mathrel{\mathop:}=} %{\coloneqq} %{:= \lr{a,b_1}\otimes b_2\otimes \cdots\otimes b_n\in A^{\otimes(n+1)},
\\
\lr{a,b}_R&\mathrel{\mathop:}=} %{\coloneqq} %{:= b_1\otimes\cdots\otimes b_{n-1}\otimes\lr{a,b_n}\in A^{\otimes (n+1)}.
\label{extension-left}
\end{aligned}
\end{equation}
\begin{definition}[\cite{VdB08}, Definition 2.3.2]
A double bracket $\lr{-,-}$ on a unital associative $\kk$-algebra $A$ is a \emph{double Poisson bracket} if
\begin{equation}
0=\lr{a,\lr{b,c}}_L+\sigma_{(123)}\lr{b,\lr{c,a}}_L+\sigma_{(132)}\lr{c,\lr{a,b}}_L.
\label{double-Jacobi-Poisson}
\end{equation}
\label{def:double-Poisson algebra}
\end{definition}
The equation \eqref{double-Jacobi-Poisson} is called the \emph{double Jacobi identity}.
In \cite[Remark 2.2]{DSKV15}, De Sole--Kac--Valeri rewrote \eqref{double-Jacobi-Poisson} in a more convenient way by extending the double brackets in the first argument as well.
If $a\in A$ and $b=b^{\prime}\otimes b^{\prime\prime}\in A\otimes A$; by
\eqref{jumping-notation},
\begin{equation}
\begin{aligned}
\lr{b^\prime\otimes b^{\prime\prime},a}_L&\mathrel{\mathop:}=} %{\coloneqq} %{:= \lr{b^\prime,a}\otimes_1 b^{\prime\prime}=\lr{b^\prime,a}^{\prime}\otimes b^{\prime \prime}\otimes\lr{b^\prime,a}^{\prime\prime},
\\
\lr{b^\prime\otimes b^{\prime\prime},a}_R&\mathrel{\mathop:}=} %{\coloneqq} %{:= b^\prime\otimes_1\lr{b^{\prime\prime},a}=\lr{b^{\prime\prime},a}^\prime\otimes b^{\prime}\otimes \lr{b^{\prime\prime},a}^{\prime\prime}.
\end{aligned}
\label{notation-Kac}
\end{equation}
Then by \eqref{notation-Kac} and \eqref{eq:double-bracket.a}, the double Jacobi identity \eqref{double-Jacobi-Poisson} can be equivalently written as
\begin{equation}
\lr{a,\lr{b,c}}_L=\lr{\lr{a,b},c}_L+\lr{b,\lr{a,c}}_R.
\label{Jacobi-Kac}
\end{equation}
\subsection{Pairings}
\label{sec:pairings}
Let $A$ be an associative $\kk$-algebra, and $E$ be an $A$-bimodule. Following \cite[\S 3.1]{VdB08a} (see also \cite[\S A.3]{Ke11}), a map
\begin{equation}
\ii{-,-}\colon E\otimes E \longrightarrow A\otimes A
\label{eq:pairings-def}
\end{equation}
is called a \emph{pairing} on $E$ if for all $e\in E$, the maps
\begin{align*}
\ii{e,-}\colon E&\longrightarrow (A\otimes A)_{\out}
\\
\ii{-,e}\colon E&\longrightarrow (A\otimes A)_{\inn}
\end{align*}
are $A$-bimodule morphisms.
The pairing is called \emph{non-degenerate} if $E$ is a finitely generated projective $A$-bimodule and $\ii{-,-}$ induces an isomorphism $E\simeq E^\vee$ --- recall \eqref{eq:bidual-def}.
Using Sweedler's notation to omit the summation symbol, we will write
\[
\ii{e_1,e_2}=\ii{e_1,e_2}'\otimes \ii{e_1,e_2}'', \quad\text{with }\ii{e_1,e_2}',\ii{e_1,e_2}''\in A.
\]
A pairing is called \emph{symmetric} if $\ii{e_1,e_2}=\ii{e_2,e_1}^\sigma$, for all $e_1,e_2\in E$ or, equivalently,
\begin{equation}
\ii{e_1,e_2}'\otimes \ii{e_1,e_2}''= \ii{e_2,e_1}''\otimes \ii{e_2,e_1}'.
\label{eq:Sweedler-pairing-symm}
\end{equation}
Finally, similarly to \eqref{extension-left} and \eqref{notation-Kac}, we extend the pairing $\ii{-,-}$ to $E\otimes A\oplus A\otimes E$ using the inner and the outer tensor products for the first and the second entry, respectively. To do that, using \eqref{jumping-notation} we define the maps
\begin{equation}
\label{eq:extension-pairing-first-second}
\begin{aligned}
\ii{e_1,-}_L\colon E\otimes A &\longrightarrow A^{\otimes 3},\quad & e_2\otimes a &\longmapsto \ii{e_1,e_2}\otimes a,
\\
\ii{e_1,-}_R\colon A\otimes E &\longrightarrow A^{\otimes 3},\quad & a\otimes e_2 &\longmapsto a\otimes \ii{e_1,e_2},
\\
\ii{-,e_2}_L\colon E\otimes A &\longrightarrow A^{\otimes 3},\quad & e_1\otimes a &\longmapsto \ii{e_1,e_2}\otimes_1 a=\ii{e_1,e_2}'\otimes a\otimes \ii{e_1,e_2}'',
\\
\ii{-,e_2}_R\colon A\otimes E &\longrightarrow A^{\otimes 3},\quad &a\otimes e_1&\longmapsto a\otimes_1 \ii{e_1,e_2}=\ii{e_1,e_2}'\otimes a\otimes \ii{e_1,e_2}'',
\end{aligned}
\end{equation}
Furthermore, we require that
\begin{equation}
\ii{e_1,a\otimes e_2}_L=\ii{e_1,e_2\otimes a}_R=\ii{a\otimes e_1,e_2}_L=\ii{e_1\otimes a,e_2}_R=0,
\label{eq:ext-pairing-zero}
\end{equation}
for all $a\in A$ and $e_1,e_2\in E$.
\section{Double Poisson vertex algebras}
\label{sec:DPVA}
In \cite{DSKV15}, De Sole, Kac and Valeri developed a theory of integrable systems on noncommutative associative algebras and, remarkably, they introduced double Poisson vertex algebras, which we present in this section.
A \emph{differential algebra} \cite[\S 3.1]{DSKV15} $(\mathcal{V},\partial)$ is a pair consisting of an associative algebra $\mathcal{V}$, and a derivation $\partial\in\operatorname{Der}\mathcal{V}$.
Given a polynomial $P(\lambda)=\sum a_1\otimes\dots \otimes a_n\lambda^{k_1}_1\cdots\lambda^{k_{n-1}}_{n-1}\in \mathcal{V}^{\otimes n}[\lambda_1,\dots,\lambda_{n-1}]$, we define
\begin{align*}
(e^{\partial\partial_{\lambda_i}}f)&*_iP(\lambda_1,\dots,\lambda_i,\dots\lambda_{n-1})
\\
&=\sum_i\sum^{k_i}_{j=0}
\binom{k_i}{j}
\big(a_1\otimes\cdots\otimes a_i\otimes (\partial^j f)a_{i+1}\otimes \cdots\otimes a_n\,\lambda^{k_1}_1\cdots\lambda^{k_i-j}_i\cdots \lambda^{k_{n-1}}_{n-1}.
\end{align*}
That is, $e^{\partial\partial_\lambda}$ replaces $\lambda$ by $\lambda+\partial$, and the parentheses point out that $\partial$ is applied to $f$. Moreover, we have
\begin{align*}
P(\lambda_1,\dots,\lambda_i&+\partial,\dots,\lambda_{n-1})_{\to}*_{n-i}f
\\
&=\sum_i\sum^{k_i}_{j=0}\binom{k_i}{j}a_1\otimes \cdots\otimes a_i(\partial^jf)\otimes \cdots\otimes a_n\,\lambda^{k_1}_1\cdots\lambda^{k_i-k}_i\cdots\lambda^{k_{n-1}}_{n-1}.
\end{align*}
In other words, the arrow means that $\partial$ is applied to $f$.
\begin{definition}[\cite{DSKV15}, p. 1058]
A \emph{double $\lambda$-bracket} on a differential algebra $(\mathcal{V},\partial)$ is a linear map
\begin{equation}
\lr{-{}_\lambda-}\colon\mathcal{V}\otimes\mathcal{V}\longrightarrow (\mathcal{V}\otimes \mathcal{V})[\lambda]
\label{eq:doublelambda-bracket-def}
\end{equation}
satisfying, for all $a,b,c\in\mathcal{V}$, the sesquilinearity conditions:
\begin{subequations}
\begin{align}
\label{eq:sesquilinearity-vertex.a}
\lr{\partial a{}_\lambda b}&=-\lambda\lr{a{}_\lambda b},
\\
\label{eq:sesquilinearity-vertex.b}
\lr{a{}_\lambda \partial b}&=(\lambda+\partial)\lr{a{}_\lambda b},
\end{align}
\end{subequations}
and the Leibniz rules:
\begin{subequations}
\begin{align}
\label{eq:Leibniz-vertex.a}
\lr{a{}_\lambda bc}&=b\lr{a{}_\lambda c}+\lr{a{}_\lambda b}c,
\\
\label{eq:Leibniz-vertex.b}
\lr{ab{}_\lambda c}&=\lr{a{}_{\lambda+\partial}c}_{\to}*_1 b+(e^{\partial \partial_\lambda}a)*_1\lr{b{}_\lambda c}.
\end{align}
\end{subequations}
\label{def:double-Poisson-vertex-alg}
\end{definition}
Recall that in \eqref{notation-Kac} we introduced suitable extensions
of the double bracket $\lr{-,-}$ to define the double Jacobi identity \eqref{Jacobi-Kac}.
Similarly, to define a suitable Jacobi identity for double $\lambda$-brackets, we need to extend \eqref{eq:doublelambda-bracket-def} as follows (see \cite[eq. (4.3)]{DSKV15}).
For all $a,b,c\in\mathcal{V}$,
\begin{equation}
\begin{aligned}
\lr{a{}_\lambda(b\otimes c)}_L&\mathrel{\mathop:}=} %{\coloneqq} %{:=\lr{a{}_\lambda b}\otimes c,
\\
\lr{a{}_\lambda(b\otimes c)}_R&\mathrel{\mathop:}=} %{\coloneqq} %{:= b\otimes\lr{a{}_\lambda c},
\\
\lr{(a\otimes b){}_\lambda c}_L&\mathrel{\mathop:}=} %{\coloneqq} %{:=\lr{a{}_{\lambda+\partial}c}_\to\otimes_1 b,
\end{aligned}
\label{eq:extension-vertex-double-bracket}
\end{equation}
where the notation $\otimes_1$ was introduced in \eqref{jumping-notation}.
\begin{definition}[\cite{DSKV15}, Definition 2.2]
A \emph{double Poisson vertex algebra} $(\mathcal{V},\lr{-{}_\lambda-})$ is a differential algebra $\mathcal{V}$, with derivation $\partial\colon\mathcal{V}\to\mathcal{V}$, endowed with a double $\lambda$-bracket
$\lr{-{}_\lambda-}\colon \mathcal{V}\otimes\mathcal{V}\to(\mathcal{V}\otimes\mathcal{V})[\lambda]$, satisfying the following axioms:
\begin{enumerate}
\item [\textup{(i)}]
Skewsymmetry:
\begin{equation}
\lr{a{}_\lambda b}=-\lr{b{}_{-\lambda-\partial}a}^\sigma,
\label{vertex-skew}
\end{equation}
where $-\lambda-\partial$ in the right-hand side is moved to the left, acting on the coefficients.
\item [\textup{(ii)}]
Jacobi identity:
\begin{equation}
\lr{a{}_\lambda\lr{b{}_\mu c}}_L=\lr{b{}_\mu\lr{a{}_\lambda c}}_R+\lr{\lr{a_{\lambda} b}{}_{\lambda+\mu}c}_L.
\label{Jacobi-vertex}
\end{equation}
\end{enumerate}
\label{DPVA}
\end{definition}
It is easy to check that if $\partial=0$, and $\lambda=\mu=0$, the axioms of double Poisson vertex algebras become the axioms of double Poisson algebras. So, as emphasized in \cite{DSKV15}, a double Poisson vertex algebra is a cross between a double Poisson algebra and a Poisson vertex algebra. Furthermore, it is straightforward to check that given a double Poisson vertex algebra, the skewsymmetry \eqref{vertex-skew} and the left Leibniz rule \eqref{eq:Leibniz-vertex.a} imply the right Leibniz rule \eqref{eq:Leibniz-vertex.b} (the authors acknowledge the help of Daniele Valeri on this point); as a consequence, \eqref{eq:Leibniz-vertex.b} is redundant in Definition \ref{DPVA} and thus we shall omit below.
Finally, following \cite[p. 1074]{DSKV15}, given $a,b,c\in\mathcal{V}$,we can write the double $\lambda$-bracket as
\begin{equation}
\lr{a{}_{\lambda}b}=\sum_{\textbf{p}\in\mathbb{Z}_+}\big((a_{\textbf{p}}b)^{\prime}\otimes (a_{\textbf{p}}b)^{\prime\prime}\big)\lambda^\textbf{p}.
\label{eq:notation-vector-bracket-sumatorio}
\end{equation}
Then, we can rewrite the Jacobi identity \eqref{Jacobi-vertex} more explicitly in the following way:
\begin{equation}
\begin{aligned}
0&=\sum_{\textbf{p},\textbf{q}\in\mathbb{Z}_+}\Big((a_{\textbf{p}}(b_{\textbf{q}}c)^{\prime})^{\prime}\otimes (a_{\textbf{p}}(b_{\textbf{q}}c)^{\prime})^{\prime\prime}\otimes (b_{\textbf{q}}c)^{\prime\prime}-(a_{\textbf{p}}c)^{\prime}\otimes (b_{\textbf{q}}(a_{\textbf{p}}c)^{\prime\prime})^{\prime}\otimes (b_{\textbf{q}}(a_{\textbf{p}}c)^{\prime\prime})^{\prime\prime}\Big)\lambda^{\textbf{p}}\mu^{\textbf{q}}
\\
&\qquad - \sum_{\textbf{p},\textbf{q}\in\mathbb{Z}_+}\Big(((a_{\textbf{p}}b)^{\prime}_{\textbf{q}}c)^{\prime}\otimes (\lambda+\mu+\partial)^{\textbf{q}}(a_{\textbf{p}}b)^{\prime\prime}\otimes ((a_{\textbf{p}}b)^{\prime}_{\textbf{q}}c)^{\prime\prime}\Big)\lambda^{\textbf{p}}.
\end{aligned}
\label{eq-Jacobi-vertex-tensor}
\end{equation}
\section{Double Courant--Dorfman algebras}
\label{sec:CD-algebras}
\allowdisplaybreaks
Roytenberg \cite{Roy09} introduced an algebraic analogue of Courant algebroids called \emph{Courant--Dorfman algebras}. Their relationship to Courant algebroids is analogous to that of Lie--Rinehart algebras to Lie algebroids: Courant--Dorfman algebras are algebraic generalizations of Courant algebroids.
In this section, based on \cite{ACF17} (see also \cite{Fer16}), we define a noncommutative version of Courant--Dorfman algebras called \emph{double Courant--Dorfman algebras} by adapting Roytenberg's definition to the noncommutative setting introduced in \S\ref{sub:basics}.
We start by recalling the familiar definition of Courant--Dorfman algebras \cite[Definition 2.1]{Roy09}:
\begin{definition}
A \emph{Courant--Dorfman algebra} consists of the following data:
\begin{enumerate}
\item
a commutative $\kk$-algebra $C$,
\item [(2)]
a $C$-module $N$,
\item [(3)]
a symmetric bilinear form $\langle-,-\rangle\colon N\otimes N\to C$,
\item [(4)]
a derivation $\partial\colon C\to N$,
\item [(5)]
a Dorfman bracket $[-,-]\colon N\otimes N\to N$.
\end{enumerate}
This data is required to satisfy the following conditions:
\begin{subequations}
\label{eq:CD-comm}
\begin{align}
[n_1,cn_2]&= c [n_1,n_2]+\langle n_1,\partial c\rangle n_2, \label{eq:CD-comm.a}
\\
\langle n_1,\partial \langle n_2,n_3\rangle\rangle&= \langle [n_1,n_2],n_3\rangle+\langle n_2,[n_1,n_3]\rangle, \label{eq:CD-comm.b}
\\
\partial\langle n_1,n_2\rangle&=[n_1,n_2]+[n_2,n_1], \label{eq:CD-comm.c}
\\
[n_1,[n_2,n_3]]&=[[n_1,n_2],n_3]+[n_2,[n_1,n_3]], \label{eq:CD-comm.d}
\\
0&=[\partial c,n], \label{eq:CD-comm.e}
\\
0&=\langle\partial c,\partial d\rangle, \label{eq:CD-comm.f}
\end{align}
\end{subequations}
for all $n,n_1,n_2,n_3\in N$ and $c,d\in C$.
\label{def:CD-algebra-def-comm}
\end{definition}
As Roytenberg pointed out, a $\kk$-module $N$ equipped with a bracket $[-,-]$ satisfying condition \eqref{eq:CD-comm.d} above is called a ($\kk$-)Leibniz algebra. Moreover, if the pairing $\langle-,-\rangle$ is nondegenerate, the action of $N$ on $C$ completely determines the derivation $\partial$, so \eqref{eq:CD-comm.e} and \eqref{eq:CD-comm.f} are redundant. In this case, the notion of a Courant--Dorfman algebra coincides with that of a Courant algebroid (see \cite{Roy09} and references therein).
\begin{definition}
Let $A$ be an associative $\kk$-algebra and $E$ be an $A$-bimodule. Let $T_A E$ be the tensor algebra of $E$ over $A$, with $E$ placed in degree 1.
We consider a derivation $\partial\colon A\to E$ and a pairing $\ii{-,-}\colon E\otimes E\to A\otimes A$ (cf. \S\ref{sec:pairings}).
A \emph{double Courant--Dorfman bracket} on $E$ is a linear map
\[
\cc{-,-}\colon T_AE\otimes T_AE\longrightarrow T_AE\otimes T_AE,
\]
of degree -1 with respect to the natural grading of the tensor algebra $T_AE$,
such that
\[
\cc{e,a}=\ii{e,\partial a},\quad \cc{a,e}=-\ii{\partial a,e},\quad \cc{a,b}=0,
\]
for all $e\in E$ and $a,b\in A$.
\label{def:double-Courant-Dorfman-bracket}
\end{definition}
For later purposes, we need to decompose the double Courant--Dorfman bracket $\cc{-,-}$ when restricted to $E\otimes E$.
Given $e,f\in E$,
\begin{equation}
\label{eq:decomposition-Courant-Dorfman-def}
\cc{e,f}=\cc{e,f}_l+\cc{e,f}_r\in E\otimes A\oplus A\otimes E.
\end{equation}
Using Sweedler's notation, we can write
\begin{equation}
\label{eq:decomposition-Courant-Dorfman-def-1}
\cc{e,f}_l=\cc{e,f}^{\prime}_l\otimes \cc{e,f}^{\prime\prime}_l,\quad \cc{e,f}_r= \cc{e,f}^{\prime}_r\otimes \cc{e,f}^{\prime\prime}_r,
\end{equation}
with
$\cc{e,f}^{\prime}_l,\cc{e,f}^{\prime\prime}_r\in E$ and $\cc{e,f}^{\prime}_r,\cc{e,f}^{\prime\prime}_l\in A$.
In view of Definitions \ref{def:CD-algebra-def-comm} and \ref{def:double-Courant-Dorfman-bracket}, to state the definition of Courant--Dorfman algebras in the noncommutative context, we need a suitable Jacobi identity that will require an extension of the double Courant--Dorfman bracket.
Inspired by \eqref{extension-left}, \eqref{notation-Kac} and \eqref{eq:extension-vertex-double-bracket}, for all $a\in A$ and $e,f\in E$, we define
\begin{equation}
\begin{aligned}
\cc{e,-}_L\colon E\otimes A&\longrightarrow E\otimes A\otimes A\oplus A\otimes E\otimes A,& \; f\otimes a&\longmapsto \cc{e,f}\otimes a,
\\
\cc{e,-}_L\colon A\otimes E&\longrightarrow A\otimes A\otimes E ,&\; a\otimes f&\longmapsto \ii{e,\partial a}\otimes f,
\\
\cc{e,-}_R\colon E\otimes A&\longrightarrow E\otimes A\otimes A,&\; f\otimes a&\longmapsto f\otimes\ii{e,\partial a}
\\
\cc{e,-}_R\colon A\otimes E&\longrightarrow A\otimes E\otimes A\oplus A\otimes A\otimes E,&\; a\otimes f&\longmapsto a\otimes \cc{e,f},
\\
\cc{-,f}_L\colon E\otimes A&\longrightarrow E\otimes A\otimes A\, + c.p,&\; e\otimes a&\longmapsto \cc{e,f}\otimes_1a+\ii{e,f}\otimes_1e,
\\
\cc{-,f}_L\colon A\otimes E &\longrightarrow A\otimes E\otimes A,&\; a\otimes e&\longmapsto -\ii{\partial a,f}\otimes_1 e,
\end{aligned}
\label{eq:extension-CD-bracket}
\end{equation}
where, $E\otimes A\otimes A\,+ c.p.$ is a shorthand for $E\otimes A\otimes A\oplus A\otimes E\otimes A\oplus A\otimes A\otimes E$, and the notation $\otimes_1$ was introduced in \eqref{jumping-notation}.
\begin{definition}
Let $A$ be an associative $\kk$-algebra and $E$ be an $A$-bimodule.
A \emph{double Courant--Dorfman algebra} is a 5-tuple $(A,E,\ii{-,-},\partial,\cc{-,-})$, where
\[
\ii{-,-}\,\colon E\otimes E\longrightarrow A\otimes A,
\]
is a symmetric pairing,
\[
\partial\,\colon A\longrightarrow E,
\]
is a derivation, and
\begin{equation}
\cc{-,-}\colon T_AE\otimes T_AE\longrightarrow T_AE\otimes T_AE.
\label{eq:Courant-Dorfman.bracket-def}
\end{equation}
is a double Courant--Dorfman bracket. This data must satisfy the following axioms:
\begin{subequations}
\label{eq:CD}
\begin{align}
\partial\ii{e,f}&=\cc{e,f}+\cc{f,e}^\sigma; \label{eq:CD.a}
\\
0&=\cc{\partial a, e}; \label{eq:CD.b}
\\
0&=\ii{\partial a,\partial b}; \label{eq:CD.c}
\\
\cc{e,fa}&=\cc{e,f}a+f\ii{e,\partial a}; \label{eq:CD.d}
\\
\cc{e,af}&=a\cc{e,f}+\ii{e,\partial a}f; \label{eq:CD.e}
\\
\cc{e,\cc{f,g}}_L&=\cc{f,\cc{e,g}}_R+\cc{\cc{e,f},g}_L; \label{eq:CD.f}
\\
\ii{e,\partial\ii{f,g}}_L&=\ii{f,\cc{e,g}}_R+\ii{\cc{e,f},g}_L, \label{eq:CD.g}
\end{align}
\end{subequations}
for all $a,b\in A$ and $e,f,g\in E$.
\label{CD-Definition}
\end{definition}
We observe that, in \eqref{eq:CD.a} and \eqref{eq:CD.g}, the derivation $\partial$ acts on $\ii{-,-}$ by the Leibniz rule; for all $e_1,e_2\in E$,
\begin{equation}
\partial\ii{e_1,e_2}=\partial\ii{e_1,e_2}^\prime\otimes\ii{e_1,e_2}^{\prime\prime}+\ii{e_1,e_2}^\prime\otimes\partial \ii{e_1,e_2}^{\prime\prime}.
\label{eq:partialLeibniz-rule}
\end{equation}
In \eqref{eq:CD.g} we use the extensions of $\ii{-,-}$ introduced in \eqref{eq:extension-pairing-first-second}; thus this identity is in $A^{\otimes 3}$. Moreover, in \eqref{eq:CD.d} and \eqref{eq:CD.e} the products use the outer bimodule structure.
\begin{remark}
In several places of the article, we will need to write \eqref{eq:CD.f} as explicit as possible. Since \eqref{eq:CD.f} takes place in $E\otimes A\otimes A+c.p.$,
we can project onto each summand. Hence, using \eqref{eq:decomposition-Courant-Dorfman-def-1}, \eqref{eq:partialLeibniz-rule}, \eqref{eq:extension-CD-bracket} and \eqref{eq:ext-pairing-zero}, we have that \eqref{eq:CD.f} is equivalent to the following three identities:
\begin{subequations}
\label{eq:CD-double-Jacobi-explicit}
\begin{align}
\begin{split}
\cc{e,\cc{f,g}^{\prime}_l}_l\otimes\cc{f,g}^{\prime\prime}_l&=\cc{e,g}^{\prime}_l\otimes\ii{f,\partial\cc{e,g}^{\prime\prime}_l}
+\cc{\cc{e,f}^{\prime}_l,g}_l\otimes_1 \cc{e,f}^{\prime\prime}_l,
\label{eq:CD-double-Jacobi-explicit.a}
\end{split}
\\
\begin{split}
\cc{e,\cc{f,g}^{\prime}_l}_r\otimes\cc{f,g}^{\prime\prime}_l&=\cc{e,g}^{\prime}_r\otimes\cc{f,\cc{e,g}^{\prime\prime}_r}_l
+\ii{\cc{e,f}^{\prime}_l,g}\otimes_1\partial\cc{e,f}^{\prime\prime}_l
\\
&\qquad -\ii{\partial\cc{e,f}^{\prime}_r,g}\otimes_1\cc{e,f}^{\prime\prime}_r,
\label{eq:CD-double-Jacobi-explicit.b}
\end{split}
\\
\begin{split}
\ii{e,\partial\cc{f,g}^{\prime}_r}\otimes \cc{f,g}^{\prime\prime}_r&=\cc{e,g}^{\prime}_r\otimes\cc{f,\cc{e,g}^{\prime\prime}_r}_r
+\cc{\cc{e,f}^{\prime}_l,g}_r\otimes_1\cc{e,f}^{\prime\prime}_l,
\label{eq:CD-double-Jacobi-explicit.c}
\end{split}
\end{align}
\end{subequations}
where \eqref{eq:CD-double-Jacobi-explicit.a} is in $E\otimes A\otimes A$, \eqref{eq:CD-double-Jacobi-explicit.b} in $A\otimes E\otimes A$, and \eqref{eq:CD-double-Jacobi-explicit.c} takes place in $A\otimes A\otimes E$.
\label{rem:explicit-Jacobi-Courant-Dorfman-remark}
\end{remark}
\section{The equivalence theorem}
\label{sec:equivalence}
In this section we prove the main theorem of this article, which is a
non-commutative version of \cite[Prop. 4.3]{Hel09} in the case of Courant
algebroids:
\begin{theorem}
There exists a one--to--one correspondence between double Poisson vertex algebras $(\mathcal{V},\partial, \lr{-{}_\lambda-}) $ freely generated in weights 0 and 1, and double Courant--Dorfman algebras $(E, \ii{-,-},\partial,\cc{-,-})$.
\label{theorem-CD-DPVA}
\end{theorem}
The rest of this section is devoted to prove Theorem \ref{theorem-CD-DPVA}, in
Section \S\ref{V-CD} we construct a double Courant--Dorfman algebra associated
with a double Poisson vertex algebra. In Section \S\ref{sec:DCD-DPVA} we provide the inverse
construction.
\subsection{From double Poisson vertex algebras to double Courant--Dorfman algebras}
\label{V-CD}
\allowdisplaybreaks
In this subsection, we shall show that the structure of a double Poisson vertex
algebra --- see Definition \ref{DPVA}, which is freely generated in weights 0 and
1, induces a structure of a double Courant--Dorfman algebra, as in Definition
\ref{CD-Definition} in its degree $0$ and $1$ parts.
Let $(\mathcal{V},\partial, \lr{-{}_\lambda-}) $ be a double Poisson vertex algebra such that $\mathcal{V}=\bigoplus_{k\geq 0}\mathcal{V}_k$ whose product has weight 0, the derivation $\partial$ has weight 1, and the double $\lambda$-bracket has weight -1. More precisely, given $v\in\mathcal{V}_i$, $w\in\mathcal{V}_j$ ,and $\lr{v{}_{\lambda}w}=\sum_{\textbf{q}\in\mathbb{Z}_+}\big((v{}_{\textbf{q}}w)^{\prime}\otimes(v{}_{\textbf{q}}w)^{\prime\prime}\big)\lambda^{\textbf{q}}$, we have $(v{}_{\textbf{q}}w)^{\prime}\otimes(v{}_{\textbf{q}}w)^{\prime\prime}\in(\mathcal{V}\otimes\mathcal{V})_{i+j-\textbf{q}-1}$, for all $\textbf{q}$.
Since the product in $\mathcal{V}$ has weight 0, note that if $a,b\in\mathcal{V}_0$ the product $ab\in\mathcal{V}_0$. Consequently, $A\mathrel{\mathop:}=} %{\coloneqq} %{:=\mathcal{V}_0$ has the structure of an associative (not necessarily commutative) $\kk$-algebra. Next, the multiplication of an element of $\mathcal{V}_0$, both on the right and on the left, by an element $\mathcal{V}_1$ gives an element of $\mathcal{V}_1$. Thus, $E:=\mathcal{V}_1$ carries an $A$-bimodule structure.
From now on, we fix $e,f,g\in E $ and $a,b\in A$. Observe that since the double $\lambda$-bracket has weight -1, if $e,f\in E$, the bracket restricted to $E\otimes E$ has the following expression:
\begin{equation}
\lr{e{}_\lambda f}=\cc{e,f}+\lambda\ii{e,f}\in((\mathcal{V}\otimes\mathcal{V})[\lambda])_1=E\otimes A\oplus A\otimes E\oplus (A\otimes A)\lambda,
\label{ansatz}
\end{equation}
where
\begin{enumerate}[label=(\roman{*}), ref=(\roman{*})]
\item
\label{item:lambda1}
The operations
\begin{align*}
\cc{-,-}\,\colon E\otimes E&\longrightarrow E\otimes A\oplus A\otimes E,
\\
\ii{-,-}\,\colon E\otimes E&\longrightarrow A\otimes A
\end{align*}
are $\kk$-linear maps.
\item
\label{item:lambda2}
We have the derivation
\[
\partial\,\colon A\longrightarrow E
\]
of weight 1 that, by convention, acts on tensor products by the Leibniz rule; see \eqref{eq:partialLeibniz-rule}.
\item
\label{item:lambda3}
Since $\lvert\lr{-{}_{\lambda}-}\rvert=-1$, $\lvert e\rvert=1$ and $\lvert a\rvert=0$, we have an action
\begin{equation}
\star\colon E\otimes A\longrightarrow A\otimes A,\quad e\otimes a\longmapsto e\star a=\lr{e{}_\lambda a}=\ii{e,\partial a};
\label{bracket-A-f}
\end{equation}
\item
\label{item:lambda4}
By \ref{item:lambda3} and \eqref{vertex-skew}, we get
\begin{equation}
\lr{a{}_\lambda e}=-\ii{\partial a,e}.
\label{lambda-bracket-f-A}
\end{equation}
\item
\label{item:lambda5}
Finally, since $a,b\in A$, $\lvert a\rvert=\lvert b\rvert=0$. Thus,
\begin{equation}
\lr{a{}_\lambda b}=0.
\label{bracket-f-g}
\end{equation}
\end{enumerate}
Now, we will use the axioms of double Poisson vertex algebras expressed in Definition \ref{DPVA} to deduce the axioms \eqref{eq:CD.a}--\eqref{eq:CD.g} that characterize double Courant--Dorfman algebras.
To prove \eqref{eq:CD.a}, by \eqref{vertex-skew} we have the identity
\begin{equation}
\lr{e{}_\lambda f}=-\lr{f{}_{-\lambda-\partial}e}^{\sigma},
\label{eq:skew-inicial-5.3a}
\end{equation}
whose left-hand side, by \eqref{ansatz}, can be rewritten as
\begin{equation}
\lr{e{}_\lambda f}=\cc{e,f}+\lambda\ii{e,f},
\label{eq:skew-inicial-5.3a-LHS}
\end{equation}
and at the right-hand side of \eqref{eq:skew-inicial-5.3a}, we obtain
\begin{equation}
-\lr{f{}_{-\lambda-\partial}e}^{\sigma}=-\cc{f,e}^\sigma+\lambda\ii{f,e}^\sigma+\big(\partial\ii{f,e}\big)^\sigma.
\label{eq:skew-inicial-5.3a-RHS}
\end{equation}
The terms with different coefficients of $\lambda$ do not interact each other; thus comparing the coefficients of $\lambda$ appearing in \eqref{eq:skew-inicial-5.3a-LHS} and \eqref{eq:skew-inicial-5.3a-RHS}, we obtain
\begin{equation}
\ii{e,f}=\ii{f,e}^\sigma.
\label{eq:ii-symmetric}
\end{equation}
Moreover, since the derivation $\partial$ acts on tensor products by the Leibniz rule (see \ref{item:lambda2} above), \eqref{eq:ii-symmetric} implies
\begin{equation}
\partial \ii{e,f}=\big(\partial \ii{f,e}\big)^\sigma.
\label{eq:ii-symmetric-con-partial}
\end{equation}
Hence, comparing the terms without $\lambda$ in \eqref{eq:skew-inicial-5.3a-LHS} and \eqref{eq:skew-inicial-5.3a-RHS}, we have
\[
\cc{e,f}
=-\cc{f,e}^{\sigma}+\partial \ii{e,f},
\]
which is tantamount to \eqref{eq:CD.a}.
Next, the sesquilinearity of the double $\lambda$-bracket (see \eqref{eq:sesquilinearity-vertex.a}) and \ref{item:lambda4} above, give the identity
\begin{equation}
\lr{\partial a{}_\lambda e}=-\lambda\lr{a{}_\lambda e}=\lambda\ii{\partial a,e}.
\label{eq:sesquilin-5.3b-1}
\end{equation}
On the other hand, using \eqref{ansatz}, we have
\begin{equation}
\lr{\partial a{}_\lambda e}=\cc{\partial a,e}+\lambda\ii{\partial a, e}.
\label{eq:sesquilin-5.3b-2}
\end{equation}
If we compare the terms without $\lambda$ in \eqref{eq:sesquilin-5.3b-1} and \eqref{eq:sesquilin-5.3b-2}, we finally obtain the identity
\begin{equation}
\cc{\partial a, e}=0,
\label{Axioma5}
\end{equation}
which is \eqref{eq:CD.b}.
To prove \eqref{eq:CD.c}, by \eqref{eq:sesquilinearity-vertex.a} and \eqref{eq:sesquilinearity-vertex.b}
\begin{equation}
\lr{\partial a{}_\lambda\partial b}=-\lambda\lr{a{}_\lambda\partial b}=-\lambda(\lambda+\partial)\lr{a{}_\lambda b}=0,
\label{eq:sesquilin-5.3c-1}
\end{equation}
where the last identity is a consequence of \ref{item:lambda5} above.
On the other hand, by \eqref{ansatz},
\begin{equation}
\lr{\partial a{}_\lambda\partial b}=\cc{\partial a,\partial b}+\lambda\ii{\partial a,\partial b}.
\label{eq:sesquilin-5.3c-2}
\end{equation}
Again, comparing the obtained expressions in \eqref{eq:sesquilin-5.3c-1} and \eqref{eq:sesquilin-5.3c-2}, we obtain
\begin{equation}
\ii{\partial a,\partial b}=0,
\label{Axioma6}
\end{equation}
which corresponds to \eqref{eq:CD.c}.
To establish the Leibniz identity \eqref{eq:CD.d}, by \eqref{eq:Leibniz-vertex.a} and \eqref{ansatz}, we set
\begin{equation}
\lr{e{}_\lambda fa}=\lr{e{}_\lambda f}a+f\lr{e{}_\lambda a}
=\cc{e,f}a+\lambda (\ii{e,f}a)+f\ii{e,\partial a}.
\label{eq:Leibniz-5.3-d-1}
\end{equation}
Now, by \eqref{ansatz}, we get
\begin{equation}
\lr{e{}_\lambda fa}=\cc{e,fa}+\lambda\ii{e,fa}.
\label{eq:Leibniz-5.3-d-2}
\end{equation}
Identifying the terms without $\lambda$ in \eqref{eq:Leibniz-5.3-d-1} and \eqref{eq:Leibniz-5.3-d-2}, we finally obtain:
\begin{equation}
\cc{e,fa}=\cc{e,f}a+f\ii{e,\partial a},
\label{Axioma1.a}
\end{equation}
which is \eqref{eq:CD.d}.
Using similar arguments, the reader can prove \eqref{eq:CD.e}.
Interestingly, note that comparing the coefficients of $\lambda$ in \eqref{eq:Leibniz-5.3-d-1} and \eqref{eq:Leibniz-5.3-d-2}, we get $\ii{e,fa}=\ii{e,f}a$. Similarly, after proving \eqref{eq:CD.e}, we obtain $\ii{e,af}=a\ii{e,f}$, meaning that $\ii{-,-}$ is $A$-bilinear in its second argument with respect to the outer bimodule structure. Moreover, applying \eqref{vertex-skew}, \eqref{eq:Leibniz-vertex.a}, \eqref{ansatz} and \eqref{eq:ii-symmetric},
\begin{align*}
\lr{ae{}_{\lambda}f}&=-\lr{f{}_{-\lambda-\partial}ae}^{\sigma}=-\big(a\lr{f{}_{-\lambda-\partial}e}+\lr{f{}_{-\lambda-\partial}a}e\big)^{\sigma}
\\
&=-a*\cc{f,e}^\sigma+\lambda(a*\ii{e,f})+a*(\partial\ii{e,f})-\ii{\partial a,f}*e
\\
&=a*\cc{e,f}+\lambda(a*\ii{e,f})-\ii{\partial a,f}*e.
\end{align*}
By \eqref{ansatz}, $\lr{ae{}_{\lambda}f}=\cc{ae,f}+\lambda\ii{ae,f}$. Thus, comparing the obtained expressions, we have
\[
\ii{ae,f}=a*\ii{e,f};
\]
in other words, $\ii{-,-}$ is $A$-linear in its first argument with respect to the inner bimodule structure.
By similar considerations on $\lr{ea{}_{\lambda}f}$, the identity $\ii{ea,f}=\ii{e,f}*a$ holds. These identities together \eqref{eq:ii-symmetric} imply that $\ii{-,-}$ is a symmetric pairing on $E$, as defined in \S\ref{sec:pairings}.
Finally, we want to obtain \eqref{eq:CD.f} and \eqref{eq:CD.g}.
Firstly, by \eqref{Jacobi-vertex}, we have
\begin{equation}
0=\lr{e{}_\lambda\lr{f{}_\mu g}}_L-\lr{f{}_\mu\lr{e{}_\lambda g}}_R-\lr{\lr{e{}_{\lambda} f}{}_{\lambda+\mu}g}_L
\label{Jacobi-inicio-DPV-CD},
\end{equation}
and using \eqref{ansatz}, \ref{item:lambda3}--\ref{item:lambda5}, \eqref{eq:decomposition-Courant-Dorfman-def-1} and \eqref{eq:extension-vertex-double-bracket}, we will rewrite each term. To start with, we focus on the first summand of \eqref{Jacobi-inicio-DPV-CD}:
\begin{equation}
\begin{aligned}
&\lr{e{}_\lambda\lr{f{}_\mu g}}_L
\\
&=\lr{e{}_{\lambda}\big(\cc{f,g}+\ii{f,g}\mu\big)}_L
\\
&=\lr{e{}_{\lambda}\cc{f,g}^{\prime}_l}\otimes \cc{f,g}^{\prime\prime}_l
+\lr{e{}_{\lambda}\cc{f,g}^{\prime}_r}\otimes \cc{f,g}^{\prime\prime}_r
+\big(\lr{e{}_{\lambda}\ii{f,g}^{\prime}}\otimes \ii{f,g}^{\prime\prime}\big)\mu
\\
&=\big(\cc{e,\cc{f,g}^{\prime}_l}+\ii{e,\cc{f,g}^{\prime}_l}\lambda\big)\!\otimes \!\cc{f,g}^{\prime\prime}_l
+\ii{e,\partial \cc{f,g}^{\prime}_r}\!\otimes \!\cc{f,g}^{\prime\prime}_r
+(\ii{e,\partial \ii{f,g}^{\prime}}\!\otimes\! \ii{f,g}^{\prime\prime})\mu
\\
&=\cc{e,\cc{f,g}^{\prime}_l}_l\otimes \cc{f,g}^{\prime\prime}_l
+\cc{e,\cc{f,g}^{\prime}_l}_r\otimes \cc{f,g}^{\prime\prime}_l
+\ii{e,\partial \cc{f,g}^{\prime}_r}\otimes \cc{f,g}^{\prime\prime}_r
\\
&\qquad +(\ii{e,\cc{f,g}^{\prime}_l}\otimes \cc{f,g}^{\prime\prime}_l)\lambda
+(\ii{e,\partial \ii{f,g}^{\prime}}\otimes \ii{f,g}^{\prime\prime})\mu.
\end{aligned}
\label{eq:Jacobi-CD-DPVA-general-1}
\end{equation}
Next, in the second summand of \eqref{Jacobi-inicio-DPV-CD}:
\begin{equation}
\begin{aligned}
&\lr{f{}_\mu\lr{e{}_\lambda g}}_R=
\\
&=\lr{f{}_\mu\big(\cc{e,g}+\ii{e,g}\lambda\big)}_R
\\
&= \cc{e,g}^{\prime}_l\otimes \lr{f{}_\mu\cc{e,g}^{\prime\prime}_l}
+\cc{e,g}^{\prime}_r\otimes \lr{f{}_\mu\cc{e,g}^{\prime\prime}_r}
+ \big(\ii{e,g}^{\prime}\otimes\lr{f{}_\mu\ii{e,g}^{\prime\prime}}\big)\lambda
\\
&=\cc{e,g}^{\prime}_l\!\otimes\! \ii{f,\partial \cc{e,g}^{\prime\prime}_l}
+\cc{e,g}^{\prime}_r\!\otimes \!\big(\cc{f,\cc{e,g}^{\prime\prime}_r}+\ii{f,\cc{e,g}^{\prime\prime}_r}\mu\big)
+ (\ii{e,g}^{\prime}\!\otimes\!\ii{f,\partial \ii{e,g}^{\prime\prime}})\lambda
\\
&=\cc{e,g}^\prime_l\otimes \ii{f,\partial \cc{e,g}^{\prime\prime}_l}+\cc{e,g}^\prime_r\otimes \cc{f,\cc{e,g}^{\prime\prime}_r}_l+\cc{e,g}^\prime_r\otimes \cc{f,\cc{e,g}^{\prime\prime}_r}_r
\\
&\qquad + (\ii{e,g}^{\prime}\otimes\ii{f,\partial \ii{e,g}^{\prime\prime}})\lambda+ (\cc{e,g}^\prime_r\otimes\ii{f,\cc{e,g}^{\prime\prime}_r})\mu.
\end{aligned}
\label{eq:Jacobi-CD-DPVA-general-2}
\end{equation}
Finally, in the third summand of \eqref{Jacobi-inicio-DPV-CD}, we obtain
\begin{equation}
\begin{split}
&\lr{\lr{e{}_{\lambda} f}{}_{\lambda+\mu}g}_L
\\
&=\lr{\big(\cc{e,f}+\ii{e,f}\lambda\big){}_{\lambda+\mu}g}_L
\\
&= \lr{\cc{e,f}^{\prime}_l{}_{\lambda+\mu+\partial}g}_{\to}\!\otimes_1\!\cc{e,f}^{\prime\prime}_l
+\lr{\cc{e,f}^{\prime}_r{}_{\lambda+\mu+\partial}g}_{\to}\!\otimes_1\!\cc{e,f}^{\prime\prime}_r
\\
&\qquad +(\lr{\ii{e,f}^{\prime}{}_{\lambda+\mu+\partial}g}_{\to}\!\otimes_1\!\ii{e,f}^{\prime\prime}\big)\lambda
\\
&=\cc{\cc{e,f}^\prime_l,g}^\prime_l\otimes \cc{e,f}^{\prime\prime}_l\otimes\cc{\cc{e,f}^\prime_l,g}^{\prime\prime}_l
+\cc{\cc{e,f}^\prime_l,g}^\prime_r\otimes \cc{e,f}^{\prime\prime}_l\otimes\cc{\cc{e,f}^\prime_l,g}^{\prime\prime}_r
\\
&\qquad +\ii{\cc{e,f}^\prime_l}^{\prime}\otimes (\lambda+\mu+\partial)\cc{e,f}^{\prime\prime}_l\otimes \ii{\cc{e,f}^\prime_l}^{\prime\prime} -\ii{\partial\cc{e,f}^\prime_r,g}^\prime\otimes\cc{e,f}^{\prime\prime}_r\otimes \ii{\partial\cc{e,f}^\prime_r,g}^{\prime\prime}
\\
&\qquad -(\ii{\partial\ii{e,f}^\prime,g}^\prime\otimes\ii{e,f}^{\prime\prime}\otimes\ii{\partial \ii{e,f}^{\prime},g}^{\prime\prime})\lambda.
\end{split}
\label{eq:Jacobi-CD-DPVA-general-3}
\end{equation}
Then, collecting the terms obtained in \eqref{eq:Jacobi-CD-DPVA-general-1}--\eqref{eq:Jacobi-CD-DPVA-general-3}, from \eqref{Jacobi-inicio-DPV-CD} we finally obtain:
\begin{subequations}
\label{eq:Jacobi-CD-DPVA-general-expression-final}
\begin{align}
0&=\lr{e{}_\lambda\lr{f{}_\mu g}}_L-\lr{f{}_\mu\lr{e{}_\lambda g}}_R-\lr{\lr{e{}_{\lambda} f}{}_{\lambda+\mu}g}_L\nonumber
\\
\begin{split}
&=\cc{e,\cc{f,g}^{\prime}_l}_l\otimes \cc{f,g}^{\prime\prime}_l
-\cc{e,g}^{\prime}_l\otimes \ii{f,\partial \cc{e,g}^{\prime\prime}_l}
- \cc{\cc{e,f}^{\prime}_l,g}_l\otimes_1 \cc{e,f}^{\prime\prime}_l
\label{eq:Jacobi-CD-DPVA-general-expression-final.a}
\end{split}
\\
\begin{split}
&\quad +\cc{e,\cc{f,g}^{\prime}_l}_r \otimes \cc{f,g}^{\prime\prime}_l
-\cc{e,g}^{\prime}_r\otimes\cc{f,\cc{e,g}^{\prime\prime}_r}_l
-\ii{\cc{e,f}^{\prime}_l,g}\otimes_1 \partial\cc{e,f}^{\prime\prime}_l
\\
&\qquad +\ii{\partial \cc{e,f}^{\prime}_r,g}\otimes_1\cc{e,f}^{\prime\prime}_r
\label{eq:Jacobi-CD-DPVA-general-expression-final.b}
\end{split}
\\
\begin{split}
&\quad + \ii{e,\partial \cc{f,g}^{\prime}_r}\otimes \cc{f,g}^{\prime\prime}_r
-\cc{e,g}^{\prime}_r\otimes\cc{f,\cc{e,g}^{\prime\prime}_r}_r
- \cc{\cc{e,f}^{\prime}_l,g}_r\otimes_1 \cc{e,f}^{\prime\prime}_l
\label{eq:Jacobi-CD-DPVA-general-expression-final.c}
\end{split}
\\
\begin{split}
&\quad +\Big(\ii{e,\cc{f,g}^{\prime}_l}\otimes \cc{f,g}^{\prime\prime}_l
- \ii{e,g}^{\prime}\otimes\ii{f,\partial \ii{e,g}^{\prime\prime}}
+\ii{\partial \ii{e,f}^{\prime},g}\otimes_1 \ii{e,f}^{\prime\prime}
\\
&\qquad- \ii{\cc{e,f}^{\prime}_l,g}\otimes_1\cc{e,f}^{\prime\prime}_l\Big)\lambda
\label{eq:Jacobi-CD-DPVA-general-expression-final.d}
\end{split}
\\
\begin{split}
&\quad +\Big(\ii{e,\partial \ii{f,g}^{\prime}}\otimes \ii{f,g}^{\prime\prime}
-\cc{e,g}^{\prime}_r\otimes\ii{f,\cc{e,g}^{\prime\prime}_r}
- \ii{\cc{e,f}^{\prime}_l,g}\otimes_1\cc{e,f}^{\prime\prime}_l\Big)\mu.
\label{eq:Jacobi-CD-DPVA-general-expression-final.e}
\end{split}
\end{align}
\label{eq:Jacobi-CD-DPVA-general-1b}
\end{subequations}
Hence, \eqref{eq:Jacobi-CD-DPVA-general-expression-final.a} agrees with \eqref{eq:CD-double-Jacobi-explicit.a}, \eqref{eq:Jacobi-CD-DPVA-general-expression-final.b} with \eqref{eq:CD-double-Jacobi-explicit.b}, and \eqref{eq:Jacobi-CD-DPVA-general-expression-final.c} is tantamount to \eqref{eq:CD-double-Jacobi-explicit.c}.
Thus, by Remark \ref{rem:explicit-Jacobi-Courant-Dorfman-remark}, \eqref{eq:CD.f} follows.
Finally, using $\partial$ acts on tensor products by the Leibniz rule, as well as \eqref{eq:extension-pairing-first-second} and \eqref{eq:ext-pairing-zero}, it is clear that \eqref{eq:Jacobi-CD-DPVA-general-expression-final.e} gives \eqref{eq:CD.g}.
So, to sum up, in this subsection we proved that a double Poisson vertex algebra freely generated in weights 0 and 1 gives rise to a double Courant--Dorfman, as stated in Definition \ref{CD-Definition}.
\subsection{From double Courant--Dorfman algebras to double Poisson vertex algebras}
\label{sec:DCD-DPVA}
\allowdisplaybreaks
Given a double Courant--Dorfman algebra $(A,E,\ii{-,-},\partial,\cc{-,-})$, as
in Definition \ref{CD-Definition}, in this subsection we prove that it induces a
double Poisson vertex algebra $(\mathcal{V},\partial,\lr{-{}_{\lambda}-})$
freely generated in weights 0 and 1. We let $\mathcal{V}'$ be the free graded associative
associative $A$ algebra with a derivation $T$ of degree $1$ generated by $E$ in
degree $1$. Let $\mathcal{V}$ be its quotient by the differential ideal generated by $(T -
\partial) a$ for all $a \in A$. This is a graded differential associative
algebra, generated by $A$ in degree $0$ and $E$ in degree $1$. We denote by
$\partial$ the derivation of degree
$1$ obtained by reduction from $T$ on $\mathcal{V}'$.
We define an operation
\[
\lr{-{}_{\lambda}-}\colon\,\mathcal{V}\otimes\mathcal{V}\longrightarrow (\mathcal{V}\otimes\mathcal{V})[\lambda]
\]
of weight -1, given by
\begin{subequations}
\label{eq:vertex-bracket}
\begin{align}
\lr{e {}_\lambda f}&=\cc{e,f}+\ii{e,f}\lambda, \label{eq:vertex-bracket.a}
\\
\lr{e{}_\lambda a}&=\ii{e,\partial a}, \label{eq:vertex-bracket.b}
\\
\lr{a{}_\lambda e}&=-\ii{\partial a,e}, \label{eq:vertex-bracket.c}
\\
\lr{a{}_\lambda b}&=0, \label{eq:vertex-bracket.d}
\end{align}
\end{subequations}
for all $a,b\in A$ and $e,f\in E$ and extend this to all of $\mathcal{V}$ by the
Leibniz property \eqref{eq:Leibniz-vertex.a}. We need to prove that the bracket
\eqref{eq:vertex-bracket} is well defined and is a double $\lambda$-bracket (\emph{i.e.}, $\lr{-{}_\lambda-}$ satisfies the properties of sesquilinearity \eqref{eq:sesquilinearity-vertex.a} and \eqref{eq:sesquilinearity-vertex.b}, as well as the left Leibniz property \eqref{eq:Leibniz-vertex.a}), further satisfying \eqref{vertex-skew} and \eqref{Jacobi-vertex}.
From now on, we fix $a,b\in A$ and $e,f,g\in E$.
By \eqref{eq:vertex-bracket}, it is clear that \eqref{eq:sesquilinearity-vertex.a} holds on $A\otimes A$, $E\otimes A$ and $A\otimes E$. Next, by \eqref{eq:vertex-bracket.a}, \eqref{eq:CD.b} and \eqref{eq:vertex-bracket.c}, we have
\[
\lr{\partial a{}_\lambda e}=\cc{\partial a,e}+\lambda \ii{\partial a, e}=\lambda \ii{\partial a,e}=-\lambda \lr{a {}_{\lambda} e},
\]
and \eqref{eq:sesquilinearity-vertex.a} follows.
Again, by \eqref{eq:vertex-bracket}, it is clear that \eqref{eq:sesquilinearity-vertex.b} holds on $A\otimes A$, $E\otimes A$ and $A\otimes E$. To prove that this identity holds on $E\otimes E$, we have
\[
\lr{e{}_{\lambda}\partial a}=\cc{e,\partial a}+\lambda\ii{e,\partial a}=\partial\ii{e,\partial a}-\cc{\partial f, A}^{\sigma}+\lambda\ii{e,\partial a}=(\lambda+\partial)\lr{e{}_{\lambda} a},
\]
where we used \eqref{eq:vertex-bracket.a}, \eqref{eq:CD.a}, \eqref{eq:CD.b} and \eqref{eq:vertex-bracket.b}.
Next, by \eqref{eq:vertex-bracket.d}, the skewsymmetry \eqref{vertex-skew} on $A\otimes A$ is trivial.
By \eqref{eq:vertex-bracket.c}, the symmetry of $\ii{-,-}$ and \eqref{eq:vertex-bracket.c}, we have
\[
-\lr{a{}_{-\lambda-\partial}e}^{\sigma}=\ii{\partial a,e}^{\sigma}=\ii{e,\partial a}=\lr{e{}_{\lambda}a}.
\]
The proof of the identity $-\lr{e{}_{-\lambda-\partial}a}^{\sigma}=\lr{a{}_{\lambda}e}$ is completely similar and we leave it to the reader.
Next, by \eqref{eq:ii-symmetric-con-partial}, \eqref{eq:vertex-bracket.d} and \eqref{eq:CD.a}, we have:
\begin{align*}
-\lr{f{}_{-\lambda-\partial}e}^{\sigma}&=-\big(\cc{f,e}-(\lambda+\partial)\ii{f,e}\big)^{\sigma}
\\
&=-\cc{f,e}^{\sigma}+\lambda\ii{e,f}+\partial\ii{e,f}
\\
&=\cc{e,f}+\lambda\ii{e,f}=\lr{e{}_{\lambda}f},
\end{align*}
where we used \eqref{eq:Sweedler-pairing-symm}.
The left Leibniz property \eqref{eq:Leibniz-vertex.a} trivially holds when the three elements are in $A$.
If $a,b\in A$, $e\in E$, by \eqref{eq:vertex-bracket.c} and \eqref{eq:vertex-bracket.d}, we get
\[
\lr{a{}_{\lambda}eb}-e\lr{a{}_{\lambda}b}-\lr{a{}_{\lambda}e}b=-\ii{\partial a,eb}+\ii{\partial a,e}b=0.
\]
Similarly, we can prove that $\lr{a{}_{\lambda}be}-b\lr{a{}_{\lambda}e}-\lr{a{}_{\lambda}b}e=0$. However, if the element of $E$ is in the first argument, the computation is slightly different:
\[
\lr{e{}_{\lambda}ab}-a\lr{e{}_{\lambda}b}-\lr{e{}_{\lambda}a}b=\ii{e,\partial(ab)}-a\ii{e,\partial b}-\ii{e,\partial a}b=0,
\]
where we used \eqref{eq:vertex-bracket.c}, \eqref{eq:partialLeibniz-rule} and the $A$-linearity of the pairing. The most interesting case occurs when only one element belongs to $A$. By \eqref{eq:vertex-bracket.a}, \eqref{eq:vertex-bracket.b} and \eqref{eq:CD.d} we can write
\begin{align*}
&\lr{e{}_{\lambda}fa}-f\lr{e{}_{\lambda}a}-\lr{e{}_{\lambda}f}a
\\
&=\cc{e,fa}+\lambda\ii{e,fa}-f\ii{e,\partial a}-(\cc{e,f}+\lambda\ii{e,f})a=0
\end{align*}
Similarly, based on the same arguments, it is straightforward to see that $\lr{e{}_{\lambda}af}-a\lr{e{}_{\lambda}f}-\lr{e{}_{\lambda}a}f=0$. Consequently, $\lr{-{}_{\lambda}-}$ as defined in \eqref{eq:vertex-bracket} satisfies the left Leibniz property \eqref{eq:Leibniz-vertex.a}.
\medskip
To prove the Jacobi identity \eqref{Jacobi-vertex}, we shall distinguish different cases, depending on the number of elements in $A$ and $E$. By \eqref{eq:vertex-bracket.d}, \eqref{Jacobi-vertex} trivially holds if there are two or three elements belonging to $A$.
If $e,f\in E$ and $a\in A$, by \eqref{eq:vertex-bracket.a}, \eqref{eq:vertex-bracket.b} and \eqref{eq:vertex-bracket.d}, we have
\begin{align*}
&\lr{e{}_{\lambda}\lr{f{}_{\mu}a}}_L
-\lr{f{}_{\mu}\lr{e{}_{\lambda}a}}_R
-\lr{\lr{e{}_{\lambda}f}{}_{\lambda+\mu}a}_L
\\
&=\lr{e{}_{\lambda}\ii{f,\partial a}}_L-\lr{f{}_{\mu}\ii{e,\partial a}}_R-\lr{\big(\cc{e,f}+\ii{e,f}\lambda\big){}_{\lambda+\mu}a}_L.
\intertext{By \eqref{eq:extension-vertex-double-bracket} and \eqref{eq:vertex-bracket}, we have}
&=\lr{e{}_{\lambda}\ii{f,\partial a}^{\prime}}\otimes \ii{f,\partial a}^{\prime\prime}-\ii{e,\partial a}^{\prime}\otimes\lr{f{}_{\mu}\ii{e,\partial a}^{\prime\prime}}-\lr{\cc{e,f}^{\prime}_l{}_{\lambda+\mu+\partial}a}_{\to}\otimes_1\cc{e,f}^{\prime\prime}_l
\\
&=\ii{e,\partial\ii{f,\partial a}^{\prime}}^{\prime}\otimes \ii{e,\partial\ii{f,\partial a}^{\prime}}^{\prime\prime}\otimes\ii{f,\partial a}^{\prime\prime}
\\
&\quad-\ii{e,\partial a}^{\prime}\otimes \ii{f,\partial\ii{e,\partial a}^{\prime\prime}}^{\prime}\otimes \ii{f,\partial\ii{e,\partial a}^{\prime\prime}}^{\prime\prime}
\\
&\quad -\ii{\cc{e,f}^{\prime}_l,\partial a}^{\prime}\otimes\cc{e,f}^{\prime\prime}_l\otimes \ii{\cc{e,f}^{\prime}_l,\partial a}^{\prime\prime},
\intertext{which, using \eqref{eq:partialLeibniz-rule}, \eqref{eq:extension-pairing-first-second} and \eqref{eq:ext-pairing-zero}, can be equivalently written as}
&=\ii{e,\partial\ii{f,\partial a}}_L- \ii{f,\partial\ii{e,\partial a}}_R-\ii{\cc{e,f},\partial a}_L.
\intertext{By \eqref{eq:CD.g}, \eqref{eq:CD.a} and \eqref{eq:CD.b}, we finally obtain}
&=\ii{e,\partial\ii{f,\partial a}}_L-\ii{f,\partial\ii{e,\partial a}}_R+\ii{f,\cc{e,\partial a}}_R-\ii{e,\partial \ii{f,\partial a }}_L=0,
\end{align*}
as we wished.
Similarly, the reader can prove that
\[
\lr{e{}_{\lambda}\lr{a{}_{\mu}f}}_L-\lr{a{}_{\mu}\lr{e{}_{\lambda}f}}_R-\lr{\lr{e{}_{\lambda}a}{}_{\lambda+\mu}f}_L=0.
\]
However, the remaining case is slightly different. So, for the reader's convenience, we show it explicitly.
By \eqref{eq:vertex-bracket} and \eqref{eq:decomposition-Courant-Dorfman-def-1}, we have
\begin{equation}
\begin{split}
&\lr{a{}_{\lambda}\lr{e{}_{\mu}f}}_L-\lr{e{}_{\mu}\lr{a{}_{\lambda}f}}_R-\lr{\lr{a{}_{\lambda}e}{}_{\lambda+\mu}f}_L
\\
&=\lr{a{}_{\lambda}\big(\cc{e,f}+\ii{e,f}\mu\big)}_L+\lr{e{}_{\mu}\ii{\partial a,f}}_R+\lr{\ii{\partial a,e}{}_{\lambda+\mu}f}_L
\\
&=\lr{a{}_{\lambda}\cc{e,f}^{\prime}_l}\otimes\cc{e,f}^{\prime\prime}_l+\ii{\partial a,f}^{\prime}\otimes\lr{e{}_{\mu}\ii{\partial a,f}^{\prime\prime}}+\lr{\ii{\partial a,e}^{\prime}{}_{\lambda+\mu+\partial}f}_{\to}\otimes_1\ii{\partial a,e}^{\prime\prime}
\\
&=-\ii{\partial a,\cc{e,f}^{\prime}_l}^{\prime}\otimes \ii{\partial a,\cc{e,f}^{\prime}_l}^{\prime\prime}\otimes\cc{e,f}^{\prime\prime}_l
\\
&\quad +\ii{\partial a,f}^{\prime}\otimes\ii{e,\partial\ii{\partial a,f}^{\prime\prime}}^{\prime} \otimes\ii{e,\partial\ii{\partial a,f}^{\prime\prime}}^{\prime\prime}
\\
&\quad -\ii{\partial\ii{\partial a,e}^{\prime},f}^{\prime}\otimes \ii{\partial a,e}^{\prime\prime}\otimes \ii{\partial\ii{\partial a,e}^{\prime},f}^{\prime\prime}
\\
&=-\ii{\partial a,\cc{e,f}}_L+\ii{e,\partial\ii{\partial a,f}}_R -\ii{\partial\ii{\partial a,e},f}_L,
\end{split}
\label{eq:Jacobi-caso-intermedio-auxiliar}
\end{equation}
where once again we used \eqref{eq:partialLeibniz-rule}, \eqref{eq:extension-pairing-first-second} and \eqref{eq:ext-pairing-zero}.
Applying Lemma \ref{lem:lema-tecnico-cuatro-terminos} to the first term, we finally obtain that \eqref{eq:Jacobi-caso-intermedio-auxiliar} vanishes, as required.
\allowdisplaybreaks
Finally, the most delicate case occurs when $e,f,g\in E$. The idea is to use \eqref{eq:vertex-bracket} and \eqref{eq-Jacobi-vertex-tensor} to rewrite each term in \eqref{Jacobi-vertex}.
Its first summand of \eqref{Jacobi-vertex} can be written as
\allowdisplaybreaks
\begin{equation}
\begin{split}
&\lr{e{}_\lambda\lr{f{}_\mu g}}_L
\\
&=\lr{e{}_{\lambda}\big(\cc{f,g}+\ii{f,g}\mu\big)}_L
\\
&=\lr{e{}_{\lambda}\cc{f,g}^{\prime}_l}\otimes \cc{f,g}^{\prime\prime}_l
+\lr{e{}_{\lambda}\cc{f,g}^{\prime}_r}\otimes \cc{f,g}^{\prime\prime}_r
+\big(\lr{e{}_{\lambda}\ii{f,g}^{\prime}}\otimes \ii{f,g}^{\prime\prime}\big)\mu
\\
&=\big(\cc{e,\cc{f,g}^{\prime}_l}+\ii{e,\cc{f,g}^{\prime}_l}\lambda\big)\!\otimes \!\cc{f,g}^{\prime\prime}_l
+\ii{e,\partial \cc{f,g}^{\prime}_r}\!\otimes\! \cc{f,g}^{\prime\prime}_r
+\big(\ii{e,\partial \ii{f,g}^{\prime}}\!\otimes\! \ii{f,g}^{\prime\prime}\big)\mu
\\
&=\cc{e,\cc{f,g}^\prime_l}_l\otimes\cc{f,g}^{\prime\prime}_l
+\cc{e,\cc{f,g}^\prime_l}_r\otimes\cc{f,g}^{\prime\prime}_l
+\ii{e,\partial\cc{f,g}^\prime_r}\otimes \cc{f,g}^{\prime\prime}_r
\\
&\qquad (\ii{e,\cc{f,g}^\prime_l}\otimes\cc{f,g}^{\prime\prime}_l)\lambda
+(\ii{e,\partial\ii{f,g}^{\prime}}\otimes\ii{f,g}^{\prime\prime})\mu.
\end{split}
\label{eq:Jacobi-CD-DPVA-general-bis-1}
\end{equation}
Using \eqref{eq:vertex-bracket} and \eqref{eq:extension-vertex-double-bracket}, the second summand in \eqref{Jacobi-vertex} reads as
\begin{equation}
\begin{split}
&\lr{f{}_\mu\lr{e{}_\lambda g}}_R
\\
&=\lr{f{}_\mu\big(\cc{e,g}+\ii{e,g}\lambda}_R
\\
&=\cc{e,g}^{\prime}_l\otimes \lr{f{}_\mu\cc{e,g}^{\prime\prime}_l}
+\cc{e,g}^{\prime}_r\otimes \lr{f{}_\mu\cc{e,g}^{\prime\prime}_r}
+\big( \ii{e,g}^{\prime}\otimes\lr{f{}_\mu\ii{e,g}^{\prime\prime}}\big)\lambda
\\
&=\cc{e,g}^{\prime}_l\!\otimes \!\ii{f,\partial \cc{e,g}^{\prime\prime}_l}
+\cc{e,g}^{\prime}_r\!\otimes\!(\cc{f,\cc{e,g}^{\prime\prime}_r}+\ii{f,\cc{e,g}^{\prime\prime}_r}\mu)
+(\ii{e,g}^{\prime}\!\otimes\!\ii{f,\partial \ii{e,g}^{\prime\prime}})\lambda
\\
&=\cc{e,g}^{\prime}_l\otimes \ii{f,\partial\cc{e,g}^{\prime\prime}_l}
+\cc{e,g}^\prime_r\otimes\cc{f,\cc{e,g}^{\prime\prime}_r}_l
+\cc{e,g}^\prime_r\otimes\cc{f,\cc{e,g}^{\prime\prime}_r}_r
\\
&\qquad+ (\ii{e,g}^\prime\otimes\ii{f,\partial\ii{e,g}^{\prime\prime}})\lambda
+(\cc{e,g}^{\prime}_r\otimes\ii{f,\cc{e,g}^{\prime\prime}_r})\mu.
\end{split}
\label{eq:Jacobi-CD-DPVA-general-bis-2}
\end{equation}
Finally, we can rewrite the third summand of \eqref{eq:vertex-bracket}:
\begin{equation}
\begin{split}
&\lr{\lr{e{}_{\lambda} f}{}_{\lambda+\mu}g}_L
\\
&=\lr{\big(\cc{e,f}+\ii{e,f}\lambda\big){}_{\lambda+\mu}g}_L
\\
&= \lr{\cc{e,f}^\prime_l{}_{\lambda+\mu+\partial}g}_{\to}\otimes_1\cc{e,f}^{\prime\prime}_l
+\lr{\cc{e,f}^{\prime}_r{}_{\lambda+\mu+\partial}g}_{\to}\otimes_1\cc{e,f}^{\prime\prime}_r
\\
&\qquad +(\lr{\ii{e,f}^{\prime}{}_{\lambda+\mu+\partial}g}_{\to}\otimes_1\ii{e,f}^{\prime\prime})\lambda
\\
&=\cc{\cc{e,f}^\prime_l,g}^{\prime}_l\otimes\cc{e,f}^{\prime\prime}_l\otimes\cc{\cc{e,f}^{\prime}_l,g}^{\prime\prime}_l
+\cc{\cc{e,f}^{\prime}_l,g}^\prime_r\otimes\cc{e,f}^{\prime\prime}_l\otimes\cc{\cc{e,f}^{\prime}_l,g}^{\prime\prime}_r
\\
&\qquad +\ii{\cc{e,f}^\prime_l,g}^{\prime}\otimes (\lambda+\mu+\partial)\cc{e,f}^{\prime\prime}_l\otimes \ii{\cc{e,f}^{\prime}_l,g}^{\prime\prime}
\\
&\qquad -\ii{\partial\cc{e,f}^{\prime}_r,g}^\prime\!\otimes\!\cc{e,f}^{\prime\prime}_r\!\otimes\! \ii{\partial\cc{e,f}^{\prime}_r,g}^{\prime\prime}
-(\ii{\partial\ii{e,f}^{\prime},g}^{\prime}\!\otimes\!\ii{e,f}^{\prime\prime}\!\otimes\!\ii{\partial\ii{e,f}^\prime,g}^{\prime\prime})\lambda.
\end{split}
\label{eq:Jacobi-CD-DPVA-general-bis-3}
\end{equation}
So, at this point, collecting the terms obtained in \eqref{eq:Jacobi-CD-DPVA-general-bis-1}--\eqref{eq:Jacobi-CD-DPVA-general-bis-3}, we finally obtained:
\begin{align*}
\lr{e{}_\lambda \lr{f{}_\mu g}}_L&-\lr{f{}_\mu\lr{e{}_\lambda g}}_R-\lr{\lr{e{}_\lambda f}{}_{\lambda+\mu}g}_L
\\
&=\circled{A}+\circled{B}+\circled{C}+\circled{D}\lambda+\circled{E}\mu,
\end{align*}
where
\begin{subequations}
\label{eq:Jacobi-collecting-terms-final-e-f-g}
\begin{align*}
\begin{split}
\circled{A}&:= \cc{e,\cc{f,g}^\prime_l}_l\otimes\cc{f,g}^{\prime\prime}_l-\cc{e,g}^\prime_l\otimes\ii{f,\partial\cc{e,g}^{\prime\prime}_l}-\cc{\cc{e,g}^{\prime}_l,g}_l\otimes_1\cc{e,f}^{\prime\prime}_l,
\end{split}
\\
\begin{split}
\circled{B}&:=\cc{e,\cc{f,g}^{\prime}_l}_r\otimes \cc{f,g}^{\prime\prime}_l-\cc{e,g}^\prime_r\otimes \cc{f,\cc{e,g}^{\prime\prime}_r}_l-\ii{\cc{e,f}^\prime_l,g}\otimes_1\partial \cc{e,f}^{\prime\prime}_l
\\
&\qquad -\ii{\partial\cc{e,f}^\prime_r,g}\otimes_1\cc{e,f}^{\prime\prime}_r,
\end{split}
\\
\circled{C}&:=\ii{e,\partial\cc{f,g}^\prime_r}\otimes\cc{f,g}^{\prime\prime}_r-\cc{e,g}^\prime_r\otimes \cc{f,\cc{e,g}^{\prime\prime}_r}_r-\cc{\cc{e,f}^{\prime}_l,g}_r\otimes_1\cc{e,f}^{\prime\prime}_l,
\\
\circled{D}&:=\ii{e,\cc{f,g}^\prime_l}\otimes \cc{f,g}^{\prime\prime}_l -\ii{e,g}^\prime\otimes\ii{f,\partial\ii{e,g}^{\prime\prime}}
\\
&\qquad -\ii{\cc{e,f}^\prime_l,g}\otimes_1\cc{e,f}^{\prime\prime}_l+\ii{\partial\ii{e,f}^{\prime},g}\otimes_1\ii{e,f}-\cc{e,g}^{\prime\prime},
\\
\circled{E}&:=\ii{e,\partial\ii{f,g}^\prime}\otimes\ii{f,g}^{\prime\prime}-\cc{e,g}^\prime_r\otimes\ii{f,\cc{e,g}^{\prime\prime}_r}-\ii{\cc{e,f}^\prime_l,g}\otimes_1\cc{e,f}^{\prime\prime}_l.
\end{align*}
\end{subequations}
Now, by Remark \ref{rem:explicit-Jacobi-Courant-Dorfman-remark}, it is clear that $\circled{A}+\circled{B}+\circled{C}=0$ by \eqref{eq:CD.f}.
Also, since \eqref{eq:CD.g} holds, $\circled{E}=0$.
Finally, Lemma \ref{lem:lema-tecnico-cuatro-terminos} gives the vanishing of $\circled{D}$.
To sum up, in this subsection, given a double Courant--Dorman algebra we proved that it gives rise to a double Poisson vertex algebra freely generated in weights 0 and 1 via \eqref{eq:vertex-bracket}. Consequently, Theorem \ref{theorem-CD-DPVA} follows.
\section{The Kontsevich--Rosenberg principle for double Courant--Dorfman algebras}
\label{sec:KR-double-CD}
\subsection{The Kontsevich--Rosenberg principle for double Poisson (vertex) algebras}
\label{sec:KR-DPVA-SECTION}
The Kontsevich--Rosenberg principle \cite{Kon94a,KR00} states that a noncommutative algebro-geometric structure on $A$ should naturally induce the corresponding algebro-geometric structure on the representation scheme ${\operatorname{Rep}}(A,N)$. Here, if $d=(d_1,\dots,d_n)\in\mathbb{N}^n$ denotes a dimension vector, we set $N:=\sum^n_{i=1}d_i$.
Then, if $M_N(\kk)$ stands for the algebra of $N\times N$-matrices, we define the functor
\[
{\operatorname{Rep}}_N(A)\colon\texttt{CommAlg}_\kk\longrightarrow \texttt{Sets},\quad B\longmapsto \operatorname{Hom}\big(A,M_N(B)\big).
\]
By classical results of Bergman \cite{Be74} and Cohn \cite{Co79}, it is well-known that this functor is representable, since we have the adjunction
\[
\operatorname{Hom}(A_N,B)=\operatorname{Hom}\big(A,M_N(B)\big),
\]
and consequently the representation scheme ${\operatorname{Rep}}(A,N)$ is $\operatorname{Spec}(A_N)$; equivalently, $A_N\mathrel{\mathop:}=} %{\coloneqq} %{:=\kk[{\operatorname{Rep}}(A,N)]$.
As Van den Bergh \cite{VdB08} pointed out, the algebra $A_N$ is isomorphic to the commutative algebra on symbols $\{a_{ij}\mid a\in A,\; 1\leq i,j\leq N\}$ subject to the relations
\begin{equation}
(\alpha a)_{ij}=\alpha a_{ij},\quad (a+b)_{ij}=a_{ij}+b_{ij},\quad (ab)_{ij}=a_{it}b_{tj},\quad 1_{ij}a_{uv}=\delta_{ju}a_{uv},
\label{eq:KR-relations-defining-AV}
\end{equation}
for all $a,b\in A$, $\alpha\in\kk$, $i,j,t,u,v\in\{1,\dots, N\}$, where from now on we sum over repeated indices. In fact, he proved \cite[Proposition 7.5.2]{VdB08} that if $(A,\lr{-,-})$ is a double Poisson algebra (see Definition \ref{def:double-Poisson algebra}), then $(A_N,\{-,-\})$ is a Poisson algebra, with bracket given by
\begin{equation}
\{a_{ij},b_{uv}\}=\lr{a,b}^{\prime}_{uj}\lr{a,b}^{\prime\prime}_{iv}.
\label{eq:KR-double-Poisson-algebras}
\end{equation}
This arrangement of indices in \eqref{eq:KR-double-Poisson-algebras} is called the \emph{standard index convention}. It will repeatedly appear below.
Given a differential algebra $(\mathcal{V},\partial)$ (cf. \S\ref{sec:DPVA}), we consider the pair $(\mathcal{V}_N,\partial_N)$, where $\mathcal{V}_V$ is as above, and the derivation $\partial_N$ is defined by
\begin{equation}
\partial_N(a_{ij})\mathrel{\mathop:}=} %{\coloneqq} %{:=(\partial a)_{ij},
\label{eq:KR-derivation-partial}
\end{equation}
where $a\in A$ and $i,j\in\{1,\dots, N\}$. One of the main results in \cite{DSKV15} is the following result that states that double Poisson vertex algebras satisfy the Kontsevich--Rosenberg principle:
\begin{proposition}[\cite{DSKV15}, Proposition 3.20 \and Theorem 3.22]
Let $\mathcal{V}$ be a differential algebra endowed with a double $\lambda$-bracket $\lr{-{}_\lambda-}$, written as $\lr{a{}_\lambda b}=\sum_{\textbf{p}\in\mathbb{Z}_+}\big((a_{\textbf{p}}b)^\prime\otimes (a_{\textbf{p}}b)^{\prime\prime}\big)\lambda^{\textbf{p}}$, for all $a,b\in\mathcal{V}$. Then, for every $N\geq 1$, there exists a well-defined $\lambda$-bracket on $\mathcal{V}_N$, given by
\begin{equation}
\{a_{ij}{}_\lambda b_{hl}\}=\sum_{\textbf{p}\in\mathbb{Z}_+}(a_{\textbf{p}}b)^\prime_{hj}(a_{\textbf{p}}b)^{\prime\prime}_{il}\lambda^{\textbf{p}},
\label{lambda-bracket-comm}
\end{equation}
where $a_{ij},b_{hk}\in\mathcal{V}_N$.
In addition, if $\mathcal{V}$ is a double Poisson vertex algebra, then $\mathcal{V}_N$ is a Poisson vertex algebra with $\lambda$-bracket defined by \eqref{lambda-bracket-comm}.
\label{prop:KR-double-Poisson-vertex-algebras}
\end{proposition}
\subsection{The Kontsevich--Rosenberg principle for double Courant--Dorfman algebras}
\label{sec:KR-double-CD-section}
\allowdisplaybreaks
Van den Bergh \cite{VdB08a} also studied how bimodules and their noncommutative structures are reflected into representation spaces.
If $E$ is an $A$-bimodule, we define $E_N$ as the (left) $A_N$-module generated by symbols $\{e_{ij}\mid 1\leq i,j\leq N\}$ that are linear in $e\in E$ and satisfy
\begin{subequations}
\label{eq:KR-relaciones-bimodules}
\begin{align}
(ae)_{ij}&=a_{it }e_{t j},
\label{eq:KR-relaciones-bimodules.a}
\\
(ea)_{ij}&=a_{t j}e_{i t},
\label{eq:KR-relaciones-bimodules.b}
\end{align}
\end{subequations}
for $i,j,t\in\{1,\dots, N\}$.
\begin{remark}
Van den Bergh \cite[\S 3.3]{VdB08a} formalized the Kontsevich--Rosenberg principle by introducing the additive functor (now called the \emph{Van den Bergh functor})
\begin{equation}
(-)_N\colon \texttt{Bimod}(A)\longrightarrow \texttt{mod}(A_N),
\label{eq:VdB-functor-def}
\end{equation}
which is right exact and sends finitely generated projective $A$-bimodules to finitely generated projective $A_N$-modules --- see also \cite[\S4.3]{Ye19}. By \cite[Lemma 3.3.1]{VdB08a}, we consider ${\operatorname{M}}_N(A_N)$ as an $A$-bimodule via the $\kk$-algebra morphism $A\to {\operatorname{M}}_N(A_N)$, $a\mapsto a_{ij}$. Also, we consider $\text{M}_N(A_N)$ as an $A_N$-module via the diagonal embedding $A_N\to {\operatorname{M}}_N(A_N)$. Then there exists a natural isomorphism
\[
P_N\simeq P\otimes_{A^{\e}}{\operatorname{M}}_N(A_N).
\]
This functor allows us to prove the key isomorphisms (see \cite[Proposition 3.3.4]{VdB08a})
\[
(\Omega^1_{{\operatorname{nc}}}A)_N\simeq \Omega^1_{A_N},\quad (\D A)_N\simeq \operatorname{Der} A_N,
\]
providing $A$ is smooth (\emph{i.e.} $A$ is finitely generated as a $\kk$-algebra and $\Omega^1_{{\operatorname{nc}}}A$ is a projective $A^{\e}$-module).
In this article, it will be more convenient to follow the more explicit approach given by \eqref{eq:KR-relaciones-bimodules}.
Nevertheless, the study of the Van den Bergh functor \eqref{eq:VdB-functor-def} has given rise to insightful developments; for instance, \cite{BKR13} and \cite{Ye19} (see also references therein).
\end{remark}
The following result states how pairings (in the sense of \S\ref{sec:pairings}) induce usual bilinear forms on the representation schemes:
\begin{lemma}[\cite{VdB08a}, Lemma 3.3.3]
If $\ii{-,-}\colon E\otimes E\to A\otimes A$ is a non-degenerate pairing then the corresponding pairing between $E_N$ (obtained by applying the standard index convention)
\[
\langle-,-\rangle\colon E_N\otimes E_N\longrightarrow A_N, \quad (e_{ij},f_{uv})\longmapsto \ii{ e,f}^{\prime}_{uj} \ii{ e,f}^{\prime\prime}_{iv}
\]
is non-degenerate as well.
\label{lem:KR-pairings}
\end{lemma}
At this point, we can can prove that the Kontsevich--Rosenberg principle holds for double Courant--Dorfman algebras:
\begin{theorem}
Let $(A,E,\ii{-,-},\partial,\cc{-,-})$ be a double Courant--Dorfman algebra.
For all $N\geq 1$, we define
\begin{subequations}
\label{eq:KR-CD-thm}
\begin{align}
\partial_N a_{ij}&\mathrel{\mathop:}=} %{\coloneqq} %{:= (\partial a)_{ij}, \label{eq:KR-CD-thm.a}
\\
\langle e_{ij},f_{uv}\rangle&\mathrel{\mathop:}=} %{\coloneqq} %{:= \ii{e,f}^{\prime}_{uj}\ii{e,f}^{\prime\prime}_{iv}, \label{eq:KR-CD-thm.b}
\\
[e_{ij},f_{uv}]&\mathrel{\mathop:}=} %{\coloneqq} %{:= \cc{e,f}^{\prime}_{l,uj} \cc{e,f}^{\prime\prime}_{l,iv}+ \cc{e,f}^{\prime}_{r,uj} \cc{e,f}^{\prime\prime}_{r,iv}, \label{eq:KR-CD-thm.c}
\end{align}
\end{subequations}
for all $a\in A$ and $e,f\in E$.
Then the 5-tuple $(A_N,E_N,\langle-,-\rangle,\partial_N, [-,-])$ is a Courant--Dorfman algebra in the sense of Definition \ref{def:CD-algebra-def-comm}.
\label{thm:KR-double-Courant-Dorfman}
\end{theorem}
\begin{proof}
It is clear that $\partial_N$, as defined in \eqref{eq:KR-CD-thm.a}, is a derivation of $A_N$, as well as $\langle-,-\rangle$ in \eqref{eq:KR-CD-thm.b} is a bilinear form on $E_N$ by Lemma \ref{lem:KR-pairings}.
Now, we extend \eqref{eq:KR-CD-thm.c} to a bilinear form $[-,-]\colon E_N\times E_N\to E_N$ that satisfies \eqref{eq:CD-comm.a}. To see that this form is well-defined, we shall verify the compatibility with the defining relations of $E_N$ written in \eqref{eq:KR-relaciones-bimodules}. The linearity of $[-,-]$ follows from the linearity of $\cc{-,-}$.
To verify the compatibility with \eqref{eq:KR-relaciones-bimodules.a}, using \eqref{eq:decomposition-Courant-Dorfman-def-1}, we can decompose \eqref{eq:CD.e} as follows:
\begin{equation}
\begin{aligned}
\cc{e,af}^{\prime}_l\otimes \cc{e,af}^{\prime\prime}_l&=a \cc{e,f}^{\prime}_l\otimes \cc{e,f}^{\prime\prime}_l\in E\otimes A,
\\
\cc{e,af}^{\prime}_r\otimes \cc{e,af}^{\prime\prime}_r&=a \cc{e,f}^{\prime}_r\otimes \cc{e,f}^{\prime\prime}_r+\ii{e,\partial a}^{\prime}\otimes \ii{e,\partial a}^{\prime\prime}f\in A\otimes E,
\end{aligned}
\label{eq:KR-Leibniz-1}
\end{equation}
Then, by \eqref{eq:KR-CD-thm.c} and \eqref{eq:KR-Leibniz-1} we have
\begin{align*}
[e_{ij},(af)_{uv}]&=\cc{e,af}^{\prime}_{l,uj}\cc{e,af}^{\prime\prime}_{l,iv}+\cc{e,af}^{\prime}_{r,uj}\cc{e,af}^{\prime\prime}_{r,iv}
\\
&=(a\cc{e,f}^{\prime}_{l})_{uj}\cc{e,f}^{\prime\prime}_{l,iv}+ (a\cc{e,f}^{\prime}_{r})_{uj}\cc{e,f}^{\prime\prime}_{r,iv}+\ii{e,\partial a}^{\prime}_{uj}(\ii{e,\partial a}^{\prime\prime}f)_{iv}
\intertext{Now, by \eqref{eq:KR-relaciones-bimodules.a}, \eqref{eq:KR-relations-defining-AV}, \eqref{eq:KR-CD-thm.a} and \eqref{eq:KR-derivation-partial}}
&=a_{ut}\big(\cc{e,f}^{\prime}_{l,tj}\cc{e,f}^{\prime\prime}_{l,iv} + \cc{e,f}^{\prime}_{r,tj}\cc{e,f}^{\prime\prime}_{r,iv}\big)+\ii{e,\partial a}^{\prime}_{uj}\ii{e,\partial a}^{\prime\prime}_{it} f_{tv}
\\
&=a_{ut}[e_{ij},f_{tv}]+\langle e_{ij},(\partial a)_{ut}\rangle f_{tv}
\\
&=a_{ut}[e_{ij},f_{tv}]+\langle e_{ij},\partial_N a_{ut}\rangle f_{tv}=[e_{ij},a_{ut}f_{tv}],
\end{align*}
for all $i,j,u,v,t\in \{1,\dots,N\}$. Similarly, we can check the compatibility with \eqref{eq:KR-relaciones-bimodules.b} as follows. By \eqref{eq:CD.d}, we can write
\begin{equation}
\begin{aligned}
\cc{e,fa}^{\prime}_l\otimes \cc{e,fa}^{\prime\prime}_l&= \cc{e,f}^{\prime}_l\otimes \cc{e,f}^{\prime\prime}_la +f\ii{e,\partial a}^{\prime}\otimes \ii{e,\partial a}^{\prime\prime}\in E\otimes A,
\\
\cc{e,fa}^{\prime}_r\otimes \cc{e,fa}^{\prime\prime}_r&= \cc{e,f}^{\prime}_r\otimes \cc{e,f}^{\prime\prime}_ra\in A\otimes E.
\end{aligned}
\label{eq:KR-Leibniz-2}
\end{equation}
Hence,
\begin{align*}
[e_{ij},(fa)_{uv}]&=\cc{e,fa}^{\prime}_{l,uj} \cc{e,fa}^{\prime\prime}_{l,iv}+ \cc{e,fa}^{\prime}_{r,uj} \cc{e,fa}^{\prime\prime}_{r,iv}
\\
&=\cc{e,f}^{\prime}_{l,uj}(\cc{e,f}^{\prime\prime}_{l}a)_{iv}+(f\ii{e,\partial a}^{\prime})_{uj}\ii{e,\partial a}^{\prime\prime}_{iv}+\cc{e,f}^{\prime}_{r,uj}(\cc{e,f}^{\prime\prime}_ra)_{iv}
\\
&=\cc{e,f}^{\prime}_{l,uj}a_{tv}\cc{e,f}^{\prime\prime}_{l,it}+\ii{e,\partial a}^{\prime}_{tj} f_{ut}\ii{e,\partial a}^{\prime\prime}_{iv}+\cc{e,f}^{\prime}_{r,uj}a_{tv}\cc{e,f}^{\prime\prime}_{r,it}
\\
&=a_{tv}\big( \cc{e,f}^{\prime}_{l,uj}\cc{e,f}^{\prime\prime}_{l,it} + \cc{e,f}^{\prime}_{r,uj}\cc{e,f}^{\prime\prime}_{r,it}\big)+\ii{e,\partial a}^{\prime}_{tj} \ii{e,\partial a}^{\prime\prime}_{iv} f_{ut}
\\
&=a_{tv}[e_{ij},f_{ut}]+\langle e_{ij},(\partial f)_{tv}\rangle f_{ut}
\\
&=a_{tv}[e_{ij},f_{ut}]+\langle e_{ij},\partial_Nf_{tv}\rangle f_{ut}=[e_{ij},a_{tv}f_{ut}],
\end{align*}
as we wished. Moreover, from these computations, \eqref{eq:CD-comm.a} follows.
To prove \eqref{eq:CD-comm.e}, by \eqref{eq:KR-derivation-partial}, \eqref{eq:KR-CD-thm.a} and \eqref{eq:KR-CD-thm.c}, we have
\[
[\partial_N a_{ij}, e_{uv}]= [(\partial a)_{ij}, e_{uv}]=\cc{\partial a, e}^{\prime}_{l,uj} \cc{\partial a, e}^{\prime\prime}_{l,iv} + \cc{\partial a, e}^{\prime}_{r,uj} \cc{\partial a, e}^{\prime\prime}_{r, iv}=0,
\]
where the last identity is due to \eqref{eq:CD.b}.
Similarly, by \eqref{eq:KR-CD-thm.a} and \eqref{eq:KR-CD-thm.b}, we can write
\[
\langle \partial_N a_{ij},\partial_N b_{uv}\rangle= \langle (\partial a)_{ij}, (\partial b)_{uv}\rangle=\ii{\partial a,\partial b}^{\prime}_{uj} \ii{\partial a,\partial b}^{\prime\prime}_{iv}=0,
\]
where we used \eqref{eq:CD.c} in the last identity.
Now, since $\partial$ acts on tensor products by the Leibniz identity --- see \eqref{eq:partialLeibniz-rule}, we can decompose \eqref{eq:CD.a} as follows
\begin{subequations}
\label{eq:KR-skew-symm}
\begin{align}
\partial\ii{e,f}^{\prime}\otimes \ii{e,f}^{\prime\prime}&=\cc{e,f}^{\prime}_l\otimes \cc{e,f}^{\prime\prime}_l+\cc{f,e}^{\prime\prime}_r\otimes \cc{f,e}^{\prime}_r\in E\otimes A,
\label{eq:KR-skew-symm.a}
\\
\ii{e,f}^{\prime}\otimes \partial \ii{e,f}^{\prime\prime}&=\cc{e,f}^{\prime}_r\otimes \cc{e,f}^{\prime\prime}_r+\cc{f,e}^{\prime\prime}_l\otimes \cc{f,e}^{\prime}_l\in A\otimes E. \label{eq:KR-skew-symm.b}
\end{align}
\end{subequations}
So, using \eqref{eq:KR-CD-thm.c} and \eqref{eq:KR-skew-symm}, we have:
\begin{align*}
&[e_{ij},f_{uv}]+[f_{uv},e_{ij}]
\\
&=\cc{e,f}^{\prime}_{l,uj} \cc{e,f}^{\prime\prime}_{l,iv}+ \cc{e,f}^{\prime}_{r,uj} \cc{e,f}^{\prime\prime}_{r,iv}+\cc{f,e}^{\prime}_{l,iv}\cc{f,e}^{\prime\prime}_{l,uj}+ \cc{f,e}^{\prime}_{r,iv}\cc{f,e}^{\prime\prime}_{r,uj}
\\
&=(\partial\ii{e,f}^{\prime})_{uj}\ii{e,f}^{\prime\prime}_{iv}+\ii{e,f}^{\prime}_{uj}(\partial\ii{e,f}^{\prime\prime})_{iv}
\\
&=\partial_N(\ii{e,f}^{\prime}_{uj})\ii{e,f}^{\prime\prime}_{iv}+\ii{e,f}^{\prime}_{uj}\partial_N(\ii{e,f}^{\prime\prime})_{iv}
\\
&=\partial_N(\ii{e,f}^{\prime}_{uj}\ii{e,f}^{\prime\prime}_{iv})=\partial_N\langle e_{ij},f_{uv}\rangle,
\end{align*}
where we used \eqref{eq:KR-CD-thm.a} and \eqref{eq:KR-CD-thm.b}, and thus \eqref{eq:CD-comm.c} follows.
To prove \eqref{eq:CD-comm.b}, we will need the following result:
\begin{claim}
Let $(A,E,\ii{-,-},\partial, \cc{-,-})$ be a double Courant--Dorfman algebra. Then, for all $e,f,g\in E$, the following identity holds:
\begin{equation}
\begin{split}
\ii{f,g}^{\prime}&\otimes \ii{e,\partial\ii{f,g}^{\prime\prime}}
= \ii{f,\cc{e,g}^{\prime}_l}\otimes \cc{e,g}^{\prime\prime}_l
+ \ii{\cc{e,f}^{\prime\prime}_r,g}\otimes_1 \cc{e,f}^{\prime}_r.
\end{split}
\label{eq:auxiliar-CD7-KR}
\end{equation}
\label{claim:auxiliar-CD7-KR}
\end{claim}
\begin{proof}
By \eqref{eq:CD.g} and \eqref{eq:Sweedler-pairing-symm}, we have
\begin{align*}
\ii{e,\partial\ii{f,g}^{\prime\prime}}^{\prime}&\otimes \ii{e,\partial\ii{f,g}^{\prime\prime}}^{\prime\prime}\otimes\ii{f,g}^{\prime\prime}
\\
&=\cc{e,f}^{\prime}_r\otimes\ii{\cc{e,f}^{\prime\prime}_r,g}^{\prime\prime}\otimes \ii{\cc{e,f}^{\prime\prime}_r,g}^{\prime}
+\ii{f,\cc{e,g}^{\prime}_l}^{\prime\prime}\otimes\cc{e,g}^{\prime\prime}_l\otimes \ii{f,\cc{e,g}^{\prime}_l}^{\prime}.
\end{align*}
Applying the permutation $\sigma_{(123)}$, \eqref{eq:auxiliar-CD7-KR} follows.
\end{proof}
Now, using \eqref{eq:KR-CD-thm} and the $A_N$-bilinearity of $\langle-,-\rangle$, the left-hand side of \eqref{eq:CD-comm.b} can be written as
\begin{equation}
\begin{aligned}
& \big\langle e_{ij},\partial_N\langle f_{uv}, g_{pq} \rangle\big\rangle
\\
&=\langle e_{ij},\partial_N (\ii{f,g}^{\prime}_{pv}\ii{f,g}^{\prime\prime}_{uq})\rangle
\\
&=\langle e_{ij}, \partial_N(\ii{f,g}^{\prime})_{pv}\rangle\ii{f,g}^{\prime\prime}_{uq}+\ii{f,g}^{\prime}_{pv}\langle e_{ij},(\partial_N\ii{f,g}^{\prime\prime})_{uq}\rangle
\\
&=\ii{e,\partial \ii{f,g}^{\prime}}^{\prime}_{pj} \ii{e,\partial \ii{f,g}^{\prime}}^{\prime\prime}_{iv}\ii{f,g}^{\prime\prime}_{uq}
+\ii{f,g}^{\prime}_{pv}\ii{e,\partial\ii{f,g}^{\prime\prime}}^{\prime}_{uj} \ii{e,\partial\ii{f,g}^{\prime\prime}}^{\prime\prime}_{iq}
\\
&=\cc{e,g}^{\prime}_{r,pj}\ii{f,\cc{e,g}^{\prime\prime}_r}^{\prime}_{iv} \ii{f,\cc{e,g}^{\prime\prime}_r}^{\prime\prime}_{uq} +\ii{\cc{e,f}^{\prime}_l,g}^{\prime}_{pj}\cc{e,f}^{\prime\prime}_{l,iv}\ii{\cc{e,f}^{\prime}_l,g}^{\prime\prime}_{uq}
\\
&\quad + \ii{f,\cc{e,g}^{\prime}_l}^{\prime}_{pv} \ii{f,\cc{e,g}^{\prime}_l}^{\prime\prime}_{uj}\cc{e,g}^{\prime\prime}_{l,iq}+\ii{\cc{e,f}^{\prime\prime}_r,g}^{\prime}_{pv}\cc{e,f}^{\prime}_{r,uj}\ii{\cc{e,f}^{\prime\prime}_r,g}^{\prime\prime}_{iq},
\end{aligned}
\label{eq:KR-CD7-LHS}
\end{equation}
where the last identity is due to \eqref{eq:CD.g} and Claim \ref{claim:auxiliar-CD7-KR}. On the other hand, by \eqref{eq:KR-CD-thm}, we can rewrite the right-hand side of \eqref{eq:CD-comm.b} in the following way:
\begin{equation}
\begin{aligned}
&\langle[e_{ij},f_{uv}],g_{pq}\rangle+\langle f_{uv},[e_{ij},g_{pq}]\rangle
\\
&=\langle \cc{e,f}^{\prime}_{l,uj}\cc{e,f}^{\prime\prime}_{l,iv},g_{pq}\rangle+ \langle \cc{e,f}^{\prime}_{r,uj}\cc{e,f}^{\prime\prime}_{r,iv},g_{pq}\rangle
\\
&\quad +\langle f_{uv},\cc{e,g}^{\prime}_{l,pj}\cc{e,g}^{\prime\prime}_{l,iq}\rangle+\langle f_{uv},\cc{e,g}^{\prime}_{r,pj}\cc{e,g}^{\prime\prime}_{r,iq}\rangle
\\
&=\langle \cc{e,f}^{\prime}_{l,uj},g_{pq}\rangle \cc{e,f}^{\prime\prime}_{l,iv}+ \cc{e,f}^{\prime}_{r,uj} \langle \cc{e,f}^{\prime\prime}_{r,iv},g_{pq}\rangle
\\
&\quad +\langle f_{uv},\cc{e,g}^{\prime}_{l,pj}\rangle \cc{e,g}^{\prime\prime}_{l,iq}+ \cc{e,g}^{\prime}_{r,pj} \langle f_{uv},\cc{e,g}^{\prime\prime}_{r,iq}\rangle
\\
&=\ii{\cc{e,f}^{\prime}_{l},g}^{\prime}_{pj} \ii{\cc{e,f}^{\prime}_{l},g}^{\prime\prime}_{uq} \cc{e,f}^{\prime\prime}_{l,iv}
+ \cc{e,f}^{\prime}_{r,uj} \ii{\cc{e,f}^{\prime\prime}_{r},g}^{\prime}_{pv} \ii{\cc{e,f}^{\prime\prime}_{r},g}^{\prime\prime}_{iq}
\\
&\quad +\ii{ f,\cc{e,g}^{\prime}_{l}}^{\prime}_{pv} \ii{ f,\cc{e,g}^{\prime}_{l}}^{\prime\prime}_{uj}\cc{e,g}^{\prime\prime}_{l,iq}
+ \cc{e,g}^{\prime}_{r,pj} \ii{ f,\cc{e,g}^{\prime\prime}_{r}}^{\prime}_{iv} \ii{ f,\cc{e,g}^{\prime\prime}_{r}}^{\prime\prime}_{uq}.
\end{aligned}
\label{eq:KR-CD7-RHS}
\end{equation}
Since we are working in the commutative setting, \eqref{eq:KR-CD7-LHS} and \eqref{eq:KR-CD7-RHS} agree, and \eqref{eq:CD-comm.b} is satisfied.
Finally, we need to prove that the Jacobi identity \eqref{eq:CD-comm.d} holds:\begin{equation}
\label{eq:KR-Jacobi-inicial}
[e_{ij},[f_{uv},g_{pq}]]=[[e_{ij},f_{uv}],g_{pq}]+[f_{uv},[e_{ij},g_{pq}]].
\end{equation}
Firstly, applying \eqref{eq:KR-skew-symm.a} (resp. \eqref{eq:KR-skew-symm.b}) in the last summand of \eqref{eq:CD-double-Jacobi-explicit.a} (resp. \eqref{eq:CD-double-Jacobi-explicit.c}), we obtain the identities
\begin{subequations}
\label{eq:auxiliar-Jacobi-doble-1}
\begin{align}
\cc{e,\cc{f,g}^{\prime}_l}^{\prime}_l&\otimes \cc{e,\cc{f,g}^{\prime}_l}^{\prime\prime}_l\otimes\cc{f,g}^{\prime\prime}_l
=\cc{e,g}^{\prime}_l\otimes\ii{f,\partial\cc{e,g}^{\prime\prime}_l}^{\prime} \otimes\ii{f,\partial\cc{e,g}^{\prime\prime}_l}^{\prime\prime}\nonumber
\\
& +\partial\ii{\cc{e,f}^{\prime}_l,g}^{\prime}\otimes \cc{e,f}^{\prime\prime}_l\otimes \ii{\cc{e,f}^{\prime}_l,g}^{\prime\prime}
-\cc{g,\cc{e,f}^{\prime}_l}^{\prime\prime}_r\otimes\cc{e,f}^{\prime\prime}_l\otimes\cc{g,\cc{e,f}^{\prime}_l}^{\prime}_r,
\label{eq:auxiliar-Jacobi-doble-1.a}
\\
\ii{e,\partial\cc{f,g}^{\prime}_r}^{\prime}&\otimes \ii{e,\partial\cc{f,g}^{\prime}_r}^{\prime\prime}\otimes\cc{f,g}^{\prime\prime}_r=\cc{e,g}^{\prime}_r\otimes\cc{f,\cc{e,g}^{\prime\prime}_r}^{\prime}_r \otimes\cc{f,\cc{e,g}^{\prime\prime}_r}^{\prime\prime}_r\nonumber
\\
&+\ii{\cc{e,f}^{\prime}_l,g}^{\prime}\otimes \cc{e,f}^{\prime\prime}_l\otimes \partial\ii{\cc{e,f}^{\prime}_l,g}^{\prime\prime}
-\cc{g,\cc{e,f}^{\prime}_l}^{\prime\prime}_l\otimes \cc{e,f}^{\prime\prime}_l\otimes \cc{g,\cc{e,f}^{\prime}_l}^{\prime}_l.
\label{eq:auxiliar-Jacobi-doble-1.b}
\end{align}
\end{subequations}
Next, by \eqref{eq:KR-CD-thm} and \eqref{eq:CD-comm.a}, we rewrite the term in the left-hand side of \eqref{eq:KR-Jacobi-inicial} in the following way:
\begin{equation}
\begin{aligned}
&[e_{ij},[f_{uv},g_{pq}]]
\\
&=[e_{ij},\cc{f,g}^{\prime}_{l,pv}\cc{f,g}^{\prime\prime}_{l,uq}]+[e_{ij},\cc{f,g}^{\prime}_{r,pv}\cc{f,g}^{\prime\prime}_{r,uq}]
\\
&=\cc{f,g}^{\prime\prime}_{l,uq}[e_{ij},\cc{f,g}^{\prime}_{l,pv}]+\langle e_{ij},\partial_N\cc{f,g}^{\prime\prime}_{l,uq}\rangle\cc{f,g}^{\prime}_{l,pv}
\\
&\quad + \cc{f,g}^{\prime}_{r,pv}[e_{ij},\cc{f,g}^{\prime\prime}_{r,uq}]+\langle e_{ij},\partial_N\cc{f,g}^{\prime}_{r,pv}\rangle \cc{f,g}^{\prime\prime}_{r,uq}
\\
&=\cc{f,g}^{\prime\prime}_{l,uq}\cc{e,\cc{f,g}^{\prime}_l}^{\prime}_{l,pj} \cc{e,\cc{f,g}^{\prime}_l}^{\prime\prime}_{l,iv}
+ \cc{f,g}^{\prime\prime}_{l,uq}\cc{e,\cc{f,g}^{\prime}_l}^{\prime}_{r,pj} \cc{e,\cc{f,g}^{\prime}_l}^{\prime\prime}_{r,iv}
\\
&\quad +\ii{e,\partial\cc{f,g}^{\prime\prime}_l}^{\prime}_{uj} \ii{e,\partial\cc{f,g}^{\prime\prime}_l}^{\prime\prime}_{iq}\cc{f,g}^{\prime}_{l,pv}
+\cc{f,g}^{\prime}_{r,pv}\cc{e,\cc{f,g}^{\prime\prime}_r}^{\prime}_{l,uj}\cc{e,\cc{f,g}^{\prime\prime}_r}^{\prime\prime}_{l,iq}
\\
&\quad + \cc{f,g}^{\prime}_{r,pv}\cc{e,\cc{f,g}^{\prime\prime}_r}^{\prime}_{r,uj}\cc{e,\cc{f,g}^{\prime\prime}_r}^{\prime\prime}_{r,iq}
+ \ii{e,\partial\cc{f,g}^{\prime}_r}^{\prime}_{pj} \ii{e,\partial\cc{f,g}^{\prime}_r}^{\prime\prime}_{iv}\cc{f,g}^{\prime\prime}_{r,uq}.
\end{aligned}
\label{eq:Jacobi-KR-first-1}
\end{equation}
By using \eqref{eq:auxiliar-Jacobi-doble-1.a}, \eqref{eq:CD-double-Jacobi-explicit.b}, \eqref{eq:ecuaciones-Jacobi-doble-skew.c}, \eqref{eq:ecuaciones-Jacobi-doble-skew.a}, \eqref{eq:ecuaciones-Jacobi-doble-skew.b} and \eqref{eq:auxiliar-Jacobi-doble-1.b}, we can rewrite the six terms obtained in \eqref{eq:Jacobi-KR-first-1} as
\begin{equation}
\begin{aligned}
&=\cc{e,g}^{\prime}_{l,pj}\ii{f,\partial\cc{e,g}^{\prime\prime}_l}^{\prime}_{iv} \ii{f,\partial\cc{e,g}^{\prime\prime}_l}^{\prime\prime}_{uq}
+\big(\partial\ii{\cc{e,f}^{\prime}_l,g}^{\prime}\big)_{pj}\cc{e,f}^{\prime\prime}_{l,iv}\ii{\cc{e,f}^{\prime}_l,g}^{\prime\prime}_{uq}
\\
&\qquad -\cc{g,\cc{e,f}^{\prime}_l}^{\prime\prime}_{r,pj}\cc{e,f}^{\prime\prime}_{l,iv}\cc{g,\cc{e,f}^{\prime}_l}^{\prime}_{r,uq}
\\
&\quad +\cc{e,g}^{\prime}_{r,pj}\cc{f,\cc{e,g}^{\prime\prime}_r}^{\prime}_{l,iv}\cc{f,\cc{e,g}^{\prime\prime}_r}^{\prime\prime}_{l,uq} +\ii{\cc{e,f}^{\prime}_l,g}^{\prime}_{pj}\big(\partial\cc{e,f}^{\prime\prime}_{l}\big)_{iv} \ii{\cc{e,f}^{\prime}_l,g}^{\prime\prime}_{uq}
\\
&\qquad -\ii{g,\partial\cc{e,f}^{\prime}_r}^{\prime\prime}_{pj}\cc{e,f}^{\prime\prime}_{r,iv}\ii{g,\partial\cc{e,f}^{\prime}_r}^{\prime}_{uq}
\\
&\quad +\cc{e,f}^{\prime}_{r,uj}\ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime}_{iq}\big(\partial \ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime}\big)_{pv}
+\cc{f,\cc{e,g}^{\prime}_l}^{\prime\prime}_{l,uj}\cc{e,g}^{\prime\prime}_{l,iq}\cc{f,\cc{e,g}^{\prime}_l}^{\prime}_{l,pv}
\\
&\qquad -\cc{e,f}^{\prime}_{r,uj}\cc{g,\cc{e,f}^{\prime\prime}_r}^{\prime}_{r,iq} \cc{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime}_{r,pv}
\\
&\quad +\big(\partial \cc{e,f}^{\prime}_r\big)_{uj}\ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime}_{iq}\ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime}_{pv}
+\cc{f,\cc{e,g}^{\prime}_l}^{\prime\prime}_{r,uj}\cc{e,g}^{\prime\prime}_{l,iq} \cc{f,\cc{e,g}^{\prime}_l}^{\prime}_{r,pv}
\\
&\qquad -\cc{e,f}^{\prime}_{l,uj}\ii{g,\partial\cc{e,f}^{\prime\prime}_l}^{\prime}_{iq} \ii{g,\partial\cc{e,f}^{\prime\prime}_l}^{\prime\prime}_{pv}
\\
&\quad +\cc{e,f}^{\prime}_{r,uj}\big(\partial\ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime}\big)_{iq}\ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime}_{pv}
+\ii{f,\partial\cc{e,g}^{\prime}_r}^{\prime\prime}_{uj}\cc{e,g}^{\prime\prime}_{r,iq}\ii{f,\partial\cc{e,g}^{\prime}_r}^{\prime}_{pv}
\\
&\qquad -\cc{e,f}^{\prime}_{r,uj}\cc{g,\cc{e,f}^{\prime\prime}_r}^{\prime}_{l,iq} \cc{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime}_{l,pv}
\\
&\quad + \cc{e,g}^{\prime}_{r,pj}\cc{f,\cc{e,g}^{\prime\prime}_r}^{\prime}_{r,iv} \cc{f,\cc{e,g}^{\prime\prime}_r}^{\prime\prime}_{r,uq}
+\ii{\cc{e,f}^{\prime}_l,g}^{\prime}_{pj}\cc{e,f}^{\prime\prime}_{l,iv}\big(\partial \ii{\cc{e,f}^{\prime}_l,g}^{\prime\prime}\big)_{uq}
\\
&\qquad -\cc{g,\cc{e,f}^{\prime}_l}^{\prime\prime}_{l,pj}\cc{e,f}^{\prime\prime}_{l,iv}\cc{g,\cc{e,f}^{\prime}_l}^{\prime}_{l,uq}.
\end{aligned}
\label{eq:KR-Jacobi-1-sumando}
\end{equation}
Note that in the sixth summand we used \eqref{eq:Sweedler-pairing-symm}.
Next, to deal with the first summand at the right-hand side of \eqref{eq:KR-Jacobi-inicial}, we use \eqref{eq:KR-CD-thm} to obtain:
\begin{equation}
\label{eq:KR-Jacobi-2-sumando-1}
\begin{split}
[g_{pq},[&e_{ij},f_{uv}]]
\\
&=[g_{pq},\cc{e,f}^{\prime}_{l,uj}\cc{e,f}^{\prime\prime}_{l,iv}]+[g_{pq},\cc{e,f}^{\prime}_{r,uj}\cc{e,f}^{\prime\prime}_{r,iv}]
\\
&= \cc{e,f}^{\prime\prime}_{l,iv}[g_{pq},\cc{e,f}^{\prime}_{l,uj}]
+\langle g_{pq},\partial_N(\cc{e,f}^{\prime\prime}_{l,iv})\rangle\cc{e,f}^{\prime}_{l,uj}
\\
&\qquad + \cc{e,f}^{\prime}_{r,uj}[g_{pq},\cc{e,f}^{\prime\prime}_{r,iv}]
+\langle g_{pq},\partial_N(\cc{e,f}^{\prime}_{r,uj})\rangle\cc{e,f}^{\prime\prime}_{r,iv}
\\
&=\cc{e,f}^{\prime\prime}_{l,iv}\cc{g,\cc{e,f}^{\prime}_l}^{\prime}_{l,uq}\cc{g,\cc{e,f}^{\prime}_l}^{\prime\prime}_{l,pj}
+\cc{e,f}^{\prime\prime}_{l,iv}\cc{g,\cc{e,f}^{\prime}_l}^{\prime}_{r,uq}\cc{g,\cc{e,f}^{\prime}_l}^{\prime\prime}_{r,pj}
\\
&\qquad +\ii{g,\partial\cc{e,f}^{\prime\prime}_l}^{\prime}_{iq} \ii{g,\partial\cc{e,f}^{\prime\prime}_l}^{\prime\prime}_{pv}\cc{e,f}^{\prime}_{l,uj}
+\cc{e,f}^{\prime}_{r,uj}\cc{g,\cc{e,f}^{\prime\prime}_r}^{\prime}_{l,iq} \cc{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime}_{l,pv}
\\
&\qquad +\cc{e,f}^{\prime}_{r,uj}\cc{g,\cc{e,f}^{\prime\prime}_r}^{\prime}_{r,iq} \cc{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime}_{r,pv}
+\ii{g,\partial\cc{e,f}^{\prime}_r}^{\prime}_{uq} \ii{g,\partial\cc{e,f}^{\prime}_r}^{\prime\prime}_{pj}\cc{e,f}^{\prime\prime}_{r,iv}.
\end{split}
\end{equation}
Similarly, using \eqref{eq:KR-CD-thm}, the fact that $\partial_N$ is a derivation of $A_N$, and \eqref{eq:KR-derivation-partial}, we have
\begin{equation}
\begin{aligned}
\partial_N\langle[e_{ij},&f_{uv}],g_{pq}\rangle
\\
&=\partial_N\langle\big(\cc{e,f}^{\prime}_{l,uj} \cc{e,f}^{\prime\prime}_{l,iv}+\cc{e,f}^{\prime}_{r,uj} \cc{e,f}^{\prime\prime}_{r,iv}\big),g_{pq}\rangle
\\
&=\partial_N\Big(\cc{e,f}^{\prime}_{r,uj}\langle\cc{e,f}^{\prime\prime}_{r,iv},g_{pq}\rangle+\langle\cc{e,f}^{\prime}_{l,uj},g_{pq}\rangle\cc{e,f}^{\prime\prime}_{l,iv}\Big)
\\
&=\partial_N\Big(\cc{e,f}^{\prime}_{r,uj}\ii{\cc{e,f}^{\prime\prime}_r,g}^{\prime}_{pv}\ii{\cc{e,f}^{\prime\prime}_r,g}^{\prime\prime}_{iq}+\ii{\cc{e,f}^{\prime}_l,g}^{\prime}_{pj}\ii{\cc{e,f}^{\prime}_l,g}^{\prime\prime}_{uq}\cc{e,f}^{\prime\prime}_{l,iv}\Big)
\\
&=(\partial\cc{e,f}^{\prime}_r)_{uj}\ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime}_{pv} \ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime}_{iq}
+\cc{e,f}^{\prime}_{r,uj}(\partial\ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime})_{pv}\ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime}_{iq}
\\
&\qquad + \cc{e,f}^{\prime}_{r,uj}\ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime}_{pv} (\partial \ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime})_{iq}
+(\partial\ii{\cc{e,f}^{\prime}_l,g}^{\prime})_{pj}\ii{\cc{e,f}^{\prime}_l,g}^{\prime\prime}_{uq}\cc{e,f}^{\prime\prime}_{l,iv}
\\
&\qquad + \ii{\cc{e,f}^{\prime}_l,g}^{\prime}_{pj}(\partial\ii{\cc{e,f}^{\prime}_l,g}^{\prime\prime})_{uq}\cc{e,f}^{\prime\prime}_{l,iv}
+\ii{\cc{e,f}^{\prime}_l,g}^{\prime}_{pj}\ii{\cc{e,f}^{\prime}_l,g}^{\prime\prime}_{uq}(\partial\cc{e,f}^{\prime\prime}_l)_{iv},
\end{aligned}
\label{eq:KR-Jacobi-2-sumando-2}
\end{equation}
where in the first three summands in the last line, we applied the symmetry of $\ii{-,-}$.
Therefore, by \eqref{eq:CD-comm.a}, \eqref{eq:KR-Jacobi-2-sumando-1} and \eqref{eq:KR-Jacobi-2-sumando-2}, we can rewrite the first summand at the right-hand side of \eqref{eq:KR-Jacobi-inicial} as
\begin{equation}
\label{eq:KR-Jacobi-2-sumando}
\begin{aligned}
&[[e_{ij},f_{uv}],g_{pq}]
\\
&=\partial_N\langle[e_{ij},f_{uv}],g_{pq}\rangle-[g_{pq},[e_{ij},f_{uv}]]
\\
&=(\partial\cc{e,f}^{\prime}_r)_{uj}\ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime}_{pv} \ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime}_{iq}
+\cc{e,f}^{\prime}_{r,uj}(\partial\ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime})_{pv}\ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime}_{iq}
\\
&\qquad + \cc{e,f}^{\prime}_{r,uj}\ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime}_{pv} (\partial \ii{g,\cc{e,f}^{\prime\prime}_r}^{\prime})_{iq}
+(\partial\ii{\cc{e,f}^{\prime}_l,g}^{\prime})_{pj}\ii{\cc{e,f}^{\prime}_l,g}^{\prime\prime}_{uq}\cc{e,f}^{\prime\prime}_{l,iv}
\\
&\qquad + \ii{\cc{e,f}^{\prime}_l,g}^{\prime}_{pj}(\partial\ii{\cc{e,f}^{\prime}_l,g}^{\prime\prime})_{uq}\cc{e,f}^{\prime\prime}_{l,iv}
+\ii{\cc{e,f}^{\prime}_l,g}^{\prime}_{pj}\ii{\cc{e,f}^{\prime}_l,g}^{\prime\prime}_{uq}(\partial\cc{e,f}^{\prime\prime}_l)_{iv}
\\
&\qquad -\cc{e,f}^{\prime\prime}_{l,iv}\cc{g,\cc{e,f}^{\prime}_l}^{\prime}_{l,uq}\cc{g,\cc{e,f}^{\prime}_l}^{\prime\prime}_{l,pj}
-\cc{e,f}^{\prime\prime}_{l,iv}\cc{g,\cc{e,f}^{\prime}_l}^{\prime}_{r,uq}\cc{g,\cc{e,f}^{\prime}_l}^{\prime\prime}_{r,pj}
\\
&\qquad -\ii{g,\partial\cc{e,f}^{\prime\prime}_l}^{\prime}_{iq} \ii{g,\partial\cc{e,f}^{\prime\prime}_l}^{\prime\prime}_{pv}\cc{e,f}^{\prime}_{l,uj}
-\cc{e,f}^{\prime}_{r,uj}\cc{g,\cc{e,f}^{\prime\prime}_r}^{\prime}_{l,iq} \cc{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime}_{l,pv}
\\
&\qquad -\cc{e,f}^{\prime}_{r,uj}\cc{g,\cc{e,f}^{\prime\prime}_r}^{\prime}_{r,iq} \cc{g,\cc{e,f}^{\prime\prime}_r}^{\prime\prime}_{r,pv}
-\ii{g,\partial\cc{e,f}^{\prime}_r}^{\prime}_{uq} \ii{g,\partial\cc{e,f}^{\prime}_r}^{\prime\prime}_{pj}\cc{e,f}^{\prime\prime}_{r,iv}.
\end{aligned}
\end{equation}
Finally, interchanging $f$ and $g$ in \eqref{eq:KR-Jacobi-2-sumando-2} (and consequently the indices $u\leftrightarrow p$ and $v\leftrightarrow q$), it is straightforward to check that the second summand at the right-hand side of \eqref{eq:KR-Jacobi-inicial} is:
\begin{equation}
\label{eq:KR-Jacobi-3-sumando}
\begin{aligned}
[f_{uv},&[e_{ij},g_{pq}]]
\\
&=\cc{e,g}^{\prime\prime}_{l,iq}\cc{f,\cc{e,g}^{\prime}_l}^{\prime}_{l,pv}\cc{f,\cc{e,g}^{\prime}_l}^{\prime\prime}_{l,uj}+\cc{e,g}^{\prime\prime}_{l,iq}\cc{f,\cc{e,g}^{\prime}_l}^{\prime}_{r,pv}\cc{f,\cc{e,g}^{\prime}_l}^{\prime\prime}_{r,uj}
\\
&\qquad +\ii{f,\partial\cc{e,g}^{\prime\prime}_l}^{\prime}_{iv} \ii{f,\partial\cc{e,g}^{\prime\prime}_l}^{\prime\prime}_{uq}\cc{e,g}^{\prime}_{l,pj}
+\cc{e,g}^{\prime}_{r,pj}\cc{f,\cc{e,g}^{\prime\prime}_r}^{\prime}_{l,iv}\cc{f,\cc{e,g}^{\prime\prime}_r}^{\prime\prime}_{l,uq}
\\
&\qquad + \cc{e,g}^{\prime}_{r,pj}\cc{f,\cc{e,g}^{\prime\prime}_r}^{\prime}_{r,iv}\cc{f,\cc{e,g}^{\prime\prime}_r}^{\prime\prime}_{r,uq}
+\ii{f,\partial\cc{e,g}^{\prime}_r}^{\prime}_{pv}\ii{f,\partial\cc{e,g}^{\prime}_r}^{\prime\prime}_{uj}\cc{e,g}^{\prime\prime}_{r,iq}.
\end{aligned}
\end{equation}
Then, by the commutativity of $A_N$, the sum of \eqref{eq:KR-Jacobi-2-sumando} and \eqref{eq:KR-Jacobi-3-sumando} is \eqref{eq:KR-Jacobi-1-sumando}. Consequently, the Jacobi identity \eqref{eq:CD-comm.d} holds, and Theorem \ref{thm:KR-double-Courant-Dorfman} follows. \qedhere
\end{proof}
|
1,116,691,497,772 | arxiv | \section{\label{sec:level1} Introduction}
The simplest explanation for the observed acceleration of the universe
is Einstein's cosmological constant, $\Lambda$. However, the value
that explains the acceleration is many orders of magnitude smaller
than that expected from quantum field theory. We are then left either to
determine a method to set the cosmological constant to a small
value or to consider $\Lambda$ an environmental variable varying from place to place in the multiverse.
Following the environmental approach numerous authors (many inspired
by the pioneering work of Weinberg\cite{Weinberg:1987dv}), have sought to explain the observed value of $\Lambda$ by postulating that the most likely universe to be observed would be
that which contains the largest potential to contain
observers. However, such ``anthropic'' approaches can become burdened by
complicated assumptions on the nature of observers. In their ``Causal
Entropic Principle'' (CEP) Bousso { \it et al.} \cite{Bousso:2007kq}
took this reasoning in a simple and elegant direction by associating
observers with entropy increase. Initial applications of this approach
have successfully predicted our value of
$\Lambda$\cite{Bousso:2007kq,Cline:2007su}. The CEP has added
appeal because there has been long standing (if not
universal\cite{FeynmanMess}) acceptance of the idea that entropy increase
would need to be imposed as a condition specific to observers rather
than a global and eternal property of the Universe\cite{Boltzmann:1895xx,Albrecht:2002uz,Dyson:2002pf,Albrecht:2004ke}.
Specifically, the CEP gives a weight to each set of cosmological
parameters proportional to the entropy produced within a causal
diamond in the corresponding cosmology. In addition to the original
work\cite{Bousso:2007kq} which found our value of $\Lambda$ to be
within one sigma of the peak of their predicted probability
distribution, the CEP was further developed by Cline {\it et
al.} \cite{Cline:2007su}, exploring constraints on other cosmological
values such as density contrast, baryon fraction, matter abundance,
and dark matter annihilation rate.
In this paper we develop this method further by using CEP to
jointly predict the values of curvature and $\Lambda$ most likely to
be observed. Since Cline {\it et al.}
\cite{Cline:2007su} did not find
significant features in their extended parameter space, we chose to vary only
$\rho_k$ and $\rho_{\Lambda}$ for this work and hold all other
parameters fixed. However, given the tail in the probability
distribution for positive curvature, an interesting extension to this
work would be to also vary additional parameters such as the density
contrast.
Anthropic constraints on curvature are interesting in their own right,
in the context of the ``flatness
problem''\cite{Dicke:1979ig,Guth:1980zm} which suggests that in the
absence of something like cosmic
inflation\cite{Guth:1980zm,Linde:1981mu,Albrecht:1982wi} the ``most
natural'' realization of big bang cosmology would be highly dominated
by curvature. A number of authors have already considered anthropic
bounds on
curvature\cite{Vilenkin:1996ar,Garriga:1998px,Freivogel:2005vv}, but
we believe this is the first work to apply the CEP to curvature. We
find that our results place an upper limit on the allowed negative curvature as
expected, but one that is looser than those works mentioned above due to a more lenient tolerance on
the sizes of structure that are allowed to form. Further, we find that
our peak probability for open universes to be away from the upper edge of our probability distribution, not dictated by it, as would be the case in the other work. We also consider solutions that allow for just
positive curvature and for both positive and negative curvature. In
these cases we find a tail in the probability distribution that allows
for a wide range of allowed $\Lambda$ and $\rho_k$, a large fraction
of which are significantly larger than our measured values. We find
that our universe is not ruled out in any scenario, but, depending on
one's choice of priors on curvature, disfavored to a certain
degree.
In section \ref{sec:Sec2} we review the CEP and the cosmology we will be considering. We then review the star formation model we will use in section \ref{sec:Sec2b}. We discuss our results in section \ref{sec:Sec3} and our conclusions in section \ref{sec:Sec4}.
\section{\label{sec:Sec2} The Causal Entropic Principle}
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=0.32\textwidth]{Prob_1d_upper.eps}
\includegraphics[width=0.32\textwidth]{Vc_pos_jan.eps}
\includegraphics[width=0.32\textwidth]{SFR_pos_jan.eps}
\includegraphics[width=0.32\textwidth]{Prob_1d_lower.eps}
\includegraphics[width=0.32\textwidth]{Vc_neg50_jan.eps}
\includegraphics[width=0.32\textwidth]{SFR_neg50_jan.eps}
\caption{Top Left Panel: Probability density for $\rho_\Lambda$ with
fixed curvature of $\rho_k = 10\rho_m$ (dashed) and $\rho_k =
0$ (solid). Top Center Panel: The comoving
volume (in units of $10^{12} Mpc^3$) for $\rho_{\Lambda} =
10^{-123}$ (larger) and $\rho_{\Lambda} = 10^{-122}$ (smaller) are
shown for $\rho_k = 10\rho_m$ (dashed) and $\rho_k = 0$
(solid). Top Right Panel: The star formation rate in units of
$\frac{M_{\odot}}{Mpc^3 yr}$. The upper curve for
each value of curvature represented in solid and dashed respectively is
$\rho_{\Lambda} = 10^{-122}$ and the lower is $\rho_{\Lambda} =
10^{-123}$. The bottom row is the probability density, comoving
volume, and star formation rate for $\rho_k = -50\rho_m$. The
blue/dashed curve is $\rho_{\Lambda} = 10^{-119}$ and the
black/solid curve is $\rho_{\Lambda} = 10^{-120}$.
}
\label{fig:rhoandprob}
\end{center}
\end{figure*}
According to the CEP the probability distribution for $\Lambda$ is given by the equation:
\begin{equation}
\label{eq: prob}
\frac{d^2P}{d\rho_{\Lambda} d\rho_k} = P_0 \times w(\rho_{\Lambda},\rho_k) \times \frac{d^2p}{d\rho_{\Lambda} d\rho_k}
\end{equation}
where $w(\rho_{\Lambda},\rho_k)$ is a weighting factor, $P$ is the total
probability, $P_0$ is a normalization factor, and $p$ is the total
prior. We will assume the joint prior probability of
$p(\rho_{\Lambda},\rho_k)$ to be independent giving
$p(\rho_{\Lambda},\rho_k) = p(\rho_{\Lambda}) \times
p(\rho_k)$. The prior for $\Lambda$ is an expression of how the
``multiverse'' is populated by physics with different values of
$\Lambda$. Here we use the standard form (sometimes motivated
by the string theory landscape) taken by previous
authors\cite{Weinberg:1987dv,Bousso:2007kq}: $\frac{dp}{d\rho_{\Lambda}} = {\rm constant}$. For simplicity we also take
$\frac{dp}{d\rho_k} = {\rm constant}$, which will enable a discussion of
the flatness problem later in the paper. These ``flat'' priors mean the
largest allowed ``cutoff'' values of $\rho_k$ and $\rho_\Lambda$ set the typical
values for the prior. The value of the cutoff turns out to be
unimportant because for flat priors $w(\rho_\Lambda,\rho_k)$ dictates
the shape of the final probability
distribution.
In the CEP framework we set $w(\rho_{\Lambda},\rho_k) = \Delta S$, where
$\Delta S$ is the total entropy produced within a causal
diamond. After considering numerous astrophysical sources for entropy
production, Bousso {\it et al.} \cite{Bousso:2007kq} find that the dominant
form of entropy production is star light reradiated by
dust. As in \cite{Bousso:2007kq}, $\Delta S $ is given by.
\begin{equation}
\label{eq: totentopy}
\Delta S = \int_{t_i}^\infty \frac{d^2S}{dV_c dt} V_c dt
\end{equation}
where $V_c$ is the total comoving volume of an observer's causal
patch. A causal patch is defined by a future light cone taken at an
initial point, such as reheating following inflation, intersected by a
past light cone at a late time point, which in the case of a universe dominated by a
cosmological constant is bounded by a de Sitter horizon and in the case of a universe dominated by positive curvature the late time event is the crunch.
We use the metric:
\begin{equation}
ds^2 = -dt^2 + a(t)^2 R_0^2 [d\chi^2 + S_k(\chi)^2d\Omega^2]
\end{equation}
where $S_k(\chi) = \sin(\chi)$ for positive curvature, $S_k(\chi) = \sinh(\chi)$ for negative curvature, and $S_k(\chi) = \chi$ for no curvature. The causal diamond is then given by $R_0 \chi = \frac{ \Delta \tau}{2} - |\frac{ \Delta \tau}{2} + \tau|$, where $\tau = \int \frac{dt}{a(t)}$. The comoving volume is then:
\begin{equation}
V_c = \left\{
\begin{array}{l l}
2 \pi R_0^3 [\chi -\frac{1}{2}\sin(2 \chi)] & \quad \mbox{for k = +1}\\
\frac{4\pi}{3} R_0^3 \chi^3 & \quad \mbox{for k = 0}\\
2 \pi R_0^3 [\frac{1}{2}\sinh(2 \chi) - \chi] & \quad \mbox{for k = -1}\\
\end{array} \right.
\end{equation}
The scale factor $a(t)$ can be found by solving the Friedmann equation:
\begin{equation}
H^2 = \frac{8 \pi}{3} (\frac{\rho_m}{a^3} + \rho_{\Lambda} + \frac{\rho_k}{a^2})
\end{equation}
where $\rho_{\Lambda} = {\Lambda}/{8 \pi}$, $\rho_k = {-3
k}/{8 \pi R_0^2}$, and $k = \{-1,0,1\}$ for a negative, flat, and
positively curved universe respectively. The value of the matter
density today
($a=1$) is set at $\rho_m = 5.2\times10^{-124}$ in Planck units which we use throughout
unless otherwise noted. Following previous work in the topic we neglect radiation. In this work we hold $\rho_m$ fixed and allow the curvature today, $\rho_k$, and $\rho_\Lambda$ to vary.
The other part of Eqn. \ref{eq: totentopy}, $\frac{d^2S}{dV_c dt}$, is the
entropy produced per comoving volume per time, which is calculated by
the convolution
\begin{equation}
\frac{d^2S}{dV_c dt}(t)= \int_0^{t} \frac{d^2S}{dMdt'}(t -t') \dot{\rho_{\star}}(t')dt'
\end{equation}
where $\frac{d^2S}{dMdt'}(t -t')$ is the entropy production rate per stellar mass
at time $t$ due to stars born at an earlier time, $t'$, and $\dot{\rho_{\star}}(t')$ is the
star formation rate at $t'$. The entropy rate per stellar mass is found by calculating
\begin{equation} \label{eq:stellarmass}
\frac{d^2S}{dMdt'}(t -t') = \frac{1}{\left<M\right>}\int_{0.08M_{\odot}}^{M_{max}(t -t')} \frac{d^2s}{dN_{\star}dt} \xi_{IMF}(M) dM
\end{equation}
where $\xi_{IMF}(M)$ is the initial mass function and
$\frac{d^2s}{dN_{\star}dt}$ is the entropy production rate for a
single star. The latter is given by the stellar luminosity divided by
the effective temperature. The number of photons emitted by a star is
dominated by the half that are reprocessed by dust at an effective
temperature of $20 \text{mev}$. This is given by:
\begin{equation}
\frac{d^2s}{dN_{\star}dt} = \frac{L_{\star}}{T_{eff}} = \frac{1}{2} (\frac{M}{M_{\odot}})^{3.5} \, 3.7\times10^{54} yr^{-1}.
\end{equation}
The prefactor in Eqn. \ref{eq:stellarmass} is the average initial mass,
$\left<M\right> = 0.48 M_{\odot}$. The lower limit of Eqn. \ref{eq:stellarmass} is
the minimum mass of a star that can support nuclear burning.
Following Bousso, we take the upper limit to be
\begin{equation}
M_{max}(t -t') = \left\{
\begin{array}{l l}
100M_{\odot} & \quad \mbox{for $t - t' < 10^5$ yr}\\
(\frac{10^{10} yr}{t-t'})^{0.4} M_{\odot} & \quad \mbox{for $t - t' > 10^5$ yr}\\
\end{array} \right.
\end{equation}
\section{\label{sec:Sec2b} Star Formation}
A key aspect to this work is how well the star formation rate is
modeled. While Bousso {\it et al.} \cite{Bousso:2007kq} use star formation
rates of Nagamine \cite{Nagamine:2006kz} {\it et al.} and Hopkins and Beacom
\cite{Hopkins:2006bw} with some simple modifications to extend these
models to include different cosmological constant values, we will
follow Cline {\it et al.} \cite{Cline:2007su} who use a
model proposed by Hernquist and Springel (HS)
\cite{Hernquist:2002rg}. The HS model was found to produce similar
results to those found in
Bousso {\it et al.} but was more straightforward to extrapolate to the
case were a larger number cosmological parameters are varied. The HS
star formation model is given by this equation:
\begin{equation}
\label{eq: SFR}
\dot{\rho_{\star}} = \rho_{m} s_0 q(t) (1 - \text{erf}(\sqrt{\frac{a}{2}} \frac{\delta_c}{\sigma_4}))
\end{equation}
where $\delta_c = 1.6868$, $a = 0.707$, $s_0 = 3.7995 \times 10^{-63}$ (taken from \cite{Hernquist:2002rg}), and $\rho_m$ (same as above) are constants and $\sigma_4$
is the root-mean-square density fluctuation that corresponds to the
mass scale that virializes at a temperature of $T = 10^4 K$.
The star
formation efficiency, $q(t)$, encompasses the rate and efficiency of
radiative cooling within a collapsing object that leads to star
formation. HS model this process with:
\begin{equation}
q(t) = (\frac{\chi(t) \tilde{\chi}}{(\chi(t)^m + \tilde{\chi}^m)^{\frac{1}{m}}})^p
\end{equation}
where $\chi(t) = (\frac{H}{H_0})^{2/3}$. $\tilde{\chi} = 4.6$, $m =
6$, and $p = 2.72$ are constants fit from numerical simulations and
$H_0 = 70 \frac{km/s}{Mpc}$.
For universes with positive curvature that end
in a crunch, the star formation rate of Eqn. \ref{eq: SFR} continues up until the crunch. We therefore needed to
place a bound on late time star formation when we no longer trust
our model. We set $\dot{\rho_{\star}} = 0$ when $\rho_r =\rho_m$ in the
collapsing phase, where $\rho_r = 1.5\times10^{-127}$. This choice allows for an exploration of the CEP properties without a strong limiting effect put in by hand.
Star formation rates for several different values of curvature and
$\Lambda$ are shown in the right panel of Fig.
\ref{fig:rhoandprob}. Their corresponding probability curves are shown
in the left panel of Fig. \ref{fig:rhoandprob}.
\section{\label{sec:Sec3} Results and Discussion}
\begin{figure}
\includegraphics[width=0.45\textwidth]{upper_feb.eps}
\caption{The 68.27\% (dark grey), 95.44\% (light grey), 99.73\% (inner white), and 99.99\% (outer white) contours for a multiverse that only admits negative curvature universes. The solid blue line that cuts through the other 4 contours is the anthropic bound from \cite{Freivogel:2005vv} for smaller sized galaxies.}
\label{fig:justnegcurv}
\end{figure}
\begin{figure}
\includegraphics[trim = 30mm 30mm 0mm 30mm, clip, width=0.57\textwidth]{lowfeb6.eps}
\caption{The 68.27\% (dark grey), 95.44\% (light grey), 99.73\% (inner white), and 99.99\% (outer white) contours for a multiverse that only admits positive curvature universes.}
\label{fig:justposcurv}
\end{figure}
\begin{figure}
\includegraphics[width=0.45\textwidth]{tot_feb.eps}
\caption{The 68.27\% (dark grey), 95.44\% (light grey), 99.73\% (inner white), and 99.99\% (outer white) contours for a multiverse that admits both positive and negative curvature universes.}
\label{fig:posandnegcurv}
\end{figure}
Using the equations of the previous section we solve for the
probability distribution over a wide range of open and closed
universes. To enable the most general discussion we consider both
open and closed cosmologies together and separately. (It is commonly \cite{Freivogel:2005vv}
but not universally \cite{Buniy:2006ed} thought that the string theory
landscape only leads to the open case.)
If we were to only consider open universes, then
Fig. \ref{fig:justnegcurv} depicts the resulting probability density distribution
in $\log\rho_{\Lambda}$-$\log\rho_k$
space. Values of curvature of $\rho_k > 40\rho_m$ fall outside of
the $99.99\%$ CL. Smaller values of $\Lambda$ and curvature lead to
larger causal diamonds and therefore have the most total entropy
production, as depicted in Fig. \ref{fig:rhoandprob}. This balances
the majority of vacua having larger values of both $\Lambda$ and
curvature, giving a peak value at $\rho_{\Lambda} = 7.9\times
10^{-123}$ and $\rho_k = 4.3 \rho_m$. Using the upper bound ($95\%$
CL) on negative curvature from WMAP+HST \cite{Komatsu:2008hk}, our
universe of $\rho_{\Lambda} = 1.25 \times 10^{-123}$ and $\rho_k =
0.016 \rho_m$ is in the $99.73\%$ CL. Fig. \ref{fig:rhoandprob}
illustrates that for a fixed value of negative curvature the
distribution for $\Lambda$ remains roughly unchanged from the
distribution for a flat universe.
Now considering only positively curved universes, there is a clear
correlation between $\Lambda$ and curvature which comes from
competition between positive curvature and $\Lambda$.
This can cause the universe to ``loiter''\cite{Sahni:1991ks} in a
state with little cosmic expansion but plenty of structure growth.
These conditions conspire to create both a larger causal
diamond and enhanced linear growth. This results in the tail on the
bottom right of Fig. \ref{fig:justposcurv} where there is ridge between collapsing
regions to the left and non-collapsing regions to the right in the
$68\%$ CL. For a fixed value of positive curvature, small values of
$\Lambda$ lead to a universe that will recollapse before there is
significant star formation. As $\Lambda$ is increased the recollapse
is delayed allowing for more star formation and a larger causal
diamond, giving more total entropy produced and therefore a more
likely universe within the CEP framework. This continues until
$\Lambda$ is large enough to allow for a non-collapsing universe, at
which point larger values of $\Lambda$ begin to suppress growth. The
narrowing of the tail comes from deviations from the ridge having
large energy densities that lead to either a rapid recollapse or early
$\Lambda$ domination. Using the upper bound ($95\%$ CL) on positive
curvature from WMAP+HST \cite{Komatsu:2008hk}, our universe of
$\rho_{\Lambda} = 1.25 \times 10^{-123}$ and $\rho_k = -0.06 \rho_m$
is in the $99.99\%$ CL.
Fig. \ref{fig:posandnegcurv} shows the 2
dimensional probability density distribution, $\frac{d^2P}{d\log\rho_{\Lambda} d\rho_k}$, in $\log\rho_{\Lambda}$-$\rho_k$
space for both positive and negative curvature.
The full span range of curvature allowed by WMAP+HST
\cite{Komatsu:2008hk} ($-0.06 \rho_m \le \rho_k \le 0.016 \rho_m$) is
in the $95.44\%$ CL. A significant fraction of the
values within the $68.27\%$ CL are positively curved
universes of both large amounts of curvature and dark energy compared
with our universe due to the competing effects mentioned above leading
to a similar tail on the lower right.
A recent paper by Bousso and Leichenauer \cite{Bousso:2008bu} has
argued that the asymptotic behavior of the star formation model shown
in upper right panel of Fig. \ref{fig:rhoandprob} may be
unphysical. Since the CEP framework depends on an accurate accounting
of star formation in universes far different than ours, a careful
study of different models is an important aspect of developing this
work further. However, we suspect that the asymptotic behavior leads
to a subdominant effect on the final probability distributions since
it coincides with a decreasing comoving volume that will diminish the
contribution it will make to the total entropy contribution.
On the other hand, the bottom right portion of the tail in
Figs. \ref{fig:justposcurv} and \ref{fig:posandnegcurv} is an area
where we have little confidence in our star formation model as the
duration of matter domination is increasingly smaller as we move
further out onto the tip of the distribution. A different cut on late
time star formation from the one we chose above or another star
formation model may find the bottom right tip less favored or ruled
out.
We have extended the CEP to include curvature and found that
regardless of whether one considers both positive and negative
curvature or just one of the two options, a non-zero value of
curvature appears to be preferred. We have also found that our
universe is not ruled out in any scenario considered here, but is
somewhat disfavored in some scenarios. The favored values for an open
universe are just a few orders of
magnitude larger than values favored by modern data, so to the extent
that the flatness puzzle is about why the curvature is not given by the
Planck scale, the CEP seems to put a significant dent in the
flatness puzzle. This is not dissimilar to anthropic arguments of
curvature, where structure formation is cut off
by excessive curvature, however the CEP offers a less restrictive
initial assumption. In Fig. \ref{fig:justnegcurv} we also plot the
bound on negative curvature calculated by Freivogel et
al. \cite{Freivogel:2005vv} (which are similar to those of Vilenkin
and Winitzki \cite{Vilenkin:1996ar} and Garriga et
al. \cite{Garriga:1998px}) by demanding that structures at least as
large as a small sized galaxy form. Our plot allows for somewhat more
curvature than is allowed by these
methods. Setting a structure formation limit based on smaller galactic
masses brings the curvature limit closer to ours. Ultimately our rough
reproduction of the anthropic cutoff is unsurprising as our
main entropy source, star formation, cuts off along with structure
formation. However, our actual prediction for curvature is not
against a cutoff for structure formation as would be the case for a
simple bound. The causal entropic weighting provides additional
rewards for smaller curvatures in the form of increased star formation
and entropy production.
\section{\label{sec:Sec4} Conclusions}
Anthropic constraints on observable parameters are interesting when
considering implications for the multiverse. The CEP has appealing
advantages over previous anthropic attempts. We find that the CEP
places upper limits on the amount of curvature that is observable and
while not ruling out our universe, the CEP finds larger curvature
preferable to our measured value. We also find an intriguing feature
in the probability space for positive curvature of an elongated tail
stretching into regions of large curvature. Our results for negatively
curved universes are broadly consistent with previous anthropic bounds
on curvature but less constraining due to a more lenient tolerance for
the minimum mass of structure allowed. Still, like the previous work we
find that anthropic considerations seem to offer cosmology
considerable relief from the flatness problem.
\begin{acknowledgements} We thank
Lloyd Knox, Damien Martin, and especially James Cline for
very helpful discussions. We also thank Tony Tyson for computing
resources as well as Perry Gee and Jim Bosch for technical computing support. This work was
supported by DOE grant DE-FG03-91ER40674.
\end{acknowledgements}
|
1,116,691,497,773 | arxiv | \section{Introduction}
\footnote{This paper is an extension of three workshop papers that were presented at the DeCoDeML workshops at ECML/PKDD 2019 and 2020: Sophie Burkhardt, Nicolas Wagner, Johannes F\"urnkranz, Stefan Kramer: Extracting Rules with Adaptable Complexity from Neural Networks using K-Term DNF Optimization; Nicolas Wagner, Sophie Burkhardt, Stefan Kramer: A Deep Convolutional DNF Learner; Sophie Burkhardt, Jannis Brugger, Zahra Ahmadi and Stefan Kramer: A Deep Convolutional DNF Learner.}Neural Networks (NNs) are commonly seen as black boxes, which makes their application in some areas still problematic (e.g., in safety-critical applications or applications in which DL is intended to support communication with a human user). Logical statements, however, are arguably much easier to grasp by humans than the main building blocks of NNs (e.g., nonlinearities, matrix multiplications, or convolutions). In general, learning of logical rules cannot be done using gradient-based algorithms as they are not differentiable. Even if we find rules that exactly describe a neural network, they might still be too complex to be understandable. To solve this problem, we propose {\textit convolutional rules} for which the complexity is not related to the dimensionality of the input but only to the dimensionality of the convolutional filters.
It is a wide-spread belief that shorter rules are usually better than longer rules, a principle known as Occam's razor. This common assumption was recently challenged again by \cite{Stecher:2016}, who revive the notion of so-called {\em characteristic rules} instead. As they show, shorter rules are often discriminative rules that help to differentiate different output classes, but are not necessarily descriptive of the data. In this work, we can confirm this observation and show how characteristic rules are produced for high-dimensional input data such as images.
Specifically, based on recent developments in deep learning with binary neural networks (BNNs) \citep{Hubara:2016}, we propose an algorithm for {\em decompositional rule extraction} \citep{Andrews:1995}, called {\em Deep Convolutional DNF Learner (DCDL)}. A BNN takes binary input and only produces binary outputs in all hidden layers as well as the output layer. Some BNNs also restrict weights to be binary, however, this is not necessary for our approach. We then approximate each layer using rules and combine these rules into one rule to approximate the whole network. As our empirical results show, this allows for better approximation than with an approach that considers the neural network to be a black box --- a so-called {\em pedagogical rule extraction} approach.
Moreover, we show how the convolutional rules can be used to visualize what the network has actually learned and that this visualization is more interpretable than visualizing the convolutional filters directly.
To sum up, our contributions are as follows:
\begin{itemize}
\item We formally define first-order convolutional rules (Section \ref{sec:first-order}) to describe a neural network using rules that are less complex than the original input.
\item We show that the decompositional rule extraction approach performs better than the approach that considers the network as a black box in terms of approximating the functionality of the neural network.
\item We show how the convolutional rules produce characteristic visualizations of what the neural network has learned.
\end{itemize}
We proceed as follows. We start off by touching upon related work on binary neural networks, rule extraction, and interpretability in convolutional networks in Section \ref{sec:related}. Deep Convolutional DNF Learner (DCDL), consisting of the specification of first-order convolutional rules and the SLS algorithm, is then introduced in Section \ref{sec:method}. Our experimental results on similarity, accuracy, and the visualization are presented and discussed in Section \ref{sec:results}.
\section{Related Work}
\label{sec:related}
Our work builds upon binary neural networks, rule extraction, and visualization of convolutional neural networks.
\subsection{Binary Neural Networks}
Binary Neural Networks (BNNs) are neural networks that restrict the target of activation functions and weights to binary values $\{-1, 1\}$. The original motivation for BNNs is to reduce the memory footprint of NNs and accelerate inference \citep{Hubara:2016}. BNNs can be stored more efficiently because binary values can be stored in 1-bit instead of 32 bits or 64 bits. Also, binary representations can avoid computationally expensive floating-point operations by using less expensive, bitwise operations, leading to a speed-up at inference time \citep{Rastegari:2016}.
However, by construction, the activation functions of BNNs lack differentiability and have less representational power due to their limitation to binary output values. Research on BNNs focuses on alleviating these two limitations.
A breakthrough for BNNs was the straight-through estimator (STE) introduced
in Hinton’s lectures \citep{Hinton:2012}. The STE calculates the gradient of the Heaviside step function $H$ as if it was the identity function. By using the STE in combination with the sign function $B(x) = 2 \cdot H(x) - 1$ instead of $H$, \cite{Hubara:2016} demonstrate the general capabilities of BNNs. They maintain real-valued weights while using binarized weights only for inference and calculation of gradients. Training updates are applied to the real-valued weights. They adapt the STE to better fit $B$ by clipping the identity function at -1 and 1 (Clipped STE). Nevertheless, the sole usage of the (Clipped) STE does not compensate for the lack of representational power of BNNs. Therefore, further improvements were proposed \citep{Rastegari:2016,Lin:2017}.
\subsection{Rule Extraction}
Rule extraction algorithms are commonly divided into decompositional and pedagogical approaches \citep{Andrews:1995}. Pedagogical (or model-agnostic) approaches view the neural network as a black box and approximate its global function using rules, whereas decompositional methods make use of the individual components of the network in order to construct the set of rules. Our work follows a decompositional approach, allowing a better approximation as compared to the pedagogical approach that we compare to in our experiments.
State-of-the-art pedagogical approaches include validity interval analysis (VIA), sampling, and reverse engineering. VIA \citep{Thrun:1993,Thrun:1995} searches for intervals in the input data within which the NN produces the same output. The found intervals can be transformed into rules. Approaches using sampling \citep{Craven:1995,Taha:1999,Sethi:2012,Schmitz:1999} try to let the NN label especially important instances in order to learn better rules. For instance, sampling can be beneficial to learn rules on parts of the unknown label function which are not covered well by the training instances. The reverse engineering approach by \cite{Augasta:2011} prunes the NN before the rules are extracted. As a result, the extracted rules are more comprehensible. \cite{Setiono:2000} use a similar technique to identify the relevant perceptrons of a NN.
Among others, decompositional algorithms use search techniques to find input combinations that activate a perceptron \citep{Fu:1994,Tsukimoto:2000}. Some search techniques provably run in polynomial time \citep{Tsukimoto:2000}. More recently, \cite{Zilke:2016} proposed an algorithm that extracts decision trees per layer which can be merged into one rule set for the complete NN. \cite{Gonzalez:2017} improve on this algorithm by polarizing real-valued activations and pruning weights through retraining. Both rely on the C4.5 \citep{quinlan:2014} decision tree algorithm for rule extraction. \cite{Kane:1993} use a similar idea but cannot retrain an arbitrary already existing NN. Right from the beginning, they train NNs having (almost) binary activations or perceptrons, which are only capable of representing logical AND, OR, or NOT operations. Rules are extracted by constructing truth tables per perceptron.
Unfortunately, the existing decompositional rule extraction algorithms have no principled theoretical foundations in computational complexity and computational learning theory. The runtime of the search algorithm developed by Tsukimoto \citep{Tsukimoto:2000} is a polynomial of the number of input variables. However, this holds only if the number of literals that constitute a term of an extracted logical rule is fixed.
The other presented search algorithms exhibit an exponential runtime. Additionally, all mentioned search algorithms lack the possibility to fix the number of terms per extracted logical rule. Although the more recent decision tree-based approaches \citep{Zilke:2016} apply techniques to reduce the complexity of the extracted rules, they cannot predetermine the maximum model complexity. The only strict limitation is given by the maximum tree depth, which corresponds to the maximum number of literals per term of a logical rule. In general, the C4.5 algorithm does not take complexity restrictions into account while training. Directly being able to limit complexity, in particular the number of terms of a DNF in rule extraction, is desirable to fine-tune the level of granularity of a requested approximation.
In addition to pedagogical and decompositional approaches, there are approaches for local explanations, which explain a particular output, and visualization \cite{Ribeiro:2016}. Also, there are approaches that create new models that are assumed to be more interpretable than the neural network \citep{Odense:2019}. As this is not our goal, we focus on the decompositional approach in our work.
\subsection{Convolutional Networks and Interpretability}
Concerning the interpretability of convolutional neural networks, existing work can be divided into methods that merely visualize or analyze the trained convolutional filters \citep{Zeiler:2014,Simonyan:2013,Mahendran:2015,Zhou:2016} and methods that influence the filters during training in order to force the CNN to learn more interpretable representations \citep{Hu:2016,Stone:2017,Ross:2017}. Our work can be situated in between those two approaches. While we do change the training procedure by forcing the CNN to use binary inputs and generating binary outputs, we also visualize and analyze the filters after training by approximating the network with logical rules.
In order to make convolutional neural networks more interpretable, \cite{Zhang:2018} propose a method to learn more semantically meaningful filters. This method prevents filters from matching several different object parts (such as the head and the leg of a cat) and instead leads to each filter only detecting one specific object part (e.g., only the head), thus making the filters more interpretable. In contrast, our approach allows for different object parts being represented in one filter, but then uses the approximation with rules to differentiate between different object parts. One term in a k-term DNF (see Section \ref{sec:first-order} for the definition of k-term DNFs) might correspond to one specific object part.
\section{Deep Convolutional DNF Learner (DCDL)}
\label{sec:method}
Our approach draws inspiration from recent work on binary neural networks, which are able to perform almost on par with non-binary networks in many cases \citep{Lin:2017,Liu:2018}. These networks are built of components that can provably be transformed into logical rules. The basic building blocks of any NN are variants of perceptrons. To ensure that a perceptron can be represented by a logical expression, we need to restrict the input as well as the output to binary values. This allows to transform perceptrons into truth tables. For hidden layers, we have to ensure that the output is binary leading to binary input for subsequent layers. For the input layer, we need to establish a binarization mechanism for categorical and a discretization mechanism for continuous features. The binarization of a categorical feature with $n$ possible values is done in a canonical way by expanding it into $n$ binary features. For the discretization of continuous features we use dithering. In particular, we use the Floyd-Steinberg algorithm\footnote{Implemented in \url{https://python-pillow.org/}.} to dither the gray scale images to black and white images and dither the individual channels of RGB images. We tested Floyd-Steinberg, Atkinson, Jarvis-Judice-Ninke, Stucki, Burkes, Sierra-2-4a, and Stevenson-Arce dithering\footnote{Based on \url{https://github.com/hbldh/hitherdither}, we also used the given error diffusion matrix.}, which are based on error diffusion, and found no statistically significant differences in the performance of the neural network using a corrected resampled t-test \citep{Nadeau:2003}.
A standard perceptron using the heaviside-function satisfies the requirement of binary outputs, but is not differentiable (unless using the delta-distribution). However, the straight-through estimator \citep{bengio:2013} calculates gradients by replacing the heaviside-function with the identity function and thus allows to backpropagate the gradients.
To be able to regularize complexity, we employ an adaptation of the stochastic local search (SLS) algorithm \citep{rueckert:2003} to extract logical expressions with $k$ terms in disjunctive normal form (k-term DNF). SLS can be parameterized with the number of terms to learn and thereby limit the maximum complexity. As the SLS algorithm is run after an NN has been trained, we do not limit the complexity at training time. \cite{know:2015} have already shown that this can be advantageous.
Convolutional Neural Networks (CNNs) are important architectures for deep neural networks \citep{dqn:2013,seq:2017,deepvar:2018}. Although convolutional layers can be seen as perceptrons with shared weights, logical expressions representing such layers need to be invariant to translation, too. However, logical expressions are in general fixed to particular features. To overcome this issue, we introduce a new class of logical expressions, which we call {\textit convolutional logical rules}. Those rules are described in relative positions and are not based on the absolute position of a feature. For inference, convolutional logical rules are moved through data in the same manner as convolutional filters. This ensures interpretability and lowers the dimensionality of extracted rules.
Pooling layers are often used in conjunction with convolutional layers, and max-pooling layers guarantee binary outputs given binary inputs. Fortunately, binary max-pooling can easily be represented by logical expressions in which all input features are connected by a logical OR. The algorithms for training and testing DCDL are summarized in Algorithms \ref{alg:train_dcdl} and \ref{alg:test_dcdl}
\begin{algorithm}
\begin{algorithmic}[1]
\Procedure{Train DCDL}{number of layers $L$}
\State $\phi\leftarrow \emptyset$
\For{layer $l=1,\hdots,L$}
\If{Convolutional layer}
\State $\psi \leftarrow \emptyset$
\For{Convolutional filter $f$}
\State $\psi_f\leftarrow$ SLS on input and output of NN for this filter
\State $\psi\leftarrow \psi\cup \psi_f$
\EndFor
\State $\phi \leftarrow \phi\cup (l,\psi)$
\ElsIf{Max pooling}
\State No training required
\ElsIf{Dense}
\State $\psi\leftarrow$ SLS on input and output of NN
\State $\phi \leftarrow \phi\cup (l,\psi)$
\EndIf
\EndFor
\Return{$\phi$}
\EndProcedure
\end{algorithmic}
\caption{}
\label{alg:train_dcdl}
\end{algorithm}
\begin{algorithm}
\begin{algorithmic}[1]
\Procedure{Test DCDL}{input data, trained SLS models $\phi$}
\State input$_1\leftarrow$ input data
\For{$(l,\psi)\in\phi$}
\If{Convolutional layer}
\State $\lambda\leftarrow\emptyset$
\For{rule $\psi_i\in \psi$}
\State $\lambda_{i}\leftarrow $ evaluate rule $\psi_i$ on input$_l$
\State $\lambda\leftarrow\lambda\cup \lambda_{i}$
\EndFor
\ElsIf{Max pooling}
\For{Output neuron}
\State Combine values in each pool with or-operation
\EndFor
\ElsIf{Dense}
\State $\lambda\leftarrow$ evaluate $\psi$ on input$_l$
\EndIf
\State input$_{l+1}\leftarrow \lambda$
\EndFor
\Return{prediction of DCDL $\lambda$}
\EndProcedure
\end{algorithmic}
\caption{}
\label{alg:test_dcdl}
\end{algorithm}
\subsection{Introduction of First-Order Convolutional Rules}
\label{sec:first-order}
This section provides the formal underpinnings and the introduction of the convolutional rules. We start with propositional k-DNF formulas and then move on to use first-order logic to take advantage of variable assignments (variables representing relative pixel positions) as we shift the filter across the image.
A k-term DNF combines Boolean variables $\{x_0,x_1,\hdots,x_{n-1}\}$ as $k$ disjunctions of conjunctions
\begin{equation}
\bigvee\limits_{i=0}^{k-1}\bigwedge\limits_{j=0}^{m_i}x_{i,j},
\end{equation}
where $x_{i,j}\in \{x_0,\neg x_0,x_1,\neg x_1,\hdots,x_{n-1},\neg x_{n-1}\}$ and $m_i\in \{0,1,\hdots,n-1\}$. An example with $k=3$ and 3 input variables could look like $(x_1\land x_2)\lor(x_3)\lor(\neg x_1\land \neg x_2)$.
In general, a rule is described in relation to a fixed set of input variables. Unfortunately, for image data this is not sufficient. The success of CNNs in image classification arguably stems from the translation invariance of filters. Thus, we propose that logical rules for image classification need to be invariant to translation as well. In the following we assume all pixels are binary.
Using propositional logic, the definition of convolutional logical rules is relatively complex. For a first-order convolutional logical rule definition the rule itself is straightforward and the complexity is shifted to the definition of the predicates and the environment with respect to which the predicates are evaluated. Due to the variability of the environment that is inherent in first-order logic, we can naturally account for the translation invariance of the rule. In other words, the rule stays the same, only the mapping of the variables to the concrete values in the universe is changed as we move the rule over the image. Propositional logic on the other hand does not have variables and is thus not amenable to the translational invariance.
A k-term first-order convolutional logical rule (FCLR) is defined as follows: We define our model $\mathcal{M}$ as the tuple $(\mathcal{F},\mathcal{P})$ consisting of a set of functions $\mathcal{F}$ and a set of predicates $\mathcal{P}$ \citep{Huth:2004}. For a non-empty set $U$, the universe of concrete values, each predicate $P\in \mathcal{P}$ is a subset $P^{\mathcal{M}}\subseteq U^{a}$ of tuples over $U$, where $a$ is the number of arguments of predicate $P$. The universe $U$ of concrete values is defined as the set of concrete pixels in the image $p_1,\hdots,p_n\in \{0,1\}$.
Now, a {\textit first-order convolutional logical rule} with one term is now defined as
\begin{equation}
\phi_{conv}:=\exists x_1,\hdots,x_a: P_{conv}(x_1,\hdots,x_a),
\label{eq:conv_rule}
\end{equation}
where $P_{conv}\in \mathcal{P}$ is the convolutional predicate and $a$ is the size of the convolutional filter or the number of elements in the filter matrix. The convolutional predicate is defined as
\begin{equation}
P_{conv}:=\{(u_1,\hdots,u_a)|u_i \mbox{ are consecutive pixels in accordance with the convolutional filter}\}.
\label{eq:def}
\end{equation}
In order to evaluate our convolutional rule, we now need to specify the environment $l$ (the look-up table) \citep{Huth:2004} with respect to which our model satisfies (or not) the convolutional rule, i.e. $\mathcal{M}\models_l \phi$. The environment depends on the position at which we evaluate our convolutional rule. Evaluating the rule at position $t$ means that the variables $x_1,\hdots,x_a$ are mapped to the corresponding pixels in the input image,
$l_t[x_1\rightarrow p_{t}], l_t[x_2\rightarrow p_{t+1}],\hdots$, where $p_1,\hdots,p_n$
are the input image pixels (for 2D images, the indices have to be adjusted to account for the change to the next line, for simplicity, here the indices correspond only to 1D input). Thus, we are able to evaluate the convolutional rule in Equation \ref{eq:conv_rule} at each position of the image.
Since we specify the rules as $k$-term DNFs, we have $k$ convolutional predicates, one for each of the $k$ terms. Therefore, we can expand Equation \ref{eq:conv_rule} to
\begin{equation}
\phi_{conv}:=\exists x_1,\hdots,x_a: P_{conv}^{(1)}(x_1,\hdots,x_a)\lor \hdots \lor P_{conv}^{(k)}(x_1,\hdots,x_a).
\label{eq:conv_rule_k}
\end{equation}
This concludes the definition of first-order convolutional rules.
\subsubsection{Example}
\label{sec:example}
The logical rules found by SLS may be displayed graphically, if the input for SLS is image data. For each image position $t$ the variables of the convolutional predicates are mapped to the appropriate pixels using environment $l_t$. Thereby, each term of the k-term DNF can be visualized as an image. The whole convolutional rule can be output as a series of $k$ gray scale images by displaying positive literals as white, negative literals as black and literals that do not influence the truth value as gray. \autoref{fig:graphical_representation_of_logical_formula} shows an example of such a visualization with a rule that has two convolutional predicates and a filter size of $3\times 3$.\\
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth]{figures/graphical_representation_of_logical_formula.png}
\caption{Graphical representation of the logical formula $P^{(1)}_{\mbox{conv}}(x)\lor P_{\mbox{conv}}^{(2)}(x)$, where $P_{\mbox{conv}}^{(1)}(x):=\{(x_1,\hdots,x_m)|x_6 \land \neg x_8 \mbox{ is true}\} $ and $ P_{\mbox{conv}}^{(2)}:=\{(x_1,\hdots,x_m)| x_9 \mbox{ is true}\}$. The first convolutional predicate is displayed on the left side, the second one on the right side. Variables which have to be true in order for the predicate to evaluate to true are marked white. Variables which have to be false in order for the predicate to evaluate to true are marked black. Variables that have no influence on the evaluation are marked gray.}
\label{fig:graphical_representation_of_logical_formula}
\end{figure}
\subsection{Stochastic Local Search (SLS)}
We implemented the SLS rule learner\footnote{\url{https://github.com/kramerlab/DCDL}} and extended it for the purpose of pedagogical rule extraction (Algorithm \ref{alg:SLS}). Since we apply SLS to predictive tasks, we adjust SLS to return the candidate that achieved the lowest score on the validation set (line 8). Scores used for the decision rule are still calculated on the training set. Calculation of scores is computationally expensive and SLS needs to evaluate the decision rule in every iteration. Therefore, we calculate scores batchwise. We introduce an adaptation that is theoretically motivated. One can always correct a term that falsely covers an instance by adding one literal, but the same does not hold in the case of an uncovered instance. We account for this by adjusting SLS to remove all literals in a term that differ from an instance (line 22).
Any SLS algorithm starts by evaluating a random solution candidate. It then selects the next candidate from a neighborhood of the former candidate. This procedure is repeated until a solution is found. If no solution is found and no improvement is found for 600 steps, we restart the search with a different random formula. Therefore, one has to define a candidate space, a scoring function to evaluate a candidate solution, a neighborhood of a candidate solution, as well as a decision rule for selecting the next candidate out of a neighborhood.
In SLS, the candidate space consists of all applicable k-term DNFs. The scoring function is defined as the number of misclassified instances by a given k-term DNF. The neighborhood of a candidate is given by all k-term DNFs that differ in one literal to the candidate. The next candidate is selected in accordance with a randomly drawn misclassified training instance (line 12). If the instance has a positive training label, with probability $p_{g1}$ a random term is modified (line 14), otherwise the term which differs least from the misclassified instance. With the probability $p_{g2}$ the modification is done by deleting a random literal (line 20). In the case of a negative training label (line 25), any term that covers the considered instance is chosen. In contrast to before, a literal not in accordance with the misclassified instance is added with a probability of $p_{s}$. Otherwise, a literal whose addition decreases the score over the training set most is appended. In the end, SLS returns the candidate that achieves the lowest score on the training set.
\begin{algorithm}[t]
\begin{algorithmic}[1]
\Procedure{SLSearch}{$k, maxIteration, p_{g1}, p_{g2}, p_{s}, batchSize, trainingStop, validationSet$}
\State $formula$ $\leftarrow$ a randomly generated $k$-term DNF formula
\State $optimalFormula \leftarrow formula$
\State $iteration \leftarrow 0$
\State $minScore \leftarrow \infty$
\While{ $iteration < maxIteration $ and $minScore > 0$}
\State $iteration \leftarrow iteration +1$
\State $newScore \leftarrow score(validationSet)$
\If{$newScore < minScore$}
\State $minScore \leftarrow newScore$
\State $optimalFormula \leftarrow formula$
\EndIf
\State $missedInstance \leftarrow$ random misclassified instance
\If{$missedInstance$ has positive label}
\State \textbf{with probability} $p_{g1}$
\State\hspace{\algorithmicindent} $term \leftarrow$ a term uniformly drawn from $formula$
\State \textbf{otherwise}
\State\hspace{\algorithmicindent} $term \leftarrow$ the term in $formula$ that differs in the smallest
\State\hspace{\algorithmicindent}\hspace{\algorithmicindent} number of literals from $missedInstance$
\State \textbf{with probability} $p_{g2}$
\State\hspace{\algorithmicindent} $literals \leftarrow$ a literal uniformly drawn from $term$
\State \textbf{otherwise}
\State\hspace{\algorithmicindent} $literals \leftarrow$ all
literals in $term$ that differ from \State\hspace{\algorithmicindent}\hspace{\algorithmicindent} $missedInstance$
\State $formula \leftarrow formula$ with $literals$ removed from $term$
\ElsIf{$missedInstance$ has negative label}
\State $term \leftarrow$ a term in $formula$ that covers $missedInstance$
\State \textbf{with probability} $p_{s}$
\State\hspace{\algorithmicindent} $literal \leftarrow$ a literal uniformly drawn from all possibilities
\State \textbf{otherwise}
\State\hspace{\algorithmicindent} $batch \leftarrow$ uniformly pick $batchSize$ many training instances
\State\hspace{\algorithmicindent} $literal \leftarrow$ a literal whose addition to $term$ reduces \State\hspace{\algorithmicindent}\hspace{\algorithmicindent} $score(batch)$ the most
\State $formula \leftarrow formula$ with $literal$ added to $term$
\EndIf
\EndWhile
\Return $optimalFormula
\EndProcedure
\end{algorithmic}
\caption{}
\label{alg:SLS}
\end{algorithm}
\section{Experimental Evaluation}
\label{sec:results}
The code for the following tests can be found on github\footnote{\url{https://github.com/kramerlab/DCDL}}. The parameters $p_{g1}$, $p_{g2}$ and $p_s$ are set to 0.5 for maximal randomness. We perform one-against-all testing so that the ground-truth-labels, which encode the classes of the data as one-hot vectors, are mapped to two classes. One class contains all images with the searched label, the other class contains all other labels. To prevent the neural network from predicting only the majority label, we balanced the labels in the training and test datasets such that one class comprises half of the dataset and the rest of the classes are randomly sampled so that each class is equally represented. We use three commonly used datasets with their predefined train-test splits: MNIST, FASHION-MNIST, and CIFAR10. For each dataset we used 5,000 samples as a holdout set for early stopping of the network and the rest for training. Each dataset has a designated test set with 10,000 samples.
\subsection{DCDL -- Similarity}
In this section we compare our DCDL approach against the black-box SLS algorithm. We look at their ability to model the behavior of a multilayer neural network for the datasets MNIST, FASHION-MNIST and CIFAR10. An overview of the experimental setup is given in \autoref{fig:test_setup_accuracy_comparison}.
First a neural network is trained, which consists of two convolutional layers followed by a max pooling layer and a sign layer. The last layer is a dense layer with dropout.
As soon as the neural net is trained, the output of the sign layers is used as a label for the training of DCDL. The sign layers transform the outputs to binary values for the SLS algorithm. The two convolutional layers and the dense layer are each approximated with Boolean formulas, which are generated by the SLS algorithm. The dithered images are initially used as input to the SLS. After the first SLS, the input in the following SLS runs is the output of the previous formula. The intermediate results of the NN serve as labels.
The approximation of the convolutional operation is, in contrast to the dense layer, not straightforward, so we will explain this process in detail here. Figure \ref{fig:aproximation_convolution} shows the process graphically.
In a convolutional layer, the input images are subsampled, and the samples are processed with the learned filters. Each sample is mapped to a value. This mapping creates a new representation of the images. Each filter gives its own representation. They are stacked as different channels. With the help of the sign layer, the representations are mapped to binary values.
The process of subsampling also takes place for the input of the DCDL approach. These samples are the input for the SLS algorithm. As labels serves the channel output of the sign layer belonging to the filter which is being approximated with the help of the SLS. Thus, each filter will be approximated by a logical formula. Using this procedure, DCDL approximates the operation of the NN with Boolean formulas.
In the black-box SLS approach, only the input images and the corresponding label predicted by the NN are provided to the algorithm. The architecture of the NN and its functionality are not taken into account. It is evaluated using two different methods. In the black-box prediction approach, the prediction of the neural network is used as a label for training. In the true label approach, the true labels of the images are used for training.
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth]{figures/test_setup_accuracy_comparison_IV.png}
\caption{Experimental setup for comparing the neural network, the DCDL and the black-box SLS approach. $c$ depends on the dataset and is the number of color channels. The content of the dotted box is shown in more detail in Figure \ref{fig:aproximation_convolution}.}
\label{fig:test_setup_accuracy_comparison}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth]{figures/Figure_Convolution.png}
\caption{This figure illustrates the approximation of a convolutional layer by the SLS algorithm. It is part of the whole experimental setup in Figure \ref{fig:test_setup_accuracy_comparison} as shown by the dotted box.}
\label{fig:aproximation_convolution}
\end{figure}
We first focus on the question whether DCDL can better approximate the prediction of the neural network than the black-box approaches. Our results in \autoref{deep_rule_set_compare_black_box} show that DCDL outperforms the black-box approach on the MNIST and Fashion-MNIST dataset and still provide a slight advantage on the CIFAR dataset. We hypothesize that the performance of the neural network itself has an influence on the ability of DCDL to extract good rules. CIFAR is a more complex dataset, and the neural network performs worse than on MNIST. This makes it also more difficult for DCDL to consistently model the network's behavior.
To calculate the similarity of the labels predicted by the neural network with the labels predicted by the SLS algorithm, we calculate
\begin{equation}
sim = \frac{ \sum_{i= 1 }^{ n } \mathbbm{1}[y'_i = y''_i]}{ n }
\label{eq:equation_similarity}
\end{equation}
with $n$ as the number of labels, $\mathbbm{1}$ the indicator function, $y'_i$ as the prediction of the investigated approach, and $y''_i$ as the label calculated by the neural network
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{figures/similarity.png}
\caption{Difference in similarity between the prediction of the DCDL and the black-box approach for $k=40$. The similarity is calculated with respect to the prediction of the neural network on the test set. The DCDL approach has higher similarity across all three datasets. The standard deviation is calculated from 30 runs overall, 3 for each class.}
\label{deep_rule_set_compare_black_box}
\end{figure}
\subsection{DCDL -- Accuracy}
The above section shows that our method performs well in terms of similarity with the neural network. However, clearly, similarity does not necessarily correlate with accuracy. For example, it would be possible that the rule learner only models the errors that the network makes, leading to a high similarity but a bad performance on the actual labels. Therefore, we also compare the accuracy on the true labels of the predictions for the methods DCDL, black-box SLS, and the neural network. For the black-box SLS algorithm, we differentiate between the method that was trained on the labels as predicted by the NN (BB prediction) and the method that was trained on the true labels (BB label). Again we use \autoref{eq:equation_similarity} to calculate the accuracy, but use the true labels of the test data instead of the labels predicted by the neural network.
Our results in \autoref{fig:accuracy_all_models} show that the neural network outperforms the other methods. DCDL clearly outperforms the black-box approach on the MNIST and Fashion-MNIST datasets and still provides a slight advantage on the CIFAR dataset. The poor performance of the classifiers on the CIFAR dataset is most likely partly caused by the dithering. The network architecture might also play a role. This will be investigated further in our future work.
As shown in Table \ref{tab:pvalue}, we also evaluated the statistical significance using a corrected resampled t-test \citep{Nadeau:2003} with $\alpha=0.05$ of the results to show that the performance of DCDL is similar to that of the neural network. For the MNIST and Fashion MNIST datasets, the neural network is not significantly better than DCDL, whereas the black box approach is significantly worse than the neural network. For CIFAR, all rule learning approaches perform worse than the neural network. The reason for this lies in the relative complexity of the dataset. We hypothesize that the network itself is not able to learn a good representation for the dataset, which in turn makes the task for the rule learners harder.
\setlength{\tabcolsep}{1.5em}
\begin{table}
\centering
\caption{{\em p}-values of the corrected resampled t-test \citep{Nadeau:2003} for MNIST, FASHION-MNIST and CIFAR for the accuracy values shown in Figure \ref{fig:accuracy_all_models}. Gray-shaded are the pairs for which the null hypothesis is rejected with significance level $\alpha = 0.05$ using a corrected resampled t-test \citep{Nadeau:2003}. }
\begin{tabular}{|l|c|c|c|}
\hline
& BB prediction& BB label& Neural network~\\
\hline
\multicolumn{4}{|c|}{MNIST}\\
\hline
DCDL & 0.59 & 0.27 & {\cellcolor[rgb]{0.753,0.753,0.753}}0.00 \\
\hline
BB prediction & -& 0.07 &{\cellcolor[rgb]{0.753,0.753,0.753}}0.00 \\
\hline
BB label & - & - & {\cellcolor[rgb]{0.753,0.753,0.753}}0.02 \\
\hline
\multicolumn{4}{|c|}{Fashion}\\
\hline
DCDL& {\cellcolor[rgb]{0.753,0.753,0.753}}0.03 &0.10 & {\cellcolor[rgb]{0.753,0.753,0.753}}0.00 \\
\hline
BB prediction & -& 0.81&{\cellcolor[rgb]{0.753,0.753,0.753}}0.00 \\
\hline
BB label & -& -& {\cellcolor[rgb]{0.753,0.753,0.753}}0.01 \\
\hline
\multicolumn{4}{|c|}{Cifar}\\
\hline
DCDL& 0.08& {\cellcolor[rgb]{0.753,0.753,0.753}}0.04& {\cellcolor[rgb]{0.753,0.753,0.753}}0.00 \\
\hline
BB prediction & - & 0.90& {\cellcolor[rgb]{0.753,0.753,0.753}}0.00 \\
\hline
BB label&-& - & {\cellcolor[rgb]{0.753,0.753,0.753}}0.00 \\
\hline
\end{tabular}
\label{tab:pvalue}
\end{table}
\begin{figure}[htb]
\centering
\includegraphics[width=\textwidth]{figures/accuracy.png}
\caption{Accuracy for different datasets of our DCDL approach, the black-box SLS, black-box SLS with true labels, and the neural network. Top left: MNIST, top right: FASHION-MNIST, bottom: CIFAR. For SLS we used $k=40$. The standard deviation is calculated from 30 runs overall, 3 for each class.}
\label{fig:accuracy_all_models}
\end{figure}
\subsection{Visualization of logical formulas}
We already showed an example for the visualization of a simple formula in Section \ref{sec:example}. Now, we want to look at the visualization of more complex formulas that are found by our algorithm. If the rule search is conducted with a small $k$, the visualized rules tend to be discriminative and often highlight only a single pixel, thus making them hard to interpret. To counter this, we set $k$ to higher values in order to learn rules that are more characteristic. However, when visualizing $k$ predicates for high $k$, this produces a lot of images. Therefore, we add a reduce step that sums and renormalizes the visualizations for all $k$ predicates. As Figure \ref{fig:one_against_all_test_results} shows, this leads to clear visualizations that look almost like probability densities or prototypes. The comparison to the convolutional filters shows that this procedure leads to visualizations that are more easily identifiable by the human eye.
The architecture of the neural network is shown in \autoref{fig:test_setup_one_against_all}. It consists of a convolutional layer followed by a sign layer and a dense layer. The dense layer converts the scalar output of the sign layer into a one-hot vector. The weights of the dense layer were set to [1,0].
The dithered images are the input for the SLS algorithm and the output of the sign layer is the label for the SLS algorithm. The visualization in \autoref{fig:one_against_all_test_results} was done on the MNIST dataset with a filter size that is equal to the size of the image. Note that for the case of MNIST, smaller filter sizes do not result in interpretable visualizations. However, in principle we can also choose filter sizes much smaller than the image itself if the images consist of complex scenes where the number itself is only a small part of the image for example. The selection of a filter size that leads to an interpretable visualization is left for future work. To sum up, we showed with the help of a simple example how the individual predicates as well as the complete convolutional rule may be visualized.
\begin{figure}[htb]
\centering
\includegraphics[width=\textwidth]{figures/one_against_all_test_results_III.png}
\caption{Results of visualizing convolutional rules rules with $k = 150$. The convolutional filter by the NN is shown for comparison.}
\label{fig:one_against_all_test_results}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth]{figures/test_setup_one_against_all_II.jpg}
\caption{Test setup for approximating convolution operations with logical formulas.}
\label{fig:test_setup_one_against_all}
\end{figure}
\section{Conclusion}
We investigated how convolutional rules enable the extraction of interpretable rules from binary neural networks. We showed the successful visualization by means of an example. Additionally, the similarity to the functionality of the neural network was measured on three different datasets and found to be higher for the decompositional approach than the black-box approach. We think there is potential in decompositional approaches for the extraction and visualization of characteristic rules. Although the logical formulas are large for human visual inspection on real-world data, their representation makes deep learning models, in principle, amenable to formal verification and validation.
In future research, we aim to incorporate further state-of-the-art components of NNs while preserving the ability of the network to be transformed into (convolutional) logical rules. Our work suggests that the combination of binary NNs and k-DNF is promising combination. To this end, one should develop a differentiable version of DCDL based on, e.g., differentiable sub modular maximization \citep{Tschiatschek:2018} or differentiable circuit SAT \citep{Powers:2018}. Generally, one should explore DCDL as a new prespective on neuro-symblic AI \citep{Garcez:2020}.
\section*{Permission to Reuse and Copyright}
Figures, tables, and images will be published under a Creative Commons CC-BY licence, and permission must be obtained for use of copyrighted material from other sources (including re-published/adapted/modified/partial figures and images from the internet). It is the responsibility of the authors to acquire the licenses, to follow any citation instructions requested by third-party rights holders, and cover any supplementary charges.
\section*{Conflict of Interest Statement}
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
\section*{Author Contributions}
The experiments were done by JB. The initial code basis was due to NW, the writing of the paper was mainly done by SB with input from JB and NW. ZA and SK were involved in the development of ideas, the polishing of the paper, and discussions throughout. KK gave feedback to the paper and helped with the writing of the submitted manuscript.
\section*{Funding}
The work was funded by the RMU Initiative Funding for Research by the Rhine Main universities (Johannes Gutenberg University Mainz, Goethe University Frankfurt and TU Darmstadt) within the project ``RMU Network for Deep Continuous-Discrete Machine Learning (DeCoDeML)''.
\section*{Acknowledgments}
Part of this research was conducted using the supercomputer Mogon offered by Johannes Gutenberg University Mainz (hpc.uni-mainz.de), which is a member of the AHRP (Alliance for High Performance Computing in Rhineland Palatinate, www.ahrp.info) and the Gauss Alliance e.V.
|
1,116,691,497,774 | arxiv | \section{Introduction}
\label{sec:intro}
While there is null result from direct searches for new physics at the Large Hadron Collider, recent LHCb measurements of lepton flavor universality in $B$ physics have shown deviations from the standard model (SM) predictions.
The measurement of $R_{K^{*0}} = {\mathcal B}(B^0 \to K^{*0} \mu^+\mu^-)/{\mathcal B}(B^0 \to K^{*0} e^+e^-)$ in two different kinematic regions by LHCb collaboration gives~\cite{Aaij:2017vbb}:
\begin{eqnarray}
R_{K^{*0}} = \left\{\begin{array}{ll}
0.66^{+0.11}_{-0.07} \pm 0.03 & (2m_\mu)^2 < q^2 < 1.1~ \ensuremath{\mathrm{GeV}}^2 \\[2mm]
0.69^{+0.11}_{-0.07} \pm 0.05& 1.1 ~ \ensuremath{\mathrm{GeV}}^2 < q^2 < 6 ~ \ensuremath{\mathrm{GeV}}^2 .
\end{array}
\right.
\end{eqnarray}
These values are compatible with the SM predictions \cite{Bordone:2016gaq,Descotes-Genon:2015uva,Capdevila:2016ivx,Capdevila:2017ert,Serra:2016ivr,Dvan,Straub:2015ica,Altmannshofer:2017fio,flavio,Jager:2014rwa} within $2.1-2.3\sigma$ and $2.4-2.5\sigma$, respectively.
The ratio of $R_K={\mathcal B}(B^+ \to K^{+} \mu^+\mu^-)/{\mathcal B}(B^+ \to K^{+} e^+e^-)$ is also given by LHCb collaboration \cite{Aaij:2014ora} which shows 2.6$\sigma$ deviation from the SM predictions \cite{Bordone:2016gaq}:
\begin{eqnarray}
R_{K} = \begin{array}{ll}
0.745^{+0.090}_{-0.074} \pm 0.036& 1 ~ \ensuremath{\mathrm{GeV}}^2 < q^2 < 6 ~ \ensuremath{\mathrm{GeV}}^2 .
\end{array}
\end{eqnarray}
Furthermore, charged current decays of $\bar{B}\to D^{(*)}l^-\bar{\nu}_l$, which have been measured by the BaBar \cite{Lees:2012xj,Lees:2013uzd}, Belle \cite{Huschle:2015rga, Sato:2016svk, Hirose:2016wfn} and LHCb \cite{Aaij:2015yra}, also indicate discrepancies in the ratios of $R_{D^{(*)}}={\mathcal B}(\bar{B}\to D^{(*)}\tau^-\bar{\nu}_{\tau})/{\mathcal B}(\bar{B}\to D^{(*)}l^-\bar{\nu}_{l})~(l=e,\mu)$. The SM predictions are estimated in \cite{Fajfer:2012vx,Bigi:2017jbd,Jaiswal:2017rve} for $R(D^*)$ and \cite{Lattice:2015rga,Na:2015kha,Bigi:2016mdz} for $R(D)$, and the current experimental average values are roughly $4\sigma$ deviations from the SM values.
In recent years, these anomalies have been paid more attentions, and possibilities for the explanation in various extensions of the SM are discussed: $Z'$ models \cite{Altmannshofer:2016jzy,Ko:2017lzd,Datta:2017pfz,Datta:2017ezo, Bhattacharya:2016mcc, Cline:2017ihf,Faisel:2017glo}, leptoquark \cite{Bhattacharya:2016mcc, Hiller:2014yaa, Duraisamy:2016gsd, Chen:2017hir, Aloni:2017ixa, Calibbi:2017qbu} for the $R_K$ anomaly, and the charged Higgs \cite{Bailey:2012jg, Celis:2012dk, Tanaka:2012nw, Crivellin:2012ye, Crivellin:2015hha, Cline:2015lqp, Alonso:2016oyd, Iguro:2017ysu}, $W^{\prime}$ models \cite{He:2012zp,Boucenna:2016qad} and leptoquark \cite{Chen:2017hir, Calibbi:2017qbu, Fajfer:2012jt, Sakaki:2013bfa, Dorsner:2013tla, Li:2016vvp, Crivellin:2017zlb} for the $R_{D}$ anomaly.
We propose a hidden gauged $U(1)_H$ model to explain the $R_{K}$ anomaly, in which a $Z'$ gauge boson interacts with the SM particles only through mediator particles. The mediators are vector-like doublet fermions, whose masses are assumed to be around $O(1)~$TeV. The $Z^{\prime}$ couplings to the SM particles appear with symmetry breaking of the hidden gauged $U(1)_H$, which results in $b \to s \mu\mu$ transition through the tree-level $Z^{\prime}$ exchange. Furthermore, introducing an another mediator, we can have a relevant $b\to c$ transition to the $R_D$ anomaly. The role of the additional mediator is played by an inert-Higgs doublet, and the charged Higgs component of the inert Higgs can induce the $b\to c$ transition at tree level. Since some essential couplings for the $b\to s \mu\mu$ are also involved in $b\to c \tau^- \bar{\nu}$, the former process can affect a possibility of explaining the $R_D$ anomaly.
Typically, the $b\to s$ transition is tightly constrained by various observables such as $B_s-\bar{B}_s$, $B_s \to \mu\mu$, etc. While in the lepton sector, the $Z^{\prime}$ coupling to the muon contributes to neutrino trident production as discussed in \cite{Altmannshofer:2014cfa, Altmannshofer:2014pba}.
At the same time, we carefully examine that there is no significant FCNC $Z$ couplings, although the mixing between the mediator and the SM fermions could be significantly large. In addition to these constraints, we also address direct searches of the $Z'$ and the charged Higgs at collider experiments.
To avoid flavor constraints from the neutral Higgses, we take degenerate masses on the inert Higgs spectra.
We utilize flavio package \cite{flavio} to perform a comprehensive analysis on various flavor observables, and obtain the global fit to the Wilson coefficients on the $R_K$ and $R_D$ anomalies.
Applying these results to our hidden gauged model, we scrutinize the favored parameter space on explaining these anomalies.
The structure of this paper is as follows. We introduce the hidden gauged extension of the SM in Sec. II. In Sec. III, the mixing between the vector-like and SM fermions and the $Z^{\prime}$ couplings are presented.
Using the flavio, we perform a global fit on the relevant Wilson coefficients to the $R_K$ and $R_D$ anomalies in Sec. IV.
Subsequently, various experimental searches and constraints are discussed in section VI. In Sec. V, we apply the obtained results in previous sections to the model parameters. Finally, Sec. VII is devoted to conclusions.
\section{The Model}
\label{sec:model}
We consider a hidden sector extension of the SM under the hidden gauged $U(1)_H$ symmetry~\cite{Yu:2016lof}. Although all the SM particles are not charged under the $U(1)_H$ symmetry, the $U(1)_H$ gauge boson can couple to them through mediators, such as new vector-like fermions or scalars. The mediators have both the $U(1)_H$ and the SM gauge charges. The schematic framework of this model is described in Fig.~\ref{fig:model}.
For the $R_K$ anomaly, essential mediators are the vector-like fermions. These fermions
could be $SU(2)_L$ singlet, doublet, or multiplets, with one or more generations.
Here, we focus on one minimal fermion assignments: all the vector-like fermions are
$SU(2)_L$ doublets, with only one generation~\footnote{
Another minimal choice is to introduce only vector-like singlet fermions with one generation.
Since our purpose is to illustrate how the $U(1)_H$ works for the $R_K$ anomaly, the
fermion assignment is not so important. We expect that the analysis is quite similar to different fermion assignments.}.
To be free of gauge anomaly, the new vector-like fermions need to possess appropriate quantum numbers.
We assign the SM quantum numbers on the vector-like fermions such that they mix with the SM fermions at tree level.
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{model.pdf}
\end{center}
\caption{The schematic description of the hidden gauged $U(1)_H$ model.}
\label{fig:model}
\end{figure}
Moreover, if we introduce additional mediators such as a charged scalar,
the $R_D$ anomaly could also be explained as a bonus.
In this work, we add a second scalar doublet $H^{\prime}$ as an inert type~\cite{Ma:2006km, Yu:2016lof} to keep our setup simple.
{\renewcommand{\arraystretch}{1.5}
\begin{table}[ht]
\begin{center}
\begin{tabular}{c| c c c | c }
\hline \hline
& $SU(3)_C$ & $SU(2)_L$ & $U(1)_Y$ & $U(1)_{H}$ \\
\hline
$ Q_{L,R}=(\tilde{U}_{L,R}~\tilde{D}_{L,R})^T$ & {\bf 3 } & {\bf 2} &$+\frac{1}{6}$ & $+1$ \\
$ L_{L,R}=(\tilde{N}_{L,R},~\tilde{L}_{L,R})^T$ & {\bf 1 } & {\bf 2} &$-\frac{1}{2}$ & $+1$ \\
\hline
$\Phi$ & {\bf 1 } & {\bf 1} &$ 0$ & $+1$ \\
\hline
$H'$ (optional) & {\bf 1 } & {\bf 2} &$ \frac{1}{2}$ & $+1$ \\
\hline
$\chi$ (optional) & {\bf 1 } & {\bf 1} &$0$ & $-1$ \\
\hline \hline
\end{tabular}
\end{center}
\caption{
The new particle contents and their quantum numbers in the gauge group $SU(2)_L \times U(1)_Y \times U(1)_{H}$.
Here $Q_{L,R}$ and $L_{L,R}$ denote the vector-like representation under the gauge group.
All the SM particles are not charged under $U(1)_{H}$. The $H'$ doublet is optional:
without $H'$ added, only $R_K$ anomaly is addressed; with $H'$, we could also explain the $R_D$ anomaly.
The vector-like fermion $\chi$ is also optional, which is $Z_2$-odd under $U(1)_{H} \to Z_2$ symmetry breaking,
and thus it is the dark matter candidate.
}
\label{table1}
\end{table}
The new particle contents and their quantum numbers are shown in Tab.~\ref{table1}.
Under this charge assignments, the model is free of gauge anomaly.
The $U(1)_H$ symmetry spontaneously breaks when a singlet scalar $\Phi$, which is only charged under the $U(1)_H$, obtains a nonzero vacuum expectation value (VEV) $v_{\Phi}$. They are cast into the form
\bea
\Phi = \frac{s + i a}{\sqrt2}, \hspace{1cm} \langle \Phi \rangle = \frac{v_{\Phi}}{\sqrt{2}},
\eea
and the Lagrangian is written as
\bea
{\mathcal L}_{\rm kin} = D_\mu \Phi^* D_\mu\Phi - V(H, \Phi, H'),
\eea
with the covariant derivative $D_\mu \Phi = (\partial_\mu + i g' Z'_\mu)\Phi$.
Here, $g'$ is the gauge coupling of the $U(1)_H$, and the gauge boson mass is given by $m_{Z'}=g'v_{\Phi}$. In the above potential, $H$ describes the SM $SU(2)$ doublet, and the two Higgs doublets are parametrized by
\bea
H = \left(\begin{array}{c} G^+ \\ \frac{v + h + i G^0}{\sqrt2}\end{array}\right), \hspace{2cm}
H' = \left(\begin{array}{c} H^{\prime +} \\ \frac{H' + i A'}{\sqrt2}\end{array}\right).
\eea
The potential $V(H, \Phi, H')$ is not relevant to our flavor study, and for simplicity, it is taken as
\bea
V(H, \Phi, H') &=& -\mu_{\Phi}^2 |\Phi|^2 + \lambda_{\Phi} |\Phi|^4 + \mu_{H^{\prime}}^2 |H^{\prime}|^2 + \lambda_{H^{\prime}} |H^{\prime}|^4 \nn
+ \lambda_{H\Phi} |H|^2|\Phi|^2 + \lambda_{H^{\prime}\Phi} |H'|^2 |\Phi|^2,
\eea
which respects a $Z_2(H') \times Z_2(\Phi)$ symmetry. The $\mu_{H^{\prime}}^2$ parameter is positive, and thus $H^{\prime}$ does not get the VEV.
The vector-like fermions are defined by
\bea
Q_{L,R} = \left(\begin{array}{c} \tilde{U}_{L,R} \\ \tilde{D}_{L,R} \end{array}\right), \hspace{2cm}
L_{L,R} = \left(\begin{array}{c} \tilde{N}_{L,R} \\ \tilde{L}_{L,R}\end{array}\right),
\eea
and their Dirac masses are
\bea
-{\mathcal L}_{\rm mass} = m_Q \bar{Q}_L Q_R + m_L \bar{L}_L L_R +h.c..
\eea
All the new fermions are charged under $U(1)_H$, as shown in Table~\ref{table1}.\footnote{These new fermions contribute to the kinetic mixing between $U(1)_Y$ and $U(1)_H$ at the one loop order. The mixing parameter is estimated to be roughly $\frac{eg'}{(4\pi)^2} \ln \frac{m_Q}{m_L}$. Here for simplicity, we take small or zero mass splitting between the vector-like quarks and leptons, and thus the kinetic mixing is very tiny and negligible.}
The right-handed vector-like fermions can couple to the left-handed SM fermions through the Yukawa interactions of $\Phi$
\bea
-{\mathcal L}_{\rm Yukawa} &=& \Phi^{\dagger}\sum_i\left( \overline{q}_{Li} Y_{q_iQ_R} Q_R + \overline{\ell}_{Li} Y_{l_iL_R} L_R\right) + {\rm h.c.},\nonumber\\
&\supset &\frac{v_{\Phi}}{\sqrt{2}}\left( Y_{d_i\tilde D_R}\bar{d}_{Li}{\tilde D}_R
+ Y_{u_i\tilde U_R}\bar{u}_{Li}{\tilde U}_R
+Y_{l_i \tilde L_R}\bar{e}_{Li}\tilde{L}_R
+Y_{\nu_i \tilde N_R}\bar{\nu}_{Li}\tilde N_R+ {\rm h.c}.\right),
\label{Yukawa_phi}
\eea
where the $SU(2)$ symmetry imposes a relation\footnote{If the vector-like fermions are the $SU(2)$ singlet, in general, they
introduce more free Yukawa couplings than the doublet case here. }
\bea
Y_{d_i\tilde D_R}=V^{\dagger}_{\rm CKM}Y_{u_i\tilde{U}_R}, \hspace{1cm}
Y_{l_i\tilde L_R}=V^{\dagger}_{\rm PMNS}Y_{\nu_i\tilde{N}_R} \label{Yukawa_relation}
\eea
with a generation index $i = 1,2,3$, Cabbio-Kobayashi-Maskawa (CKM) matrix $V_{\rm CKM}$ and Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix $V_{\rm PMNS}$. Similar situations are studied in \cite{Altmannshofer:2014cfa, Fuyuto:2015gmk} for the quark sectors and \cite{Poh:2017tfo} for the lepton sectors. The interactions in Eq.~(\ref{Yukawa_phi}) describe the mixing between the vector-like and SM fermions, which yields $\bar{s}bZ^{\prime}$ and $\bar{\mu}\mu Z^{\prime}$ couplings after diagonalization of mass matrices. These $Z^{\prime}$ couplings are originated from three Yukawa components, $Y_{b\tilde D_R},~Y_{s\tilde D_R}$ and $Y_{\mu \tilde L_R}$.
Apart from the right-handed vector-like fermions, the left-handed vector-like fermions have couplings to the right-handed SM fermions through the inert doublet
\bea
-{\mathcal L}'_{\rm Yukawa} &=& {\rm y}'_{Q_Lu_i} \overline{Q_L} \tilde{H}' u_R + {\rm y}'_{Q_Ld_i} \overline{Q_L} H' d_R
+ {\rm y}'_{L_Le_i} \overline{L_L} H' e_R +{\rm h.c.},
\label{Yukawa_H}
\eea
with $\tilde H^{\prime}=i\sigma_2H^{\prime *}$.
Unlike the Yukawa couplings in Eq.~(\ref{Yukawa_phi}), the above interactions do not contribute to the mass matrices of the fermions. $b\to c \tau \nu$ transition is induced by the charged scalar in the inert doublet, in which the vector-like fermions play a role in yielding $\bar{c}bH^{+}$ and $\bar{\nu}\tau H^{+}$ couplings.
Since we introduce a hidden gauge sector, it is likely to have hidden particles, which are not charged under the SM gauge group at all.
If so, the lightest un-colored one would be the dark matter candidate.
Here we assume it is the WIMP dark matter.
When we study the phenomenology of the hidden sector particles, typically only the lightest dark matter candidate plays an important role.
For simplicity, we introduce a vector-like fermion $\chi$, which only carries the $U(1)_H$ charge
\bea
{\mathcal L} = \bar{\chi} i\gamma^\mu D_\mu \chi, \quad D_\mu = \partial_\mu - i g' Z'_\mu.
\eea
To make it a dark matter candidate, we assign a $Z_2$ charge for the $\chi$.
Under $U(1)_H$ breaking down to the $Z_2$ symmetry, the $\chi$ is $Z_2$-odd while all other mediators and the SM particles are $Z_2$ even.
\section{Mass Matrices and Couplings}
\label{sec:modelconstraint}
The interactions in Eq. (\ref{Yukawa_phi}) imply off-diagonal parts in fermion mass matrices, which are described by
\begin{eqnarray}
\left(\begin{array}{cc}
\overline{f}_{i,L} &\overline{\tilde{F}}_L
\end{array}\right)
\left(
\begin{array}{cc}
\frac{v}{\sqrt{2}}y_{i} &\frac{v_{\Phi}}{\sqrt{2}}Y_{f_i,\tilde{F}_R} \\
0 & M_{F}
\end{array}
\right)
\left(
\begin{array}{c}
f_{i,R}
\\
\tilde{F}_R
\end{array}
\right)+h.c. \equiv
\overline{f}_{La} {\mathcal M}^f f_{Ra} + h.c.,
\end{eqnarray}
with $(f_{i},\tilde{F})^T_{L,R}=(u_{i},\tilde{U})^T_{L,R}$, $(d_{i},\tilde{D})^T_{L,R}$, $(l_{i},\tilde{L})^T_{L,R}$, $i=1, \dots, 3$ and $a = 1, \dots, 4$. $y_i$ is the SM Higgs Yukawa coupling, and its VEV is $v\simeq 246~$GeV. The matrix ${\cal M}^f$ is diagonalized by unitary matrices $U^f_L$ and $U^f_R$
\bea
U_{L}^{f\dagger} {\mathcal M}^f U_{R}^{f} = \textrm{diag}(m_{f_1},m_{f_2},m_{f_3},m_{f_4}),
\eea
which gives relationships between the original gauge eigenstate $f_a$ and new mass eigenstate $f^{\prime}_a$ as $f^{\prime}_{R}=(U^{f}_R)^{\dagger}f_{R}$ and $f^{\prime}_L=(U^f_L)^{\dagger}f_L$.
This diagonalization can affect original gauge interactions such as the $W$ and $Z$ boson couplings. However, it should be noted that, since the left-handed vector-like fermions have the same SM charges as those of the SM fermions, the $Z$ boson couplings to the left-handed fermions do not change. We will see this situation later.
In the charged-lepton sector, the mixing between the second generation lepton $\mu$ and vector-like lepton $\tilde{L}$ is essential for the $b\to s \mu\mu$ transition. Assuming there is only one mixing, namely, $Y_{\mu \tilde L_R}\neq 0$, the matrices $U_{L}^{\ell}$ and $U_{R}^{\ell}$ can be described by\footnote{Here, for simplicity, complex phases are not written down. }
\bea
U_{L}^{\ell\dagger} =
\left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & c_{\alpha_L} & 0 & -s_{\alpha_L} \\
0 & 0 & 1 & 0 \\
0 & s_{\alpha_L} & 0 & c_{\alpha_L}
\end{array}
\right)
,\hspace{1cm} U_{R}^{\ell} =
\left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & c_{\alpha_R} & 0 & s_{\alpha_R} \\
0 & 0 & 1 & 0 \\
0 & -s_{\alpha_R} & 0 & c_{\alpha_R}
\end{array}
\right)
\eea
where $s_{\alpha}\equiv \sin\alpha$ and $c_{\alpha}\equiv \cos\alpha$. The mixing angle $\alpha_L$ and $\alpha_R$ are approximately given by
\bea
s_{\alpha_L} \simeq \frac{ v_{\Phi}Y_{\mu \tilde{L}_R}/ \sqrt{2} }{\sqrt{v^2_{\Phi}Y^2_{\mu \tilde{L}_R}/2 + M_L^2}}, \hspace{1cm}
s_{\alpha_R} \simeq \frac{m_\mu}{\sqrt{v^2_{\Phi}Y^2_{\mu \tilde{L}_R}/2 + M_L^2}} s_{\alpha_L}.
\eea
Since we suppose that $v_{\Phi}\sim O(100)~$GeV, $Y_{\mu\tilde L}\sim O(0.1-1)$ and $M_{L}\sim O(1)~$TeV, it turned out that the angle $\alpha_L \gg \alpha_R$.
For the down-quark sector, the Yukawa couplings $Y_{b\tilde D_R}$ and $Y_{s\tilde D_R}$ are necessary for the $R_K$ anomaly, which implies the nonzero values of $Y_{t\tilde U_R}$ and $Y_{c\tilde U_R}$ with the same order of magnitude. Although, apart from the charged-lepton sector, the diagonalization is somewhat complicated due to the two mixing parameters, we obtain similar situation
\bea
|1-U^{u\dagger}_{L} |_{ij} \gg |1-U^u_R|_{ij}, \hspace{1cm} |1-U^{d\dagger}_{L}|_{ij} \gg |1-U^{d}_{R} |_{ij}.
\eea
Again, the deviations of the right-handed mixing matrices from identity are suppressed by order $\frac{m_{q}}{m_Q}$.
For the neutrino parts, its Lagrangian contains
\begin{eqnarray}
{\cal L}_{\rm neutrino} \supset -M_L\overline{\tilde{N}}_L \tilde{N}_R - \frac{v_{\Phi}}{\sqrt{2}}Y_{\nu_i,\tilde{N}_R}\overline{\nu}_{Li} \tilde{N}_R +h.c. \label{neutrino}
\end{eqnarray}
As explained in the previous section, the second term is related to that of the charged lepton due to the $SU(2)$. As long as we keep only nonzero $Y_{\mu \tilde L_R}$, the coupling is given by $Y_{\nu_i\tilde N_R}=(V_{\rm PMNS})_{i2}Y_{\mu \tilde L_R}$.
For sake of simplicity, we assume that the $V_{\rm PMNS}$ is a unit matrix, which results in only nonzero $Y_{\nu_{\mu}\tilde N_R}$ with more direct relationship of $Y_{\nu_{\mu}\tilde N_R}=Y_{\mu\tilde L_R}$. As in the same way of the charged-lepton sector, if we rotate the neutrino fields with
\bea
\left(\begin{array}{c}
\nu^{\prime}_{Li} \\
\nu^{\prime}_{L4}
\end{array}
\right) =
U^{\nu\dagger}_L
\left(
\begin{array}{c}
\nu_{Li} \\
\tilde{N}_L
\end{array}
\right), \hspace{1cm}
U^{\nu\dagger}_L = \left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & c_{\beta_L} & 0 & s_{\beta_L} \\
0 & 0 & 1 & 0 \\
0 & -s_{\beta_L} & 0 & c_{\beta_L}
\end{array}
\right)
\eea
where
\begin{eqnarray}
s_{\beta_L} = \frac{v_{\Phi}Y_{\mu \tilde{L}_R}/ \sqrt{2} }{\sqrt{M_L^2+v_{\Phi}^2Y^2_{\mu,\tilde{L}_R}/2}} \simeq s_{\alpha_L}, \label{neutrino_mixing}
\end{eqnarray}
the Lagrangian in Eq. (\ref{neutrino}) becomes
\begin{align}
{\cal L}_{\rm neutrino}=-\sqrt{M^2_L+v^2_{\Phi}Y^2_{\mu\tilde L_R}/2}~\bar{\nu}^{\prime}_{L4}\tilde{N}_R+h.c.
\end{align}
We find that $U^{\nu}_L \simeq U_{L}^{\ell}$ due to the approximate $SU(2)$ symmetry, which deviates once the lepton masses are taken into account.
After diagonalizing the mass matrices, the $Z$ boson couplings to the lepton sector are
\bea
{\mathcal L}_Z \supset \frac{e}{s_W c_W} Z_\mu &\bigg[\bar{e}^{\prime}_{L} \gamma^\mu U^{\ell \dagger}_L(T^3_{L} + s_W^2 ) U^{\ell}_L e^{\prime}_{L}
+ \bar{e}^{\prime}_{R} \gamma^\mu U^{\ell \dagger}_R (T^3_{R} + s_W^2 ) U^{\ell}_R e^{\prime}_{R}\nonumber\\
&- \bar{\nu}^{\prime}_L\gamma^{\mu}U^{\nu\dagger}_LT^3_{L}U^{\nu}_L\nu_L^{\prime} +\frac{1}{2} \bar{\tilde N}_R\gamma^{\mu}\tilde N_R
\bigg],
\eea
where $T^3_{L}$ and $T^3_{R}$ are defined as $T^3_{L} = -\frac{1}{2} {\rm{diag}}(1,1,1,1)$ and $T^3_{R} = -\frac{1}{2}{\rm{diag}}(0, 0,0 ,1) $.
It is seen that the left-handed $Z$ couplings are diagonal, while the flavor-violating $Z$ couplings to the right-handed charged leptons exist. However, the right-handed mixing angle is tiny, which results in the small flavor-violating couplings.
Similarly, in the quark sectors, while the $Z$ couplings to the left-handed quarks do not receive any corrections from the field definitions, the right-handed couplings have the flavor-violating parts but they are suppressed.
The $W$ couplings to the leptons are given by
\bea
{\mathcal L}_W \supset \frac{e}{\sqrt2 s_W } W^-_{\mu} \left[\bar{e}^{\prime}_{L} \gamma^\mu U^{\ell \dagger}_L {\rm{diag}}(1,1,1,1) U^{\nu}_L \nu^{\prime}_{L}
+ \bar{e}^{\prime}_{Ra} \gamma^\mu (U^{\ell \dagger}_R)_{a4} \tilde{N}_{R} \right] + h.c.,
\eea
Because of the approximate $SU(2)$ symmetry, the left-handed couplings have $U^{\ell \dagger}_L {\rm{diag}}(1,1,1,1) U^{\nu}_L \simeq 1$. The right-handed interactions are also present with a suppression factor $U^{\ell \dagger}_R$. On the other hand, the quark sector has a somewhat different situation due to the CKM structure. The model parameters $Y_{d_i\tilde D_R}, v_{\Phi}$ and $m_Q$, which decide $U^{u}_{L,R}$ and $U^{d}_{L,R}$, should be chosen to realize the measured CKM values. In our analysis, we use moderate values of the model parameters which satisfy the CKM values within error bars.
\begin{figure}
\begin{center}
\includegraphics[width=0.7 \textwidth]{eff_Zp.pdf}
\end{center}
\caption{Examples of the left-handed $Z^{\prime}$ couplings induced by mixing between the vector-like and the SM fermions with mass insertion notation.
Here the double line represents the vector-like fermions.
\label{fig:eff_Zp}}
\end{figure}
One of our main interests is the $Z'$ couplings, which are given by
\bea
{\mathcal L}_{Z^{\prime}} &\supset& -g' Z'_\mu \left[\bar{e}^{\prime}_{L} \gamma^\mu U^{\ell \dagger}_L {\rm{diag}}(0,0,0,1) U^{\ell}_L e^{\prime}_{L} +\bar{d}^{\prime}_{L} \gamma^\mu U^{d \dagger}_L {\rm{diag}}(0,0,0,1) U^{d}_L d^{\prime}_{L} \right].
\eea
These interactions can be described by integrating out the heavy fermions
\bea
\left(U^{\ell \dagger}_L {\rm{diag}}(0,0,0,1) U^{\ell}_L\right)_{ij}\simeq & \frac{v^2_{\Phi}}{2m^2_{L}}Y_{l_i\tilde L_R}Y^*_{l_j\tilde L_R},\\
\left(U^{d \dagger}_L {\rm{diag}}(0,0,0,1) U^{d}_L\right)_{ij}\simeq & \frac{v^2_{\Phi}}{2m^2_{Q}}Y_{d_i\tilde L_R}Y^*_{d_j\tilde L_R}.
\eea
and this situation is schematically drawn in Fig. \ref{fig:eff_Zp}.
If we focus on only relevant Yukawa couplings $Y_{b\tilde D_R},~Y_{s\tilde D_R}$ and $Y_{\mu\tilde L_R}$ to the $R_K$ anomaly, they yield
\begin{align}
{\cal L}_{Z^{\prime}} \supset &-Z^{\prime}_{\mu}\bigg(g^L_{sb}~\bar{s}_L\gamma^{\mu}b_L+g^L_{\mu\mu}~\bar{\mu}_L\gamma^{\mu}\mu_L+g^L_{bs}~\bar{b}_L\gamma^{\mu}s_L+g^L_{ss}~\bar{s}_L\gamma^{\mu}s_L+g^L_{bb}~\bar{b}_L\gamma^{\mu}b_L
\bigg), \label{eff_Zp}
\end{align}
where
\begin{align}
g^L_{d_id_j}=g^{\prime}\frac{v^2_{\Phi}}{2m^2_Q}Y_{d_i\tilde D_R}Y^*_{d_j\tilde D_R},
\hspace{1cm}
g^L_{\mu\mu}=g^{\prime}\frac{v^2_{\Phi}}{2m^2_L}Y_{\mu\tilde L_R}Y^*_{\mu\tilde L_R}, \label{eff_Zp_gL}
\end{align}
proportional to the mixing angles,
and we omit the prime in the fermion fields.
The first two terms in Eq. (\ref{eff_Zp}) lead to the $b\to s \mu\mu$ transition, and
the three Yukawa couplings also produce the $Z^{\prime}$ couplings to the up quark and neutrino sectors, such as $\bar{t}tZ^{\prime}$ and $\bar{\nu}_{\mu}\nu_{\mu}Z^{\prime}$, due to the $SU(2)$ symmetry. Note that the right-handed $Z^{\prime}$ couplings are also present, however, they are induced by the inert scalar loop and are numerically suppressed. Therefore, our current study does not take them into account.
\begin{figure}
\begin{center}
\includegraphics[width=0.7 \textwidth]{eff_Hp.pdf}
\end{center}
\caption{Examples of the charged scalar couplings through mixing between the vector-like and the SM fermions with mass insertion notation.
Here the double line represents the vector-like fermions.
\label{fig:eff_Hp}}
\end{figure}
The other intriguing interactions are the Yukawa couplings between the inert-Higgs doublet and the SM fermions, in particular, the charged Higgs couplings $\bar{c}bH^+$ and $\bar{\nu}\tau H^+$. Using mass-insertion method as in the $Z^{\prime}$ coupling, we obtain the Yukawa couplings
\begin{align}
{\cal L}_{H^{\prime}}\supset&-\left({\cal Y}_{c_{R}b_{L}}\bar{c}_Rb_L+{\cal Y}_{c_{L}b_{R}}\bar{c}_Lb_R\right)H^{\prime +}-{\cal Y}_{\nu_{\mu}\tau_{R}}\bar{\nu}_{L\mu}\tau_R H^{\prime +}+{\rm h.c.}, \label{eff_Yukawa}
\end{align}
with
\begin{align}
{\cal Y}_{c_{R}b_{L}}&=-\frac{v_{\Phi}}{\sqrt{2}m_Q}{\rm y}^{\prime *}_{Q_Lc}Y^*_{b\tilde D_R},\hspace{0.5cm}
{\cal Y}_{c_{L}b_{R}}=\frac{v_{\Phi}}{\sqrt{2}m_Q}Y_{c\tilde D_R}{\rm y}^{\prime}_{Q_Lb},\hspace{0.5cm}
{\cal Y}_{\nu_{\mu}\tau_{R}}=\frac{v_{\Phi}}{\sqrt{2}m_L}Y_{\nu_{\mu}\tilde N_R}{y}^{\prime}_{L_L\tau}.
\end{align}
These interactions are originated from the mixing between the vector-like and SM fermions as seen in Fig. \ref{fig:eff_Hp}. Their dependences on $v_{\Phi}/m_{Q,L}$ are different from those in Eq. (\ref{eff_Zp_gL}).
The neutral components in the inert Higgs also induce the Yukawa couplings as in Eq. (\ref{eff_Yukawa}). In our study, we assume that their masses are degenerate. This assumption produces a simple situation where some scalar operators disappear due to the degeneracy as mentioned later.
As we will see in the next section, while the explanation of $R_K$ anomaly needs nonzero $g^L_{sb}$ and $g^L_{\mu\mu}$, the $R_D$ anomaly requires nonzero ${\cal Y}_{cb}$ and ${\cal Y}_{\nu\tau}$. From the phenomenological point of view, we consider a minimal setup in which these relevant couplings are exclusively focused on. Simultaneously, it implies nonzero $Y_{b\tilde D_R},~Y_{s\tilde D_R},~Y_{\mu\tilde L_R}$ and ${\rm y}^{\prime}_{Q_Lb},~{\rm y}^{\prime}_{Q_Lc},~{\rm y}^{\prime}_{L_L\tau}$. From now on, we denote them as
\bea
Y_b \equiv Y_{b\tilde D_R}, \quad Y_s \equiv Y_{s\tilde D_R}, \quad Y_\mu \equiv Y_{\mu\tilde L_R},
\eea
and
\bea
y'_\tau \equiv {\rm y}'_{L_L\tau}, \quad y'_b \equiv {\rm y}'_{Q_Lb}, \quad y'_c \equiv {\rm y}'_{Q_Lc}.
\eea
We also assume that the Yukawa couplings are real, and a flavor hierarchy $|Y_b|\gg |Y_s|$.
\section{Rare $B$ Decay Anomalies}
\label{sec:banomaly}
The relevant effective Hamiltonians to $b\to s \mu^+\mu^-$ and $b\to c\tau^-\bar{\nu}$ are given by \cite{Altmannshofer:2013foa,Descotes-Genon:2013wba,Beaujean:2013soa}
\begin{align}
{\cal H}_{\rm eff}(b\to s\mu^+ \mu^-)&=-\frac{4G_F}{\sqrt{2}}V_{tb}V^*_{ts}\frac{e^2}{16\pi^2}\bigg[C_9(\bar{s}\gamma^{\mu}P_Lb)(\bar{\mu}\gamma_{\mu}\mu)+C_{10}(\bar{s}\gamma^{\mu}P_Lb)(\bar{\mu}\gamma_{\mu}\gamma_5\mu) \bigg],\\
{\cal H}_{\rm eff}(b\to c\tau^-\bar{\nu})&=\frac{4G_F}{\sqrt{2}}V_{cb}\bigg[C_Sm_b(\bar{c}P_Rb)(\bar{\tau}P_L\nu)+C_S^{\prime}m_b(\bar{c}P_Lb)(\bar{\tau}P_L\nu) \bigg].
\end{align}
The Wilson coefficients of $C_9 $ and $C_{10}$ are induced by the $Z^{\prime}$ interactions in Fig. \ref{fig:eff_Zp}. Integrating out the $Z'$ boson, the Wilson coefficients are given by
\begin{align}
C_9=-C_{10}&=-\frac{1}{2m^2_{Z'} C^{bs}_{\rm SM}}g^L_{sb}g^L_{\mu\mu}
=-\frac{v^2_{\Phi}}{8C^{bs}_{\rm SM}m^2_Qm^2_L}Y_{s}Y_{b}Y_{\mu}^2, \label{C9_10}
\end{align}
with $C^{bs}_{\rm SM}=\frac{4G_F}{\sqrt{2}}V_{tb}V^*_{ts}\frac{e^2}{16\pi^2}$. The final expression in Eq.~(\ref{C9_10}) is obtained by using the couplings defined in Eq. (\ref{eff_Zp_gL}). Although $m_{Z^{\prime}}$ dependence appears in the middle of the Eq.~(\ref{C9_10}), it is canceled out by those in $g^L_{sb}$ and $g^L_{\mu\mu}$. Thus, $v_{\Phi}$ is finally left only in the numerator.
The nonzero Yukawa, $Y_b,~Y_s$ and $Y_{\mu}$, lead to $Y_q=(V_{\rm CKM})_{qs}Y_s+(V_{\rm CKM})_{qb}Y_b$ for $q=u,c,t$ and $Y_{\nu_i}=(V_{\rm PMNS})_{\nu_i \mu}Y_{\mu}$ for $i=e,\mu,\tau$.
For the $R_D$ anomaly, neglecting the neutrino flavor textures which is irrelevant to our study, we simply assume that the PMNS matrix is a unit matrix, which results in $Y_{\nu_{\mu}}=Y_{\mu}$. The Yukawa coupling allows the flavor-violating charged Higgs coupling $\bar{\tau}P_L\nu_{\mu}H^-$. As seen in \cite{Bhattacharya:2016mcc}, such a new flavor-violating process would also contribute to the $R_D$ anomaly. Therefore, the charged scalar interactions in Fig. \ref{fig:eff_Hp} produce
\begin{align}
C_S^{\prime}&=-\frac{1}{C^{bc}_{\rm SM}m^2_{H^{\prime}}m_b}{\cal Y}_{c_Rb_L}{\cal Y}^*_{\nu_{\mu }\tau_R}
=\frac{v^2_{\Phi}}{2C^{bc}_{\rm SM}m^2_{H^{\prime}}m_bm_Qm_L}{y}^{\prime }_{c}{y^{\prime }}_{\tau}Y_{b}Y_{\mu },\label{Csp}\\
C_S&=-\frac{1}{C^{bc}_{\rm SM}m^2_{H^{\prime}}m_b}{\cal Y}_{c_Lb_R}{\cal Y}^*_{\nu_{\mu }\tau_R}=-\frac{v^2_{\Phi}}{2C^{bc}_{\rm SM}m^2_{H^{\prime}}m_bm_Qm_L}{y}^{\prime}_{b}{y}^{\prime}_{\tau}Y_{c}Y_{\mu}, \label{Cs}
\end{align}
with $C^{bc}_{\rm SM}=\frac{4G_F}{\sqrt{2}}V_{cb}$. The third terms are written by Eq. (\ref{eff_Yukawa}). It is seen that the model parameters appearing in $C_9$ and $C_{10}$ in the Eq. ({\ref {C9_10}}) are correlated to $C_S^{\prime}$ and $C_S$ except $m_{H^{\prime}}$ and $y^{\prime}_{b,c,\tau}$.
Therefore, in our model, the $R_D$ anomaly is related to the $R_K$ anomaly through the common parameters.
In order to determine the possible sizes of these Wilson coefficients, we use flavio 0.21 \cite{flavio}:
\begin{itemize}
\item{$C_9$ and $C_{10}$}
The observables and the corresponding experimental measurements listed in the Appendix in Ref.~\cite{Altmannshofer:2017fio} are used, and Table~\ref{tab:rk} in the last page summarizes them. We take into account all known correlations
among observables and approximate the uncertainties as Gaussian. From Eq.~(\ref{C9_10}), our model holds the relation $C_9=-C_{10}$, therefore essentially only one Wilson coefficient needs to be fitted. Assuming all the UV parameters are real, we define the following real parameters in the global fit:
\begin{eqnarray}
C_{9,\rm new} = C_9 \cdot \exp({i\arg{(C^{bs}_{\rm SM})}}) = -C_{10,\rm new} = -C_{10} \cdot \exp({i\arg(C^{bs}_{\rm SM})}) \label{C9_new}
\end{eqnarray}
The Bayesian method is employed in the global fit with following procedures: we first obtain the likelihood function with a single argument $C_{9,\rm new}$ from flavio FastFit class, and assume a uniform prior probability of $C_{9,\rm new}$ ranges from -3 to 3.
We then use pymultinest~\cite{Buchner:2014nha} to implement a Monte Carlo sampling, and finally obtain the posterior probability shown in Fig.~\ref{fig:c9fit} with corresponding 1$\sigma$ and 2$\sigma$ uncertainties, which is plotted using Superplot~\cite{Fowlie:2016hew}. In the figure, the best-fit point is indicated by a star, and $1\sigma$ and 2$\sigma$ regions are represented by blue and green lines on the top of curve. The obtained values are in agreement with the current observed branching ratio for $B^{0,\pm}\to K^{*0,\pm} \mu\mu,~B_s\to \phi \mu\mu,~B\to X_s\mu\mu$ and $B_s\to\mu\mu$.
\begin{figure}
\begin{center}
\includegraphics[width=0.4 \textwidth]{c9_-c10_new_delta.pdf}
\end{center}
\caption{The likelihood as function of the $C_{9,\rm new}$ from the Bayesian fit. The blue and green lines on the top of curve represent the $1\sigma$ and $2\sigma$ regions. The star at the bottom is the best-fit point.
\label{fig:c9fit}}
\end{figure}
\item{$C_S$ and $C_S^{\prime}$}
For the $R_{D^*}$, measurements in Ref.~\cite{Sato:2016svk,Abdesselam:2016xqt,Aaij:2015yra} are used, while those in Ref.~\cite{Lees:2013uzd,Huschle:2015rga} are taken into account for the $R_D$ . We use FastFit in flavio to obtain the 2-D global fit for $C_S$ and $C'_S$, which is shown in Fig.~\ref{fig:cscsp}. The four blue regions are $1\sigma,~2\sigma$ and 3$\sigma$ allowed regions by fitting with the $R_D$ and $R_{D^*}$ measurements mentioned above. The light and dark red region are excluded by {{the current LHC limit ${\rm BR}(B^-_c\to\tau^-\bar{\nu})<30\%$~\cite{Alonso:2016oyd}, and recasted LEP limit}} ${\rm BR}(B^-_c\to\tau^-\bar{\nu})<10\%$~\cite{Akeroyd:2017mhr}, respectively.
We find that among four favored regions, only two of them are more preferred after considering the constraints from the ${\rm BR}(B^-_c\to\tau^-\bar{\nu})$ limits.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{RD_w_Bc.pdf}
\end{center}
\caption{The two dimensional contour for $C_S$ and $C'_S$ as a result of the FastFit. The blue regions corresponds to 1$\sigma$, 2$\sigma$ and 3$\sigma$ regions in agreement with $R_D$ and $R_{D^*}$ measurements. The red and light red regions are excluded {{by the current LHC limit ${\rm BR}(B^-_c\to\tau^-\bar{\nu})<30\%$ \cite{Alonso:2016oyd}, and recasted LEP limit}} ${\rm BR}(B^-_c\to\tau^-\bar{\nu})<10\%$ \cite{Akeroyd:2017mhr}, respectively.
\label{fig:cscsp}}
\end{figure}
For later use, we project the two dimensional contour to a 1-dimensional fit by setting $C_S=0$.
The Bayesian fit for $C'_S$ alone is obtained using the same strategy as in the fitting of $C_{9,\rm new}$. For this fit, we choose prior probability as uniformly distribute between $-2$ and $0$, the fitting results is shown in Fig.~\ref{fig:cspfit}. As in Fig.~\ref{fig:c9fit}, the best fit value is represented by the star at the bottom, and $1\sigma$ and $2\sigma$ regions are blue and green lines above curve.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{RD_Csp_Cs=0.pdf}
\end{center}
\caption{The likelihood as function of $C'_{S}$ with $C_S=0$ based on the Bayesian fit. The star indicates the best fit value, and the blue and green lines are $1\sigma$ and $2\sigma$ regions.
\label{fig:cspfit}
}
\end{figure}
\end{itemize}
Finally, we summarize the obtained numerical values in Table. \ref{table:general_analysis} and \ref{table:model_analysis}, where
those of $C_S$ and $C_S^{\prime}$ without the constraint from $B^-_c\to\tau^-\bar{\nu}$ transition are listed.
\renewcommand{\arraystretch}{1.0}
\begin{table}[h]
\begin{tabular}{l|c|c|c}
& $C_9=-C_{10}$ & $C_S$ & $C_S^{\prime} $ \\
\hline
\hline
~$1\sigma$~ & $[-0.742,~-0.643]$ & $[-0.344,~0.074]$ & $[-0.821,~-0.461]$\\
& & $[0.180,~0.517]$ & $[-0.444,~-0.068]$\\
& & $[-1.045,~-0.627]$ & $[0.755,~1.109]$\\
& & $[-1.498,~-1.153]$ & $[0.351,~0.734]$\\
\hline
$2\sigma$ & $[-0.767,~-0.618]$ & $[-0.389,~0.128]$ & $[-0.863,~-0.430]$ \\
& & $[-0.123,~0.547]$ & $[-0.482,~0.100]$ \\
& & $[-0.605,~-0.083]$ & $[0.712,~1.162]$ \\
& & $[-1.022,~-0.497]$ & $[0.182,~0.770]$ \\
\hline
$3\sigma$ & $-$ & $[-0.087,~0.684]$ & [-0.919,~-0.234] \\
& & $[0.462,~1.078]$ & [-0.529,~0.175] \\
& & $[-0.672,~0.110]$ & [0.520,~1.234] \\
& & $[-1.045,~-0.431]$ & [0.113,~0.828] \\
\end{tabular}
\caption{Suitable sizes of the Wilson coefficients $C_9,C_{10}$ and $C_S,C_S^{\prime}$ for the $R_K$ and $R_D$ anomalies. Here, the restrictions from $Br(B^-_c\to\tau^-\bar{\nu})$ are not taken into account. }
\label{table:general_analysis}
\end{table}
\begin{table}[h]
\begin{tabular}{c|c|c}
& $C_{9, \rm new}=-C_{10, \rm new}$ & $C_S^{\prime}$ with $C_S=0$ \\
\hline
\hline
~$1\sigma$~ & $[0.515,~-0.72]$ & $[-0.788,~-0.717]$ \\
\hline
$2\sigma$ & $[0.433,~0.843]$ & $[-0.824,~-0.682]$
\end{tabular}
\caption{Desired values of $C_{9,\rm new}$ defined in Eq. (\ref{C9_new}) and $C_S^{\prime}$ on the line of $C_S=0$. }
\label{table:model_analysis}
\end{table}
\section{Experimental Searches and Constraints}
\label{sec:constraints}
The hidden gauge boson $Z'$ encounters direct constraints from collider searches.
Depending on the $Z'$ mass, $m_{Z'} = g' v_\Phi$, there are different limits on the signal rate of the $Z'$ production. Here, we discuss the following cases: light $Z'$ case ($m_{Z'} < m_Z$) and heavy $Z'$ case ($m_{Z'} > m_Z$).
\begin {itemize}
\item When the $Z'$ is lighter than the $Z$ mass, some viable parameter regions exist.
Since the $Z'$ has no coupling to the electron, LEP searches cannot provide
direct constraint on the light $Z'$.
Furthermore, the Tevatron~\cite{Aaltonen:2011gp,Abazov:2010ti} and LHC~\cite{Aaboud:2017buh,Khachatryan:2016zqb} searches for $Z'$ to dilepton final state only apply to the case of $m_{Z^{\prime}}>100~$GeV.
The relevant limit to the light $Z'$ case comes from the LHC searches at $ p p \to Z\to 4\mu$. Its typical SM process is through the off-shell $Z$ mediated $2\mu$ decay, while the
light $Z'$ could be on-shell when {{$m_{Z'} < m_{Z} - 2 m_\mu$}}.
A detailed analysis on how to recast the current LHC search limit has been done in Ref.~\cite{Altmannshofer:2014pba}.
Mapping their analyses to our model, we obtain the constraint shown in Fig.~\ref{fig:zprime4l}. In the figure, the orange line indicates the $g^{\prime}$ coupling as a function of $m_{Z^{\prime}}$ with $v_\Phi=700$ GeV. The excluded region corresponds to the blue region. For $v_\Phi=700$ GeV, we find that the region of $m_{Z'}\lesssim 10$~GeV and $50~{\rm GeV}\lesssim m_{Z'}$ still have some spaces for $10^{-3}<g^{\prime}<0.14$. However, one should note that for $m_{Z'}\lesssim 10$~GeV, we are not able to integrate out the $Z'$ particle when deriving the Wilson coefficients $C_9$ and $C_{10}$ in Eq.\ref{C9_10}.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{zprime4l.pdf}
\end{center}
\caption{Constraint on low mass $Z'$ from $Z\to 4\mu$ search at LHC. The blue region is excluded by the current LHC searches. The orange line represents our benchmark model with $v_\Phi=700$ GeV.
\label{fig:zprime4l}}
\end{figure}
\item In the case of the heavy $Z^{\prime}$, the LHC searches in the di-muon final states put the tightest constraints on its mass.
The current limit on $m_{Z^{\prime}}$ is around $3\sim4$ TeV ~\cite{Aaboud:2017buh,Khachatryan:2016zqb}. In order to apply this limit to our case, we should first take into account the suppressed coupling to the first generation quarks by the small mixing with vector-like fermions, which results in small production cross section.
Moreover, although the LHC searches assume that the decay branching ratio to the muon or electron lepton is $100\%$, our $Z'$ can also decay into the SM quarks,
and other light $U(1)_H$ charged particles. Therefore, the branching ratio could be much smaller than the assumed value.
We estimate $pp\to Z'$ production cross section with Madgraph~\cite{Alwall:2014hca}, and demonstrate the constraint on $\sigma(pp\to Z'\to \mu\mu)$ in Fig.~\ref{fig:zprime_heavy} with benchmark values: $v_\Phi=700$ GeV, $Y_b=-1$ and $Y_s=0.0184$. In addition to the muons, another decay processes of the bottom, top, neutrino, charged scalar, scalar $(s, H)$ and pseudo-scalar ($A$) are also present. A naive estimation implies that the $Z'$ decay branching ratio to the muons is roughly $1/10.5$.
The cross section times branching ratio is shown in the red curve. According to the figure, the $Z'$ mass can be as low as $1.5$ TeV.
Furthermore, the branching ratio can be reduced if additional decay channels come from
unknown hidden gauge sector particles, which are irrelevant to our flavor study.
However, if we allow opening additional decay channels, the constraint could be significantly relaxed.
For example, taking the branching ratio to be ${\rm BR}(Z^{\prime}\to\mu\mu)=1/20$, which is shown in the green curve, the $Z'$ mass can be lower than 750 GeV.
\end {itemize}
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{zprime_constraint.pdf}
\end{center}
\caption{Constraint on $Z'$ with $v_\Phi=$ 700 GeV, $Y_b=-1$ and $Y_s=0.0184$. The blue line is the $95\%$ C.L exclusion limit by ATLAS experiments~\cite{Aaboud:2017buh}. The red and green curves correspond to the branching ratio of ${\rm BR}(Z'\to \mu\mu)=1/10.5$ and ${\rm BR}(Z'\to \mu\mu)=1/20$ (including new hidden particles), respectively.
\label{fig:zprime_heavy}}
\end{figure}
Next, we briefly comment on the constraints on the mediators. In \cite{Aaboud:2017zfn}, the vector-like quark search is discussed with its decay into $Wb$, and the current limit is $m_{Q} > 1.1$ TeV. On the other hand, studies about the multi-lepton final state~\cite{Dermisek:2014qca} show that the limit on the vector-like lepton is $m_{L} > 500$ GeV. In our numerical analysis, we take the benchmark point $m_{Q}=m_{L}=1.4$ TeV, which is consistent with these limits.
Regarding the inert-Higgs doublet, the charged scalar receives the tightest bounds.
The LHC searches for $H^\pm$ are all associated with the top quark production through $tbH^\pm$ coupling. Therefore, we can evade any constraints from the LHC by setting $y'_t=y'_b=0$ such that the $tbH^\pm$ coupling vanishes, and it follows that $C_S=0$. Actually, we investigate the scenario that $y'_t$ and $y'_b$ are nonzero and calculate the production cross section for the process $pp\to \bar{t} bH^+$ with Madgraph for a benchmark point: $y'_t=y'_b=1$, $m_Q=1.4$ TeV, $v_\Phi=700$ GeV, $Y_b=-1$ and $Y_s=0.0184$. It is found that $m_{H^\pm}>300$ GeV is consistent with the ATLAS experiments~\cite{Aad:2015typ,Aaboud:2016dig} provided that ${\rm BR}(H^+\to\tau\nu)<80$\%. Again, if taking into account the $H^{\prime\pm}$ decay channel, $H^{\prime +}\to t\bar{b}$, $H^+\to c\bar{b}$, $H^{\prime+}\to c\bar{s}$ and $H^{\prime +}\to \tau\nu$, we find that the requirement ${\rm BR}(H^+\to\tau\nu)<80$\% is easily satisfied.
In the following, we consider various constraints from flavor physics and low energy observables:
\begin{itemize}
\item{$B_s-\bar{B_s}$ mixing}
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{CLLV_Bsmixing_th.pdf}
\end{center}
\caption{The normalized likelihood as function of the $C_V^{LL}$ coefficient using the Bayesian fit.
\label{fig:cvllfit}}
\end{figure}
The $\bar{s}bZ^{\prime}$ coupling induces a relevant operator to the $B_s-\bar{B_s}$ mixing
\begin{align}
{\cal H}_{\rm eff}=-C_V^{LL}(\bar{s}_L\gamma^{\mu}b_L)(\bar{s}_L\gamma_{\mu}b_L)
\end{align}
with $C^{LL}_V=-\left(g^L_{sb} \right)^2/m^2_{Z^{\prime}}$. We calculate the allowed region of this operator using the measurements of $\Delta M_s$ for $B_s-\bar{B}_s$ mixing in Ref.~\cite{Amhis:2014hma}. In our analysis, the Bayesian fit is also employed. The likelihood function is approximated by Gaussian probability distribution function with mean as difference of the theoretical prediction and experimental central value, and the variance as the the quadratic sum of the theoretical uncertainty and the experimental uncertainty. The prior probability distribution is taken as uniform distribution between $-3\times 10^{-10}$ to $3\times 10^{-10}$. The fitting results are shown in Fig.~\ref{fig:cvllfit}, and it is found that $|C_V^{LL}| < 1.29\times 10^{-11}$ at 1~$\sigma$. Note that, although scalar operators ${\cal O}^{LL}_S$ and ${\cal O}^{RR}_S$ listed in \cite{Buras:2012fs} are caused by neutral and CP-odd scalars in the inert-Higgs doublets, they vanish as long as the two scalar masses are degenerate. Also, we do not consider an operator ${\cal O}^{LR}_S$ since its Wilson coefficient is proportional to $y^{\prime}_s$ that is irrelevant to our current study.
\item{Neutrino trident production}
Neutrino trident production gives an upper limit on $g^L_{\mu\mu}/m_{Z^{\prime}}=v_{\Phi}Y^2_{\mu L_R}/2m^2_L$. Its cross section is given by \cite{Altmannshofer:2014pba, Alok:2017jgr, Alok:2017sui}
\begin{align}
\frac{\sigma_{\rm SM+NP}}{\sigma_{\rm SM}}=\frac{1}{1+(1+4s^2_W)^2}\left[\left\{1+v^2\left(\frac{g^L_{\mu\mu}}{m_{Z^{\prime}}}\right)^2\right\}^2+\left\{1 +4s^2_W+v^2\left(\frac{g^L_{\mu\mu}}{m^2_{Z^{\prime}}}\right)^2 \right\}^2 \right]. \label{NT}
\end{align}
The experimental measurement~\cite{Mishra:1991bv} shows ${\sigma_{\rm exp}}/{\sigma_{\rm SM}}=0.82\pm0.28$. For $m_L=1.4~{\rm TeV}$ and $v_{\Phi}=700~{\rm GeV}$, the $1\sigma$ region gives an upper bound on the Yukawa coupling $|Y_{\mu L_R}|\lesssim2.1$.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth]{bsnunu.pdf}
\end{center}
\caption{The $b\to s \nu\nu$ transition induced by the $Z^{\prime}$ exchanges.
Here the double line represents the vector-like fermions. \label{fig:bsnunu}}
\end{figure}
\item{$B\to K \nu\bar{\nu}$}
In the same way as $b\to s \mu\mu$ process, $b\to s \nu\bar{\nu}$ transition is also induced as shown in Fig.~\ref{fig:bsnunu}. The effective Hamiltonian is \cite{Bhattacharya:2016mcc}
\begin{align}
{\cal H}_{\rm eff}(b\to s\nu_i\bar{\nu}_j)=-\frac{\alpha G_F}{\sqrt{2}\pi}V_{tb}V^*_{ts}C^{ij}_L(\bar{c}\gamma^{\mu} P_Lb)(\bar{\nu}_i\gamma_{\mu}(1-\gamma_5)\nu_j).
\end{align}
Our current setup produces only the 2nd generation neutrino at final states, therefore the Wilson coefficient is described by
\begin{align}
C^{22}_L=-\frac{1}{\sqrt{2}m^2_{Z^{\prime}}}\frac{\pi}{\alpha G_FV_{ts}V^*_{ts}}g^L_{sb}g^L_{\nu_{\mu}\nu_{\mu}},
\end{align}
with $g^L_{\nu_{\mu}\nu_{\mu}}=g^L_{\mu\mu}$.
Following the bound on the Wilson coefficient obtained in \cite{Bhattacharya:2016mcc}, we find that $-13{\rm Re} [C^{22}_L]+|C^{22}_L|^2 \leq 473$.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{bsgamma.pdf}
\end{center}
\caption{$b\to s \gamma$ process through the charged scalar loop. The double lines represent the vector-like fermions $\tilde U$ or $\tilde D$. The left diagram is proportional to $y_b^{\prime}y_c^{\prime}$, while the right diagram contains $y_c^{\prime}y_s^{\prime}$. \label{fig:bsgamma}}
\end{figure}
\item{$b\to s\gamma$}
The Yukawa couplings in Eq. (\ref{eff_Yukawa}) contribute to $b\to s\gamma$ process through the charged scalar loop. In Fig. \ref{fig:bsgamma}, two possible diagrams are drawn. It is seen that while the left diagram contains the product $y^{\prime}_by^{\prime}_c$ related to $C_S$ and $C_S^{\prime}$, the diagram on the right is proportional to $y^{\prime}_cy^{\prime}_s$. Therefore, we include only the left diagram which contains two relevant parameters to $b\to c$ transition. The effective operators are given by \cite{Borzumati:1998tg}
\bea
{\cal H}_{\rm eff}=-\frac{4G_F}{\sqrt{2}}V_{tb}V^*_{ts}\bigg(C_7{\cal O}_7+C_8{\cal O}_8 \bigg)
\eea
with
\bea
{\cal O}_7&=&\frac{e}{16\pi^2}m_b\bar{s}_L\sigma^{\mu\nu}b_RF_{\mu\nu}\\
{\cal O}_8&=&\frac{e}{16\pi^2}m_b\bar{s}_LT^a\sigma^{\mu\nu}b_RG^A_{\mu\nu},
\eea
The Wilson coefficients in our model are
\bea
C_7&=&\frac{v^2}{V_{tb}V_{ts}^*}{\cal Y}_{c_Lb_R}{\cal Y}^*_{c_Rs_L}\left( \frac{C^0_{7,XY}(x_c)}{m_bm_c}+\frac{C^0_{7,YY}(x_c)}{m^2_c}\right),\\
C_8&=&\frac{v^2}{V_{tb}V_{ts}^*}{\cal Y}_{c_Lb_R}{\cal Y}^*_{c_Rs_L}\left( \frac{C^0_{8,XY}(x_c)}{m_bm_c}+\frac{C^0_{8,YY}(x_c)}{m^2_c}\right),
\eea
where $x_c=m^2_c/m^2_{H^{\prime}}$ and the loop functions $C^0_{7,8}$ are listed in \cite{Borzumati:1998tg}. The Wilson coefficients $C_7$ and $C_8$ are proportional to a product of $Y_sY_cy_b^{\prime}y_c^{\prime}$. While $Y_s$ and $Y_c$ are predicted by the explanation of the $R_{K}$ anomaly, $y_b^{\prime}$ and $y^{\prime}_c$ are related to $C^{\prime}_S$ and $C_S$ in Eqs. ({\ref {Csp}}) and ({\ref {Cs}}).
According to a global fit of the Wilson coefficients in the $b\to s$ transition in \cite{Descotes-Genon:2015uva}, contributions from new physics to $C_7(\mu_b=5~{\rm GeV})$ are allowed between $-0.04\leq C_7(\mu_b)\leq 0.0$. Taking that the charged scalar mass is $300$ GeV, we find that the loop functions are roughly $O(10^{-5} \sim 10^{-6})$. In addition, the Wilson coefficients receive suppression proportional to $Y_sY_c\sim10^{-4}$. Later, we will check this constraint numerically. It should be noted that, as in the $B_s-\bar{B}_s$ mixing, the neutral scalars do not contribute to the process due to cancellations between $H^{\prime}$ and $A^{\prime}$ contributions for the degenerate mass spectra in the inert Higgs.
\end{itemize}
In addition to the above constraints, lepton-flavor-violating processes, such as $\tau\to 3\mu$ and $\tau\to\mu\gamma$ can also be induced. However, these processes also need irrelevant parameters to the $B$ anomalies, we do not deal with them here.
Finally, we briefly discuss the hidden particles which are only charged under the $U(1)_H$ group.
Although these hidden particles are not relevant to the $B$ physics observables, it provides additional signatures.
We assume that there is a WIMP dark matter candidate in the hidden sector, and the dark matter particle $\chi$
should annihilate into SM particles during freeze-out at early universe.
In the non-relativistic approximation,
the annihilation cross section times relative velocity $\langle \sigma v \rangle$ can be decomposed as
\bea
\label{eq:xsecvab}
\langle \sigma v \rangle = a + b v^2 + {\cal O}(v^4),
\eea
{{where $a$ and $b$ are the $s$-wave and $p$-wave cross sections. }}
We only take the dominant contributions into account.
When $m_\chi < m_{Z'}$, the dominant annihilation channel is through the $s$-channel process $\bar\chi \chi \to f\bar f$, where $ f = \mu , \nu_\mu, b, t$:
\bea
\langle \sigma v \rangle_{\rm s-channel}(\bar\chi \chi \to f\bar f)
= \frac{g'^2 g_{Z'ff}^2}{2\pi} \frac{2 m_\chi^2 + m_f^2}{4 m_\chi^2 - m_{Z'}^2} \sqrt{1 - \frac{m_f^2}{m_\chi^2}},
\eea
where $g_{Z'ff}$ denotes the $Z'$ coupling to the fermion $f$.
When $m_\chi > m_{Z'}$, additional $t$-channel $\bar\chi \chi \to Z' Z' \to f\bar f f\bar f$ {{appears}}. The annihilation cross section is
\bea
\langle \sigma v \rangle_{\rm t-channel}(\bar\chi \chi \to Z' Z')
= \frac{g'^4}{4\pi } \frac{m_\chi^2 }{(2 m_\chi^2 - m_{Z'}^2)^2} \left(1 - \frac{m_{Z'}^2}{m_\chi^2} \right)^{3/2}.
\eea
The thermal relic density is written as
\bea
\Omega_{\rm DM} h^2 = \frac{m_\chi n_\chi}{\rho_{\rm crit}/h^2} = \frac{ s_0 h^2}{\sqrt{\frac{\pi }{45}}\rho_{\rm crit}} \frac{1} {M_{\rm{pl}} \sqrt{g_{\rm eff}} {\mathcal I}(x_f) },
\eea
where $s_0$ is the entropy density of the present universe, $\rho_{\rm crit}$ the critical density,
$n_\chi$ the dark matter number, $h$ the hubble parameter, $g_{\rm eff}$ the effective degree of freedom during freeze-out, and ${\mathcal I}(x_f) = \frac{a + 3 b/x_f}{x_f}$, with $x_f = m_\chi/T_f$ and $T_f$ the temperature during freeze-out.
There are also constraints from the direct detection measurements. However, we expect the limit is not so tight for the light $Z'$, because the size of the gauge coupling $g'$ is typically smaller than $0.1$.
Furthermore, in our model the dark matter only dominantly couples to the third generation quarks inside nucleons. This kinds of scenarios
have been studied in Ref.~\cite{Sage:2016uxt}. From their study, we find that for the light $Z'$, if $g' \sim 0.05$, the direct detection constraints could be escaped.
We will perform detailed study on the dark matter sector in future.
\section{Results}
\label{sec:results}
\begin{figure}
\begin{center}
\includegraphics[width=0.43 \textwidth]{RK_anomaly.pdf}
\includegraphics[width=0.395 \textwidth]{RK_anomaly_ym.pdf}
\end{center}
\caption{[Left] $1\sigma$ and $2\sigma$ regions of $C_{9, \rm new}(=-C_{10,\rm new})$ are drawn with black solid and dashed lines, respectively. $v_{\Phi}$ is fixed at $700~$GeV. The best fitted value for the $R_K$ anomaly is indicated by the green line, and the red line represents $-X_bX_s =2.3\times 10^{-3}$. The blue region is excluded by the $B_s-\bar{B_s}$ mixing. [Right] The size of $C_{9, \rm new}$ as a function of $Y_{\mu}$ with the fixed value $-X_bX_s=2.3\times 10^{-3}$. We take $m_Q=m_L=1.4~$TeV.}
\label{fig:RK_anomaly}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.4 \textwidth]{B0K0mumu.pdf}
\includegraphics[width=0.4 \textwidth]{B0Kstarmumu.pdf}\\
\includegraphics[width=0.4 \textwidth]{BpKpmumu.pdf}
\includegraphics[width=0.4 \textwidth]{BpKstarmumu.pdf}
\end{center}
\caption{The $B$ decay branching ratios ${\rm dBR}(B^0\to K^0 \mu^+ \mu^-)/{ dq^2}$ (upper left), ${\rm dBR}(B^0\to K^{*0} \mu^+ \mu^-)/{dq^2}$ (upper right), ${\rm dBR}(B^\pm\to K^\pm \mu^+ \mu^-)/{dq^2}$ (lower left), ${\rm dBR}(B^+\to K^{*+} \mu^+ \mu^-)/{dq^2}$ (lower right) for different energy bins. The experimental data are expressed by dots with uncertainties. The purple and green bands represent the theoretical predictions of the SM and our model with the best fitted $C_9$, respectively. The theoretical uncertainties are calculated by varying the input parameters with the Gaussian uncertainties.}
\label{fig:RK_data}
\end{figure}
We first consider the effective operators for the $R_{K}$ anomaly in terms of $X_q \equiv v_{\Phi}Y_{q}/\sqrt{2}m_{Q}~(q=b,s)$ and $X_\mu \equiv v_{\Phi}Y_{\mu}/\sqrt{2}m_{L}$. With these expressions, the Eq. (\ref{C9_new}) is described by
\begin{align}
C_{9, \rm new}(=-C_{10, \rm new})=-\frac{1}{2v^2_{\Phi}|C^{bs}_{\rm SM}|}X_bX_sX^2_{\mu}.
\end{align}
This expression consists of only relevant model parameters through the combinations of $X_q$ and $X_{\mu}$ that are related to the mixing angles in the quark and lepton sectors. It should be noted that the value of $C_{9, \rm new}$ in Table. \ref{table:model_analysis} is positive, therefore either $X_b$ or $X_s$ should be negative to compensate for the minus sign in the above expression. The left plot in Fig. \ref{fig:RK_anomaly} shows $1$$\sigma$ and $2$$\sigma$ regions with black solid and dashed lines in $(-X_bX_s,~X_{\mu}^2)$ plane with the fixed value of $v_{\Phi}=700~$GeV.
The best fitted value is drawn by a green line, and the blue region is excluded by $B_s-\bar{B_s}$ mixing. We also consider the constraint from the $B\to K\nu\nu$ process, however, the present parameter region can evade the restriction. It is seen that $X_bX_s$ is tightly constrained by the $B_s-\bar{B_s}$ mixing and the allowed size is roughly $O(10^{-3})$ at most. And, it follows that a somewhat large value of $X_{\mu}^2\sim O(0.1)$ is necessary to achieve the required Wilson coefficients for the $R_{K}$.
Such a large value can be obtained by taking a large $Y_{\mu}$ or a small $m_L$. Here, we fix the mass $m_L$ at 1.4 TeV and see how large Yukawa coupling, $Y_{\mu}$, is needed at one benchmark region of $-X_bX_s=2.3\times 10^{-3}$. The region is indicated by a red line in the left panel in Fig. \ref{fig:RK_anomaly}, and the benchmark value results in $Y_bY_s=-0.0184$ if $m_Q=m_L$. Since our assumption is $|Y_b|\gg |Y_s|$, we simply take $Y_b=-1$ and $Y_s=0.0184$. In the right plot of Fig. \ref{fig:RK_anomaly}, the size of $C_{9,\rm new}$ is shown as a function of $Y_{\mu}$. Here, instead of the mass-insertion method, we use $g^L_{sb}$ and $g^L_{\mu\mu}$ obtained after diagonalization of the mass matrices. All lines in the right figure correspond to those in the left one. The $2\sigma$ region requires $1.2\leq Y_{\mu}\leq 1.9$, and the best fitted value stays around $Y_{\mu}\sim 1.5$. The values of $Y_b=-1,~Y_s=0.0184$ and $Y_{\mu}=1.5$ yield $g^L_{sb}/g^{\prime}=-2.17\times 10^{-3}$ and $g^L_{\mu\mu}/g^{\prime}=0.22$, which leads to $C_{9,\rm new}=0.6$. The current size of $g^L_{\mu\mu}/m_{Z^{\prime}}$ gives the cross section of the neutrino trident production in Eq. (\ref{NT}), $\sigma_{\rm SM+NP}/\sigma_{\rm SM}=1.0$, which is within the $1\sigma$ error. Later, we also use these Yukawa couplings as the benchmark values for the discussion of the $R_D$ anomaly.
Figure~\ref{fig:RK_data} shows the experimental data and theoretical predictions on the $B$ decay branching ratios in different energy bins for ${\text{dBR}}(B^0\to K^0 \mu^+ \mu^-)/{dq^2}$ (upper left), ${\rm dBR}(B^0\to K^{*0} \mu^+ \mu^-)/{dq^2}$ (upper right), ${\rm dBR}(B^\pm\to K^\pm \mu^+ \mu^-)/{dq^2}$ (lower left), ${\rm dBR}(B^+\to K^{*+} \mu^+ \mu^-)/{dq^2}$ (lower right). The experimental results obtained by CDF~\cite{CDF:2012qwd}, LHCb~\cite{Aaij:2016flj}, CMS~\cite{Chatrchyan:2013cda,Khachatryan:2015isa} are plotted.
The purple bands correspond to the SM predictions with uncertainties obtained by varying the input parameters with the Gaussian distribution. The green bands represent the best fitted value of $C_{9,\rm new}$, which is indicated by the green line in Fig. \ref{fig:RK_anomaly}, with the uncertainties obtained by the same way as the purple bands. The width of the purple and green bands are not the same, and their ratios are roughly proportional to $|C^{bs}_{\rm SM}/(C^{bs}_{\rm SM}+C_i)|^2~(i=9,10)$. It is seen that our model provides a better fit on the data than the SM.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.4 \textwidth]{RD_anomaly.pdf}
\end{center}
\caption{$1\sigma$ and $2\sigma$ regions of $C_S^{\prime}$ with black solid and dashed lines, respectively. It is taken that $m_{H'}=300~$GeV, and the benchmark parameters
for the $R_K$ anomaly: $v_{\Phi}=700~$GeV,~$m_Q=m_L=1.4~$TeV, $Y_b=-1,~Y_s=0.018$ and $Y_{\mu}=1.5$.}
\label{fig:RD_anomaly}
\end{figure}
Finally, we discuss the possibility of explaining the $R_{D}$ anomaly using the parameter space which could explain the $R_K$ anomaly.
Once $Y_b$ and $Y_s$ are fixed, $Y_c$ is also given due to the $SU(2)$ symmetry. Using the benchmark values, $Y_b=-1,~Y_s=0.018$, we find that $Y_c=-0.02$ where the $Y_b$ part dominantly gives a contribution to $Y_c$ since $(V_{\rm CKM})_{cb} Y_b>Y_s$. As seen in Fig. \ref{fig:cscsp}, positive $C_S^{\prime}$ is disfavored by the upper limit on the $B^-_c\to \tau^- \bar{\nu}$ process, and the absolute values of $C_S$ and $C_S^{\prime}$ should be $O(0.1)$. In the current setup, the negative $C_S^{\prime}$ can be realized by $Y_b=-1$. However, a naive estimation implies that $C_S^{\prime} \gg C_S$ since $C_S$ is proportional to $Y_c$ which is two orders of magnitude smaller than $Y_b$. Therefore, we exclusively focus on a specific situation with $C_S^{\prime}\neq 0$ but $C_S=0$ as listed in Table \ref{table:model_analysis}.
Figure \ref{fig:RD_anomaly} shows the $1\sigma$ and $2\sigma$ regions of $C_S^{\prime}$ in $(y^{\prime}_c,y^{\prime}_{\tau})$ plane. Some of the model parameters
are correlated with the $R_K$ anomaly, and they are fixed at $Y_b=-1,~Y_s=0.018,~Y_{\mu}=1.5,~v_{\phi}=700~$GeV and $m_Q=m_L=1.4~$TeV. The charged scalar mass is taken as $300~$GeV. As we can expect from the required value of $C_S^{\prime}$, it is found that the large Yukawa couplings $y_c^{\prime}$ and $y_{\tau}^{\prime}$ are needed for the explanation of the $R_D$ anomaly. As long as we focus on only $C_S^{\prime}$, the size of $y_b^{\prime}$ is not predicted. Therefore, our current setup does not affect $b\to s\gamma$ process. However, it is found that,
even if we take $y_c^{\prime}=2$ and $y_b^{\prime}=1$, $C_7(\mu_b)$ roughly becomes $1.6\times 10^{-3}$ which is consistent with the current observed value.
\section{Conclusion}
\label{sec:conclude}
We have studied a hidden gauged $U(1)_H$ extension of the SM including vector-like fermion (and scalar) mediators and investigated possibilities of explanations for the existing $R_K$ (and $R_D$) anomalies in $B$ physics.
The vector-like fermions have the same SM charges as the SM doublet fermions, and they mix with the left-handed SM fermions.
The new scalar is an inert doublet, which possesses the Yukawa interaction between the left-handed vector-like fermions and the right-handed SM fermions.
While the vector-like fermions play a role in inducing $b\to s \mu\mu$ transition through the $Z^{\prime}$ exchange, the inert doublet yields significant charged scalar couplings for $b\to c\tau\nu$ transition.
In order to determine desired sizes of the Wilson coefficients, we utilized the flavio package and performed a global Bayesian fit including several observables, $B^{0, \pm}\to K^{*0,\pm}\mu\mu, B_s\to\phi\mu\mu, B_s\to X_s\mu\mu $ and $ B_s\to \mu\mu$. At the same time, we also considered various flavor constraints, $B_s-\bar{B_s}$ mixing, $B \to K \nu\bar{\nu}$, neutrino trident productions, $b \to s \gamma$, and $B^-_c\to\tau^-\bar{\nu}$. It is found that the absolute values of $C_9 (= - C_{10})$, $C_{S}^{\prime}$ and $C_{S}$ are required to be $O(0.5)$ roughly. Also, it turned out that the positive values of $C_{S}^{\prime}$ are almost excluded by $B^-_c\to\tau^- \bar{\nu}$ if we impose its severe limit on the branching ratio, ${\rm BR}(B^-_c\to\tau^-\bar{\nu})<10\%$.
The required sizes of $C_9$ and $C_{10}$ can be achieved by somewhat large $Y_\mu$, which describes the mixing parameter between the vector-like and SM leptons, since those in the quark sector $Y_b$ and $Y_s$ are highly constrained by the $B_s-\bar{B}_s$ mixing. Our benchmark point with $m_Q=m_L=1.4~$TeV, $v_{\Phi}=700~$GeV and $Y_bY_s=-0.018$ needs $Y_{\mu} \sim 1.5$ for the successful scenario of the $R_K$ anomaly. This benchmark leads to a consistent situation in which $C_S$ is suppressed due to tiny $Y_c$ and negative $C_S^{\prime}$ is obtained by taking $Y_b<0$. The possible region of $C_S^{\prime}$ requires $O(1)$ couplings of $y_c^{\prime}$ and $y_{\tau}^{\prime}$.
Since this model contains new particles, there are promising collider and astrophysical signatures. Current LHC searches on $pp \to \mu\bar{\mu}$ and $pp \to 4 \mu$ put strong constraints on the $Z'$ masses, although the light region of $M_{Z'} < 100$ GeV is still allowed. Furthermore, since the $U(1)$ gauge symmetry is hidden, it is likely that other hidden particles also exist, such as dark matter candidate, in the hidden sector.
Although the hidden sector particles do not contribute to the $B$ physics observables,
they are able to contribute to the dark matter relic density and possible indirect detections,
and would relax the constraints on the heavy $Z'$. On the other hand,
the charged Higgs is not tightly constrained due to its Yukawa texture.
We anticipate that the future collider searches will explore larger mass regions for the $Z'$ and charged Higgs, and these collider signatures and
potential dark matter signatures could be able to validate this model.
\renewcommand{\arraystretch}{0.7}
\begin{table}[h]
\begin{center}
\small\begin{tabular}{||c|c|c||}
\hline\hline
\multicolumn{3}{|c|}{\textbf{Branching ratios}} \\
\hline\hlin
\cline{1-3}
{{Observable}} & {{$[q^{2}_{\text{min}},q^{2}_{\text{max}}]$ [GeV$^{2}$]}} & {{Experiments}} \\ \hline
$\frac{d}{dq^{2}}\text{BR}(B\to X_{s} \mu\mu)$ & [1, 6],[14.2, 25] & BaBar~\cite{Lees:2013nxa} \\
\hline
\multirow{2}{*}{$\frac{d}{dq^{2}}\text{BR}(B^{+}\to K^{*+}\mu\mu)$} & [0.0, 2.0], [2.0, 4.3]&CDF~\cite{CDF:2012qwd}\\ \cline{2-3}
&[2,4], [4,6], [15,19] &LHCb~\cite{Aaij:2014pli}\\ \hline
\multirow{2}{*}{$\frac{d}{dq^{2}}\text{BR}(B^{\pm}\to K^{\pm}\mu\mu)$} & [0.0, 2.0], [2.0, 4.3]&CDF~\cite{CDF:2012qwd}\\ \cline{2-3}
&[1.1, 2], [2, 3], [3, 4], [4, 5], [15,22] &LHCb~\cite{Aaij:2014pli}\\ \hline
\multirow{3}{*}{$\frac{d}{dq^{2}}\text{BR}(B^{0}\to K^{*0}\mu\mu)$} & [0.0, 2.0], [2.0, 4.3]&CDF~\cite{CDF:2012qwd}\\ \cline{2-3}
&[1.1, 2.5], [2.5, 4], [4, 6], [15, 19] &LHCb~\cite{Aaij:2016flj}\\
\cline{2-3}
&[1, 2], [2, 4.3] & CMS~\cite{Chatrchyan:2013cda,Khachatryan:2015isa}\\
\hline
\multirow{2}{*}{$\frac{d}{dq^{2}}\text{BR}(B^{0}\to K^{0}\mu\mu)$} & [0.0, 2.0], [2.0, 4.3]&CDF~\cite{CDF:2012qwd}\\ \cline{2-3}
&[2.5, 4], [4, 6], [15,22] &LHCb~\cite{Aaij:2014pli}\\ \hline
\multirow{2}{*}{$\frac{d}{dq^{2}}\text{BR}(B_s\to \phi\mu\mu)$} & [1, 6] &CDF~\cite{CDF:2012qwd}\\ \cline{2-3}
&[1, 6], [15, 19]&LHCb~\cite{Aaij:2015esa}\\ \hline
$R_{K^*}$ & [0.045, 1.1], [1.1, 6.0] & LHCb~\cite{Aaij:2017vbb}\\
\hline
$R_{K}$ & [1.0, 6.0] & LHCb~\cite{Aaij:2014ora}\\
\hline
$BR(B_s\to\mu\mu),~ BR(B^0\to \mu\mu)$ & -----& LHCb\cite{Tolk:2017dqr}\\
\hline
\multicolumn{3}{|c|}{\textbf{Angular Observables}} \\
\hline\hline
\multirow{3}{*}{$\langle A_{FB} \rangle (B^{0}\to K^{*0}\mu\mu)$} & [0.0, 2.0], [2.0, 4.3]& CDF~\cite{CDF:2012qwd}\\ \cline{2-3}
&[1.1, 2.5], [2.5, 4], [4, 6], [15, 19]&LHCb~\cite{Aaij:2015oid}\\
\cline{2-3}
&[1, 2], [2, 4.3], [4.3, 6] & CMS~\cite{CMS:2017ivg,Chatrchyan:2013cda,Khachatryan:2015isa}\\
\hline
\multirow{4}{*}{$\langle F_{L} \rangle (B^{0}\to K^{*0}\mu\mu)$} & [0.0, 2.0], [2.0, 4.3]& CDF~\cite{CDF:2012qwd}\\ \cline{2-3}
&[1.1, 2.5], [2.5, 4], [4, 6], [15, 19]&LHCb~\cite{Aaij:2015oid}\\
\cline{2-3}
&[1, 2], [2, 4.3], [4.3, 6] & CMS~\cite{CMS:2017ivg,Chatrchyan:2013cda,Khachatryan:2015isa}\\
\cline{2-3}
& [0.04, 2.0], [2.0, 4.0], [4.0, 6.0]& ATLAS~\cite{ATLAS:2017dlm}\\
\hline
\multirow{2}{*}{$\langle S_{3} \rangle (B^{0}\to K^{*0}\mu\mu)$}
&[1.1, 2.5], [2.5, 4], [4, 6], [15, 19]&LHCb~\cite{Aaij:2015oid}\\
\cline{2-3}
&[1, 2], [2, 4.3], [4.3, 6] & CMS~\cite{CMS:2017ivg,Chatrchyan:2013cda,Khachatryan:2015isa}\\
\hline
\multirow{2}{*}{$\langle S_{4} \rangle (B^{0}\to K^{*0}\mu\mu)$}
&[1.1, 2.5], [2.5, 4], [4, 6], [15, 19]&LHCb~\cite{Aaij:2015oid}\\
\cline{2-3}
&[1, 2], [2, 4.3], [4.3, 6] & CMS~\cite{CMS:2017ivg,Chatrchyan:2013cda,Khachatryan:2015isa}\\
\hline
\multirow{2}{*}{$\langle S_{5} \rangle (B^{0}\to K^{*0}\mu\mu)$}
&[1.1, 2.5], [2.5, 4], [4, 6], [15, 19]&LHCb~\cite{Aaij:2015oid}\\
\cline{2-3}
&[1, 2], [2, 4.3], [4.3, 6] & CMS~\cite{CMS:2017ivg,Chatrchyan:2013cda,Khachatryan:2015isa}\\
\hline
\multirow{3}{*}{$\langle P_{1} \rangle (B^{0}\to K^{*0}\mu\mu)$} & [0.04, 2.0], [2.0, 4.0], [4.0, 6.0]& ATLAS~\cite{ATLAS:2017dlm}\\ \cline{2-3}
&[1.1, 2.5], [2.5, 4], [4, 6], [15, 19]&LHCb~\cite{Aaij:2015oid}\\
\cline{2-3}
&[1, 2], [2, 4.3], [4.3, 6] & CMS~\cite{CMS:2017ivg,Chatrchyan:2013cda,Khachatryan:2015isa}\\
\hline
\multirow{2}{*}{$\langle P'_{4} \rangle (B^{0}\to K^{*0}\mu\mu)$} & [0.04, 2.0], [2.0, 4.0], [4.0, 6.0]& ATLAS~\cite{ATLAS:2017dlm}\\ \cline{2-3}
&[1.1, 2.5], [2.5, 4], [4, 6], [15, 19]&LHCb~\cite{Aaij:2015oid}\\
\cline{2-3}
\hline
\multirow{3}{*}{$\langle P'_{5} \rangle (B^{0}\to K^{*0}\mu\mu)$} & [0.04, 2.0], [2.0, 4.0], [4.0, 6.0]& ATLAS~\cite{ATLAS:2017dlm}\\ \cline{2-3}
&[1.1, 2.5], [2.5, 4], [4, 6], [15, 19]&LHCb~\cite{Aaij:2015oid}\\
\cline{2-3}
&[1, 2], [2, 4.3], [4.3, 6] & CMS~\cite{CMS:2017ivg,Chatrchyan:2013cda,Khachatryan:2015isa}\\
\hline
$\langle \overline{F_L} \rangle (B^{0}\to K^{*0}\mu\mu)$ & &\\
$\langle \overline{S_3} \rangle (B^{0}\to K^{*0}\mu\mu)$& [2, 5], [15, 19] &LHCb~\cite{Aaij:2015oid} \\
$\langle \overline{S_4} \rangle (B^{0}\to K^{*0}\mu\mu)$ & &\\
\hline
\end{tabular}
\end{center}
\caption{List of observable used in global fit of $R_{K^{(*)}}$ anomaly.
\label{tab:rk}}
\end{table}%
\normalsize
\begin{acknowledgments}
HL thanks David Straub for helpful discussions about flavio and Aniket Jorglekar for valuable discussion on the diagonalization of fermion mass matrix. JHY and KF are supported by DOE Grant DE-SC0011095.
\end{acknowledgments}
|
1,116,691,497,775 | arxiv | \section{Introduction}
In this paper we study the problem of designing state feedback controllers for linear time invariant systems with probabilistic system parameters. Such systems in continuous time can be defined as
\begin{align}
\dot{\x} &= \A(\param)\x + \B(\param)\u, \eqnlabel{contiDyn}
\end{align}
where $\param\in\mathbb{R}^d$ is a vector of uncertain parameters, with joint probability density function $\pdfp$. Matrices $\A(\param)\in\mathbb{R}^{n\times n}$, $\B(\param) \in\mathbb{R}^{n\times m}$ are system matrices that depend on $\param$. Consequently, the solution $\x:=\x(t,\param)\in\mathbb{R}^n$ also depends on $\param$. The objective is to design a state-feedback law in the form $\u = \K\x$, which stabilizes the system in some suitable sense, where $\K\in\mathbb{R}^{m\times n}$. Thus, we are looking to obtain a constant deterministic gain $\K$ that stabilizes the systems in \eqn{contiDyn}, with probabilistic uncertainty in $\param$.
Stability of dynamical systems of the type
\begin{align}
\dot{\x} = \A(\param)\x, \eqnlabel{simpleSys}
\end{align}
have been extensively studied in the framework of stochastic dynamical systems. Depending on the nature of $\param$, two approaches are commonly used. If $\param$ is Gaussian white noise, the solution process is a diffusion process and is analyzed using theory of Markov processes \cite{dynkin2012theory,bharucha2012elements} and Ito calculus \cite{itocalc}. If $\param$ is not Gaussian white noise, and has well defined samples properties, the system in \eqn{simpleSys} can be analyzed by ordinary rules of calculus. Such systems can be considered to be a collection of ordinary deterministic differential equations upon which a probability measure has been induced by the parameter process $\param$. The second class of systems are often what we actually encounter in engineering problems. The Gaussian white noise case is a mathematical abstraction and admits richer set of analysis tools. However, this should not be the motivation for assuming white noise process for systems with random parameters as it is a non-trivial matter and should be considered carefully \cite{eugene1965relation}.
Early work on stability analysis of linear systems with randomly time-varying parameters was done by Rosenbloom \cite{rosenbloom1954analysis}, where he studied first order linear systems with stationary Gaussian parameter process. His work focussed on stability properties in terms of the moments. Due to the stationary properties of $\Delta$, the moments and hence the asymptotic properties of system could be studied explicitly. This work was extended by Bertram and Sarachik \cite{bertram1959stability} for linear diagonal systems and were first to provide a Lyapunov framework for studying such systems with random parameters. The parameters were also considered to be Gaussian and recovered Rosebloom's result, with a more general approach. They also considered the problem where $\param$ is random but piecewise constant. At the same time, Kats and Krosovskii \cite{kats1960stability} independently provided a Lyapunov framework for analyzing stability in probability and moments, for systems where the $\param$ process is a stationary Markov process with finite number of states. Bharucha \cite{bharucha1961stability} extended the work by Kats \textit{et al. } by exploiting the fact that a stationary Markov parameter process admits a piecewise constant property in the linear system. Bharucha also showed that for such systems, asymptotic stability implies exponential stability. Palmer \cite{palmer1966sufficient} significantly extended the work by Kats \textit{et al. } and Bharucha on linear systems with Markov coefficients by applying Markov chain theory and exploiting the induced piecewise constant property. The piecewise constant property was also exploited in the works of Morozan \cite{morozan1967stability} and Soeda \textit{et al. } \cite{soeda1966stability}. The above described literature focussed on systems for which the solution can be obtained in closed form. More general linear systems of the form $\dot{\x} = (\A_0 + \A_1(\param))\x$, where $\A_0$ is a stability matrix and $\A_1(\param)$ is matrix whose non-zero coefficients are stationary, ergodic processes with almost surely continuous sample functions, were studied by Kozin \cite{kozin1963almost}. He provided sufficient conditions for asymptotic stability with probability one, for such systems. This work was refined further \cite{caughey1965almost,ariaratnam1967dynamic, morozan1967stability,wang1966almost, infante1968stability, gray1967frequency}. For related work when $\param$ is Gaussian white noise, please refer to \cite{khasminskii2011stochastic, kushner} and references therein.
In this paper, we focus on systems where $\param$ is a vector of random variables, which is a simpler problem than randomly time-varying parameters. The problem of analyzing systems with uncertain, but constant, parameters have also been addressed in the robust control literature. In that approach, the support of $\param$ is assumed to be polytopic, and stability is analyzed for parameter combinations along the vertices of the polytope \cite{bernussou1989linear,Boydetal:1994, boyd1989structured}. This is the so called ``worst-case'' approach for stability analysis. In this paper, we present new stability analysis and control design methods, which ensure exponential mean square stability for systems with probabilistic system parameters. These are developed in the polynomial chaos framework \cite{wiener}. A comparison with worst-case quadratic stability approach \cite{corless1994robust}, for systems with uniformly distributed $\param$ \cite{barmish3} is also performed, which highlights the conservativeness of the worst-case approach.
The paper is organized as follows. First, we present a brief background on polynomial chaos theory and its application to linear systems with random parameters. This is followed by the main contribution of this paper, captured in propositions 1, 2, and 3; and theorems 1, 2, and 3. For benchmarking, we also present randomized algorithms for system analysis and design. We demonstrate the computational efficiency of polynomial chaos framework with an example based on an F-16 aircraft model. The paper ends with a summary section.
\section{Polynomial Chaos Theory}
Polynomial chaos is a non-sampling based method to determine evolution of uncertainty in dynamical system with probabilistic system parameters \cite{pcFEM}. Very briefly, a general second order process $X(\omega)\in
\mathcal{L}_2(\Omega,\mathcal{F},P)$ can be expressed by polynomial
chaos as
\begin{equation}
\eqnlabel{gPC}
X(\omega) = \sum_{i=0}^{\infty} x_i\phi_i({\param}(\omega)),
\end{equation}
where $\omega$ is the random event and $\phi_i({\param}(\omega))$
denotes the polynomial chaos basis of degree $p$ in terms of the random variables
$\param(\omega)$. $(\Omega,\mathcal{F},P)$ is a probability space, where $\Omega$
is the sample space, $\mathcal{F}$ is the $\sigma$-algebra of the
subsets of $\Omega$, and $P$ is the probability measure. According to Cameron and Martin \cite{CameronMartin} such an expansion converges in the $\mathcal{L}_2$ sense for any arbitrary stochastic process with finite second moment. In practice, the infinite series is truncated and $X(\omega)$ is approximated by
\[
X(\omega) \approx \hat{X}(\omega) = \sum_{i=0}^{N} x_i\phi_i({\param}(\omega)).
\] The functions $\{\phi_i\}$ are a family of
orthogonal basis in $\mathcal{L}_2(\Omega,\mathcal{F},P)$ satisfying
the relation
\begin{equation}
\Exp{\phi_i\phi_j}:= \int_{\mathcal{D}_{\param}}\hspace{-0.1in}{\basis{i}{\param}\basis{j}{\param} \pdfp
\,d\param} = h_i^2\delta_{ij}, \eqnlabel{basisFcn}
\end{equation}
where $\delta_{ij}$ is the Kronecker delta, $h_i$ is a constant
term corresponding to $\int_{\mathcal{D}_{\param}}{\phi_i^2\pdfp\,d\param}$,
$\mathcal{D}_{\param}$ is the domain of the random variable $\param(\omega)$, and
$\pdfp$ is a probability density function for $\param$.
Polynomial chaos theory is becoming an useful framework to study control systems with random parameters \cite{hover2006application,pctrajgen,fisher2009linear,bhattacharya2012linear,kim2012generalized,ulissi2013control,templeton2012probabilistic}.
\subsection{Application to Dynamical Systems with Random Parameters}
With respect to the dynamical system defined in \eqn{contiDyn}, the solution can be approximated by the polynomial chaos expansion as
\begin{align}
\x(t,\param) \approx \hat{\x}(t,\param) = \sum_{i=0}^N \x_i(t)\basis{i}{\param},
\end{align}
where the polynomial chaos coefficients $\x_i \in \mathbb{R}^n$. Define $\mo{\Phi}(\param)$ to be
\begin{align}
\mo{\Phi} &\equiv \mo{\Phi}(\param) := \begin{pmatrix}\basis{0}{\param} & \cdots & \basis{N}{\param}\end{pmatrix}^T, \text{ and } \\
\mo{\Phi}_n &\equiv \mo{\Phi}_n(\param) := \mo{\Phi}(\param) \otimes \I{n},
\end{align}
where $\I{n}\in\mathbb{R}^{n\times n}$ is identity matrix. Also define matrix $\X\in\mathbb{R}^{n\times(N+1)}$, with polynomial chaos coefficients $\x_i$, as
\[ \X = \begin{bmatrix} \x_0 & \cdots & \x_N \end{bmatrix}.\]
This lets us define $\hat{\x}(t,\param)$ as
\begin{align}
\hat{\x}(t,\param) := \X(t)\mo{\Phi}(\param) \eqnlabel{compactX}.
\end{align}
Noting that $\hat{\x} \equiv \vec{\hat{\x}}$, we obtain an alternate form for \eqn{compactX},
\begin{align}
\hat{\x} \equiv \vec{\hat{\x}} & = \vec{\X\mo{\Phi}} = \vec{\I{n}\X\mo{\Phi}} \nonumber \\
& = (\mo{\Phi}^T\otimes \I{n})\vec{\X} = \mo{\Phi}_n^T\xpc, \eqnlabel{compactxpc}
\end{align}
where $\xpc := \vec{\X}$, and $\vec{\cdot}$ is the vectorization operator \cite{horn2012matrix}.
Since $\hat{\x}$ from \eqn{compactxpc} is an approximation, substituting it in \eqn{simpleSys} we get equation error $\vo{e}$, which is given by
\begin{align}
\vo{e} &:= \dot{\hat{\x}} - \A(\param)\hat{\x}\\
& = \mo{\Phi}_n^T\xpcdot - \A(\param)\mo{\Phi}_n^T\xpc.
\end{align}
Best $\set{L}_2$ approximation is obtained by setting
\begin{align}
\inner{\vo{e}\phi_i} := \Exp{\vo{e}\phi_i}=0, \text{ for } i = 0,1,\cdots,N.
\end{align}
Upon simplification, we get a set of $n(N+1)$ \textit{deterministic} ordinary differential equations in $\xpc$,
\begin{align}
\xpcdot = \Exp{\mo{\Phi}\otimes\mo{\Phi}_n^T}^{-1}\Exp{\mo{\Phi}\otimes\left(\A\mo{\Phi}_n^T\right)}\xpc. \eqnlabel{pcDynamics}
\end{align}
Using the following properties of Kronecker product
\begin{align*}
A\otimes (B \otimes C) &= (A\otimes B) \otimes C, \\
(A\otimes B)(C\otimes D) &= (AC)\otimes(BD),
\end{align*}
we have the following propositions.\\[5mm]
\begin{proposition} For $\mo{\Phi}_n$, and $\mo{\Phi}$ as defined earlier
\begin{equation}
\mo{\Phi}\otimes\mo{\Phi}^T_n = (\mo{\Phi}\mo{\Phi}^T)\otimes \I{n}. \eqnlabel{id1}
\end{equation}
\end{proposition}
\begin{proof}
\begin{align*} \mo{\Phi}\otimes\mo{\Phi}^T_n &= \mo{\Phi}\otimes(\mo{\Phi}^T\otimes \I{n}) = (\mo{\Phi}\otimes\mo{\Phi}^T)\otimes \I{n} \\
&= \left[(\mo{\Phi} \cdot 1) \otimes (1 \cdot \mo{\Phi}^T)\right] \otimes \I{n}\\
&= \left[(\mo{\Phi} \otimes 1)(1 \otimes \mo{\Phi}^T)\right] \otimes \I{n} \\
&= (\mo{\Phi} \mo{\Phi}^T)\otimes \I{n}
\end{align*}
\end{proof}
\begin{corollary}
\begin{equation}
\Exp{\left(\mo{\Phi}\mo{\Phi}^T\right)\otimes \I{n}} = \boldsymbol{\mathsf{diag}}\left(\Exp{\phi_i^2}\right)\otimes \I{n}. \eqnlabel{cor1}
\end{equation}
\end{corollary}
\begin{proof} Straight forward using \eqn{basisFcn}.\end{proof}
\begin{proposition}
For any matrix $\A \in \mathbb{R}^{n\times n}$ and $\mo{\Phi}_n$, $\mo{\Phi}$ as defined earlier
\begin{equation}
\mo{\Phi} \otimes \left(\A \mo{\Phi}_n^T\right) = \left(\mo{\Phi}\mo{\Phi}^T\right) \otimes \A. \eqnlabel{id2}
\end{equation}
\end{proposition}
\begin{proof}
\begin{align*}
\mo{\Phi} \otimes \left(\A \mo{\Phi}_n^T\right) & = \left(\I{N+1}\mo{\Phi}\right) \otimes \left(\A \mo{\Phi}_n^T\right) \\
& = (\I{N+1} \otimes \A)(\mo{\Phi}\otimes\mo{\Phi}_n^T) \\
& = (\I{N+1} \otimes \A)\left((\mo{\Phi}\mo{\Phi}^T)\otimes \I{n}\right) \\
& = (\I{N+1} \mo{\Phi}\mo{\Phi}^T) \otimes (\A \I{n})\\
& = (\mo{\Phi}\mo{\Phi}^T) \otimes \A.
\end{align*}
\end{proof}
With \eqn{id1} and \eqn{id2}, we can write \eqn{pcDynamics} as
\begin{align}
\xpcdot &= \Apc \xpc, \eqnlabel{pcDynamics:final}
\end{align}
where
\begin{align}
\Apc & = \Exp{\left(\mo{\Phi}\mo{\Phi}^T\right)\otimes \I{n}}^{-1} \Exp{\left(\mo{\Phi}\mo{\Phi}^T\right) \otimes \A}, \eqnlabel{Apc}
\end{align}
and $\mo{\Phi}$, $\A$ depend on $\param$ as defined earlier.
Let us also introduce notations $\mo{\bar{A}}:=\Exp{\mo{\Phi}_n\A\mo{\Phi}_n^T}$, $\mo{\bar{B}}:=\Exp{\mo{\Phi}_n\B\mo{\Phi}_m^T}$, $\mo{\bar{Q}}:=\Exp{\mo{\Phi}_n\mo{Q}\mo{\Phi}_n^T}$, and $\mo{\bar{R}}:=\Exp{\mo{\Phi}_m\mo{R}\mo{\Phi}_m^T}$, to compactly present the results below.
\section{Stability}
In this section, we study the exponential stability of the second mean for the dynamical system in \eqn{simpleSys}. The equilibrium solution is said to possess exponential stability of the $m^{\text{th}}$ mean if $\exists\,\delta > 0$ and constants $\alpha>0,\beta>0$ such that $\|\x_0\| < \delta$ implies $\forall t\geq t_0$
\begin{align}
\Exp{\|\x(t;\x_0,t_0)\|^m_m} \leq \beta \Exp{\|\x_0\|_m^m} e^{-\alpha (t-t_0)}.
\end{align}
It can be trivially shown that the dynamical system in \eqn{simpleSys}, with random variables $\param$, is exponentially stable in the $2^{\text{nd}}$ mean, or exponentially stable in the mean square sense (EMS-stable), if $\exists$ a Lyapunov function $V(\x):=\x^T\P\x$, with $\P=\P^T>0$, and $\alpha>0$ such that
\begin{align}
\Exp{\dot{V}} \leq -\alpha \Exp{V}. \eqnlabel{stab_condition}
\end{align}
As the sample properties are well defined, the derivation is obtained by mimicking the proof for deterministic systems. \Eqn{stab_condition} implies
\begin{align*}
\oderiv{\Exp{V}}{t} &\leq -\alpha \Exp{V}, \\
\Rightarrow \Exp{\x^T\P\x} &\leq \Exp{\x_0^T\P\x_0}e^{-\alpha (t-t_0)}.
\end{align*}
Recall for $\P=\P^T$,
\[
\lambda_{\text{min}}(\P) \| \x \|_2^2 \leq \x^T\P\x \leq \lambda_{\text{max}}(\P) \| \x \|_2^2,
\]
\begin{align*}
\Rightarrow &\lambda_{\text{min}}(\P) \Exp{\|\x\|_2^2} \leq \lambda_{\text{max}}(\P) \Exp{\| \x_0 \|_2^2} e^{-\alpha (t-t_0)},\\
\Rightarrow & \Exp{\|\x\|_2^2} \leq \kappa(\P) \Exp{\| \x_0 \|_2^2} e^{-\alpha (t-t_0)}.
\end{align*}
EMS-stability condition in the polynomial chaos framework is presented next.
\begin{proposition}
For any matrix $\mo{M}\in\mathbb{R}^{m\times n}$ and $\mo{\Phi}_n$ as defined earlier
\begin{equation}
\mo{M}\mo{\Phi}_n^T = \mo{\Phi}^T_m (\I{N+1}\otimes\mo{M}). \eqnlabel{prop3}
\end{equation}
\end{proposition}
\begin{proof}
\begin{align*}
\mo{M}\mo{\Phi}_n^T & = (1 \otimes \mo{M})(\mo{\Phi}^T \otimes \I{n}) \\ &= \mo{\Phi}^T \otimes \mo{M} = (\mo{\Phi}^T\I{N+1}) \otimes (\I{m}\mo{M})\\
&= (\mo{\Phi}^T \otimes \I{m})(\I{N+1}\otimes \mo{M}) \\ & = \mo{\Phi}^T_m (\I{N+1}\otimes\mo{M}).
\end{align*}
\end{proof}
\begin{corollary}
\begin{equation}
\mo{\Phi}_n\mo{M}^T = (\I{N+1}\otimes\mo{M}^T)\mo{\Phi}_m. \eqnlabel{cor2}
\end{equation}
\end{corollary}
\begin{proof}
Take transpose of \eqn{prop3}.
\end{proof}
\vspace{5mm}
\begin{theorem}
The dynamical system in \eqn{simpleSys} is EMS-stable if $\exists$ $\P=\P^T>0$ and $\alpha > 0$ such that
\begin{align}
\mo{\bar{A}}^T\set{P} + \set{P}\mo{\bar{A}} + \alpha\Exp{\mo{\Phi}_n\mo{\Phi}_n^T}\set{P}\leq 0, \eqnlabel{pc_stab}
\end{align}
where $\set{P}:=\I{N+1}\otimes \P$.
\end{theorem}
\begin{proof}
With $V(\x):=\x^T\P\x$, and $\P = \P^T > 0$,
\begin{align*}
\dot{V}&= \dot{\x}^T\P\x + \x^T\P\dot{\x}\\
& = \x^T\left(\A^T\P + \P\A\right)\x \\
&= \xpc^T\left(\mo{\Phi}_n\A^T\P\mo{\Phi}_n^T + \mo{\Phi}_n\P\A\mo{\Phi}_n^T\right)\xpc.
\end{align*}
Using \eqn{prop3} and \eqn{cor2}, replace $\P\mo{\Phi}_n^T$ and $\mo{\Phi}_n\P$ by $\mo{\Phi}_n^T\set{P}$ and $\set{P}\mo{\Phi}_n$ respectively. This gives us
\begin{align*}
\dot{V} = \xpc^T\left(\mo{\Phi}_n\A^T\mo{\Phi}_n^T\set{P} + \set{P} \mo{\Phi}_n\A^T\mo{\Phi}_N^T \right)\xpc
\end{align*}
Similarly,
\[V = \x^T\P\x = \xpc^T\mo{\Phi}_n\P\mo{\Phi}_n^T\xpc = \xpc^T\mo{\Phi}_n\mo{\Phi}_n^T\set{P}\xpc.\]
$\Exp{\dot{V}} \leq -\alpha \Exp{V}$ is equivalent to
\begin{align*}
\mo{\bar{A}}^T\set{P} + \set{P}\mo{\bar{A}} + \alpha\Exp{\mo{\Phi}_n\mo{\Phi}_n^T}\set{P}\leq 0.
\end{align*}
\end{proof}
\Eqn{pc_stab} is a convex constraint in $\P$ and $\alpha$ (pg. 11, \cite{Boydetal:1994}).
In our previous work \cite{fisher2009linear}, we presented stability conditions with Lyapunov function defined as $V(\xpc):=\xpc^T \P \xpc$, and studied stability of the deterministic system in \eqn{pcDynamics:final}. The dynamics and the Lyapunov function were both deterministic and standard Lyapunov arguments were used to analyze stability. Stability of \eqn{simpleSys} was then inferred by showing that
\[\lim_{t\rightarrow \infty} \xpc(t)\rightarrow 0 \Rightarrow \Exp{\|\x\|^m_m} \rightarrow 0.\] Here we analyze the stability of \eqn{simpleSys} directly, with Lyapunov functions of the type $V(\x):=\x^T \P \x$.
\section{Controller Synthesis}
In this section, condition for a state feedback controller $\K$ that achieves EMS-stability for the system in \eqn{contiDyn}, is presented. Let $V(\x):=\x^T\P\x$, where $\P = \P^T > 0$, be the Lyapunov function that certifies this. The closed-loop system with control $\u = \K\x$ is therefore
\begin{align}
\xdot = \Bigl(\A(\param) + \B(\param)\K\Bigr)\x \eqnlabel{clp}.
\end{align}
\begin{theorem}
Closed loop system in \eqn{clp} is EMS-stable if
\begin{align}
\set{Y}\mo{\bar{A}}^T + \mo{\bar{A}}\set{Y} + \set{W}^T\mo{\bar{B}}^T +\mo{\bar{B}}\set{W} + \alpha\set{Y}\Exp{\mo{\Phi}_n\mo{\Phi}_n^T}\leq 0, \eqnlabel{lmi}
\end{align}
where $\set{W} := \I{N+1}\otimes \W$, $\set{Y} := \I{N+1}\otimes \Y$, $\Y := \P^{-1}$, and $\W:=\K\Y$.
\end{theorem}
\begin{proof}
With $V(\x):=\x^T\P\x$, where $\P = \P^T > 0$,
\begin{align*}
\dot{V} &= \x^T\left[\A^T\P +\P\A + \K^T\B^T\P + \P\B\K \right]\x, \\
& = \xpc^T \left[\mo{\Phi}_n\A^T\P\mo{\Phi}^T_n + \mo{\Phi}_n\P\A\mo{\Phi}^T_n + \right. \\ & \left.\mo{\Phi}_n\K^T\B^T\P\mo{\Phi}^T_n + \mo{\Phi}_n\P\B\K\mo{\Phi}^T_n \right]\xpc.
\end{align*}
Using \eqn{prop3} and \eqn{cor2}, we get
\begin{align*}
\dot{V} &= \xpc^T \left[\mo{\Phi}_n\A^T\mo{\Phi}^T_n\set{P} + \set{P}\mo{\Phi}_n\A\mo{\Phi}^T_n + \set{K}^T\mo{\Phi}_n\B^T\mo{\Phi}\set{P}^T_n + \set{P}\mo{\Phi}_n\B\mo{\Phi}^T_n\set{K} \right]\xpc,
\end{align*}
where $\set{K} := \I{N+1}\otimes \K$. Therefore, $\Exp{\dot{V}} \leq -\alpha \Exp{V}$ is equivalent to
\begin{align}
\mo{\bar{A}}^T\set{P} + \set{P}\mo{\bar{A}} + \set{K}^T\mo{\bar{B}}^T\set{P} + \set{P}\mo{\bar{B}}\set{K} + \alpha\Exp{\mo{\Phi}_n\mo{\Phi}_n^T}\set{P}\leq 0. \eqnlabel{bmi}
\end{align}
The above equation is nonconvex in $\set{P}$ and $\set{K}$ and can be convexified using the well known substitutions \cite{bernussou1989linear} $\Y := \P^{-1}$, and $\W:=\K\Y$. These substitutions can be written in terms of $\set{P}, \set{K}, \set{Y}$, and $\set{W}$ as
\begin{align*}
\set{W} & = \I{N+1}\otimes \W = \I{N+1}\I{N+1}\otimes \K\Y \\
& = (\I{N+1}\otimes \K)(\I{N+1}\otimes \Y) \\
& = \set{K}\set{Y}.
\end{align*}
It is also straightforward to show $\set{P} = \set{Y}^{-1}$ and $\set{K} = \set{W}\set{Y}^{-1}$. Substituting these in \eqn{bmi}, and pre-post multiplying by $\set{Y}$, we get the following convex inequality in $\Y, \W$ and $\alpha$
\begin{align*}
&\set{Y}\mo{\bar{A}}^T + \mo{\bar{A}}\set{Y} + \set{W}^T\mo{\bar{B}}^T+\mo{\bar{B}}\set{W} + \alpha\set{Y}\Exp{\mo{\Phi}_n\mo{\Phi}_n^T}\leq 0.
\end{align*}
\end{proof}
The performance parameter $\alpha$ can be maximized, along with \eqn{lmi}, as a generalized eigen-value problem (GEVP) (pg. 11, \cite{Boydetal:1994}).
\begin{theorem}
A fixed gain $\K$ that minimizes
\begin{equation}
\Exp{\int_0^\infty (\x^T\mo{Q}\x + \u^T\mo{R}\u)dt}
\eqnlabel{cost}
\end{equation}
subject to $\u = \K\x$ and dynamics given by \eqn{contiDyn} can be synthesized in the polynomial chaos framework by solving the following convex optimization problem
\[
\max \textbf{tr} \Y
\]
subject to
\[
\begin{bmatrix}
\set{Y}\mo{\bar{A}}^T + \mo{\bar{A}}\set{Y} + \set{W}^T\mo{\bar{B}}^T + \mo{\bar{B}}\set{W} & \set{Y} & \set{W}^T \\
\set{Y} & -\mo{\bar{Q}}{-1} & \mo{0} \\
\set{W} & \mo{0} & -\mo{\bar{R}}^{-1}
\end{bmatrix} \leq 0.
\]
\end{theorem}
\begin{proof}
The problem is equivalent to solving
$ \min_{\P,\K} \textbf{tr} \P$, such that $\oderiv{}{t} \Exp{\x^T\P\x} \leq -\Exp{\x^T(\mo{Q} + \K^T\mo{R}\K)\x}$.
In the polynomial chaos framework, the constraint can be written as
\begin{align*}
&\Exp{\mo{\Phi}_n(\A + \B\K)^T\P\mo{\Phi}_n^T} + \Exp{\mo{\Phi}_n\P(\A + \B\K)\mo{\Phi}_n^T} +\Exp{\mo{\Phi}_n (\mo{Q} + \K^T\mo{R}\K) \mo{\Phi}_n^T} \leq 0
\end{align*}
Using \eqn{prop3} and \eqn{cor2}, we get
\begin{align*}
& \mo{\bar{A}}^T\set{P} + \set{P}\mo{\bar{A}} + \set{K}^T\mo{\bar{B}}^T\set{P} + \set{P} \mo{\bar{B}} \set{K} + \mo{\bar{Q}} + \set{K}^T\mo{\bar{R}}\set{K} \leq 0.
\end{align*}
Substituting $\Y = \P^{-1}$ and $\mo{W} = \K\Y$, or equivalently $\set{Y} = \set{P}^{-1}$ and $\set{W} = \set{K}\set{Y}$, and pre-post multiplying by $\set{Y}$ we get
\begin{align*}
& \set{Y}\mo{\bar{A}}^T + \mo{\bar{A}}\set{Y} + \set{W}^T\mo{\bar{B}}^T+ \mo{\bar{B}}\set{W} + \set{Y}\mo{\bar{Q}}\set{Y} + \set{W}^T\mo{\bar{R}}\set{W} \leq 0.
\end{align*}
By redefining the cost function in terms of $\Y$ and making use of the Schur complement, we arrive at the result.
\end{proof}
In our previous work \cite{fisher2009linear}, four controller synthesis algorithms were presented, among which was the formulation $\u=\K\x$, with constant deterministic gain $\K$. The synthesis condition for this case was presented in terms of a bilinear matrix inequality, which could not be convexified. Here, we present a convex formulation for the same synthesis problem.
Next we present Monte-Carlo based algorithms for stability analysis and controller synthesis. These will serve as benchmarks for the polynomial chaos based algorithms. Let $\{\param_s\}_{s=1}^{S} \sim p(\param)$ be samples drawn from distribution $p(\param)$, and the sample trajectories be $\x(t,\param_s)$. Define $\X_{mc}:=[\x(t,\param_1)\; \cdots \; \x(t,\param_S)]$, and $\x_{mc}:=\vec {\X_{mc}}$. Recall that for a function $F(\param)$, the expected value $\Exp{F(\param)}$ can be approximated as
\begin{align}
& \Exp{F(\param)}:= \int_{\domain{\param}} F(\param)p(\param)d\param \nonumber \\
& \approx \frac{1}{S} \sum_{s=1}^{S} F(\param_s), \text{ with } \param_s \sim p(\param). \eqnlabel{mcApprox}
\end{align}
Therefore, EMS-stability condition with Lyapunov function $V(\x(t,\param)) := \x^T(t,\param)\P\x(t,\param)$, $\x_s:=\x(t,\param_s)$ and $\A_s := \A(\param_s)$, can be approximated as
\begin{align*}
& \frac{1}{S}\sum_{s=1}^S \dot{V}(\x_s) \leq -\alpha \frac{1}{S}\sum_{s=1}^S V(\x_s), \\
\implies & \frac{1}{S}\sum_{s=1}^S \x_s^T(\A_s^T \P + \P \A_s) \x_s \leq -\alpha \frac{1}{S}\sum_{s=1}^S \x_s^T \P \x_s,\\
\implies & \frac{1}{S}\sum_{s=1}^S \x_s^T(\A_s^T \P + \P \A_s + \alpha \P ) \x_s \leq 0, \\
\implies & \frac{1}{S} \x_{mc}^T \boldsymbol{\mathsf{diag}}(\A_s^T \P + \P \A_s + \alpha \P)_{s=1}^{S} \x_{mc} \leq 0,\\
\end{align*}
or
\begin{equation}
\A_s^T \P + \P \A_s + \alpha \P \leq 0, \text{ for } s=1,\cdots,S.
\eqnlabel{mcStability}
\end{equation}
Similarly, a feasible gain $\K$ can be obtained by satisfying
\begin{align}
\Y \A_s^T + \A_s \Y + \W^T \B_s^T + \B_s \W + \alpha \Y \leq 0,
\eqnlabel{mc:feasible}
\end{align}
for $s=1,\cdots,S$, with $\B_s:=\B(\param_s)$. Finally, the optimal gain for cost function given by \eqn{cost}, can be obtained by solving the following optimization problem
\[
\max \textbf{tr} \Y
\]
subject to
\begin{equation}
\small
\begin{bmatrix}
\Y\A_s^T + \A_s\Y + \W^T \B_s^T + \B_s\W & \Y & \W^T \\
\Y & -\mo{Q}^{-1} & \mo{0} \\
\W & \mo{0} & -\mo{R}^{-1}
\end{bmatrix} \leq 0,
\eqnlabel{mc:optimal}
\end{equation}
for $s=1,\cdots,S$. Conditions in \eqn{mc:feasible} and \eqn{mc:optimal} are derived by approximating the expectation integral using \eqn{mcApprox} and following the argument presented in the derivation of \eqn{mcStability}.
\section{Example}
The plant considered here is an F-16 aircraft at high angle of attack \cite{fenwu:f16}, with states $x=[V\;\;\alpha\;\;q\;\;\theta\;\;T]^T$, where $V$ is the velocity, $\alpha$ the angle of attack, $q$ the pitch rate, $\theta$ is the pitch angle, and $T$ is the thrust. The controls, $u=[\delta_e\;\delta_{th}]^T$, are the elevator deflection $\delta_e$, and the throttle $\delta_{th}$. This linear model is obtained about $\alpha=35^o$, where the moderately high angle of attack causes inaccurate modeling of aerodynamic coefficients and thus results in parametric uncertainty. The $A(\param)$ and $B$ matrices are given by
\begin{equation} \eqnlabel{f16}
A(\param)=\left[\begin{array}{ccccc}0.1658&-13.1013& -7.2748(1+0.4\Delta) &-32.1739&0.2780\\
0.0018&-0.1301&0.9276(1+0.4\Delta)&0&-0.0012\\
0&-0.6436&-0.4763&0&0\\
0&0&1&0&0\\ 0&0&0&0&-1
\end{array}\right], \;
B=\left[\begin{array}{cc}0&-0.0706\\0&-0.0004\\0&-0.0157\\0&0\\64.94&0
\end{array}\right].
\end{equation}
Similar to the model in \cite{fenwu:f16}, the terms in parenthesis in the $A(\param)$ matrix are assumed to be uncertain and are functions of a \textit{single} random variable $\param$, uniformly distributed over $[-1,1]$. This uncertainty is due to the uncertainty in the damping term $C_{xq}$, which is difficult to model in high angle of attack. The uncertainty is assumed to be distributed uniformly by $\pm 40\%$ about the nominal values $-7.2748$ and $0.9276$ respectively. \Fig{result} shows the variation of $\|\P^\ast\|$ obtained by solving the optimization problem outlined above for both Monte-Carlo and polynomial chaos based formulation. The objective function is defined by $\mo{Q}:=\boldsymbol{\mathsf{diag}}([1E-1, 1E2, 1E-2, 1E0, 1E-3])$, and $\mo{R}:= \boldsymbol{\mathsf{diag}}([1E2, 5E-3])$. In the Monte-Carlo approach, the problem is solved with sample sizes $5, 10, 50, 100, 200, 500, 1000, 1500,$ and $2000$ respectively. To capture the confidence in the solution for each sample size, the optimization problem is solved 100 times with new samples. \Fig{result} shows the mean $\|\P^\ast\|$ and its standard deviation. We can observe that the mean $\|\P^\ast\|$ converges and the confidence in the solution also improves as we increase the sample size. \Fig{result} also shows the convergence of $\|\P^\ast\|$ with increasing order of polynomial chaos approximation, solved with approximation order $1,\cdots,20$. Since the polynomial chaos approach is a deterministic framework, there is no variability in the answer for a given approximation order. The $x$-axis in \fig{result} is the actual number variables in the optimization problem that is solved and is obtained from \texttt{CVX}\cite{grant2008cvx}/\texttt{SDPT3}\cite{toh1999sdpt3}. This is proportional to the number of samples in Monte-Carlo approach and the order of approximation in polynomial chaos approach. Comparing the two plots we see that the trend with polynomial chaos is consistent with mean trend from Monte-Carlo. However, to get a high confidence solution, the Monte-Carlo approach requires to solve a problem with an order of magnitude higher in size. With polynomial chaos approach we can get close to the converged solution with fairly low order approximation, and thus offers a computational advantage for the class of problems considered here. Although, \fig{result} shows this trend for the specific system, we can expect to see this advantage for general systems as well. This is because, in general, polynomial chaos is computationally more efficient than Monte-Carlo for characterizing uncertainty in dynamical systems.
\begin{figure}[h!]
\includegraphics[width=\textwidth]{performance.eps}
\caption{Variation of $\|\P^\ast\|$ with optimization problem size, for Monte-Carlo (mean and standard deviation) and polynomial chaos based controller synthesis algorithms. Problem size is proportional to the number of samples in Monte-Carlo approach and the order of approximation in polynomial chaos approach. }
\figlabel{result}
\end{figure}
\section{Summary}
In this paper we presented new convex conditions for EMS-stability of linear dynamical systems with probabilistic uncertainty in system parameters. These results were obtained using the polynomial chaos framework and are similar to the well known results for deterministic systems. We applied this framework to design EMS-stabilizing controllers for an F-16 aircraft model and demonstrated the computational advantage of polynomial chaos based approach over Monte-Carlo based approach for analyzing dynamical systems with random parameters.
\bibliographystyle{IEEEtran}
|
1,116,691,497,776 | arxiv |
\section{Introduction}
In this section,
motivation of hadron production measurements,
types of targets for measurements,
and the NA61/SHINE experimental apparatus will be discussed.
\subsection{Hadron Production}
In modern long-baseline neutrino oscillation experiments, neutrino beams are created by colliding protons with a light nuclear target, such as carbon or beryllium.
Secondary hadrons are produced via primary interactions of beam protons and their decays contribute to the neutrino flux.
Neutrinos mostly come from decays of charged pions, while kaon contributions to the flux are getting larger at higher energy region.
In addition, a significant fraction of neutrinos come from re-interactions of secondary hadrons (pions, kaons, protons, and so on)
in target, aluminum magnetic horn, or with other beamline materials.
Therefore, precise knowledge on hadron production for both primary and secondary interactions is necessary.
In long-baseline neutrino oscillation analyses, number of observed neutrino events at the near and far detectors can be written as proportional to the neutrino flux and cross-section:
\begin{eqnarray}
N_{ND}&\propto&\int \Phi_{ND} \cdot \sigma \,\, dE_\nu \nonumber \\
N_{FD}&\propto&\int \Phi_{FD} \cdot \sigma \cdot P_{osc} \,\, dE_\nu \nonumber \\
&\propto&\int R_{\frac{FD}{ND}} \cdot \Phi_{ND} \cdot \sigma \cdot P_{osc} \,\, dE_\nu \nonumber
\end{eqnarray}
where $\Phi_{ND(FD)}$, $\sigma$, $P_{osc}$, and $R_{\frac{FD}{ND}}$ denote neutrino flux at the near (far) detector, neutrino cross-section on target materials, neutrino oscillation probability, and far to near neutrino flux ratio, respectively.
Here, near detectors can not fully constrain flux at the far site and large uncertainty remains.
For instance at the T2K experiment, to achieve desired sensitivity for the observation of CP violation,
total systematic uncertainty needs to be below 4\%.
On the contrary, current T2K has about 6\% total uncertainty on the $\nu_e$ appearance channel including 3\% uncertainty on the $\Phi \cdot \sigma$ term.
Similar situation will be expected for the future Hyper-K and future DUNE experiments.
An example of flux uncertainty for the DUNE experiment is shown in Figure~\ref{Fig:DuneFlux}.
In addition, neutrino cross-section and other near detector measurements rely heavily on a precise flux prediction.
For instance, the flux prediction of the MINER$\nu$A experiment in the NuMI beamline at Fermilab relies on former hadron production measurements~\cite{Aliaga:2016oaz}, however, its coverage is not enough and they assign large uncertainties.
Therefore, hadron production is the leading source of uncertainty and its reduction is desired.
\begin{figure}[htb]
\centering
\includegraphics[height=53mm]{./DuneFluxUncertainty.pdf}
\caption{Flux uncertainty prediction of the DUNE experiment~\cite{Laura-NA61Beyond2020}.}
\label{Fig:DuneFlux}
\end{figure}
\subsection{Thin and Replica Target Measurement}
There are two types of measurements to constrain hadron production uncertainty: thin and replica targets.
Thin targets (a few \% of proton-nuclear interaction length, $\lambda$) are used to study primary and secondary interactions.
Proton ($p$), pion ($\pi^\pm$), and kaon ($K^\pm$) beams strike nuclear targets (such as carbon, beryllium, aluminum)
and secondary hadrons are produced.
To constrain the uncertainty on production, measurements of total inelastic and production cross-sections as well as differential cross-sections ($\frac{d^2\sigma}{dpd\theta}$) are made.
A typical thin graphite target is shown in Figure~\ref{Fig:target} (Left).
Replica targets (often also called thick targets) are used to study hadrons exiting target and to measure
production cross-section via beam survival probability:
$P_{\textrm{\footnotesize{survival}}} = e^{-Ln\sigma_{\textrm{\scriptsize{prod}}}}$.
An example of a T2K replica graphite target is shown in Fig~\ref{Fig:target} (Right).
\begin{figure}[htb]
\centering
\includegraphics[height=50mm]{./ThinTarget.pdf} \hspace{5mm}
\includegraphics[height=50mm]{./T2KReplicaTarget.pdf}
\caption{(Left) A 1.5cm long thin graphite target ($\sim$3\% of $\lambda$). (Right) A 90cm long T2K replica graphite target ($\sim$1.9 $\lambda$).
}
\label{Fig:target}
\end{figure}
\subsection{The NA61/SHINE experiment}
The NA61/SPS Heavy Ion and Neutrino Experiment (NA61/SHINE)
is a fixed-target experiment at the CERN SPS, which studies hadron production in hadron-nucleus and nucleus-nucleus collisions for various physics goals.
The NA61/SHINE apparatus is a large acceptance spectrometer for charged particles.
Figure~\ref{Fig:NA61} (Left) shows the NA61/SHINE experimental setup.
Main tracking detectors are four large TPCs (VTPC-1, VTPC-2, MTPC-L, MTPC-R), where
two of them sit inside the super-conducting dipole magnets with a combined maximum bending power of 9\,Tm,
and other two are located downstream of magnets symmetrically with respect to the beamline.
To fill the gap on the beam direction, GapTPC is positioned along the beam axis between two VTPCs.
Moreover, forward TPCs (FTPC 1, 2, 3) have been installed in 2017 (Figure~\ref{Fig:NA61} (Right)) and huge improvement on the forward acceptance has been achieved (Figure~\ref{Fig:FTPCacceptance}).
This facility update is particularly important to understand forward proton and pion productions for NuMI and LBNF beamlines.
These TPCs provides good momentum reconstruction and particle identification capabilities.
Scintillator-based time of flight detectors (ToF) are located downstream of TPCs, which give complementary particle identification capability especially for the region where the Bethe-Bloch dE/dx curves overlap.
The Projectile Spectator Detector (PSD), a forward hadron calorimeter, sits downstream of the ToF system.
\begin{figure}[htb]
\centering
\includegraphics[height=49mm]{./NA61.pdf} \hspace{2mm}
\includegraphics[height=49mm]{./FTPCs.pdf}
\caption{(Left) The schematic top-view layout of the NA61/SHINE experiment. (Right) Constructed FTPC 2 and FTPC 3 chambers.
}
\label{Fig:NA61}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[height=55mm]{./acc_FTPC.pdf}
\caption{ NA61/SHINE acceptance as a function of production angle and momentum. (Left) without FTPCs. (Right) with FTPCs.
}
\label{Fig:FTPCacceptance}
\end{figure}
\section{Results and Plans}
In this section, NA61/SHINE latest results and ongoing measurements for the T2K and Fermilab neutrino program will be presented.
\subsection{Measurements for the T2K flux prediction}
NA61/SHINE performed thin and replica target measurements to reduce the uncertainty on the T2K flux prediction.
To constrain primary proton beam interactions on carbon target,
proton beam at 31 GeV/$c$ was shot on a thin carbon target (2cm long, $\sim$4\%\,$\lambda$, denote as $p$+C).
These interactions were used for measurements of total cross-sections and spectra measurements of $\pi^\pm$, $p$, $K^+$,$K^0_S$, and $\Lambda^0$.
NA61/SHINE has published these results in \cite{Abgrall:2011ae, Abgrall:2011ts, Abgrall:2013wda, Abgrall:2015hmv}.
These thin target measurements successfully improved flux uncertainty down to 10\% (Figure~\ref{Fig:T2Ktuning} (Left), black dotted line).
To further constrain flux uncertainty,
NA61/SHINE measured $\pi^\pm$ spectra using proton beam at 31 GeV/$c$ on the T2K replica target~\cite{Abgrall:2012pp, Abgrall:2016jif} (Figure~\ref{Fig:T2Ktuning} (Left), black solid line).
Recently, NA61/SHINE has performed improved measurements with five times larger statistics~\cite{Berns:2018tap}.
Yields of hadrons ($\pi^\pm$, $p$, $K^\pm$) exiting the surface of the T2K replica target has been measured and compared with several model predictions.
Figure~\ref{Fig:T2Ktuning} (Right) shows expected improvement on the flux uncertainty and
Figure~\ref{Fig:T2KReplicaYields} shows an example of observed differential yields of hadrons.
\begin{figure}[htb]
\centering
\includegraphics[height=52mm]{./T2KfluxUnc.pdf}
\caption{ T2K flux uncertainty at the far detector. (Left) Black dotted and solid lines are flux uncertainty tuned with thin target measurements and replica target measurement ($\pi^\pm$ yields only), respectively.
(Right) Expected flux uncertainty improvement with the latest replica target measurements.
}
\label{Fig:T2Ktuning}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[height=64mm]{./T2KReplicaYields.pdf}
\caption{ Yield of hadrons T2K replica target. From left to right, $\pi^+$, $K^+$, and proton yields are shown, respectively. NA61/SHINE data points (green points) are compared with predictions of NuBeam (red line) and QGSP\_BERT (blue line) predictions with Geant4 version 10.03.
}
\label{Fig:T2KReplicaYields}
\end{figure}
\subsection{Measurements for the Fermilab neutrino program}
Measurements for the Fermilab neutrino beamlines, NuMI and LBNF, are ongoing with NA61/SHINE.
In 2015, superconducting magnets were not operational and spectral measurements were not possible.
Therefore, experimental setup was optimized to measure inelastic and production cross-sections,
which they are defined in terms of the inelastic $\sigma_\mathrm{inel}$ and coherent elastic $\sigma_\mathrm{el}$ cross-sections:
\begin{eqnarray}
\sigma_\mathrm{total} = \sigma_\mathrm{inel} + \sigma_\mathrm{el}. \nonumber
\end{eqnarray}
The inelastic cross-section $\sigma_\mathrm{inel}$ is defined as the sum of all processes due to strong interactions
other than coherent elastic scattering.
In case new hadrons are produced, process is taken into account the production $\sigma_\mathrm{prod}$ cross-section.
Considering that the quasi-elastic scattering is a part of the inelastic scattering, the production cross-section $\sigma_\mathrm{prod}$ can be difined in terms of the quasi-elastic scattering $\sigma_\mathrm{qe}$:
\begin{eqnarray}
\sigma_\mathrm{prod} = \sigma_\mathrm{inel} - \sigma_\mathrm{qe}. \nonumber
\end{eqnarray}
Measurements of six different interactions on thin carbon and aluminum targets using $\pi^+$ and $K^+$ beams have been recently published~\cite{Aduszkiewicz:2018uts}.
Results are summarized in Figure~\ref{Fig:NA61totalSigma}.
Precision of these measurements are typically 2\,--\,3\%, while current NuMI beam for the MINER$\nu$A experiment assumes 5\% (10\,--\,30\%) for pions (kaons)~\cite{Aliaga:2016oaz}.
Thus, these measurements will significantly reduce the uncertainty on the neutrino flux prediction in NuMI.
\begin{figure}[htb]
\centering
\includegraphics[height=54mm]{./prodSigma_vs_momentum_pub.eps} \hspace{-9mm}
\includegraphics[height=54mm]{./inelSigma_vs_momentum_pub.eps}
\caption{ Results of cross-section measurements with the 2015 NA61/SHINE data set. (Left) Production cross-section. (Right) Inelastic cross-section.
}
\label{Fig:NA61totalSigma}
\end{figure}
In 2016, $p$+C/Be/Al at 60 GeV/$c$, $\pi^+$+C/Be at 60 GeV/$c$, and $p$+C/Be at 120 GeV/$c$ interactions were measured.
Spectra measurements are ongoing and an example of V$_0$ candidates found in $\pi^+$+C interaction is shown in Figure~\ref{Fig:V0candidates}.
In 2017, measurements continued including FTPCs and data calibration is ongoing.
In 2018, a replica target of the NuMI beamline has been built and measurements with proton beam at 120 GeV/$c$ has been performed.
\begin{figure}[htb]
\centering
\includegraphics[height=70mm]{./V0candidates.pdf}
\caption{ V$_0$ candidates found in $\pi^+$+C at 60 GeV/$c$ interactions in the 2016 data sets.
Plot with regard to transverse momentum of oppositely charged decay products with respect to the V$_0$ ($p_\mathrm{T}$) vs longitudinal momentum asymmetry ($\alpha = \frac{p_\mathrm{l}^+ - p_\mathrm{l}^-}{p_\mathrm{l}^+ + p_\mathrm{l}^-}$)
}
\label{Fig:V0candidates}
\end{figure}
\section{Prospect: NA61 beyond 2020}
In this section, plans and possible measurements in the NA61/SHINE facility after CERN long shutdown 2 will be presented.
The next generation neutrino oscillation experiments are being proposed, such as LBNF/DUNE
and successor experiments of T2K (T2K Phase 2 and Hyper-Kamiokande experiments).
The LBNF beamline will shoot protons with momentum somewhere between 60 and 120\,GeV/$c$ on carbon or beryllium target, similarly as the NuMI beamline with 120\,GeV/$c$ protons.
Because primary protons have high momentum, secondary protons tend to be produced beam-forward direction and their re-interactions contribute a lot to the neutrino flux for the LBNF and NuMI beamlines.
Therefore, it is very important to measure forward proton productions to further improve the neutrino flux prediction.
The T2K beamline will be re-used for its successor experiments with the same beam momentum and upgraded beam intensity, and there exists a possibility to re-design the target for future operations.
For all the next generation experiments, hadron production measurements with replica targets are highly desirable once their design is fixed.
The NA61/SHINE collaboration is preparing program extension proposal after CERN Long Shutdown 2 (2019-2020).
Significant facility modifications are planned including
TPC electronics upgrade which significantly increases readout rate up to 1000\,Hz,
installation of modern silicon-based tracking detectors surrounding the target which improves vertex reconstruction precision drastically,
and new time of flight detectors with improved timing resolution ($\sigma_{\rm time} \sim$50\,ps) which increases particle identification resolution, in addition to the existing large acceptance TPCs.
These plans are summarized in ~\cite{NA61Addendum}.
Therefore, it is a great opportunity for the next generation neutrino experiments to perform precise measurements of hadron production based on their demands at the upgraded NA61/SHINE facility.
\section{Summary}
Precision hadron production measurements are essential to reduce the leading systematic uncertainty on the neutrino flux prediction.
NA61/SHINE thin and replica target measurements have improved and will further improve the flux prediction in T2K.
NA61/SHINE measurements for the Fermilab neutrino programs are ongoing.
NA61/SHINE is proposing program extension after Long Shutdown 2, which significantly upgrades their facility and is a great opportunity to pursue important measurements.
\bigskip \bigskip \begin{center} \begin{large
We would like to thank the CERN EP, BE and EN Departments for the strong support of NA61/SHINE.
This work was supported by the U.S. Department of Energy.
|
1,116,691,497,777 | arxiv | \section{Introduction: multiferroic hexagonal manganites}
Materials with coexisting magnetic and electric order, the so-called multiferroics, have been
attracting a lot of attention since it was recognized that they can display gigantic
magnetoelectric coupling effects: Magnetic properties can be controlled by electric fields and
vice versa.\cite{Kimura03a,Hur04a,Fiebig05a,Cheong07a} In this context the hexagonal (h-)
manganites h-$R$MnO$_3$ with $R$ = Sc, Y, Dy--Lu play an exceptional role because they offer
great flexibility for tuning such magnetoelectric correlations and studying the coupling between
spin, charge, and lattice degrees of freedom in multiferroics. Crystals of the h-$R$MnO$_3$ family
are structurally equivalent and display ferroelectric order below about 1000~K with a spontaneous
polarization of $\approx 5.6$~$\mu$C/cm$^2$ along the hexagonal
$c$-axis.\cite{Coeure66a,Pauthenet70a,Kumagai12a} The $R^{3+}$ ions vary in size and magnetic
moment and due to the interaction with the Mn$^{3+}$ ions the variety of magnetic phases and
magnetoelectric interaction phenomena emerging below about 100~K is particularly rich. This
includes contributions to the magnetization induced by ferroelectric poling, giant magneto-elastic
coupling effects, and a coupling between magnetic and ferroelectric domain
walls.\cite{Lottermoser04a,Lee08a,Fiebig02b}
A discussion of the magnetoelectric coupling phenomena in the h-$R$MnO$_3$ compounds with $R$ =
Dy--Yb invariably involves the magnetic $4f$ moments. Yet, investigations of the rare-earth order
had remained scarce for a long time. Only recently, earlier magnetization measurements have been
complemented by a structural analysis of the $R^{3+}$ order in h-HoMnO$_3$, h-YbMnO$_3$, and
h-DyMnO$_3$ by neutron or resonant x-ray diffraction. The studies revealed that the $3d$--$4f$
interaction in h-\textit{R}MnO$_3$ is more complex than previously
assumed\cite{Fabreges08a,Harikrishnan09a,Lonkai02a,Sugie02a,Munawar06a,Tyson10a} and that the
magnetic Mn$^{3+}$ and $R^{3+}$ lattices can have a different space
symmetry.\cite{Wehrenfennig10a} Yet, with very few exceptions\cite{Fabreges08a} little is known about the $4f$--$4f$ exchange interaction
between $R^{3+}$ moments occupying \textit{different} Wyckoff positions, i.e., the 4b and 2a sites
of the hexagonal unit cell (see inset of Fig.~\ref{fig:fig3} (a) for a schematic
illustration). A detailed knowledge, however, is indispensable for understanding
the complex magnetic, multiferroic, and magnetoelectric coupling processes in h-$R$MnO$_3$. It
becomes even more important in view of the current intensive studies addressing the domain
walls\cite{Meier12a,Meier12b,Wu12a,Du11a} and their magnetic properties.\cite{Geng12a} Here, any
statement about the \textit{local} magnetic properties initially requires a precise knowledge of
the \textit{global} bulk spin structure. Consequently, the first goal is to develop a model
explaining the magnetic phase diagram of the h-$R$MnO$_3$ series in general.
For our study we have chosen h-ErMnO$_3$ --- its magnetic Mn$^{3+}$ phase diagram is similar to
that of h-TmMnO$_3$ and h-YbMnO$_3$ which establishes it as a prototypical compound within the
h-$R$MnO$_3$ series. Based on magnetoelectric and magnetization measurements a magnetic Er$^{3+}$
order was proposed\cite{Iwata98a,Sugie02a} but not verified or uniquely related to the coexisting
Mn$^{3+}$ order.
In this Report we clarify the spin structure of the Mn$^{3+}$ and Er$^{3+}$ sublattices and
introduce a microscopic model coherently explaining the magnetic phase diagram of h-ErMnO$_3$ that
can be projected onto the h-$R$MnO$_3$ system in general. For this purpose we performed
complementary magnetization, second harmonic generation (SHG), and neutron-diffraction
measurements at temperatures down to 30~mK. Below the N\'{e}el temperature we find
antiferromagnetic order of the Mn$^{3+}$ moments triggering the magnetic order of the
Er$^{3+}$(4b) site according to the same representation ($\Gamma_4^{\rm{Er,Mn}}$) while the
Er$^{3+}$(2a) sites remain disordered. In contrast, the ground state toward 0~K is ferrimagnetic
with \textit{all} spins ordered according to the same representation ($\Gamma_2^{\rm{Er,Mn}}$).
The intermediary phase transition ($\Gamma_4^{\rm{Er,Mn}}\to\Gamma_2^{\rm{Er,Mn}}$) occurs via a
transient breakdown of the magnetic order on the Mn$^{3+}$ and Er$^{3+}$(4b) sites so that the
highest possible symmetry is maintained. The study advances our understanding of the different
roles the Mn$^{3+}$, $R^{3+}$(2a), and $R^{3+}$(4b) moments play in establishing the magnetic
phase diagram of the h-\textit{R}MnO$_3$ system.
\section{Experimental results}
\subsection{Temperature- and field-dependent magnetization measurements} \label{magnetization}
In order to clarify the spin structure in h-ErMnO$_3$ we scrutinized the magnetic phase diagram
and extended it towards the milli-Kelvin regime to capture the actual ground state. Our
magnetization measurements were performed at the LaMMB-MagLab of the Helmholtz-Zentrum Berlin
using a h-ErMnO$_3$ single crystal of 20~mg grown by the floating-zone technique. For measurements
above 1.8~K a standard vibrating sample magnetometer was used, whereas cantilever magnetometry was
applied to study magnetic transitions below 1.8~K.
Figure~\ref{fig:fig1}(a) shows magnetization data for $M(T)$ taken with increasing temperature
$dT/dt=+0.2$~K/min after field cooling ($H\,\|\,c$). Note that identical curves were obtained for
$dT/dt<0$ (not shown) indicating non-hysteretic behavior. For the remainder of this work we
therefore restrict the discussion to measurements with $dT/dt>0$.
\begin{figure}[h]
\centering
\includegraphics[width=8.5cm,keepaspectratio,clip]{Figure1.eps}
\caption{\label{fig:fig1} (a) Temperature dependence of the magnetization $M(T)$ in h-ErMnO$_3$
measured after field cooling. The data set, recorded with $dT/dt=+0.2$~K/min in magnetic fields
applied parallel to the $c$-axis, reveals two different anomalies indicated by gray triangles and
black squares, respectively. As guide to the eye dashed lines indicate the magnetic-field
dependence of the two transitions (b) Magnetic-field dependence of the magnetization $M(H)$ at
80~mK. The curve shows two transistions that manifest as a step-like increase in $M(H)$. (c)
HT-phase diagram of h-ErMnO$_3$. Triangles and squares correspond to transitions deduced form
$M(T)$ (see (a)) while filled circles represent transitions observed in $M(H)$. Blue open circles
are additional data points taken from Ref.~\onlinecite{Yen07a}.}
\end{figure}
As seen in Fig.~\ref{fig:fig1}(a), at 0.01~T a magnetic moment of $4\cdot 10^{-3}$~Am$^2$/g
is measured at 1.8~K. It decreases rapidly with increasing temperature showing a
minimum in its derivative $dM/dT$ around 2.2~K. The result is in agreement with earlier
publications where the magnetization was attributed to long range magnetic order in the Er$^{3+}$
sublattice.\cite{Sugie02a} For increasing magnetic fields the associated change in $M(T)$ becomes
less pronounced and the minimum in $dM/dT$ shifts to higher temperature (see gray triangles in
Fig.~\ref{fig:fig1}(a)). The same trend is observed in magnetic torque measurements revealing a
pronounced change in response between 1~K and 4~K (not shown). A second anomaly manifests at
0.75~T as a step-like increase in $M(T)$. This anomaly shifts continuously to higher temperatures
when the magnetic field is increased further (see black circles in Fig.~\ref{fig:fig1}(a)).
Figure~\ref{fig:fig1}(c) summarizes the results by showing the magnetic phase diagram in the
temperature/magnetic-field plane. In order to trace the phase boundaries we investigated the
magnetic-field dependence of the magnetization $M(H)$ down to the milli-Kelvin range as
exemplified by Fig.~\ref{fig:fig1}(b). All low-temperature magnetic-field scans clearly indicate
two consecutive anomalies around 0.03~T and 0.6~T, respectively. Hence, based on our magnetization
measurements we have to distinguish four different magnetically ordered states which are denoted
as AFM, FIM$_1$, FIM$_2$, and FM in the phase diagram in Fig.~\ref{fig:fig1}(c).
The detailed analysis of the nature of these states, i.e.\ the corresponding spin structures and
interactions, will be the topic of the following sections. We will distinguish four
representations (magnetic space symmetries): $\Gamma_1$ ($P6_3cm$), $\Gamma_2$
($P6_3\underline{cm}$), $\Gamma_3$ ($P\underline{6}_3c\underline{m}$), and $\Gamma_4$
($P\underline{6}_3\underline{c}m$). As we will see, lower symmetries like $P6_3$ or
$P\underline{6}_3$ do not have to be considered. For the sake of clarity we will first focus on
the magnetic order of the Mn$^{3+}$ lattice, followed by a study of the Er$^{3+}$ order at the 4b
and 2a sites. In the third part we will associate these spin structures to the magnetic phases
identified in Fig.~\ref{fig:fig1}(c). An investigation of the intermediary transitions connecting
the different magnetic phases follows. It will allow us to describe the sequence of magnetic phase
transitions in a comprehensive model based on the coupling between the magnetic sublattices in the
h-ErMnO$_3$ system with an outlook to the h-$R$MnO$_3$ system in general.
\subsection{SHG and neutron-diffraction study of the Mn$^{3+}$ spin structure} \label{Mn-order}
The magnetic Mn$^{3+}$ moments order antiferromagnetically at $T_{\rm N}\approx
80$~K.\cite{Koehler64a} The resulting structure has been discussed for more than five decades.
Based on the extensive literature on h-ErMnO$_3$ the analysis and discussion can already be
restricted to $\Gamma_2^{\rm{Mn}}$ and $\Gamma_4^{\rm{Mn}}$ as only possible magnetic
representations for the Mn$^{3+}$ order (see Fig.~\ref{fig:fig4} for a schematic
illustration).\cite{Fiebig02a,Sekhar05a,Park02a,Lee06a} In order to scrutinize the emergence of
the magnetic order according to either of these representations we performed complementary
experiments by optical SHG and by neutron diffraction on selected site-specific reflections. The
neutron-diffraction experiments were performed at the E2 beamline of the Helmholtz-Zentrum Berlin.
A h-ErMnO$_3$ single crystal of $2\times 3\times 5$~mm$^3$ (180~mg) was cut from the same batch as
the crystal studied by magnetometry. The sample was mounted in a $^3$He/$^4$He dilution insert and
investigated in the $h0l$ plane at a wavelength of {2.39~\AA}. Since neutron-diffraction
experiments alone prohibit a unique distinction between the two aforementioned space
groups,\cite{Park02a} SHG measurements were also conducted in order to uniquely determine the
magnetic Mn$^{3+}$ order. SHG is described by the equation
$P_i(2\omega)=\epsilon_0\chi_{ijk}E_j(\omega)E_k(\omega)$. A light field $\vec{E}$ at frequency
$\omega$ is incident onto a crystal, inducing a dipole oscillation $\vec{P}(2\omega)$, which acts
as source of a frequency-doubled light wave. The susceptibility $\chi_{ijk}$ couples incident
light fields with polarizations $j$ and $k$ to a SHG contribution with polarization $i$. The
magnetic and crystallographic symmetry of a compound is uniquely related to the set of nonzero
components $\chi_{ijk}$ and, therefore, allows to distinguish $\Gamma_2^{\text{Mn}}$ and
$\Gamma_4^{\text{Mn}}$.\cite{Fiebig05b} Note that SHG has been applied earlier for investigating
the magnetic structure of the Mn$^{3+}$ sublattice in h-ErMnO$_3$ but never before neutron
diffraction and SHG were applied to the \textit{same} sample and verified for the consistency of
the two techniques.
\begin{figure}[h]
\centering
\includegraphics[width=8.5cm,keepaspectratio,clip]{Figure2.eps}
\caption{\label{fig:fig2} (a) Comparison of optical SHG measurements and neutron-diffraction
experiments. Both the $\chi_{xxx}$ component of the nonlinear susceptibility tensor and the
magnetic (101) reflection arise at $T_{\rm N}=80$~K. Between 1~K and 4~K the (101) peak intensity
displays a dip which is shown in detail in the inset to Fig.~\ref{fig:fig2}(a). (b) SIMREF2.6
simulation of the relative (101) peak intensity assuming Mn$^{3+}$ order according to
$\Gamma_2^{\rm{Mn}}$ and $\Gamma_4^{\rm{Mn}}$. (c), (d) Time dependence of the SHG signal from
$\chi_{xxx}$ after decreasing the temperature from 5 to 1.8~K. The steady gradual decrease
observed in (c) is supported by a minuscule magnetic field $H\,\|\,c$ ramped from 0 to 0.05~T
linearly with time in (d). (e) Temperature dependence of the (103) reflection probing the
Mn$^{3+}$ order as corroborated by the simulations presented in (f). The gray dotted line retraces
the SHG data shown in (a) serving as guide to the eye. In agreement with (a) the (103) peak
intensity transiently breaks down between 1~K and 4~K.}
\end{figure}
Figure~\ref{fig:fig2}(a) shows a comparison of temperature-dependent neutron-diffraction and SHG
intensities, the former for the (101) reflection and the latter for the $\chi_{xxx}$ component of
the nonlinear susceptibility tensor.\cite{Fiebig02a,Park02a} Both signals arise below $T_{\rm N}$
and exhibit the same temperature dependence emphasizing that SHG is coupling to the
antiferromagnetic order of the Mn$^{3+}$ moments. The SHG measurement proves that the Mn$^{3+}$
moments of h-ErMnO$_3$ order according to the $\Gamma_4^{\text{Mn}}$ representation in the
temperature range $10~{\rm K}<T<T_{\rm N}$, clearly discarding the $\Gamma_2^{\text{Mn}}$ symmetry
on the basis of selection rules: Only $\Gamma_4^{\text{Mn}}$ allows SHG from $\chi_{xxx}$ whereas
$\Gamma_2^{\text{Mn}}$ does not.\cite{Fiebig05b,Kordel09a}
Below about 2~K the SHG intensity from $\chi_{xxx}$ begins to drop with time. According to
Fig.~\ref{fig:fig2}(c) a decrease by 20\% is obtained during the first 90~s after reducing the
temperature from 5 to 1.8~K. The decrease slows down afterwards. Note that this
behavior is reproducible and not caused by changes of the linear optical properties during
cooling. A minuscule magnetic field accelerates the decrease of the SHG signal.
Figure~\ref{fig:fig2}(d) shows that it steadily drops to zero within 30~s when a magnetic field is
ramped from 0 to 0.05~T linearly with time. The gradual decrease of the $\chi_{xxx}$ component
from its full value to zero is characteristic for a $\Gamma_4^{\text{Mn}}\to\Gamma_2^{\text{Mn}}$
transition of the Mn$^{3+}$ sublattice.\cite{Fiebig02a,Wehrenfennig10a,Fiebig00a,Fiebig05c} This
is contrasted by only a minor dip in the neutron diffraction at the (101) peak between 1~K and
4~K. The persistence of the (101) reflection is expected as $\Gamma_4^{\text{Mn}}$ and
$\Gamma_2^{\text{Mn}}$ representations lead to (101) peak intensities differing by only 7\% (see also
Fig.~\ref{fig:fig2}(b)).\cite{Park02a,Lee06a} The dip emerging at the intermediary transition,
however, cannot be explained on the basis of symmetry.
We therefore verified its occurrence by repeating the temperature dependent measurement on a
second magnetic reflection coupling to the Mn$^{3+}$ order. For this purpose we chose the (103)
reflection as SIMREF2.6 simulations\cite{Ritter99a} (see Fig.~\ref{fig:fig2}(f)) reveal no
intermixing with Er$^{3+}$ contributions.\cite{footnote1} Figure~\ref{fig:fig2}(e) shows a dip
between 1~K and 4~K that is even more pronounced compared to the one seen in
Fig.~\ref{fig:fig2}(a). It points to a transient breakdown of the Mn$^{3+}$ order that will be
discussed in Section~\ref{model}. Note that the dip in the Mn$^{3+}$-related intensity was not
observed in earlier neutron measurements on the same sample where the base temperature of the
experiment was limited to 1.8~K.\cite{Tomuta03a} This emphasizes the importance of establishing a
well-defined ground state, here by entering the milli-Kelvin regime, for performing an accurate
analysis of the magnetic order in this strongly frustrated system.
\subsection{Neutron-diffraction study of the Er$^{3+}$ spin structure} \label{Er-order}
Analogous to the case of the Mn$^{3+}$ moments four different representations denoted
$\Gamma_1^{\text{Er}}$ to $\Gamma_4^{\text{Er}}$ have to be distinguished for the Er$^{3+}$
order.\cite{Fabreges08a,Munoz00a,Nandi08a,Fiebig03a} However, in contrast to the Mn$^{3+}$ lattice
no SHG contributions coupling to the rare-earth system were found so that our symmetry analysis is
entirely based on neutron data.
\begin{figure}[h]
\centering
\includegraphics[width=8.5cm,keepaspectratio,clip]{Figure3.eps}
\caption{\label{fig:fig3} (a) Temperature dependence of the (100) peak intensity and schematic
illustration of the Er$^{3+}$(2a) and Er$^{3+}$(4b) positions in the hexagonal unit cell of
h-ErMnO$_3$. (b) SIMREF2.6 simulation reveal that a (100) reflection can emerge only for
Er$^{3+}$ order according to the $\Gamma_4^{\rm{Er}}$ or $\Gamma_2^{\rm{Er}}(\rm{FIM})$
representation with the latter one leading to the stronger reflection. (c) $M(T)$ measurement
taken at 10~mT exhibiting a similar temperature dependence as the (100) peak intensity.}
\end{figure}
In Fig.~\ref{fig:fig3}(a) we present the temperature dependence of the (100) reflection of
h-ErMnO$_3$. Since neither $\Gamma_4^{\rm{Mn}}$ order nor $\Gamma_2^{\rm{Mn}}$ order allows
Mn$^{3+}$ contributions to the (100) peak, any contribution to this peak has to be attributed to
the magnetic order of the Er$^{3+}$ moments.\cite{Wehrenfennig10a} The (100) peak emerges at 80~K.
Its intensity increases monotonously with decreasing temperature down to about 10~K where a change
of slope occurs. Thus, according to Fig.~\ref{fig:fig3}(a) the Er$^{3+}$ sublattice does not order
at 3~K as reported before,\cite{Sugie02a} but already at $T_{\rm N}$. Yet, the
temperature dependence of the Er$^{3+}$-related signal is different from those related to
Mn$^{3+}$ in Fig.~\ref{fig:fig2}. This is reminiscent of the situation in h-DyMnO$_3$ and
h-YbMnO$_3$ and indicates that the magnetic Er$^{3+}$ order is related to the antiferromagnetic
Mn$^{3+}$ order by the magnetic triggering mechanism identified in
Ref.~\onlinecite{Wehrenfennig10a}. The triggering mechanism describes a biquadratic
order-parameter coupling between two subsystems of a compound. Ordering in one of them can induce
ordering in the other at the same temperature whenever the associated coupling term lowers the
ground state energy.\cite{Holakowsky73} Because of the biquadratic nature of the coupling the
Er$^{3+}$ and Mn$^{3+}$ sublattices can order according to different magnetic space groups at
$T_{\rm N}$. Thus, in spite of the known magnetic order of the Mn$^{3+}$ system, the
magnetic Er$^{3+}$ system may order according to any of the four representations
$\Gamma_1^{\rm{Er}}$ to $\Gamma_4^{\rm{Er}}$.
Supplementary simulations by SIMREF2.6, however, reveal that for Mn$^{3+}$ order according to
$\Gamma_{2,4}^{\rm{Mn}}$ the experimentally observed (100) reflection only arises if the Er$^{3+}$
sublattice orders according to the $\Gamma_2^{\rm{Er}}$ or $\Gamma_4^{\rm{Er}}$ representation. In
turn it is forbidden for $\Gamma_1^{\rm{Er}}$ and $\Gamma_3^{\rm{Er}}$ as shown by the simulations
in Fig.~\ref{fig:fig3}(b). We further note that $\Gamma_2^{\rm{Er}}$ involves ferromagnetic order
on the 2a and 4b sites and is therefore associated to a macroscopic magnetization which can be
excluded for temperatures between 10~K and $T_{\rm N}$ based on the magnetization
measurement in Fig.~\ref{fig:fig3}(c). We therefore conclude ``compatible'' magnetic order of the
Er$^{3+}$ and Mn$^{3+}$ sublattices according to the same representation, $\Gamma_4^{\rm{Er,Mn}}$,
at $T_{\rm N}$ with an antiferromagnetically ordered Er$^{3+}$(4b) site and a disordered
Er$^{3+}$(2a) site as sketched in Fig.~\ref{fig:fig4} (AFM-phase). This is qualitatively the same
scenario met in h-YbMnO$_3$,\cite{Fabreges08a} but different from
h-DyMnO$_3$,\cite{Wehrenfennig10a} and h-HoMnO$_3$.\cite{Lottermoser04a,Vajk05a}
\begin{figure}[h]
\centering
\includegraphics[width=8.5cm,keepaspectratio,clip]{Figure4.eps}
\caption{\label{fig:fig4} Schematic illustration of the spin structure in the different
magnetically ordered phases of h-ErMnO$_3$. Three phases have to be distinguished: The
antiferromagnetic (AFM) phase, the ferrimagnetic (FIM) phase (magnetic ground state), and the
magnetic-field-induced ferromagnetic (FM) phase. Note that depending on the strength of the
magnetic-field $H||c$ the crystal can either be in a single-domain (FIM$_2$) or a multi-domain
(FIM$_1$) state in the FIM phase. Black arrows between the different phases indicate possible
temperature / magnetic-field driven transitions.}
\end{figure}
Below 10~K the pronounced increase of the (100) peak intensity indicates a change in the magnetic
Er$^{3+}$ order. This change is matched by the increase in $M(T)$ seen in Fig.~\ref{fig:fig3}(c).
The only possible transition that is in accordance with the neutron-diffraction and the
magnetization measurements is a $\Gamma_4^{\rm{Er}}\to\Gamma_2^{\rm{Er}}$ transition of the
Er$^{3+}$ sublattices. First of all, as mentioned before, the (100) reflection exclusively occurs
for the representations $\Gamma_2^{\rm{Er}}$ and $\Gamma_4^{\rm{Er}}$. Regarding the two remaining
representations only $\Gamma_2^{\rm{Er}}$ allows for the magnetization observed at low
temperature. This narrows the number of possible magnetic structures down to two, having either an
antiparallel ($\Gamma_2^{\rm{Er}}(\rm{FIM})$) or parallel ($\Gamma_2^{\rm{Er}}(\rm{FM})$)
alignment of the Er$^{3+}$ moments on the 4b and 2a sites. The SIMREF2.6 simulations summarized in
Fig.~\ref{fig:fig3}(b), however, clearly discard Er$^{3+}$ order according to
$\Gamma_2^{\rm{Er}}(\rm{FM})$ because in this case destructive interference of the Er$^{3+}$(4b)
and Er$^{3+}$(2a) contributions suppress the (100) reflection. In contrast, constructive
interference occurs for an antiparallel orientation of the Er$^{3+}$(4b) and Er$^{3+}$(2a) spins
and explains the drastic increase of the (100) peak intensity seen in Fig.~\ref{fig:fig3}(a). We
thus conclude that the ground state of the Er$^{3+}$ order is ferrimagnetic,
$\Gamma_2^{\rm{Er}}(\rm{FIM})$, with antiparallel orientation of the Er$^{3+}$ moments on the 4b
sites relative to the 2a sites (FIM-phase in Fig.~\ref{fig:fig4}). In terms of representations,
the magnetic order of h-ErMnO$_3$ at low temperature is again the same as in
h-YbMnO$_3$\cite{Fabreges08a} and different from h-DyMnO$_3$,\cite{Wehrenfennig10a} and
h-HoMnO$_3$.\cite{Lottermoser04a,Vajk05a}
\subsection{Microscopic magnetic structure of the Er$^{3+}$ and Mn$^{3+}$ sublattices} \label{structure}
Based on the magnetization, SHG, and neutron-diffraction experiments we can now derive a coherent
model describing the different magnetic states indicated in the phase diagram in
Fig.~\ref{fig:fig1}(c). We have already seen that the Mn$^{3+}$ moments and the Er$^{3+}$ moments
on the 4b sites order antiferromagnetically in the AFM-phase while the Er$^{3+}$(2a) spins remain
disordered. The order corresponds to the $\Gamma_4^{\rm{Er,Mn}}$ representation and is
schematically depicted in Fig.~\ref{fig:fig4}. The FIM$_1$ phase encountered toward 0~K in zero
magnetic field is characterized by antiferromagnetic Mn$^{3+}$ order and ferrimagnetic Er$^{3+}$
order, both according to the $\Gamma_2^{\rm{Er,Mn}}$ representation.
Additional information on the nature of this phase can be extracted from the $M(H)$ measurement
presented in Fig.~\ref{fig:fig1}(b). The pronounced response to small magnetic fields ($H\lesssim
0.03$~T) points to the formation of a ferrimagnetic multi-domain state in zero magnetic field with
an $1:1$ ratio of domains with $+M_z$ and $-M_z$. Consequently, the FIM$_2$ phase denotes the
ferrimagnetic single-domain state with a coercive field of $\approx$0.03~T toward 0~K as boundary
between the FIM$_1$ and FIM$_2$ states. The ferrimagnetic nature of this state is further
reflected by the change in signal occurring at the FIM$_2\to$~FM transition in
Fig.~\ref{fig:fig1}(b). By linearly extrapolating the $M(H)$ data gained within the two phases and
comparing the corresponding $M(0)$ values we find that the magnetization almost triples across the
transition. This behavior can be understood in terms of a change from an antiparallel to a
parallel arrangement of the Er$^{3+}$(4b) spins with respect to the Er$^{3+}$(2a) spins, i.e., a
change of the magnetic moment per unit cell from (4-2)$\cdot\mu_{\rm Er^{3+}}$ to
(4+2)$\cdot\mu_{\rm Er^{3+}}$. The transition to the ferromagnetic Er$^{3+}$ order does not
involve a change of magnetic symmetry and is therefore still described by the
$\Gamma_2^{\rm{Er,Mn}}$ representation.
Note that earlier studies did not distinguish between the FIM$_1$ and FIM$_2$ region of the
ferrimagnetic phase. In contrast, the ferrimagnetic rare-earth order was believed to be suppressed
by small magnetic-fields of only 0.05~T which would misleadingly imply weak exchange between the
Er$^{3+}$ spins.\cite{Yen07a}
After the unique determination of the microscopic magnetic structure of h-ErMnO$_3$ we can now
turn to the intermediary states encountered during the transitions between the AFM, FIM, and FM
phases. These transitions will allow us to draw further conclusions about the coupling between the
$4f$ moments at the Er$^{3+}$(2a) and Er$^{3+}$(4b) sites and the $3d$ moments of Mn$^{3+}$.
\subsection{Magnetic interaction of the Er$^{3+}$ and Mn$^{3+}$ sublattices} \label{transitions}
The comparison of Figs.~\ref{fig:fig2} and~\ref{fig:fig3}(a) already revealed an interaction
between the Mn$^{3+}$ and Er$^{3+}$(4b) sublattices: The former triggers the order in the latter
at $T_{\rm N}$. Below 10~K the Er$^{3+}$(2a) order supplements the Er$^{3+}$(4b) order and
additional transitions occur as seen in the phase diagram in Fig.~\ref{fig:fig1}(c).
Figure~\ref{fig:fig3}(a) reveals that, unlike the Er$^{3+}$(4b) order, the Er$^{3+}$(2a) is not
triggered by the Mn$^{3+}$ order. The Er$^{3+}$ spins at the 2a sites begin to order at higher
temperature ($\sim 10$~K) compared to the observed reorientation of the Mn$^{3+}$ spins ($\sim
2$~K). This precludes the triggering mechanism as it would require identical reorientation
temperatures for the two sublattices.
Hence, we propose that here the Er$^{3+}$(2a) order drives the reordering of the Mn$^{3+}$
sublattice through non-biquadratic triggered (and thus linear) order-parameter coupling. This
assumption is supported by three observations: (i) The magnetic rare-earth order continues to
strengthen toward lower temperature whereas the Mn$^{3+}$ order is already saturated. Therefore
the Mn$^{3+}$ order cannot be responsible for the Er$^{3+}$(2a) order. (ii) The reordering of the
Mn$^{3+}$ sublattice ``follows'' the Er$^{3+}$(2a) order at lower temperature. Therefore the
latter order, which is continuously strengthening with decreasing temperature, must be guiding the
reorientation. (iii) The Mn$^{3+}$ sublattice adopts the emerging $\Gamma_2$-like order of the
Er$^{3+}$(2a) sublattice by undergoing a $\Gamma_4^{\rm{Mn}}\to\Gamma_2^{\rm{Mn}}$ transition.
\begin{figure}[h]
\centering
\includegraphics[width=8.5cm,keepaspectratio,clip]{Figure5.eps}
\caption{\label{fig:fig5} (a) Temperature dependence of the (102) reflection. An increase in
intensity occurs below $T_{\rm N}=80$~K with consecutive breakdowns below 10~K. (b) Decomposition
of contributions to the (102) peak intensity based on SIMREF2.6 simulations. Below $T_N$ the
crystallographic contribution (cryst) to the (102) reflection is supplemented by magnetic
contributions originating from Mn$^{3+}$ and Er$^{3+}$. (c) Sketch illustrating the mutual
induction of the magnetic $3d$ and $4f$ order. Below $T_{\rm N}$ the Mn$^{3+}$ order triggers
magnetic long range order on the 4b sites as indicated by the white arrow. In contrast, magnetic
moments on the 2a sites only order below 10~K inducing a reorientation of Er$^{3+}$(4b) moments
which is followed by a reorientation of the Mn$^{3+}$ spins.}
\end{figure}
The data shown so far do not reveal the role of the Er$^{3+}$(4b) in this interaction: Once the 2a
site orders it dominates the (100) reflection in Fig.~\ref{fig:fig3}(a) to an extent that the
response from the Er$^{3+}$(4b) site is obscured. We therefore investigated various $h0l$
diffraction peaks by SIMREF2.6 simulation in order to identify other magnetic peaks with a similar
weight of contributions from the Mn$^{3+}$, the Er$^{3+}$(2a), and the Er$^{3+}$(2a) that would
allow us to analyzing the interplay between Er$^{3+}$ moments on the 2a and 4b sites. As revealed
by Fig.~\ref{fig:fig5}(b), the (102) reflection fulfills this condition. We find that the
intensity of the (102) reflection experiences striking breakdowns around 2.5~K and 7~K revealing
magnetic reorientations with a transient breakdown of the magnetic order at these
temperatures.\cite{Lonkai04a,Helton11a} The breakdown at 2.5~K is related to the Mn$^{3+}$
sublattice because it is also observed in the (101) and (103) reflexes (see Fig.~\ref{fig:fig2})
which are entirely determined by the Mn$^{3+}$ order. In turn, the additional breakdown at 7~K can
only indicate a magnetic reorientation of the Er$^{3+}$(4b) sublattice. We conclude that, like the
Mn$^{3+}$ order, the Er$^{3+}$(4b) order toward 0~K is driven by the Er$^{3+}$(2a) order because
it follows the same criteria (i-iii) as the Mn$^{3+}$ reordering.
In summary, toward 0~K h-ErMnO$_3$ undergoes a transformation with a change of the representation
according to $\Gamma_4^{\rm{Er,Mn}}\to\Gamma_2^{\rm{Er,Mn}}$. However, the $\Gamma_2$ phase is
assumed at a different temperature for the Er$^{3+}$(2a), the Er$^{3+}$(4b), and the Mn$^{3+}$
sublattices.
\section{Comprehensive model for the magnetic phase transitions of hexagonal E\symbol{114}M\symbol{110}O$_3$} \label{model}
By combining the analysis of the magnetic phase diagram in Sections~\ref{magnetization} to
\ref{structure} and the analysis of the magnetic transitions between these phases in
Section~\ref{transitions} we are now able to present a comprehensive scenario of the magnetic
interactions and the resulting phase transitions in h-ErMnO$_3$ with a projection on the
h-$R$MnO$_3$ series as a whole. We distinguish two fundamentally different temperature ranges as
illustrated in Fig.~\ref{fig:fig5}(c):
\textit{Between the N\'{e}el temperature and $\sim 10$~K:} In this range the magnetic structure is
determined by the ordering of the Mn$^{3+}$ sublattice. The Mn$^{3+}$ spins order
antiferromagnetically at $T_{\rm N}$ and promote rare-earth ordering on the Er$^{3+}$(4b) site.
The Er$^{3+}$(4b) order is induced at the same temperature, $T_{\rm N}$, via a triggering
mechanism with biquadratic Mn$^{3+}$--Er$^{3+}$(4b) order-parameter coupling. Both the Mn$^{3+}$
and Er$^{3+}$(4b) lattice order according to the same representation, $\Gamma_4^{\rm{Er,Mn}}$.
However, because of the biquadratic coupling this compatibility is not mandatory. Indeed,
Mn$^{3+}$ and $R^{3+}$(4b) order according to different representations in h-DyMnO$_3$. In any
case, the Er$^{3+}$(2a) site remains disordered in this temperature range.
\textit{Below $\sim 10$~K:} In this range the magnetic structure is determined by the ordering of
the Er$^{3+}$(2a) sublattice. The spins arrange uniformly at $\sim 10$~K according to the
representation $\Gamma_2^{\rm{Er}}$ with a macroscopic magnetization per domain and a nonzero net
magnetization once the magnetic field lifts the degeneracy between oppositely oriented FIM domain
states. When the Er$^{3+}$(2a) order strengthens toward low temperature, the Er$^{3+}$(2a)
ordering at first induces $\Gamma_4^{\rm{Er}}\to\Gamma_2^{\rm{Er}}$ reordering on the
Er$^{3+}$(4b) site ($\sim 7$~K) followed by $\Gamma_4^{\rm{Mn}}\to\Gamma_2^{\rm{Mn}}$ reordering
of the Mn$^{3+}$ sublattice ($\sim 2.5$~K). The coupling to the Er$^{3+}$(2a) sublattice is linear
and therefore not guided by the triggering mechanism. This is evidenced by the difference between
the (re-) ordering temperatures and the accordance of the representations describing the magnetic
order in the different sublattices. The Er$^{3+}$(2a) and Er$^{3+}$(4b) sites maintain an
antiparallel spin orientation that can be overcome in an external magnetic field in the order of
magnitude of 1~T driving a transition from ferrimagnetic to ferromagnetic Er$^{3+}$ order. A
similar situation is met in all rare-earth h-$R$MnO$_3$ compounds except in h-HoMnO$_3$ where the
Ho$^{3+}$ ground state is antiferromagnetic, yet with similar sublattice correlations as in the
other h-$R$MnO$_3$ compounds.
\section{Summary}
In conclusion, the combination of magnetization, SHG, and neutron-diffraction experiments at
temperatures down to the milli-Kelvin regime reveals important features in the magnetic phase
diagram of h-ErMnO$_3$. We identify the spin structure in the respective magnetic phases, the
magnetic coupling between the Mn$^{3+}$, Er$^{3+}$(4b), and Er$^{3+}$(2a) sublattices, and the
resulting phase transitions establishing the phase diagram. We find a high-temperature range above
10~K where the Mn$^{3+}$ sublattice induces magnetic Er$^{3+}$ order and a low-temperature range
below 10~K where the Er$^{3+}$ sublattice induces magnetic Mn$^{3+}$ order. Most of all, we find
that the ordering on the 4b and 2a sites of the Er$^{3+}$ sublattices play a strikingly
independent and different role in establishing the magnetic order in h-ErMnO$_3$. The
comprehensive model for the phase diagram of the h-ErMnO$_3$ developed in this work can be
projected onto the other rare-earth h-$R$MnO$_3$ compounds with only small variations. Thus, we
are now able to understand the complex magnetic phases of the h-$R$MnO$_3$ system on a universal
basis.
D.M. acknowledges support by the Alexander von Humboldt Foundation and the NSF Science and Technology Center (E3S). M.F. thanks the DFG (SFB 608) for subsidy.
|
1,116,691,497,778 | arxiv | \section{Introduction}
\label{sec:introduction}
With the ever rising concern regarding privacy and security in the modern digital
world, we have seen an increased interest in hardware trusted execution
environments (also known as \textbf{secure enclaves}) from
computer architects, software developers, and security researchers. %
Such enclaves provide hardware enforced security guarantees that protect the
enclaved code from outside interference, including interference from the
operating system (OS) or untrusted hardware found in the system. %
This is achieved through a variety of security measures such as memory
encryption and attestation. %
These measures come at a performance (and energy) cost, so enclaves are only
typically used for applications where security, instead of efficiency, is the
paramount concern. %
Unfortunately, even though enclaves are able to prevent malicious system
components from directly interfering with the enclaved code, there are still
numerous attacks that are made possible through side-channels. %
By exploiting architectural and microarchitectural behavior of the system in
unintended ways, these attacks (e.g., Zombieload~\cite{schwarz_zombieload_2019} or
TLBleed~\cite{gras_tlbleak_2018}) create covert communication channels and leak
secret information. %
Thankfully, enclaves have remained effective, as side-channels are typically
very noisy communication channels and often require several iterations of the
same attack before being able to reliably leak any information. %
While it is not hard to imagine cases where the attacker is targeting specific
immutable data and the enclaved code can be arbitrarily triggered (e.g., to
encrypt some data), in a lot of cases (e.g., SGX implementations of
Tor~\cite{WWWtor,kim_sgx-tor_2018}, secure database implementations, or systems
secured against rollbacks~\cite{matetic+:SECURITY2017rote}) the attacker targets
transient execution data (e.g., Tor traffic) and only has one opportunity to
perform the attack and leak information. %
In these cases, the majority of the available side-channels are not effective,
as it is not possible to distinguish a single iteration of the attack from
system noise~\cite{lyu_survey_2018, aldaya_port_2019, maurice_hello_2017}. %
MicroScope, and \emph{microarchitectural replay attacks} in general, as
introduced by Skarlatos et al.~\cite{skarlatos_microscope:_2019},
enable an attacker to ``trap'' the execution of an enclaved application and
force it to re-execute specific regions of code ad infinitum. %
With MicroScope, this is achieved by abusing a combination of speculative
execution and page fault handling, in cases where the latter is still delegated
to the (malicious/compromised) OS. %
Under typical execution, regardless if the code is executed in an enclave or
not, if an address translation misses in the translation lookaside buffer (TLB)
a page table walk is triggered. %
While the page walk is happening, the application is able to continue executing
speculatively, as long as the instructions do not depend on the faulting memory
instruction. %
If during the page walk it is determined that the page is not available and that
the OS needs to be invoked, then the speculatively executed instructions are
squashed and execution restarts from the faulting instruction. %
This time another page walk might be needed, as the translation still does not
exist in the TLB, but since the operating system has now mapped the page, the
page walk typically succeeds. %
MicroScope takes advantage of this behavior by having the OS signal that it has
mapped the page without actually doing so. %
This traps the victim application in a loop where a memory instruction (referred
to as \textbf{``the handle''}) misses in the TLB, triggers a page walk and
continues executing speculatively, triggers a page fault, squashes the
speculatively executed instructions, re-executes the faulting instruction and
misses in the TLB again. %
By triggering these loops at specific parts of the code, just before the
instructions that cause the side-channel information leakage, the attacker can
repeat the side-channel until all the underlying noise is filtered out, making
even the least reliable side-channels easy to exploit. %
While MicroScope focuses specifically on abusing the page handling mechanism
found on some enclaves, other sources of speculation and re-execution can be
similarly abused. %
For this reason, in this work we aim to solve the problem of microarchitectural
replay attacks in general, and not just MicroScope. %
Specifically:
\begin{itemize}
\item We expand from re-execution brought on by page faults to re-execution
brought on by any form of speculation in modern processors, some of which
do not require a malicious OS. %
\item We also extend the method developed by Skarlatos et al. to be
applicable to cases where a single handle cannot by itself trigger a large
number of re-executions, by introducing a method that utilizes multiple
handles. %
\item Finally, we introduce \textbf{Delay-on-Squash}, a solution to this
critical issue of microarchitectural replay attacks, which can
transparently provide protection for MicroScope and future attacks, while
avoiding significant impact on performance, energy, area, and
implementation complexity. %
\end{itemize}
A very simple but na\"ive solution would be to simply disallow speculative
execution after a page walk miss, but this would only protect against MicroScope
itself and not against microarchitectural replay attacks that utilize other
handle types or multiple handles (\autoref{sec:beyond-microscope}). %
At the same time, we cannot just disable speculation every time a squash
happens, as that would be detrimental to the performance of the system
(\autoref{sec:evaluation}). %
Furthermore, the purpose of a replay attack (primarily in secure enclaves) is to
amplify side-channel instructions. %
Speculative side-channel defences that track the \emph{data dependencies} of
side-channel instructions, such as NDA~\cite{weisse_nda:_2019},
STT~\cite{yu_speculative_2019, sdo20}, and others~\cite{barber_specshield_2019,
conditional19, fustos_spectreguard:_2019, barber20}, do not work in
this context, because the side-channel instructions may actually be in the
\emph{correct path of execution} and \emph{can also be fed with non-speculative
data coming from before the point of misspeculation} that is used for replay. %
Broader defences (not restricted to data dependencies) that could work in such a
case, e.g., InvisiSpec~\cite{yan_invisispec:MICRO2018},
Delay-on-Miss~\cite{dom19, dom20}, and many others~\cite{ghosts, safespec_2019,
Saileshwar19, revice20, muontrap20, kiriansky_dawg:MICRO2018,
taram_context-sensitive_2019}, not only focus on a small subset of side-channels
but also incur much heavier penalties. %
Our goal is to propose a highly-efficient new defence to effectively prevent
replay attacks with minimum performance cost and hardware overhead. %
Delay-on-Squash selectively delays the speculative execution of instructions
when it detects that they might be used as part of a microarchitectural replay
attack. %
We observe that if an attack requires microarchitectural replay to be
successful, then we do not need to restrict all speculation but rather only
repeating speculation interleaved with misspeculation. %
To achieve this, we use a lightweight mechanism that tracks (using Bloom
filters) which instructions have been squashed and later re-issued under unsafe
conditions, and prevents them from executing speculatively. %
Our evaluation shows that a fully secure configuration of Delay-on-Squash,
requiring little storage and overall overhead, can achieve $97\%$ of the
performance of a baseline insecure out-of-order CPU. %
\section{Side-channels}
\label{sec:side-channels}
In order to provide compatibility across different hardware implementations,
modern CPU architectures separate the visible \emph{architectural} behavior and
state of the system from the underlying \emph{microarchitectural}
implementation. %
For each visible architectural state, there exist one or more corresponding
hidden microarchitectural states ($\mu$-states)~\cite{mcilroy_spectre_2019}. %
The users/applications interact with the architectural state but have limited or
no access to the underlying $\mu$-state. %
However, while it is usually not possible to observe the $\mu$-state directly,
it is possible to infer it based on observable side-effects. %
Microarchitectural side-channel attacks take advantage of the $\mu$-state of
modern CPUs to
leak information under conditions where it is not
possible to do so on the architectural level. %
For example, cache side-channels take advantage of the difference in timing
between a hit or a miss in the cache to encode
information~\cite{osvik_cache_2006, yarom_flush+_2014, lyu_survey_2018}, by
indirectly manipulating and probing the state of the cache through normal memory
operations. %
Similar timing side-channels can also be constructed in other parts of the
system~\cite{ge_survey_2018}, such as by utilizing functional unit (FU)
contention. %
Finally, non-timing side-channels are also possible, exploiting side-effects of
the execution such as power consumption~\cite{kocher+:CRYPTO1999dpa} or EMF
radiation~\cite{gandolfi+2001emf}. %
As these side-channels are not purposely designed communication channels but
rather side-effects of the normal architectural and microarchitectural behavior
of the system, they are inherently noisy and unreliable. %
For example, when using a cache-based side-channel, there is nothing to prevent
the cacheline(s) being used for the side-channel from being evicted by a third
process in the system. %
Similarly, interrupts, context switches, and other interruptions in the
application execution can also disrupt the side-channel. %
Since the system does not provide any architectural mechanisms for synchronizing
the transmitter and the receiver during side-channel operations, these also have
to be constructed using the side-channel itself or other mechanisms. %
All these issues make exploiting side-channels harder, but not impossible. %
After all, these issues can be found in conventional communication
channels as well, especially in the layers closest to the actual hardware and
transmission mediums. %
Similarly to how modern communication protocols are designed with the
underlying channel characteristics in mind, so can protocols for side-channels
be designed. %
For example, noise on a side-channel can be filtered out using error detection
and correction codes, combined with statistical methods. %
Maurice et al.~\cite{maurice_hello_2017} have presented one such protocol where
they were able to implement \texttt{ssh} communications over a cache-based
side-channel. %
The question then becomes \emph{not if a side-channel can be exploited but
rather how easy, reliable, and fast the channel is}. %
As many of the underlying issues can be resolved by simply repeating the
transmission of information through the side-channel until successful, whether a
side-channel can be \emph{practically} exploited becomes a function of the delay
between each retransmission (i.e., how fast can the side-channel be repeated)
and the average number of retransmissions necessary (i.e., how many times does
the side-channel have to be repeated). %
Under the conditions described by Maurice et al., both the transmitter and the
receiver are under the full control of the attacker. %
This enables the attacker to not only control when the side-channel transmission
takes place, to better synchronize the transmitter and the receiver, but to also
repeat the transmission as many times as necessary. %
However, this is not always the case. %
For example, sometimes the transmitter is not a purposely designed application
but a targeted victim, such as a cryptographic application running in a secure
enclave. %
The attacker then uses side-channels to monitor the behavior of the application
under normal execution and infer sensitive information, such as cryptographic
keys. %
In some cases, the attacker is able to either directly execute or trigger the
execution of the victim application at will, repeating the execution as many
times as necessary to extract the keys. %
This ability to \emph{replay} the victim code multiple times is crucial in being
able to reliably exploit the utilized side-channel, due to the issues we have
already discussed. %
This is where MicroScope, a groundbreaking \emph{microarchitectural replay
attack} developed by Skarlatos et al.~\cite{skarlatos_microscope:_2019} comes
into play. %
\section{Microarchitectural Replay Attacks}
\label{sec:mra}
Microarchitectural replay attacks, as introduced by Skarlatos et
al.~\cite{skarlatos_microscope:_2019}, are not by themselves a side-channel
attack, speculative or otherwise. %
Instead, they can be seen as a tool to \emph{amplify the effects of
side-channel attacks}, enabling the attacker to mount a successful attack under
conditions where it would not be possible otherwise. %
This is why microarchitectural replay can be so dangerous: even the smallest,
most innocuous amount of information leakage can be amplified and
abused. %
This is particularly dangerous when applied to secure enclaves, where the
applications are security sensitive and the programmers themselves are typically expected
to manage the risk of side-channels. %
In addition, even though they exploit speculative execution, microarchitectural
replay attacks \emph{are not the same} as speculative side-channel attacks. %
Whereas the latter target the wrong execution path, effectively bypassing
software and hardware barriers to access information illegally,
microarchitectural replay attacks can amplify even the \emph{correct} path of
execution. %
This means that defences that stop speculative data
transmission~\cite{weisse_nda:_2019, yu_speculative_2019, sdo20} are irrelevant
since a replay attack can also amplify side-channel instructions that are on the
correct path. %
\subsection{MicroScope}
\label{sec:microscope}
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{0.2960\textwidth}
\includegraphics[width=\textwidth]{figs/replay-attacks-1.pdf}
\caption{Single Handle}
\label{fig:replay-attacks-1}
\end{subfigure}\hfill%
\begin{subfigure}[t]{0.2960\textwidth}
\includegraphics[width=\textwidth]{figs/replay-attacks-2.pdf}
\caption{Serial Handles}
\label{fig:replay-attacks-2}
\end{subfigure}\hfill%
\begin{subfigure}[t]{0.3105\textwidth}
\includegraphics[width=\textwidth]{figs/replay-attacks-3.pdf}
\caption{Nested Handles}
\label{fig:replay-attacks-3}
\end{subfigure}
\caption{Three different microarchitectural replay attack patterns. The
instructions in the reorder buffer are marked with an `H'~for the replay
handles, `S' for the side-channel instructions, and `X' for other instructions
not specific to the attack. The currently acquired handles are denoted in
bold.
}
\label{fig:replay-attacks}
\end{figure*}
Many modern CPUs offer secure execution contexts referred to as \emph{trusted
execution environments} (sometimes also referred to as ``secure enclaves'') that
protect the executed code from outside interference, including interference from
the operating system (OS) or the hypervisor. %
The characteristics of these enclaves differ for each architecture, but they
typically include encrypted memory for applications running in the enclave, code
verification to prevent malicious code from being executed in the enclave, and
hardware-enforced isolation of the enclaved execution context from any other
execution context in the system, including the OS. %
These measures are meant to protect sensitive code and data, such as
cryptographic functions and their keys, from any attacker that might have
compromised other parts of the system, including the OS or the hypervisor. %
MicroScope targets exactly this case, focusing specifically on Intel's Secure
Guard Extensions (SGX) enclave as a use case, although the underlying
exploitable concept is not limited to SGX. %
MicroScope exploits the fact that under SGX the page management of the
application is still delegated to the OS\footnote{This is considered secure
because the memory accessed under SGX execution is cryptographically encrypted
and verified, preventing even the OS from accessing or manipulating it.}, to
capture the execution of the application and force the application to be
re-executed as many times as necessary for the side-channel attack to be
successful. %
Specifically, MicroScope takes advantage of how page faults are handled during
execution and of the
out-of-order capabilities of modern CPUs. %
Assuming that the victim code is known and a compromised OS, MicroScope works as
follows:
\begin{enumerate}
\item The OS manipulates the page table of the victim application to ensure
that a load instruction will cause a page fault. %
This instruction is referred to as \emph{a handle}, and the operation is
referred to as \emph{``acquiring a handle.''} %
\item When the victim application reaches the handle (\textbf{H} in
\autoref{fig:replay-attacks-1}), a TLB miss occurs and a page walk
is triggered. %
The exact details of what happens during the page walk are beyond the
scope of this work, but the two main points of interest are that
\begin{enumerate*}[label=(\roman*)]
\item it will take some time before the page fault is resolved and
\item during this time the execution of the victim application will
continue speculatively.
\end{enumerate*}
\item While executing speculatively, the victim application executes the
instructions that follow the load, as long as they are \emph{not} dependent on
the value of the load.%
\footnote{Unlike speculative side-channel attacks, these instructions are
slated to execute and commit. If fed with non-speculative data from before
the handle \textbf{H}, they cannot even be delayed by proposed defences,
e.g.,~\cite{weisse_nda:_2019,yu_speculative_2019,sdo20}.} %
In the case of MicroScope, the handle is selected so that the instructions
that are speculatively executed following the page fault unwillingly
form the transmitter for the side-channel (\textbf{S}~in
\autoref{fig:replay-attacks-1}, step \circled{1}). %
\item While the victim is executing the side-channel instructions, the page
walker will conclude that the page is not mapped and trigger a page fault,
delegating the page handling to the OS. %
This causes all of the instructions after the load, including the
transmission code that has already been executed once speculatively, to be
squashed and later re-executed
(step \circled{2} in \autoref{fig:replay-attacks-1}). %
\item The OS updates the page table and invalidates the relevant TLB
entries, signaling the victim application that the load is ready to be
executed. %
However, the OS does this in a way that
ensures that the handle \emph{will page fault again}, and execution will
restart from the second step.
\item By repeating this procedure, the OS is able to keep the victim
application in a speculative loop, where the handle will keep faulting,
causing the instructions that follow it (the transmission code) to be
speculative executed again and again. %
The only way for the loop to be broken is for the attacker to
\emph{release} the handle, allowing the victim to successfully service the
TLB miss. %
\end{enumerate}
Essentially, by abusing the speculative execution mechanism of squashing and
re-executing, MicroScope is able to trap the execution of the victim application
in a loop for an arbitrary number of \emph{replay iterations}, until the
attacker is able to reliably denoise the side-channel. %
The receiver in this case can be a typical side-channel receiver, trying to
detect the side-effects caused by the transmitter, such as changes in the cache
or FU contention. %
We will not go into details about the receiver, as our solution focuses on
disrupting the transmitter; instead we point any curious readers to the original
MicroScope paper~\cite{skarlatos_microscope:_2019}. %
It is also possible to use more than one handle, as seen in
\autoref{fig:replay-attacks-2}. %
The attacker simply releases the first handle and then acquires a second handle,
assuming that more than one handle can be found in the
region of code between the first handle and the side-channel instructions. %
Under the conditions described in MicroScope using multiple handles in this way
is not useful, but it can be abused by yet unknown attacks. %
\subsection{Replay Attacks Beyond MicroScope}
\label{sec:beyond-microscope}
MicroScope exploits page faults and a specific behavior found in some secure
enclaves (delegating page faults to the OS), but that does not mean that this is
the only behavior that can be exploited. %
Specifically, under MicroScope, a single instruction that page faults is used as
a handle to replay a set of instructions indefinitely
(\autoref{fig:replay-attacks-1}). %
This is possible because
\begin{enumerate*}[label=(\roman*)]
\item page faults are a specific type of misspeculation that can be
repeated indefinitely, and
\item there is nothing in the architecture that prevents an instruction
from misspeculating several times in a row.
\end{enumerate*}
However, as the authors of MicroScope allude to in their work, it is conceivable
that other forms of speculation can be used as handles, such as branch
prediction or even transactional memory\footnote{Transactional memory is not
necessarily implemented as conventional speculative execution, confined within
the ROB, but it does have similar characteristics.}, neither of which can be
repeated indefinitely. %
Once a branch has been executed, the correct path is known and the
incorrect path will not be misspredicted a second time. %
Similarly, transactions usually abort after a number of tries, at which point
they follow a fallback path. %
By using multiple handles (\textbf{H1}~and \textbf{H2} in
\autoref{fig:replay-attacks-2}), an attacker could extend the duration of the
attack for each case, especially if handles of different types are used
together. %
In addition, if the system restricted the number of times each instruction is
allowed to misspeculate and then be re-executed speculatively, e.g., as a simple
but na\"ive solution to MicroScope, an attacker would still be able to use
multiple handles to force a finite but tangible number of replays. %
We can see that multiple handles acquired and released one after the other can
be abused by an attacker to bypass some of the restrictions posed by different
kinds of speculation and system restrictions. %
However, using handles \emph{serially} like this only allows for a limited
number of replays, with the total number being the sum of the replays of
each handle. %
In contrast, \autoref{fig:replay-attacks-3} shows a way of \emph{nesting}
handles where each handle (\textbf{H1} in the figure) amplifies the number of
replays of the next handle (\textbf{H2}). %
This works by
\begin{enumerate*}[label=(\roman*)]
\item having the inner handle \textbf{H2} cause as many replays as
possible before releasing it (\circled{2} and \circled{3} in
~\autoref{fig:replay-attacks-3}),
\item using the outer handle to squash and then re-execute the inner handle
(\circled{1} and \circled{4}),
\item re-acquiring the new (as far as dynamic instructions are concerned)
inner handle, and
\item repeating indefinitely. %
\end{enumerate*}
With this technique, the total number of replays is not the sum of each
individual handle's replays, but instead the product, growing exponentially
with each handle. %
This can make a huge difference in the number of replays. %
For example, if the attacker is able to acquire five handles and use each only
once (i.e., one replay each), with the serial handles the total number of attack
iterations would amount to ten, while with the nested handles it would amount to
32. %
Of course, nesting handles has its own challenges, including the fact that not
all forms of speculation can be used; in some cases (e.g., page faults) the
misspeculation is not handled until after the instruction has reached the head
of the ROB. %
On the other hand, modern microarchitectures allow branch prediction to go
several levels deep, resulting in the possibility of multiple outstanding
(unresolved) predictions at any one time. %
By arranging for \emph{older} branches to depend on \emph{longer latency}
operations (e.g., misses deeper in the memory hierarchy) and thus resolve slower
than younger branches, an effective nested multiple-handle attack can be
mounted. %
Similarly, if \emph{closed nesting} of transactions is
supported~\cite{haines1994composing}, the same effect can be easily achieved by
controlling the abort of the inner versus outer transactions. %
In closed nesting, an abort of the inner transaction does not abort the outer
transaction which will then proceed to repeat the inner transaction. %
Our goal is not to simply stop the currently known attacks (namely MicroScope)
but to secure speculative execution from as many present and future
microarchitectural replay attacks as possible. %
Hence, we assume that all of the three cases presented here are possible and we
have designed and evaluated our solution accordingly. %
\section{Threat Model}
\label{sec:threat-model}
MicroScope, and microarchitectural replay attacks in general, make sense only
under certain conditions. %
For example, in the specific case of MicroScope, the assumption is that the OS
is restricted from accessing the victim's execution state and memory. %
If this is not the case, then the OS can manipulate the victim directly, without
the need for MicroScope. %
At the same time, even if direct manipulation is not possible, there are cases
where it is possible to repeat the execution of the side-channel without using a
microarchitectural replay attack. %
For example, a digital rights management (DRM) system that accepts encrypted
data and decrypts them with a secret key could be fed the same data repeatedly
by a malicious attacker, allowing for enough iterations to reliably leak the
secret key. %
This is possible because, in this case, the secret is immutable, regardless of
the input data. %
On the other hand, as a counter example, there exist applications where the
secrets consist of transient data, such as SGX implementations of
Tor~\cite{kim_sgx-tor_2018} (where the sensitive data are arbitrary network
packets) or applications where countermeasures against rollback attacks have
been taken~\cite{matetic+:SECURITY2017rote}. %
In such cases, the attacker only has a single opportunity to successfully leak
the data. %
These are exactly the cases where MicroScope and microarchitectural replay
attacks in general become useful. %
Coincidentally, these are also the cases we expect to be the least protected
against side-channel attacks, as, before MicroScope, they were very hard to
exploit. %
While MicroScope focuses on secure enclaves, one can imagine how a
microarchitectural replay attack can be performed from a non-privileged context
as well. %
For example, if the branch predictor is abused to create handles, it should be
possible to do a microarchitectural replay attack from an unprivileged SMT
thread. %
We observe that for a microarchitectural replay attack to be necessary, two
conditions have to apply:
\begin{enumerate}
\item The side-channel used in the attack is ineffective unless it is repeated
numerous times, as otherwise there would be no reason to replay the attack
code. %
\item The attacker cannot arbitrarily execute the victim code. %
If this was not the case then there would be no need for a microarchitectural
replay attack as the attacker could simply re-execute the victim code as
many times as necessary. %
\end{enumerate}
These two conditions combined mean that it is completely safe, under our current
threat model, to allow a single execution of a side-channel attack while under
speculation, as it will not be effective. %
Any additional attempts will be blocked by our replay-blocking defence
mechanism, Delay-on-Squash. %
\section{Delay-on-Squash}
\label{sec:dos}
The principle behind Delay-on-Squash is simple: If an instruction is issued,
then squashed due to a replay handle, and then re-appears in the pipeline, it
is not allowed to be re-issued, as it might constitute part of a
microarchitectural replay attack. %
In the next part of this section, we have split the description of the
Delay-on-Squash mechanism into two subsections: %
First, we discuss the concept on a high level, with no regard for any
implementation constraints (\autoref{sec:dos-concept}), and then we discuss
the practicalities of an actual implementation
(\autoref{sec:dos-implementation}). %
\subsection{Conceptual Description}
\label{sec:dos-concept}
To keep track of instructions that have been issued, squashed, and re-issued
under the same handle, Delay-on-Squash needs to keep track of all the potential
handles in the ROB. %
Due to the serial and nested cases that we described in
\autoref{sec:beyond-microscope}, it is not enough to keep track of only the
handle that caused the last squash, instead Delay-on-Squash
takes into consideration all the handles that affect each squashed
instruction. %
To prevent interference from other contexts, this information needs to
accessible only by the enclave and should be securely stored and restored on a
context switch, much like the rest of the execution state (e.g., registers). %
To track all the handles, Delay-on-Squash utilizes a FIFO queue where all the dynamic
instructions that can cause misspeculation and squashing are inserted during
dispatch. %
While in the queue, these instructions are considered as potential handles and,
by definition, are considered as ``unsafe.'' %
Instructions can only be removed from the head of the queue\footnote{Even if the
instructions are squashed in the ROB, they still need to remain in the handle
queue until they reach the head.} and only if it can be determined that they are
``safe,'' i.e., that they can no longer act as a handle, which happens only when
they have \emph{moved outside the window of speculation}. %
We will discuss the exact conditions for this in \autoref{sec:dos-shadows}, but
for now let us say that this happens when the instructions are ready to be
committed. %
As the queue is a FIFO, and handles can only be removed from the head of the
queue, for a handle to become safe all the older handles
in the queue need to also become safe. %
This is a very important condition, as it prevents the serial and nested replay
patterns presented in \autoref{sec:beyond-microscope}. %
With this queue of potential handles, it is now possible to keep
track of squashed instructions on a per-handle basis. %
Specifically, every time a misspeculation occurs and instructions are squashed,
Delay-on-Squash records the PCs for all the instructions that have been issued
and are now being squashed. %
The \emph{youngest} handle (in ROB order) is retrieved from the queue and is
associated to the squashed PCs. %
These PCs then remain stored until their corresponding handle is determined to
be safe, i.e., when the handle is removed from the queue. %
Using the youngest handle, instead of the actual misspeculating instruction that
caused the squash, is important, once again to prevent the nested replay
pattern. %
Essentially, by storing the PCs of the squashed instructions until the youngest
(at the point of the squash) handle is safe, Delay-on-Squash ensures that the
record of the squashed PCs will remain stored until all the handles that were
present in the window of speculation during the squash have left the window and
are thus safe. %
With this guarantee, all that remains is to check these records before issuing
an instruction. %
If an instruction PC matches one in the stored records, it means that this
instruction has previously been issued and squashed, and that \emph{the handles
that preceded it are still in the ROB and are still considered unsafe}. %
Such instructions are prevented from being issued until the relevant handles are
deemed to be safe and the records are removed. %
With this mechanism in place, we can instantly detect when an instruction has
been issued, squashed, and then re-issued, even in more complex cases, such as
when the attacker might be utilizing nested handles. %
This enables us to detect and prevent replay iterations of the code, instantly
stopping any microarchitectural replay attacks in their tracks. %
This mechanism does have one disadvantage: %
The pattern of ``issue, squash, and re-issue'' can also happen under normal
speculative execution (i.e., when not under attack), for example when a load is
squashed due to a memory order violation, as in such cases the execution path
remains the same. %
In addition, if we are executing a loop that is small enough to fit several loop
iterations in the window of speculation, a squash in one of them will cause all
the iterations that follow (within the window) to be delayed, as the
instructions at each iteration all share the same PC. %
Fortunately, as we will see in the evaluation (\autoref{sec:evaluation}), the
actual number of cases affected by this are very small and come at a negligible
performance cost ($1\%$). %
\subsection{Efficient Implementation}
\label{sec:dos-implementation}
While the Delay-on-Squash mechanism, as conceptually described in the previous
section, can be implemented into an actual hardware solution, it would require
large storage and expensive content addressable memories for keeping
\emph{exact sets} of squashed PCs. %
This would lead to prohibitively large area, latency, and energy overheads. %
Instead, we use Bloom filters~\cite{10.1145/362686.362692} to represent the sets
of PCs (of squashed instructions) that are temporarily prevented from being
issued speculatively. %
With this approach, we can store the PCs of tens of squashed instructions with
only a few bits of storage. %
\subsubsection{Bloom Filters}
\label{sec:dos-bloom}
Bloom filters are hash-based, probabilistic data structures used to test if an
element (in our case, a PC) is part of a set. %
They work by hashing the element with a number of hash functions, each of which
indicates a position in the filter that has to be set to ``one''. %
While it is possible to get a false positive when checking the filter, it is not
possible to get a false negative, which is an important property for us, as
false negatives would lead to unsafe replay iterations. %
False positives, on the other hand, only manifest as reduced opportunities for
speculation, in our case causing a negligible performance overhead, when
compared against the overheads of existing security mechanisms in secure
enclaves~\cite{zhao16}, which are the primary target for Delay-on-Squash. %
\fakeheader{Bloom Filter Implementation}
In Delay-on-Squash we use the simplest, most efficient form of binary Bloom
filters where the only way of erasing elements from the filter is to clear the
whole filter by bulk-resetting it. %
Other approaches also exist~\cite{cohen2003spectral, fan2000summary}, but they come at increased overheads and
they would not offer significant performance benefits for our use case. %
For our implementation, we assume that all the hash functions of the PC of a
particular dynamic instruction are precomputed during dispatch and kept in its
ROB entry. %
The precomputed hash functions are then used (in the case of a squash) to index
the Bloom filter and set the corresponding ``ones,'' in parallel with multiple
other ROB entries. %
We also assume that the whole process is hidden behind the back-end recovery
latency following a squash~\cite{gonzalez2010processor}. %
\fakeheader{Rolling Bloom Filters}
In the conceptual description, for each squash, we associate each set of
squashed PCs with a handle. %
Observe, however, that in the case of a replay attack, the set of PCs would
hardly change from squash to squash. %
Maintaining the same \emph{redundant} information across multiple Bloom filters
is, clearly, a waste of resources. %
Instead, we could consolidate the information in a single rolling Bloom filter
spanning several squashes, holding all the PCs of all the squashed instructions.
In each squash, we insert the squashed PCs in the filter and then we
\emph{associate the Bloom filter with the youngest handle in existence in the
handle queue.} %
While this is an effective way of keeping all the information we need to prevent
replay attacks, it makes the clearing of the Bloom filter difficult. %
Recall that we have opted for a simple binary Bloom filter, where it is not
possible to erase individual instructions. %
Also, we only associate the youngest seen handle with the Bloom filter, since it
is not possible to remove individual PCs anyway. %
The condition for clearing the Bloom filter is then for the associated handle to
leave the window of speculation and become safe, i.e., all PCs in the Bloom
filter are safe. %
Unfortunately, as at each squash we re-associate the Bloom filter with the
\emph{youngest} handle during the squash, the lifetime of the filter can be
extended an arbitrary number of times and the clearing can be deferred for an
arbitrarily long time. %
\begin{figure}[t]
\centering
\includegraphics[width=0.65\columnwidth]{figs/dos-example.pdf}
\caption{Step-by-step example of the instruction tracking mechanism in
Delay-on-Squash.}
\label{fig:example}
\end{figure}
Instead, we use two rolling Bloom filters and switch between them
periodically. %
At each point in time, one of the filters is designated as \emph{active}, while
the other is inactive, \emph{waiting to be cleared.} %
On a squash, the PCs of the squashed instructions are inserted in the currently
active filter, and the filter is associated with the youngest handle. %
Meanwhile, the inactive filter is waiting for its associated handle to leave the
window of speculation in order to be cleared with a bulk-reset. %
As soon as the inactive filter is cleared, then it can take over the role of the
active filter, letting the previous active filter become inactive and eventually
be cleared as well. %
In our implementation of Delay-on-Squash, we have also included a mechanism that
checks if a filter is starting to become saturated (based on the number of bits
that are set to ``one'') before switching to a new active filter, to better
space out the filters over the window of speculation. %
Of importance here is that
instructions that are about to be issued need to check both filters, the active
and the inactive, to see if they can be issued safely. %
The approach can naturally be extended to a cyclical list of several Bloom
filters, out of which one is active and the rest are inactive. %
The Bloom filters (as well as the shadow tracking information) are
context-specific, i.e., each execution context has its own set of filters, which
is securely stored and reloaded on a context switch. %
This prevent other contexts, including the OS, from saturating or otherwise
manipulating Delay-on-Squash. %
\fakeheader{A Working Example}
Figure~\ref{fig:example} contains a step-by-step example of how Delay-on-Squash
tracks handles and how the Bloom filters ($BF_A$ and $BF_B$ in the
figure) are used: %
\begin{enumerate}
\item The ROB contains three potential handle instructions, $H1$, $H2$, and
$H3$. %
The example also contains the instructions $X$ and $S$, with $S$ being
actual side-channel instructions, and $X$ being some other
instruction. %
The exact types of all these instructions are not important for this
example. %
In the first step, we see how all the potential handle instructions are
inserted into the handle queue (HQ).
\item In the second step, we detect that $H2$ was misspeculated and needs to
be re-executed. %
For simplicity, we assume that the instruction that caused the squash
remains in the ROB. %
All the younger instructions that follow it (and have already been issued)
are hashed and inserted into the active Bloom filter ($BF_A$). %
The filter is then marked with the youngest potential handle, which can be
found at the tail of the handle queue. %
Handles can only be removed from the handle queue when they reach the
head so all handles that are squashed are marked accordingly (5$_S$). %
\item Execution restarts from $H2$, this time following a slightly different
path, issuing all the same instructions as before, as well as the
additional instruction $Y$. %
For the example, we assume that $H2$ is still not considered safe. %
$S$ and $H3$ hit in $BF_A$ and are delayed, while $Y$ does not hit and is
allowed to execute speculatively. %
We have now successfully prevented the side-channel instructions $S$ from
being replayed, with the unwanted side-effect of also delaying the
innocuous $H3$ instruction. %
However, we will see in the evaluation section that delaying some
additional instructions introduces only negligible performance overheads.
%
\item Now we detect that $H1$ was misspeculated and all younger instructions
need to be squashed. %
$X$, $H2$, and $Y$ are the only younger instructions that were actually
issued (since the rest were delayed), so we insert them in the Bloom
filter. %
Since $BF_A$ is already more than half full, we instead update the $BF_B$
filter, as seen in the figure. %
\item Finally, in the last step, we see that all handles, except $H1$ have
been marked as squashed. %
However, since $H1$ remains, we cannot reset any of the filters, as they
are marked with instructions younger than $H1$. %
Because of this, $H1$ can cause an infinite number of squashes (e.g., by
going back to the ROB state in step \circled{1} in the example) and
repetitions, but the side-channel instructions $S$ will not be replayed
again. %
\item Once the $H1$ handle is resolved it is removed from the HQ upon which
the squashed handles reaches the head and can trigger the Bloom filters to
be cleared (not shown). %
\end{enumerate}
In some rare cases, it is possible that after squashing the handle queue is left
empty, as all instructions are either squashed or deemed safe (e.g., if $H_1$ is
safe). %
In such cases we can run into a corner case where the squashed handles ($H_2$)
are re-introduced into the window of speculation but the Bloom filters have all
been cleared, enabling the attacker to perform several replay iterations. %
We handle this conservatively by delaying the clearing of the Bloom filters by
the length of the dynamic instruction window, to ensure that the handles are not
re-introduced. %
This happens very rarely and has no measurable effect on performance. %
\subsubsection{Handles and the Window of Speculation}
\label{sec:dos-shadows}
We have talked about handles that can cause misspeculation and squashing and
when such handles can be considered as safe. %
In the most na\"ive approach, we can consider handles to be safe when they reach
the head of the ROB and are retired, but this is awfully pessimistic. %
Instead, we draw from the existing research on speculative
execution~\cite{mehdi_ooo, yan_invisispec:MICRO2018, ghosts, dom19, dom20,
invarspec20, bell_lipasti:ISPASS2004} and consider handles as safe when they can
no longer cause squashing, regardless of their position in the ROB. %
Specifically, we have adopted the approach of using speculative shadows by
Sakalis et al.~\cite{ghosts, dom19, dom20}, a mechanism for detecting the
earliest point at which an instruction is no longer speculative. %
While these shadows are designed to work with speculative side-channel defences,
which do not necessarily work against microarchitectural replay attacks
(\autoref{sec:related-work}), the underlying principle can still be used. %
According to Sakalis et al., any speculative instruction that can cause
squashing is referred to as a ``shadow-casting'' instruction. %
Depending on the type of speculation, Sakalis et al. have defined four different
types of shadows~\cite{ghosts, dom19}, but these can be extended to include
other types of speculation as well, such as the transactional memory case
described earlier. %
\begin{itemize}
\item \textbf{E-Shadows} are cast by exceptions, as is the case of the page
faults used in MicroScope. %
\item \textbf{C-Shadows} are cast by control flow instructions such as
branches. %
\item \textbf{D-Shadows} are cast by stores with unknown addresses, as they
might modify values of speculatively executed~loads. %
\item \textbf{M-Shadows} are cast by speculative reordering that might violate
the memory model of the system. %
\end{itemize}
Once an instruction
\begin{enumerate*}[label=(\roman*)]
\item is no longer shadowed by another instruction and
\item no longer casts any shadows itself,
\end{enumerate*}
i.e., when there is no reason for said instruction to be squashed, the
instruction is considered non-speculative. %
At this point, the instruction has left the window of speculation and, assuming
that the instruction was a potential handle, it can be considered as safe. %
The advantage of this approach is that it is possible for potential handles to
reach the safe state a lot earlier than if we were waiting for them to retire. %
In addition, Sakalis et al. also describe a hardware implementation to track
speculation based on a FIFO queue~\cite{dom19}, where younger shadow-casting
instructions are only resolved once all older shadow-casting instructions have
also been resolved. %
This design fits well with Delay-on-Squash, as the handle queue has similar
characteristics. %
Note that while using the speculative shadows is not necessary and the use of
alternative methods is possible (e.g., using the head and tail of the ROB),
doing so in a na\"ive manner can lead to significant performance
degradation~\cite{ghosts, invarspec20}. %
\subsubsection{Security Implications}
As discussed in (\autoref{sec:threat-model}), Delay-on-Squash only targets
microarchitectural replay attacks, where more than one iteration is necessary to
leak sensitive information. %
Sometimes, it is possible to leak information with a single iteration of an
attack, although this is not easy to achieve~\cite{lyu_survey_2018,
aldaya_port_2019, maurice_hello_2017}, especially when the attacker does not
have full control of the victim. %
Delay-on-Squash does not protect against such attacks, as they are
indistinguishable (and are, in fact, part of) normal execution. %
Instead, if such protections are required, Delay-on-Squash can be combined with
other defence mechanisms, such as defences against speculative side-channel
attacks. %
We direct the reader to out Related Work (\autoref{sec:related-work}) for more
details. %
In addition, we want to ensure that Delay-on-Squash does not introduce any new
side-channels into the system. %
As already discussed, the state kept by the mechanism, i.e., the bloom filters
and the handle tracking information, need to be kept isolated from other
contexts and stored/restored on a context switch. %
This prevents the attacker from manipulating Delay-on-Squash either to ``make it
forget`` a replayed instruction or to introduce unnecessary overheads in the
victim application. %
At the same time, the attacker cannot in any way probe the information that the
Delay-on-Squash mechanism has of the victim, as that would allow the attacker to
ascertain which instructions the victim has executed. %
These can be enforced by the hardware by isolating the mechanism between
contexts and storing it in an encrypted manner on a context switch, much like
the rest of the context (e.g., the register file) is already stored. %
Finally, as Delay-on-Squash prevents instructions from executing under certain
conditions, it raises the question of whether it can be used to mount a
Denial-of-Service attack on the victim. %
While Delay-on-Squash can affect the performance of the victim negatively, as
long as instructions are being committed from the head of the reorder buffer
(which is never delayed by Delay-on-Squash) then execution will not stall. %
Execution can only stall if the victim is caught into an execute-squash-replay
loop (due to a microarchitectural replay attack), in which case no forward
progress would have been made even if Delay-on-Squash was not present. %
In practice, when execution is slowed down for any significant amount of time,
due to an attack, the hardware should notify the enclaved software, allowing it
to take appropriate measures to prevent the leakage and ensure that the system
does not stall. %
\section{Evaluation}
\label{sec:evaluation}
\begin{table}
\centering
\caption{The simulated system parameters.}
\label{tbl:params}
\begin{tabular}{ l | l }
\hline
Parameter & Value\\
\hline
Technology node & 22nm @ 3.4GHz \\
Issue / Execute / Commit width & 8 \\
Cache line size & 64 bytes \\
L1 private cache size & 32KiB, 8-way \\
L1 access latency & 2 cycles \\
L2 shared cache size & 1MiB, 16-way \\
L2 access latency & 20 cycles \\
\hline
Number of Bloom filters & 2 \\
Number of bits, hash functions per filter & 64, 2 \\
\hline
\end{tabular}
\end{table}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{figs/normalized_ipc.pdf}
\caption{Performance (instructions per cycle) normalized to the baseline.}
\label{fig:ipc}
\end{figure*}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{figs/false_positive_rate.pdf}
\caption{Rate of Bloom filter false positives over the total number of
\emph{executed} (i.e., including squashed) instructions.}
\label{fig:false-rate}
\end{figure}
We use the Gem5 simulator~\cite{binkert_gem5:CANEWS2011} together with the SPEC
CPU2006 benchmark suite~\cite{spec:cpu06} for the performance evaluation,
combined with McPAT~\cite{li_mcpat:MICRO2009} and
CACTI~\cite{li_CACTI:ICCAD2011} for the energy evaluation. %
We skip the first three billion instructions and then simulate in detail
for another three billion. %
The main parameters of the simulated system can be seen in
~\autoref{tbl:params}. %
To avoid mixing different energy models, we estimate the energy expenditure of
the additional handle and squash tracking structures as existing similar
structures in McPAT, appropriately sized. %
Specifically, the handle/shadow tracking mechanism is modelled as an additional
ROB structure, the Bloom filters as register files, and the hash functions for
the Bloom filters as an additional integer arithmetic unit. %
We evaluate the following configurations in detail: %
\begin{itemize}
\item \textbf{baseline:} The insecure out-of-order CPU in Gem5. %
\item \textbf{Delay-All:} A lower-bound, non-speculative configuration where
all instructions are delayed until no unsafe handles precede them. %
To make the comparison with the Delay-on-Squash configurations fair,
Delay-All utilizes the same handle-tracking mechanism as Delay-on-Squash to
detect when a handle is safe (\autoref{sec:dos-shadows}). %
\item \textbf{Delay-on-Squash (Perfect):} The ideal implementation of
Delay-on-Squash, as described in \autoref{sec:dos-concept}. %
We assume that we have unlimited space for storing all the squashed
instructions for as long as necessary. %
\item \textbf{Delay-on-Squash (Bloom):} The actual implementation of
Delay-on-Squash with Bloom filters, as described in
\autoref{sec:dos-implementation}, with the Bloom filter parameters seen in
\autoref{tbl:params}. %
We chose these parameters empirically to balance performance, area, and
energy usage. %
\end{itemize}
As the SPEC2006 applications do not contain any enclaved code regions, we apply
Delay-on-Squash during the whole execution of the application. %
In a real use case, Delay-on-Squash will only be applied to the regions of the
application running in the enclave, while regions (and applications) running
outside the enclave (as is the majority of applications on modern systems), will
not be affected. %
\subsection{Performance}
\label{sec:eval-performance}
\autoref{fig:ipc} contains the number of instructions per cycle (IPC),
normalized to the insecure out-of-order baseline. %
Overall, implementing Delay-All, which we are only including as a lower-bound
configuration, would lead to $41\%$ lower performance, when compared to the
baseline. %
On the other hand, implementing Delay-on-Squash with a perfect filter would
eliminate almost all of the performance cost, reaching $99\%$ of the baseline
performance. %
Only three applications see any perceptible performance degradation,
\texttt{Xalan} ($-8\%$), \texttt{sphinx} ($-6\%$), and \texttt{mcf} ($-3\%$),
all of which misspeculate and squash more often than average. %
In some cases, the same static instructions exist in the ROB in multiple dynamic
instances, for example when a loop is dynamically unrolled in the ROB
(\autoref{sec:dos-concept}). %
If a squash happens in such a situation, all instances of the static instructions
will be conservatively treated by Delay-on-Squash as potential replay attacks
and speculation will be restricted in a large part of the dynamic instruction
window. %
In contrast, applications like \texttt{cactusADM} and \texttt{lbm}, which rarely
misspeculate, are not affected at all. %
However, the amount of misspeculation in the application is not the only
parameter that affects its performance with the perfect filter, instead the
applications sensitivity to instruction and memory level parallelism (ILP and
MLP) is also important. %
Restricting the speculative execution of an application which relies on ILP and
MLP to achieve high performance can lead to more severe performance penalties,
when compared with an application that does not rely on either~\cite{dom20}. %
Finally, we have the realistic Delay-on-Squash implementation, based on the
Bloom filters, with the parameters described in \autoref{tbl:params}. %
Alongside the IPC, in \autoref{fig:false-rate} we can also see the false
positive rate of the applications, i.e., the number of false positives in the
Bloom filter over the number of instructions executed. %
On average, we observe one false positive every 174 instructions (a rate of
roughly $0.6\%$), but this number differs significantly between all the
application. %
Applications that have higher than average false positive rates (most
pronounced in: \texttt{mcf}, \texttt{leslie3d}, and \texttt{milc})
also tend to have lower performance than the baseline. %
However, once more, this false positive rate is not the only parameter affecting
the performance. %
The application whose performance is affected the most is \texttt{libquantum}
($-12\%$), which is not one of the applications with the highest false positive
rates. %
Instead, \texttt{libquantum} is a streaming benchmark that relies heavily on
MLP, so it is heavily affected by the false positives. %
In contrast, \texttt{omnetpp}, which is the application with the fourth largest
false positive rate, sees no performance degradation at all. %
Overall, for Delay-on-Squash with the Bloom filters, we observe a mean
performance of $97\%$ of the baseline, two percentage points lower than the
ideal version. %
Given that, by choice, applications running in secure enclaves exchange
performance for higher security guarantees, a $3\%$ performance degradation is
practically negligible. %
\subsection{Energy Usage}
\label{sec:eval-energy}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{figs/normalized_energy_usage.pdf}
\caption{Energy usage normalized to the baseline. Each bar considers of four
different parts, representing the dynamic CPU energy usage (bottom, light),
the static CPU energy usage (middle, shaded), the DRAM energy usage (middle,
light), and the overhead of the Delay-on-Squash tracking structures (top,
shaded).
}
\label{fig:energy}
\end{figure*}
The energy usage of the different configurations can be seen in
\autoref{fig:energy}. %
Each bar in the figure is split into four parts, representing (from bottom to
top) dynamic, static, DRAM, and overhead energy. %
The overhead contains both the dynamic and static energy usage of the squash
tracking mechanism and the Bloom filters, and is only included in the Bloom
filter configuration, as that is the only configuration where we have a
realistic hardware design. %
Overall, there are two factors that affect the energy usage in the various
configurations: Number of executed instructions and execution time. %
While we simulated all benchmarks for the same number of \emph{committed}
instructions, the number of (dynamically) executed instructions varies depending
on the amount of speculation and squashing. %
Specifically, instructions that are executed speculatively and are squashed
still require energy, but this energy is not used for useful work. %
For this reason, the dynamic energy usage of the applications is reduced in the
strict Delay-All version.
However, while restricted speculation improves the dynamic energy usage, the
static and DRAM energy usage are instead increased, because both depend on the
execution time, which is increased. %
Specifically, unless power- or clock-gated, the CPU has the same amount of
leakage regardless if instructions are executed or not. %
Then, the longer the CPU has to be active, the more the static energy
increases. %
Similarly, DRAM cells need to be refreshed at regular intervals, so the DRAM
energy increases with the execution time. %
Overall, these affects overshadow the energy gains provided by the reduced
misspeculation, so the Delay-All version has an overall energy usage that is
$23\%$ higher than the baseline. %
In contrast, Delay-on-Squash with a perfect filter, has identical
characteristics as the baseline. %
The performance difference between the two versions is very small, so there are
no significant differences in the static and DRAM energy, and speculation is not
restricted enough to lead to any energy usage reductions. %
As we are not modelling the filter mechanism, there is no overhead, but it is
important to note that storing and accessing all the squashed instructions in
the window of speculation would introduce very large overheads. %
On the other hand, the Bloom filter version has a $4\%$ energy overhead over the
baseline, out of which half (2 percentage points) is due to the increase in
execution time and half is due to the overhead of the mechanism. %
This would lead to an overall increase in the energy overhead, as the gains (due
to the reduction in the execution time) would be overshadowed by the increased
overhead due to the larger structures. %
\section{Related Work}
\label{sec:related-work}
\fakeheader{Related Enclave Attacks and Defences}
Intel's SGX is a popular target for attacks, presumably due to the popularity of
the platform. %
Controlled-channel attacks\cite{xu_controlled-channel_2015} and Sneaky Page
Monitoring (SPM)~\cite{wang_leaky_2017} are two side-channels aimed at SGX that
take advantage of page handling, much like MicroScope does. %
However, instead of using page faults to trigger replays of the attack code,
they instead use the fact that the page management is delegated to the OS to
monitor the access patterns of the victim directly and leak information. %
As the OS has full control over the page system, these two attacks can have low
to zero noise and do not need to be repeated several times, so no
microarchitectural replay is necessary. %
CacheZoom~\cite{moghimi_cachezoom_2017} is another attack that uses fine-grained
control to perform a side-channel but only targets cache side-channels. %
As per our threat model (\autoref{sec:threat-model}), we consider such attacks
outside the scope of our work, as when such attacks are possible there is no
reason for an attacker to employ microarchitectural replay.
We will, however, note that there exist solutions for such side-channel attacks,
such as Klotski by Zhang et al.~\cite{zhang_klotski_2020}, but they come with
steep performance and area overheads (up to $10\times$ in execution time,
depending on the configuration, for Klotski). %
There also exist speculative side-channel attacks that abuse speculative
execution and can illegally gain access to SGX data, such as
Zombieload~\cite{schwarz_zombieload_2019} or TLBleed~\cite{gras_tlbleak_2018}. %
Such attacks are affected from the same noise issues as any other side-channel
attack, and they could be used in conjunction with microarchitectural replay. %
Finally, Skarlatos et al.~\cite{skarlatos_microscope:_2019} allude, in their
MicroScope work, to possible MicroScope defences, such as fences, speculative
defences, or page fault protection defences, but they are not evaluated in
detail. %
\cs{ASPLOS abstract}
\fakeheader{Side-Channel Defences}
There are numerous works on side-channel attacks and
defences~\cite{ge_survey_2018}, particularly when it comes to cache
side-channels~\cite{lyu_survey_2018}. %
There are two main research umbrellas under which such defences generally fall
under, obfuscation or isolation.
Obfuscation-based solutions try to hide the execution patterns that lead to the
side-channel. %
Examples include prefetching data to introduce noise in the cache-access
patterns~\cite{fuchs_disruptive_2015} or oblivious RAM (ORAM) to hide memory
access patterns~\cite{zhang_klotski_2020}. %
On the other hand, isolation-based solutions try to isolate the
resources used by the application from the attacker. %
These include examples such as using cache partitioning to prevent the attacker
from interfering with the victim's cache lines~\cite{wang_new_2007,
domnitser_non-monopolizable_2012, page_partitioned_2005,
kong_deconstructing_2008}. %
Each of these solutions tries to prevent or limit the creation of specific
side-channels and comes with different costs, overheads, and limitations. %
To the best of our knowledge, there exists no published work evaluating the
total cost of implementing all the different solutions that would have been
necessary to prevent all the side-channels that microarchitectural replay
attacks, such as MicroScope, can amplify. %
Also, only perfect solutions would be effective, as otherwise even the smallest
observable side-effects can be amplified by MicroScope. %
\fakeheader{Speculative Side-Channel Defences}
Since microarchitectural replay attacks abuse speculative execution, we can also
look into speculative side-channel defences that prevent the leakage of
information while executing speculatively. %
However, there is a critical difference between microarchitectural replay and
speculative side-channel attacks: %
The latter abuse speculative execution to \emph{illegally} gains access to
information while on \emph{the wrong execution path}, while, on the other hand,
microarchitectural replay attacks can amplify \emph{the correct path} and leak
information that has been accessed \emph{legally}. %
Defences and optimizations that only try to block the wrong execution
path~\cite{kim_pact20, invarspec20, speccfi20} will not work against Microscope
or other microarchitectural replay attacks. %
Furthermore, during a speculative side-channel attack, there is a \emph{data
dependence chain} between the illegal access and the information-leaking
instructions, while in a microarchitectural replay attack the handle and the
side-channel instructions have to be independent. %
Hence, speculative side-channel defences do not always work against
microarchitectural replay attacks, while also incurring higher performance
overheads. %
For example, state-of-the-art defences that block the transmission of
speculative accessed data to potential side-channel instructions, such as
NDA~\cite{weisse_nda:_2019}, STT~\cite{yu_speculative_2019, sdo20}, and
others~\cite{barber_specshield_2019, conditional19, fustos_spectreguard:_2019,
barber20}, are ineffective in this context, because, under
microarchitectural replay, the side-channel instructions may actually be in the
\emph{correct path of execution} and \emph{can also be fed with non-speculative
data}. %
At the same time, other defences that are restricted on data dependencies, such
as InvisiSpec~\cite{yan_invisispec:MICRO2018}, Delay-on-Miss~\cite{dom19,
dom20}, and many others~\cite{ghosts, safespec_2019, Saileshwar19, revice20,
muontrap20, kiriansky_dawg:MICRO2018, taram_context-sensitive_2019}, only focus
on specific side-channels, under the assumption that not all side-channels can
be exploited as easily. %
However, the effectiveness of microarchitectural replay attacks comes precisely
from the fact that they can be used to amplify and successfully mount hard to
exploit attacks, also making these speculative side-channel defences unfit for
our purposes. %
\section{Conclusion}
Microarchitectural side-channel attacks rely on observable $\mu$-state changes
caused by the victim application under attack. %
Such observations are commonly noisy, for example, due to the extremely short
lifetime of transient $\mu$-state or due to system interference that causes
persistent $\mu$-state to change. %
To successfully launch an attack, the side-channel has to be amplified and
denoised by being repeated several times. %
This is the specific purpose of MicroScope and microarchitectural replay
attacks in general, which trap the victim application in a loop, causing it to
execute the side-channel over and over again. %
MicroScope specifically, has proven to be successful in leaking information from
secure enclaves, but microarchitectural replay attacks, in general, may have
broader applicability. %
At the same time, while these attacks do abuse speculative execution, existing
speculative side-channel defences are unable to mitigate them. %
We make the observation that such replay attacks rely on repeated squashes of
replay handles and that the instructions causing the side-channel must reside in
the same ROB window as the handles. %
Based on this observation, we propose Delay-on-Squash, which is an efficient
technique for tracking squashed instructions and preventing them from being
replayed. %
We also propose an efficient implementation of Delay-on-Squash, using Bloom
filters and speculative shadow tracking~\cite{dom19}.
We evaluate several configurations with different parameters, and we show that a
fully secure system against microarchitectural replay attacks (according to the
threat model) comes at a mere $3\%$ performance degradation when compared to a
baseline, insecure out-of-order, CPU, while also keeping the energy cost low, at
$4\%$ over the baseline. %
\bibliographystyle{IEEEtranS}
|
1,116,691,497,779 | arxiv | \section*{References}\list
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1,116,691,497,780 | arxiv | \section{\bf{Introduction}}
In mathematical analysis and in particular from functional analysis viewpoint, convolution on Euclidean space is a linear map on the function spaces related to Euclidean spaces, for more details see \cite{FollR}, also convolution is similar to cross-correlation.
In classical harmonic analysis convolution is an integral which interpret the quantity of overlap of a function as it is shifted over another function, see \cite{FollH} or \cite{HR1}. It has applications that include statistics methods, image and signal processing, electrical engineering, and also differential equations.
The convolution can be defined for functions on groups other than Euclidean space, see \cite{FollH} or \cite{HR1}. In particular, the circular convolution can be defined for periodic functions (that is, functions on the circle group), and the discrete convolution can be defined for functions on the set of integers. On the other hand one of the most important concepts in Fourier theory, Gabor theory, Wavelet theory, Multiresolution analysis and also in crystallography, is that of a convolution.
Many non-abelian groups in harmonic analysis can be considered as a semi-direct products of groups.
In this paper we want to introduce a new approach to the convolution on the semi-direct products of groups and also we study the basic properties of this theory.
The principal role played by convolutions in classical harmonic analysis is in evidence throughout \cite{BH},\cite{JEW},\cite{50},\cite{Zyg}.
Since convolution plays an important role in general theory of harmonic analysis, we focuss on the convolution theory and we define a convolution on the functions spaces related to semidirect products of groups.
More precisely many classes of non-abelian groups can be written as a semidirect product of groups in which one of the groups is abelian. In this case the semi-direct product group is non-abelian and so the standard convolution is noncommutative and so there is a lack to study the harmonic analysis on semi-direct products of groups. Thus it maybe worthwhile if we define a new convolution such that it's commutativity behavior depends only on one of the groups. In fact we are looking for a convolution on these classes of groups in which allow us to studying and analyzing the convolution with respect to the second group in the semi-direct product.
Throughout this article which contains 4 section we assume that $H$ and $K$ are locally compact topological groups and $\tau:H\to Aut(K)$ is a continuous homomorphism and $G_\tau=H\ltimes_\tau K$ is the semi-direct product of $H$ and $K$ with respect to $\tau$. Section 2 is devoted to fix notations and also a brief summary on the classic properties of the semi-direct products of groups. In section 3 first we define left and right $\tau$-convolution on $L^1(G_\tau)$ which makes $L^1(G_\tau)$ into a Banach algebra. Then we define the $\tau$-convolution as a linear combination of left and right convolution and also we define a $\tau$-involution on $L^1(G_\tau)$ and we show that $L^1(G_\tau)$ is a non-associative Banach $*$-algebra with respect to the $\tau$-convolution and the $\tau$-involution. We prove that the $\tau$-convolution is commutative if and only if $K$ is abelian. We also show that this new algebra structure is completely different from the standard algebra on the $L^1$-function algebra related to each locally compact group.
Recall that our main idea on this theory is that when $H$ is the trivial group we get the classical convolution on the locally compact group $K$. Also we prove that the $\tau$-convolution coincides with the standard convolution of $L^1(G_\tau)$ if and only if $H$ is the trivial group.
Finally, in section 4 as an application for $p>1$ we make $L^p(G_\tau)$ into a left Banach $L^1_{\tau_l}(G_\tau)$-module.
\section{\bf{Preliminaries and notations}}
A non-associative algebra $\mathcal{B}$ is a linear space $\mathcal{B}$ over field of complex or real numbers endowed with a bilinear map $(x,y)\mapsto xy$ from $\mathcal{B}\times\mathcal{B}$ into $\mathcal{B}$ and also a Jordan algebra is a commutative non-associative algebra $\mathcal{B}$ whose product satisfies the Jordan identity $(xy)x^2=x(yx^2)$ for all $x,y\in \mathcal{B}$.
Note that the term non-associative stands for not necessarily associative. More precisely we use the term non-associative algebra in order to emphasize that the associativity of the product is not being assumed.
A non-associative Banach algebra is a non-associative algebra $\mathcal{B}$ over the field of complex or real numbers, whose underlying linear space is a Banach space with respect to a norm $\|.\|$ satisfying $\|xy\|\le\|x\|\|y\|$ for all $x,y\in \mathcal{B}$.
Let $X$ be locally compact Hausdorff space. By $\mathcal{C}_c(X)$ we mean the space of all continuous complex valued functions on $X$ with compact supports. If $\mu$ is a positive Radon measure on $X$, for each $1\le p<\infty$ the Banach space of equivalence classes of $\mu$-measurable complex valued functions $f:X\to\mathbb{C}$ such that
$$\|f\|_p=\left(\int_X|f(x)|^pd\mu(x)\right)^{1/p}<\infty$$
is denoted by $L^p(X,\mu)$ which contains dense subspace $\mathcal{C}_c(X)$.
When $G$ is a locally compact group with left Haar measure $dx$ and modular function $\Delta_G$
for each $p\ge 1$ we mean by $L^p(G)$ the Banach space $L^p(G,dx)$. If $p=1$ standard convolution of $f,g\in L^1(G)$ defined via
\begin{equation}\label{0.0}
f\ast g(x)=\int_Gf(y)g(y^{-1}x)dy
\end{equation}
and also the standard involution of $f\in L^1(G)$ is given by $f^*(x)=\Delta_G(x^{-1})\overline{f(x)}$
which makes $L^1(G)$ into a Banach $*$-algebra. We recall that the Banach $*$-algebra $L^1(G)$ with respect to the standard convolution given in (\ref{0.0}) has an approximate identity (see Proposition 2.42 of \cite{FollH}) and also the Banach $*$-algebra $L^1(G)$ is commutative if and only if $G$ is abelian.
Let $H$ and $K$ be locally compact groups with identity elements $e_H$ and $e_K$ respectively and left Haar measures $dh$ and $dk$ respectively and also let $\tau:H\to Aut(K)$ be a homomorphism such that the map $(h,k)\mapsto \tau_h(k)$ is continuous from $H\times K$ to $K$. In this case we say the homomorphism $\tau:H\to Aut(K)$ is continuous.
The semidirect product $G_\tau=H\ltimes_\tau K$ is a locally compact topological group with underlying set $H\times K$ which is equipped with product topology and group operation is defined by
\begin{equation}\label{}
(h,k)\ltimes_\tau(h',k')=(hh',k\tau_h(k'))\hspace{0.5cm}{\rm and}\hspace{0.5cm}(h,k)^{-1}=(h^{-1},\tau_{h^{-1}}(k^{-1})).
\end{equation}
The left Haar measure of $G_\tau$ is $d\mu_{G_\tau}(h,k)=\delta(h)dhdk$ and the modular function of $G_\tau$ is $\Delta_{G_\tau}(h,k)=\delta(h)\Delta_H(h)\Delta_K(k)$, where the positive continuous homomorphism $\delta:H\to(0,\infty)$ is given by
$dk=\delta(h)d(\tau_h(k))$ and also $\Delta_H$ and $\Delta_K$ are modular functions of locally compact groups $H$ and $K$ respectively, for more details see Theorem 15.29 of \cite{HR1}.
\section{\bf{$\tau$-convolution and $\tau$-involution}}
In the following we define a $\tau$-convolution and a $\tau$-involution on $L^1(G_\tau)$ which is different from the usual convolution and involution of $L^1(G_\tau)$. We note that our idea on this extension is that only when $H$ is the trivial group this algebra structure coincides with the standard algebra structure of $L^1(K)$.
For $f\in L^1(G_\tau)$, let $\widetilde{f}\in L^1(K)$ be given by
\begin{equation}\label{9}
\widetilde{f}(k):=\int_Hf_{t}(k)\delta(t)dt,
\end{equation}
where for each $t\in H$ the function $f_t$ is defined for a.e. $k$ in $K$ via $f_t(k)=f(t,k)$. Then the integral defined in (\ref{9}) converges. In fact we have
\begin{align*}
\|\widetilde{f}\|_{L^1(K)}
&=\int_K|\widetilde{f}(k)|dk
\\&\le\int_K\left(\int_H|f(t,k)|\delta(t)dt\right)dk=\|f\|_{L^1(G_\tau)}.
\end{align*}
For $f,g\in L^1(G_\tau)$, we define the right $\tau$-convolution on $L^1(G_\tau)$ by
\begin{equation}\label{10.r}
f\rst g(h,k):=\int_Hf_{h}\ast g_{t}(k)\delta(t)dt.
\end{equation}
where $f_h\ast g_t$ is the standard convolution on $L^1(K)$. We recall that according to the Fubini-Toneli Theorem, for each $f\in L^1(G_\tau)$ we have $f_h\in L^1(K)$ for a.e. $h$ in $H$. The integral defined in (\ref{10.r}) converges and also for each $f,g\in L^1(G_\tau)$ we have $\|f\rst g\|_{L^1(G_\tau)}\le\|f\|_{L^1(G_\tau)}\|g\|_{L^1(G_\tau)}$. Indeed by using Fubini's Theorem and Proposition 2.39 of \cite{FollH} we get
\begin{align*}
\|f\rst g\|_{L^1(G_\tau)}
&=\int_H\int_K\left|\int_Hf_{h}\ast g_{t}(k)\delta(t)dt\right|\delta(h)dkdh
\\&\le\int_H\int_K\int_H|f_{h}\ast g_{t}(k)|\delta(h)\delta(t)dtdkdh
\\&\le\int_H\int_H\left(\int_K|f_{h}\ast g_{t}(k)|dk\right)\delta(h)\delta(t)dtdh
\\&=\int_H\int_H\|f_h\ast g_t\|_{L^1(K)}\delta(h)\delta(t)dtdh
\\&\le\int_H\int_H\|f_h\|_{L^1(K)}\|g_t\|_{L^1(K)}\delta(h)\delta(t)dtdh=\|f\|_{L^1(G_\tau)}\|g\|_{L^1(G_\tau)}.
\end{align*}
The right $\tau$-convolution for a.e $(h,k)\in G_\tau$ satisfies $f\rst g(h,k)=f_h\ast\widetilde{g}(k)$.
This is because for a.e. $(h,k)\in G_\tau$ we have
\begin{align*}
f\rst g(h,k)
&=\int_Hf_h\ast g_t(k)\delta(t)dt
\\&=\int_H\left(\int_Kf(h,s)g(t,s^{-1}k)ds\right)\delta(t)dt
\\&=\int_Kf(h,s)\left(\int_Hg(t,s^{-1}k)\delta(t)dt\right)ds=f_h\ast\widetilde{g}(k).
\end{align*}
Similarly the left $\tau$-convolution on $L^1(G_\tau)$ is defined by
\begin{equation}\label{10.l}
f\lst g(h,k):=\int_Hf_{t}\ast g_{h}(k)\delta(t)dt,
\end{equation}
which analogy for a.e. $(h,k)\in G_\tau$ satisfies $f\lst g(h,k)=\widetilde{f}\ast g_h(k)$ and also
$\|f\lst g\|_{L^1(G_\tau)}\le\|f\|_{L^1(G_\tau)}\|g\|_{L^1(G_\tau)}.$
Now the $\tau$-convolution of $f,g\in L^1(G_\tau)$ is defined as
\begin{equation}\label{11}
f\st g=2^{-1}\left(f\rst g+f\lst g\right).
\end{equation}
Then clearly $f\ast g\in L^1(G_\tau)$. More precisely we have
\begin{align*}\label{12}
\|f\st g\|_{L^1(G_\tau)}
&=2^{-1}\|f\rst g+f\lst g\|_{L^1(G_\tau)}
\\&\le2^{-1}\left(\|f\rst g\|_{L^1(G_\tau)}+\|f\lst g\|_{L^1(G_\tau)}\right)\le \|f\|_{L^1(G_\tau)}\|g\|_{L^1(G_\tau)}.
\end{align*}
Note that the $\tau$-convolution of $f,g\in L^1_\tau(G_\tau)$ defined in (\ref{11}), for a.e. $(h,k)\in G_\tau$ can be rewritten in the following form
\begin{equation}\label{13}
f\st g(h,k)=2^{-1}\left(f_h\ast\widetilde{g}(k)+\widetilde{f}\ast g_h(k)\right).
\end{equation}
\begin{theorem}\label{11.1}
Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$.
The right $\tau$-convolution defined in (\ref{10.r}) makes $L^1(G_\tau)$ into a Banach algebra.
\end{theorem}
\begin{proof}
Let $f,g,u$ in $L^1(G_\tau)$. Using the associativity of the standard convolution on $L^1(K)$ for a.e. $(h,k)\in G_\tau$ we have
\begin{align*}
(f\rst g)\rst u(h,k)
&=(f\rst g)_h\ast\widetilde{u}(k)
\\&=\int_K(f\rst g)_h(s)\widetilde{u}(s^{-1}k)ds
\\&=\int_K\left(\int_Hf_h\ast g_t(s)\delta(t)dt\right)\widetilde{u}(s^{-1}k)ds
\\&=\int_H\left(\int_Kf_h\ast g_t(s)\widetilde{u}(s^{-1}k)ds\right)\delta(t)dt
\\&=\int_H(f_h\ast g_t)\ast\widetilde{u}(k)\delta(t)dt
\\&=\int_Hf_h\ast(g_t\ast\widetilde{u})(k)\delta(t)dt
\\&=\int_H\left(\int_Kf_h(s)(g_t\ast\widetilde{u})(s^{-1}k)ds\right)\delta(t)dt
\\&=\int_H\left(\int_Kf_h(s)(g\rst u)_t(s^{-1}k)ds\right)\delta(t)dt
\\&=\int_Hf_h\ast(g\rst u)_t(k)\delta(t)dt
=f\rst(g\rst u)(h,k).
\end{align*}
\end{proof}
Next theorem shows the same result as in Theorem \ref{11.1} for the left $\tau$-convolution.
\begin{theorem}\label{11.1.2}
Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$.
The left $\tau$-convolution defined in (\ref{10.l}) makes $L^1(G_\tau)$ into a Banach algebra.
\end{theorem}
\begin{proof}
We show that the left $\tau$-convolution is associative. Let $f,g,u\in L^1(G_\tau)$. Then, for a.e. $(h,k)\in G_\tau$ we have
\begin{align*}
(f\lst g)\lst u(h,k)
&=\int_H(f\lst g)_t\ast u_h(k)\delta(t)dt
\\&=\int_H(\widetilde{f}\ast g_t)\ast u_h(k)\delta(t)dt
\\&=\int_H\widetilde{f}\ast(g_t\ast u_h)(k)\delta(t)dt
=f\lst(g\lst u)(h,k).
\end{align*}
\end{proof}
But the following proposition guarantees that the $\tau$-convolution is not associative in general.
\begin{proposition}\label{11.2}
{\it The $\tau$-convolution defined in (\ref{11}), for each $f,g,u\in L^1(G_\tau)$ satisfies
\begin{equation}\label{11.2.1}
(f\st g)\st u-f\st (g\st u)=f\rst g\rst u-f\lst g\lst u.
\end{equation}
}\end{proposition}
\begin{proof}
Let $f,g,u\in L^1(G_\tau)$. Using Theorem \ref{11.1} and also Theorem \ref{11.1.2}, left and right $\tau$-convolutions are associative.
Thus it can be easily checked that for a.e. $(h,k)\in G_\tau$ we have
\begin{equation}\label{11.2.a}
f\rst (g\lst u)(h,k)=f\rst(g\rst u)(h,k)=(f\rst g)\rst u(h,k),
\end{equation}
\begin{equation}\label{11.2.b}
(f\rst g)\lst u(h,k)=f\lst (g\lst u)(h,k)=(f\lst g)\lst u(h,k).
\end{equation}
Also, using definition of the $\tau$-convolution and also (\ref{11.2.a}) and (\ref{11.2.b}) we have
\begin{align*}
(f\st g)\st u
&=2^{-1}\left((f\st g)\rst u+(f\st g)\lst u\right)
\\&=2^{-2}\left((f\rst g+f\lst g)\rst u+(f\rst g+f\lst g)\lst u\right)
\\&=2^{-2}\left((f\rst g)\rst u+(f\lst g)\rst u+(f\rst g)\lst u+(f\rst g)\rst u\right),
\end{align*}
and also similarly
\begin{align*}
f\st (g\st u)
&=2^{-1}\left(f\rst (g\st u)+f\lst (g\st u)\right)
\\&=2^{-2}\left(f\rst (g\rst u+g\lst u)+f\lst (g\rst u+g\lst u)\right)
\\&=2^{-2}\left(f\rst (g\rst u)+f\rst (g\lst u)+f\lst (g\rst u)+f\lst (g\lst u)\right).
\end{align*}
Now a straight forward calculation implies (\ref{11.2.1}).
\end{proof}
For $f\in L^1(G_\tau)$ let the $\tau$-involution be denoted by $f^{*^\tau}$ and defined for a.e. $(h,k)\in G_\tau$ by
\begin{equation}\label{14}
f^{*^\tau}(h,k):=(f_h)^*(k),
\end{equation}
where $(f_h)^*$ is the standard involution of $f_h$ in $L^1(K)$ given by $$(f_h)^*(k)=\Delta_K(k^{-1})\overline{f_h(k^{-1})},$$
which satisfies $\|(f_h)^*\|_{L^1(K)}=\|f_h\|_{L^1(K)}$ and $(f_h)^{*^*}=f_h$. Then, we have $f^{*^\tau}\in L^1(G_\tau)$. More precisely the linear map $*^\tau:L^1(G_\tau)\to L^1(G_\tau)$ is an isometry, because
\begin{align*}
\|f^{*^\tau}\|_{L^1(G_\tau)}&=\int_H\int_K|f^{*^\tau}(h,k)|\delta(h)dkdh
\\&=\int_H\left(\int_K|f_h^*(k)|dk\right)\delta(h)dh
\\&=\int_H\|f_h\|_{L^1(K)}\delta(h)dh=\|f\|_{L^1(G_\tau)}.
\end{align*}
Thus we prove the following theorem.
\begin{theorem}\label{11.3}
Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$. $L^1(G_\tau)$ is a non-associative Banach $*$-algebra with respect to the $\tau$-convolution defined in (\ref{11}) and also the $\tau$-involution defined in (\ref{14}).
\end{theorem}
\begin{proof}
It remains to show that the $\tau$-involution is an anti-homomorphism. Let $f,g\in L^1(G_\tau)$. Using the anti-homomorphism property of the involution on $L^1(K)$, for a.e. $(h,k)\in G_\tau$ we have
\begin{align*}
(f\st g)^{*_\tau}(h,k)
&=(f\rst g)^{*_\tau}(h,k)+(f\lst g)^{*_\tau}(h,k)
\\&=\{(f\rst g)_h\}^*(k)+\{(f\lst g)_h\}^*(k)
\\&=(f_h\ast\widetilde{g})^*(k)+(\widetilde{f}\ast g_h)^*(k)
\\&=(\widetilde{g})^*\ast f_h^*(k)+g_h^*\ast(\widetilde{f})^*(k)=g^{*_\tau}\st f^{*_\tau}(h,k).
\end{align*}
\end{proof}
From now on, the notation $L^1_\tau(G_\tau)$ stands for the non-associative Banach $*$-algebra mentioned in Theorem \ref{11.3} and also we use the notations $L^1_{\tau_l}(G_\tau)$ and $L^1_{\tau_r}(G_\tau)$ for the Banach algebra mentioned in Theorem \ref{11.1} respectively Theorem \ref{11.1.2}.
A left(right) $\tau_l$-ideal of $L^1_{\tau_l}(G_\tau)$ is a subspace $\mathcal{I}$ of $L^1(G_\tau)$ such that for each $f\in \mathcal{I}$ and $g\in L^1(G_\tau)$ we have $g\lst f\in \mathcal{I}$ ($f\lst g\in \mathcal{I}$) and by a closed left(right) $\tau_l$-ideal we mean a $\|.\|_{L^1(G_\tau)}$-closed left(right) $\tau_l$-ideal in $L^1_\tau(G_\tau)$. Note that the same definitions can be considered for left, right or two sided $\tau_r$-ideals of $L^1_{\tau_r}(G_\tau)$.
In the following, we introduce a tool which transfer elements of $L^1(K)$ into the elements of $L^1(G_\tau)$. In fact we define a $*$-homomorphism form $L^1(K)$ into $L^1_\tau(G_\tau)$.
For $\Phi$ in $L^1(G_\tau)$ and $\psi$ in $L^1(K)$, let
\begin{equation}\label{15}
\Phi(\psi)(h,k):=\psi(k)\int_K\Phi(h,s)ds.
\end{equation}
Then $\Phi(\psi)\in L^1_\tau(G_\tau)$ with $\|\Phi(\psi)\|_{L^1_\tau(G_\tau)}\le\|\Phi\|_{L^1_\tau(G_\tau)}\|\psi\|_{L^1(K)}$. Indeed we have
\begin{align*}
\int_H\int_K|\Phi(\psi)(h,k)|\delta(h)dkdh
&=\int_H\int_K\left|\psi(k)\int_K\Phi(h,s)ds\right|dk\delta(h)dh
\\&\le\int_H\int_K\int_K|\psi(k)||\Phi(h,s)|dsdk\delta(h)dh
\\&=\left(\int_K|\psi(k)|dk\right)\left(\int_H\int_K|\Phi(h,s)|ds\delta(h)dh\right)
=\|\Phi\|_{L^1_\tau(G_\tau)}\|\psi\|_{L^1(K)}.
\end{align*}
We consider $L_1^+$ as the set of all nonnegative $\Phi$ in $L^1(G_\tau)$ with $\|\Phi\|_{L^1(G_\tau)}=1$.
\begin{proposition}\label{19}
{\it Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$. Then, for each $\Phi\in L^+_1$, $f\in L^1(G_\tau)$ and also $\psi\in L^1(K)$ we have
\begin{enumerate}
\item $\displaystyle\widetilde{f}\ast\psi(k)=\int_Hf_t\ast\psi(k)\delta(t)dt$\ for\ a.e. $k\in K$,
\item $\displaystyle\psi\ast\widetilde{f}(k)=\int_H\psi\ast f_t(k)\delta(t)dt$\ for \ a.e. $k\in K$,
\item $\Phi(\psi)\st f(h,k)=2^{-1}\left(\|\Phi_h\|_{L^1(K)}\psi\ast\widetilde{f}(k)+\psi\ast f_h(k)\right)$\ for \ a.e. $(h,k)\in G_\tau$,
\item $f\st\Phi(\psi)(h,k)=2^{-1}\left(f_h\ast\psi(k)+\|\Phi_h\|_{L^1(K)}\widetilde{f}\ast\psi(k)\right)$\ for \ a.e. $(h,k)\in G_\tau$
\end{enumerate}
}\end{proposition}
\begin{proof} (1) Let $f\in L^1(G_\tau)$ and $\psi\in L^1(K)$. Then for a.e. $k\in K$ we have
\begin{align*}
\widetilde{f}\ast\psi(k)&=\int_K\widetilde{f}(s)\psi(s^{-1}k)ds
\\&=\int_K\left(\int_Hf(t,s)\delta(t)dt\right)\psi(s^{-1}k)ds
\\&=\int_H\left(\int_Kf(t,s)\psi(s^{-1}k)ds\right)\delta(t)dt=\int_Hf_t\ast\psi(k)\delta(t)dt.
\end{align*}
Similarly for a.e $k\in K$ we have
$$\displaystyle\psi\ast\widetilde{f}(k)=\int_H\psi\ast f_t(k)\delta(t)dt.$$
(3) Let $\Phi,f\in L^1(G_\tau)$ and $\psi\in L^1(K)$. Using (\ref{13}) for a.e. $(h,k)\in G_\tau$ we have
\begin{align*}
\Phi(\psi)\st f(h,k)&=2^{-1}\left(\left(\Phi(\psi)\right)_h\ast\widetilde{f}+\widetilde{\Phi(\psi)}\ast f_h(k)\right)
\\&=2^{-1}\left(\psi\ast\widetilde{f}(k)\|\Phi_h\|_{L^1(K)}+\psi\ast f_h(k)\right).
\end{align*}
Similarly for a.e $(h,k)\in G_\tau$ we have
$$f\st\Phi(\psi)(h,k)=2^{-1}\left(\widetilde{f}\ast\psi(k)\|\Phi_h\|_{L^1(K)}+f_h\ast\psi(k)\right).$$
\end{proof}
Next theorem shows that each $\Phi\in L^+_1$ defines a $*$-homomorphism from $L^1(K)$ into $L^1_\tau(G_\tau)$.
\begin{theorem}\label{30}
Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$ also let $\Phi\in L^+_1$.
Then, the linear map $\lambda_\Phi:L^1(K)\to L^1(G_\tau)$ defined by $\psi\mapsto \lambda_\Phi(\psi):=\Phi(\psi)$ is an isometric $*$-homomorphism.
\end{theorem}
\begin{proof}
According to the preceding result and also (\ref{15}) if $\Phi\in L^+_1$ then for each $\psi\in L^1(K)$ we have
$$\|\Phi(\psi)\|_{L^1(G_\tau)}=\|\Phi\|_{L^1(G_\tau)}\|\psi\|_{L^1(K)}
=\|\psi\|_{L^1(K)}.$$
Let $\Phi\in L^+_1$ and also $\psi,\phi\in L^1(K)$. Then for a.e. $(h,k)\in G_\tau$ we have
\begin{equation}\label{30.1}
\Phi(\psi)\st\Phi(\phi)(h,k)=\psi\ast\phi(k)\|\Phi_h\|_{L^1(K)}.
\end{equation}
Because
\begin{align*}
\Phi(\psi)\rst\Phi(\phi)(h,k)
&=\int_H\Phi(\psi)_h\ast\Phi(\phi)_t(k)\delta(t)dt
\\&=\int_H\left(\int_K\Phi(\psi)(h,s)\Phi(\phi)(t,s^{-1}k)ds\right)\delta(t)dt
\\&=\int_H\left(\int_K\psi(s)\phi(s^{-1}k)\left(\int_K\Phi(h,s')ds'\right)\left(\int_K\Phi(t,s'')ds''\right)ds\right)\delta(t)dt
\\&=\int_H\left(\int_K\Phi(h,s')ds'\right)\left(\int_K\Phi(t,s'')ds''\right)\left(\int_K\psi(s)\phi(s^{-1}k)ds\right)\delta(t)dt
\\&=\psi\ast\phi(k)\|\Phi_h\|_{L^1(K)}\int_H\left(\int_K\Phi(t,s'')ds''\right)\delta(t)dt=\psi\ast\phi(k)\|\Phi_h\|_{L^1(K)}.
\end{align*}
Similarly for a.e. $(h,k)\in G_\tau$ we have $\Phi(\psi)\lst\Phi(\phi)(h,k)=\psi\ast\phi(k)\|\Phi_h\|_{L^1(K)}$ which implies (\ref{30.1}).
Now our pervious result (\ref{30.1}) implies that for each $\psi,\phi\in L^1(K)$ and also for a.e. $(h,k)\in G_\tau$ we have
$$\Phi(\psi\ast\phi)(h,k)
=\psi\ast \phi(k)\int_K\Phi(h,s)ds
=\Phi(\psi)\st\Phi(\phi)(h,k).$$
Also for each $\psi\in L^1(K)$ and for a.e. $(h,k)\in G_\tau$ we have
\begin{align*}
\Phi(\psi^*)(h,k)&=\psi^*(k)\int_K\Phi(h,s)ds
\\&=\left(\Phi(\psi)_h\right)^*(k)=\Phi(\psi)^{*_\tau}(h,k).
\end{align*}
\end{proof}
According to Theorem \ref{30}, for each $\Phi\in L^+_1$ the set $\lambda_\Phi(L^1(K))=\{\Phi(\psi):\psi\in L^1(K)\}$ is a closed $*$-subalgebra of $L^1_{\tau_l}(G_\tau)$ and also $L^1_{\tau_r}(G_\tau)$.
The following corollary is a consequence of the theorem \ref{30}.
\begin{corollary}
{\it Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$ and also let $\Phi\in L_1^+$. If $\mathcal{J}$ is a closed left(right) ideal of $L^1(K)$ then $\lambda_\Phi(\mathcal{J})$ is a closed left(right) $\tau_r$-ideal($\tau_l$-ideal) of $\lambda_\Phi(L^1(K))$.}
\end{corollary}
Next theorem shows that $f\mapsto\widetilde{f}$ is a norm decreasing $*$-homomorphism from $L^1_\tau(G_\tau)$ onto $L^1(K)$.
\begin{theorem}\label{31}
The map defined by $f\mapsto S_K^\tau(f)=\widetilde{f}$ is a norm decreasing $*$-homomorphism from $L^1_\tau(G_\tau)$ onto $L^1(K)$.
\end{theorem}
\begin{proof}
Let $f,g\in L^1_\tau(G_\tau)$. Using Fubini's theorem for a.e. $k$ in $K$ we have
\begin{align*}
\widetilde{f\rst g}(k)
&=\int_{H}f\ast_\tau g(h,k)\delta(h)dh
\\&=\int_H\left(\int_Hf_h\ast g_t(k)\delta(t)dt\right)\delta(h)dh
\\&=\int_H\int_H\left(\int_Kf(h,s)g(t,s^{-1}k)ds\right)\delta(h)\delta(t)dtdh
\\&=\int_K\left(\int_Hf(h,s)\delta(h)dh\right)\left(\int_Hg(t,s^{-1}k)\delta(t)dt\right)ds
\\&=\int_K\widetilde{f}(s)\widetilde{g}(s^{-1}k)ds
=\widetilde{f}\ast\widetilde{g}(k).
\end{align*}
Similarly for a.e. $k\in K$ we have $\widetilde{f\lst g}(k)=\widetilde{f}\ast\widetilde{g}(k).$ Thus we have
$$\widetilde{f\st g}=2^{-1}\left(\widetilde{f\rst g}+\widetilde{f\lst g}\right)=\widetilde{f}\ast\widetilde{g}.$$
Also for a.e. $k$ in $K$ we have
\begin{align*}
\widetilde{f^{*_\tau}}(k)
&=\int_Hf^{*_\tau}(t,k)\delta(t)dt
\\&=\int_Hf_t^*(k)\delta(t)dt
\\&=\Delta_K(k^{-1})\int_H\overline{f_t(k^{-1})}\delta(t)dt={\left(\widetilde{f}\right)}^{*}(k).
\end{align*}
Note that since $\|\widetilde{f}\|_{L^1(K)}\le \|f\|_{L^1(G_\tau)}$ so the map $f\mapsto \widetilde{f}$ is norm-decreasing. Now let $\Phi\in L^+_1$ and $\psi$ in $L^1(K)$ be arbitrary. Then for a.e. $k\in K$ we have
\begin{align*}
\widetilde{\Phi(\psi)}(k)&=\int_H\Phi(\psi)(t,k)\delta(t)dt
\\&=\int_H\psi(k)\left(\int_K\Phi(t,s)ds\right)\delta(t)dt
\\&=\psi(k)\int_H\int_K\Phi(t,s)ds\delta(t)dt=\psi(k).
\end{align*}
Which implies that $\widetilde{L^1_\tau(G_\tau)}=\{\widetilde{f}:f\in L^1_\tau(G_\tau)\}=L^1(K)$.
\end{proof}
Also we can conclude the following corollary. Note that the same result holds for $L^1_{\tau_l}(G_\tau)$.
\begin{corollary}\label{}
{\it The map $f\mapsto S_K^\tau(f)=\widetilde{f}$ is a norm decreasing $*$-homomorphism from $L^1_{\tau_r}(G_\tau)$ onto $L^1(K)$.}
\end{corollary}
Let $\mathcal{J}_\tau^1:=\{f\in L^1_\tau(G_\tau):S_K^\tau(f)=\widetilde{f}=0\}$. Theorem \ref{31} implies that $\mathcal{J}_\tau^1$ is a closed two sided $\tau$-ideal in $L^1_\tau(G_\tau)$.
As an immediate application of the theorem \ref{31} we show that if $\mathcal{I}$ is a closed left(right) $\tau$-ideal of $L^1_\tau(G_\tau)$ with $\mathcal{J}_\tau^1\subseteq\mathcal{I}$ then $\widetilde{\mathcal{I}}=\{\widetilde{f}:f\in \mathcal{I}\}$ is a closed left(right) ideal of $L^1(K)$.
\begin{corollary}\label{32}
{\it Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$. If $\mathcal{I}$ is a closed left(right) $\tau$-ideal of $L^1_\tau(G_\tau)$ with $\mathcal{J}_\tau^1\subseteq\mathcal{I}$ then $\widetilde{I}$ is a closed left(right) ideal of $L^1(K)$.}
\end{corollary}
\begin{proof}
In general setting using Proposition \ref{31} if $\mathcal{I}$ is left(right) $\tau$-ideal of $L^1_\tau(G_\tau)$ then $\widetilde{\mathcal{I}}$ is a left(right) ideal of $L^1(K)$. Now let $\mathcal{I}$ be closed in $L^1_\tau(G_\tau)$. We show that $\widetilde{\mathcal{I}}$ is closed in $L^1(K)$. Let $\psi\in L^1(K)$ and $\{f_n\}\subset \mathcal{I}$ with $\psi=\|.\|_{L^1(K)}-\lim_{n}\widetilde{f_n}$. If $\Phi\in L_1^+$ we have $\Phi(\psi)=\|.\|_{L^1_\tau(G_\tau)}-\lim_n\Phi(\widetilde{f_n})$. Since $\mathcal{J}_\tau^1\subseteq\mathcal{I}$ for each $f\in\mathcal{I}$ we have $\Phi(\widetilde{f})\in\mathcal{I}$. Using proposition \ref{31} and also since for each $n$ we have $\Phi(\widetilde{f_n})\in \mathcal{I}$ and $\mathcal{I}$ is closed we get
$$\Phi(\psi)=\|.\|_{L^1(G_\tau)}-\lim_n\Phi(\widetilde{f_n})\in\mathcal{I}.$$
So we have $\psi=\widetilde{\Phi(\psi)}\in\widetilde{\mathcal{I}}$.
\end{proof}
In the following corollary we show that if $L^1_\tau(G_\tau)$ has an identity then $K$ should be discrete.
\begin{proposition}\label{}
{\it Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$. If Banach algebra $L^1_{\tau_r}(G_\tau)$ or $L^1_{\tau_l}(G_\tau)$ has identity, then $K$ is discrete.}
\end{proposition}
\begin{proof}
It is enough to prove the result for the left $\tau$-convolution. Let $\epsilon_{\tau_l}$ be an identity for the left $\tau$-convolution. Using Theorem \ref{31} for each $f\in L^1_{\tau_l}(G_\tau)$ and also for a.e. $k\in K$ we have
\begin{align*}
\widetilde{f}\ast\widetilde{e_\tau}(k)
&=\widetilde{f\ast(e_\tau)}(k)
\\&=\int_Hf\st e_\tau(h,k)\delta(h)dh
\\&=\int_Hf(h,k)\delta(h)dh=\widetilde{f}(k).
\end{align*}
Now since the linear map $S_K^\tau:L^1_\tau(G_\tau)\to L^1(K)$ is surjective, $\widetilde{e_\tau}$ is an identity for $L^1(K)$. Thus Theorem 19.19 and also Theorem 19.20 of \cite{HR1} imply that $K$ is discrete.
\end{proof}
\begin{corollary}\label{33}
{\it Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$. If $\tau$-convolution has identity, then $K$ is discrete.}
\end{corollary}
In the next proposition we show that the map $S_K^\tau:L^1_\tau(G_\tau)\to L^1(K)$ given by $S_K^\tau(f)=\widetilde{f} $ is not injective in the general settings. More precisely we prove that $\mathcal{J}_\tau^1=\{0\}$ if and only if $H$ be the trivial group.
\begin{proposition}\label{33.1}
{\it Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$. The linear map $S_K^\tau:L^1_\tau(G_\tau)\to L^1(K)$ is injective if and only if $H$ is trivial group.}
\end{proposition}
\begin{proof}
If $H$ be the trivial group then clearly $\widetilde{f}=0$ implies $f=0$ and so the linear map $S_K^\tau$ is injective.
To prove the converse note that for each nonnegative function $\varphi\in L^1(H)$ with $\|\varphi\|_{L^1(H)}=1$ and also for each $\psi\in L^1(K)$ we can define ${\psi}_\varphi(h,k)=\delta(h^{-1})\varphi(h)\psi(k)$ for a.e $(h,k)\in G_\tau$. Then ${\psi}_\varphi$ belongs to $L^1_\tau(G_\tau)$ and also for a.e. $k\in K$ we have
$\widetilde{{\psi}_\varphi}(k)=\psi(k)$.
Now if $S_K^\tau$ is injective and also $H$ is not the trivial group then for a fixed $\psi\in L^1(K)$ we have
$$\{0\}\subset\{\psi_\varphi-\psi_{\varphi'}:\varphi,\varphi'\in \mathcal{C}_c^+(H), \|\varphi\|_{L^1(H)}=\|\varphi'\|_{L^1(H)}=1,\varphi\not=\varphi'\}\subseteq \mathcal{J}_\tau^1,$$
which contradicts injectivity of the linear map $S_K^\tau$.
\end{proof}
As an applications of Proposition \ref{33.1} we can prove that $L^1_{\tau}(G_\tau)$ is an associative Banach $*$-algebra if and only if $H$ is trivial group.
\begin{corollary}
{\it Let $K$ be seconde countable and $\tau:H\to Aut(K)$ be a continuous homomorphism and also let $G_\tau=H\ltimes_\tau K$. The $\tau$-convolution defined in (\ref{11}) is associative if and only if $H$ is the trivial group.}
\end{corollary}
\begin{proof}
Obviously when $H$ is the trivial group, the $\tau$-convolution is associative. Conversely suppose the $\tau$-convolution is associative and $\widetilde{f}=0$. Let $\{\psi_n\}_{n=1}^\infty$ be a sequence approximate identity for $L^1(K)$ according to Proposition 2.42 of \cite{FollH} and $\Phi\in L^+_1$. Using associativity of the $\tau$-convolution for a.e. $(h,k)\in G_\tau$ we have
\begin{align*}
f(h,k)&=\lim_nf_h\ast\psi_n(k)
\\&=\lim_n\lim_mf_h\ast(\psi_n\ast\psi_m)(k)
\\&=\lim_n\lim_mf_h\ast\left(\widetilde{\Phi(\psi_n)}\ast\widetilde{\Phi(\psi_m)}\right)(k)
\\&=\lim_n\lim_m f\rst\Phi(\psi_n)\rst\Phi(\psi_m)(h,k)
\\&=\lim_n\lim_m f\lst\Phi(\psi_n)\lst\Phi(\psi_m)(h,k)
\\&=\lim_n\lim_m \widetilde{f}\ast\widetilde{\Phi(\psi_n)}\ast(\psi_m)_h(k)=0.
\end{align*}
Now proposition \ref{33.1} guarantee that $H$ is the trivial group.
\end{proof}
Next corollary shows that when $K$ is second countable, the right $\tau$-convolution is commutative if and only if $K$ is abelian and $H$ is the trivial group.
\begin{corollary}\label{33.2}
{\it Let $K$ be seconde countable and $\tau:H\to Aut(K)$ be a continuous homomorphism and also let $G_\tau=H\ltimes_\tau K$. The right $\tau$-convolution is commutative if and only if $K$ is abelian and $H$ is the trivial group.}
\end{corollary}
\begin{proof}
Clearly when $H$ is the trivial group the right $\tau$-convolution coincides with the standard convolution on $L^1(K)$ and so if $K$ is abelian the left $\tau$-convolution is commutative. Now let right $\tau$-convolution be commutative and also $f,g\in L^1_{\tau_r}(G_\tau)$. Using Proposition \ref{31} for a.e. $k\in K$ we have
\begin{align*}
\widetilde{f}\ast\widetilde{g}(k)=\widetilde{f\rst g}(k)=\widetilde{g\rst f}(k)=\widetilde{g}\ast\widetilde{f}(k).
\end{align*}
Now since $\{\widetilde{f}:f\in L^1_{\tau_r}(G_\tau)\}=L^1(K)$ we get that $L^1(K)$ is commutative and so $K$ is abelian. Also since the right $\tau$-convolution is commutative, using (\ref{13}) for each $f,g\in L^1_{\tau_r}(G_\tau)$ and a.e. $(h,k)\in G_\tau$ we have $f_h\ast\widetilde{g}(k)=g_h\ast \widetilde{f}(k)$. To show that $H$ is the trivial group, assume that $\{\psi_n\}_{n=1}^\infty$ is an approximate identity for $L^1(K)$ and also $\widetilde{f}=0$. Then if $\Phi\in L^+_1$ for a.e. $(h,k)\in G_\tau$ we have
\begin{align*}
f(h,k)&=\lim_nf_h\ast\psi_n(k)
\\&=\lim_nf_h\ast\widetilde{\Phi(\psi_n)}(k)
\\&=\lim_n\Phi(\psi_n)_h\ast \widetilde{f}(k)=0.
\end{align*}
\end{proof}
The same result for the left $\tau$-convolution can be obtained by the similar argument.
In the next theorem we show that $\tau$-convolution is commutative whenever $K$ is abelian and also we prove that the converse is true.
\begin{theorem}\label{34}
{\it Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$. The $\tau$-convolution is commutative if and only if $K$ is abelian.}
\end{theorem}
\begin{proof}
When the $\tau$-convolution is commutative, similar method as we use in Corollary \ref{33.2} works and implies that $K$ is abelian.
Now let $K$ be an abelian group. Thus the Banach $*$-algebra $L^1(K)$ is commutative and so according to the definition of the $\tau$-convolution, for each $f,g\in L^1_\tau(G_\tau)$ and also for a.e. $(h,k)\in G_\tau$ we have
\begin{align*}
f\st g(h,k)
&=2^{-1}\int_H(f_h\ast g_t(k)+f_t\ast g_h(k))\delta(t)dt
\\&=2^{-1}\int_H(g_t\ast f_h(k)+g_h\ast f_t(k))\delta(t)dt=g\st f(h,k).
\end{align*}
\end{proof}
As an immediate consequence of Theorem \ref{34} we show that when $K$ is abelian the $\tau$-convolution and $\tau$-involution makes $L^1(G_\tau)$ into a Jordan Banach $*$-algebra.
\begin{corollary}
{\it Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$ with $K$ abelian. $L^1(G_\tau)$ with respect to the $\tau$-convolution and $\tau$-involution is a Jordan Banach $*$-algebra.}
\end{corollary}
\begin{proof}
If $K$ is abelian, Theorem \ref{34} guarantee that $\tau$-convolution is commutative. According to Theorem \ref{11.3} it is sufficient to show that the $\tau$-convolution satisfies the Jordan identity. Now let $f,g\in L^1(G_\tau)$. Using Proposition \ref{11.2} and also commutativity and associativity of the standard convolution on $L^1(K)$ for a.e $(h,k)\in G_\tau$ we have
\begin{align*}
(f\st g)\st (f\st f)(h,k)-f\st\left(g\st (f\st f)\right)(h,k)
&=(f\rst g)\rst (f\rst f)(h,k)-f\lst\left(g\lst (f\lst f)\right)(h,k)
\\&=(f\rst g)_h\ast\widetilde{(f\rst f)}(k)-\widetilde{f}\ast\left(g\lst (f\lst f)\right)_h(k)
\\&=(f\rst g)_h\ast\left(\widetilde{f}\ast\widetilde{f}\right)(k)-\widetilde{f}\ast\left(\widetilde{g}\ast(f\lst f)_h\right)(k)
\\&=(f_h\ast\widetilde{ g})\ast\left(\widetilde{f}\ast\widetilde{f}\right)(k)-\widetilde{f}\ast\left(\widetilde{g}\ast(\widetilde{f}\ast f_h)\right)(k)=0.
\end{align*}
\end{proof}
\begin{remark}\label{35}
The standard convolution on $L^1(G_\tau)$ is commutative if and only if $G_\tau$ is abelian. Theorem \ref{34} guarantee that the $\tau$-convolution has some differences with standard convolution which can be defined on the function algebra $L^1(G)$ for each arbitrary locally compact group $G$. But in the sequel we show
that $\tau$-convolution on $L^1(G_\tau)$ coincides with the standard convolution of $L^1(G_\tau)$ only when $H$ is the trivial group which implies that the $\tau$-convolution on $L^1(G_\tau)$ is completely different form the standard convolution of $L^1(G_\tau)$.
\end{remark}
Let $\mathcal{A}$ be a closed $*$-subalgebra of $L^1_\tau(G_\tau)$. A sequence $\{u_n\}_{n=1}^\infty$ in $\mathcal{A}$ is called a $\tau$-sequence approximate identity for the closed $*$-subalgebra $\mathcal{A}$ if for each $f\in \mathcal{A}$ we have
\begin{equation}\label{}
\lim_{n}\|u_n\st f-f\|_{L^1(G_\tau)}=\lim_{n}\|f\st u_n -f\|_{L^1(G_\tau)}=0.
\end{equation}
Next theorem shows us that the non-associative Banach $*$-algebra $L^1_\tau(G_\tau)$ has a $\tau$-sequence approximate identity if and only if $H$ is the trivial group.
\begin{theorem}\label{36}
Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$. The $\tau$-convolution admits a $\tau$-sequence approximate identity if and only if $H$ be the trivial group.
\end{theorem}
\begin{proof}
If $H$ is the trivial group $\{e\}$ then for each continuous homomorphism $\tau:H\to Aut(K)$ we have
$$G_\tau=\{e\}\ltimes_\tau K=K.$$
which implies $L^1(G_\tau)=L^1(K)$. Thus the $\tau$-convolution coincides with the standard convolution of $L^1(K)$ and so any standard approximate identity for $L^1(K)$ is a $\tau$-sequence approximate identity for $L^1(G_\tau)$. Conversely assume that the $\tau$-convolution admits a $\tau$-sequence approximate identity $\{u_n\}_{n=1}^\infty$. Let $f\in\mathcal{J}^1_\tau$, we show that $f(h,k)=0$ for a.e. $(h,k)\in G_\tau$.
By Theorem \ref{31}, $\{\widetilde{u_n}\}_{n=1}^\infty$ is a sequence of approximate identity for $L^1(K)$. Using (\ref{13}) for a.e. $(h,k)\in G_\tau$ we have
\begin{align*}
f(h,k)&=\lim_{n}f\st u_n(h,k)
\\&=2^{-1}\lim_{n}\left(f_h\ast\widetilde{u_n}(k)+\widetilde{f}\ast (u_n)_h(k)\right)
\\&=2^{-1}\lim_{n}f_h\ast\widetilde{u_n}(k)=2^{-1}f(h,k).
\end{align*}
Thus we get $f(h,k)=0$ for a.e. $(h,k)\in G_\tau$ and so we have $\mathcal{J}^1_\tau=\{0\}$.
Now Proposition \ref{33.1} works and implies that $H$ is the trivial group.
\end{proof}
As an immediate consequence of Theorem \ref{36} we achieve that when $H$ and $K$ are second countable locally compact groups the $\tau$-convolution defined coincides with the standard convolution of $L^1(G_\tau)$ if and only if $H$ is the trivial group. Which complectly guarantee that our extension is worthwhile and also not trivial.
\begin{corollary}\label{37}
{\it Let $H$ and $K$ be second countable locally compact groups and $\tau:H\to Aut(K)$ be a continuous homomorphism and also let $G_\tau=H\ltimes_\tau K$. The $\tau$-convolution defined in (\ref{11}) coincides with the standard convolution of $L^1(G_\tau)$ if and only if $H$ is the trivial group.}
\end{corollary}
\begin{proof}
Clearly if $H$ is the trivial group the $\tau$-convolution coincides with the standard convolution of $L^1(G_\tau)$. Conversely if the $\tau$-convolution coincides with the standard convolution of $L^1(G_\tau)$ for each $f,g\in L^1(G_\tau)$ we have $f\st g=f\ast g$. Since the underlying topological space of $G_\tau$ is second countable, the standard convolution on $L^1(G_\tau)$ possess a sequence approximate identity thus the $\tau$-convolution admits a $\tau$-sequence approximate identity and using Theorem \ref{36} the result holds.
\end{proof}
Let $\mathcal{A}$ be a closed $*$-subalgebra of $L^1_\tau(G_\tau)$. A family $\{u_\alpha\}_{\alpha\in I}$ in $\mathcal{A}$ is called a $\tau$-approximate identity for the $*$-subalgebra $\mathcal{A}$ if for each $f\in \mathcal{A}$ we have
\begin{equation}\label{}
\lim_{\alpha\in I}\|u_\alpha\st f-f\|_{L^1_\tau(G_\tau)}=\lim_{\alpha\in I}\|f\st u_\alpha -f\|_{L^1_\tau(G_\tau)}=0.
\end{equation}
Although according to the Theorem \ref{36}, when $H$ and $K$ are second countable if the $\tau$-convolution on $L^1_\tau(G_\tau)$ admits a $\tau$-approximate identity then automatically $H$ should be the trivial group but in the following theorem we show that in general settings if $\Phi\in L^+_1$ the closed
$*$-subalgebra $\lambda_\Phi\left(L^1(K)\right)$ possesses a special kind of $\tau$-approximate identity.
\begin{theorem}\label{40}
{\it Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$ and also let $\Phi\in L^+_1$.
The closed $*$-subalgebra $\lambda_\Phi\left(L^1(K)\right)$ admits a bounded $\tau$-approximate identity.}
\end{theorem}
\begin{proof}
Let $\{\psi_\alpha\}_{\alpha\in I}$ be a bounded approximate identity for $L^1(K)$ with $\|\psi_\alpha\|_{L^1(K)}=1$, $\psi_\alpha=\psi_\alpha^*$ and $\psi_\alpha\ge0$ for each $\alpha\in I$. Thus for each $\phi\in L^1(K)$ we have
\begin{equation}\label{40.1}
\lim_{\alpha\in I}\|\psi_\alpha\ast\phi-\phi\|_{L^1(K)}=\lim_{\alpha\in I}\|\phi\ast\psi_\alpha-\phi\|_{L^1(K)}=0.
\end{equation}
Now let $\Phi\in L^+_1$ and also for each $\alpha\in I$ let $u_\alpha(h,k):=\Phi(\psi_\alpha)(h,k)$. Then for each $\alpha\in I$ we have $u_\alpha\in L^1_\tau(G_\tau)$ with $\|u_\alpha\|_{L^1_\tau(G_\tau)}=1$ and also $u_\alpha=u_\alpha^{*_\tau}$. We show that $\{u_\alpha\}_{\alpha\in I}$ is a $\tau$-approximate identity for $\Phi\left(L^1(K)\right)$. Using Theorem \ref{30} and also (\ref{40.1}) for each $f\in \lambda_\Phi\left(L^1(K)\right)$ with $f=\Phi(\phi)$ we have
\begin{align*}
\lim_{\alpha\in I}\|u_\alpha\st f-f\|_{L^1_\tau(G_\tau)}
&=\lim_{\alpha\in I}\|\Phi(\psi_\alpha)\st\Phi(\phi)-\Phi(\psi)\|_{L^1_\tau(G_\tau)}
\\&=\lim_{\alpha\in I}\|\Phi\left(\psi_\alpha\ast\phi-\psi\right)\|_{L^1_\tau(G_\tau)}
\\&=\lim_{\alpha\in I}\|\psi_\alpha\ast\phi-\psi\|_{L^1(K)}=0.
\end{align*}
Similarly we have
$$\lim_{\alpha\in I}\|f\st u_\alpha-f\|_{L^1_\tau(G_\tau)}=0.$$
\end{proof}
\section{{\bf $L^p(G_\tau)$ as a left Banach $L^1_{\tau_l}(G_\tau)$-module}}
The left $\tau$-convolution defined in (\ref{10.l}) can be extended from $L^1(G_\tau)$ to other $L^p(G_\tau)$ spaces with $1\le p\le\infty$. In this section we make $L^p(G_\tau)$ into a left Banach $L^1(G_\tau)$-module. First we define a module action.
Let the left module action $\lst_{(p)}:L^1_{\tau_l}(G_\tau)\times L^p(G_\tau)\to L^p(G_\tau)$ defined via $(f,u)\mapsto f\lst_{(p)}u$ where $f\lst_{(p)}u$ is given by
\begin{equation}\label{51}
f\lst_{(p)}u(h,k):=\int_Hf_t\ast u_h(k)\delta(t)dt.
\end{equation}
The left module action defined in (\ref{51}) for a.e. $(h,k)\in G_\tau$ can be written in the form
\begin{equation}\label{52}
f\st_{(p)}u(h,k)=\widetilde{f}\ast u_h(k).
\end{equation}
Because using Fubini's theorem for a.e. $(h,k)\in G_\tau$ we have
\begin{align*}
f\st_{(p)}u(h,k)
&=\int_Hf_t\ast u_h(k)\delta(t)dt
\\&=\int_H\left(\int_Kf(t,s)u(h,s^{-1}k)ds\right)\delta(h)dh
\\&=\int_K\left(\int_Hf(t,s)\delta(t)dt\right)u(h,s^{-1})kds=\widetilde{f}\ast u_h(k).
\end{align*}
The module action defined in (\ref{51}) is converges and also belongs to $L^p(G_\tau)$. Because using Fubini's theorem and also
Proposition 2.39 of \cite{FollH} we have
\begin{align*}
\|f\lst_{(p)}u\|_{L^p(G_\tau)}^p
&=\int_K\int_H|f\lst_{(p)}u(h,k)|^p\delta(h)dhdk
\\&=\int_K\int_H|\widetilde{f}\ast u_h(k)|^p\delta(h)dhdk
\\&=\int_H\left(\int_K|\widetilde{f}\ast u_h(k)|^pdk\right)\delta(h)dh
\\&=\int_H\|\widetilde{f}\ast u_h\|_{L^1(K)}^p\delta(h)dh
\\&\le\|\widetilde{f}\|_{L^1(K)}^p\int_H\|u_h\|_{L^p(K)}^p\delta(h)dh
=\|f\|_{L^1_{\tau_l}(G_\tau)}^p\|u\|_{L^p(G_\tau)}^p.
\end{align*}
\begin{remark}\label{}
When $p=1$ the left module action defined in (\ref{51}) coincides with the left $\tau$-convolution defined in (\ref{10.l}). Also when $H$ is the trivial group the left module action defined in (\ref{51}) coincides with the left module action defined on $L^p(K)$ via Proposition 2.39 of \cite{FollH}.
\end{remark}
In the next theorem we show that the left module action defined in (\ref{51}) is associative and so makes $L^p(G_\tau)$ into a left Banach $L^1_{\tau_l}(G_\tau)$-module.
\begin{theorem}\label{53}
Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$ and also let $1\le p\le\infty$. $L^p(G_\tau)$ with respect to the left module action $\st_{(p)}$ defined in (\ref{51}) is a left Banach $L^1_{\tau_l}(G_\tau)$-module.
\end{theorem}
\begin{proof}
It is sufficient to show that the left module action (\ref{51}) is associative. Let $f,g\in L^1_{\tau_l}(G_\tau)$ and $u\in L^p(G_\tau)$. Using Theorem \ref{31} and also (\ref{52}) for a.e. $(h,k)\in G_\tau$ we have
\begin{align*}
\left(f\lst g\right)\st_{(p)}u(h,k)
&=\widetilde{(f\lst g)}\ast u_h(k)
\\&=(\widetilde{f}\ast\widetilde{g})\ast u_h(k)
\\&=\widetilde{f}\ast(\widetilde{g}\ast u_h)(k)
\\&=\widetilde{f}\ast(g\lst_{(p)} u)_h(k)=f\lst_{(p)}\left(g\lst_{(p)}u\right)(h,k).
\end{align*}
\end{proof}
A sequence $\{f_n\}_{n=1}^\infty$ in $L^1_{\tau_l}(G_\tau)$ is called a left $\tau_l$-sequence approximate identity, if for each $p\ge1$ and $u\in L^p(G_\tau)$ satisfies
\begin{equation}\label{54}
\lim_{n}\|f_n\lst u-u\|_{L^p(G_\tau)}=0.
\end{equation}
In the following theorem we show that when $K$ is a second countable locally compact group the left Banach $L^1_{\tau_l}(G_\tau)$-module $L^p(G_\tau)$ admits a left $\tau_l$-sequence approximate identity.
\begin{theorem}\label{55}
Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$ with $K$ second countable. The Banach algebra $L^1_{\tau_l}(G_\tau)$ possesses a left $\tau_l$-sequence approximate identity.
\end{theorem}
\begin{proof}
Let $\{\psi_n\}_{n=1}^\infty$ be an approximate identity for $L^1(K)$ according to Proposition 2.42 of \cite{FollH}. Thus for each $p\ge1$ and also $\varphi\in L^p(K)$ we have
\begin{equation}\label{55.1}
\lim_{n}\|\psi_n\ast\varphi-\varphi\|_{L^p(G_\tau)}=0.
\end{equation}
Let $\Phi\in L_1^+$ and for all $n$ put $f_n(h,k):=\Phi(\psi_n)(h,k)$ for a.e. $(h,k)\in G_\tau$. We show that $\{f_n\}_{n=1}^\infty$ a left $\tau_l$-sequence approximate identity. Let $p\ge 1$ and also $u\in L^p(G_\tau)$. Using (\ref{51}), (\ref{55.1}) and also Theorem \ref{31} and The Dominated Convergence Theorem
we achieve
\begin{align*}
\lim_{n}\|f_n\lst u-u\|_{L^p(G_\tau)}^p
&=\lim_{n}\int_H\left(\int_K|f_n\lst u(h,k)-u(h,k)|^pdk\right)\delta(h)dh
\\&=\lim_{n}\int_H\left(\int_K|\Phi(\psi_n)\lst u(h,k)-u(h,k)|^pdk\right)\delta(h)dh
\\&=\lim_{n}\int_H\left(\int_K|\widetilde{\Phi(\psi_n)}\ast u_h(k)-u_h(k)|^pdk\right)\delta(h)dh
\\&=\lim_{n}\int_H\left(\int_K|\psi_n\ast u_h(k)-u_h(k)|^pdk\right)\delta(h)dh
\\&=\lim_{n}\int_H\left(\|\psi_n\ast u_h-u_h\|_{L^p(K)}^p\right)\delta(h)dh
\\&=\int_H\left(\lim_n\|\psi_n\ast u_h-u_h\|_{L^p(K)}^p\right)\delta(h)dh=0.
\end{align*}
\end{proof}
As an immediate consequence of Theorem \ref{55} we have the following corollary.
\begin{corollary}\label{}
{\it Let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$ with $K$ second countable. The Banach algebra $L^1_{\tau_l}(G_\tau)$ possesses a $\tau_l$-sequence approximate identity.}
\end{corollary}
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1,116,691,497,781 | arxiv | \section{Introduction}
Given a dynamical system $T:X \to X$ we generally define a \emph{speedup} of $(X,T)$ to be another dynamical system $S:Y \to Y$ conjugate to $T':X\rightarrow X$ where $T'(x)=T^{p(x)}(x)$ for some function $p:X\rightarrow\mathbb{Z}^{+}$. We will refer to $p$ in the preceding definition as the ``jump function.'' Speedups can be viewed in both the topological and measure-theoretic category. While this paper considers topological speedups, the study of speedups has a longer history in the measure-theoretic category. Of note are two papers of Neveu \cite{Neveu1, Neveu2}. The first paper proves a structure theorem relating jump functions for speedups to ``stopping times.'' His second paper extends Abramov's formula on the entropy of induced transformation to calculate the entropy of speedups with a measurable jump function. The most general theorem in this category came in a $1985$ paper of Arnoux, Ornstein, and Weiss, where they showed: for any ergodic measure preserving transformation $(X,\mathscr{B},\mu,T)$ and any aperiodic $(Y,\mathscr{C},\nu, S)$ there is a $\mathscr{B}-$measurable function $p:X\rightarrow\Z^{+}$ such that $T'(x)=T^{p(x)}(x)$ is invertible $\mu$-a.e. and $(X,\mathscr{B},\mu,T')$ is isomorphic to $(Y,\mathscr{C},\nu, S)$ \cite{AOW}. Finding a topological analogue to this theorem was the original impetus for this paper.
While the result of Arnoux, Ornstein, and Weiss shows that almost no assumptions are needed to realize one measure-theoretic system as the speedup of another, there are restrictions that arise in the topological category. The situation closely mirrors that for orbit equivalence. Recall that two invertible dynamical systems $T:X \to X$ and $S:Y \to Y$ are \emph{orbit equivalent} if $(Y,S)$ is conjugate to $T':X \to X$ where $T$ and $T'$ have the same orbits.
Dye's theorem says that any two ergodic transformations on non-atomic Lebesgue probability spaces are measure-theoretically orbit equivalent \cite{Dye}. Over 35 years later Giordano, Putnam, and Skau gave a characterization of when two minimal Cantor systems $(X,T)$ and $(Y,S)$ are topologically orbit equivalent \cite{GPS}.
Here we briefly describe the dimension groups used in their characterization (see Section \ref{SectionDimGrp} for much more). The triple $\mathcal{G}(X,T)$ is defined as \linebreak $(G(X,T),G(X,T)_+,[\mchi_X])$ where $G(X,T)$ is the group of continuous integer-valued functions modulo functions of the form\linebreak $f-fT^{-1}$, $G(X,T)_+$ is the semi-group of equivalence classes of non-negative functions, and the distinguished order unit $[\mchi_X]$ is the equivalence class of the constant function 1 on $X$. The subgroup $\Inf(\mathcal{G}(X,T))$ is the set of functions which integrate to $0$ against every $T$-invariant Borel probability measure.
\begin{theorem}\textbf{\cite[Theorem $2.2$]{GPS}:}\label{GPS Theorem} Let $(X,T)$ and $(Y,S)$ be minimal Cantor systems. The following are equivalent:
\begin{enumerate}
\item $(X,T)$ and $(Y,S)$ are orbit equivalent.
\item The dimension groups $\mathcal{G}(X,T)/\Inf(\mathcal{G}(X,T))$ and\linebreak $\mathcal{G}(Y,S)/\Inf(\mathcal{G}(Y,S))$, are order isomorphic by a map preserving the distinguished order unit.
\item There exits a homeomorphism $F:X\rightarrow Y$ carrying the $T$-invariant probability measures onto the $S$-invariant probability measures.
\end{enumerate}
\end{theorem}
In this paper, we show an analogous result but for speedups. Given unital ordered groups $\mathcal{G}$ and $\mathcal{H}$, we say that an epimorphism $\varphi:\mathcal{H} \twoheadrightarrow \mathcal{G}$ is \emph{exhaustive} if for every $h \in H_+$ and
$g \in G$ satisfying $0 < g < \varphi(h)$, there is a $h' \in H$ such that $\varphi(h')=g$ and $0<h'<h$.
\begin{main}\label{main} Let $(Y,S)$ and $(X,T)$ be minimal Cantor systems. The following are equivalent:
\begin{enumerate}
\item $(Y,S)$ is a speedup of $(X,T)$.
\item There exists an exhaustive epimorphism\linebreak $\varphi:\mathcal{G}(Y,S)/\Inf(G(Y,S))\twoheadrightarrow\mathcal{G}(X,T)/\Inf(G(X,T))$ which preserves the distinguished order unit.
\item There exists homeomorphism $F:X\rightarrow Y$, such that $F_{*}:M(X,T)\hookrightarrow M(Y,S)$ is an injection.
\end{enumerate}
\end{main}
The unital ordered groups in both of these results arise from the study of $C^*$-crossed product algebras associated with the minimal Cantor systems. In particular, $(G(X,T),G(X,T)_+)$ is the ordered $K^0$-group for the crossed product algebra $C(X) \underset{T}{\ltimes} \Z$.
Giordano, Putnam and Skau provided a dynamical characterization of the isomorphism of these objects as well, via a relation called \emph{strong} orbit equivalence. Two minimal Cantor systems $(X,T)$ and $(Y,S)$ are strong orbit equivalent if there exists a homeomorphism $F:X \to Y$ such that
$$FT(x) = S^{n(x)}F(x) \qquad \text{ and } \qquad FT^{m(x)}(x) = SF(x)$$
where $m,n:X \to \Z$ have at most one point of discontinuity each.
\begin{theorem}{\bf \cite[Theorem ]{GPS}}\label{GPS Theorem Strong} Let $(X,T)$ and $(Y,S)$ be minimal Cantor systems. The following are equivalent:
\begin{enumerate}
\item $(X,T)$ and $(Y,S)$ are strongly orbit equivalent.
\item The dimension group $\mathcal{G}(X,T)$ is order isomorphic to $\mathcal{G}(Y,S)$ by a map preserving distinguished order units.
\item $C(X) \underset{T}{\ltimes} \Z$ is isomorphic to $C(Y)\underset{S}{\ltimes} \Z.$
\end{enumerate}
\end{theorem}
In our second main theorem, we prove the following a speedup analog of Theorem~\ref{GPS Theorem Strong}. A \emph{strong speedup} is a speedup where the jump function has at most one point of discontinuity.
\begin{main}\label{mainstrong}
Let $(Y,S)$ and $(X,T)$ be minimal Cantor systems. The following are equivalent:
\begin{enumerate}
\item $(Y,S)$ is a strong speedup of $(X,T)$.
\item There is an exhaustive epimorphism $\varphi :\mathcal{G}(Y,S) \twoheadrightarrow \mathcal{G}(X,T)$ preserving the distinguished order unit.
\item There is a homeomorphism $h:X \to Y$ such that if $f \in \cob(Y,S)$ then $fh \in \cob(X,T)$.
\end{enumerate}
\end{main}
The paper is organized as follows. In Section 2 we provide relevant background on minimal Cantor systems, including Kakutani-Rohlin towers and dimension groups. In Section 3, we prove some basic facts about speedups and the jump function associated to them. In Section 4 we prove Main Theorem 2 where we characterize strong speedups, and in Section 5 we prove Main Theorem 1 for general speedups. The proofs for the two Main Theorems generally follow the same format. We prove Main Theorem 2 first as it requires fewer preliminary lemmas than Main Theorem 2. The proof of Main Theorem 1 follows the same format and we will highlight the differences and analogs statements. For the proofs, we follow many of the same techniques as in \cite{GPS} and \cite{Glasner-Weiss}, adapted to the speedup situation.
In Section 6 we define and explore the relation of \emph{speedup equivalence}. That is, when two minimal Cantor systems are speedups of one another. It follows from our main theorem that if two minimal Cantor system are orbit equivalent then they are speedup equivalent. We show in Theorem \ref{speedupequivalence} that the converse is true when the systems are have finitely many ergodic invariant measures (in particular when they are uniquely ergodic). However, Julien Melleray's forthcoming paper \emph{Dynamical Simplicies and Fra\"iss\'e Theory} showed speedup equivalence and orbit equivalence are different relations in general \cite{Melleray}.
\section{Acknowledgments}
The authors thank the referee for their close reading of our paper and for their insightful suggestions. Additionally, the first author dedicates this paper to Laura Brade. Your unwavering belief in me and support of me made this paper possible. Thank you.
\section{Preliminaries}
\subsection{Minimal Cantor systems}
Throughout this paper $X$ will always be taken to be a Cantor space, that is a compact, metrizable, perfect, zero-dimensional space. A \emph{Cantor system} will consist of a pair $(X,T)$ where $X$ is a Cantor space and $T:X\rightarrow X$ is a homeomorphism. For any $x \in X$, we denote the \emph{orbit} of $x$ by
$\mathcal{O}_{T}(x)=\{T^{n}(x):x\in\Z\}$ and the \emph{forward orbit} of $x$ by
$\mathcal{O}^+_{T}(x)=\{T^{n}(x):x\in\Z^+\}$. We will restrict our attention in this paper to
\emph{minimal Cantor systems}, i.e., Cantor systems $(X,T)$ in which every orbit is dense.
It is well-known (see \cite{Walters}) that for a Cantor system $(X,T)$, the density of all orbits $\mathcal{O}_{T}(x)$ is equivalent to the density of all forward orbits $\mathcal{O}^+_{T}(x)$.
Fix a minimal Cantor system $(X,T)$ and let $M(X)$ denote the collection of all Borel probability measures on $X$. We are interested in the set $M(X,T) \subset M(X)$ of those measures $\mu \in M(X)$ which are $T$-invariant, i.e., where
$\mu(T^{-1}(A))=\mu(A)$ for every Borel subset $A$. Because we are working with minimal Cantor systems, all $\mu \in M(X,T)$ have full support, and every $\mu \in M(X,T)$ is non-atomic.
The Bogolioubov-Krylov Theorem implies $M(X,T)\neq\emptyset$. Moreover, the set $M(X,T)$ is a compact, convex subset of $M(X)$. Further, the Ergodic Decomposition Theorem states every measure $\mu\in M(X,T)$ can be uniquely represented as an integral against a measure $\tau$ which is fully supported on the ergodic measures of $M(X,T)$.
\subsection{Kakutani-Rohlin towers}
We will consider minimal Cantor systems through a refining sequence of Kakutani-Rokhlin tower partitions. These tower partitions, defined below, were instrumental in relating minimal Cantor systems to Bratteli diagrams, and hence dimension groups, and $C^*$-algebras \cite{HPS, GPS,Glasner-Weiss}. We include proofs of some basic facts here for completeness.
\begin{defn}
A \emph{{Kakutani-Rokhlin tower partition}} (or \emph{KR-partition}) of a minimal Cantor system $(X,T)$ is a clopen partition $\mathcal{P}$ of $X$ of the form
$$
\mathcal{P}=\{T^{j}C(i): 1 \leq i\leq I,\, 0\le j<h(i)\}
$$
where $I \geq 1$, and for each $1 \leq i \leq I$, $C(i)$ is a clopen set and $h(i)$ is a positive integer.
\end{defn}
Fixing $i \in \{1,\ldots ,I\}$, we refer to the collection of sets $\{T^{j}C(i):0\le j<h(i)\}$ as the $i^{th}$ \emph{column} of the partition
$\mathcal{P}$, and $h(i)$ as the height of this column. The set $T^{j}C(i)$ is the $j^{th}$ \emph{level} of the $i^{th}$ column.
Furthermore, we refer to the union $\Union C(i)$ as the base of $\mathcal{P}$,
$$
\Base (\mathcal{P}) =\Union_{1 \leq i\leq I}C(i)
$$
and the union $\Union T^{h(i)-1}C(i)$ as the top of $\mathcal{P}$,
$$\Top (\mathcal{P}) = \Union_{0 \leq i \leq I}T^{h(i)}C(i)
$$
as the \emph{top} of $\mathcal{P}$.
Thus in the $i^{th}$ column of a KR-partition, if $0\leq j<h(i)$ then $T$ maps the $j^{th}$ level of the column onto the $(j+1)^{st}$ level of that same column. Moreover, $T$ maps $\Top (\mathcal{P})$ homeomorphically onto $\Base (\mathcal{P})$.
The following are three basic facts about the construction of KR-partitions which we will use extensively.
\begin{lemma} \label{returntime}
Let $(X,T)$ be a minimal Cantor system and let $N$ be given. There is an $\varepsilon>0$ such that if $A$ is a clopen set with $\diam (A) < \varepsilon$ then for all $x \in A$, $r_A(x)>N$.
\end{lemma}
\begin{proof}
Suppose not. Then for all $k \geq 1$, there is a clopen set $A_k$ with $\diam(A_k)<\frac{1}{k}$ and $x_k$ with $1 \leq r_{A_k}(x_k) \leq N$. By passing to a subsequence, we may assume there is an $1 \leq r\leq N$ such that $r_{A_k}(x_k)=r$ for all $k \geq 1$. Thus we have $d(T^r(x_k),x_k)< \frac{1}{k}$ for all $k \geq 1$. Let $x$ be a limit point for the sequence $\{x_k\}$, the continuity of $T^r$ implies $T^r(x)=x$. But $T$ has no periodic points, a contradiction.
\end{proof}
\begin{lemma}\label{simpletower}
Let $(X,T)$ be a minimal Cantor system and let $C \subset X$ be a nonempty clopen subset of $X$. Let $\mathcal{Q}$ be any clopen partition of $X$. Then there is a clopen KR-partition $\mathcal{P}$ of $X$ which refines $\mathcal{Q}$ and where $\Base(\mathcal{P}) = C$.
\end{lemma}
\begin{proof}
For $x \in C$, define the first return time of $x$ to $C$ as follows $r(x) = \inf \{n>0 : T^n(x) \in C\}$.
Since $T$ is minimal, $r(x)$ is a well-defined positive integer for all $x \in X$ and the function
$r: C \to \mathbb{N}$ is uniformly bounded by an integer $N\geq 1$.
By the continuity of $T^n$ for each $n \geq 1$, each set $r^{-1}\{n\}$ is closed. Therefore,
$C$ is partitioned into finitely many nonempty closed sets $r^{-1}\{n\}$, which therefore must be clopen.
For each $n \in \{1,\ldots ,N\}$, partition $r^{-1}\{n\}$ into clopen sets by considering the intersection of $r^{-1}\{n\}$ with elements of the partition $\bigvee_{j=0}^{n-1} T^{-j} \mathcal{Q}$. This results in a finite collection of nonempty, pairwise disjoint clopen sets $C(1), \ldots , C(I)$ whose union is $C$.
Fix $i \in \{1,\ldots,I\}$. The set $C(i)$ is a subset of $r^{-1}\{n\}$ for some $n \geq 1$; set $h(i)=n$.
Since $C(i)$ is a subset of an element of $\bigvee_{j=0}^{n-1} T^{-j} \mathcal{Q}$, if
$0 \leq j < h(i)$, then $T^jC(i)$ is a subset of an element of $\mathcal{Q}$. Therefore,
$$\mathcal{P}=\{T^jC(i) : 1 \le i\le I, 0 \leq j < h(i)\}$$ is the desired partition.
\end{proof}
\begin{lemma} \label{refinetower}
Let $(X,T)$ be a minimal Cantor System and let
$\mathcal{P} = \{T^j C(i) : 1 \le i \le I, 0 \leq j < h(i)\}$
be a KR-partition for $(X,T)$. Suppose $x_0 \in T^{h(1)-1}C(1)$
and $T(x_0) \in C(2)$ and $\varepsilon>0$. Then there is a KR-partition
$\mathcal{Q} = \{T^j B(i) : 1 \le i \le I', 0 \leq j < k(i)\}$ such that
\begin{enumerate}
\item $\mathcal{Q}$ refines $\mathcal{P}$,
\item $\diam(T^j B(i))<\varepsilon$ for all $i,j$,
\item $\Base(\mathcal{Q}) \subset C(2)$ and $\Top(\mathcal{Q}) \subset T^{h(1)-1}C(1)$
\item $x_0 \in T^{h(1)-1}B(1)$ and $T(x_0) \in B(2)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $U, V$ be clopen sets such that $x_0 \in U \subset T^{h(1)-1}C(1)$ and
$T(x_0) \in V \subset C(2)$. Set $B = V \cap T(U)$. Then
$T(x_0) \in B \subset C(2)$ and $x_0 \in T^{-1}B \subset T^{h(1)-1}C(1)$.
Let $r>0$ be the first return time of $T(x_0)$ to $B$. Since $T^r(x_0) \neq x_0$,
there is a clopen set $D \subset B$ containing $T(x_0)$ such that $x_0 \not \in T^{r-1}(D)$.
Let $\mathcal{R}$ be any partition with diameter less than $\varepsilon$.
Apply Lemma \ref{simpletower} to the set $B$ and the partition $\mathcal{P} \vee \{D,D^c\} \vee \mathcal{R}$ to obtain a KR-partition $\mathcal{Q}$ with base $B$. Properties (1)-(3) for $\mathcal{Q}$ follow. Because the $\mathcal{Q}$ refines $\{D,D^c\}$, the column that contains the point $T(x_0)$ does not also contain $x_0$. Therefore, after re-indexing the sets we can assume Property (4) as well.
\end{proof}
\subsection{Dimension groups} \label{SectionDimGrp}
Like the Giordano, Putnam and Skau results, our main theorems are expressed in terms of the unital dimension group $\mathcal{G}(X,T)$. These partially ordered groups are examples of dimension groups, direct limits of ordered groups $(\Z^{n_k},(\Z_+)^{n_k})$. Here we will give an abstract definition of dimension groups, which as shown by Effros, Handelman and Shen is equivalent \cite{EHS}. The relevant definitions and facts are briefly summarized, see \cite{HPS},\cite{GPS},\cite{CANT} for much more.
\begin{defn} A \emph{{partially ordered group}} is a countable Abelain group $G$ together with a special subset denoted $G_{+}$, referred to as the \emph{{positive cone}}, satisfying the following:
\begin{enumerate}
\item $G_{+}+G_{+}\subseteq G_{+}$
\item $G_{+}-G_{+}=G$
\item $G_{+}\intersect(-G_{+})=\{0\}$
\end{enumerate}
\end{defn}
Given $a,b\in G$ we will write
$$
a\le b\text{ if } b-a\in G_{+} \qquad \text{ and } \qquad
a < b\text{ if } b-a\in G_{+}\setminus\{0\}.
$$
\begin{defn}
We will further require that our partially ordered Abelian groups be \emph{unperforated} by which we mean: if $a\in G$ and $na\in G_{+}$ for some $n\in\Z^{+}$ then $a\in G_{+}$.
\end{defn}
\begin{defn}
A partially ordered group is said to satisfy the \emph{{Riesz interpolation property}} if given $a_{1},a_{2},b_{1},b_{2}\in G$ with $a_{i}\le b_{j}$ for $i,j=1,2$, then there exists $c\in G$ such that
$$
a_{i}\le c\le b_{j} \text{ for }i,j=1,2.
$$
\end{defn}
\begin{note}
We will take the following as a definition, but it is really a theorem of Effros, Handelman and Shen, \cite{EHS}.
\end{note}
\begin{defn}
A \emph{{dimension group}} is an unperforated partially ordered group $(G,G_{+})$ which satisfies the Riesz interpolation property.
\end{defn}
\begin{defn}
An \emph{order unit} of a dimension group $(G,G_{+})$ is an element $u$ of $G_{+}$ with the property that for all positive $g\in G$ there exists a positive integer $N$ such that $g\le Nu$.
\end{defn}
\begin{defn}
A dimension group is \emph{simple} if every nonzero positive element is an order unit.
\end{defn}
\begin{defn}
Given two dimension groups $(G,G_+)$ and $(H,H_+)$, and order units $u \in G$ and $v \in H$,
by a \emph{{unital dimension group isomorphism}} from $(G,G_+,u)$ to $(H,H_+,v)$, we mean a group isomorphism
$\varphi:G \to H$ such that
\begin{enumerate}
\item $\varphi(G_+) = H_+$
\item $\varphi(u) = v$.
\end{enumerate}
\end{defn}
While unital ordered isomorphisms of $\mathcal{G}(X,T)$ and $\mathcal{G}(Y,S)$ play an important role in the characterization of minimal Cantor systems up to orbit equivalence, the relevant relation for us is the existence of an \emph{epimorphism} from one unital ordered group to another.
\begin{defn}
Given two dimension groups $(G,G_+)$ and $(H,H_+)$, and order units $u \in G$ and $v \in H$,
by a \emph{{unital dimension group epimorphism}} from $(G,G_+,u)$ onto $(H,H_+,v)$, we mean
a group epimorphism $\varphi:G \to H$ such that
\begin{enumerate}
\item $\varphi(G_+) \subseteq H_+$
\item $\varphi(u) = v$.
\end{enumerate}
\end{defn}
As it turns out, we will require one additional order property for the unital dimension group epimorphisms that arise in this paper.
\begin{defn}
Let $\mathcal{G}=(G,G_+,u)$ and $\mathcal{H}=(H,H_+,v)$ be unital ordered groups.
We say that a unital dimension group epimorphism from $\mathcal{G}$ to $\mathcal{H}$ is
\emph{exhaustive} if for every $g \in G_+$ and
$h \in H$ satisfying $0 < h < \varphi(g)$, there is a
$g' \in G$ such that $\varphi(g')=h$ and $0<g'<g$.
\end{defn}
Note that every dimension group isomorphism is an exhaustive epimorphism. Further, a unital dimension group epimorphism having the exhaustive property implies $\varphi(G_+)=H_+$.
We finish this subsection by providing several examples to illustrate the notion of an exhaustive epimorphism. The examples will be of the following forms. Any ordered group of these forms represent a simple dimension group.
\begin{enumerate}
\item $(G_1,(G_1)_+)$ where $G_1$ is a countable dense subgroup of $\mathbb{R}^2$, and $(G_1)_+ = \{ (x,y) \in G_1 : x>0 \text{ and } y>0\} \cup \{(0,0)\}$.
\item $(G_2,(G_2)_+)$ where $G_2$ is a countable dense subgroup of $\mathbb{R}^2$, and $(G_2)_+ = \{ (x,y) \in G_2 : y>0\} \cup \{(0,0)\}$.
\item $(G_3,(G_3)_+)$ where $G_3$ is a countable dense subgroup of $\mathbb{R}$, and $(G_3)_+ = \{ x \in \mathbb{R} : x \geq 0\}$.
\end{enumerate}
If the example is of the type $(G_1,(G_1)_+)$ or $(G_3,(G_3)_+)$ above, we will say that the subset of $\R^2$ or $\R$ has the usual ordering. If it is of the type $(G_2,(G_2)_+)$ above, we will say that the ordering is given by the second coordinate. Because these are all simple, any positive element can play the role of the distinguished order unit.
\begin{example}\label{exh:ex1}
Consider the group $G = \Q \times \Q$ with the usual ordering and the map $\varphi: (G,G_+) \to (G,G_+)$ where $\varphi(x,y) = \left( \frac{2}{3} x + \frac{1}{3} y,\frac{1}{3} x + \frac{2}{3} y\right)$. This is a bijection from $G$ to itself with $\varphi(G_+) \subsetneq G_+$, and thus not exhaustive.
\end{example}
\begin{example}\label{exh:ex2}
Consider the group $G_1 = \Q \times \Q$ with the usual ordering and $G_2 = \Q \times \Q$ with ordering given by the second coordinate. Consider the map $\varphi: (G_1,(G_1)_+) \to (G_2,(G_2)_+)$ where $\varphi(x,y) = \left( x-y,x+y\right)$. This is a bijection from $G_1$ to $G_2$, and $\varphi((G_1)_+) \subset (G_2)_+$. However, this is not exhaustive. For example, $(-1,1) \in (G_2)_+$, but $\varphi^{-1}(-1,1) = (0,1) \notin (G_1)_+$.
On the other hand, let us consider these same ordered groups modulo their respective infinitesimal subgroups, which we denote as $(H_1,(H_1)_+)$ and $(H_2,(H_2)_+)$. Then $(G_1,(G_1)_+) = (H_1,(H_1)_+)$, but $(H_2,(H_2)_+)$ is isomorphic to $(\Q , \Q_+)$ as any element of the form $(x,0)$ is an infinitesimal in $(G_2,(G_2)_+)$. Then the induced map from $\varphi$ from $(H_1,(H_1)_+)$ to $(\Q,\Q_+)$, which we denote $\hat{\varphi}$, has the formula $\hat{\varphi}(x,y) = x+y$, and is an exhaustive epimorphism. So in particular, one cannot deduce the exhaustive property of a group epimorphism from the induced map on the groups modulo infinitesimals.
\end{example}
\begin{example}\label{exh:ex3}
Suppose $\alpha, \beta$ are irrational numbers that are not rationally related.
Consider the group $G_1 = \{ (a \alpha + b, a \alpha + c \beta + d) \in \mathbb{R}^2 : a,b,c,d \in \Z \}$
with the usual ordering. Let $\varphi : G_1 \to \R$ be projection onto the second coordinate, let $G_3 = \varphi(G_1) \subset \R$ with the usual ordering. Note that in this case $\varphi ((G_1)_+) = (G_3)_+$.
Now fix $c,d \in \Z$ such that $0 < c \beta + d < 1$. Let $h = c\beta +d \in G_3$, and $g = (1,1) \in G_1$. Then we have $0 < h < \varphi(g)$. However, $\varphi^{-1}\{h\} = \{(b,c \beta + d) : b \in \Z \}$. If $(b,c \beta + d) > (0,0)$ then $b>0$, but if $(b,c \beta + d) < (1,1)$ then $b<1$. Therefore, there is no element $g'$ in $G_1$ satisfying both $\varphi(g') = h$ and $0<g'<g$.
\end{example}
\begin{remark}
Note that in Example \ref{exh:ex3} the kernel of $\varphi$ is $\{0\} \times \Z$. In particular, $\ker \varphi$ is a discrete subset of $\R^2$. For the setting where $G_1$ is a dense subset of $\R^2$ and $G_3$ is a dense subset of $\R$ each with the usual ordering, one can prove that a group epimorphism $\varphi:G_1\to G_3$ is exhaustive if and only if $\ker \varphi$ is a dense subset of a line in $\R^2$. There are certainly generalizations of this fact, but a complete characterization of when an ordered group epimorphism is exhaustive appears to be subtle and is beyond the scope of this paper. It could be that the key to such a characterization is to consider these ordered groups modulo infinitesimals as dense subsets of affine maps on compact sets as suggested by the referee of this paper. With this perspective, an epimorphism of ordered groups is the restriction of an affine map on a space to a subset.
\end{remark}
\subsection{Dynamical dimension groups}
For a minimal Cantor system $(X,T)$, consider $C(X,\Z)$, the group of continuous integer-valued functions on $X$. Let $\cob(X,T)$ denote the subgroup of \emph{$T$-coboundaries}, i.e. $\cob(X,T)=\{f-f\circ T:f\in C(X,\Z)\}$.
Furthermore, define
$$
G(X,T)=C(X,\Z)/\cob(X,T).
$$
Next we define the \emph{positive cone} $G(X,T)_+ \subset G(X,T)$ to be
$$
G(X,T)_{+}=\{[f]: f\in C(X,\Z), f(X) \subset \Z_+\}.
$$
and the distinguished order unit $[\mchi_X]$ to be the equivalence class of the constant function one.
The triple $\mathcal{G}(X,T) = (G(X,T),G(X,T)_+,[\mchi_X])$ forms a unital dimension group and further,
every simple, unital dimension group except $(\Z,\Z_+,1)$ arises as $(G(X,T),G(X,T)_+,[\mchi_X])$ for some minimal Cantor system $(X,T)$ as shown by Giordano, Putnam, and Skau, \cite{GPS}.
The second unital dimension group we will consider is a quotient of $\mathcal{G}(X,T)$ above.
Set $$\Inf(X,T) = \left\{f \in C(X,\Z) : \int f d \mu = 0 \text{ for all } \mu \in M(X,T) \right\}$$
and notice $ \cob(X,T)\subset \Inf(X,T)$. We obtain another unital dimension group by considering the group
$G(X,T)/\Inf(X,T)$, the positive cone given by the equivalence classes of nonnegative functions, and the
distinguished order unit given by the equivalence class of the constant function one. We will denote this
unital dimension group by $\mathcal{G}(X,T)/\Inf(X,T)$.
\section{Speedup Basics}
Here we consider some basic properties of a speedup of a minimal Cantor system.
\begin{defn}
Given minimal Cantor systems $(X,T)$ and $(Y,S)$, we say that $(Y,S)$ is a \emph{speedup} of $(X,T)$ if there is a function $p:X\rightarrow\mathbb{Z}^{+}$ such that $(Y,S)$ is topologically conjugate to $(X,T')$ where $T'(x)=T^{p(x)}(x)$ for all $x \in X$.
\end{defn}
We will refer to the function $p$ above a jump function for the speedup. While we place no continuity conditions on the jump function in the definition above, the assumption that $(X,T')$ is conjugate to a minimal Cantor system $(Y,S)$ implies that $(X,T')$ is also a minimal Cantor system, which implies that $p$ is lower semicontinuous as seen below.
\begin{prop} Suppose $(X,T)$ is a minimal Cantor system. Let $p:X\rightarrow\mathbb{Z}^{+}$ and suppose that $T'(x)=T^{p(x)}(x)$ is continuous. Then $p$ is lower semicontinuous.
\end{prop}
\begin{proof}
It suffices to show that for every $n\in\Z^{+}$ $p^{-1}(\{n\})$ is closed. The lower semicontinuity of $p$ will then follow since for any $\alpha \in \R$, $\{x:p(x)>\alpha\}$ is the complement of union of finitely many sets of the form $p^{-1}(\{n\})$.
To this end,
let $n\in\mathbb{Z}^{+}$, $\{x_{m}\}_{m\ge 1}\subseteq p^{-1}(\{n\})$ and $x\in X$ such that $x_{m}\rightarrow x$. Since both $T'$ and $T^{n}$ are continuous, we have
$$
T'(x_{m})\rightarrow T'(x)\text{ and } T^{n}(x_{m})\rightarrow T^{n}(x).
$$
Since for every $m$, $T'(x_{m})=T^{n}(x_{m})$ we have that
$$
T'(x)=T^{n}(x).
$$
We may conclude $p(x)=n$ as a result of $T$ being aperiodic.
\end{proof}
It follows from the above that if $p$ is bounded then each of the sets $p^{-1}(\{n\})$ is open as well. Thus we have the following.
\begin{coro}
Let $(X,T)$ be a minimal Cantor system. Suppose $(X,T')$ is a minimal Cantor system, where $p:X\rightarrow\mathbb{Z}^{+}$, and $T'(x)=T^{p(x)}(x)$ for all $x \in X$.
Then $p$ is continuous if and only if $p$ is bounded.
\end{coro}
In the case where $(Y,S)$ is a speedup of $T$ with a bounded jump function, we call $S$ a \emph{bounded speedup} of $T$. Bounded speedups can be viewed as a topological analogue of the speedups studied by Neveu, where the jump function is integrable. Bounded speedups are interesting in their own right and are the subject of the paper \cite{AAO}.
As in the case for orbit equivalence, strong topological speedups, where the jump function has a single point of discontinuity deserve special attention.
\begin{defn}
Given a minimal Cantor systems $(X,T)$ and $(Y,S)$, we say that $(Y,S)$ is a \emph{strong speedup} of $(X,T)$ if there is a function $p:X\rightarrow\mathbb{Z}^{+}$ with at most one point of discontinuity in $X$ such that $(Y,S)$ is topologically conjugate to $(X,T')$ where $T'(x)=T^{p(x)}(x)$ for all $x \in X$.
\end{defn}
\section{Proof of Main Theorem \ref{mainstrong}}
In this section we prove Main Theorem \ref{mainstrong} which is our characterization theorem for strong speedups. We will prove the various implications as separate theorems in the subsequent three subsections.
\subsection{Proof of (1) $\implies$ (3)}
In Section 3 we defined $\cob(X,T)=\{f-f\circ T:f\in C(X,\Z)\}$. Note that by replacing $f$ with $g=-fT$ in $f-f\circ T$, we obtain $f-f\circ T = g- g \circ T^{-1}$, so
$\cob(X,T)$ can be written as $\{g-g\circ T^{-1}:g\in C(X,\Z)\}$.
\begin{theorem}
Let $(Y,S)$ and $(X,T)$ be minimal Cantor systems. If $(Y,S)$ is a strong speedup of $(X,T)$ then there is a homeomorphism $h:X \to Y$ such that if $f \in \cob(Y,S)$ then $fh \in \cob(X,T)$.
\end{theorem}
\begin{proof}
Suppose $(X,T)$ and $(Y,S)$ are minimal Cantor systems and $(Y,S)$ is conjugate to $(X,T')$ where $T'(x) = T^{p(x)}(x)$ for all $x \in X$ and $p:X \to \Z$ has a single point of discontinuity $x_0$.
Let $h:X \to Y$ denote the conjugacy between $(Y,S)$ and $(X,T')$. Then $h$ is a homeomorphism. It remains to show that if $f \in \cob(Y,S)$ then $fh \in \cob(X,T)$.
Suppose $f \in \cob(Y,S)$. Then $f=g-gS^{-1}$ for $g \in C(Y,\Z)$. Note that for any constant $k$, if $g'=g+k$ then $f=g'-g'S^{-1}$. Therefore, we may assume without loss of generality that $g(h(x_0))=0$.
Since $g \in C(Y,\Z)$, we have $g=\sum_{i=1}^I n_i \mathbbm{1}_{B_i}$ for some nonzero integers $n_i$ and clopen sets $B_i \subset Y$. Since $g(h(x_0))=0$, we may assume $h(x_0) \not \in \cup_{i=1}^I B_i$. Set $A_i = h^{-1}B_i$ and $A_i(k) = A_i \cap p^{-1}\{k\}$. Since $p$ is continuous away from $x_0$, we have that each $A_i(k)$ is clopen and there are only finitely many sets $A_i(k)$. Compiling all of this information, we see
\begin{align*}
gh-gS^{-1}h & = \sum_i n_i\mathbbm{1}_{B_i}h -n_i\mathbbm{1}_{SB_i}h \\
& = \sum_i n_i\mathbbm{1}_{A_i} -n_i\mathbbm{1}_{h^{-1}SB_i} \\
& = \sum_{i,k} k\mathbbm{1}_{A_i(k)} -k\mathbbm{1}_{T^k A_i(k)} \\
& = \sum_{i,k} k\mathbbm{1}_{A_i(k)} -k\mathbbm{1}_{A_i(k)} T^{-k} \in \cob(X,T).
\end{align*}
\end{proof}
\subsection{Proof of (3) $\implies$ (2)}
We begin with some preliminary lemmas.
\begin{lemma} \label{towercob}
Let $f \in C(X,\mathbb{Z})$. Then $f \in \cob(X,T)$ if and only if there is an $\varepsilon>0$ such that for any clopen set $A$,
if $\diam(A) < \varepsilon$ then for all $x \in A$, $$ \sum_{j=0}^{r_A(x)-1} fT^j(x) = 0.$$
\end{lemma}
\begin{proof}
Suppose $f \in \cob(X,T)$. Then $f = gT-g$ where $g$ is uniformly continuous. Therefore there is an $\varepsilon>0$ such that if $d(x,y)<\varepsilon$, $g(x)=g(y)$. Let $A$ be any clopen set with $\diam(A)< \varepsilon$.
Let $r_A : A \to \N$ denote the first return time function to $A$.
Then for any $x \in A$ we have $gT^{r_A(x)}(x)-g(x) =0$ and thus
$$
\sum_{j=0}^{r_A(x)-1} fT^j(x) = \sum_{j=0}^{r_A(x)-1} (gT - g )T^j(x)= gT^{r_A(x)}(x)-g(x) =0
$$
Conversely, suppose there is a clopen set $A$ such that for all $x \in A$,
$ \sum_{j=0}^{r_A(x)-1} fT^j(x) = 0$. Fix $a \in A$. Set $g(a)=0$, and
for $1 \leq j < r_A(a)$, set $$gT^j(a) = \sum_{k=0}^{j-1}fT^k(a)$$
This defines $g$ on all of $X$ since every point in $X$ is equal to
$T^j(a)$ for some $a \in A$ and $0 \leq j < r_A(a)$.
If $x \in A$ then $$gT(x)-g(x) = f(x).$$
Now suppose $x = T^j(a)$ where $0<j<r_A(x)-1$. Then
$$gT(x)-g(x) = \sum_{k=0}^{j}fT^k(a) - \sum_{k=0}^{j-1}fT^k(a) = fT^j(a) = f(x).$$
Finally, assume $x = T^{r_A(a)-1}(a)$. Then $T(x) \in A$, so
$$gT(x)-g(x) = - \sum_{k=0}^{r_A(a)-2}fT^k(a).$$
But since $ \sum_{k=0}^{r_A(a)-1}fT^k(a) =0$, we have
$$ - \sum_{k=0}^{r_A(a)-2}fT^k(a) =fT^{r_A(a)-1}(a) = f(x). $$
Thus $gT-g=f$ on all of $X$.
\end{proof}
\begin{lemma} \label{towerpos}
Let $(X,T)$ be a minimal Cantor system and suppose $[f] \in G(X,T)$. Then $[f] >0$ if and only if there is an $\varepsilon>0$ such that for any clopen set $A$, if $\diam(A) < \varepsilon$ then for all $x \in A$, $$ \sum_{j=0}^{r_A(x)-1} fT^j(x) > 0.$$
\end{lemma}
\begin{proof}
Suppose $[f] >0$. Then $f$ is equal to a nonzero, nonnegative function $g$ plus an element $k$ of $\cob(X,T)$.
There is a nonempty clopen set $C$ such that for $x \in C$, $g(x)>0$. Since $T$ is minimal, there is an $r>1$ such that for all $x \in X$, $\{x,T(x), \ldots T^{r-1}(x) \} \cap C \neq \emptyset$. Therefore, if $n > 2r$, we have
$$\sum_{j=0}^{n-1} gT^j(x) \geq \frac{1}{r+1} n>0.$$
By Lemmas \ref{towercob} and \ref{returntime}, there is an $\varepsilon>0$ such that for any clopen set $A$, if $\diam(A) < \varepsilon$ then for all $x \in A$, $r_A(x) > 2r$ and
$$ \sum_{j=0}^{r_A(x)-1} kT^j(x) = 0. $$ Therefore, we have
$$ \sum_{j=0}^{r_A(x)-1} fT^j(x) = \sum_{j=0}^{r_A(x)-1} gT^j(x) > 0.$$
Conversely, suppose there is a clopen set $A$ such that for all $x \in A$, $$ \sum_{j=0}^{r_A(x)-1} fT^j(x) > 0.$$
For $x \in A$, set
$$f'(x) = \sum_{j=0}^{r_A(x)-1} fT^j(x)$$ otherwise set $f'(x)=0$.
Then $f'(x) \geq 0$ for all $x \in X$ and $f'$ is a nonzero function.
Further, we have
$$ \sum_{j=0}^{r_A(x)-1} fT^j(x) = \sum_{j=0}^{r_A(x)-1} f'T^j(x)$$
which by Lemma \ref{towercob} implies $[f]=[f']>0$.
\end{proof}
\begin{lemma} \label{functionineq}
Let $(X,T)$ be a minimal Cantor system. Suppose $[f], [g] \in G(X,T)$ such that $0<[f]<[g]$. Then there is an $f'\in C(X,\Z)$ such that $0 \leq f'(x) \leq g(x)$ for all $x \in X$ and $[f']=[f]$.
\end{lemma}
\begin{proof}
Both $[f]$ and $[g-f]$ are in $G(X,T)_+$. Take $\varepsilon$ to be smaller than $\varepsilon_f$ and $\varepsilon_{g-f}$ from Lemma \ref{towerpos}. Use Lemma \ref{refinetower} to create a KR-partition
$\mathcal{P} = \{T^jA(i) : 1 \leq i \leq I, 0 \leq j < h(i)\}$ over a set $A$ with diameter less than $\varepsilon$ such that $g$ and $f$ are constant on every level of $\mathcal{P}$.
Then for any $x \in A(i)$, we have
$$\sum_{j=0}^{h(i)-1} gT^j(x) \geq \sum_{j=0}^{h(i)-1} fT^j(x) \geq 0.$$
Therefore, it is possible to select values for a new function $f'$ on levels of the $i$th column of $\mathcal{P}$ so that $g(x) \geq f'(x) \geq 0$ for all $x$ and so that $$\sum_{j=0}^{h(i)-1} f'T^j(x) = \sum_{j=0}^{h(i)-1} fT^j(x).$$
By Lemma \ref{towercob}, $[f']=[f]$.
\end{proof}
By setting $g$ equal to the indicator function of a clopen set in Lemma \ref{functionineq}, we immediately obtain the following.
\begin{coro} \label{clopensubset}
Let $(X,T)$ be a minimal Cantor system and $A \subset X$ a clopen set. Suppose $[f] \in G$ such that $0<[f]<[\mathbbm{1}_A]$. Then there is a clopen set $C \subset A$ such that $[\mathbbm{1}_C]=[f]$.
\end{coro}
We are now ready to prove (3) $\implies$ (2) in Main Theorem \ref{mainstrong}.
\begin{theorem}
Let $(X,T)$ and $(Y,S)$ be minimal Cantor systems. Suppose there is a homeomorphism $h:X \to Y$ such that if $f \in \cob(Y,S)$ then $fh \in \cob(X,T)$. Then there is an exhaustive epimorphism $\varphi :\mathcal{G}(Y,S) \twoheadrightarrow \mathcal{G}(X,T)$ preserving the distinguished order unit.
\end{theorem}
\begin{proof}
Suppose there is a homeomorphism $h:X\to Y$ such that if $f \in \cob(Y,S)$ then $fh \in \cob(X,T)$. Define $\varphi:G(Y,S) \to G(X,T)$ via $\varphi:[f] \mapsto [fh]$. Then $\varphi$ is a homomorphism, $\varphi(G(Y,S)_+)=G(X,T)_+$, and $\varphi[\mathbbm{1}_Y]=[\mathbbm{1}_X]$. Moreover, $\varphi$ is onto since $\varphi[fh^{-1}]=[f]$.
To see the exhaustive property, suppose that $0<[f]<\varphi[g]$ where $[g]\in G(Y,S)_+$ and $[f]\in G(X,T)$. Then $0<[f]<[gh]$ so by Lemma \ref{functionineq}, there is an $f' \in C(X,\Z)$ such that $0 \leq f'(x) \leq gh(x)$ for all $x \in X$ and $[f']=[f]$. Now consider $[f'h^{-1}] \in G(Y,S)$. Since $f'h^{-1}(y) \geq 0$ for all $y \in Y$ we have $[f'h^{-1}]\in G(Y,S)_+$ and $\varphi[f'h^{-1}]=[f']=[f]<[gh]$. Since $[f']=[f] \neq [0]$, $f'$ is a nonzero function, thus $0<[f]<[gh]$ as desired.
\end{proof}
\subsection{Proof of (2) $\implies$ (1)}
This is the most complicated of the implications, and thus we will need multiple preliminary lemmas. A central theme of these lemmas is to convert relations between unital dimension groups to relations between sets.
\begin{lemma}\label{clopenpartition}
Let $(X,T)$ and $(Y,S)$ be minimal Cantor systems. Assume there is an exhaustive epimorphism $\varphi :\mathcal{G}(Y,S) \twoheadrightarrow \mathcal{G}(X,T)$. Let $A \subset X$ and $B \subset Y$ be clopen sets such that $\varphi[\mathbbm{1}_B]=[\mathbbm{1}_A]$. Then for any clopen partition $\{A_1,A_2,\ldots,A_n\}$ of $A$ there is a clopen partition $\{B_1,B_2,\ldots,B_n\}$ of $B$ such that $\varphi[\mathbbm{1}_{B_i}]=[\mathbbm{1}_{A_i}]$ for all $1 \leq i \leq n$.
Moreover, given any $n$ distinct
points $y_1,y_2,\ldots ,y_n \in B$
we may select $B_i$ to contain $y_i$ for $i=1,2,\ldots,n$.
\end{lemma}
\begin{proof}
We work recursively, set $A_0=\emptyset$ and $B_0=\emptyset$.
Now suppose $1 \leq k \leq n$ and we have defined disjoint clopen sets $B_0,B_1,\ldots B_{k-1}$ such that $B_i\subset B$ and $\varphi[\mathbbm{1}_{B_i}]=[\mathbbm{1}_{A_i}]$ for $0\leq i < k$. There are two cases: Either $k<n$ or $k=n$.
If $k<n$, then $$0 < [\mathbbm{1}_{A_k}] < [\mathbbm{1}_A]-\sum_{i=0}^{k-1} [\mathbbm{1}_{A_i}]
=\varphi[\mathbbm{1}_B]-\sum_{i=0}^{k-1} \varphi [\mathbbm{1}_{B_i}].$$
Since $\varphi$ is exhaustive, there is an $[f]\in G(Y,S)$ such that $$0<[f]<[\mathbbm{1}_B]-\sum_{i=0}^{k-1} [\mathbbm{1}_{B_i}]$$ and $\varphi[f]=[\mathbbm{1}_{A_k}]$. By Corollary \ref{clopensubset}, there is a clopen set $B_k \subset B\setminus \cup_{i=0}^{k-1} B_i$ such that $[f]=[\mathbbm{1}_{B_k}]$ so $\varphi[\mathbbm{1}_{B_k}]=[\mathbbm{1}_{A_k}]$.
For $k=n$, set $B_n=B\setminus \cup_{i=0}^{n-1} B_i$. It follows that
$$\varphi[\mathbbm{1}_{B_n}]=\varphi[\mathbbm{1}_B]-\sum_{i=0}^{n-1} \varphi [\mathbbm{1}_{B_i}]
=[\mathbbm{1}_A]-\sum_{i=0}^{n-1} [\mathbbm{1}_{A_i}]
=[\mathbbm{1}_{A_n}].$$
For the ``Moreover...'' part, assume that $y_1 \in B_i$. By minimality there is a $k \in \mathbb{Z}$ such that $S^k(y_1) \in B_1$ (in fact there are infinitely many).
Furthermore, there is a clopen set $U \subset B_i$ containing $y_1$ such that $S^k(U) \subset B_1$. Set $B_1' = (B_1\setminus S^kU) \cup U$, and set
$B_i' = (B_i\setminus U) \cup S^k(U)$. Then $y_1 \in B_1'$ and it is not difficult to see that $[\mathbbm{1}_{B_1}]=[\mathbbm{1}_{B_1'}]$ and $[\mathbbm{1}_{B_i}]=[\mathbbm{1}_{B_i'}]$.
To arrange $y_i \in B_i$ for $i>1$, we apply the same idea. We simply need to take care for these points that we choose $U$ and $k \in \mathbb{Z}$ so that $U \cap S^k(U)$ does not contain $y_j$ for $j<i$.
\end{proof}
\begin{lemma} \label{clopenpartition2}
Let $(X,T)$ and $(Y,S)$ be minimal Cantor systems. Assume there is an epimorphism $\varphi :\mathcal{G}(Y,S) \twoheadrightarrow \mathcal{G}(X,T)$. Let $A \subset X$ and $B \subset Y$ be clopen sets such that $\varphi[\mathbbm{1}_B]=[\mathbbm{1}_A]$. Then for any clopen partition $\{B_1,B_2,\ldots,B_n\}$ of $B$ there is a clopen partition $\{A_1,A_2,\ldots,A_n\}$ of $A$ such that $\varphi[\mathbbm{1}_{B_i}]=[\mathbbm{1}_{A_i}]$ for all $1 \leq i \leq n$.
Moreover, given any $n$ distinct points $x_1,x_2,\ldots ,x_n \in A$ we may select $A_i$ to contain $x_i$ for $i=1,2,\ldots,n$.
\end{lemma}
The proof of Lemma \ref{clopenpartition2} is nearly identical to Lemma \ref{clopenpartition} except we do not need the exhaustive property, only that $\varphi(G(Y,S)_+)=G(X,T)_+$.
To say a bit more, note that if $0 < [\mathbbm{1}_{B_i}] < [\mathbbm{1}_B]$ then it follows that $0<\varphi [\mathbbm{1}_{B_i}] < [\mathbbm{1}_A]$ and thus there is a clopen set $A_i \subset A$ with $[\mathbbm{1}_{A_i}]=\varphi [\mathbbm{1}_{B_i}]$ by Corollary \ref{clopensubset}.
Given an exhaustive epimorphism from one unital dimension group to another
$\varphi:\mathcal{G}(Y,S) \to \mathcal{G}(X,T)$, Lemma \ref{clopenpartition} above will allow us to take a KR-partition for $(Y,S)$ and mirror that partition in $X$ in a way that respects $\varphi$. While the partition in $X$ need not be a KR-partition, it will be KR-like in the following sense.
\begin{defn}
Let $(X,T)$ be a minimal Cantor system. We say a clopen partition
$\{A(i,j): 1 \leq i \leq I, 0\leq j < h(i) \}$ is \emph{KR-like} provided for each $i$ and $0\leq j,j'<h(i)$, $[\mathbbm{1}_{A(i,j)}]=[\mathbbm{1}_{A(i,j')}]$.
\end{defn}
The following indicates how we will use the above lemmas to mirror a KR-partition for one minimal Cantor system $(Y,S)$ in a KR-like partition for another minimal Cantor system $(X,T)$.
\begin{lemma}\label{copytower}
Suppose there is an exhaustive epimorphism $\varphi: \mathcal{G}(Y,S) \twoheadrightarrow \mathcal{G}(X,T)$. Fix $x_0 \in X$.
Let $\mathcal{Q}=\{S^jB(i) : 1 \leq i \leq I, 0 \leq j < h(i)\}$ be a clopen KR-partition of $Y$ with $y_0 \in S^{h(1)-1}B_1$ and
$S(y_0) \in B_2$.
Then there is a KR-like partition $\mathcal{P} = \{A(i,j): 1 \leq i \leq I, 0 \leq j < h(i)\}$ of $X$ such that:
\begin{enumerate}
\item $x_0 \in A(1,h(1)-1)$ and $T(x_0) \in A(2,0)$.
\item For every $(i,j)$, $\varphi([\mathbbm{1}_{S^jB_i}])=[\mathbbm{1}_{A(i,j)}]$.
\end{enumerate}
\end{lemma}
\begin{proof}
Since $\varphi$ is unital, we have $\varphi[\mathbbm{1}_{Y}]=[\mathbbm{1}_{X}]$. Hence, by Lemma~\ref{clopenpartition} there exists clopen partition $\mathcal{P}=\{A(i,j): 1 \leq i \leq I, 0 \leq j < h(i)\}$ such that for each $(i,j)$
$$
\varphi([\mathbbm{1}_{S^jB(i)}])=[\mathbbm{1}_{A(i,j)}].
$$
Notice that the KR-like property of $\mathcal{P}$ follows from the above relation.
Additionally, Lemma~\ref{clopenpartition} ensures that $x_0\in A(1,h(1)-1)$ and $T(x_0)\in A(2,0)$.
\end{proof}
Once a KR-like partition in $X$ is created, we may use the following lemma to begin to define a speedup $T'$ of $T$.
\begin{lemma}[Strong Speedup Lemma]\label{speedup}
Let $(X,T)$ be a minimal Cantor system and let $A,C$ be nonempty disjoint clopen subsets of $X$. If $\mathbbm{1}_{A}-\mathbbm{1}_{C}\in\cob(X,T)$, then there exists a continuous $p:A\rightarrow \Z^{+}$ such that $T'$, defined by $T'(x)= T^{p(x)}(x)$, is a homeomorphism from $A$ to $C$.
\end{lemma}
\begin{proof}
Let $A$ and $C$ be as above. Since $\mathbbm{1}_{A}-\mathbbm{1}_{C}\in\cob(X,T)$, there exists a KR-partition $\mathcal{P}=\{T^{j}D(i):1\le i\le n,\, 0\le j<h(i)\}$, such that
$\mathbbm{1}_{A}$ and $\mathbbm{1}_{C}$ are constant on all levels of $\mathcal{P}$ and $\mathbbm{1}_{A}-\mathbbm{1}_{C}$ sums to zero on all columns of $\mathcal{P}$.
Fix $i$ and consider the $i^{th}$ column of $\mathcal{P}$. By the above construction, there exists a bijection
$$\Gamma: \{j : 0 \leq j < h(i), T^jD(i) \subset A\}
\to \{j : 0 \leq j < h(i), T^jD(i) \subset C\}.$$
For $x \in T^jD(i) \subset A$, set
$$p(x) = \min\{n > 0 : T^n(x) \in T^{\Gamma(j)}D(i) \}.$$
That $p(x)$ is well-defined follows from the fact that the forward orbit of $x$ is dense. That $p$ is continuous on $T^jD(i)$ follows from the fact that both $T^jD(i)$ and $T^{\Gamma(j)}D(i)$ are clopen.
If $\Gamma(j)>j$ then $p$ is simply the constant $\Gamma(j)-j$ on $T^jD(i)$, and
$T^{\Gamma(j)-j}$ yields a homeomorphism from $T^jD(i)$ to $T^{\Gamma(j)}D(i)$.
Now suppose $\Gamma(j)<j$. Then $p(x) = r(x) - (j-\Gamma(j))$ where $r(x)$ is the first return time of $x$ to $T^jD(i)$. It follows from the fact that the first return map $x \mapsto T^{r(x)}(x)$ is a homeomorphism from $T^jD(i)$ to itself that $x \mapsto T^{p(x)}(x)$ is a homeomorphism from $T^jD(i)$ to $T^{\Gamma(j)}D(i)$.
\end{proof}
With these basic lemmas established, we need to show how to consistently define $T'$ on a refining sequence of KR-like partitions of $X$, thus defining $T'$ on more and more of the space.
\begin{lemma}[Refinement of mirrored partitions]\label{refinemirror}
Suppose $(X,T)$ is a minimal Cantor system with KR-like clopen partition $\mathcal{P} = \{A(i,j): 1 \leq i \leq I, 0 \leq j < h(i) \}$ and $\varepsilon>0$, then there exists another KR-like clopen partition $\mathcal{P}' =\{A'(k,j): 1 \leq k \leq K, 0 \leq j < h(i)\}$ with the following properties:
\begin{enumerate}
\item For all $k,j$, $\diam(A'(k,j))<\varepsilon$.
\item $\mathcal{P}'$ refines $\mathcal{P}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Fix $i \in \{1,\ldots ,I\}$ and $0 \leq j <h(i)$.
Partition $A(i,j)$ into finitely many sets, $A(i,1,j), \ldots , A(i,L,j)$ each of diameter less than $\varepsilon$. Now for $j' \neq j$ and $0\leq j' < h(i)$, we have $\varphi [\mathbbm{1}_{A(i,j)}]=[\mathbbm{1}_{A(i,j')}]$. So, applying Lemma~\ref{clopenpartition2} to the identity map there exist clopen sets
$A(i,1,j'), \ldots , A(i,L,j')$ such that
$$
\varphi [\mathbbm{1}_{A'(i, \ell ,j')}]=[\mathbbm{1}_{A'(i, \ell ,j)}]
$$
for $1 \leq \ell \leq L$.
Continue in the same manner for every pair $(i,j)$. After finitely many iterations and re-indexing, the result is a KR-like clopen partition $\mathcal{P}'=\{A'(k,j): 1 \leq k \leq K, 0 \leq j < h(i)\}$ which refines $\mathcal{P}$ and satisfies $\diam (A'(k,j)) < \varepsilon$ for all $k,j$.
\end{proof}
\begin{lemma}\label{refineKR}
Suppose there is an exhaustive epimorphism\linebreak $\varphi:\mathcal{G}(Y,S)\twoheadrightarrow \mathcal{G}(X,T)$. If $\mathcal{Q}$ is a KR-partition of $Y$ and $\mathcal{P}$ is a ``mirrored'' clopen partition of $X$ ($\mathcal{P}$ satisfies Lemma~\ref{copytower}), then any tower-preserving refinement of $\mathcal{Q}$, and can be mirrored onto $\mathcal{P}$ while still preserve Lemma~\ref{copytower}.
\end{lemma}
\begin{proof}
Let $\mathcal{Q}=\{S^jB_i:1\le i\le I,0\le j<h(i)\}$ be a KR-partition of $Y$ and $\mathcal{P}=\{A(i,j)\}$ be as in Lemma~\ref{copytower}. Suppose $\mathscr{P}=\{P_n\}_{n=1}^{m}$ is a finite, clopen partition of $Y$. Consider $P_1$ and assume for some fixed $i,j$ that
$$
P_1\cap S^jB(i)\neq \emptyset \text{ and } S^j_2B(i)\nsubseteq P_1.
$$
In this case we split $S^j_2B(i)$ into two clopen sets:
$$
P_1\cap S^jB(i) \text{ and } P_1^c\cap S^jB(i).
$$
We now split the entire tower $\{S^{\ell}B(i)\}$ into the following two towers:
$$
\{S^{k-j}(P_1\cap T^j_2B(i))\}_{k=0}^{h(i)-1}\text{ and } \{S^{k-j}(P_1^c\cap S^jB(i))\}_{k=0}^{h(i)-1}.
$$
Observe, for each $k$ we have,
$$S^kB(i)=S^{k-j}(P_1\cap S^j_2B(i))\cup S^{k-j}(P_1^c\cap S^jB(i)),$$
thus by Lemma~\ref{clopenpartition2} there exist disjoint clopen subsets $A_1(i,k),A_c(i,k)$ such that
$$
\varphi[\mathbbm{1}_{S^{k-j}(P_1\cap S^jB(i))}]=[\mathbbm{1}_{A_1(i,k)}]\text{ and } \varphi[\mathbbm{1}_{S^{k-j}(P_1^c\cap S^jB(i))}]=[\mathbbm{1}_{A_c(i,k)}].
$$
Continuing in this same manner for each $P_n\in\mathscr{P}$ and every pair $(i,j).$ After finitely many steps we both can refine $\mathcal{Q}$ to $\mathcal{Q}'$ so that $\mathcal{Q}'=\{S^jB'(i):1\le i\le I', 0\le j<h(i)\}$ and $\mathcal{Q}'$ is finer than $\mathscr{P}$. In addition, we can subdivide $\mathcal{P}$, into $\mathcal{P}'=\{A'(i,j)\}$ is a way such that:
\begin{enumerate}
\item For each $(i,j)\, \varphi[\mathbbm{1}_{S^jB'(i)}]=[\mathbbm{1}_{A'(i,j)}]$
\item For each $(i,j)$ there exists $(\alpha,\beta)$ such that $A'(i,j)\subseteq A(\alpha,\beta)$.
\end{enumerate}
Additionally, Lemma~\ref{clopenpartition2} ensures that $x_0\in A'(1,h(1)-1)$ and $T(x_0)\in A'(2,0)$. Finally, a similar construction as done in the ``moreover'' part of Lemma~\ref{clopenpartition} ensures that $y_0\in S^{h(1)-1}B'(1)$ and $S(y_0)\in B'_2$.
\end{proof}
\begin{proof}[{\bf Proof of (2) $\Rightarrow$ (1)}]
Assume there is an exhaustive epimorphism $\varphi: \mathcal{G}(Y,S) \twoheadrightarrow \mathcal{G}(X,T)$ and fix points $x_0 \in X$ and $y_0$ in $Y$. Let $\{\varepsilon_k\}_{k\geq 0}$ be a sequence of positive numbers which decrease to zero such that $\varepsilon_0$ and $\varepsilon_1$ are larger than the diameters of both $X$ and $Y$.
We will recursively define a sequence of KR-partitions $\{\mathcal{Q}^n\}$ of $Y$ and KR-like partitions
$\{\mathcal{P}^n\}$ of $X$. By requiring that every element of $\mathcal{Q}^n$ and $\mathcal{P}^n$ have diameter less than $\varepsilon_{n}$, we will insure that these partitions generate the topologies of $X$ and $Y$.
\underline{Initial Step:}
First we define KR-partitions of height 1,
$\mathcal{P}^1$ and $\mathcal{Q}^1$.
Find a clopen set $B_1(1) \subset Y$ which contains $y_0$ but not $S(y_0)$. Set $B_1(2) = B_1(1)^c$ and
$\mathcal{Q}^1 = \{B_1(1),B_1(2)\}$.
Apply Lemma \ref{clopenpartition2} to create a clopen partition
$\{A_1(1,0),A_1(2,0)\}$ of $X$ such that
$x_0 \in A_1(1,0)$, $T(x_0) \in A_1(2,0)$, and
$\varphi[\mchi_{B_1(i,0)}] = [1_{A_1(i,0)}]$ for $i = 1,2$.
\noindent
\underline{Recursive Hypothesis:}
Now suppose $n \geq 1$ and we have a refining sequence of clopen KR-partitions
$\mathcal{Q}^1 \prec \mathcal{Q}^2 \prec \cdots \prec \mathcal{Q}^n$ of $Y$ where
$$\mathcal{Q}^k =
\{ S^j B_k(i) : 1 \leq i \leq I_k, 0 \leq j < h_k(i) \}$$
and a refining sequence of clopen KR-like partitions
$\mathcal{P}^1 \prec \mathcal{P}^2 \prec \cdots \prec \mathcal{P}^n$ of $X$
where
$$\mathcal{P}^k =
\{A_k(i,j): 1 \leq i \leq I_k, 0 \leq j < h_k(i) \}$$
with the following properties for all $k = 1, \ldots ,n $ and
for all $1 \leq i \leq I_k, 0 \leq j < h_k(i)$:
\begin{enumerate}
\item $ \varphi[\mchi_{S^jB_k(i)}]=[\mchi_{A_k(i,j)}] $
\item $\diam(A_k(i,j))<\varepsilon_k$,
\item $\diam(S^j B_k(i))<\varepsilon_k$,
\item $\diam \left( \cup_i A_k(i,0)\right)<\varepsilon_{k-1}$ and $\diam \left( \cup_i A_k(i,h(i)-1)\right)<\varepsilon_{k-1}$,
\item $\diam \left( \cup_i B_k(i)\right)<\varepsilon_{k-1}$, and $\diam \left( \cup_i S^{h(i)-1}B_k(i)\right)<\varepsilon_{k-1}$,
\item $x_0 \in A_k(1,h_k(1)-1)$ and $T(x_0) \in A_k(2,0)$,
\item $y_0 \in S^{h(1)-1}B_k(1)$ and $S(y_0) \in B_k(2)$.
\end{enumerate}
Further assume that for $1 \leq k \leq n$,
we have defined continuous functions $p_k: X \setminus \Top(\mathcal{P}^k) \to \mathbb{N}$ such that
\begin{enumerate}
\item $p_k = p_{k-1}$ on $X \setminus \Top(\mathcal{P}^{k-1})$,
\item the map $T_k'$ defined as $T_k'(x) = T^{p(x)}(x)$ gives a homeomorphism from
$A_k(i,j)$ to $A_k(i,j+1)$ for all $i$ and $0\leq j < h_k(i)-1$.
\end{enumerate}
\noindent
\underline{Recursive Step:} The recursive step is broken down into four parts.
\textbf{Step 1}:
Use Lemma \ref{refinetower} to create initial KR-partition for $(Y,S)$
$$\mathcal{Q} = \{S^j B(i) : 1 \leq i \leq I, 0 \leq j < h(i)\}$$
of $Y$ refining $\mathcal{Q}^n$ with the following properties for all $i,j$:
\begin{itemize}
\item $\diam(S^j B(i))<\varepsilon_{n+1}$,
\item $ \cup_i B(i) \subset B_n(2)$, and
$\cup_i S^{h(i)-1}B(i) \subset S^{h_n(1)-1}B_n(1)$,
\item $y_0 \in S^{h(1)-1}B(1)$ and $S(y_0) \in B(2)$.
\end{itemize}
Note that by the recursive hypothesis, the second item above implies
$\diam \left( \cup_i B(i)\right)<\varepsilon_n$, and $\diam \left( \cup_i S^{h(i)-1}B(i)\right)<\varepsilon_n$.
\textbf{Step 2}: Fix $i$ and consider the partition of $B_n(i)$ into elements
$S^{j_1} B(i_1), S^{j_2} B(i_2), \ldots , S^{j_m} B(i_m)$ of $\mathcal{Q}$.
By the recursive hypothesis, we know $\varphi[\mchi_{B_n(i)}]=[\mchi_{A_n(i,0)}]$.
Use Lemma \ref{clopenpartition2} to partition $A_n(i,0)$ into clopen sets
$A(i_1,j_1), A(i_2,j_2), \ldots , A(i_m, j_m)$ such that
$\varphi[\mchi_{S^{j_k}B(i_k)}]=[\mchi_{A(i_k,j_k)}]$ for all $k$.
In the particular case of $i=2$, use the ``Moreover...'' part of Lemma \ref{clopenpartition2} to insure $T(x_0) \in A(2,0)$.
Set $z_0 = (T_n')^{-h_n(1)+1}(x_0) \in A_n(1,0)$. We know from Step 1 that $y_0 \in S^{h(1)-1} B(1)$. Therefore, one of the sets in the partition of $A(1,0)$ is $A(1,h(1)-h_n(1))$. Use the ``Moreover...'' part of Lemma \ref{clopenpartition2} to insure $z_0 \in A(1,h(1)-h_n(1))$.
Consider any set $A(i,j)$ where $A(i,j) \subset A_n(k,0)$.
For $1 \leq l < h_n(i)$, define $A(i,j+l)=(T_n')^lA(i,j)$.
If $j+l < h(i)$, we have the following
$$[\mchi_{A(i,j+l)}] = \varphi[\mchi_{S^{j+l}B(i)}] = \varphi[\mchi_{S^{j+l+1}B(i)}] = [\mchi_{A(i,j+l+1)}].$$
Therefore, the union of the partitions of the individual sets in $\mathcal{P}^n$ results in a KR-like partition
$$\mathcal{P} = \{A(i,j) : 1 \leq i \leq I, 0 \leq j < h(i)\}$$
satisfying
\begin{itemize}
\item $\varphi[\mchi_{S^jB(i)}]=[\mchi_{A(i,j)}] $,
\item $ \cup_i A(i,0) \subset A_n(2,0)$, and
$\cup_i A(i,h(i)-1) \subset A_n(1,h_n(i)-1)$,
\item $x_0 \in A(1,h(1)-1)$ and $T(x_0) \in A(2,0)$.
\end{itemize}
Again by the recursive hypothesis, the second item above implies \\
$\diam \left( \cup_i A(i,0)\right)<\varepsilon_{n}$ and $\diam \left( \cup_i A(i,h(i)-1)\right)<\varepsilon_n$.
Let $A(i,j) \in \mathcal{P}$ be a level with $A(i,j) \subset \Top (\mathcal{P}^n)$ and $0 \leq j < h(i)$.
Because $\mathcal{P}$ is a KR-like partition, we may apply Lemma \ref{speedup} to define a continuous function $p_{n+1}$ on $A(i,j)$ so that $x \mapsto T^{p_{n+1}}(x)$ gives a homeomorphism from $A(i,j)$ to $A(i,j+1)$. For any $A(i,j) \in \mathcal{P}$ with $A(i,j) \subset X \setminus \Top (\mathcal{P}^n)$, set $p_{n+1} = p_n$ on $A(i,j)$. Define $T'(x) = T^{p_{n+1}(x)}(x)$ on $X \setminus \Top (\mathcal{P})$.
Thus the partitions $\mathcal{Q}$ and $\mathcal{P}$ satisfy all of the conditions (1) and (3)-(7). The next two steps are required to guarantee (2).
\textbf{Step 3}: Use Lemma \ref{refinemirror} to refine $\mathcal{P}$ to create a KR-like partition
$\mathcal{P}^{n+1} = \{A_{n+1}(i,j) : 1 \leq i \leq I_{n+1}, 0 \leq j < h_{n+1}(i)\}$
where the diameter of every set $A_{n+1}(k,j)$ is less than $\varepsilon_{n+1}$, and
$A_{n+1}(k,j) \subset A(i,j)$. If necessary, renumber the first index
so that $x_0 \in A_{n+1}(1,h_{n+1}(1)-1)$ and $T(x_0) \in A_{n+1}(2,0)$.
\textbf{Step 4}: In a manner similar to Step 2,
we will use Lemma \ref{clopenpartition} to refine $\mathcal{Q}^{n}$ to
create $\mathcal{Q}^{n+1}$ to mirror $\mathcal{P}^{n+1}$. Note that this is the one step of the proof that requires that the epimorphism be exhaustive.
Fix $i,j$ and consider the partition of $A(i,j)$ into elements $\{A_{n+1}(k,j)\}$ of $\mathcal{P}^{n+1}$. Use Lemma \ref{clopenpartition} to partition $S^j B(i)$ into clopen sets\linebreak $\{S^j B_{n+1}(k)\}$ such that
$\varphi[\mchi_{S^jB_{n+1}(k)}]=[\mchi_{A_{n+1}(k,j)}]$ for all $k,j$.
In the particular cases of $S^{h(1)-1} B(1)$ and $B(2)$, use the ``Moreover...'' part of Lemma \ref{clopenpartition} to insure that $y_0 \in S^{h(1)-1}B_{n+1}(1)$ and $S(y_0) \in B_{n+1}(2)$.
\noindent
\underline{Endgame:}
With this recursive construction, we establish the existence of an infinite sequence of
refining sequence of clopen KR-partitions $\{\mathcal{Q}^k\}_{k\geq 1}$ and clopen KR-like partitions $\{\mathcal{P}^k\}_{k\geq 1}$ satisfying properties (1)-(7) for all $k \geq 1$. For any $x \in \setminus \cap \Top(\mathcal{P}^k) = \{x_0\}$, we have defined $p(x) = \lim_{n \to \infty}p_n(x)$ and $T'(x) = T^{p(x)}(x)$. We complete the definition of $p$ by noting that property (4) implies $\cap \Top(\mathcal{P}^k) = \{x_0\}$ and
defining $p(x_0)=1$. Because $p_k$ is continuous on $X \setminus \Top(\mathcal{P}^k)$, the point $x_0$ is the only possible point of discontinuity for $p$.
Setting $T'(x)=T^{p(x)}(x)$ completes the definition of a strong speedup of $T$.
We now define a set map
$h : \cup \mathcal{P}^k \to \cup \mathcal{Q}^k$ so that
$h(A_k(i,j)) = S^jB_k(i)$. This set map respects subsets by construction of the sets
$A_k(i,j)$ and $S^jB_k(i)$. By properties (2) and (3) the sequences of partitions $\{\mathcal{Q}^k\}$ and $\{\mathcal{P}^k\}$ separate points in the spaces $X$ and $Y$ respectively; therefore, $h$ gives a homeomorphism from $X$ to $Y$ which we will also denote by $h:X\to Y$.
Note that property (6) and (7) implies $h(x_0) = y_0$ and $h(T(x_0)) = S(y_0)$ since the partition elements that contain those points are always paired by the map $h$.
Since $T'A_n(i,j)=A_n(i,j+1)$ for any $A_n(i,j) \in \mathcal{P}^n$ with $0\leq j < h_n(i)-1$,
we have $$h(T'A_n(i,j)) = h(A_n(i,j+1)) = S^{j+1}B_n(i) = Sh(A_n(i,j)).$$
The point $x_0$ is the only point in $X$ which is in $A_n(i, h_n(i)-1)$ for all $n\geq 1$ and
$T(x_0)$ is the only point in $X$ which is in $A_n(i, 0)$ for all $n\geq 1$, and we have
$$h(T'(x_0)) = h(T(x_0)) = S(y_0) = Sh(y_0).$$
It follows that $T'$ and $S$ are conjugate, and that $S$ is a strong speedup of $T$, completing the proof.
\end{proof}
\section{Topological Speedups}
In this section we work with general speedups and prove Main Theorem \ref{main}. We will follow the same overall strategy as in the previous section and prove several analogous results. Note that here, the notation $[f]$ denotes an equivalence class in $G/Inf$ instead of $G$.
\subsection{Proof of (1) $\Rightarrow$ (3)}
\begin{theorem}
Let $(X,T)$ and $(Y,S)$ be minimal Cantor systems. If $S$ is a speedup of $T$ then there exists homeomorphism $h:X\rightarrow Y$, such that $h_{*}:M(X,T)\hookrightarrow M(Y,S)$ is an injection.
\end{theorem}
\begin{proof}
Let $p:X\rightarrow\mathbb{Z}^{+}$ be such that $T'(x)=T^{p(x)}(x)$ is a minimal homeomorophism of $X$ and let $\mu\in M(X,T)$.
Because $T$ is invertible, $M(X,T)=M(X,T^{-1})$ thus it suffices to
show that for any Borel set $A\subset X$ and any $\mu \in M(X,T)$ we have $\mu(T'A) = \mu (A)$. We obtain this by partitioning $A$ into sets according to the value of $p$. Since $p$ is lower semi-continuous the result follows. More specifically we have the following where $\sqcup$ denotes disjoint union:
\begin{align*}
\mu(T'(A))&=\mu\left(T'\left(\displaystyle\bigsqcup_{n\in\mathbb{Z}^{+}}A\intersect p^{-1}(\{n\})\right)\right) \\
&=\mu\left(\displaystyle\bigsqcup_{n\in\mathbb{Z}^{+}}T'(A\intersect p^{-1}(\{n\}))\right) \\
&=\mu\left(\disUnion_{n\in\Z^{+}}T^{n}(A\intersect p^{-1}(\{n\})) \right)\\
&=\displaystyle\sum_{n\in\mathbb{Z}^{+}}\mu(T^{n}(A\intersect p^{-1}(\{n\}))) \\
&=\displaystyle\sum_{n\in\mathbb{Z}^{+}}\mu(A\intersect p^{-1}(\{n\}))\text{ as $\mu\in M(X,T)$.} \\
&=\mu(A).
\end{align*}
Now suppose $(X,T')$ and $(Y,S)$ are conjugate through $F:X \to Y$.
Then by the above argument, the map $F_*: \mu \to \mu \circ F^{-1}$ provides an injection $F_*:M(X,T) \to M(Y,S)$.
\end{proof}
Note that if $F_*$ above provides a bijection, then by Theorem \ref{GPS Theorem}, $T$ and $S$ would be orbit equivalent.
\subsection{Proof of (3) $\Rightarrow$ (2)}
The following is an analog of Lemma \ref{towercob}.
\begin{lemma} \label{inflemma}
Suppose $(X,T)$ is a minimal Cantor system and $f \in C(X,\Z)$. Then $f \in \Inf(X,T)$ if and only if $\frac{1}{N} \sum_{j=0}^{N-1} fT^{j}$ converges uniformly to $0$.
\end{lemma}
\begin{proof}
Suppose $\frac{1}{N}\sum_{j=0}^{N-1}fT^{j}$ does not converge uniformly to $0$; then there exists $\varepsilon>0$, increasing sequence $\{N_{k}\}$, and corresponding sequence $\{x_k\}$ for which $$\left|\frac{1}{N_k}\sum_{j=0}^{N_{k}-1}fT^j(x_k)\right|>\varepsilon.$$
Define $\mu_{k}=\sum_{j=0}^{N_{k}-1}\delta_{T^{j}x_k}$, where $\delta_z$ represents the Dirac delta measure at $z$. Because the set of Borel probability measures on $X$ form a compact set in the weak$^*$-topology, the sequence $\{\mu_{k}\}$ has a convergent subsequence, which will rename $\{\mu_{k}\}$, whose limit $\mu$ is $T$-invariant. Hence,
\begin{align*}
\left|\int f\,d\mu\right|&=\lim_{k\rightarrow\infty}\left|\int f\, d\mu_{k}\right|\\
&=\lim_{k\rightarrow\infty}\frac{1}{N_{k}}\left|\sum_{j=0}^{N_{k}-1}fT^{j}(x_k)\right|\\
&\ge\varepsilon.
\end{align*}
Therefore, $f\notin \Inf(X,T)$.
Conversely, suppose $\frac{1}{N} \sum_{j=0}^{N-1} fT^{j}$ converges uniformly to $0$ and let $\mu \in M(X,T)$ where $\mu$ is ergodic. Then by the Ergodic Theorem there exists $x \in X$ such that
$$\int f d \mu =\lim_{N \to \infty} \frac{1}{N} \sum_{j=0}^{N-1} fT^{j}(x)$$
which implies $\int f d \mu=0$. By the Ergodic Decomposition Theorem, $f \in \Inf(X,T)$.
\end{proof}
The following is an analog of Lemma \ref{towerpos}.
\begin{lemma}\label{towerpos2}
Let $(X,T)$ be a minimal Cantor system and suppose $[f] \in G(X,T)/\Inf G(X,T)$. Then $[f] \in (G(X,T)/\Inf G(X,T))_+ \setminus \{0\}$
if and only if there is an $\varepsilon>0$ such that for any clopen set $A$, if $\diam(A)< \varepsilon$ then for all $x \in A$,
$$\sum_{j=0}^{r_A(x)-1} fT^j(x) > 0.$$
\end{lemma}
\begin{proof}
Suppose $[f] \in (G(X,T)/\Inf G(X,T))_+\setminus \{0\}$, then $f$ is equal to a nonzero, nonnegative function $g$ plus an element $[k]$ of $\Inf G(X,T)$. There is a nonempty clopen set $C$ such that for $x \in C$, $g(x)>0$. Since $T$ is minimal, there is an $r>1$ such that for all $x \in X$, $\{x,T(x), \ldots T^{r-1}(x) \} \cap C \neq \emptyset$. Therefore, if $n > 2r$, we have
$$\sum_{j=0}^{n-1} gT^j(x) \geq \frac{1}{r+1} n.$$
Also, by Lemma \ref{inflemma}, there is an $N$ such that if $x \in X$ and $n > N$,
$$\left| \sum_{j=0}^{n-1} kT^j(x) \right| < \frac{1}{r+1} n.$$
Finally, by Lemma \ref{returntime}, there is an $\varepsilon>0$ such that if $A$ is a clopen set with $\diam (A) < \varepsilon$ then for all $x \in A$, $r_A(x)>\max \{N, 2r \}$, which implies
$$\sum_{j=0}^{r_A(x)-1} fT^j(x) > 0.$$
The converse follows from Lemma~\ref{towerpos}.
\end{proof}
\begin{lemma}
\label{posbound}
Let $(X,T)$ be a minimal Cantor system and suppose $[f] \in G(X,T)/\Inf G(X,T)$. If $[f] \in (G(X,T)/\Inf G(X,T))_+ \setminus \{0\}$
then there is a $c>0$ such that
$$\int f d \mu \geq c$$
for all $\mu \in M(X,T)$.
\end{lemma}
\begin{proof}
Suppose $[f] \in (G(X,T)/\Inf G(X,T))_+\setminus \{0\}$, then since\linebreak $G(X,T)/\Inf G(X,T)$ is simple, there is an $n\geq 1$ such that
$n [f] > [1]$. Therefore
$$n \int f d \mu \geq 1$$
for all $\mu \in M(X,T)$.
\end{proof}
The following is an analog of Lemma \ref{functionineq}.
\begin{lemma}\label{functionineq2}
Suppose $[f],[g],\in G(X,T)/\Inf(G(X,T))$ with $0<[f]<[g]$. Then there exists $f'\in C(X,\Z)$ such that $0\le f'(x)\le g(x)$ for every $x\in X$ and $[f']=[f]$.
\end{lemma}
\begin{proof}
Both $[f]$ and $[g-f]$ are in $(G(X,T)/\Inf G(X,T))_+\setminus \{0\}$. Take $\varepsilon$ to be smaller than the minimum of the two numbers that exist for $f$ and $g-f$ from Lemma \ref{towerpos2}. Use Lemma \ref{refinetower} to create a KR-partition
$\mathcal{P} = \{T^jA(i) : 1 \leq i \leq I, 0 \leq j < h(i)\}$ over a set $A$ with diameter less than $\varepsilon$ such that $g$ and $f$ are constant on every level of $\mathcal{P}$.
Then for any $x \in A(i)$, we have
$$\sum_{j=0}^{h(i)-1} gT^j(x) \geq \sum_{j=0}^{h(i)-1} fT^j(x) \geq 0.$$
Therefore, it is possible to select values for a new function $f'$ on levels of the $i$th column of $\mathcal{P}$ so that $g(x) \geq f'(x) \geq 0$ for all $x$ and so that $$\sum_{j=0}^{h(i)-1} f'T^j(x) = \sum_{j=0}^{h(i)-1} fT^j(x).$$
By Lemma \ref{towercob}, $[f']=[f]$.
\end{proof}
The following is an analog of Corollary \ref{clopensubset}.
\begin{coro}[\cite{Glasner-Weiss}]\label{Glasner-Weiss}
Let $(X,T)$ be a minimal Cantor system. Let $f\in C(X,\Z)$ satisfy $0<\int fd\mu <1$ for every $\mu\in M(X,T)$. Then there exists a clopen subset $A$ in $X$ such that $\int fd\mu=\mu(A)$ for every $\mu\in M(X,T)$. That is $[\mathbbm{1}_A]=[f]$ in $G(X,T)/\Inf(G(X,T))$.
\end{coro}
\begin{theorem}
Let $(X,T)$ and $(Y,S)$ be minimal Cantor systems. Suppose there exists homeomorphism $F:X\rightarrow Y$, such that $F_{*}:M(X,T)\hookrightarrow M(Y,S)$ is an injection.
Then there exists an exhaustive epimorphism $\varphi:\mathcal{G}(Y,S)/\Inf(G(Y,S))\twoheadrightarrow\mathcal{G}(X,T)/\Inf(G(X,T))$ which preserves the distinguished order units.
\end{theorem}
\begin{proof}
As in the proof for $(3) \implies (2)$ in Main Theorem \ref{mainstrong}, we define
$\varphi:\mathcal{G}(Y,S)/\Inf(G(Y,S)) \to \mathcal{G}(X,T)/\Inf(G(X,T))$ by $\varphi[f] = [fF]$. Having all the analogous lemmas as in the strong speedup case, the proof is the same.
\end{proof}
\subsection{Proof of $(2)\Rightarrow (1)$}
Our strategy for this section is the same as the previous two. That is, prove the equivalent of all lemmas in the strong speedups section and use the same proof technique. Throughout, the notation $[f]$ here denotes an equivalence class in \linebreak $G(X,T)/\Inf (G(X,T))$ or $G(Y,S)/\Inf (G(Y,S))$.
Below are analogues of Lemma \ref{clopenpartition} and \ref{clopenpartition2}. The proofs are the same except we use Corollary \ref{Glasner-Weiss} in place of Corollary \ref{clopensubset}.
\begin{lemma}\label{clopenpartition-2}
Let $(X,T)$ and $(Y,S)$ be minimal Cantor systems. Assume there is an exhaustive epimorphism $\varphi :\mathcal{G}(Y,S) \twoheadrightarrow \mathcal{G}(X,T)$. Let $A \subset X$ and $B \subset Y$ be clopen sets such that $\varphi[\mathbbm{1}_B]=[\mathbbm{1}_A]$. Then for any clopen partition $\{A_1,A_2,\ldots,A_n\}$ of $A$ there is a clopen partition $\{B_1,B_2,\ldots,B_n\}$ of $B$ such that $\varphi[\mathbbm{1}_{B_i}]=[\mathbbm{1}_{A_i}]$ for all $1 \leq i \leq n$.
Moreover, given any $n$ distinct
points $y_1,y_2,\ldots ,y_n \in B$
we may select $B_i$ to contain $y_i$ for $i=1,2,\ldots,n$.
\end{lemma}
\begin{lemma} \label{clopenpartition2-2}
Let $(X,T)$ and $(Y,S)$ be minimal Cantor systems. Assume there is an epimorphism $\varphi :\mathcal{G}(Y,S) \twoheadrightarrow \mathcal{G}(X,T)$. Let $A \subset X$ and $B \subset Y$ be clopen sets such that $\varphi[\mathbbm{1}_B]=[\mathbbm{1}_A]$. Then for any clopen partition $\{B_1,B_2,\ldots,B_n\}$ of $B$ there is a clopen partition $\{A_1,A_2,\ldots,A_n\}$ of $A$ such that $\varphi[\mathbbm{1}_{B_i}]=[\mathbbm{1}_{A_i}]$ for all $1 \leq i \leq n$.
Moreover, given any $n$ distinct points $x_1,x_2,\ldots ,x_n \in A$ we may select $A_i$ to contain $x_i$ for $i=1,2,\ldots,n$.
\end{lemma}
Unlike the previous facts, the Speedup Lemma, which would be an analogue to the Strong Speedup Lemma \ref{speedup} will take a bit more work to establish. The following lemma serves as a precursor to the key lemma and is instrumental for its proof.
\begin{lemma}[Partial Speedup Lemma]\label{pkey lemma}
Let $(X,T)$ be a minimal Cantor system, and let $A,B$ be nonempty, disjoint, clopen subsets of $X$. If for all $\mu\in M(X,T),\, \mu(A)<\mu(B),$ then there exists $p:A\rightarrow\Z^{+}$ such that $T':A\rightarrow B$ defined as $S(x)=T^{p(x)}(x)$ is a homeomorphism onto its image.
\end{lemma}
\begin{proof}
Define $f=\mathbbm{1}_{B}-\mathbbm{1}_{A}$. With the given hypothesis, \linebreak
$[f] \in (G(X,T)/\Inf G(X,T))_+\setminus \{0\}$. Using Lemma \ref{towerpos2}, we may find a clopen set $C$ such that
$$\sum_{j=0}^{r_C(x)-1} fT^j(x) > 0.$$
Now use Lemma \ref{refineKR} to create a KR-partition $\mathcal{P} = \{ T^j C(i) : 0 \leq i \leq I, 0 \leq j < h(i) \}$ with $\Base (\mathcal{P}) = C$ and with $\mathbbm{1}_{B}$ and $\mathbbm{1}_{A}$ constant on all levels of $\mathcal{P}$.
Now fix $i \in \{1,\ldots, I\}$. Since the number of levels in the $i$th column which are subsets of $B$ is greater than the number of levels that are subsets of $A$ there is an injection
$$\Gamma: \{j : 0 \leq j < h(i), T^jC(i) \subset A\}
\to \{j : 0 \leq j < h(i), T^jC(i) \subset B\}.$$
For $x \in T^jC_i \subset B$, set
$$p(x) = \min\{n > 0 : T^n(x) \in T^{\Gamma(j)}C(i) \}.$$
As in the case of Lemma \ref{speedup}, $p$ is well-defined and continuous and $x \mapsto T^{p(x)}(x)$ is a homeomorphism from $T^jC(i)$ to $T^{\Gamma(j)}C(i)$. By working similarly in each column, the map $x \mapsto T^{p(x)}(x)$ yields a homeomorphism from $A$ to a clopen subset of $B$.
\end{proof}
We will use our partial speedup lemma to inductively prove a full speedup lemma.
\begin{lemma}\label{clopen1}
Let $(X,T)$ be a minimal Cantor system. Then for every $\varepsilon>0$ there exists $\delta>0$ such that for every clopen set $A \subset X$ with $\emph{\diam}(A)<\delta$ and every $\mu\in M(X,T),$ we have $\mu(A)<\varepsilon$.
\end{lemma}
\begin{proof}
Fix any $\mu \in M(X,T)$.
Choose $N \geq 1$ such that $1/N < \varepsilon$. Use Lemma \ref{returntime} to find $\delta>0$ such that if $\diam(A) < \delta$ then for all $x \in A$, $r_A(x) >N$. Since $r_A(x)>N$ for all $x \in A$, the sets
$A, TA, \ldots T^{N-1}A$ are all disjoint and of equal $\mu$-measure. Therefore $\mu(A)< 1/N < \varepsilon$.
\end{proof}
We will use induction on our previous lemma to prove our key Lemma.
\begin{lemma}[Speedup Lemma - Infinitesimal Case]\label{Key Lemma}
Let $(X,T)$ be a minimal Cantor system and let $A,B$ be nonempty disjoint, clopen subsets of $X$. If for all $\mu\in M(X,T),\, \mu(A)=\mu(B)$, then there exists $p:A\rightarrow\Z^{+}$ such that $T':A\rightarrow B$, defined as $T'(x)=T^{p(x)}(x)$, is a homeomorphism onto $B$.
\end{lemma}
\begin{proof}
Let $A$ and $B$ be nonempty, disjoint, clopen subsets of $X$ and $x_0\in A$. Since $T$ is minimal there exists $k\in\Z^{+}$ such that $T^{k}(x)\in B$, let $y_0=T^{k}(x_0)$.
We will use induction to find a decreasing sequences of sets $\{A_{n}\}_{n\ge 0}$ and $\{B_{n}\}_{n\ge 0}$ such that
$$
\Intersect_{n\ge 0}A_{n}=\{x_0\} \text{ and }\Intersect_{n\ge 0}B_{n}=\{y_0\}
$$
all while defining $T'$ on larger and larger parts of $A$.
First find a clopen set $C_1\subsetneq A$ containing $x_0$ such that $\diam (C_1) < \varepsilon_1$. Set $A_1 = A \setminus C_1$.
Since every $\mu \in M(X,T)$ is full, $\mu(C_1)>0$ for all $\mu \in M(X,T)$ and for some $\eta_1>0$, $\mu(A_1)<\mu(B)-\eta_1$ for all $\mu \in M(X,T)$. Use Lemma \ref{clopen1} to find $D_1 \subset B$ containing $y_0$ with $\mu(D_1)< \eta_1$ for all $\mu \in M(X,T)$.
Therefore, we have $\mu(A_1)<\mu(B\setminus D_1)$ for all $\mu \in M(X,T)$.
Then by Lemma \ref{pkey lemma}, we can define a continuous $p: A_1 \to \Z^+$ so that $T'(x) = T^{p(x)}(x)$ is a homeomorphism from $A_1$ into $B \setminus D_1$. Set $B_1 = T'(A_1)$ and continue. Note that $x_0 \in A \setminus A_1$ and $y_0 \in B \setminus B_1$. We can find a $D_1 \subset B$ containing $y_0$ with $\mu(D_1)< \eta_1$ for all $\mu \in M(X,T)$.
For the next step, we start in the set $B$. We find a clopen set $D_2 \subsetneq B \setminus B_1$ of $y_0$ such that $\diam(D_2) < \varepsilon_2$. Set $B_2 = B \setminus (B_1 \cup D_2)$ Then using an argument similar to the above, we can apply Lemma \ref{clopen1} to find $C_2 \subset A\setminus A_1$ containing $x_0$ such that $\mu(A \setminus C_2)>\mu(B_2)$ for all $\mu$. Apply Lemma \ref{pkey lemma} to $T^{-1}$, $B_2$ and $A \setminus C_2$ we can define a continuous
$q: A_1 \to \Z^+$ so that
$y \mapsto T^{-q(y)}(y)$ is a homeomorphism from $B_2$ into $A \setminus C_2$.
Set $A_2$ equal to the image of this homeomorphism. If $x\in A_2$ then set $p(x) = q(y)$ where $x$ is the image of $y$. Then $T'(x) = T^{p(x)}(x)$ is a homeomorphism from $A_2$ to $B_2$.
We continue recursively to define sets $A \supset A_1 \supset A_2 \supset \cdots$ and $B \supset B_1 \supset B_2 \supset \cdots $ such that
\begin{itemize}
\item $\diam(A_n) < \varepsilon_n$ for $n$ odd,
\item $\diam(B_n) < \varepsilon_n$ for $n$ even,
\item $x_0 \in A_n$ for all $n$,
\item $y_0 \in B_n$ for all $n$,
\item $\mu(A_n) = \mu(B_n)$ for all $n$.
\end{itemize}
Moreover, as we are defining $A_n$ and $B_n$ we are defining $p:A_n \setminus A_{n-1} \to \N$ such that $T': (A \setminus A_n) \to (B \setminus B_n)$ defined by $T'(x) = T^{p(x)}(x)$ is a homeomorphism. We complete the definition of $T'$ on $A$ by setting $p(x_0) = k$ so that $T'(x_0) = T^k(x_0)=y_0$.
Note that $T'$ is continuous at $x_0$ by the following argument. Let $\varepsilon>0$ be given. Find $n$ such that $\varepsilon_n < \varepsilon$.
Choose $\delta>0$ such that $B(x_0, \delta) \subset A_{n}$. Then we have
$$T'(B(x_0,\delta)) \subset T'(A_{n}) = B_n \subset B(y_0,\epsilon_n)
\subset B(y_0,\epsilon).$$ This finishes the proof.
\end{proof}
With the key lemma now extended to speedups in general, we may exactly mimic the proof of $(2)\Rightarrow (1)$ from the strong speedups sections.
\begin{theorem}
Let $(X,T)$ and $(Y,S)$ be minimal Cantor systems. If there exists an exhaustive epimorphism $\varphi:\mathcal{G}(Y,S)/\Inf(G(Y,S))\twoheadrightarrow\mathcal{G}(X,T)/\Inf(G(X,T))$ which preserves the distinguished order units then $(Y,S)$ is a speedup of $(X,T)$.
\end{theorem}
\begin{proof}
The proof follows exactly the same argument as for (2) $\implies$ (1) in Section 4. The only differences are the following:
\begin{itemize}
\item Lemma~\ref{clopenpartition-2} is used in place of Lemma~\ref{clopenpartition}
\item Lemma~\ref{clopenpartition2-2} is used in place of Lemma~\ref{clopenpartition2}
\item the Speedup Lemma (Lemma~\ref{Key Lemma}) is used in place of the Strong Speedup Lemma (Lemma~\ref{speedup})
\item Lemma~\ref{refinemirror} is true using Lemma~\ref{clopenpartition2-2} instead of Lemma~\ref{clopenpartition2}.
\end{itemize}
Note that the argument from (2) $\implies$ (1) in Section 4 which discusses the continuity properties of $p$ does not follow here because Lemma~\ref{Key Lemma} does not imply that $p$ is continuous on elements of the clopen KR-partitions. However, for the conclusion of this theorem, we do not need to control for the continuity of the function $p$.
\end{proof}
\section{Speedup Equivalence}
A natural question arises in light the results of speedups and orbit equivalence. If two minimal Cantor systems are speedups of one another, are they orbit equivalent? To discuss this further, define minimal Cantor systems $(X_{1},T_{1})$ and $(X_{2},T_{2})$ to be \emph{{speedup equivalent}} if $T_{1}$ is a speedup of $T_{2}$ and $T_{2}$ is a speedup of $T_{1}$. Combining \cite[Thm 2.2]{GPS} with our Main Theorem 2 we obtain the following corollary.
\begin{coro}\label{Cor.1}
Let $(X_{1},T_{1})$ and $(X_{2},T_{2})$ be minimal Cantor systems. If $(X_{1},T_{1})$ and $(X_{2},T_{2})$ are orbit equivalent, then $(X_{1},T_{1})$ and $(X_{2},T_{2})$ are speedup equivalent.
\end{coro}
Rephrasing Corollary~\ref{Cor.1} above, as equivalence relations orbit equivalence is contained in speedup equivalence. This leads us to a fundamental question: are orbit equivalence and speedup equivalence the same equivalence relation? In this section, we prove the orbit equivalence and speedup equivalence are the same when each minimal Cantor system has finitely many ergodic measures. In general, orbit equivalence and speedup equivalence are not the same as shown by Julien Melleray's forthcoming paper \emph{Dynamical Simplicies and Fra\"iss\'e Theory} \cite{Melleray}. Before we proceed, we will need the following proposition.
\begin{prop}\label{MutuallySingular}
Let $(X_{i},T_{i})$ be minimal Cantor systems and $\varphi:X_{1}\rightarrow X_{2}$ be a homeomorphism. If $\varphi_{*}:M(X_{1},T_{1})\hookrightarrow M(X_{2},T_{2})$ is an injection, then $\varphi_{*}$ preserves pairs of mutually singular measures.
\end{prop}
We now use Proposition~\ref{MutuallySingular} to prove the following theorem.
\begin{theorem}\label{speedupequivalence}
Let $(X_{i},T_{i}),\, i=1,2,$ be minimal Cantor systems each with finitely many ergodic measures. If $(X_{1},T_{1})$ and $(X_{2},T_{2})$ are speedup equivalent, then $(X_{1},T_{1})$ and $(X_{2},T_{2})$ are orbit equivalent.
\end{theorem}
\begin{proof}
Since $T_{1}$ and $T_{2}$ are speedup equivalent, combining part $(3)$ of Main Theorem 1 and Proposition~\ref{MutuallySingular} it follows immediately that
$$
|\partial_{e}(M(X_{1},T_{1}))|=|\partial_{e}(M(X_{2},T_{2}))|
$$
and without loss of generality we may assume $|\partial_{e}(M(X_{1},T_{1}))|=n$ for some $n\in\Z^{+}$. Every measure $\mu$ in $M(X_{2},T_{2})$ is a convex combination of ergodic measures in a unique way. With this in mind for $\mu\in M(X_{2},T_{2})$, let $E(\mu)$ denote the collection of all ergodic measures of $M(X_{2},T_{2})$ which have a positive coefficient in the unique ergodic decomposition of $\mu$. Observe if $\mu_{1},\mu_{2}\in M(X_{2},T_{2})$ with $\mu_{1}\neq\mu_{2}$ and $\mu_{1}\perp\mu_{2}$, then
$$
E(\mu_{1})\intersect E(\mu_{2})=\emptyset.
$$
Now as $T_{2}$ is a speedup of $T_{1}$,
there exists $\varphi:X_{1}\rightarrow X_{2}$, a homeomorphism, such that
$$
\varphi_{*}:M(X_{1},T_{1})\hookrightarrow M(X_{2},T_{2})
$$
is an injection. Since $|\partial_{e}(M(X_{1},T_{1}))|=|\partial_{e}(M(X_{2},T_{2}))|=n$ and $\varphi_{*}$ is injective, we see that $\{E(\varphi_{*}(\nu_{i}))\}_{i=1}^{n}$, where $\{\nu_{i}\}_{i=1}^{n}=\partial_{e}(M(X_{1},T_{1}))$, is a collection of $n$ pairwise disjoint sets, as distinct ergodic measures are mutually singular. It follows that for each $i=1,2,\dots,n$, $E(\nu_{i})$ is a distinct singleton, and therefore $\varphi_{*}(\partial_{e}(M(X_{1},T_{1})))=\partial_{e}(M(X_{2},T_{2}))$. Coupling the facts that $\varphi_{*}$ is an affine map and a bijection on extreme points, we may conclude that $\varphi_{*}$ is a bijection, and hence is an affine homeomorphism between $M(X_{1},T_{1})$ and $M(X_{2},T_{2})$ arising from a space homeomorphism. Therefore, by \cite[Thm. $2.2$]{GPS} $(X_{1},T_{1})$ and $(X_{2},T_{2})$ are orbit equivalent.
\end{proof}
\begin{note}
In \cite{AAO} the authors give an example of a unique ergodic systems whose square is minimal and has more than one invariant measure. Hence, speedups can leave not only the conjugacy class but orbit equivalence class as well.
\end{note}
|
1,116,691,497,782 | arxiv | \section{Introduction}
Optimization for complex systems often involves fitting a system prediction model to estimate how a system performs and then optimizing the decisions based on the system prediction model as shown in Figure~\ref{fig:ocs_fc}. Two major tasks in optimization of complex systems include training or meta-modeling a statistical or system model and optimizing input or decisions based on statistical model.
\begin{figure}[h]
\begin{center}
\includegraphics[width=2.5in]{figures/p2/fc5.pdf}
\caption{Optimization of complex systems}
\label{fig:ocs_fc}
\end{center}
\end{figure}
In real world complex systems, underlying relationships are commonly unknown and are approximated
from data using empirical models. \citet{wu2004travel} applied support vector regression in travel time prediction and proved support vector regression was applicable in traffic data analysis. \cite{lv2014traffic} applied a deep learning approach with autoencoders in traffic flow prediction. \cite{ohlmacher2003using} used logistic regression to generate a landslide-hazard map to predict landslide hazards. \cite{leathwick2005using} used multivariate adaptive regression splines to predict the distributions of freshwater diadromous fish.
If one seeks to optimize a complex system, the optimization
method would need to be able to handle the data-driven approximation models. Given the wide range
of possible approximation models, such as machine learning algorithms, the most commonly
employed optimization approach in these situations is a heuristic approach, such as an evolutionary
algorithm, that cannot guarantee global optimality. Rather than having the approximation model
dictate the need for a heuristic optimization method, the research in this paper seeks a balance that
utilizes a flexible approximation model with structure that can be exploited to enable
true global optimization. In other words, the ``best of both worlds'' is sought, by achieving global
optimality while still maintaining a flexible approximation model.
The approxmation model of choice in this paper is multivariate adaptive regression splines (MARS),
introduced by machine learning pioneer Jerome Friedman in
1991~\citep{friedman1991multivariate}. The structure of MARS
is based on a linear statistical spline model and provides a flexible fit to data while also
achieving a parsimonious model.
The desire to conduct global optimization is seen in many applications, and there are a number
of approaches classified as global optimization methods~\citep{horst2000introduction}. The primary
challenge in achieving global optimality is that many real world applications involve multiple
local optima. Finding a global optimum requires sifting through the local optima and recognizing
when one is suboptimal. The vast majority of applications employ heuristic search algorithms
seek to overcome the challenge of local optima, but do not guarantee global optimality.
Examples include heuristics based on evolutionary algorithms~\citep{morra2018optimization,russo2018knowledge,dahi2018stop},
particle swarm optimization~\citep{alswaitti2018density},
the grasshopper optimization method~\citep{ewees2018improved}, and
the weighted superposition attraction method~\citep{baykasouglu2018dynamic}.
In order to guarantee global optimality, the approach in this paper takes advantage of well-known
properties of mixed integer and quadratic programming (MIQP)~\citep{bliek1u2014solving}.
Some recent applications in which MARS has been employed for empirical modeling include
a water pollution prediction problem~\citep{kisi2016application},
the head load in a building~\citep{roy2018estimating},
the estimation of landfill leachate~\citep{bhatt2017estimating},
and the damage identification for web core composite bridges~\citep{mukhopadhyay2018multivariate},
In optimization problems, MARS has been employed as the empirical model to approximate
unknown relationships in a variety of applications. For stochastic dynamic programming,
the use of MARS to approximate the value function was introduced by~\cite{chen1999applying}.
Since then, the MARS value function approximation approach has been used to numerically solve
a 30-dimensional water reservoir management problem~\citep{cervellera2006optimization},
a 20-dimensional wastewater treatment system~\citep{tarun2011optimizing,tsai2004stochastic,tsai2005flexible},
and a 524-dimensional nonstationary ground-level ozone pollution control problem~\citep{yang2009decision}.
In revenue management, MARS was employed to estimate upper and lower bounds for the
value function of a Markov decision problem~\citep{chen2003solving,siddappa2007refined},
and MARS was used to represent the revenue function in airline overbooking
optimization~\cite{siddappa2008optimising}.
In two-stage stochastic programming, MARS was used to efficiently represent the
expected profit function for an airline fleet assignment problem~\citep{pilla2008statistical}.
This fleet assignment research was extended to utilize a cutting plane method with MARS
to conduct the optimization~\citep{pilla2012multivariate}.
The contribution of this current work extends the approach of
\cite{martinez2017global},
who developed a piece-wise linear MARS structure and formulated a mixed integer and
linear programming problem to globally optimize vehicle design parameters to
improve performance in crash simulations.
The piece-wise linear MARS function may be nonconvex, and the approach of Martinez et al.
will yield a global optimum. However, restricting to piece-wise linear forms limits
the flexibility of the empirical model. Hence, in the current work, the MARS form
employed is based on the original MARS model. The primary challenge for an optimization
method is handling the nonconvex MARS interaction terms, which are products of univariate terms.
By restricting to two-way interactions, we can utilize quadratic programming methods.
In real world applications, two-way interactions are commonly sufficient for empirical modeling~\cite{kutner2005applied}
In summary, the contribution of the presented approach is a MIQP
global optimization method for a MARS model that allows up to
two-way interaction terms that are products of truncated linear univariate functions
(TITL-MARS). This approach is referred to as TITL-MARS-OPT and is compared against
a genetic algorithm for two types of cases.
First, a wind farm power distribution case study is described, and then other TITL-MARS forms
are tested.
Python code for TITL-MARS and TITL-MARS-OPT will be made available on GitHub upon
acceptance of this paper~(\url{https://github.com/JuXinglong/TITL-MARS-OPT}).
The rest of this paper is organized as follows.
Section~\ref{background} describes background on TITL-MARS.
Section~\ref{formulation} presents the MIQP formulation for TITL-MARS-OPT.
The computational study is given in Section~\ref{experiments}, and
Section~\ref{conclusion} concludes the paper.
\section{Background of two-way interaction truncated linear multivariate adaptive regression splines}\label{background}
This section introduces the two-way interaction truncated linear MARS (TITL-MARS) model.
The two-way interaction truncated linear MARS regression model with the response variable ${f}( \mathbf{x}_i)$ is to be built on the independent variable $\mathbf{x}_i$ and can be written in the form of the linear combination of the basis functions as~\cite{friedman1991multivariate}
\begin{equation}\label{eq:MARS_2way}
\begin{array}{l}
\hat{f}( \mathbf{x})=a_0+\sum_{m=1}^{M}\left\{a_m \cdot B_m(\mathbf{x}) \right\}.
\end{array}
\end{equation}
The MARS model is denoted as $\hat{f}( \mathbf{x})$, and $a_0$ is the constant term of the model. The basis function is denoted as $B_m(\mathbf{x})$, and $a_m$ is the coefficient of $B_m(\mathbf{x})$. The index of the basis function is denoted as $m$, and $M$ is the total number of basis functions. The basis function $B_m(\mathbf{x})$ using the truncated linear term has the following form
\begin{equation}\label{eq:MARS_2way_B}
\begin{array}{l}
B_m(\mathbf{x})= \prod_{k=1}^{Km}[s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}) ]_+.
\end{array}
\end{equation}
The truncated linear term is denoted as ${[s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}) ]}_+$, and the basis function $B_m(\mathbf{x})$ is the product of truncated linear terms.
The index of the truncated linear term in $B_m(\mathbf{x})$ is denoted as $k$, and $K_m$ is the total number of truncated linear terms in $B_m(\mathbf{x})$. The sign of the truncated linear term is $s_{k,m}$, which can be $+1$ or $-1$. The $v$-th component of $\mathbf{x}$ is denoted as $x_{v(k,m)}$, and $t_{v(k,m)}$ is the corresponding knot value.
\textit{TITL-MARS is the special case of MARS in which $K_m \leqslant 2$.}
\section{Formulation of two-way interaction truncated linear MARS using mixed integer quadratic programming}\label{formulation}
The general mixed integer quadratic programming problem~\citep{bliek1u2014solving} is given as
\begin{equation}
\label{eq:miqpp}
\begin{array}{ll}
\min & \frac{1}{2} \textbf{z}^T\textbf{Q}\textbf{z}+\textbf{c}^T\textbf{z} \\
{s.t.} & \textbf{A}\textbf{z}=\textbf{b} \\
{} & \textbf{l} \leqslant \textbf{z} \leqslant \textbf{u}\\
{} &\textbf{z} \in \mathbb{R}^P \times \mathbb{Z}^{D-P},\\
\end{array}
\end{equation}
while the two-way interaction truncated linear MARS is given in Section~\ref{background}. In~\eqref{eq:miqpp}, the decision variable is $\textbf{z}$, and the quadratic coefficients matrix is $\textbf{Q}$. The coefficients of the linear terms in the objective funtion are in vector $\textbf{c}$. The linear constraints are denoated as $\textbf{A}\textbf{z}=\textbf{b}$. The lower bound and upper bound of $\textbf{z}$ are $\textbf{l}$ and $\textbf{u}$, respectively. The dimension of $\textbf{z}$ is $D$. There are $P$ dimensions of real values, and $D-P$ dimensions of integers.
Problem in the form~\ref{eq:miqpp} can be solved using the CPLEX solver.
The TITL-MARS optimization prolbem is given as follows. \begin{eqnarray}
\label{eq:mars_opt_1}
\min\,\, & \hat{f}( \mathbf{x})=a_0+\sum_{m=1}^{M}\left\{a_m \cdot B_m(\mathbf{x}) \right\} \\
\label{eq:mars_opt_2}
\mbox{s.t.}\,\, & \textbf{l} \leqslant \textbf{x} \leqslant \textbf{u}\\
\label{eq:mars_opt_3}
{} &\textbf{x} \in \mathbb{R}^P \times \mathbb{Z}^{D-P}\\
\label{eq:mars_opt_4}
&B_m(\mathbf{x})= \prod_{k=1}^{Km}[s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}) ]_+\\
\label{eq:mars_opt_5}
&{{[s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}) ]}_+}= \max \{s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}),0\}
\end{eqnarray}
The objective function~\eqref{eq:mars_opt_1} is the TITL-MARS model. The constraint set~\eqref{eq:mars_opt_2} is the boundary of $\mathbf{x}$. The constraint set~\eqref{eq:mars_opt_3} specifies the data types. Constraints~\eqref{eq:mars_opt_4} and \eqref{eq:mars_opt_5} specify the basis functions and the truncated linear terms.
Let $\mathcal{M}$ denote an upper bound of $|x_{v(k,m)}-t_{v(k,m)}|$ and $|t_{v(k,m)}-x_{v(k,m)}|$. Let $y_{k,m}$ be an indicator variable for the nonnegativity of $s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)})$, an let $\eta_{k,m}$ denote the univariate truncated linear function, given as
\begin{equation}\label{eq:utlbf1}
\eta_{k,m}=[s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)})]_+=\max\{s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}),0\}.
\end{equation}
Specifically, when $s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}) \geqslant 0$, $y_{k,m}=1$ and $\eta_{k,m}=s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)})$, otherwise $y_{k,m}=0$ and $\eta_{k,m}=0$.
The TITL-MARS optimization prolbem can be formulated into a general mixed integer quadratic programming problem as follows.
\begin{eqnarray}
\label{eq:mars_opt_formu_1}
\min\,\, & a_0+\sum_{m=1}^{M}\left\{a_m \cdot \prod_{k=1}^{Km} \eta_{k,m} \right\} \\
\mbox{s.t.}\,\, & s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)})\leqslant \eta_{k,m} \leqslant s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)})+\mathcal{M}\cdot(1-y_{k,m}),\nonumber \\
\label{eq:mars_opt_formu_3}
{}&\forall k=1,\ldots, K_m, \forall m=1,\ldots, M\\
\label{eq:mars_opt_formu_4}
{}&0\leqslant \eta_{k,m}\leqslant \mathcal{M}\cdot y_{k,m},\forall k=1,\ldots, K_m, \forall m=1,\ldots, M\\
\label{eq:mars_opt_formu_5}
{}&\textbf{l} \leqslant \textbf{x} \leqslant \textbf{u}\\
\label{eq:mars_opt_formu_6}
{} &\textbf{x} \in \mathbb{R}^P \times \mathbb{Z}^{D-P}\\
\label{eq:mars_opt_formu_7}
{}&\eta_{k,m} \in \mathbb{R},\forall k=1,\ldots, K_m, \forall m=1,\ldots, M\\
\label{eq:mars_opt_formu_8}
{}&y_{k,m} \in \mathbb{B},\forall k=1,\ldots, K_m, \forall m=1,\ldots, M.
\end{eqnarray}
The objective~\eqref{eq:mars_opt_formu_1} is the TITL-MARS model. Equations~\eqref{eq:mars_opt_formu_3} -~\eqref{eq:mars_opt_formu_8} formulate the basis functions into linear constraints and specifying the boundaries and data types.
TITL-MARS-OPT is an optimization process as shown in Figure~\ref{fig:titl_mars_opt_fc}. The process has two steps. The first step is to fit TITL-MARS model, and the second step optimizes TITL-MARS model using MIQP. The benefits of the optimization process has two aspects. First, TITL-MARS can be fit using most commercial MARS software. Second, MIQP can be globally optimized using CPLEX~\citep{ilog12ibm}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=1.8in]{figures/p2/fc2.pdf}
\caption{TITL-MARS-OPT optimization process}
\label{fig:titl_mars_opt_fc}
\end{center}
\end{figure}
\section{Experiments and results}\label{experiments}
In this section, first the genetic algorithm for TITL-MARS optimization is given, and then the presented TITL-MARS-OPT is tested on wind farm power distribution TITL-MARS models and other mathematical models with the genetic algorithm (TITL-MARS-GA) as a benchmark.
\subsection{Genetic algorithm}
The genetic algorithm can also be used as an optimization method to optimize the function (TITL-MARS-GA), as given in Algorithm~\ref{alg:genetic}~\citep{michalewicz1996evolution}, where the input is the two-way interaction MARS model and the maximum generation number $M_{\max}$, and the output is an optimum and an optimum value. In the ``initialization'' step (line 1 in Algorithm~\ref{alg:genetic}), we generate a population and code the individuals from the decimal form to the binary form. In the ``fitness value'' step (line 2), we decode the individuals from the binary form to the decimal form and evaluate each of the individual's decimal values in the MARS model function to obtain the fitness value. In the ``keep the best'' step (line 3), we sort the individuals by their fitness values and store the individual with the best fitness value. The ``selection'' step (line 6) selects parents from the prior population. The ``crossover'' (line 7) chooses two parents and produces a new population. The ``mutation'' (line 8) chooses one point within an individual and changes it from 1 to 0 or from 0 to 1. In this paper, the TITL-MARS-GA algorithm is used as a benchmark compared with the TITL-MARS-OPT method.
\begin{algorithm}
\small
\caption{Genetic algorithm for TITL-MARS optimization}\label{alg:genetic}
\KwData{$\hat{f}( \mathbf{x})=a_0+\sum_{m=1}^{M}\left\{a_m \cdot \prod_{k=1}^{Km}[s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}) ]_+ \right\}, M_{\max}$}
\KwResult{$\mathbf{x}_{\max},f(\mathbf{x}_{\max})$ }
\textbf{Initialization}: Generate a population and code the individuals from decimal to binary.\\
\textbf{Fitness value}: Decode individuals from binary to decimal and get function value.\\
\textbf{Keep the best}: Store the individual with highest or lowest fitness value.\\
$\mbox{gen}=1$\\
\While{\mbox{gen}$<M_{\max}$}
{
\textbf{Selection}: Select parents from prior population.\\
\textbf{Crossover}: Choose two parents and produce a new population.\\
\textbf{Mutation}: Choose one point and $1\rightarrow0$ or $0\rightarrow1$.\\
\textbf{Fitness value}: Decode individuals from binary to decimal and get function value.\\
\textbf{Keep the best}: Store the individual with highest or lowest fitness value.\\
$\mbox{gen}=\mbox{gen}+1$
}
\end{algorithm}
The parameters of TITL-MARS-GA in this paper are from the literatures \cite{grefenstette1986optimization} and \cite{michalewicz1996evolution}, and given in Table~\ref{tab:titi_mars_ga_parameters}.
\begin{table}[h]
\centering
\small
\caption{Parameter settings of TITL-MARS-GA}
\label{tab:titi_mars_ga_parameters}
\scalebox{1.0}{
\begin{tabular}{c|c|c}
\hline
Parameter&\cite{grefenstette1986optimization}& \cite{michalewicz1996evolution}\\
\hline
Population size&30&50 \\
Maximum number of generations& 300&1000 \\
Crossover rate&0.9 & 0.8\\
Mutation rate& 0.01& 0.15\\
\hline
\end{tabular}}
\end{table}
The TITL-MARS-GA optimization process has two steps as shown in Figure~\ref{fig:titl_mars_ga_fc}. The first step fits a two-way interaction truncated linear MARS model, and the second step optimizes decisions of TITL-MARS using the genetic algorithm. The drawback of TITL-MARS-GA is that it does not guarantee global optimality.
\begin{figure}[h]
\begin{center}
\includegraphics[width=1.8in]{figures/p2/fc4.pdf}
\caption{TITL-MARS-GA optimization process}
\label{fig:titl_mars_ga_fc}
\end{center}
\end{figure}
\subsection{Experimental environment}
The experiments are run on a workstation with 64 bit Windows 10 Enterprise system. The CPU version is an Intel(R) CPU E3-1285 v6 @ 4.10GHz, and the RAM has 32 GB.The programming code is written in Python version is 3.6, and the CPLEX solver version is 12.8.
\subsection{Optimization of wind farm power distribution function}
Wind farm power is of paramount significance as a renewable energy source. In this paper, the Monte Carlo method~\citep{metropolis1949monte} is used to generate random wind farm layouts, and the TITL-MARS method is used to study the power distribution under certain wind speeds and directions. After the TITL-MARS model is generated, the TITL-MARS-OPT method is used to study the best turbine position and the worst position. We use the following steps to generate the wind farm power distribution function, as shown in Figure~\ref{fig:titl_mars_wf_fc}. First, we randomly generate $N$ wind farm layouts. Second, we calculate average power output at each location. Third, we use the data from second step to build the TITL-MARS power distribution model.
\begin{figure}[h]
\begin{center}
\includegraphics[width=1.8in]{figures/p2/fc3.pdf}
\caption{Steps to generate a TITL-MARS wind farm power distribution model}
\label{fig:titl_mars_wf_fc}
\end{center}
\end{figure}
After the wind passes through a wind turbine $j$, a part of the wind energy will be absorbed by turbine $j$ and leave the downstream wind with the reduced speed, which is called the wake effect~\citep{jensen1983note}, and the wake effect model is shown in Figure~\ref{fig:wf_wake_m}.
\begin{figure*}[ht]
\centering
\begin{subfigure}[b]{0.4\textwidth}
\centering
\includegraphics[height=1.6in]{figures/p2/wake_loss_model.png}
\caption[]%
{{\small Wind wake effect model}}
\label{fig:wf_wake_m_1}
\end{subfigure}
\hspace{0in}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[height=1.6in]{figures/p2/multi_wake_loss_model.png}
\caption{{\small Wake effect illustration}}
\label{fig:wf_wake_m_2}
\end{subfigure}
\caption[]
{\small Wake effect}
\label{fig:wf_wake_m}
\end{figure*}
Wind speed at turbine $i$ with the wake effect of turbine $j$ is $v_{i,j}$ and can be calculated as
\begin{equation}
\label{eq:rsi}
v_{i,j}=v_0 \left( 1-\frac{2}{3} \cdot \frac{R_j^2}{r_j^2} \right).
\end{equation}
$R_j$ is the radius of the wind turbine $j$, and $r_j$ is the wake radius of the wind turbine $j$. The final wind speed $v_i$ at turbine $i$ with multiple wake effects is given as
\begin{equation}
\label{eq:vi}
\begin{array}{l}
v_i=v_0 \left[ 1-\sqrt{\sum \limits_{j \in \Phi_{i}} \left(1-\frac{v_{i,j}}{v_0}\right)^2} \right],
\end{array}
\end{equation}
where $\Phi _{i}$ is the index set of the turbines which are upwind of the turbine $i$. Afterwards, the actual power of turbine $i$ can be obtained as~\cite{liu2013electric}
\begin{equation}\label{eq:powercurvef}
p(v_i) = \left\{ {\begin{array}{*{20}{ll}}
{0, } & v_{i} < 2\\
{0.3{v_i}^3,} & {2 \leqslant v_{i} < 12.8{\mkern 1mu} }\\
{629.1,} & 12.8{\mkern 1mu} \leqslant {v_{i} \leqslant 18}\\
{0, } & v_{i} > 18{\mkern 1mu},\\
\end{array}} \right.
\end{equation}
and the power curve is shown in Figure~\ref{fig:power_curve}. which is the relationship between the wind turbine power and the wind speed.
\begin{figure}[]
\begin{center}
\includegraphics[width=3in]{figures/p2/power_curve.pdf}
\caption{Relationship of wind speed with output power of a wind turbine}
\label{fig:power_curve}
\end{center}
\end{figure}
The wind farm power distribution is generated using the Monte Carlo methods for a given wind farm and a specific wind distribution.
$f_{w1}$ is generated from a wind farm where there is only one wind speed and one direction. The wind farm is divided into 41 by 41 cells, and each cell has a width of 308 m. The wind is from northeast ($\frac{\pi}{4}$) at 15 m/s.
$f_{w2}$ is generated from a wind farm where the wind has only one wind speed and four directions. The wind farm has the same dimension as that of $f_{w1}$. The wind is from north (0), south ($\pi$), east ($\frac{\pi}{2}$), and west ($\frac{3\pi}{2}$) at 15 m/s.
$f_{w3}$ is generated from a wind farm where the wind has only one wind speed at 15 m/s and six directions, 0, $\frac{\pi}{3}$, $\frac{2\pi}{3}$, $\pi$, $\frac{4\pi}{3}$, and $\frac{5\pi}{3}$.
$f_{w4}$ is generated from a wind farm where the wind has three wind speeds, 12 m/s, 10 m/s, and 8 m/s, and 12 directions, 0, $\frac{\pi}{6}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$, $\frac{2\pi}{3}$, $\pi$, $\frac{7\pi}{6}$, $\frac{4\pi}{3}$, $\frac{3\pi}{2}$, $\frac{5\pi}{3}$, and $\frac{11\pi}{6}$.
TITL-MARS-OPT and TITL-MARS-GA are used to optimize on the wind farm power distribution models to find a global maximum and a minimum, and the results are shown in Figure~\ref{fig:mwd_surfaces} and summarized in Table~\ref{tab:mars_ti_opt_cmp1}. The results are the average value of 30 executions. The table shows the optimal values derived from the TITL-MARS-OPT and TITL-MARS-GA, as well as the computation time in seconds. The result shows that the TITL-MARS-OPT method finds better solutions than the genetic algorithm and uses less time. The maximum value and minimum value are very useful before actually building the wind turbines. The maximum location indicates that it is the best location to build a wind turbine based on the given requirements. The minimum location indicates that this location is the worst location on the wind farm, and if the budget is tight, the piece of land around the minimum location can be neglected.
\begin{table}
\centering
\small
\caption{Comparison of TITL-MARS-OPT and TITL-MARS-GA on wind farm power distribution TITL-MARS models}
\label{tab:mars_ti_opt_cmp1}
\scalebox{1.0}{
\begin{tabular}{c|l|cccc}
\hline
Function&Measurement&MARS-OPT&MARS-GA 1&MARS-GA 2&MARS-GD\\
\hline
\multirow{4}{*}{$f_{w1}$}&\cellcolor{blue!25}Maximum&\cellcolor{blue!25}636.76&\cellcolor{blue!25}618.40&\cellcolor{blue!25}630.75&\cellcolor{blue!25}608.50\\
&Time(seconds)& 0.15& 1.45&9.06&0.04\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}578.30&\cellcolor{green!25}579.18&\cellcolor{green!25}578.33&\cellcolor{green!25}579.21\\
&Time(seconds)& 0.32& 1.46&9.08&0.04\\
\hline
\multirow{4}{*}{$f_{w2}$}&\cellcolor{blue!25}Maximum&\cellcolor{blue!25}588.10&\cellcolor{blue!25}569.96&\cellcolor{blue!25}585.42&\cellcolor{blue!25}582.50\\
&Time(seconds)& 0.15& 1.46&9.18&0.04\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}545.90&\cellcolor{green!25}545.88&\cellcolor{green!25}545.85&\cellcolor{green!25}545.91\\
&Time(seconds)&0.40&1.48&9.15&0.08\\
\hline
\multirow{4}{*}{$f_{w3}$} &\cellcolor{blue!25}Maximum&\cellcolor{blue!25}622.10&\cellcolor{blue!25}608.13&\cellcolor{blue!25}621.48&\cellcolor{blue!25}609.20\\
&Time(seconds)&0.15&1.49&9.36&0.08\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}599.50&\cellcolor{green!25}599.77&\cellcolor{green!25}599.51&\cellcolor{green!25}599.86\\
&Time(seconds)&0.85&1.52&9.34&0.05\\
\hline
\multirow{4}{*}{$f_{w4}$}
&\cellcolor{blue!25}Maximum&\cellcolor{blue!25}393.80&\cellcolor{blue!25}387.92&\cellcolor{blue!25}393.13&\cellcolor{blue!25}382.71\\
&Time(seconds)&0.04&1.57&9.72&0.23\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}303.21&\cellcolor{green!25}303.46&\cellcolor{green!25}303.21&\cellcolor{green!25}303.91\\
&Time(seconds)& 0.51&1.58&9.67&0.01\\
\hline
\end{tabular}}
\end{table}
\begin{figure*}[]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=2.2in]{figures/p2/mars_surface/mars_1d1s_ne_15_with_extremes_2.pdf}
\caption[]%
{{\small Case 1 : $f_{w1}$}}
\label{fig:mwd_surface_ex_1}
\end{subfigure}
\hspace{0em}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=2.2in]{figures/p2/mars_surface/mars_4d1s_15_withextremes_2.pdf}
\caption[]%
{{\small Case 2 : $f_{w2}$}}
\label{fig:mwd_surface_ex_2}
\end{subfigure}
\vspace{0\baselineskip}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=2.2in]{figures/p2/mars_surface/mars_6d1s_15_31_by_31_withextremes_2.pdf}
\caption{{\small Case 3 : $f_{w3}$}}
\label{fig:mwd_surface_ex_3}
\end{subfigure}
\hspace{0em}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=2.2in]{figures/p2/mars_surface/mars_12d3s_21_by_21_2.pdf}
\caption[]%
{{\small Case 4 : $f_{w4}$}}
\label{fig:mwd_surface_ex_4}
\end{subfigure}
\vspace{0\baselineskip}
\caption[]
{\small One run result comparison of TITL-MARS-OPT and TITL-MARS-GA on wind farm power distribution TITL-MARS model}
\label{fig:mwd_surfaces}
\end{figure*}
\subsection{Optimization of other functions}
TITL-MARS-OPT method and TITL-MARS-GA are tested to optimize other six TITL-MARS models to find the global maximum and minimum. The first two TITL-MARS models $f_1$ and $f_2$ are two-dimensional~\citep{miyata2005free}. The $f_3$ and $f_4$ are 10-dimensional TITL-MARS models. $f_5$ is 19-dimensional, and $f_6$ is 21-dimensional~\citep{ariyajunya2013adaptive}. The results are shown in Figure~\ref{fig:other_surfaces} and summarized in Table~\ref{tab:mars_ti_opt_cmp2}. The results are the average value of 30 runs and show TITL-MARS-OPT is superior to TITL-MARS-GA. The result shows that TITL-MARS-OPT achieves better solutions than TITL-MARS-GA and uses less time, which is consistent with the prior result. The results also show that TITL-MARS-OPT is robust in dealing both low-dimensional and high-dimensional TITL-MARS models.
\begin{table}
\centering
\small
\caption{Result comparison of TITL-MARS-OPT and TITL-MARS-GA on six other TITL-MARS mathematical models}
\label{tab:mars_ti_opt_cmp2}
\scalebox{1}{
\begin{tabular}{c|l|cccc}
\hline
Function&Measurement&MARS-OPT&MARS-GA 1&MARS-GA 2&MARS-GD\\
\hline
\multirow{4}{*}{$f_{1}$}&\cellcolor{blue!25}Maximum&\cellcolor{blue!25}8.30&\cellcolor{blue!25}6.23&\cellcolor{blue!25}8.27&\cellcolor{blue!25}6.60\\
&Time(seconds)& 0.70&1.50&9.25&2.38\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}-8.20&\cellcolor{green!25}-6.42&\cellcolor{green!25}-6.33&\cellcolor{green!25}-5.06\\
&Time(seconds)& 1.65&1.51 &9.18&0.57\\
\hline
\multirow{4}{*}{$f_{2}$}
&\cellcolor{blue!25}Maximum&\cellcolor{blue!25}1.81&\cellcolor{blue!25}1.02 &\cellcolor{blue!25}1.45&\cellcolor{blue!25}1.24\\
&Time(seconds)& 0.31&1.42&8.91&1.71\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}-2.20&\cellcolor{green!25}-1.38 &\cellcolor{green!25}-2.20&\cellcolor{green!25}-1.38\\
&Time(seconds)&0.32&1.42 &8.88&1.54\\
\hline
\multirow{4}{*}{$f_{3}$}
&\cellcolor{blue!25}Maximum&\cellcolor{blue!25} 5,774.08&\cellcolor{blue!25}5,092.52 &\cellcolor{blue!25}5,410.16&\cellcolor{blue!25} 5661.76\\
&Time(seconds)&0.02 &2.71 &15.74 &0.02\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25} -1126.39&\cellcolor{green!25}2,289.47 &\cellcolor{green!25}903.69&\cellcolor{green!25}-806.86 \\
&Time(seconds)&0.02 &2.67 &15.81&10.35\\
\hline
\multirow{4}{*}{$f_{4}$}
&\cellcolor{blue!25}Maximum&\cellcolor{blue!25} 48,800.26&\cellcolor{blue!25}-13,516.73 &\cellcolor{blue!25}-19,762.80&\cellcolor{blue!25}-1,891,678.92\\
&Time(seconds)&96.32&2.98 &17.67&14.24\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25} -3,952,146.24&\cellcolor{green!25}-2,605,904.95&\cellcolor{green!25} -3,599,093.85&\cellcolor{green!25}-3,119,142.40\\
&Time(seconds)&0.41 &2.95 &17.57 &0.01\\
\hline
\multirow{4}{*}{$f_{5}$}
&\cellcolor{blue!25}Maximum&\cellcolor{blue!25} 97,679.99&\cellcolor{blue!25}78,263.13 &\cellcolor{blue!25}92,458.62&\cellcolor{blue!25}80,461.36\\
&Time(seconds)&0.02 &5.20 &30.01&28.45 \\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25} -15,439.62&\cellcolor{green!25}33,298.80 &\cellcolor{green!25} 5,904.35&\cellcolor{green!25}719.07\\
&Time(seconds)&2.46 &5.21 &30.38& 28.72\\
\hline
\multirow{4}{*}{$f_{6}$}
&\cellcolor{blue!25}Maximum&\cellcolor{blue!25} 111,225.22&\cellcolor{blue!25} 63,506.19&\cellcolor{blue!25}105,645.30&\cellcolor{blue!25}88956.05\\
&Time(seconds)&0.02 &5.34 &30.62& 29.90\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25} -14,215.61&\cellcolor{green!25}17,886.61&\cellcolor{green!25}494.43&\cellcolor{green!25}-5757.07\\
&Time(seconds)&0.04 &5.33 &30.58&12.64\\
\hline
\end{tabular}}
\end{table}
\begin{table}
\centering
\small
\caption{Result comparison of true function value}
\label{tab:mars_true_value_cmp__}
\scalebox{0.6}{
\begin{tabular}{c|l|ccccccc}
\hline
Function&Measurement&OPT function(model)&MARS+GD&GA 1&GA 2&GD&PL-OPT&PL OPT+GD\\
\hline
\multirow{4}{*}{$f_{1}$}&\cellcolor{blue!25}Maximum&\cellcolor{blue!25}8.01(8.32)&\cellcolor{blue!25}8.21 &\cellcolor{blue!25}8.20&\cellcolor{blue!25}8.21&\cellcolor{blue!25}5.76&\cellcolor{blue!25}-4.76(-5.15)&\cellcolor{blue!25}2.96\\
&Time(seconds)&0.25 &0.24&0.78&7.58&0.01&0.05&0.00\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}-4.56(-8.16)&\cellcolor{green!25}-6.20&\cellcolor{green!25}-3.75&\cellcolor{green!25}-5.76&\cellcolor{green!25}-3.30&\cellcolor{green!25}-4.86(-5.58)&\cellcolor{green!25}-6.20\\
&Time(seconds)&10.57&0.73 &0.77 &7.54&0.01&0.05&0.00\\
\hline
\multirow{4}{*}{$f_{2}$}&\cellcolor{blue!25}Maximum&\cellcolor{blue!25} 0.86(1.81)&\cellcolor{blue!25}1.00&\cellcolor{blue!25}1.00&\cellcolor{blue!25}1.00&\cellcolor{blue!25}1.00&\cellcolor{blue!25}0.31(0.56)&\cellcolor{blue!25}1.00\\
&Time(seconds)&0.10&0.14 &0.59&5.25&0.10&0.24&0.00\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}-0.61(-2.20)&\cellcolor{green!25}-1.00&\cellcolor{green!25}-0.87&\cellcolor{green!25}-1.00&\cellcolor{green!25}-0.96&\cellcolor{green!25}-0.46(-0.53)&\cellcolor{green!25}-1.00\\
&Time(seconds)&2.65&0.22 & 0.54&5.27&0.10&0.13&0.00\\
\hline
\multirow{4}{*}{$f_{3}$}&\cellcolor{blue!25}Maximum&\cellcolor{blue!25} 6,029.13(5,774.08)&\cellcolor{blue!25}6132.00&\cellcolor{blue!25}5,488.40&\cellcolor{blue!25}5,647.09&\cellcolor{blue!25}5893.00&\cellcolor{blue!25}4821.00(4667.86)&\cellcolor{blue!25}6132.00\\
&Time(seconds)&0.01 &0.01&1.90&18.81&0.01&0.07&0.00\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}-826.08(-1,126.39)&\cellcolor{green!25}-923.86&\cellcolor{green!25}2,755.59&\cellcolor{green!25}742.00&\cellcolor{green!25}-525.09&\cellcolor{green!25}288.27(118.11)&\cellcolor{green!25}-123.44\\
&Time(seconds)& 0.01&0.01& 1.85&18.81&0.47&0.05&0.00\\
\hline
\multirow{4}{*}{$f_{4}$}&\cellcolor{blue!25}Maximum&\cellcolor{blue!25}-1,991.28(48,800.26)&\cellcolor{blue!25}-0.01 &\cellcolor{blue!25} -1,643,753.96&\cellcolor{blue!25}-131,067.27&\cellcolor{blue!25}-0.01&\cellcolor{blue!25}-505,071.14(-398,359.99)&\cellcolor{blue!25}-0.01\\
&Time(seconds)&107.47&112.74 &2.12&21.98&2.47&0.07&0.00\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}-4,581,942.36(-3,952,146.24)&\cellcolor{green!25}-4,616,810.03&\cellcolor{green!25}-3,651,963.86&\cellcolor{green!25}-3,834,009.84&\cellcolor{green!25}-3,161,354.60&\cellcolor{green!25}-4,433,291.39(-3,945,225.00)&\cellcolor{green!25}-4,616,810.03\\
&Time(seconds)&0.40&0.42 &2.04 &21.83&0.01&0.10&0.00\\
\hline
\end{tabular}}
\end{table}
\begin{figure*}[]
\centering
\hspace{0em}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f1_max_bp.pdf}
\caption[]%
{{\small $f_1$}}
\label{fig:f1_max}
\end{subfigure}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f2_max_bp.pdf}
\caption[]%
{{\small $f_2$}}
\label{fig:f2_max}
\end{subfigure}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f3_max_bp.pdf}
\caption[]%
{{\small $f_3$}}
\label{fig:f3_max}
\end{subfigure}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f4_max_bp.pdf}
\caption[]%
{{\small $f_4$}}
\label{fig:f4_max}
\end{subfigure}
\caption{\small maximum values boxplot}
\label{fig:max_bp}
\end{figure*}
\begin{figure*}[]
\centering
\hspace{0em}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f1_min_bp.pdf}
\caption[]%
{{\small $f_1$}}
\label{fig:f1_min}
\end{subfigure}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f2_min_bp.pdf}
\caption[]%
{{\small $f_2$}}
\label{fig:f2_min}
\end{subfigure}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f3_min_bp.pdf}
\caption[]%
{{\small $f_3$}}
\label{fig:f3_min}
\end{subfigure}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f4_min_bp.pdf}
\caption[]%
{{\small $f_4$}}
\label{fig:f4_min}
\end{subfigure}
\caption{\small minimum values boxplot}
\label{fig:min_bp}
\end{figure*}
\subsection{$f_1$, $f_2$, $f_3$ and $f_4$ functions}
\begin{flalign}
f_1({x_1,x_2})=& 3(1-x_1)^2\exp(-x_1^2-(x_2+1)^2) -10(\frac{x_1}{5}-x_1^3-x_2^5)\exp(-x_1^2-x_2^2)&&\\\nonumber
&-\frac{1}{3}\exp(-(x_1+1)^2-x_2^2)+2x_1, &&\\\nonumber
&-2\leqslant x_1 \leqslant 2,-2\leqslant x_2 \leqslant 2&&\nonumber
\end{flalign}
\begin{flalign}
f_2(x_1,x_2)=&\sin \left(\frac{\pi x_1}{12}\right) \cos \left(\frac{\pi x_2}{16}\right)&&\\&-20\leqslant x_1\leqslant20,-20\leqslant x_2\leqslant 20&&\nonumber
\end{flalign}
\begin{flalign}
f_3(\mathbf{x})=&x_1^2+x_2^2+x_1x_2-14x_1-16x_2+(x_3-10)^2-4(x_4-5)^2+(x_5-3)^2&&\\\nonumber
&+2(x_6-1)^2+5x_7^2+7(x_8-11)^2+2(x_9-10)^2+2(x_{10}-7)^2+45&&\\
&-10\leqslant x_i \leqslant 10&&\nonumber
\end{flalign}
\begin{flalign}
f_4(\mathbf{x})=&\sum\limits_{j=1}^{10}\exp(x_j)\left(c_j+x_j-\ln\sum\limits_{k=1}^{10}\exp(x_k)\right)&&\\\nonumber
&\mathbf{c}=[-0.6089,-17.164,-34.054,-5.914,-24.721,-14.986,-24.100,-10.708,&&\\\nonumber&-26.662,-22.179]&&\\
&-10\leqslant x_i \leqslant 10&&\nonumber
\end{flalign}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[height=1.7in]{figures/p2/f1.pdf}
\caption[]%
{{\small $f_1(x_1,x_2)$}}
\label{fig:f1_s}
\end{subfigure}
\hspace{0em}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[height=1.7in]{figures/p2/f2.pdf}
\caption[]%
{{\small $f_2(x_1,x_2)$}}
\label{fig:f2_surface}
\end{subfigure}
\vspace{0\baselineskip}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[height=1.7in]{figures/p2/f3.pdf}
\caption[]%
{{\small $f_3(x_1,x_2) \mbox{ other } x_i=4.0$}}
\label{fig:f3_s}
\end{subfigure}
\hspace{0em}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[height=1.7in]{figures/p2/f4.pdf}
\caption[]%
{{\small $f_4(x_1,x_2) \mbox{ other } x_i=3.0$}}
\label{fig:f4_s}
\end{subfigure}
\vspace{0\baselineskip}
\caption{\small Surfaces of dataset functions}
\label{fig:f_surfaces}
\end{figure*}
\begin{figure*}[]
\centering
\hspace{0em}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=2.2in]{figures/p2/mars_surface/mars_p8_with_extremes_2.pdf}
\caption[]%
{{\small $f_1$}}
\label{fig:f1_surface_ex}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=2.2in]{figures/p2/mars_surface/mars_p4_with_extremes_2.pdf}
\caption[]%
{{\small $f_2$}}
\label{fig:f2_surface_ex}
\end{subfigure}
\caption{\small One run result comparison of TITL-MARS-OPT and TITL-MARS-GA on $f_1$ and $f_2$ TITL-MARS models}
\label{fig:other_surfaces}
\end{figure*}
\section{Conclusion}\label{conclusion}
In this paper, a new method (TITL-MARS-OPT) is proposed to globally optimize analytically on the two-way interaction truncated linear MARS (TITL-MARS) by using mixed integer quadratic programming. We verified the presented TITL-MARS-OPT method on the wind farm power distribution TITL-MARS models and six other mathematical TITL-MARS models. The application on wind farm power distribution models gives the best location and worst location information on the wind farm. The testing TITL-MARS models are from 2-dimensions to up to 21-dimensions, and it shows the TITL-MARS-OPT method is robust in dealing with TITL-MARS models with varied dimensions. We also compared the TITL-MARS-OPT method with TITL-MARS-GA in TITL-MARS model optimization, and it shows that the new method can achieve better accuracy and time efficiency. TITL-MARS-OPT can achieve as high as 316\% and on average 46\% better solution quality and is on average 175\% faster than TITL-MARS-GA. In addition, the Python code and the testing models of this paper are made open source, and it will contribute to the study of TITL-MARS models and optimization.
\bibliographystyle{unsrtnat}
\section{Introduction}
Optimization for complex systems often involves fitting a system prediction model to estimate how a system performs and then optimizing the decisions based on the system prediction model as shown in Figure~\ref{fig:ocs_fc}. Two major tasks in optimization of complex systems include training or meta-modeling a statistical or system model and optimizing input or decisions based on statistical model.
\begin{figure}[h]
\begin{center}
\includegraphics[width=2.5in]{figures/p2/fc5.pdf}
\caption{Optimization of complex systems}
\label{fig:ocs_fc}
\end{center}
\end{figure}
In real world complex systems, underlying relationships are commonly unknown and are approximated
from data using empirical models. \citet{wu2004travel} applied support vector regression in travel time prediction and proved support vector regression was applicable in traffic data analysis. \cite{lv2014traffic} applied a deep learning approach with autoencoders in traffic flow prediction. \cite{ohlmacher2003using} used logistic regression to generate a landslide-hazard map to predict landslide hazards. \cite{leathwick2005using} used multivariate adaptive regression splines to predict the distributions of freshwater diadromous fish.
If one seeks to optimize a complex system, the optimization
method would need to be able to handle the data-driven approximation models. Given the wide range
of possible approximation models, such as machine learning algorithms, the most commonly
employed optimization approach in these situations is a heuristic approach, such as an evolutionary
algorithm, that cannot guarantee global optimality. Rather than having the approximation model
dictate the need for a heuristic optimization method, the research in this paper seeks a balance that
utilizes a flexible approximation model with structure that can be exploited to enable
true global optimization. In other words, the ``best of both worlds'' is sought, by achieving global
optimality while still maintaining a flexible approximation model.
The approxmation model of choice in this paper is multivariate adaptive regression splines (MARS),
introduced by machine learning pioneer Jerome Friedman in
1991~\citep{friedman1991multivariate}. The structure of MARS
is based on a linear statistical spline model and provides a flexible fit to data while also
achieving a parsimonious model.
The desire to conduct global optimization is seen in many applications, and there are a number
of approaches classified as global optimization methods~\citep{horst2000introduction}. The primary
challenge in achieving global optimality is that many real world applications involve multiple
local optima. Finding a global optimum requires sifting through the local optima and recognizing
when one is suboptimal. The vast majority of applications employ heuristic search algorithms
seek to overcome the challenge of local optima, but do not guarantee global optimality.
Examples include heuristics based on evolutionary algorithms~\citep{morra2018optimization,russo2018knowledge,dahi2018stop},
particle swarm optimization~\citep{alswaitti2018density},
the grasshopper optimization method~\citep{ewees2018improved}, and
the weighted superposition attraction method~\citep{baykasouglu2018dynamic}.
In order to guarantee global optimality, the approach in this paper takes advantage of well-known
properties of mixed integer and quadratic programming (MIQP)~\citep{bliek1u2014solving}.
Some recent applications in which MARS has been employed for empirical modeling include
a water pollution prediction problem~\citep{kisi2016application},
the head load in a building~\citep{roy2018estimating},
the estimation of landfill leachate~\citep{bhatt2017estimating},
and the damage identification for web core composite bridges~\citep{mukhopadhyay2018multivariate},
In optimization problems, MARS has been employed as the empirical model to approximate
unknown relationships in a variety of applications. For stochastic dynamic programming,
the use of MARS to approximate the value function was introduced by~\cite{chen1999applying}.
Since then, the MARS value function approximation approach has been used to numerically solve
a 30-dimensional water reservoir management problem~\citep{cervellera2006optimization},
a 20-dimensional wastewater treatment system~\citep{tarun2011optimizing,tsai2004stochastic,tsai2005flexible},
and a 524-dimensional nonstationary ground-level ozone pollution control problem~\citep{yang2009decision}.
In revenue management, MARS was employed to estimate upper and lower bounds for the
value function of a Markov decision problem~\citep{chen2003solving,siddappa2007refined},
and MARS was used to represent the revenue function in airline overbooking
optimization~\cite{siddappa2008optimising}.
In two-stage stochastic programming, MARS was used to efficiently represent the
expected profit function for an airline fleet assignment problem~\citep{pilla2008statistical}.
This fleet assignment research was extended to utilize a cutting plane method with MARS
to conduct the optimization~\citep{pilla2012multivariate}.
The contribution of this current work extends the approach of
\cite{martinez2017global},
who developed a piece-wise linear MARS structure and formulated a mixed integer and
linear programming problem to globally optimize vehicle design parameters to
improve performance in crash simulations.
The piece-wise linear MARS function may be nonconvex, and the approach of Martinez et al.
will yield a global optimum. However, restricting to piece-wise linear forms limits
the flexibility of the empirical model. Hence, in the current work, the MARS form
employed is based on the original MARS model. The primary challenge for an optimization
method is handling the nonconvex MARS interaction terms, which are products of univariate terms.
By restricting to two-way interactions, we can utilize quadratic programming methods.
In real world applications, two-way interactions are commonly sufficient for empirical modeling~\cite{kutner2005applied}
In summary, the contribution of the presented approach is a MIQP
global optimization method for a MARS model that allows up to
two-way interaction terms that are products of truncated linear univariate functions
(TITL-MARS). This approach is referred to as TITL-MARS-OPT and is compared against
a genetic algorithm for two types of cases.
First, a wind farm power distribution case study is described, and then other TITL-MARS forms
are tested.
Python code for TITL-MARS and TITL-MARS-OPT will be made available on GitHub upon
acceptance of this paper~(\url{https://github.com/JuXinglong/TITL-MARS-OPT}).
The rest of this paper is organized as follows.
Section~\ref{background} describes background on TITL-MARS.
Section~\ref{formulation} presents the MIQP formulation for TITL-MARS-OPT.
The computational study is given in Section~\ref{experiments}, and
Section~\ref{conclusion} concludes the paper.
\section{Background of two-way interaction truncated linear multivariate adaptive regression splines}\label{background}
This section introduces the two-way interaction truncated linear MARS (TITL-MARS) model.
The two-way interaction truncated linear MARS regression model with the response variable ${f}( \mathbf{x}_i)$ is to be built on the independent variable $\mathbf{x}_i$ and can be written in the form of the linear combination of the basis functions as~\cite{friedman1991multivariate}
\begin{equation}\label{eq:MARS_2way}
\begin{array}{l}
\hat{f}( \mathbf{x})=a_0+\sum_{m=1}^{M}\left\{a_m \cdot B_m(\mathbf{x}) \right\}.
\end{array}
\end{equation}
The MARS model is denoted as $\hat{f}( \mathbf{x})$, and $a_0$ is the constant term of the model. The basis function is denoted as $B_m(\mathbf{x})$, and $a_m$ is the coefficient of $B_m(\mathbf{x})$. The index of the basis function is denoted as $m$, and $M$ is the total number of basis functions. The basis function $B_m(\mathbf{x})$ using the truncated linear term has the following form
\begin{equation}\label{eq:MARS_2way_B}
\begin{array}{l}
B_m(\mathbf{x})= \prod_{k=1}^{Km}[s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}) ]_+.
\end{array}
\end{equation}
The truncated linear term is denoted as ${[s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}) ]}_+$, and the basis function $B_m(\mathbf{x})$ is the product of truncated linear terms.
The index of the truncated linear term in $B_m(\mathbf{x})$ is denoted as $k$, and $K_m$ is the total number of truncated linear terms in $B_m(\mathbf{x})$. The sign of the truncated linear term is $s_{k,m}$, which can be $+1$ or $-1$. The $v$-th component of $\mathbf{x}$ is denoted as $x_{v(k,m)}$, and $t_{v(k,m)}$ is the corresponding knot value.
\textit{TITL-MARS is the special case of MARS in which $K_m \leqslant 2$.}
\section{Formulation of two-way interaction truncated linear MARS using mixed integer quadratic programming}\label{formulation}
The general mixed integer quadratic programming problem~\citep{bliek1u2014solving} is given as
\begin{equation}
\label{eq:miqpp}
\begin{array}{ll}
\min & \frac{1}{2} \textbf{z}^T\textbf{Q}\textbf{z}+\textbf{c}^T\textbf{z} \\
{s.t.} & \textbf{A}\textbf{z}=\textbf{b} \\
{} & \textbf{l} \leqslant \textbf{z} \leqslant \textbf{u}\\
{} &\textbf{z} \in \mathbb{R}^P \times \mathbb{Z}^{D-P},\\
\end{array}
\end{equation}
while the two-way interaction truncated linear MARS is given in Section~\ref{background}. In~\eqref{eq:miqpp}, the decision variable is $\textbf{z}$, and the quadratic coefficients matrix is $\textbf{Q}$. The coefficients of the linear terms in the objective funtion are in vector $\textbf{c}$. The linear constraints are denoated as $\textbf{A}\textbf{z}=\textbf{b}$. The lower bound and upper bound of $\textbf{z}$ are $\textbf{l}$ and $\textbf{u}$, respectively. The dimension of $\textbf{z}$ is $D$. There are $P$ dimensions of real values, and $D-P$ dimensions of integers.
Problem in the form~\ref{eq:miqpp} can be solved using the CPLEX solver.
The TITL-MARS optimization prolbem is given as follows. \begin{eqnarray}
\label{eq:mars_opt_1}
\min\,\, & \hat{f}( \mathbf{x})=a_0+\sum_{m=1}^{M}\left\{a_m \cdot B_m(\mathbf{x}) \right\} \\
\label{eq:mars_opt_2}
\mbox{s.t.}\,\, & \textbf{l} \leqslant \textbf{x} \leqslant \textbf{u}\\
\label{eq:mars_opt_3}
{} &\textbf{x} \in \mathbb{R}^P \times \mathbb{Z}^{D-P}\\
\label{eq:mars_opt_4}
&B_m(\mathbf{x})= \prod_{k=1}^{Km}[s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}) ]_+\\
\label{eq:mars_opt_5}
&{{[s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}) ]}_+}= \max \{s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}),0\}
\end{eqnarray}
The objective function~\eqref{eq:mars_opt_1} is the TITL-MARS model. The constraint set~\eqref{eq:mars_opt_2} is the boundary of $\mathbf{x}$. The constraint set~\eqref{eq:mars_opt_3} specifies the data types. Constraints~\eqref{eq:mars_opt_4} and \eqref{eq:mars_opt_5} specify the basis functions and the truncated linear terms.
Let $\mathcal{M}$ denote an upper bound of $|x_{v(k,m)}-t_{v(k,m)}|$ and $|t_{v(k,m)}-x_{v(k,m)}|$. Let $y_{k,m}$ be an indicator variable for the nonnegativity of $s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)})$, an let $\eta_{k,m}$ denote the univariate truncated linear function, given as
\begin{equation}\label{eq:utlbf1}
\eta_{k,m}=[s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)})]_+=\max\{s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}),0\}.
\end{equation}
Specifically, when $s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}) \geqslant 0$, $y_{k,m}=1$ and $\eta_{k,m}=s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)})$, otherwise $y_{k,m}=0$ and $\eta_{k,m}=0$.
The TITL-MARS optimization prolbem can be formulated into a general mixed integer quadratic programming problem as follows.
\begin{eqnarray}
\label{eq:mars_opt_formu_1}
\min\,\, & a_0+\sum_{m=1}^{M}\left\{a_m \cdot \prod_{k=1}^{Km} \eta_{k,m} \right\} \\
\mbox{s.t.}\,\, & s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)})\leqslant \eta_{k,m} \leqslant s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)})+\mathcal{M}\cdot(1-y_{k,m}),\nonumber \\
\label{eq:mars_opt_formu_3}
{}&\forall k=1,\ldots, K_m, \forall m=1,\ldots, M\\
\label{eq:mars_opt_formu_4}
{}&0\leqslant \eta_{k,m}\leqslant \mathcal{M}\cdot y_{k,m},\forall k=1,\ldots, K_m, \forall m=1,\ldots, M\\
\label{eq:mars_opt_formu_5}
{}&\textbf{l} \leqslant \textbf{x} \leqslant \textbf{u}\\
\label{eq:mars_opt_formu_6}
{} &\textbf{x} \in \mathbb{R}^P \times \mathbb{Z}^{D-P}\\
\label{eq:mars_opt_formu_7}
{}&\eta_{k,m} \in \mathbb{R},\forall k=1,\ldots, K_m, \forall m=1,\ldots, M\\
\label{eq:mars_opt_formu_8}
{}&y_{k,m} \in \mathbb{B},\forall k=1,\ldots, K_m, \forall m=1,\ldots, M.
\end{eqnarray}
The objective~\eqref{eq:mars_opt_formu_1} is the TITL-MARS model. Equations~\eqref{eq:mars_opt_formu_3} -~\eqref{eq:mars_opt_formu_8} formulate the basis functions into linear constraints and specifying the boundaries and data types.
TITL-MARS-OPT is an optimization process as shown in Figure~\ref{fig:titl_mars_opt_fc}. The process has two steps. The first step is to fit TITL-MARS model, and the second step optimizes TITL-MARS model using MIQP. The benefits of the optimization process has two aspects. First, TITL-MARS can be fit using most commercial MARS software. Second, MIQP can be globally optimized using CPLEX~\citep{ilog12ibm}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=1.8in]{figures/p2/fc2.pdf}
\caption{TITL-MARS-OPT optimization process}
\label{fig:titl_mars_opt_fc}
\end{center}
\end{figure}
\section{Experiments and results}\label{experiments}
In this section, first the genetic algorithm for TITL-MARS optimization is given, and then the presented TITL-MARS-OPT is tested on wind farm power distribution TITL-MARS models and other mathematical models with the genetic algorithm (TITL-MARS-GA) as a benchmark.
\subsection{Genetic algorithm}
The genetic algorithm can also be used as an optimization method to optimize the function (TITL-MARS-GA), as given in Algorithm~\ref{alg:genetic}~\citep{michalewicz1996evolution}, where the input is the two-way interaction MARS model and the maximum generation number $M_{\max}$, and the output is an optimum and an optimum value. In the ``initialization'' step (line 1 in Algorithm~\ref{alg:genetic}), we generate a population and code the individuals from the decimal form to the binary form. In the ``fitness value'' step (line 2), we decode the individuals from the binary form to the decimal form and evaluate each of the individual's decimal values in the MARS model function to obtain the fitness value. In the ``keep the best'' step (line 3), we sort the individuals by their fitness values and store the individual with the best fitness value. The ``selection'' step (line 6) selects parents from the prior population. The ``crossover'' (line 7) chooses two parents and produces a new population. The ``mutation'' (line 8) chooses one point within an individual and changes it from 1 to 0 or from 0 to 1. In this paper, the TITL-MARS-GA algorithm is used as a benchmark compared with the TITL-MARS-OPT method.
\begin{algorithm}
\small
\caption{Genetic algorithm for TITL-MARS optimization}\label{alg:genetic}
\KwData{$\hat{f}( \mathbf{x})=a_0+\sum_{m=1}^{M}\left\{a_m \cdot \prod_{k=1}^{Km}[s_{k,m} \cdot (x_{v(k,m)}-t_{v(k,m)}) ]_+ \right\}, M_{\max}$}
\KwResult{$\mathbf{x}_{\max},f(\mathbf{x}_{\max})$ }
\textbf{Initialization}: Generate a population and code the individuals from decimal to binary.\\
\textbf{Fitness value}: Decode individuals from binary to decimal and get function value.\\
\textbf{Keep the best}: Store the individual with highest or lowest fitness value.\\
$\mbox{gen}=1$\\
\While{\mbox{gen}$<M_{\max}$}
{
\textbf{Selection}: Select parents from prior population.\\
\textbf{Crossover}: Choose two parents and produce a new population.\\
\textbf{Mutation}: Choose one point and $1\rightarrow0$ or $0\rightarrow1$.\\
\textbf{Fitness value}: Decode individuals from binary to decimal and get function value.\\
\textbf{Keep the best}: Store the individual with highest or lowest fitness value.\\
$\mbox{gen}=\mbox{gen}+1$
}
\end{algorithm}
The parameters of TITL-MARS-GA in this paper are from the literatures \cite{grefenstette1986optimization} and \cite{michalewicz1996evolution}, and given in Table~\ref{tab:titi_mars_ga_parameters}.
\begin{table}[h]
\centering
\small
\caption{Parameter settings of TITL-MARS-GA}
\label{tab:titi_mars_ga_parameters}
\scalebox{1.0}{
\begin{tabular}{c|c|c}
\hline
Parameter&\cite{grefenstette1986optimization}& \cite{michalewicz1996evolution}\\
\hline
Population size&30&50 \\
Maximum number of generations& 300&1000 \\
Crossover rate&0.9 & 0.8\\
Mutation rate& 0.01& 0.15\\
\hline
\end{tabular}}
\end{table}
The TITL-MARS-GA optimization process has two steps as shown in Figure~\ref{fig:titl_mars_ga_fc}. The first step fits a two-way interaction truncated linear MARS model, and the second step optimizes decisions of TITL-MARS using the genetic algorithm. The drawback of TITL-MARS-GA is that it does not guarantee global optimality.
\begin{figure}[h]
\begin{center}
\includegraphics[width=1.8in]{figures/p2/fc4.pdf}
\caption{TITL-MARS-GA optimization process}
\label{fig:titl_mars_ga_fc}
\end{center}
\end{figure}
\subsection{Experimental environment}
The experiments are run on a workstation with 64 bit Windows 10 Enterprise system. The CPU version is an Intel(R) CPU E3-1285 v6 @ 4.10GHz, and the RAM has 32 GB.The programming code is written in Python version is 3.6, and the CPLEX solver version is 12.8.
\subsection{Optimization of wind farm power distribution function}
Wind farm power is of paramount significance as a renewable energy source. In this paper, the Monte Carlo method~\citep{metropolis1949monte} is used to generate random wind farm layouts, and the TITL-MARS method is used to study the power distribution under certain wind speeds and directions. After the TITL-MARS model is generated, the TITL-MARS-OPT method is used to study the best turbine position and the worst position. We use the following steps to generate the wind farm power distribution function, as shown in Figure~\ref{fig:titl_mars_wf_fc}. First, we randomly generate $N$ wind farm layouts. Second, we calculate average power output at each location. Third, we use the data from second step to build the TITL-MARS power distribution model.
\begin{figure}[h]
\begin{center}
\includegraphics[width=1.8in]{figures/p2/fc3.pdf}
\caption{Steps to generate a TITL-MARS wind farm power distribution model}
\label{fig:titl_mars_wf_fc}
\end{center}
\end{figure}
After the wind passes through a wind turbine $j$, a part of the wind energy will be absorbed by turbine $j$ and leave the downstream wind with the reduced speed, which is called the wake effect~\citep{jensen1983note}, and the wake effect model is shown in Figure~\ref{fig:wf_wake_m}.
\begin{figure*}[ht]
\centering
\begin{subfigure}[b]{0.4\textwidth}
\centering
\includegraphics[height=1.6in]{figures/p2/wake_loss_model.png}
\caption[]%
{{\small Wind wake effect model}}
\label{fig:wf_wake_m_1}
\end{subfigure}
\hspace{0in}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[height=1.6in]{figures/p2/multi_wake_loss_model.png}
\caption{{\small Wake effect illustration}}
\label{fig:wf_wake_m_2}
\end{subfigure}
\caption[]
{\small Wake effect}
\label{fig:wf_wake_m}
\end{figure*}
Wind speed at turbine $i$ with the wake effect of turbine $j$ is $v_{i,j}$ and can be calculated as
\begin{equation}
\label{eq:rsi}
v_{i,j}=v_0 \left( 1-\frac{2}{3} \cdot \frac{R_j^2}{r_j^2} \right).
\end{equation}
$R_j$ is the radius of the wind turbine $j$, and $r_j$ is the wake radius of the wind turbine $j$. The final wind speed $v_i$ at turbine $i$ with multiple wake effects is given as
\begin{equation}
\label{eq:vi}
\begin{array}{l}
v_i=v_0 \left[ 1-\sqrt{\sum \limits_{j \in \Phi_{i}} \left(1-\frac{v_{i,j}}{v_0}\right)^2} \right],
\end{array}
\end{equation}
where $\Phi _{i}$ is the index set of the turbines which are upwind of the turbine $i$. Afterwards, the actual power of turbine $i$ can be obtained as~\cite{liu2013electric}
\begin{equation}\label{eq:powercurvef}
p(v_i) = \left\{ {\begin{array}{*{20}{ll}}
{0, } & v_{i} < 2\\
{0.3{v_i}^3,} & {2 \leqslant v_{i} < 12.8{\mkern 1mu} }\\
{629.1,} & 12.8{\mkern 1mu} \leqslant {v_{i} \leqslant 18}\\
{0, } & v_{i} > 18{\mkern 1mu},\\
\end{array}} \right.
\end{equation}
and the power curve is shown in Figure~\ref{fig:power_curve}. which is the relationship between the wind turbine power and the wind speed.
\begin{figure}[]
\begin{center}
\includegraphics[width=3in]{figures/p2/power_curve.pdf}
\caption{Relationship of wind speed with output power of a wind turbine}
\label{fig:power_curve}
\end{center}
\end{figure}
The wind farm power distribution is generated using the Monte Carlo methods for a given wind farm and a specific wind distribution.
$f_{w1}$ is generated from a wind farm where there is only one wind speed and one direction. The wind farm is divided into 41 by 41 cells, and each cell has a width of 308 m. The wind is from northeast ($\frac{\pi}{4}$) at 15 m/s.
$f_{w2}$ is generated from a wind farm where the wind has only one wind speed and four directions. The wind farm has the same dimension as that of $f_{w1}$. The wind is from north (0), south ($\pi$), east ($\frac{\pi}{2}$), and west ($\frac{3\pi}{2}$) at 15 m/s.
$f_{w3}$ is generated from a wind farm where the wind has only one wind speed at 15 m/s and six directions, 0, $\frac{\pi}{3}$, $\frac{2\pi}{3}$, $\pi$, $\frac{4\pi}{3}$, and $\frac{5\pi}{3}$.
$f_{w4}$ is generated from a wind farm where the wind has three wind speeds, 12 m/s, 10 m/s, and 8 m/s, and 12 directions, 0, $\frac{\pi}{6}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$, $\frac{2\pi}{3}$, $\pi$, $\frac{7\pi}{6}$, $\frac{4\pi}{3}$, $\frac{3\pi}{2}$, $\frac{5\pi}{3}$, and $\frac{11\pi}{6}$.
TITL-MARS-OPT and TITL-MARS-GA are used to optimize on the wind farm power distribution models to find a global maximum and a minimum, and the results are shown in Figure~\ref{fig:mwd_surfaces} and summarized in Table~\ref{tab:mars_ti_opt_cmp1}. The results are the average value of 30 executions. The table shows the optimal values derived from the TITL-MARS-OPT and TITL-MARS-GA, as well as the computation time in seconds. The result shows that the TITL-MARS-OPT method finds better solutions than the genetic algorithm and uses less time. The maximum value and minimum value are very useful before actually building the wind turbines. The maximum location indicates that it is the best location to build a wind turbine based on the given requirements. The minimum location indicates that this location is the worst location on the wind farm, and if the budget is tight, the piece of land around the minimum location can be neglected.
\begin{table}
\centering
\small
\caption{Comparison of TITL-MARS-OPT and TITL-MARS-GA on wind farm power distribution TITL-MARS models}
\label{tab:mars_ti_opt_cmp1}
\scalebox{1.0}{
\begin{tabular}{c|l|cccc}
\hline
Function&Measurement&MARS-OPT&MARS-GA 1&MARS-GA 2&MARS-GD\\
\hline
\multirow{4}{*}{$f_{w1}$}&\cellcolor{blue!25}Maximum&\cellcolor{blue!25}636.76&\cellcolor{blue!25}618.40&\cellcolor{blue!25}630.75&\cellcolor{blue!25}608.50\\
&Time(seconds)& 0.15& 1.45&9.06&0.04\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}578.30&\cellcolor{green!25}579.18&\cellcolor{green!25}578.33&\cellcolor{green!25}579.21\\
&Time(seconds)& 0.32& 1.46&9.08&0.04\\
\hline
\multirow{4}{*}{$f_{w2}$}&\cellcolor{blue!25}Maximum&\cellcolor{blue!25}588.10&\cellcolor{blue!25}569.96&\cellcolor{blue!25}585.42&\cellcolor{blue!25}582.50\\
&Time(seconds)& 0.15& 1.46&9.18&0.04\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}545.90&\cellcolor{green!25}545.88&\cellcolor{green!25}545.85&\cellcolor{green!25}545.91\\
&Time(seconds)&0.40&1.48&9.15&0.08\\
\hline
\multirow{4}{*}{$f_{w3}$} &\cellcolor{blue!25}Maximum&\cellcolor{blue!25}622.10&\cellcolor{blue!25}608.13&\cellcolor{blue!25}621.48&\cellcolor{blue!25}609.20\\
&Time(seconds)&0.15&1.49&9.36&0.08\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}599.50&\cellcolor{green!25}599.77&\cellcolor{green!25}599.51&\cellcolor{green!25}599.86\\
&Time(seconds)&0.85&1.52&9.34&0.05\\
\hline
\multirow{4}{*}{$f_{w4}$}
&\cellcolor{blue!25}Maximum&\cellcolor{blue!25}393.80&\cellcolor{blue!25}387.92&\cellcolor{blue!25}393.13&\cellcolor{blue!25}382.71\\
&Time(seconds)&0.04&1.57&9.72&0.23\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}303.21&\cellcolor{green!25}303.46&\cellcolor{green!25}303.21&\cellcolor{green!25}303.91\\
&Time(seconds)& 0.51&1.58&9.67&0.01\\
\hline
\end{tabular}}
\end{table}
\begin{figure*}[]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=2.2in]{figures/p2/mars_surface/mars_1d1s_ne_15_with_extremes_2.pdf}
\caption[]%
{{\small Case 1 : $f_{w1}$}}
\label{fig:mwd_surface_ex_1}
\end{subfigure}
\hspace{0em}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=2.2in]{figures/p2/mars_surface/mars_4d1s_15_withextremes_2.pdf}
\caption[]%
{{\small Case 2 : $f_{w2}$}}
\label{fig:mwd_surface_ex_2}
\end{subfigure}
\vspace{0\baselineskip}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=2.2in]{figures/p2/mars_surface/mars_6d1s_15_31_by_31_withextremes_2.pdf}
\caption{{\small Case 3 : $f_{w3}$}}
\label{fig:mwd_surface_ex_3}
\end{subfigure}
\hspace{0em}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=2.2in]{figures/p2/mars_surface/mars_12d3s_21_by_21_2.pdf}
\caption[]%
{{\small Case 4 : $f_{w4}$}}
\label{fig:mwd_surface_ex_4}
\end{subfigure}
\vspace{0\baselineskip}
\caption[]
{\small One run result comparison of TITL-MARS-OPT and TITL-MARS-GA on wind farm power distribution TITL-MARS model}
\label{fig:mwd_surfaces}
\end{figure*}
\subsection{Optimization of other functions}
TITL-MARS-OPT method and TITL-MARS-GA are tested to optimize other six TITL-MARS models to find the global maximum and minimum. The first two TITL-MARS models $f_1$ and $f_2$ are two-dimensional~\citep{miyata2005free}. The $f_3$ and $f_4$ are 10-dimensional TITL-MARS models. $f_5$ is 19-dimensional, and $f_6$ is 21-dimensional~\citep{ariyajunya2013adaptive}. The results are shown in Figure~\ref{fig:other_surfaces} and summarized in Table~\ref{tab:mars_ti_opt_cmp2}. The results are the average value of 30 runs and show TITL-MARS-OPT is superior to TITL-MARS-GA. The result shows that TITL-MARS-OPT achieves better solutions than TITL-MARS-GA and uses less time, which is consistent with the prior result. The results also show that TITL-MARS-OPT is robust in dealing both low-dimensional and high-dimensional TITL-MARS models.
\begin{table}
\centering
\small
\caption{Result comparison of TITL-MARS-OPT and TITL-MARS-GA on six other TITL-MARS mathematical models}
\label{tab:mars_ti_opt_cmp2}
\scalebox{1}{
\begin{tabular}{c|l|cccc}
\hline
Function&Measurement&MARS-OPT&MARS-GA 1&MARS-GA 2&MARS-GD\\
\hline
\multirow{4}{*}{$f_{1}$}&\cellcolor{blue!25}Maximum&\cellcolor{blue!25}8.30&\cellcolor{blue!25}6.23&\cellcolor{blue!25}8.27&\cellcolor{blue!25}6.60\\
&Time(seconds)& 0.70&1.50&9.25&2.38\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}-8.20&\cellcolor{green!25}-6.42&\cellcolor{green!25}-6.33&\cellcolor{green!25}-5.06\\
&Time(seconds)& 1.65&1.51 &9.18&0.57\\
\hline
\multirow{4}{*}{$f_{2}$}
&\cellcolor{blue!25}Maximum&\cellcolor{blue!25}1.81&\cellcolor{blue!25}1.02 &\cellcolor{blue!25}1.45&\cellcolor{blue!25}1.24\\
&Time(seconds)& 0.31&1.42&8.91&1.71\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}-2.20&\cellcolor{green!25}-1.38 &\cellcolor{green!25}-2.20&\cellcolor{green!25}-1.38\\
&Time(seconds)&0.32&1.42 &8.88&1.54\\
\hline
\multirow{4}{*}{$f_{3}$}
&\cellcolor{blue!25}Maximum&\cellcolor{blue!25} 5,774.08&\cellcolor{blue!25}5,092.52 &\cellcolor{blue!25}5,410.16&\cellcolor{blue!25} 5661.76\\
&Time(seconds)&0.02 &2.71 &15.74 &0.02\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25} -1126.39&\cellcolor{green!25}2,289.47 &\cellcolor{green!25}903.69&\cellcolor{green!25}-806.86 \\
&Time(seconds)&0.02 &2.67 &15.81&10.35\\
\hline
\multirow{4}{*}{$f_{4}$}
&\cellcolor{blue!25}Maximum&\cellcolor{blue!25} 48,800.26&\cellcolor{blue!25}-13,516.73 &\cellcolor{blue!25}-19,762.80&\cellcolor{blue!25}-1,891,678.92\\
&Time(seconds)&96.32&2.98 &17.67&14.24\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25} -3,952,146.24&\cellcolor{green!25}-2,605,904.95&\cellcolor{green!25} -3,599,093.85&\cellcolor{green!25}-3,119,142.40\\
&Time(seconds)&0.41 &2.95 &17.57 &0.01\\
\hline
\multirow{4}{*}{$f_{5}$}
&\cellcolor{blue!25}Maximum&\cellcolor{blue!25} 97,679.99&\cellcolor{blue!25}78,263.13 &\cellcolor{blue!25}92,458.62&\cellcolor{blue!25}80,461.36\\
&Time(seconds)&0.02 &5.20 &30.01&28.45 \\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25} -15,439.62&\cellcolor{green!25}33,298.80 &\cellcolor{green!25} 5,904.35&\cellcolor{green!25}719.07\\
&Time(seconds)&2.46 &5.21 &30.38& 28.72\\
\hline
\multirow{4}{*}{$f_{6}$}
&\cellcolor{blue!25}Maximum&\cellcolor{blue!25} 111,225.22&\cellcolor{blue!25} 63,506.19&\cellcolor{blue!25}105,645.30&\cellcolor{blue!25}88956.05\\
&Time(seconds)&0.02 &5.34 &30.62& 29.90\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25} -14,215.61&\cellcolor{green!25}17,886.61&\cellcolor{green!25}494.43&\cellcolor{green!25}-5757.07\\
&Time(seconds)&0.04 &5.33 &30.58&12.64\\
\hline
\end{tabular}}
\end{table}
\begin{table}
\centering
\small
\caption{Result comparison of true function value}
\label{tab:mars_true_value_cmp__}
\scalebox{0.6}{
\begin{tabular}{c|l|ccccccc}
\hline
Function&Measurement&OPT function(model)&MARS+GD&GA 1&GA 2&GD&PL-OPT&PL OPT+GD\\
\hline
\multirow{4}{*}{$f_{1}$}&\cellcolor{blue!25}Maximum&\cellcolor{blue!25}8.01(8.32)&\cellcolor{blue!25}8.21 &\cellcolor{blue!25}8.20&\cellcolor{blue!25}8.21&\cellcolor{blue!25}5.76&\cellcolor{blue!25}-4.76(-5.15)&\cellcolor{blue!25}2.96\\
&Time(seconds)&0.25 &0.24&0.78&7.58&0.01&0.05&0.00\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}-4.56(-8.16)&\cellcolor{green!25}-6.20&\cellcolor{green!25}-3.75&\cellcolor{green!25}-5.76&\cellcolor{green!25}-3.30&\cellcolor{green!25}-4.86(-5.58)&\cellcolor{green!25}-6.20\\
&Time(seconds)&10.57&0.73 &0.77 &7.54&0.01&0.05&0.00\\
\hline
\multirow{4}{*}{$f_{2}$}&\cellcolor{blue!25}Maximum&\cellcolor{blue!25} 0.86(1.81)&\cellcolor{blue!25}1.00&\cellcolor{blue!25}1.00&\cellcolor{blue!25}1.00&\cellcolor{blue!25}1.00&\cellcolor{blue!25}0.31(0.56)&\cellcolor{blue!25}1.00\\
&Time(seconds)&0.10&0.14 &0.59&5.25&0.10&0.24&0.00\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}-0.61(-2.20)&\cellcolor{green!25}-1.00&\cellcolor{green!25}-0.87&\cellcolor{green!25}-1.00&\cellcolor{green!25}-0.96&\cellcolor{green!25}-0.46(-0.53)&\cellcolor{green!25}-1.00\\
&Time(seconds)&2.65&0.22 & 0.54&5.27&0.10&0.13&0.00\\
\hline
\multirow{4}{*}{$f_{3}$}&\cellcolor{blue!25}Maximum&\cellcolor{blue!25} 6,029.13(5,774.08)&\cellcolor{blue!25}6132.00&\cellcolor{blue!25}5,488.40&\cellcolor{blue!25}5,647.09&\cellcolor{blue!25}5893.00&\cellcolor{blue!25}4821.00(4667.86)&\cellcolor{blue!25}6132.00\\
&Time(seconds)&0.01 &0.01&1.90&18.81&0.01&0.07&0.00\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}-826.08(-1,126.39)&\cellcolor{green!25}-923.86&\cellcolor{green!25}2,755.59&\cellcolor{green!25}742.00&\cellcolor{green!25}-525.09&\cellcolor{green!25}288.27(118.11)&\cellcolor{green!25}-123.44\\
&Time(seconds)& 0.01&0.01& 1.85&18.81&0.47&0.05&0.00\\
\hline
\multirow{4}{*}{$f_{4}$}&\cellcolor{blue!25}Maximum&\cellcolor{blue!25}-1,991.28(48,800.26)&\cellcolor{blue!25}-0.01 &\cellcolor{blue!25} -1,643,753.96&\cellcolor{blue!25}-131,067.27&\cellcolor{blue!25}-0.01&\cellcolor{blue!25}-505,071.14(-398,359.99)&\cellcolor{blue!25}-0.01\\
&Time(seconds)&107.47&112.74 &2.12&21.98&2.47&0.07&0.00\\
\cline{2-4}
&\cellcolor{green!25}Minimum&\cellcolor{green!25}-4,581,942.36(-3,952,146.24)&\cellcolor{green!25}-4,616,810.03&\cellcolor{green!25}-3,651,963.86&\cellcolor{green!25}-3,834,009.84&\cellcolor{green!25}-3,161,354.60&\cellcolor{green!25}-4,433,291.39(-3,945,225.00)&\cellcolor{green!25}-4,616,810.03\\
&Time(seconds)&0.40&0.42 &2.04 &21.83&0.01&0.10&0.00\\
\hline
\end{tabular}}
\end{table}
\begin{figure*}[]
\centering
\hspace{0em}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f1_max_bp.pdf}
\caption[]%
{{\small $f_1$}}
\label{fig:f1_max}
\end{subfigure}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f2_max_bp.pdf}
\caption[]%
{{\small $f_2$}}
\label{fig:f2_max}
\end{subfigure}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f3_max_bp.pdf}
\caption[]%
{{\small $f_3$}}
\label{fig:f3_max}
\end{subfigure}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f4_max_bp.pdf}
\caption[]%
{{\small $f_4$}}
\label{fig:f4_max}
\end{subfigure}
\caption{\small maximum values boxplot}
\label{fig:max_bp}
\end{figure*}
\begin{figure*}[]
\centering
\hspace{0em}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f1_min_bp.pdf}
\caption[]%
{{\small $f_1$}}
\label{fig:f1_min}
\end{subfigure}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f2_min_bp.pdf}
\caption[]%
{{\small $f_2$}}
\label{fig:f2_min}
\end{subfigure}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f3_min_bp.pdf}
\caption[]%
{{\small $f_3$}}
\label{fig:f3_min}
\end{subfigure}
\begin{subfigure}[b]{0.23\textwidth}
\centering
\includegraphics[height=1.4in]{figures/p2/bp/f4_min_bp.pdf}
\caption[]%
{{\small $f_4$}}
\label{fig:f4_min}
\end{subfigure}
\caption{\small minimum values boxplot}
\label{fig:min_bp}
\end{figure*}
\subsection{$f_1$, $f_2$, $f_3$ and $f_4$ functions}
\begin{flalign}
f_1({x_1,x_2})=& 3(1-x_1)^2\exp(-x_1^2-(x_2+1)^2) -10(\frac{x_1}{5}-x_1^3-x_2^5)\exp(-x_1^2-x_2^2)&&\\\nonumber
&-\frac{1}{3}\exp(-(x_1+1)^2-x_2^2)+2x_1, &&\\\nonumber
&-2\leqslant x_1 \leqslant 2,-2\leqslant x_2 \leqslant 2&&\nonumber
\end{flalign}
\begin{flalign}
f_2(x_1,x_2)=&\sin \left(\frac{\pi x_1}{12}\right) \cos \left(\frac{\pi x_2}{16}\right)&&\\&-20\leqslant x_1\leqslant20,-20\leqslant x_2\leqslant 20&&\nonumber
\end{flalign}
\begin{flalign}
f_3(\mathbf{x})=&x_1^2+x_2^2+x_1x_2-14x_1-16x_2+(x_3-10)^2-4(x_4-5)^2+(x_5-3)^2&&\\\nonumber
&+2(x_6-1)^2+5x_7^2+7(x_8-11)^2+2(x_9-10)^2+2(x_{10}-7)^2+45&&\\
&-10\leqslant x_i \leqslant 10&&\nonumber
\end{flalign}
\begin{flalign}
f_4(\mathbf{x})=&\sum\limits_{j=1}^{10}\exp(x_j)\left(c_j+x_j-\ln\sum\limits_{k=1}^{10}\exp(x_k)\right)&&\\\nonumber
&\mathbf{c}=[-0.6089,-17.164,-34.054,-5.914,-24.721,-14.986,-24.100,-10.708,&&\\\nonumber&-26.662,-22.179]&&\\
&-10\leqslant x_i \leqslant 10&&\nonumber
\end{flalign}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[height=1.7in]{figures/p2/f1.pdf}
\caption[]%
{{\small $f_1(x_1,x_2)$}}
\label{fig:f1_s}
\end{subfigure}
\hspace{0em}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[height=1.7in]{figures/p2/f2.pdf}
\caption[]%
{{\small $f_2(x_1,x_2)$}}
\label{fig:f2_surface}
\end{subfigure}
\vspace{0\baselineskip}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[height=1.7in]{figures/p2/f3.pdf}
\caption[]%
{{\small $f_3(x_1,x_2) \mbox{ other } x_i=4.0$}}
\label{fig:f3_s}
\end{subfigure}
\hspace{0em}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[height=1.7in]{figures/p2/f4.pdf}
\caption[]%
{{\small $f_4(x_1,x_2) \mbox{ other } x_i=3.0$}}
\label{fig:f4_s}
\end{subfigure}
\vspace{0\baselineskip}
\caption{\small Surfaces of dataset functions}
\label{fig:f_surfaces}
\end{figure*}
\begin{figure*}[]
\centering
\hspace{0em}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=2.2in]{figures/p2/mars_surface/mars_p8_with_extremes_2.pdf}
\caption[]%
{{\small $f_1$}}
\label{fig:f1_surface_ex}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=2.2in]{figures/p2/mars_surface/mars_p4_with_extremes_2.pdf}
\caption[]%
{{\small $f_2$}}
\label{fig:f2_surface_ex}
\end{subfigure}
\caption{\small One run result comparison of TITL-MARS-OPT and TITL-MARS-GA on $f_1$ and $f_2$ TITL-MARS models}
\label{fig:other_surfaces}
\end{figure*}
\section{Conclusion}\label{conclusion}
In this paper, a new method (TITL-MARS-OPT) is proposed to globally optimize analytically on the two-way interaction truncated linear MARS (TITL-MARS) by using mixed integer quadratic programming. We verified the presented TITL-MARS-OPT method on the wind farm power distribution TITL-MARS models and six other mathematical TITL-MARS models. The application on wind farm power distribution models gives the best location and worst location information on the wind farm. The testing TITL-MARS models are from 2-dimensions to up to 21-dimensions, and it shows the TITL-MARS-OPT method is robust in dealing with TITL-MARS models with varied dimensions. We also compared the TITL-MARS-OPT method with TITL-MARS-GA in TITL-MARS model optimization, and it shows that the new method can achieve better accuracy and time efficiency. TITL-MARS-OPT can achieve as high as 316\% and on average 46\% better solution quality and is on average 175\% faster than TITL-MARS-GA. In addition, the Python code and the testing models of this paper are made open source, and it will contribute to the study of TITL-MARS models and optimization.
\bibliographystyle{unsrtnat}
|
1,116,691,497,783 | arxiv | \section{Preliminaries}
In this section we give a brief overview of some definitions and
results from \cite{EKR,DKR3}.
We begin with the basic notions of the theory of free partially
commutative groups.
Let $\Gamma $ be a finite, undirected, simple graph. Let
$X=V(\Gamma )=\{x_1,\dots, x_n\}$ be the set of vertices of $\Gamma $ and let
$F(X)$ be the free group on $X$. Let
\[
R=\{[x_i,x_j]\in F(X)\mid x_i,x_j\in X \textrm{ and there is an edge of }
\Gamma \textrm{ joining }
x_i \textrm{ to } x_j \}.
\]
We define the {\em partially commutative group} with ({\em commutation})
{\em graph}
$\Gamma $ to be the group $G(\Gamma )$ with presentation $ \left< X\mid
R\right>$. When the underlying graph is clear from the context we
write simply $G$.
Denote by $\lg(g)$ the minimum of the lengths words that represents the
element $g$. If $w$ is a word representing $g$ and $w$ has length
$\lg(g)$ we call $w$ a {\em minimal form} for $g$.
When the meaning is clear we shall say that $w$ is a
minimal element of $G$ when we mean that $w$ is a minimal form of an
element of $G$.
We say that $w \in G$
is {\em cyclically minimal} if and only if
$$
\lg(g^{-1}wg) \ge \lg(w)
$$
for every $g \in G$. We write $u\circ w$ to express the fact that
$\lg(uw)=\lg(u)+\lg(w)$, where $u,w\in G$.
We will need the notions of a divisor and the greatest
divisor of a word $w$ with respect to a subset $Y\subseteq X$, defined
in \cite{EKR}.
Let $u$ and $w$ be elements of $G$. We say that $u$ is a {\em left }
({\em right}) {\em divisor} of $w$
if there exists $v\in G$ such that $w= u
\circ v$ ($w= v\circ u$).
We order the set of all left (right) divisors of a word $w$ as
follows. We say that $u_2$ is greater than $u_1$ if and only if
$u_1$ left (right) divides $u_2$.
It is shown in \cite{EKR} that, for any $w\in G$ and $Y\subseteq
X$, there exists a unique maximal left divisor of $w$
which belongs to the subgroup $G(Y) <G$ which is called
the {\em greatest left divisor} $\operatorname{gd}^{l}_Y(w)$ of $w$ in $Y$.
The {\em greatest right divisor} of $w$ in $Y$ is defined analogously.
We omit the indices when no ambiguity occurs.
The {\em non-commutation} graph of
the partially commutative group $G(\Gamma )$ is the graph $\Delta$,
dual to $\Gamma$, with vertex set $V(\Delta)=X$ and
an edge connecting $x_i$ and $x_j$ if and only if $\left[x_i, x_j
\right] \ne 1$. The graph $\Delta $ is a union of its connected
components $\Delta _1, \ldots , \Delta _k$ and words that depend on
letters from distinct components commute. For any graph $\Gamma $,
if $S$ is a subset
of $V(\Gamma )$ we shall write $\Gamma (S)$ for the full subgraph of $\Gamma $ with
vertices $S$. Now, if
the vertex set of
$\Delta _k$ is $I_k$ and $\Gamma _k=\Gamma (I_k)$
then
$G=
G(\Gamma _1) \times \cdots \times G(\Gamma _k)$.
For $g \in G$ let $\mathop{{\ensuremath{\alpha}}}(g)$ be the set of elements $x$ of $X$ such
that $x^{\pm 1}$ occurs in a minimal word $w$ representing $g$.
It is shown in \cite{EKR} that $\alpha (g)$ is well-defined.
Now suppose that the full subgraph $\Delta (\mathop{{\ensuremath{\alpha}}}(w))$
of $\Delta$ with vertices $\alpha (w)$ has connected components
$\Delta _1,\ldots, \Delta _l$ and let the vertex set of $\Delta _j$ be $I_j$.
Then, since $[I_j,I_k]=1$,
we can split $w$ into the product of commuting words,
$w=w_1\circ \cdots \circ w_l$, where $w_j\in G(\Gamma (I_j))$, so $[w_j,w_k]=1$
for all $j,k$.
If $w$ is cyclically minimal then
we call this expression for $w$ a
{\em block decomposition} of $w$ and
say $w_j$ a {\em block} of $w$, for $j=1,\ldots ,l$. Thus $w$ itself is
a block if and only if $\Delta (\alpha (w))$ is connected. In general let
$v$ be an element of $G$ which is not necessarily cyclically minimal. We may write
$v=u^{-1}\circ w \circ u$, where $w$ is cyclically minimal and then
$w$ has a block decomposition $w=w_1\cdots w_l$, say. Then
$w_j^u=u^{-1}\circ w_j\circ u$ and we call the expression $v=w_1^u \cdots w_l^u$
the {\em block decomposition}
of $v$ and say that $w_j^u$ is a {\em block}
of $v$, for $j=1,\ldots , l$. Note that this definition is slightly different
from that given in \cite{EKR}.
Let $Y$ and $Z$ be subsets of $X$. As in \cite{DKR3}
we define the {\em orthogonal complement}
of $Y$ {\em in} $Z$ to be
\[\cal{o}^Z(Y)=\{u\in Z|d(u,y)\le 1, \textrm{ for all } y\in Y\}.\]
By convention we set $\cal{o}^Z(\emptyset )=Z$.
If $Z=X$ we call $\cal{o}^X(Y)$ the orthogonal complement of $Y$, and if
no ambiguity arises then we write $Y^\perp$ instead
of $\cal{o}^X(Y)$ and $x^{\perp}$ for $\{x\}^\perp$.
Let
$\operatorname{CS}(\Gamma)$ be the set of all subsets $Z$ of $X$ of the form
$Y^\perp$ for some $Y\subseteq X$. The set $\operatorname{CS}(\Gamma )$ is shown in \cite{DKR3}
to
be a lattice, the {\em lattice of
closed sets} of $\Gamma$.
The {\em centraliser} of a subset $S$ of $G$ is
\[C(S)=C_G(S)=\{g\in G: gs=sg,
\textrm{ for all } s\in S\}.\]
The set ${\textswab C}(G)$ of centralisers of a group is a lattice.
An element $g \in G$ is called a {\em root element}
if $g$ is not a proper
power of any element of $G$. If $h=g^n$, where $g$ is a root element
and $n\ge 1$, then $g$ is said to be a {\em root} of $h$.
As shown in \cite{DK} every element of the partially commutative group
$G$ has a unique root, which
we denote $r(g)$.
If $w\in G$ define $A(w)=\langle Y\rangle=G(Y)$, where
$Y=\mathop{{\ensuremath{\alpha}}}(w)^\perp \setminus \mathop{{\ensuremath{\alpha}}}(w)$.
Let $w$ be a cyclically
minimal element of $G$ with block decomposition $w=w_1\cdots w_k$
and let $v_i=r(w_i)$. Then, from \cite[Theorem 3.10]{DK},
\begin{equation}\label{eq:centraliser}
C(w)=\langle v_1\rangle \times \cdots \times \langle v_k
\rangle\times A(w).
\end{equation}
We shall use \cite[Corollary 2.4]{DKR2} several times in what follows,
so for ease of reference we state it here: first
recalling the necessary notation.
It follows from
\cite[Lemma 2.2]{DKR2} that if $g$ is a cyclically minimal element of $G$
and $g=u\circ v$ then $vu$ is cyclically minimal.
For a cyclically minimal element $g\in G$ we define
$\tilde{g}=\{h\in G|h=vu, \textrm{
for some } u,v \textrm{ such that } g=u\circ v\}$. (We allow $u=1$, $v=g$
so that $g\in \tilde{g}$.)
\begin{lem}{\bf \cite[Corollary 2.4]{DKR2}}\label{cor:conj}
Let $w,g$ be (minimal forms of) elements of $G$ and
$w=u^{-1}\circ v\circ u$, where $v$ is cyclically minimal. Then
there exist minimal forms $a$, $b$, $c$, $d_1$, $d_2$ and
$e$ such that $g=a\circ b\circ c\circ d_2$, $u=d_1\circ a^{-1}$, $d=d_1\circ d_2$,
$w^g=d^{-1}\circ e\circ d$, $\tilde{e}=\tilde{v}$, $e=v^b$, $\alpha (b)\subseteq \alpha (v)$ and
$[\alpha (b\circ c),\alpha (d_1)]=[\alpha (c),\alpha (v)]=1$.
\end{lem}
Figure \ref{fig:vk} expresses the conclusion
of Lemma \ref{cor:conj} as a Van Kampen diagram. In this diagram
we have assumed that $v=b\circ f$ and so $e=f\circ b$.
The regions labelled $B$ are tessellated using relators corresponding
to the relation $[\alpha (b\circ c),\alpha (d_1)]=1$ and the region labelled $A$
with relators corresponding to $[\alpha (c),\alpha (v)]=1$.
Reading
anticlockwise from the vertex labelled $0$ the boundary
label of the exterior region is $g^{-1}wg$ and the label of the interior
region (not labelled $A$ or $B$) is $e^d$.
\begin{figure}\label{fig:vk}
\begin{center}
\psfrag{a}{$a$}
\psfrag{b}{$b$}
\psfrag{c}{$c$}
\psfrag{d1}{$d_1$}
\psfrag{d2}{$d_2$}
\psfrag{f}{$f$}
\psfrag{0}{$0$}
\psfrag{A}{$A$}
\psfrag{B}{$B$}
\includegraphics[scale=0.4]{vk.eps}
\end{center}
\caption{A Van Kampen diagram for Lemma \ref{cor:conj}}
\end{figure}
\section{Parabolic Subgroups}
\subsection{Parabolic and Block-Homogeneous Subgroups}
As usual let $\Gamma $ be a graph with vertices $X$ and $G=G(\Gamma )$. If $Y$
is a subset of $X$ denote by $\Gamma (Y)$ the full subgraph of $\Gamma $ with
vertices $Y$.
Then $G(\Gamma (Y))$ is the
free partially commutative group with graph $\Gamma (Y)$. In \cite{EKR}
it is shown that $G(\Gamma (Y))$ is the subgroup $\langle Y \rangle$ of $G(\Gamma )$
generated by $Y$. We call $G(\Gamma (Y))$ a {\em canonical parabolic}
subgroup of $G(\Gamma )$ and, when no ambiguity arises, denote it $G(Y)$.
The elements of $Y$ are termed the {\em canonical} generators of
$G(Y)$.
\begin{defn}
A subgroup $P$ of $G$ is called {\em parabolic} if it is conjugate
to a canonical parabolic subgroup $G(Y)$ for some $Y\subseteq X$.
The {\em rank} of $P$ is the cardinality $|Y|$ and $Y$ is called a
set of {\em canonical generators} for $P$.
\end{defn}
To see that the definition of rank of a parabolic subgroup is well
defined note that if $Y,Z\subseteq X$ and $G(Y)=G(Z)^g$, for some
$g\in G$, then we have $y=g^{-1}w_yg$, for some $w_y\in G(Z)$, for all $y\in Y$.
It follows, from \cite[Lemma 2.5]{EKR}, by counting the exponent
sums of letters
in a geodesic word representing $g^{-1}wg$,
that $y\in \alpha (w_y)$, so $y\in Z$. Hence
$Y\subseteq Z$ and similarly $Z\subseteq Y$ so $Y=Z$.
\begin{defn}
A subgroup $H$ is called {\em block-homogeneous} if, for all $h\in H$,
if $h$ has block decomposition $h =w_1w_2 \ldots w_k$ then
$w_i \in H$, for $i=1,\ldots, k$.
\end{defn}
\begin{lem} \label{lem:4.1}
An intersection of block-homogeneous subgroups is again a
block-homogeneous subgroup.
If $H$ is block-homogeneous and $g\in G$ then $H^g$ is block-homogeneous.
In particular
parabolic subgroups are block-homogeneous.
\end{lem}
\begin{proof}
The first statement follows
directly from the definition.
Let $H$ be block-homogeneous and $g\in G$ and let $w^g\in H^g$, where
$w\in H$. Write $w=u^{-1}\circ v\circ u$, where $v$ is cyclically reduced and
has block-decomposition $v=v_1\cdots v_k$. Then the blocks of $w$ are
$v_j^u$, so $v_j^u\in H$, for $j=1,\ldots ,k$. From Lemma \ref{cor:conj} there
exist $a, b, c, d_1, d_2, e$ such that
$g=a\circ b\circ c\circ d_2$, $u=d_1\circ a^{-1}$, $d=d_1\circ d_2$,
$w^g=d^{-1}\circ e\circ d$, $\tilde{e}=\tilde{v}$, $e=v^b$, $\alpha (b)\subseteq \alpha (v)$ and
$[\alpha (b\circ c),\alpha (d_1)]=[\alpha (c),\alpha (v)]=1$.
As $\tilde{e}=\tilde{v}$ it follows that
$\Delta (\alpha (e))=\Delta (\alpha (v))$ so $e$ has block-decomposition $e=e_1\cdots e_k$,
where $\tilde{e_j}=\tilde{v_j}$.
Therefore $w^g$ has block-decomposition $w^g=e^d=e_1^d\cdots e_k^d$.
Moreover
$e=v^b$ so $e_j=v_j^b$. Thus
\[e_j^d=e_j^{cd}=v_j^{bcd}=v_j^{d_1bcd_2}=v_j^{ug}\in H^g,
\]
which implies that $H^g$ is block-homogeneous.
It follows from \cite[Lemma 2.5]{EKR} that any canonical parabolic subgroup
is block-homogeneous and this gives the final statement.
\end{proof}
\subsection{Intersections of parabolic subgroups}
In this section we show that an intersection of parabolic subgroups is
again a parabolic subgroup. To begin with we establish
some preliminary results.
\begin{lem}\label{lem:euclid}
Let $Y,Z\subseteq X$, let $w\in G(Y)$ and let $g\in G(X)$ be such
that
\begin{equation*
\gcd{}^l_{Y}(g)=\gcd{}^r_{Z}(g)=1.
\end{equation*}
\begin{enumerate}
\item\label{it:euclid1} If $w^g\in G(Z)$ then
$g\in A(w)$ and $w\in G(Y)\cap G(Z)=G(Y\cap Z)$.
\item\label{it:euclid2}
If $Y=Z$ and $g\in C(w)$ then $g\in A(w)$.
\end{enumerate}
\end{lem}
\begin{proof}
For \ref{it:euclid1},
in the notation of Lemma \ref{cor:conj} we have $w=u^{-1}\circ v\circ u$,
$w^g=d_2^{-1}\circ d_1^{-1}\circ e\circ d_1\circ d_2$ and
$g=a\circ b\circ c\circ d_2$. Applying the conditions of this Lemma we obtain
$a=b=d_2=1$, $u=d_1$ and $e=v$ so $w^g=w$ and $g=c$. Moreover, from Lemma
\ref{cor:conj} again we obtain $[\alpha (g),\alpha (w)]=1$. If $x\in \alpha (w)\cap \alpha (g)$ this
means that $g=x\circ g^\prime$, with $x\in Y$, contradicting the hypothesis on
$g$. Hence $ \alpha (w)\cap \alpha (g)=\emptyset$ and $g\in A(w)$.
Statement \ref{it:euclid2}
follows from \ref{it:euclid1}.
\end{proof}
\begin{cor}\label{cor:parin}
Let $Y,Z\subseteq X$ and $g\in G$. If $G(Y)^g\subseteq G(Z)$ and
$\gcd^l_{Y^\perp}(g)=1$ then $Y\subseteq Z$ and $\alpha (g)\subseteq Z$.
\end{cor}
\begin{proof}
Assume first that $\gcd^l_Y(g)=\gcd_Z^r(g)=1$. Let $y\in Y$ and
$w=y$ in Lemma \ref{lem:euclid};
so $y^g\in G(Z)$ implies that $g\in A(y)$ and $y\in Z$. This holds for
all $y\in Y$ so we have $Y\subseteq Z$ and $g\in A(Y)$. Hence, in this case, $g=1$.
Now suppose that $g=g_1\circ d$, where $g_1=\gcd_Y^l(g)$. Then $\gcd_{Y^\perp}^l(g)=1$
implies that $\gcd_{Y^\perp}^l(d)=1$. Now write $d=e\circ g_2$, where
$g_2=\gcd_Z^r(d)$. Then $G(Y)^g=G(Y)^d$ and $G(Y)^d=G(Z)$ implies $G(Y)^e=G(Z)$.
As $\gcd^l_{Y^\perp}(d)=1$ the same is true of $e$ and from the above we conclude that
$e=1$ and that $Y\subseteq Z$. Now $g=g_1\circ g_2$, where $\alpha (g_1)\subseteq Y
\subseteq Z$ and $\alpha (g_2)\subseteq Z$. Thus $\alpha (g)\subseteq Z$, as required.
\end{proof}
\begin{prop} \label{prop:intpar}
Let $P_1$ and $P_2$ be parabolic subgroups. Then $P=P_1\cap P_2$ is
a parabolic subgroup. If $P_1\nsubseteq P_2$ then the rank of $P$ is
strictly smaller than the rank of $P_1$.
\end{prop}
This lemma follows easily from the next more technical result.
\begin{lem}\label{lem:formpar}
Let $Y,Z\subset X$ and $g\in G$. Then
$$
G(Y)\cap G(Z)^g=G(Y\cap Z\cap T)^{g_2},
$$
where $g=g_1\circ d\circ g_2$, $\gcd_Z^l(d)=\gcd_Y^r(d)=1$, $g_1\in G(Z)$,
$g_2\in G(Y)$
and
$T=\alpha(d)^\perp$.
\end{lem}
\begin{proof}[Derivation of Proposition \ref{prop:intpar}
from Lemma \ref{lem:formpar}]
Let $P_1=G(Y)^a$ and $P_2=G(Z)^b$, for some $a,b\in G$. Then
$P=\left(G(Y)\cap G(Z)^{ba^{-1}}\right)^a$, which is parabolic since
Lemma \ref{lem:formpar} implies that
$G(Y)\cap G(Z)^{ba^{-1}}$ is parabolic.
Assume that the rank of $P$ is
greater than or equal to the rank of $P_1$.
Let $g=ba^{-1}$. The rank of $P$ is equal to the rank of
$G(Y)\cap G(Z)^g$ and, in
the notation of Lemma \ref{lem:formpar},
$G(Y)\cap G(Z)^g=G(Y\cap Z\cap T)^{g_2}$, where
$g=g_1\circ d\circ g_2$, with $T=\alpha (d)^\perp$, $g_2\in G(Y)$ and $g_1\in G(Z)$. Therefore
$Y\subseteq Y\cap Z\cap T$ which implies $Y\subseteq Z\cap T$.
Thus $G(Y)\subseteq G(Z\cap T)=G(Z\cap T)^d$ so
$G(Y)=G(Y)^{g_2}\subseteq G(Z\cap T)^{dg_2}\subseteq G(Z)^{dg_2}=G(Z)^g$.
Hence $P_1\subseteq P_2$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:formpar}]
Let $g_1=\operatorname{gd}^l_Z(g)$ and write $g=g_1\circ g^\prime$. Let
$g_2=\operatorname{gd}^r_Y(g^\prime)$ and write $g^\prime =d\circ g_2$. Then
$g_1, g_2$ and $d$ satisfy the conditions of the lemma. Set
$T=\alpha (d)^\perp$.
As $G(Y)\cap G(Z)^g=G(Y)\cap G(Z)^{dg_2}=G(Y)^{g_2}\cap G(Z)^{dg_2}=
\left(G(Y)\cap G(Z)^d\right)^{g_2}$ it suffices to show that
$G(Y)\cap G(Z)^d=G(Y\cap Z\cap T)$.
If $d=1$ then $T=X$ and $G(Y)\cap G(Z)=G(Y\cap Z)$, so the result holds.
Assume then that $d\neq 1$.
Let $p=w^d\in G(Y)\cap G(Z)^d$, with $w\in G(Z)$. Applying
Lemma \ref{lem:euclid} to $w^d\in G(Y)$ we have
$d\in A(w)$ and $w\in G(Z)\cap
G(Y)=G(Y\cap Z)$. Thus $w\in \alpha (d)^\perp=T$ and so
$w\in G(Y\cap Z\cap T)$.
This shows that $G(Y)\cap G(Z)^d\subseteq G(Y\cap Z\cap T)$ and as
the reverse inclusion follows easily the proof is complete.
\end{proof}
\begin{prop}
The intersection of parabolic subgroups is a parabolic subgroup and
can be obtained as an intersection of a finite number of subgroups
from the initial set.
\end{prop}
\begin{proof}
In the case of two parabolic subgroups the result follows from
Proposition \ref{prop:intpar}. Consequently, the statement also holds for a
finite family of parabolic subgroups. For the general case we
use Proposition \ref{prop:intpar} again, noting that a proper intersection
of two parabolic subgroups is a parabolic subgroup of lower rank, and
the result follows.
\end{proof}
As a consequence of this Proposition we obtain: given two parabolic
subgroups $P$ and $Q$ the intersection $R$ of all parabolic
subgroups containing $P$ and $Q$ is the unique minimal parabolic
subgroup containing both $P$ and $Q$. Define $P\vee Q=R$ and
$P\wedge Q=P\cap Q$.
\begin{cor}
The parabolic subgroups of $G$ with the operations $\vee$ and $\wedge$
above form a lattice.
\end{cor}
\subsection{The Lattice of Parabolic Centralisers}
Let $Z \subseteq X$. Then the subgroup $C_G(Z)^g$ is called a
parabolic centraliser. As shown in \cite[Lemma 2.3]{DKR3} every
parabolic centraliser is a parabolic subgroup: in fact
$C_G(Z)^g=G(Z^\perp)^g$. The converse also holds as the following proposition shows.
\begin{prop}
A parabolic subgroup $G(Y)^g$, $Y\subseteq X$ is a centraliser if
and only if there exists $Z\subseteq X$ so that $Z^\perp=Y$. In this
case $G(Y)^g=C_G(Z^g)$.
\end{prop}
\begin{proof}
It suffices to prove the proposition for $g=1$ only. Suppose that
there exists such a $Z$. It is then clear that $G(Y)\subseteq
C_G(Z)$. If $w\in G$, $w$ is a reduced word and
$\alpha(w)\nsubseteq Y$ then there exists $x\in \alpha (w)$ and
$z\in Z$ so that $[x,z]\ne 1$ and consequently, by \cite[Lemma 2.4]{EKR},
$[w,z]\ne 1$. Assume further that $G(Y)$ is a
centraliser of a set of elements $w_1, \ldots, w_k$ written in a
reduced form. Since for any $y\in Y$ holds $[y,w_i] = 1$ then again,
by \cite[Lemma 2.4]{EKR}, $[y,\alpha(w_i)]=1$. Denote
$Z=\bigcup\limits_{i=1}^k \alpha(w_i)$. We have $[y,z]=1$ for all
$z\in Z$ and consequently $Y\subseteq Z^\perp$.
Conversely if
$x\in Z^\perp$
then
$x\in C_G(w_1, \ldots, w_k)$
so $x\in Y$.
\end{proof}
We now introduce the structure of a lattice on the set of all
parabolic centralisers.
As we have shown above the intersection of two parabolic subgroups
is a parabolic subgroup. So, we set $P_1 \wedge P_2=P_1\cap P_2$.
The most obvious way to define $P_1\vee P_2$ would be to set $P_1\vee P_2=
\langle P_1, P_2\rangle$. However, in this case $P_1\vee P_2$ is not
necessarily a centraliser, though it is a parabolic subgroup.
For any $S\subseteq G$ we define the $\overline S=\cap\{P: P$ is a parabolic centraliser
and $S\subseteq P\}$. Then $\ov S$ is the minimal parabolic centraliser containing
$S$; since intersections of centralisers are centralisers and intersections of parabolic
subgroups are parabolic subgroups. We now define
$P_1\vee P_2=
\overline{\langle P_1, P_2\rangle}$.
\begin{comment}
We are now interested \ilya{why are we so interested in this???} in
the case when a subgroup $H$ of $G$ admits the following
presentation $H=A\times B$, where $A$ and $B$ are block-homogeneous.
\begin{lem}
Let $H=A\times B$, where $A$ and $B$ be block-homogeneous subgroups.
Then $\overline{H}=\overline{A} \times \overline{B}$.
\end{lem}
\begin{proof}
Firstly note that conjugating, if necessary, we may assume that
$\overline{A}=G(Y)$ and $\overline{B}=G(Z)^g$ for some $g\in G$. In
order to show that $\overline{H}$ is the direct product of $G(Y)$ and
$G(Z)^g$ we need to prove
\begin{enumerate}
\item that $U=G(Y)\cap G(Z)^g$ is trivial and
\item that $G(Y)$ commutes with $G(Z)^g$.
\end{enumerate}
We first prove the claim that $U=G(Y)\cap G(Z)^g$ is trivial. To prove
this one has to consider the intersection $U$, which is described in
the proof of Proposition \ref{prop:intpar}. Consider first the case when
$g=1$. Suppose that $y\in U=G(Y\cap Z)$, where $y$ is a standard
generator. Then there exists $w\in A$ so that $y\in \alpha(w)$ and
since $A$ is a block-homogeneous subgroup we may assume that $w$ is
a block. Similarly there exists a block element $v$ in $B$ so that
$y\in \alpha(v)$. For
In the above notation we have
\[
C(w)=\langle \sqrt{w} \rangle \times A(\alpha(w))
\]
and $v\in C(w)$. Since $v\in B$, $v\in C(w)$ and so $v\in A(\alpha
(w))$. Consequently $y\in A(\alpha(w))$, a contradiction for
$y\notin A(\alpha(w))$ a $y\in \alpha(w)$.
\ilya{Case $g\ne 1$ to be considered in version 1.01, service pack
2. However, I think it is literally the same (if we have the
description of the intersection of two parabolic subgroups right).}
We next prove that $G(Y)$ commutes with $G(Z)^g$. Take $z\in \alpha
(\overline{A})$, and let $w$ be a block in $A$ so that $z\in
\alpha(w)$. Take a block element $v$ in $B$. As $w\in C(v)$, $w\in
A(\alpha(v))$ and so $[z,\alpha(v)]=1$.
\end{proof}
\begin{cor}
Let $H_1=A_1\times B_1$, $H_2=A_2\times B_2$ and $A_1,A_2,B_1, B_2$
be block-homogeneous. Then
\[
H_1\cap H_2=(H_1 \cap A_2)\times (H_1\cap B_2)\times (H_2\cap
A_1)\times (H_2\cap B_1).
\]
\end{cor}
\end{comment}
\section{Quasiparabolic subgroups}
\subsection{Preliminaries}
As before let $\Gamma$ be a finite graph with vertex set $X$
and $G=G(\Gamma)$ be
the corresponding
partially commutative group.
\begin{defn}
Let
$w$ be a
cyclically minimal root element of $G$ with block decomposition
$w=w_1 \cdots w_k$ and let $Z$ be a subset of $X$ such that
$Z\subseteq \alpha (w)^\perp$. Then the
subgroup $Q=Q(w,Z)=\langle w_1 \rangle \times \dots \times \langle
w_k\rangle \times G(Z)$ is called a
{\em canonical
quasiparabolic subgroup} of $G$.
\end{defn}
Note that we may choose $w=1$ so that canonical parabolic
subgroups are canonical quasiparabolic subgroups.
Given a canonical quasiparabolic subgroup $Q(w,Z)$, with $w$ and $Z$
as above, we may reorder the $w_i$ so that $\lg(w_i)\ge 2$, for
$i=1,\ldots ,s$ and $\lg(w_i)=1$, for $i=s+1,\ldots ,k$. Then
setting $w^\prime=w_1\cdots w_s$ and
$Z^\prime=Z\cup \{w_{s+1},\ldots, w_k\}$ we have
$Z^\prime \subseteq \alpha (w)^\perp$ and $Q(w,Z)=Q(w^\prime,Z^\prime)$.
This prompts the following definition.
\begin{defn}
We say that a canonical quasiparabolic subgroup $Q=\langle w_1
\rangle\times \dots \times \langle w_k\rangle \times G(Z)$ is written
in standard form if $|\alpha(w_i)| \ge 2$, $i=1,\dots, k$, or
$w=1$.
\end{defn}
There are two obvious advantages to the standard form which we record
in the following lemma.
\begin{lem}\label{lem:stdform}
The standard form of a canonical quasiparabolic subgroup $Q$
is unique, up to reordering of blocks of $w$. If
$Q(w,Z)$ is the standard form of $Q$ then $Z\subseteq \alpha (w)^\perp\setminus \alpha (w)$.
\end{lem}
\begin{proof}
That the standard form is unique follows from uniqueness of
roots of elements in partially commutative groups. The second statement
follows directly from the definitions.
\end{proof}
\begin{defn}
A subgroup $H$ of \/ $G$ is called quasiparabolic if it is
conjugate to a canonical quasiparabolic subgroup.
\end{defn}
Let $H=Q^g$ be a quasiparabolic subgroup of $G$, where $Q$ is the
canonical quasiparabolic subgroup of $G$ in
standard form
\[
Q=\langle w_1\rangle \times \dots \times \langle w_k \rangle \times
G(Z).
\]
We call $(|Z|,k)$ the \emph{rank} of $H$. We use the left lexicographical
order on ranks of quasiparabolic subgroups:
if $H$ and $K$ are
quasiparabolic subgroups of ranks $(|Z_H|,k_H)$ and
$(|Z_K|, k_K)$, respectively,
then $\operatorname{rank}(H) < \operatorname{rank}(K)$ if
$(|Z_H|,k_H)$ precedes $(|Z_K|, k_K)$ in left
lexicographical order.
The centraliser $C_G(g)$ of an element $g\in G$ is a
typical example of a quasiparabolic subgroup
\cite{DK}. We shall see below
(Theorem \ref{thm:2bcentr}) that
the centraliser of any set of elements of the group $G$ is
a quasiparabolic subgroup.
\begin{lem}\label{lem:qphom}
A quasiparabolic subgroup is a block-homogeneous subgroup and
consequently any intersection of quasiparabolic subgroups is again
block-homogeneous.
\end{lem}
\begin{proof}
Let $Q(w,Z)$ be a canonical quasiparabolic subgroup. Since
$w$ is a cyclically minimal root element it follows that $Q(w,Z)$ is
block-homogeneous. An application of Lemma \ref{lem:4.1} then implies
$Q(w,Z)^g$ is also block-homogeneous.
\end{proof}
We shall need the following lemma in Section \ref{sec:ht}.
\begin{lem}\label{lem:qpcont}
Let $Q_1=Q(u,Y)$ and $Q_2=Q(v,Z)$ be canonical quasiparabolic subgroups
in standard form
and let $g\in G$. If $Q_2^g\subseteq Q_1$, $g\in G(Z^\perp)$ and
$\operatorname{gd}^r_Y(g)=1$ then $Q_2^g$ is a canonical quasiparabolic subgroup.
\end{lem}
\begin{proof}
Let $u$ and $v$ have block decompositions $u=u_1\cdots u_k$ and
$v=v_1\cdots v_l$, respectively. As $g\in G(Z^\perp)$ we have
\[
Q_2^g= \langle v_1^g\rangle \times \cdots \times \langle v_l^g\rangle\times G(Z).
\]
Therefore,
for $j=1,\ldots ,l$,
either $v_j^g=u_i$ for some $i=1,\dots, k$,
or $v_j^g\in G(Y)$.
If $v_j^g=u_i$ then
$v_j^g$ is a cyclically minimal root element.
If, on the other hand, $v_j^g\in G(Y)$
then, from Lemma \ref{cor:conj}, there exist elements
$b,c,d$ and $e$ such that $g=b\circ c\circ d$, $v_j^g=d^{-1}\circ e\circ d$ and
$e=v_j^b$ is a cyclically minimal root element.
As $v_j^g\in G(Y)$ and $\operatorname{gd}_Y^r(g)=1$ we have
$d=1$ and so $v_j^g=e$ and is a cyclically minimal root element. Therefore $Q_2^g$ is a
canonical quasiparabolic subgroup.
\end{proof}
\subsection{Intersections of Quasiparabolic Subgroups}
The main result of this section is the following
\begin{thm} \label{thm:intqpsub}
An intersection of quasiparabolic subgroups is a quasiparabolic
subgroup.
\end{thm}
We shall make use of the following results.
\begin{lem} \label{lem:4.2}
Let $A=A_1\times \dots \times A_l$ and $B=B_1\times\dots \times
B_k$, $A_i$, $B_j$, $i=1,\dots, l$, $j=1,\dots, k$ be block-homogeneous subgroups of $G$ and
$C=A\cap B$. Then
\[C=\prod\limits_{\begin{tiny}\begin{array}{c}
i=1,\dots, l; \\
j=1,\dots, k
\end{array}\end{tiny}
} (A_i\cap B_j).\]
\end{lem}
\begin{proof}
If $C=1$ then the result is straightforward. Assume then that $C\ne 1$,
$w\in C$ and $w\ne 1$ and let $w=w_1\dots w_t$ be the block decomposition
of $w$. Since
$C$ is a
block-homogeneous subgroup, $w_i\in C$, $i=1,\dots, t$.
As $w_i$ is a block element we have $w_i\in A_r$ and
$w_i\in B_s$ and consequently $w_i$ lies in
$\prod_{i,j}(A_i\cap B_j)$. As it is clear that
$C\ge \prod_{i,j}(A_i\cap B_j)$
this proves the lemma.
\end{proof}
\begin{lem} \label{lem:conj}
Let $Z\subseteq X$, $w\in G(Z)$, $g\in G$. Suppose that
$u=g^{-1}wg$ is cyclically minimal and $\gcd_{\alpha(w)}^l(g)=1$,
then $g$ and $w$ commute.
\end{lem}
\begin{proof}
Let $g=d\circ g_1$, where $d=\gcd_{\alpha(w)^\perp}^l(g)$. If
$g_1=1$ then $g\in C(w)$. Suppose $g_1\ne 1$. Then
$\gcd_{\alpha(w)}^l(g_1)=1$ so we write $g_1=x \circ g_2$, where $x\in
(X\cup X^{-1}) \setminus (\alpha(w) \cup \alpha(w)^\perp)$
and thus $u=g_2^{-1}x^{-1}w x g_2$ is written in geodesic form.
This is a contradiction for $l(w)<l(u)$.
\end{proof}
\begin{lem}\label{lem:2quasi}
Let
\[
Q_1=\langle u_1 \rangle \times \dots \times \langle u_l \rangle
\times G(Y) \textrm{ and } Q_2=\langle v_1 \rangle \times \dots \times \langle v_k
\rangle
\times G(Z)
\]
be canonical quasiparabolic subgroups in standard form and let $g\in G$
such that $\operatorname{gd}^l_Z(g)=1$.
Write $g=d\circ h$, where $h=\operatorname{gd}^r_Y(g)$ and
set $T=\alpha (d)^\perp$.
Then, after reordering the $u_i$ and $v_j$ if necessary, there exist
$m, s, t$ such that
\begin{equation} \label{eq:intqp}
Q_1\cap Q_2^g=\left(\prod_{i=1}^s \langle v_i\rangle\times
\prod_{i=s+1}^t\langle v_i\rangle \times \prod_{j=s+1}^m\langle u_i\rangle
\times G(Y\cap Z\cap T)\right)^g
\end{equation}
and
\begin{enumerate}[(i)]
\item $\langle u_i\rangle =\langle v_i\rangle^g$, for $i=1,\ldots ,s$;
\item\label{item:2quasi2} $\langle v_i\rangle^g\subseteq G(Y)$, for $i=s+1,\ldots ,t$; and
\item $\langle u_i\rangle \subseteq G(Z)$, for $i=s+1,\ldots ,m$.
\end{enumerate}
\end{lem}
\begin{proof}
As $Q_1\cap Q_2^g=(Q_1\cap Q_2^d)^h$ we may assume that $h=1$ and $d=g$,
so $\operatorname{gd}^r_Y(g)=1$. As $Q_1$ and $Q_2^g$ are block-homogeneous
we may apply Lemma \ref{lem:4.2} to compute their intersection.
Therefore we consider the various possible intersections of factors of $Q_1$ and
$Q_2^g$.
\begin{enumerate}[(i)]
\item If $\langle u_i\rangle \cap {\langle v_j\rangle}^g\ne 1$ then, as
$u_i$ and $v_j$ are root elements,
$\langle u_i \rangle =\langle v_j\rangle ^g$.
Suppose that this is the case for $u_1,\dots u_s$ and
$v_1,\dots, v_s$ and that $\langle u_i\rangle \cap \langle v_j\rangle ^g=1$, if $i>s$ or $j>s$.
\item If $\langle v_j\rangle ^g\cap G(Y) \ne 1$ then, since $v_j$ is cyclically minimal, $\langle
v_j\rangle^g \subset G(Y)$. This cannot happen if $j\le s$ so suppose it
is the case for
$v_{s+1}, \dots, v_t$, and that $\langle v_j \rangle^g\cap G(Y)=1$, for $j>t$.
\item If $\langle u_i\rangle \cap {G(Z)}^g \ne 1$ then
$u_i=w^g$, $w\in G(Z)$ and by Lemma \ref{lem:conj}, $w$ and $g$
commute so does $u_i=w=u_i^g$. This cannot happen if $i\le s$ so
suppose that it's the case for
$u_{s+1},\dots,u_{m}$, and not for $i>m$.
\item Finally, using Lemma \ref{lem:formpar} and the assumption that
$\operatorname{gd}^r_Y(g)=\operatorname{gd}^l_Z(g)=1$, we have
$G(Y)\cap {G(Z)}^g={G(Y\cap Z\cap T)}={G(Y\cap Z\cap T)}^g$,
where $T=\alpha(g)^\perp$.
\end{enumerate}
Combining these intersections \eqref{eq:intqp} follows from Lemma \ref{lem:4.2}.
\end{proof}
\begin{cor}\label{cor:qprank}
Let $H_1$ and $H_2$ be quasiparabolic subgroups of $G$ then $H_1\cap H_2$ is
quasiparabolic and {\rm
$\operatorname{rank}(H_1\cap H_2) \le \min\{\operatorname{rank}(H_1), \operatorname{rank}(H_2)\}$}.
\end{cor}
\begin{proof}
Let $H_1=Q_1^f$ and $H_2=Q_2^g$, where $Q_1=Q(u,Y)$ and $Q_2=Q(v,Z)$ are quasiparabolic
subgroups in standard form, as in Lemma \ref{lem:2quasi}.
As in the proof of Proposition \ref{prop:intpar} we may assume that
$f=1$ and $\operatorname{gd}^l_Z(g)=1$ and so Lemma \ref{lem:2quasi} implies
$H_1\cap H_2$ is quasiparabolic. If $\operatorname{rank}(H_1\cap H_2)\ge \operatorname{rank}(H_1)$
then $|Y|\le |Y\cap Z\cap T|$ so $Y\subseteq Z\cap T$. In this case
\eqref{item:2quasi2} of Lemma \ref{lem:2quasi} cannot occur.
Therefore, in the notation of Lemma \ref{lem:2quasi},
$\operatorname{rank}(H_1\cap H_2)=s+m$. If $\operatorname{rank}(H_1\cap H_2)\ge \operatorname{rank}(H_1)$ then
$s+m\ge l$ which implies $m=l-s$ and so $u_i\in G(Z)^g$, for $i=s+1,\ldots ,l$.
As $u_i=v_i^g$, for $i=1,\ldots, s$ it follows that $H_1\subseteq H_2$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:intqpsub}.]
Given Corollary \ref{cor:qprank} the intersection of
an infinite collection of quasiparabolic
subgroups is equal to the intersection of a finite sub-collection.
From Corollary \ref{cor:qprank} again such an intersection is quasiparabolic
and the result follows.
\end{proof}
\subsection{A Criterion for a Subgroup to be a Centraliser} \label{sec:2bcentr}
\begin{thm} \label{thm:2bcentr}
A subgroup $H$ of $G$ is a centraliser if and only if the two
following conditions hold.
\begin{enumerate}
\item \label{cond1} $H$ is conjugate to some canonical
quasiparabolic subgroup $Q$.
\item \label{cond2} If $Q$ is written in standard form
$$
Q=\langle w_1\rangle \times \dots \times \langle w_k \rangle \times
G(Y),
$$
where
$w=w_1\dots w_k$ is the block decomposition of a cyclically
minimal element $w$, $w_i$ is a root element and
$|\alpha(w_i)|\ge 2$, $i=1,\dots, k$, then
\[
Y\in \operatorname{CS}(\Gamma )\textrm{ and }Y\in \operatorname{CS}(\Gamma _w)\textrm{
where }\Gamma _w=\Gamma (\alpha(w)^\perp\setminus\alpha (w)).\]
\end{enumerate}
\end{thm}
\begin{proof}
Let $H=C(u_1,\dots, u_l)$. Then $H=\bigcap\limits_{i=1}^k C(u_i)$ and we
may assume that each $u_i$ is a block root element. Since $C(u_i)$ is a
quasiparabolic subgroup, then by Theorem \ref{thm:intqpsub}, $H$
is also a quasiparabolic subgroup and is conjugate to a canonical
quasiparabolic subgroup $Q=\langle w_1\rangle \times \dots \times \langle w_k \rangle \times
G(Y)$ written in standard form. Thus condition \ref{cond1} is
satisfied.
Then $H=Q^g$ and, after conjugating all the $u_i$'s by
$g^{-1}$ we have a centraliser $H^{g^{-1}}=Q$. Thus we may assume that
$H=Q$.
Let $w=w_1\cdots w_k$, set $Z=\alpha(w)^\perp\setminus\alpha (w)$ and
$T=\bigcup\limits_{i=1}^l \alpha (u_i)$.
As $w$ has block decomposition $w=w_1\cdots w_k$ we have
$C(w)=\langle w_1\rangle \times \dots \times
\langle w_k \rangle \times G(Z)$.
For all $y\in Y$ we have $y\in C(u_i)$ so
and thus $y\in C(\alpha(u_i))$ and
$Y\subseteq T^\perp$.
Conversely if $y\in T^\perp$ then $y\in C(u_i)$
so $y\in Q$ and, by definition of standard form, $y\in Y$. Therefore $Y=T^\perp$.
It follows that $Y\in \operatorname{CS}(\Gamma )$
and since by Lemma \ref{lem:stdform} we have $Y\cap \alpha (w)=\emptyset $ we also
have $Y\subseteq Z$.
It remains to prove that $Y\in \operatorname{CS}(\Gamma _w)=\operatorname{CS}(\Gamma (Z))$.
Set $W=\alpha(w)$. We show that $T\cup Z\subseteq W\cup Z$. Take
$t\in T=\bigcup\limits_{i=1}^l \alpha (u_i)$, $t\notin W$ and
suppose that $t\in \alpha(u_m)$. Since $w \in C(u_i)$, we have $u_m
\in C(w)=\langle w_1\rangle \times \dots \times \langle w_k \rangle
\times G(Z)$. Now $u_m$ is a root block element and
$C(w)$ is a block-homogeneous subgroup so if
$ u_m = w_j^{\pm 1}$ for some $j$ then $t \in W=\alpha(w)$, contrary to
the choice of $t$. Therefore $u_m \in G(Z)$, so $t\in Z$ and $T \cup
Z\subseteq W\cup Z$, as claimed.
Assume now that $Y\notin \operatorname{CS}(\Gamma (Z))$. In this case there exists an element
$z\in Z\setminus Y$ such that $z\in \operatorname{cl}_Z(Y)$. Since $z\notin Y=T^\perp$,
there exists $u_m$ such that $[u_m,z]\ne 1$ and so there exists $t\in \alpha(u_m)$
such
that $[t, z]\ne 1$. As $[z,W]=1$, we have $t\notin W$ and since
$W\cup Z\supseteq T\cup Z$, we get $t\in Z$. This together with
$t\in \alpha(u_m)\subseteq Y^\perp$ implies that $t\in \cal{o}^Z(Y)$. Since
$[z,t]\ne 1$, we obtain $z\notin \operatorname{cl}_Z(Y)$, in contradiction to
the choice of $z$. Hence $\operatorname{cl}_Z(Y)=Y$ and $Y\in \operatorname{CS}(\Gamma (Z))$.
Conversely, let $Q=\langle w_1\rangle \times \dots \times \langle
w_k \rangle \times G(Y)$ be a canonical quasiparabolic subgroup
written in the standard form, $Y\in \operatorname{CS}(X)$ and $Y \in \operatorname{CS}(\Gamma (Z))$,
where $Z=\alpha (w)^\perp\setminus\alpha (w)$.
We shall prove that $Q=C(w, z_1,\dots, z_l)$, where $z_1,\dots, z_l$ are
some elements of $Z$. If $Y=Z$ then $Q=C(w)$. If $Y\subsetneq Z$
then, since $Y=\operatorname{cl}_Z(Y)$, there exist $z_1,u\in Z$ so that $z_1\in
\cal{o}^Z(Y)$ and $[z_1,u]\ne 1$. In which case $C(w,z_1)=\langle
w_1\rangle \times \dots \times \langle w_k \rangle \times G(Y_1)$,
$Y\subseteq Y_1 \subsetneq Z$ (the latter inclusion is strict for
$u\notin Y_1$). If $Y_1=Y$ then $Q=C(w, z_1)$, otherwise iterating
the procedure above, the statement follows.
\end{proof}
A centraliser which is equal to a canonical quasiparabolic subgroup
is called a {\em canonical quasiparabolic centraliser}.
\section{Height of the Centraliser Lattice}\label{sec:ht}
In this section we will give a new shorter proof of the main theorem
of \cite{DKR2}.
\begin{thm}\label{thm:height}
Let $G=G(\Gamma )$ be a free partially commutative group, let ${\textswab C}(G)$ be
its centraliser lattice and let $L=\operatorname{CS}(\Gamma )$ be the lattice of closed
sets of $\Gamma $. Then the height $h({\textswab C}(G))=m$ equals the height $h(L)$
of the lattice of closed sets $L$.
\end{thm}
In order to prove this theorem we introduce some notation for the various parts of canonical
quasiparabolic subgroups.
\begin{defn}
Let $Q=\langle w_1\rangle\times \cdots \times \langle w_k\rangle\times G(Z) $ be
a quasiparabolic
subgroup in standard form. Define the {\em block set} of $Q$ to be $\mathcal{B}(Q)=
\{w_1 \ldots, w_k\}$ and the
{\em parabolic part} of $Q$ to be $\mathcal{P}(Q)=G(Z)$.
Let $Q^\prime $ be a quasiparabolic subgroup with block set
$\langle v_1\rangle \times \cdots \times \langle v_l\rangle$ and parabolic part $G(Y)$.
Define the {\em block difference} of $Q$ and $Q^\prime$ to be
$b(Q,Q^\prime)=|\mathcal{B}(Q)\setminus \mathcal{B}(Q^\prime)|$, that is the number of blocks occurring in
the block set of $Q$ but not $Q^\prime$. Define the {\em parabolic difference}
of $Q$ and $Q^\prime$ to be $p(Q,Q^\prime)=|Z\setminus Y|$.
\end{defn}%
The following lemma is the key to the proof of the theorem above.
\begin{lem}\label{lem:cpad}
Let $C$ and $D$ be canonical quasiparabolic centralisers such that
$C>D$ and $b=b(D,C)>0$.
Then $p(C,D)>0$ and there exists a strictly descending chain
of canonical parabolic centralisers
\begin{equation}\label{eq:keychain}
\mathcal{P}(C)>C_b>\cdots >C_1>\mathcal{P}(D)
\end{equation}
of length $b+1$.
\end{lem}
\begin{proof}
Let
$C$ and $D$ have parabolic parts
$\mathcal{P}(C)=G(Y)$ and $\mathcal{P}(D)=G(Z)$, for closed subsets $Y$ and $Z$ in $\operatorname{CS}(\Gamma )$.
Let the block sets of $C$ and $D$ be
$\mathcal{B}(C)=\{u_1,\ldots ,u_k\}$ and $\mathcal{B}(D)=\{v_1,\ldots , v_l\}$. Fix
$i$ with $1\le i\le l$. As
$D<C$, either $\langle v_i\rangle=\langle u_j\rangle$, for some $j$, or $\langle v_i\rangle\subseteq
G(Y)$. As $b(D,C)>0$ there exists $i$ such that $\langle v_i\rangle\subseteq G(Y)$.
Moreover, for such $i$, we have $\alpha (v_i)\subseteq Y\setminus Z$, so
$p(C,D)>0$.
Assume that, after relabelling if necessary,
$\langle v_i\rangle =\langle u_i\rangle$, for $i=1,\ldots ,s$, and that
$\langle v_{s+1}\rangle,\ldots ,\langle v_{l}\rangle\subseteq G(Y)$, so $b=l-s$. Choose
$t_i\in \alpha (v_{s+i})$ and let $Y_i=\operatorname{cl}(Z\cup\{t_1,\ldots ,t_i\})$,
for $i=1,\ldots ,l-s=b$.
Let $C_i=G(Y_i)=C_G(Y_i^\perp)$, so $C_i$ is a canonical
parabolic centraliser. We claim that the chain \eqref{eq:keychain} is
strictly descending. To begin with, as $t_1\in Y_1\setminus Z$ we have $G(Z)<C_1$.
Now fix $i$ and $n$ such that $1\le i < n \le b$. If $a\in \alpha (v_{s+n})$ then
$a\in Z^\perp$ and $a\in \alpha (v_{s+j})^\perp\subseteq t_j^\perp$, for $1\le j<n$,
by definition of the standard form of quasiparabolic subgroups. Hence
$a\in (Z\cup\{t_1,\ldots ,t_i\})^\perp$. Thus $[a,b]=1$, for all
$b\in Y_i$. This holds for all $a\in \alpha (v_{s+n})$ so
$Y_i\subseteq \alpha (v_{s+n})^\perp$. As $v_{s+n}$ is a block of length
at least $2$ we have $\alpha (v_{s+n})\cap \alpha (v_{s+n})^\perp=\emptyset$, so
$Y_i\cap \alpha (v_{s+n})=\emptyset$. Hence $t_n\notin Y_i$ and it follows
that $C_i<C_{i+1}$, $i=1,\ldots ,b-1$. Now choose $c\in \alpha (v_{s+1})$
such that $[c,t_1]\neq 1$. Then $c\in Y$, as $\alpha (v_{s+1})\subseteq Y$,
however $c\notin Y_b$, since $t_1\in Z^\perp \cap t_1^\perp\cap \cdots
\cap t_b^\perp=Y_b^\perp$ and $Y_b=Y_b^{\perp\perp}$. As $D<C$ we have
$Z\subseteq Y$ so
$C_b=G(Y_b)<G(Y)$.
\end{proof}
We can use this lemma to prove the following about chains of
canonical quasiparabolic subgroups.
\begin{lem}\label{lem:qparac}
Let $C_0>\cdots >C_d$ be a strictly descending chain of
canonical quasiparabolic centralisers such that $C_0$ and
$C_d$ are canonical parabolic centralisers. Then there exists
a strictly descending chain $C_0>P_1>\cdots >P_{d-1}>C_d$, of
canonical parabolic centralisers.
\end{lem}
\begin{proof}
First we divide the given centraliser chain
into types depending on
block differences. Then we replace the chain with a chain of
canonical parabolic centralisers, using Lemma \ref{lem:cpad}.
A simple counting argument shows that the new chain has
length at least as great as the old one.
In detail let $I=\{0,\ldots ,d-1\}$ and
\begin{align*}
I_+ &=\{i\in I: b(C_{i+1},C_i)>0\},\\
I_0 &=\{i\in I: b(C_{i+1},C_i)=0\textrm{ and } p(C_i,C_{i+1})>0\}
\textrm{ and }\\
I_- &= \{i\in I: b(C_{i+1},C_i)= p(C_i,C_{i+1})=0\}.
\end{align*}
Then $I=I_+\sqcup I_0\sqcup I_-$. For $i\in I_+$ let $\Delta _i$ be the strictly
descending chain of canonical parabolic centralisers of length
$b(C_{i+1},C_i)+1$ from $\mathcal{P}(C_i)$ to $\mathcal{P}(C_{i+1})$, constructed in
Lemma \ref{lem:cpad}. For $i\in I_0$ let $\Delta _i$ be the length one chain
$\mathcal{P}(C_i)>\mathcal{P}(C_{i+1})$ and for $i\in I_-$ let $\Delta _i$ be the length
zero chain $\mathcal{P}(C_i)=\mathcal{P}(C_{i+1})$. This associates
a chain $\Delta _i$ of canonical parabolic centralisers to each $i\in I$
and we write $l_i$ for the length of $\Delta _i$.
If $\Delta _i=P_0>\cdots >P_{l_i}$ and
$\Delta _{i+1}=P^\prime_0>\cdots >P^\prime_{l_{i+1}}$ then
by definition $P_{l_i}=P^\prime_0$, for $i=1,\ldots ,d-1$. We may
therefore concatenate $\Delta _i$ and $\Delta _{i+1}$ to give a chain
of canonical parabolic centralisers
\[P_0> \cdots >P_{l_i}=P^\prime_0>\cdots >P^\prime_{l_{i+1}}\]
of length $l_i+l_{i+1}$. Concatenating $\Delta _1, \ldots ,\Delta _{d-1}$ in
this way we obtain a strictly descending chain of canonical parabolic centralisers
of length $l=\sum_{i=0}^{d-1} l_i$.
Moreover
\[l=\sum_{i\in I_+}b(C_{i+1},C_i)+|I_+|+|I_0|,\]
since $l_i=b(C_{i+1},C_i)+1$, for all $i\in I_+$, $l_i=1$, for all $i\in I_0$ and $l_i=0$, for all $l_i\in I_-$. As $|I|=d$ we have now
\[
l-d =\sum_{i\in I_+}b(C_{i+1},C_i)-|I_-|.\]
To complete the argument we shall show that
\[
\sum_{i\in I_+}b(C_{i+1},C_i)=|\cup_{i=0}^d\mathcal{B}(C_i)|\ge |I_-|.
\]
As $\mathcal{B}(C_0)=\emptyset$ we have $b(C_1,C_0)=|\mathcal{B}(C_0)\cup\mathcal{B}(C_1)|$.
Assume inductively that
\[
\sum_{i=0}^k b(C_{i+1},C_i)=|\cup_{i=0}^{k+1}\mathcal{B}(C_i)|,
\]
for
some $k\ge 0$. Then
\[
\sum_{i=0}^{k+1} b(C_{i+1},C_i)=
|\cup_{i=0}^{k+1}\mathcal{B}(C_i)|+|\mathcal{B}(C_{k+2})\setminus \mathcal{B}(C_{k+1})|.
\]
Moreover, if $w\in \mathcal{B}(C_{k+2})\setminus \mathcal{B}(C_{k+1})$ then
$w\in \mathcal{P}(C_j)$, for all $j\le k+1$, so $w\notin \mathcal{B}(C_j)$, for
$j=0,\ldots ,k+1$. Hence
\[
\mathcal{B}(C_{k+2})\setminus \mathcal{B}(C_{k+1})=\mathcal{B}(C_{k+2})\setminus \cup_{i=0}^{k+1}\mathcal{B}(C_i)\]
and it follows that
\[
\sum_{i=0}^{k+1} b(C_{i+1},C_i)=|\cup_{i=0}^{k+2}\mathcal{B}(C_i)|.
\]
As $b(C_{i+1},C_i)=0$ if $i\notin I_+$ it follows that
\[
\sum_{i\in I_+}b(C_{i+1},C_i)=|\cup_{i=0}^d\mathcal{B}(C_i)|,
\]
as required. If $i\in I_-$ then $b(C_{i+1},C_i)=p(C_i,C_{i+1})=0$, so
$b(C_i,C_{i+1})>0$. Therefore there is at least one element $w\in
\mathcal{B}(C_i)\setminus \mathcal{B}(C_{i+1})$. It follows that $w\notin \mathcal{B}(C_j)$, for
all $j\ge i+1$ and so $I_-\le |\cup_{i=0}^d\mathcal{B}(C_i)|$. Therefore
$l-d\ge 0$ and the proof is complete.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:height}]
Let
\begin{equation*}
G=C_0>\cdots > C_d=Z(G)
\end{equation*}
be a maximal descending chain of
centralisers of $G$. By Theorem \ref{thm:2bcentr}, each of the
$C_i$'s is a quasiparabolic subgroup.
If each
$C_i$ is canonical
then, since $G$ and $Z(G)$ are both canonical parabolic centralisers
the result follows from Lemma \ref{lem:qparac}.
Suppose that now $C_1,\dots, C_s$ are canonical quasiparabolic and
$C_{s+1}$ is not: say
$C_{s+1}=Q^{g}$, where $Q$ is a canonical
quasiparabolic subgroup.
Let $C_s=Q(u,Y)$ and $Q=Q(v,Z)$ both in standard form.
\begin{comment}
\begin{align*}
C_s &= \langle u_1\rangle \times \cdots \times \langle u_k\rangle\times G(Y)\textrm{ and }\\
Q&= \langle v_1\rangle \times \cdots \times \langle v_l\rangle\times G(Z)
\end{align*}
in standard form.
Setting $w_j=v_j^g$, we have for $j=1,\ldots ,l$,
\begin{enumerate}
\item either $w_j=u_i$ for some $i=1,\dots, k$,
or $w_j\in G(Y)$, and
\item $G(Z)^g\le G(Y)$.
\end{enumerate}
\end{comment}
Write $g=f\circ h$, where $f=\operatorname{gd}_{Z^\perp}^l(g)$ and let
$f=e\circ d$, where $d=\operatorname{gd}^r_Y(f)$, so $d \in G(Y\cap Z^\perp)$. Then
$G(Z)^h=G(Z)^{dh}= G(Z)^g\subseteq G(Y)$ and $\alpha (h)\subseteq Y$, from Corollary
\ref{cor:parin}. Hence $\alpha (d\circ h)\subseteq Y$ which implies that
$\alpha (d\circ h)\subseteq \mathcal{P}(C_s)\subseteq \cdots
\subseteq \mathcal{P}(C_0)$. It follows that $C_r^{dh}=C_r$, for $r=0,\ldots ,s$.
Therefore conjugating $C_0>C_1>\cdots >C_d$ by
$(dh)^{-1}$ we obtain a chain in which $C_0,\ldots, C_{s}$ are unchanged
and $C_{s+1}=Q^e=\langle v_1\rangle^e\times \langle v_l \rangle^e\times G(Z)$,
with $\operatorname{gd}^l_Z(e)=\operatorname{gd}^r_Y(e)=1$, $e\in G(Z^\perp)$.
\begin{comment}
and either
$v_j^e=u_i$, for some $i$, or $v_j^e\in G(Y)$. If $v_j^e=u_i$ then
$v_j^e$ is cyclically minimal. If, on the other hand, $v_j^e\in G(Y)$
then, from Lemma \ref{cor:conj}, there exist elements
$b,c,d_j$ and $e_j$ such that $e=b\circ c\circ d_j$, $v_j^e=d_j^{-1}\circ e_j\circ d_j$,
$\tilde e_j=\tilde v_j$, $e_j=v_j^b$, $\alpha (b)\subseteq \alpha (v_j)$ and
$[\alpha (c),\alpha (v_j)]=1$. As $v_j^e\in G(Y)$ we have $d_j\in G(Y)$ and so $\operatorname{gd}^r_Y(e)=1$
implies
$d_j=1$ and so $e_j=v_j^e$ which is cyclically minimal.
\end{comment}
As Lemma \ref{lem:qpcont} implies that $Q^e$ is a
canonical quasiparabolic subgroup we now have a chain in which
$C_0,\ldots, C_{s+1}$ are canonical quasiparabolic. Continuing this
way we eventually obtain a chain, of length $d$,
of canonical quasiparabolic centralisers
to which the first part of the proof may be applied.
\end{proof}
|
1,116,691,497,784 | arxiv | \section{Introduction}\label{intro}
Standard candles, in particular supernovae Ia (SNe), are one of the most important and reliable estimators
of distance in cosmology~\cite{Riess:1998cb,Perlmutter:1998np}. As is well known, the evidence for cosmological acceleration rests principally on
their properties and on their calibration. Since the discovery of acceleration, a large effort has been
devoted to testing and improving the calibration of the SNe and to correcting their light curves
in order to achieve data samples as free of systematics as possible~\cite{March:2011rv,Amendola:2012wc}.
Since their light comes from relatively high redshifts, SNe are expected to be lensed to some
extent by intervening matter along the line of sight. The correction induced by this effect is
normally subdominant but will become one of the major sources of uncertainty when richer and deeper
SN catalogs will be collected in the next years. The Large Synaptic Survey Telescope (LSST) project plans for instance to collect up to half a million
SNe in ten years~\cite{Abell:2009aa}, a huge increase from the roughly 1000 SNe known so far.
The effect of gravitational lensing will in general change the intrinsic distribution function of the SN magnitudes, increasing the scatter and introducing some non-Gaussianity, if originally absent. In part I of our present investigation~\cite{Amendola:2013twa}, we have obtained the lensing variance, skewness and kurtosis of the SN distribution via sGL, a fast simulation method developed in~\cite{Kainulainen:2009dw,Kainulainen:2010at,Kainulainen:2011zx}. The results were directly confronted to $N$-body simulations and shown to fit them very well up to a redshift of order 1.5, with the advantage of being given as a function of the relevant cosmological parameters. These fits can be employed to take into account the lensing extra scatter for any value of the cosmological parameters and also to model the lensing non-Gaussianity.
In this paper we propose instead to use the accurate determination of
the lensing moments of Ref.~\cite{Amendola:2013twa} to {\it measure} the cosmological parameters. As is often the case in cosmology, what was once a noise to be eliminated can become a signal when either data or modeling improve. Such idea in the present context has been first discussed in~\cite{Bernardeau:1996un,Hamana:1999rk,Valageas:1999ir} and later further developed in~\cite{Dodelson:2005zt}. We improve upon~\cite{Dodelson:2005zt} in two ways. First, we use not just the variance of the lensing signal but the 3rd and 4th order moments as well. Second, we do not assume that the intrinsic SN variance is fixed, but we marginalize over it at every redshift bin independently. The first step boosts the sensitivity (this was first proposed in~\cite{Bernardeau:1996un}) while the second the robustness of the method and allows us to show that a fundamental cosmological parameter, $\sigma_8$, can indeed be measured by LSST survey using SN lensing alone to within 3\%--7\%, a value that is competitive with usual methods based on cosmic shear, cosmic microwave background (CMB) or cluster abundance, and completely independent of these. In particular, it does not rely on measuring galaxy shapes (as cosmic shear) and is therefore immune to the systematics associated to the cross-correlation of intrinsic galaxy ellipticities. Also, it does not require to extrapolate the amplitude $\sigma_8$ from recombination epoch to today, as with the CMB technique, nor to make assumptions on the threshold of formation of structures that are needed when employing galaxy clusters. It is therefore a relatively direct measurement of $\sigma_8$ that can cross-check the results obtained via these other methods.
It is interesting to note that our proposal is essentially to carry out a one-point statistics on the supernova distribution on the Hubble diagram. This contrasts with other proposed methods which rely on two or higher point statistics such as that of~\cite{Cooray:2005yp}, where SN lensing and their inferred magnification was used as a tracer of dark matter clustering. In fact, by not relying on two-point statistics we avoid issues related to spatial correlations such as those arising from finite survey areas.
The main assumption that is needed for the SN lensing method is that the supernovae have an intrinsic magnitude distribution that is Gaussian, so that the entire non-Gaussianity can be attributed to lensing. In the future, this assumption can be directly tested by building a large calibrated sample of local supernovae. In principle, however, one could also include in the analysis an intrinsic non-Gaussianity and marginalize over it.
In order to make our method more directly applicable to future datasets, we will
base our estimation of $\sigma_8$ on the moments of the lensing distribution. One could use however also the full likelihood, again obtained via the sGL simulations. We will show however that a simplified likelihood based only on the first few moments is a very good approximation to the full likelihood.
The paper is organized as follows. In Section \ref{setup} we will describe the universe and lensing model adopted, and discuss the statistical properties of the lensing PDF as far as the central moments are concerned. In Section \ref{snalysis} we will examine the impact of lensing on the SN analysis, and in Section \ref{sec:constraining-s8} we will quantify how tightly can SNe constrain $\sigma_8$. Finally, we will conclude in Section \ref{conclusions}. We will discuss some more technical details in the Appendices \ref{sec:app-full-cov} and \ref{sec:lecorre}, and provide in Appendix~\ref{sec:zits} redshift-dependent fits for the second-to-fourth central moments of the lensing PDF which are simplified versions of the original fits in~\cite{Amendola:2013twa}.
\section{Lensing moments}\label{setup}
In this Section we will first describe the model we will use to compute the moments of the lensing PDF. Then we will discuss the properties of the cumulants as far as the convolution of the lensing and supernova distributions is concerned.
\subsection{Universe and lensing model} \label{sec:model}
We will calculate the second-to-fourth central moments $\mu_{2-4, \text{lens}}$ of the lensing PDF using the results of Ref.~\cite{Amendola:2013twa}.
There, accurate analytical fits as a function of $\{z,\,\sigma_{8},\,\Omega_{m0}\}$ were given for the broad ranges
$0 \leq z \leq 3,\, 0.35 \leq \sigma_{8} \leq 1.25,\, 0.1 \leq \Omega_{m0} \leq 0.52$. The dependence of the lensing moments on other parameters (such as $\Omega_{k0},\, w_{0},\, n_{s}$) was shown to be almost negligible.
The results of Ref.~\cite{Amendola:2013twa} were obtained using the stochastic gravitational lensing (sGL) method introduced in Refs \cite{Kainulainen:2009dw,Kainulainen:2010at,Kainulainen:2011zx}.
The sGL method is based on (i) the weak lensing approximation and (ii) generating stochastic configurations of inhomogeneities along the line of sight.
Regarding (ii), the matter contrast $\delta_{M}(r,t)$ is modeled according to the so-called ``halo model'' (see, for example, \cite{Neyman:1952, Peebles:1974, Scherrer:1991kk, Seljak:2000gq, Ma:2000ik, Peacock:2000qk, Scoccimarro:2000gm, Cooray:2002dia}), where the inhomogeneous universe is approximated as a collection of different types of halos whose positions obey the linear power spectrum.
The halo model assumes that on small scales the statistics of matter correlations is dominated by the internal halo density profiles, while on large scales the halos are assumed to cluster according to linear theory.\footnote{In \cite{Amendola:2013twa} correlations in the halo positions were, however, neglected. As shown in~\cite{Kainulainen:2010at,Kainulainen:2011zx}, this should be indeed a good approximation for the redshift range of $z \lesssim 1$ in which we are mainly interested in this paper.}
The halos were modeled using the halo mass function given in Ref.~\cite{Jenkins:2000bv}, which has a good degree of universality~\cite{White:2002at}. The use of other mass functions such as the one given in Ref.~\cite{Courtin:2010gx} does not change substantially the results of our analysis.
The halo profiles are modeled according to the Navarro-Frenk-White (NFW) profile \cite{Navarro:1995iw}, which is able to model both galaxy-sized halos and superclusters with an appropriately chosen concentration parameter. The concentration parameter depends on the cosmology and we use the universal and accurate model proposed in Ref.~\cite{Zhao:2008wd}.
Regarding (i), the lens convergence $\kappa$ in the weak-lensing approximation is given by the following integral evaluated along the unperturbed light path~\cite{Bartelmann:1999yn}:
\begin{equation} \label{eq:kappa}
\kappa(z_{s})=\int_{0}^{r_{s}}dr \, \rho_{MC} \, G(r,r_{s})\,\delta_{M}(r,t(r))
\end{equation}
where the quantity $\delta_{M}(r,t)$ is the local matter density contrast (which is modeled as described above), the density $\rho_{MC} \equiv a_0^3 \, \rho_{M0}$ is the constant matter density in a comoving volume, and the function $G(r,r_{s})= \frac{4\pi G}{c^2 \, a} \; \frac{f_{k}(r)f_{k}(r_{s}-r)}{f_{k}(r_{s})}$ gives the optical weight of a matter structure at the comoving radius $r$.
The functions $a(t)$ and $t(r)$ are the scale factor and geodesic time for the background FLRW model, and $r_{s}=r(z_{s})$ is the comoving position of the source at redshift $z_{s}$.
Also, $f_{k}(r)=\sin(r\sqrt{k})/\sqrt{k},\, r,\,\sinh(r\sqrt{-k})/\sqrt{-k}$ depending on the curvature $k>,=,<0$, respectively.
At the linear level, the shift in the distance modulus caused by lensing is expressed in terms of the convergence only:
\begin{equation} \label{eq:dm}
\Delta m(z) \simeq 5 \log_{10}\big [1-\kappa(z) \big] \simeq - \frac{5}{\log 10} \; \kappa(z) \,.
\end{equation}
Eq.~(\ref{eq:kappa}) connects the statistical distribution of matter to the
statistical distribution of convergences.
The sGL method for computing the lens convergence is based on generating random
configurations of halos along the line of sight and computing the associated
integral in Eq.~(\ref{eq:kappa}) by binning into a number of independent lens
planes.
A detailed explanation of the sGL method can be found
in~\cite{Kainulainen:2011zx,Kainulainen:2010at,Kainulainen:2009dw} and a
publicly available numerical implementation, the \textsc{turboGL}\ package, at
\href{http://www.turbogl.org/}{turbogl.org}.
Because of the theoretical approximations (weak lensing and halo model approximation) and modeling uncertainties (halo mass function and concentration parameter model) intrinsic in the sGL modeling, the results of Ref.~\cite{Amendola:2013twa} can be relied upon at the level of $\sim$10\%.
\subsection{Cumulants cumulate}
Observationally, the lensing PDF is convolved with the intrinsic standard-candle distribution. Now, there are fundamental statistical quantities
which are additive over convolutions: the cumulants. In other words,
if $X$ and $Y$ are two independent random variables, then the cumulants
$K_{i}(Z)$ of the convolution $Z\equiv X\star Y$ are just given
by $K_{i}(Z)=K_{i}(X)+K_{i}(Y)$.
Here and in the following, we will use $\mu_{i}',\mu_{i}$ and $K_{i}$
to denote respectively the $i$-th raw moment, central moment and cumulant.
We will abuse this notation and often refer to the first raw moment (the mean) $\mu_{1}'$ using the notation for the first central moment $\mu_{1}$ which is identically zero.
If $p(x)$ is a PDF then
by defining the generating function
\begin{equation}
\phi(t)=\left \langle e^{tx} \right \rangle=\int e^{tx}p(x)\,\textrm{d} x \,,
\end{equation}
one has that the cumulants and moments are defined as
\begin{eqnarray}
\mu'_{n} & \,\equiv\, & \left. \frac{\textrm{d}^{n}\phi}{\textrm{d} t^{n}} \right |_{t=0} \,,\\
K_{n} & \,\equiv\, & \left. \frac{\textrm{d}^{n}\log\phi}{\textrm{d} t^{n}} \right |_{t=0} \,.
\end{eqnarray}
We will initially assume that all standard candles have a distribution
which is intrinsically Gaussian. By intrinsically we mean neglecting
any systematic effect (such as lensing itself) that might
distort this distribution. A Gaussian has only two non-zero cumulants,
namely $K_{1}=\mu'_{1}$ and $K_{2}=\mu'_{2}-\mu_{1}'^{2}=\sigma^{2}$.
The weak lensing PDF, in turn, has by definition an (almost) negligible mean, and thus $K_{1, \text{lens}}=\mu'_{1, \text{lens}} \simeq 0$ so that for the lensing
PDF the moments are all central moments.
Using the relation between cumulants and central moments, to wit
\begin{align}
K_2 &= \mu _2 \label{K2} \\
K_3 &=\mu _3 \\
K_4 &=\mu _4-3 \mu _2^2
\end{align}
(where we used the fact that $\mu_{1}=0$) and the additivity of
the cumulants discussed above we can write the first central moments
of the convolved standard-candle PDF as (after straightforward manipulation)
\begin{align}
\mu_{2} & \;\equiv\;\sigma_{{\rm tot}}^{2}
\;=\;\sigma_{{\rm lens}}^{2}+\sigma_{\rm SN}^{2}\,,\label{eq:mu2}\\
\mu_{3} & \;=\;\mu_{3,{\rm lens}}\,,\label{eq:mu3}\\
\mu_{4} & \;=\;\mu_{4,{\rm lens}}+6\,\sigma_{{\rm lens}}^{2}\,\sigma_{\rm SN}^{2}
+ 3 \, \sigma_{\rm SN}^{4} \,.\label{eq:mu4}
\end{align}
Here $\sigma_{\rm SN}^{2}$ is the variance of the unlensed standard candles, which is sourced by the intrinsic dispersion $\sigma_{\rm int}$ and by the observational error $\sigma_{\rm exp}$:
\begin{align} \label{errorsb}
\sigma_{\rm SN}^{2} = \sigma_{\rm int}^{2} + \sigma_{\rm exp}^{2} \,.
\end{align}
We will assume later on that $\sigma_{\rm exp}$ is negligible as compared to
$\sigma_{\rm int}$, such that $\sigma_{\rm SN} = \sigma_{\rm int}$. This assumption has no influence
on our method and results.
There are some immediate conclusions from the above relations. First, it means
that one can directly compare a measure of $\mu_{3}$, which is the unnormalized
skewness (not to be confused with the normalized skewness, defined by
$\mu_{3}/\sigma^{3}$), with a theoretical prediction such as the fitting
function for $\mu_{3,{\rm lens}}$ provided in Ref.~\cite{Amendola:2013twa}.
Second, it means that using other moments besides just $\mu_2$ one can break the
degeneracy between the cosmological parameters (basically $\Omega_{m0}$ and
$\sigma_8$) and the nuisance parameter $\sigma_{\rm int}$. Such a degeneracy was arguably
the most important limitation of a previous work in the
literature~\cite{Dodelson:2005zt}. We will come back to this possibility in
Section~\ref{sec:constraining-s8}.
Incidentally, since (for many SNe) the mean $\mu'_1$ gives a very precise
measurement of $\Omega_{m0}$ (roughly independent of lensing), if one has
precise independent measurements of $\sigma_8$ one can use~\eqref{eq:mu2} to
make an estimate of the intrinsic dispersion $\sigma_{\rm int}(z)$ by subtracting from
$\sigma_{{\rm tot}}^{2}$ the variance due to lensing and the experimental error.
This can be an interesting result on its own, as it can help understand the
physics of SNe. We nevertheless do not explore this possibility further in this
work.
\section{Impact of lensing on SN analysis}
\label{snalysis}
The standard use of standard candles such as supernovae is to map the luminosity distance-redshift relation of the background FLRW model so as to constrain the content of the universe. From this point of view any fluctuation of the SN distance modulus, such as instrumental error and lensing, is seen as noise (completely opposite will be the approach followed in the next Section \ref{sec:constraining-s8}).
Therefore, as far as the standard SN analysis is concerned lensing has to be dealt with appropriately so as not to bias the parameter extraction of the {\em background} parameters.
It is important to account for both the skewness and the cosmology dependence of the lensing PDF~\cite{Amendola:2010ub}.
In particular, the redshift dependence of lensing can distort and rotate the confidence-level contours.
The inclusion of lensing complicates the usual analysis with the effect that a standard $\chi^{2}$ analysis is not longer possible or consistent.\footnote{Even without lensing the usual supernovae $\chi^{2}$ analysis, which is iterative and sets $\chi^{2} / {\rm d.o.f.} \equiv 1$ is not ideal as it can lead to biases and does not allow determination of $\sigma_{\rm int}$ or model-selection analysis. See \cite{Kim:2011hg,March:2011xa,Lago:2011pk} for more details.} The skewness of the lensing PDF violates indeed the Gaussian assumption, while the cosmology dependence of the normalization of the likelihood imposes that the latter is kept contrary to what is done in the usual $\chi^{2}$ analysis.
For example, if one is using a Gaussian likelihood with error depending on the theoretical parameters, then one cannot at the same time keep the parameter-dependent normalization factor and use a mock catalog with $\chi^{2}=0$. Indeed, a $\chi^{2}=0$ catalog is supposed to have the likelihood peaked at the chosen fiducial model, but this will not be the case as the parameter-dependent normalization factor will tilt the likelihood surface in a way which depends on the error.
\begin{figure*}[t!]
\begin{centering}
\includegraphics[width=2\columnwidth]{figs-sig/convolved-PDF}
\caption{ Comparison of the original SN Gaussian PDF ($P_{\rm SN}$, dashed brown) with the lensed supernova PDF ($L$ of Eq.~(\ref{Li}), green), obtained by the convolution of the lensing PDF ($P_{\rm lens}$, dot-dashed blue) with $P_{\rm SN}$. We assume $\sigma_{\rm SN}=0.12$ mag, a redshift of $z=1$ and the WMAP9-only best-fit values as fiducial model. As can be seen, the distortion of $L$ from Gaussianity is small around the peak, but it gets large in the high-magnification tail. For this plot we employ a cut $\kappa_{\rm cut} = 0.35$ (or $\Delta m = -0.94$ mag) for the lensing PDF, as discussed in Ref.~\cite{Amendola:2013twa}. See Section \ref{snalysis} for more details.
\label{fig:convolved-PDF}}
\end{centering}
\end{figure*}
A full likelihood analysis uses the lensing PDF, in our case either directly obtained from \href{http://www.turbogl.org/}{\textsc{turboGL}\ }or reconstructed using the log-normal approximation as proposed in Ref.~\cite{Amendola:2013twa}.\footnote{The lensing PDF may also be obtained using the ``universal'' lensing PDF of Ref.~\cite{Wang:2002qc}.} We denote the lensing PDF as $P_{\rm lens}(\Delta m,z, \Omega_{m0}, \sigma_8)$.
Within our approximations $P_{\rm lens}$ has negligible mean. The SN likelihood is then obtained by convolving the lensing PDF with the SN uncertainty distribution, which we assume to be Gaussian in the distance moduli and denote with $P_{\rm SN}(mean, sigma)$.
The latter uncertainty is sum of the intrinsic source brightness scatter and of the observational errors, as discussed in Eq.~(\ref{errorsb}).
The likelihood function for a single SN observation is then
\begin{align}
L_{i}( \Omega_{m0}, \sigma_8, \xi)&= \int \textrm{d} y \, P_{\rm lens}\Big(y,z_{i}, \Omega_{m0}, \sigma_8 \Big) \; \times \label{Li} \\
\times \; & P_{\rm SN}\Big ( m_{t}(z_{i},\Omega_{m0})- m_{i} +\xi-y,\,\sigma_{{\rm SN},i} \Big)\,,\nonumber
\end{align}
where $z_{i}, m_{i}, \sigma_{{\rm SN},i}$ are the $i$th SN redshift, distance modulus and uncertainty, respectively. The quantity $m_{t}$ is the predicted distance modulus for a source at redshift $z_{i}$ in a flat $\Lambda$CDM model of matter density parameter $\Omega_{m0}$:
\begin{align}
m_{t}(z, \Omega_{m0}) = 5\log_{10}\frac{d_{L}(z)}{10\,\textrm{pc}}\,, \label{modulus}
\end{align}
where luminosity distance $d_{L}$ and Hubble rate $H$ are
\begin{align}
d_{L}(z) & =(1+z)\int_{0}^{z}\frac{{\rm d}\bar{z}}{H(\bar{z})}\,, \\
\frac{H^{2}}{H^{2}_{0}} & = \Omega_{m0} (1+z)^{3}+\Omega_{\Lambda0} \,,
\end{align}
where $\Omega_{\Lambda0}=1-\Omega_{m0}$.
The quantity $\xi$ is an unknown offset sum of the supernova absolute magnitudes, of $k$-corrections and other possible systematics. Figure~\ref{fig:convolved-PDF} depicts the three distributions $P_{\rm SN}$, $P_{\rm lens}$ and $L_{i}$, where $P_{\rm lens}$ was modeled using the log-normal template proposed in Ref.~\cite{Amendola:2013twa}. As it can be seen, the distortion in $L_{i}$ from Gaussianity is small near the peak, but it gets large in the high-magnification tail. Finally, we define the total likelihood function as the product of all independent likelihood functions in the data sample, further marginalized over the unknown $\xi$:
\begin{equation}
L_{\rm tot}(\Omega_{m0}, \sigma_8, \{\sigma_{{\rm int},j}\})=\int \textrm{d} \xi \; \Pi_{i}L_{i}( \Omega_{m0}, \sigma_8, \xi)\,,\label{fooltot}
\end{equation}
where we have explicitly stressed the dependence of $L_{\rm tot}$ on the intrinsic dispersion $\sigma_{\rm int}$, which we will leave as a set of free parameters $\{\sigma_{{\rm int},j}\}$, to be marginalized over in \emph{each redshift bin} $z_{j}$. Since $\xi$ is degenerate with $\log_{10}H_{0}$ we are effectively marginalizing also over the expansion rate of the universe.
As said earlier, SN observations are usually used to determine background parameters, in our case $\Omega_{m0}$, but generally also $\Omega_{k0}, w_{0}, w_{a}$, etc. Therefore, the likelihood of Eq.~(\ref{fooltot}) has to be further marginalized over $\sigma_8$, $L_{\rm bkg}=\int \textrm{d} \sigma_8 L_{\rm tot}$, and the question is how lensing affects $L_{\rm bkg}$. What we found is that while the effect of lensing can distort, increase and rotate the confidence-level contours of $L_{\rm bkg}$, it does not substantially bias the position of its maximum. The reason is twofold. First, the skewness induced by lensing is not sizable as far as the mean magnitudes are concerned and the variance added by lensing is subdominant with respect to the intrinsic dispersion of the supernovae, as shown by Figure~\ref{fig:convolved-PDF}. Second, the cosmology dependence of lensing will become less and less important as more precise measurements will restrict the available range of cosmological parameters within which the effect of lensing can vary. Although sizable biases were found in~\cite{Amendola:2010ub}, we now trace it to the toy model employed for the matter field (an universe populated with $~10^{14} M_\odot$ NFW halos). The much more realistic current modeling of \textsc{turboGL}\ produces weaker lensing effects.
The main focus of this paper is, however, not on $L_{\rm bkg}$ but on the signal hidden in the scatter of the SN distance moduli, in particular the information relative to $\sigma_8$. Following this point of view, we will then use directly the likelihood $L_{\rm tot}$ of Eq.~(\ref{fooltot}).
\section{Constraining \boldmath $\sigma_{8}$ with SN data}
\label{sec:constraining-s8}
\subsection{The Method-of-the-Moments}
In this section we come back to the question of whether one can use
measurements of the observed distributions of standard candles at
different redshifts to constrain the statistics of matter inhomogeneities, in particular $\sigma_8$. This
idea was first proposed by Ref.~\cite{Dodelson:2005zt} for supernovae,
but in that paper the authors focused mainly on using the additional
variance due to lensing. This is problematic, as in principle one
does not know what is the value of the intrinsic dispersion $\sigma_{{\rm int}}$ or even if
it is constant in redshift or not. Thus, even in the limit of perfect
modeling of instrumental errors and no extra systematics, a measurement
of a growing $\sigma_{\rm int}$ with $z$ could be attributed to
some sort of evolution effect of the supernovae rather than to lensing. The only way out
would be to reach a very good level of physical understanding of the
explosion process (and of other systematic effects) to be able to
accurately predict what $\sigma_{\rm int}(z)$ should look like.
In this work we break this degeneracy between the cosmological parameters and the nuisance parameter $\sigma_{\rm int}$ using other moments besides just $\mu_2$. In particular, we propose to measure the cosmological parameters $\{ \Omega_{m0}, \sigma_8 \}$ and the distribution of $\sigma_{\rm int}(z)$ at the same time by using the information contained in the mean $\mu_{1}'$ and the first three central moments (which we will collectively refer to simply as $\mu_{1-4}$).
At this point we could just use the full likelihood $L_{\rm tot}$ of Eq.~(\ref{fooltot}), which automatically contains all the available lensing information.
However, we develop here an alternative approach which focuses, as outlined above, on the information carried by the lensing moments. There are mainly three reasons to do so: (i) it is computationally faster as data can be binned in redshift\footnote{The full likelihood $L_{\rm tot}$ of (\ref{fooltot}) cannot be binned in redshift without losing information about moments above the second one.} and easily implemented in numerical codes; (ii) it does not require knowledge of the full lensing PDF and instead simply needs the theoretical prediction of the second-to-fourth lensing moments (available as analytical fits in~\cite{Amendola:2013twa}), which can directly be confronted with observations; and (iii) it will be essentially a $\chi^{2}$ approach (without need of convolutions), with all its advantages such as the ability to easily include nuisance parameters (for instance, describing some intrinsic non-Gaussianity of supernovae).
The disadvantage is that the likelihood gets somewhat more complicated as a result of the correlations between the moments, and in principle requires some information on high moments.
\begin{figure*}[t!]
\begin{centering}
\includegraphics[width=2\columnwidth]{figs-sig/cov-1-2-3-4-LSST100k-green}
\caption{Comparison of the moment analysis using different combinations of moments for the case of $10^5$ supernovae in LSST and assuming as usual a fiducial $\sigma_{\rm int} = 0.12$ mag. \emph{Left:} using $\mu'_1 - \mu_2$ only; \emph{middle:} using $\mu'_1 - \mu_3$; \emph{right:} using $\mu'_1 - \mu_4$. We see that after marginalizing over $\sigma_{\rm int}$ in each redshift bin, very little information is gained from $\mu'_1 - \mu_2$ only, and one has to resort to at least the third moment. Note that although $\mu_4$ adds some extra information, most of it comes from $\mu'_1 - \mu_3$.
\label{fig:mu1-2-3-4-comparison}}
\end{centering}
\end{figure*}
The idea is very simple: we build a likelihood at each redshift bin directly for
the first four moments $\mu_{1-4}$, to be called the \emph{method-of-the-moments}
(MeMo) likelihood:
\begin{align}
&L_{\rm MeMo}(\Omega_{m0}, \sigma_8, \{\sigma_{{\rm int},j}\}) = \exp \left( - \frac{1}{2} \sum_{j}^{{\rm bins}} \chi_{j}^2 \right) \,,\label{Lmom} \\
&\chi^2_j = \big(\boldsymbol{\mu}-\boldsymbol{\mu}_{\rm fid}\big)^t \;\Sigma_j^{-1}\; \big(\boldsymbol{\mu}-\boldsymbol{\mu}_{\rm fid}\big) \,, \label{chi2mom} \\
&\boldsymbol{\mu} = \{ \mu_1',\,\mu_2,\,\mu_3,\,\mu_4 \} \,,
\end{align}
where the vector $\boldsymbol{\mu}(z_{j},\sigma_{8},\Omega_{m0},\{ \sigma_{{\rm int},j} \})$ is the theoretical prediction for the moments, and its second-to-fourth components are defined in~\eqref{eq:mu2}--\eqref{eq:mu4}. The mean $\mu_{1}'$ is the theoretical distance modulus $m_{t}(z, \Omega_{m0})$ of~\eqref{modulus}.
The quantity $\boldsymbol{\mu}_{\rm fid}(z_{j})$ is the vector of fiducial or measured (sample) moments. In the former case it is $\boldsymbol{\mu}(z_{j},\sigma_{8},\Omega_{m0},\{ \sigma_{{\rm int},j} \})$ evaluated at the fiducial model, while in the latter case its components are:
\begin{align}
\mu_{1,{\rm fid}}'(z_{j}) & = \frac{\sum_{k} m_{k,j}}{N_j} \,, \label{sample1} \\
\mu_{i,{\rm fid}}(z_{j}) & = \frac{\sum_{k} [m_{k,j}- \mu_{1,{\rm fid}}'(z_{j})]^{i}}{N_j} \,, \label{samplei}
\end{align}
where $m_{k,j}$ are the SN distance moduli observed in the redshift bin centered at $z_{j}$. The covariance matrix $\Sigma$ is built using the fiducial (or observed) moments and therefore does not depend on cosmology (but it does on $z$).
This can be understood intuitively from (\ref{chi2mom}) as in this case the $\chi^2_j$ function is minimized, as expected, by $\boldsymbol{\mu}=\boldsymbol{\mu}_{\rm fid} $.
Consequently, the normalization factor (basically the determinant of $\Sigma$) is irrelevant and we have neglected it in (\ref{Lmom}).
The number of moments to be used in this analysis is in principle arbitrary as each new moment adds information. However, as will be shown below, for supernova almost all of the information is already included using $\mu_{1-4}$ (and a very good fraction of it already in $\mu_{1-3}$).
In the Gaussian limit of $L_{\rm tot}$, the covariance matrix relative to the redshift bin $z_{j}$ is simply
\begin{align}
\Sigma_{{\rm gau},j} = \frac{1}{N_j} \left (
\begin{array}{cccc}
\sigma_j^2 & 0 & 0 & 0 \\
0 & 2 \sigma_j^4 & 0 & 12 \sigma_j^6 \\
0 & 0 & 6 \sigma_j^6 & 0 \\
0 & 12 \sigma_j^6 & 0 & 96 \sigma_j^8 \\
\end{array}
\right) \label{sigmagau}
\end{align}
where $\sigma^{2}_j$ is the variance of the dataset, fiducial or observed, at the redshift bin centered on $z_j$. In the former case it is $\sigma^{2}_j = \sigma_{{\rm int},j}^{2}+ \sigma_{{\rm lens},j}^{2}$, with the latter evaluated at the fiducial flat $\Lambda$CDM model given by the 9-year WMAP-only best-fit values~\cite{Hinshaw:2012aka}. As discussed in Section~\ref{sec:model} we will consider only the dependence of lensing on $\{ z, \sigma_{8},\Omega_{m0} \}$. The fiducial values of the latter two are $0.821$ and $0.279$, respectively. The quantity $N_j$ is the number of SNe in the $j$-th redshift bin.
$\Sigma_{\rm gau}$ is a good approximation in the limit in which the deviation from the Gaussianity induced by lensing is small; in this case the covariance will be dominated by the Gaussian sampling variance. As we discuss in Appendix~\ref{sec:lecorre}, this is the case for standard candles with $z \lesssim 0.5$.
In the case of a general distribution the covariance matrix is more complicated. It can be written in a more compact way in terms of the cumulants:
\begin{widetext}
\begin{align}
& \Sigma_j = \frac{1}{N_j} \times \label{sigmabadass} \\
&\bBigg@{4}(
\begin{array}{cccc}
K_2 & K_3 & K_4 & 6 K_2 K_3+K_5 \\
- & 2 K_2^2+K_4 & 6 K_2 K_3+K_5 & 12 K_2^3+14 K_4 K_2+6 K_3^2+K_6 \\
- & -& 6 K_2^3+9 K_4 K_2+9 K_3^2+K_6 & 72 K_3 K_2^2+18 K_5 K_2+30 K_3 K_4+K_7 \\
- & -& -& 96 K_2^4+204 K_4 K_2^2+216 K_3^2 K_2+28 K_6 K_2+34 K_4^2+48 K_3 K_5+K_8 \\
\end{array}
\bBigg@{4}), \nonumber
\end{align}
\end{widetext}
where we denote by ``--'' the symmetric terms.
As can be seen, an accurate estimation of $\Sigma_j$ requires knowledge of all central moments up to $\mu_8$ (see Appendix \ref{sec:app-full-cov} for the relation between cumulants and moments). Nevertheless, as we will show, most of the information is contained in $\mu_{1-3}$, so in practice one would only need to go up to $\mu_6$. Moreover, as explained above, $\Sigma_j$ is to be evaluated at the fiducial values, so it is only necessary to know these moments at these fiducial values, and not as a function of $\{\sigma_{8},\,\Omega_{m0}\}$ (as carried out in~\cite{Amendola:2013twa}). As discussed before, the only nonzero cumulant of the Gaussian supernova PDF is $K_2$. Therefore $\Sigma$ can be evaluated, for each redshift bin, at the fiducial model simply by using $K_{2} = \sigma_{\rm lens}^2+\sigma_{\rm int}^{2}$ [see~\eqref{K2} and \eqref{eq:mu2}] and computing all the $K_{i}$ using \eqref{K2app}--\eqref{K8app} where the $\mu_{i} \rightarrow \mu_{i,{\rm lens}}$, i.e., the moments relative to the lensing PDF. Both Eqs (\ref{sigmagau}) and (\ref{sigmabadass}) were obtained using the \textsc{Mathematica} package \textsc{mathStatica} in the limit of large number of observations $N_{j}$ in each bin.
The MeMo likelihood of Eq.~(\ref{Lmom}) depends on the intrinsic dispersions $\{\sigma_{{\rm int},j}\}$, which are let free and independent in each redshift bin. In principle one has two choices in order to make forecasts of future constraints on $\{\sigma_{8},\,\Omega_{m0}\}$: either fixing $\sigma_{{\rm int},i}$ to a fiducial value (say $\sigma_{{\rm int},i} = 0.12$ mag, constant in redshift~\cite{Bernstein:2011zf,Abell:2009aa}) or leave it as a set of free parameters, to be marginalized over in each redshift bin. The former is computationally convenient as effectively reduces the dimension of the full likelihood by one. The latter is the more appropriate approach, and is the one recommended for analyzing real data. One has to choose the appropriate marginalization intervals for the flat priors to be used in (\ref{Lmom}). Although in principle these could be set as $\sigma_{{\rm int},i} \in \{0,\, \infty\}$, for all future SN catalogs here considered, the interval $\sigma_{{\rm int},i} \in \{0.1,\, 0.14\}$ mag proved sufficient.
\begin{figure}[t!]
\begin{centering}
\includegraphics[width=.9\columnwidth]{figs-sig/cov1234-vs-lhood-sint-fix-LSST100k-v2.png}
\caption{Comparison of the full likelihood analysis (colored contours) with the MeMo-likelihood one (solid green contours), using the full 4 dimensional covariance matrix of $\mu'_1 - \mu_4$ as given in Eq.~(\ref{sigmabadass}), for the case of $10^5$ supernovae in LSST and assuming a fixed $\sigma_{\rm int} = 0.12$ mag. Also shown is the MeMo results using only the diagonal part of the covariance matrix (black dotted contours). We see that the MeMo likelihood, using the mean and the second-to-fourth central moments, correctly reproduces the results relative to the full likelihood. Note that the fiducial of the MeMo contours was slightly changed in this plot to coincide with the one of the full likelihood and allow better comparison.
\label{fig:mu1-4-cov-corr}}
\end{centering}
\end{figure}
\begin{figure}[t]
\begin{centering}
\includegraphics[width=.9\columnwidth]{figs-sig/full-vs-approx-covs}
\caption{Comparison of the full covariance matrix~\eqref{sigmabadass} constraints (green contours) with 3 possible approximations: the Gaussian matrix~\eqref{sigmagau} (black dashed contours), using only the diagonal part of~\eqref{sigmabadass} (black dotted contours) and using only $\mu'_1 - \mu_3$ (brown long-dashed contours) for the case of $10^5$ supernovae in LSST and marginalizing over all $\sigma_{{\rm int},j}$ (note that Figure~\ref{fig:mu1-4-cov-corr} in contrast assumes a fixed value of $\sigma_{\rm int}$). As can be seen, most of the information is already contained on the first 3 moments. Note also that the non-Gaussian diagonal matrix is a very good approximation after marginalization over $\sigma_{{\rm int},j}$, whereas the full Gaussian matrix underestimates $\delta \sigma_8$ to less than 50\% of the real value. \label{fig:cov-aproxs}}
\end{centering}
\end{figure}
Figure~\ref{fig:mu1-2-3-4-comparison} shows the comparison of the constraints using different combinations of moments as forecasted for the LSST 100k catalog, explained below in Section~\ref{sec:forecasts}. We see that after marginalizing over $\sigma_{\rm int}$ in each redshift bin, very little information is gained from $\mu'_1 - \mu_2$ only, and one has to resort to the third and (with marginal improvement) the fourth moment.
This was arguably the biggest caveat of the original work~\cite{Dodelson:2005zt}: in order to extract information from the variance only they had to assume the intrinsic dispersion to be well known. However, when one allows complete freedom for the different $\sigma_{{\rm int},j}$, almost no information can be collected from just the mean and the variance.
Figure~\ref{fig:mu1-4-cov-corr} confronts the full likelihood analysis with the MeMo, using $\mu'_1 - \mu_4$ assuming for computational convenience a fixed value of $\sigma_{\rm int} = 0.12$ mag in all bins. We see that by employing the full covariance matrix one can get a very good agreement between both methods. In what follows we will therefore stick to the MeMo method (using $\mu'_1 - \mu_4$) when deriving our forecasts.
The fact that, as seen from Figure~\ref{fig:mu1-2-3-4-comparison}, the fourth moment adds little constraining power to the analysis has the important consequence that the MeMo likelihood can be limited to mean, variance and skewness. This fact made more evident in Figure~\ref{fig:cov-aproxs} where we plot together both set of contours. This clearly makes the MeMo method more robust against the presence of SN outliers that could possibly bias the value of the higher moments. That being said, in what follows we will forecast results assuming all four moments are used.
\begin{figure*}[t]
\begin{centering}
\includegraphics[width=2\columnwidth]{figs-sig/full-vs-lens-cov-1-4-insets}
\caption{In green, forecast of cosmological constraints using the central moments
$\mu_{1}'$, $\mu_{2}$, $\mu_{3}$ and $\mu_{4}$ and supernovae from: \emph{(left)}
DES, if the scatter could be reduced to 0.12 mag (the predicted scatter given
in~\cite{Bernstein:2011zf} is somewhat larger);
\emph{(middle)} LSST, with $10^5$ SNe, also assuming a fiducial intrinsic scatter of 0.12 mag (as predicted
in~\cite{Abell:2009aa});
and \emph{(right)} same for LSST but for $5 \times 10^5$ SNe. The red contours are the ``lensing-only'' likelihood,
see text.}
\label{fig:full-constraints}
\end{centering}
\end{figure*}
Since, at least at first glance, the total convolved PDF is not largely dissimilar from a Gaussian (see Figure~\ref{fig:convolved-PDF}), one naturally wonders whether the full covariance matrix~\eqref{sigmabadass} can be approximated by the Gaussian one~\eqref{sigmagau}. We have carried out this test adding also a second approximation for \eqref{Lmom}, namely using only the (full, non-Gaussian) diagonal terms. Figure~\ref{fig:cov-aproxs} also compares the three $\mu'_1 - \mu_4$ covariances for the case of LSST 100k (see Section~\ref{sec:forecasts}). As can be seen, the non-Gaussian corrections to the errors are very relevant, and broaden the constraints on $\sigma_8$ by a factor greater than two. However, these lensing corrections become negligible for smaller redshift as discussed in Appendix \ref{sec:lecorre}.
We also conclude that the diagonal part of $\Sigma$ carries almost the same information as the full matrix. Due to this finding, we provide in Appendix \ref{sec:lecorre} fitting functions with respect to lensing for the diagonal components of $\Sigma$.
It is worth pointing out at this point that the formalism developed in this
section could be generalized in a straightforward way so as to include a
fiducial $\mu_{i,{\rm int}}$ for $i=3,\,4$. In other words, it would be possible
to add some non-Gaussianity -- possibly due to the SN themselves and/or to other transients -- also for the intrinsic supernovae PDF and
marginalize over these additional nuisance parameters.
\subsection{Constraints from future supernova surveys}\label{sec:forecasts}
In this Section we forecast the precision with which one can measure $\sigma_8$ with future supernova data. For this we study three different catalogs, based on 2 surveys: Dark Energy Survey (DES)~\cite{Bernstein:2011zf} and LSST~\cite{Abell:2009aa}. DES is already operational, and is expected to observe around 3000 SN during its observational cycle. LSST is expected to be operational by the end of the decade, and should observe a tantalizing amount of roughly 50000 SN per year. It is assumed in~\cite{Abell:2009aa} that these can be measured with an average scatter of $\sigma_{{\rm int},j}=0.12$ mag. We therefore assume in what follows a fiducial a scatter of $\sigma_{{\rm int},j}=0.12$ mag constant in all redshifts.\footnote{Note that only the fiducial is assumed constant. I.e., we still marginalize over $\sigma_{{\rm int},j}$ in each redshift bin, as described in Section~\ref{sec:constraining-s8}.}
The 3 cases we will consider here are (i) DES, with the distribution given in~\cite{Bernstein:2011zf} (note that the same work predicts a higher intrinsic scatter, varying in $z$ between 0.14 and 0.25 mag); (ii) LSST, with the redshift distribution given in~\cite{Abell:2009aa} and a total of $10^5$ supernovae (which we dub ``LSST 100k'' and which should correspond to 2 years of observation); and (iii) same as (ii) but for a total of $5\times10^5$ SNe (``LSST 500k'', corresponding to the full 10-year survey).
Figure~\ref{fig:full-constraints} depicts the constraints on $\{\Omega_{m0}, \sigma_8 \}$ that can be obtained with the future supernova surveys described above. From left to right, we use DES, LSST 100k and LSST 500k. The green contours are the final $1,\,2,\,3\sigma$ constraints, using~\eqref{Lmom} and \eqref{sigmabadass}. Note that LSST is thus forecasted to measure $\sigma_{8}$ with error bars of around 0.057 with $10^5$ SNe (0.023 with $5\times 10^5$), which is around the interesting $7\%$ ($3\%$) level. For DES, however, the statistical information is underwhelming, and one will not gain good knowledge on $\sigma_{8}$, even though we assumed an intrinsic scatter smaller than the one estimated by the DES collaboration~\cite{Bernstein:2011zf}. The red contours, mostly of bookkeeping interest, depict the constraints provided by the lensing effect only (without using information about the mean). In other words, from the non-Gaussianity introduced by lensing on the final PDF. More precisely, they are the constraints obtained substituting the first row and column of~\eqref{sigmabadass} by zeros. Note that the red and green contours are almost perpendicular, which shows that the final constraints can be approximated by taking independently mean (used in standard SNe analyses) and lensing effects.
\begin{figure}[t]
\begin{centering}
\includegraphics[width=0.95\columnwidth]{figs-sig/sint-posteriors-1234-100k}
\caption{The posterior distribution of $\sigma_{\rm int}$ (in magnitudes) in different redshift bins of width $\Delta z = 0.05$ using LSST 100k. Although this catalog has 18 such bins, we depict only 6 for clarity. For low-$z$ lensing is small and the posterior is wide. For intermediate $z$ the constraints are the tightest (they peak at around $z=0.5$), because that is where there are more SNe. For the highest $z$-bins the number of forecasted supernovae decreases and the peaks broaden again.}
\label{fig:sint-posterior}
\end{centering} %
\end{figure}
\begin{figure*}[t]
\begin{centering}
\includegraphics[width=\columnwidth]{figs-sig/mu-contours-tot}
\;\;\includegraphics[width=\columnwidth]{figs-sig/mu-contours-lens}
\caption{Constant central-moment contours, both for the total convoluted PDF (left) and for the lensing-only PDF (right). For each moment all contour levels are separated by a constant amount. Note that there is a near but not exact degeneracy, which explains why the higher-moment-only contours (i.e.~without the mean) have a finite area (as in the case of LSST in the range of parameters considered -- see Figure~\ref{fig:full-constraints}).}
\label{fig:mu-contours}
\end{centering} %
\end{figure*}
Besides constraints in $\sigma_{8}$, the central moments (specially $\mu_2$) also give information on the intrinsic scatter of SNe in each redshift bin. Figure~\ref{fig:sint-posterior} depicts the achievable precision in measuring $\sigma_{{\rm int},j}$ in different redshift bins of width $\Delta z = 0.05$ using LSST 100k. Although this catalog has 18 such bins, we only depict 6 of them for clarity. The posteriors are all normalized to have unit area. The precision with which one can measure the different $\sigma_{{\rm int},j}$ varies roughly from 0.01 to 0.03 mag, depending on the redshift.
To illustrate the physics behind the red (``lensing-only'') contours of Figure~\ref{fig:full-constraints}, it is useful to depict the contours of constant value for the different central moments. This is done in Figure~\ref{fig:mu-contours}, both for the total convoluted PDF (left) and for the lensing-only PDF (right). Note that there is a near but not exact degeneracy, which explains the fact that although very broad, the lensing-only contours are in fact closed.
\section{Conclusions}
\label{conclusions}
In this paper we developed a method to measure $\sigma_8$ by exploiting the
lensing effects of matter clustering along the line of sight of SNe, extending
the results of~\cite{Dodelson:2005zt}. We have shown that
one can obtain interesting constraints in a survey with one or few hundred thousands supernovae,
as in the LSST survey.
In particular, we find that $\sigma_8$ can be estimated to within 3\% if 500,000 supernovae
with average magnitude error 0.12 are collected.
This method is independent of and complementary to the standard methods using
CMB, cosmic shear or cluster abundance and bypasses the need to assume a constant intrinsic variance
as in~\cite{Dodelson:2005zt}. We still assume that the SN magnitude distribution
does not have an intrinsic non-Gaussianity, although in principle one can marginalize over this extra
parameter.
Instead of employing the full likelihood, we have shown that the method of moments (MeMo) provides
a very good approximation and is much faster to implement. Even the simplest case of diagonal approximation works with sufficient accuracy (we provide fitting functions in Appendix \ref{sec:lecorre}).
We have shown that it is enough to use the first three moments (loosely speaking, mean, variance and skewness) so as to capture the non-Gaussian information available in the full likelihood.
This should make the MeMo method robust against the presence of SN outliers that could possibly bias the value of higher moments.
The needed moments as a function of cosmological parameters have been already derived in \cite{Amendola:2013twa}.
One could at first be suspicious about the feasibility of measuring the full covariance matrix~\eqref{sigmabadass}, as it assumes we are able to measure all moments up to the $8^{\rm th}$. Nevertheless, we have two reasons to believe this is not as hard as it seems. First, similarly to what was discussed in part I~\cite{Amendola:2013twa},
using the results of~\cite{Takahashi:2011qd} we evaluated the dependence of the first 8 moments on the point at which we cut the tail of the $\kappa$PDF. The dependence is not very strong and saturates somewhere between $\kappa_{\rm cut}$ of 0.2 and 0.7, depending on the moments. This range reflects the fact that higher moments indeed require a higher value of $\kappa_{\rm cut}$ for more precise results. However: (i) even for the $8^{\rm th}$ moment using $\kappa_{\rm cut} = 0.35$ would result in only a $20\%$ error in the value of $\mu_8$; (ii) as we have shown in the text, a very good estimate of $\sigma_8$ can be achieved neglecting the $4^{\rm th}$ moment entirely, and thus dropping the need for either the $7^{\rm th}$ and $8^{\rm th}$ moment for the computation of the covariance matrix. For $z>1$ this issue could become important, not so much because one needs a slightly larger $\kappa_{\rm cut}$ but mainly because turboGL in its current form starts to deviate from the numerical simulations due to medium and strong-lensing corrections.
Finally, we note that this method can be extended to other parameters, for instance to the growth function of matter perturbations. This will be explored in future work.
\begin{figure}[t]
\begin{centering}
\includegraphics[width=.95\columnwidth]{figs-sig/Delta-n-10to7}
\caption{Value of the correction factors $\Delta_{n}$ for the variance of the moments using the WMAP9 fiducial values of $\{\sigma_8 \simeq0.821, \,\Omega_{m0} \simeq 0.279\}$. \emph{Bottom to top:} $\Delta_{2}$, $\Delta_{3}$ and $\Delta_{4}$. For supernova forecasts in the usual domain $0<z<1.2$ these corrections can be large, specially for $\sigma^2(\mu_4)$. For higher redshifts and for higher values of $\{\sigma_8, \,\Omega_{m0}\}$ they become even larger. \label{fig:Delta-n}}
\end{centering}
\end{figure}
\begin{acknowledgments}
It is a pleasure to thank Stephan Hilbert, Martin Makler, Bruno Moraes, Ribamar
Reis, Peter Schneider, Brian Schmidt and Ryuichi Takahashi for fruitful discussions. LA and VM acknowledge support from DFG through the TRR33 program ``The Dark
Universe''. MQ is grateful to Brazilian research agencies CNPq and
FAPERJ for support and to ITP, Universität Heidelberg for hospitality
during part of the development of this project.
\end{acknowledgments}
|
1,116,691,497,785 | arxiv | \section{Conclusion}
We presented a strongly-typed implementation of miniKanren\xspace for OCaml. Our implementation
passes all tests written for miniKanren\xspace (including those for disequality constraints);
in addition we implemented many interesting relational programs known from
the literature. We claim that our implementation can be used both as a convenient
relational DSL for OCaml and an experimental framework for future research in the area of
relational programming.
\section{miniKanren\xspace~--- a Short Presentation}
\label{sec:demo}
In this section we briefly describe miniKanren\xspace in its original form, using a canonical example.
miniKanren\xspace is organized as a set of combinators and macros for Scheme/Racket, designated to describe
a search for the solution of a certain \emph{goal}. There are four domain-specific constructs
to build \emph{goals}:
\begin{itemize}
\item Syntactic unification~\cite{Unification} in the form \lstinline[language=scheme]{(== $t_1$ $t_2$)}, where $t_1$, $t_2$ are
some \emph{terms}; unification establishes a syntactic basis for all other goals. If there is a unifier for
two given terms, the goal is considered satisfied, a most general unifier is kept as a partial solution, and the execution
of current branch continues. Otherwise, the current branch backtracks.
\item Disequality constraint~\cite{CKanren} in the form \lstinline[language=scheme]{($\not\equiv$ $t_1$ $t_2$)}, where
$t_1$, $t_2$ are some terms; a disequality constraint prevents all branches (starting from the current), in which the
specified terms are equal (w.r.t. the search state), from being considered.
\item Conditional construct in the form
\begin{lstlisting}[language=scheme]
(conde
[$g^1_1\;\;g^1_2\;\;\dots\;\;g^1_{k_1}$]
[$g^2_1\;\;g^2_2\;\;\dots\;\;g^2_{k_2}$]
$\ldots$
[$g^n_1\;\;g^n_2\;\;\dots\;\;g^n_{k_n}$]
)
\end{lstlisting}
where each $g^i_j$ is a goal. A \lstinline{conde} goal considers each collection of subgoals, surrounded by square brackets, as
implicit conjunction (so \lstinline[language=scheme]{[$g^i_1\;\;g^i_2\;\;\dots\;\;g^i_{k_i}$] } is considered as a
conjunction of all $g^i_j$) and tries to satisfy each of them independently~--- in other words, operates on them
as a disjunction.
\item Fresh variable introduction construct in the form
\begin{lstlisting}[language=scheme]
(fresh ($x_1\;\;x_2\;\;\dots\;\;x_k$)
$g_1$
$g_2$
$\ldots$
$g_n$
)
\end{lstlisting}
where each $g_i$ is a goal. This form introduces fresh variables $x_1\;\;x_2\;\;\dots\;\;x_k$ and
tries to satisfy the conjunction of all subgoals $g_1\;\;g_2\;\;\dots\;\;g_n$ (these subgoals may contain
introduced fresh variables).
\end{itemize}
As an example consider a list concatenation relation; by a well-established tradition, the names
of relational objects are superscripted by ``$^o$'', hence \lstinline{append$^o$}:
\begin{lstlisting}[mathescape=true,language=scheme,numbers=left,numberstyle=\small,stepnumber=1,numbersep=-5pt]
(define (append$^o$ x y xy)
(conde
[(== '() x) (== y xy)]
[(fresh (h t ty)
(== `(,h . ,t) x)
(== `(,h . ,ty) xy)
(append$^o$ t y ty))]))
\end{lstlisting}
We interpret the relation ``\lstinline{append$^o$ x y xy}'' as ``the concatenation of \lstinline{x} and \lstinline{y}
equals \lstinline{xy}''. Indeed, if the list \lstinline{x} is empty, then (regardless the content of \lstinline{y}) in order for the relation to hold
the value for \lstinline{xy} should by equal to that of \lstinline{y}~--- hence line 3. Otherwise, \lstinline{x} can be decomposed into the head
\lstinline{h} and the tail \lstinline{t}~--- so we need some fresh variables. We also need the additional variable \lstinline{ty} to designate the list
that is in the relation \lstinline{append$^o$} with \lstinline{t} and \lstinline{y}. Trivial relational reasonings complete the implementation (lines 5-7).
A goal, built with the aid of the aforementioned constructs, can be run by the following primitive:
\begin{lstlisting}[mathescape=true,language=scheme]
run $n$ ($q_1\dots q_k$) $G$
\end{lstlisting}
Here $n$ is the number of requested answers (or ``*'' for all answers), $q_i$ are fresh query variables, and $G$ is a goal, which can
contain these variables.
The \lstinline{run} construct initiates the search for answers for a given goal and returns a (finite or infinite) list
of answers~--- the bindings for query variables, which represent individual solutions for that query. For example,
\begin{lstlisting}[mathescape=true,language=scheme]
run 1 (q) (append$^o$ q '(3 4) '(1 2 3 4) )
\end{lstlisting}
\noindent returns a list \lstinline{((1 2))}, which constitutes the answer for a query variable $q$. The process of constructing
the answers from internal data structures of miniKanren interpreter is called \emph{reification}~\cite{WillThesis}.
\section{Performance Evaluation}
\label{sec:evaluation}
One of our initial goals was to evaluate what performance impact would choosing OCaml as a host language makes. In addition we spent some
effort in order to implement miniKanren\xspace in an efficient, tagless manner, and, of course, the outcome of this decision also has to be
measured. For comparison we took faster-miniKanren\footnote{\url{https://github.com/webyrd/faster-miniKanren}}~--- a full-fledged
miniKanren\xspace implementation for Scheme/Racket. It turned out that faster-miniKanren implements a number of optimizations~\cite{WillThesis, Optimizations}
to speed up the search; moreover, the search order in our implementation initially was a little bit different. In order to make the comparison
fair, we additionally implemented all these optimizations and adjusted the search order to exactly coincide with
what faster-miniKanren does.
\begin{figure}[t]
\centering
\includegraphics[scale=0.4]{graph.png}
\caption{The Results of the Performance Evaluation}
\label{eval}
\end{figure}
\FloatBarrier
For the set of benchmarks we took the following problems:
\begin{itemize}
\item \textbf{pow, logo}~--- exponentiation and logarithm for integers in binary form. The concrete tests relationally computed
$3^5$ (which is 243) and $log_3 243$ (which is, conversely, 5). The implementaion was adopted from~\cite{KiselyovArithm}.
\item \textbf{quines, twines, trines}~--- self/co-evaluating program synthesis problems from~\cite{Untagged}. The
concrete tests took the first 100, 15 and 2 answers for these problems respectively.
\end{itemize}
The evaluation was performed on a desktop computer with Intel Core i7-4790K CPU @ 4.00GHz processor and 16GB of memory.
For OCanren \mbox{ocaml-4.04.0+frame_pointer+flambda} was used, for faster-miniKanren~--- Chez~Scheme~9.4.1.
All benchmarks were executed in the natively compiled mode ten times, then average user time was taken. The results of the evaluation
are shown on Figure~\ref{eval}. The whole evaluation repository with all scripts and detailed description is accessible
from \lstinline{GitHub}\footnote{\url{https://github.com/Kakadu/ocanren-perf}}.
The first conclusion, which is rather easy to derive from the results, is that the tagless approach indeed matters. Our initial
implementation did not show essential speedup in comparison even with $\mu$Kanren (and was even \emph{slower} on the logarithm
and permutations benchmarks). The situation was improved drastically, however, when we switched to the tagless version.
Yet, in comparison with faster-miniKanren, our implementation is still lagging behind. We can conclude that the optimizations
used in the Scheme/Racket version, have a different impact in the OCaml case; we save this problem for future research.
\section{Examples}
\label{sec:examples}
In this section we present some examples of a relational specification, written with the aid of our library.
Besides miniKanren\xspace combinators themselves, our implementation contains two syntax extensions~--- one
for \lstinline{fresh} construct and another for \emph{inverse-$\eta$-delay}~\cite{MicroKanren}, which is
sometimes necessary to delay recursive calls in order to prevent infinite looping. In addition, we included a
small relational library of data structures like lists, numbers, booleans, etc. This library is written
completely on the user level using techniques described in Section~\ref{sec:injection} with no utilization
of any unsafe features. The examples given below illustrate the usage of all these elements as well.
\subsection{List Concatenation and Reversing}
List concatenation and reversing are usually the first relational programs considered, and we do not wish
to deviate from this tradition. We've already considered the implementation of \lstinline{append$^o$} in
original miniKanren\xspace in Section~\ref{sec:demo}. In our case, the implementation looks familiar:
\begin{lstlisting}
let rec append$^o$ x y xy =
(x === nil ()) &&& (y === xy) |||
(fresh (h t)
(x === h
(fresh (ty)
(h
(append$^o$ t y ty)
)
)
let rec revers$^o$ a b =
conde [
(a === nil ()) &&& (b === nil ());
(fresh (h t)
(a === h
(fresh (a')
(append$^o$ a' !< h b)
(revers$^o$ t a')
)
)
]
\end{lstlisting}
Here we make use of our implementation of relational lists, which provides convenient shortcuts for
standard functional primitives:
\begin{itemize}
\item ``\lstinline{nil ()}'' corresponds to ``\lstinline{[]}'';
\item ``\lstinline{h
\item ``\lstinline{a
\end{itemize}
In our implementation the basic miniKanren\xspace primitive ``\lstinline{conde}'' is implemented as a
disjunction of a list of goals, not as a built-in syntax construct. We also make use of explicit
conjunction and disjunction infix operators instead of nested bracketed structures which, we
believe, would look too foreign here.
\subsection{Relational Sorting and Permutations}
For the next example we take list sorting; specifically, we present a sorting for lists of natural numbers
in Peano form since our library already contains built-in support for them. However, our example can be
easily extended for arbitrary (but linearly ordered) types.
List sorting can be implemented in miniKanren\xspace in a variety of ways~--- virtually any existing algorithm can
be rewritten relationally. We, however, try to be as declarative as possible to demonstrate the
advantages of the relational approach. From this standpoint, we can claim that the sorted version of an empty list is an
empty list, and the sorted version of a non-empty list is its smallest element, concatenated with a sorted
version of the list containing all its remaining elements.
The following snippet literally implements this definition:
\begin{lstlisting}
let rec sort$^o$ x y =
conde [
(x === nil ()) &&& (y === nil ());
fresh (s xs xs')
(y === s
(sort$^o$ xs xs')
(smallest$^o$ x s xs)
]
\end{lstlisting}
The meaning of the expression ``\lstinline{smallest$^o$ x s xs}'' is ``\lstinline{s} is the smallest element of a (non-empty) list \lstinline{x}, and \lstinline{xs} is the
list of all its remaining elements''. Now, \lstinline{smallest$^o$} can be implemented using a case analysis (note, ``\lstinline{l}'' here is a non-empty list):
\begin{lstlisting}
let rec smallest$^o$ l s l' =
conde [
(l === s
fresh (h t s' t' max)
(l' === max
(l === h
(minmax$^o$ h s' s max)
(smallest$^o$ t s' t')
]
\end{lstlisting}
Finally, we implement a relational minimum-maximum calculation
primitive:
\begin{lstlisting}
let minmax$^o$ a b min max =
conde [
(min === a) &&& (max === b) &&& (le$^o$ a b);
(max === a) &&& (min === b) &&& (gt$^o$ a b)
]
\end{lstlisting}
Here ``\lstinline{le$^o$}'' and ``\lstinline{gt$^o$}'' are built-in comparison goals for natural numbers in Peano form.
Having relational \lstinline{sort$^o$}, we can implement sorting for regular integer lists:
\begin{lstlisting}
let sort l =
run q (sort$^o$ (inj_nat_list l))
(fun qs -> from_nat_list ((Stream.hd qs)#prj) )
\end{lstlisting}
Here \lstinline{Stream.hd} is a function which takes a head from a lazy stream of answers,
\lstinline{inj_nat_list}~--- an injection from regular integer lists into logical lists of logical Peano numbers,
\lstinline{from_nat_list}~--- a projection from lists of Peano numbers to lists of integers.
It is interesting, that since \lstinline{sort$^o$} is
relational, it can be used to calculate a list of all \emph{permutations}
for a given list. Indeed, sorting each permutation results in the same list.
So, the problem of finding all permutations can be relationally reformulated into
the problem of finding all lists which are converted by sorting into the given one:
\begin{lstlisting}
let perm l = map (fun a -> from_nat_list a#prj)
(run q (fun q -> fresh (r)
(sort$^o$ (inj_nat_list l) r)
(sort$^o$ q r)
)
(Stream.take ~n:(fact (length l))))
\end{lstlisting}
Note, for sorting the original list we used exactly the same primitive. Note also,
we requested exactly \lstinline{fact @@ length l} answers; requesting more
would result in an infinite search for non-existing answers.
\subsection{Type Inference for STLC}
Our final example is a type inference for Simply Typed Lambda Calculus~\cite{Lambda}. The problem and
solution themselves are rather textbook examples again~\cite{TRS, WillThesis}; however, with this example
we show once again the utilization of generic programming techniques we described in Section~\ref{sec:injection}.
As a supplementary generic programming library here we used object-oriented generic transformers\footnote{\url{https://github.com/dboulytchev/GT}};
we presume, however, that any other framework could equally be used.
We first describe the type of lambda terms and their logic representation:
\begin{lstlisting}
module Term = struct
module T = struct
@type ('varname, 'self) t =
| V of 'varname
| App of 'self * 'self
| Abs of 'varname * 'self
with gmap
let fmap f g x = gmap(t) f g x
end
include T
include FMap2(T)
let v s = inj (distrib (V s))
let app x y = inj (distrib (App (x, y)))
let abs x y = inj (distrib (Abs (x, y)))
end
\end{lstlisting}
Now we have to repeat the work for the type of simple types:
\begin{lstlisting}
module Type = struct
module T = struct
@type ('a, 'b) t =
| P of 'a
| Arr of 'b * 'b
with gmap
let fmap f g x = gmap(t) f g x
end
include T
include FMap2(T)
let p s = inj (distrib (P s))
let arr x y = inj (distrib (Arr (x, y)))
end
\end{lstlisting}
Note, the ``relational'' part is trivial, boilerplate and short (and could even be generated
using a more advanced framework).
The relational type inferencer itself rather resembles the original implementation. The only
difference (besides the syntax) is that instead of data constructors we use their logic
counterparts:
\begin{lstlisting}
let rec lookup$^o$ a g t =
fresh (a' t' tl)
(g === (inj_pair a' t')
(conde [
(a' === a) &&& (t' === t);
(a' =/= a) &&& (lookup$^o$ a tl t)
])
let infer$^o$ expr typ =
let rec infer$^o$ gamma expr typ =
conde [
fresh (x)
(expr === v x)
(lookupo x gamma typ);
fresh (m n t)
(expr === app m n)
(infer$^o$ gamma m (arr t typ))
(infer$^o$ gamma n t);
fresh (x l t t')
(expr === abs x l)
(typ === arr t t')
(infer$^o$ ((inj_pair x t)
]
in
infer$^o$ (nil()) expr typ
\end{lstlisting}
\section{Streams, States, and Goals}
\label{sec:goals}
This section describes a top-level framework for our implementation. Even though it contains
nothing more than a reiteration of the original implementation~\cite{MicroKanren, CKanren}
in OCaml, we still need some notions to be properly established.
The search itself is implemented using a backtracking lazy stream monad~\cite{KiselyovBacktracking}:
\begin{lstlisting}
type $\alpha$ stream
val mplus : $\alpha$ stream -> $\alpha$ stream -> $\alpha$ stream
val bind : $\alpha$ stream -> ($\alpha$ -> $\beta$ stream) -> $\beta$ stream
\end{lstlisting}
Monadic primitives describe the shape of the search, and their implementations may
vary in concrete miniKanren\xspace versions.
An essential component of the implementation is a bundle of the following types:
\begin{lstlisting}
type env = $\dots$
type subst = $\dots$
type constraints = $\dots$
type state = env * subst * constraints
\end{lstlisting}
Type \lstinline{state} describes a point in a lazily constructed search tree: type \lstinline{env} corresponds
to an \emph{environment}, which contains some supplementary information (in particular, an environment is needed to
correctly allocate fresh variables), type \lstinline{subst} describes a substitution, which keeps current bindings
for some logical variables, and, finally, type \lstinline{constraints} represents disequality constraints,
which have to be respected. In the simplest case \lstinline{env} is just a counter for the number of the next free
variable, \lstinline{subst} is a map-like structure and \lstinline{constraints} is a list of substitutions.
\begin{comment}
This check is not needed in faster-miniKanren or any other untyped approach to implement
relational DSL. In case of OCanren it is dangerous to allow reification of a logic variable
(attributed by a type one the first state) in the context of an another state where it is attributed by another type.
Cases like this one can trigger undefined behavior. In the current implementation of logical variables created in the foreign
state are not treated as logic variables. This restriction is difficult to express in types, so the check was moved
from compile-time to runtime. This design decision is unfortunate, however this check doesn't affect perfomance in
any significant way and we don't know any useful program where such a mistake can happen.
\end{comment}
The next cornerstone element is the \emph{goal} type, which is considered as a transformer of a state into
a lazy stream of states:
\begin{lstlisting}
type goal = state -> state stream
\end{lstlisting}
In terms of the search, a goal nondeterministically performs one step of the search: for a given
node in a search tree it produces its immediate descendants. On the user level the type \lstinline{goal}
is abstract, and states are completely hidden.
Next to last, there are a number of predefined combinators:
\begin{lstlisting}
val (&&&) : goal -> goal -> goal
val (|||) : goal -> goal -> goal
val call_fresh : ($v$ -> goal) -> goal
....
\end{lstlisting}
Conjunction ``\lstinline{&&&}'' combines the results of its argument goals using \lstinline{bind},
disjunction ``\lstinline{|||}'' concatenates the results using \lstinline{mplus}, abstraction
primitive \lstinline{call_fresh} takes an abstracted goal and applies it to a freshly created
variable. Type $v$ in the last case designates the type for a fresh variable, which we leave
abstract for now. These combinators serve as the bricks for the implementation of conventional
miniKanren\xspace constructs and syntax extensions (\lstinline{conde}, \lstinline{fresh}, etc.)
Finally, there are two primitive goal constructors:
\begin{lstlisting}
val (===) : $t$ -> $t$ -> goal
val (=/=) : $t$ -> $t$ -> goal
\end{lstlisting}
The first one is a unification, while the other is a disequality constraint. Here, we again left
the type of terms $t$ abstract; it will be instantiated later.
In the implementation of miniKanren\xspace both of these goals are implemented using unification~\cite{CKanren}; this
is true for us as well. However, due to a drastic difference between the host languages, the implementation of
efficient polymorphic unification itself leads to a number of tricks with typing and data representation, which are
absent in the original version.
In this setting, the run primitive is represented by the following function:
\begin{lstlisting}
val run : goal -> state stream
\end{lstlisting}
This function creates an initial state and applies a goal to it. The states in the return stream describe
various solutions for the goal. As the stream is constructed lazily, taking elements one by one makes
the search progress.
To discover concrete answers, the state has to be queried for its contents. As a rule, a few variables
are reified in a state, i.e. their bindings in the corresponding substitution are retrieved.
Disequality constraints for free variables have to be reified additionally (e.g. represented as a list of
``forbidden'' terms). As forbidden terms can contain free variables, the constraint reification
process is recursive.
In our case, the reification is a subtle part, since, as we will see shortly, it can not be implemented in a
type-safe fragment of the language.
\section{Term Representation and Injection}
\label{sec:injection}
Polymorphic unification, considered in the previous section, works for the values of any type under the assumption that we
are capable of identifying logical variables. The latter depends on the term representation. In the original
implementation all terms are represented as a conventional S-expressions, while logical variables (in a simplest case)~--- as one-element vectors; it's an end user responsibility to respect this convention and refrain from operating
with vectors as a user data.
In our case we want to preserve both strong typing and type inference. Since we have chosen to use polymorphic
unification, it is undesirable to represent logical variables of different types differently (while technically
possible, it would compromise the lightweight approach we used so far). This means that terms with logical
variables have to be typed differently from user-defined data~--- otherwise it would be possible to use
terms in contexts where logical variables are not handled properly. At the same time we do not want term types
to be completely different from user-defined types --- for example, we would like to reuse user-defined constructors, etc.
This considerations boil down to the idea of \emph{logical representation} for a user-defined type. Informally,
a logical representation for the type $t$ is a type $\rho_t$ with a couple of conversion functions:
$$
\begin{array}{rcl}
\uparrow \;: t \to \rho_t & - & \mbox{injection}\\
\downarrow\;: \rho_t \to t & - & \mbox{projection}
\end{array}
$$
The type $\rho_t$ repeats the structure of $t$, but can contain logic variables. So, the injection is total,
while the projection is partial.
It is important to design representations as instances of some generic scheme (otherwise, miniKanren\xspace combinators
could not be properly typed). In addition it is desirable to provide a generic way to build
injection/projection pair in a uniform manner (and, even better, automatically) to lift the burden of
their implementation off the end user shoulders and improve the reliability of the solution. Finally,
the representation must provide a way to detect logic variable occurrences.
In this part we consider two approaches to implementing logical representations. The first is rather easy to
develop and implement; unfortunately, the implementation demonstrates a poor performance for a number of
important applications. In order to fix this deficiency, we develop a more elaborate technique which
nevertheless reuses some components from the previous one. In Section~\ref{sec:evaluation}
we present the results of performance evaluation for both approaches.
\subsection{Tagged Logical Values}
The first natural solution is to use tagging for representing logical representations.
We introduce the following polymorphic type $[\alpha]$\footnote{In concrete syntax called ``$\alpha\;$\lstinline{logic}''}, which
corresponds to a logical representation of the type $\alpha$:
\begin{lstlisting}
type $[\alpha]$ = Var of int | Value of $\alpha$
\end{lstlisting}
Informally speaking, any value of type $[\alpha]$ is either a value of type $\alpha$, or a free
logic variable. Note, the constructors of this type cannot be disclosed to an end user, since the only possible way to create a logic variable
should still be by using the ``\lstinline{fresh}'' construct; thus the logic type is abstract in the interface.
Now, we may redefine the signature of abstraction, unification and disequality primitives in the
following manner
\begin{lstlisting}
val call_fresh : ($[\alpha]$ -> goal) -> goal
val (===) : $[\alpha]$ -> $[\alpha]$ -> goal
val (=/=) : $[\alpha]$ -> $[\alpha]$ -> goal
\end{lstlisting}
Both unification and disequality constraint would still use the same polymorphic unification; their external, visible type,
however, is restricted to logical types only.
Apart from variables, other logical values can be obtained by injection; conversely, sometimes a logical value can be projected to
a regular one. We supply two basic functions\footnote{In concrete syntax called ``\lstinline{inj}'' and ``\lstinline{prj}''.}
for these purposes
\begin{lstlisting}[mathescape=true]
val ($\uparrow_\forall$) : $\alpha$ -> $[\alpha]$
val ($\downarrow_\forall$) : $[\alpha]$ -> $\alpha$
let ($\uparrow_\forall$) x = Value x
let ($\downarrow_\forall$) = function Value x -> x | _ -> failwith $\mbox{``not a value''}$
\end{lstlisting}
\noindent which can be used to perform a \emph{shallow} injection/projection. As expected, the injection is total, while the projection is partial.
The shallow pair works well for primitive types; to implement injection/projection for arbitrary types we exploit the
idea of representing regular types as fixed points of functors~\cite{ALaCarte}. For our purposes it is desirable to make
the functors fully polymorphic~--- thus a type, in which we can place a logical variable into arbitrary position,
can be easily manufactured. In addition this approach makes it possible to refactor the existing code to use relational
programming with only minor changes.
To illustrate this approach, we consider an iconic example~--- the list type. Let us have a conventional definition
for a regular polymorphic list in OCaml:
\begin{lstlisting}
type $\alpha$ list = Nil | Cons of $\alpha$ * $\alpha$ list
\end{lstlisting}
For this type we can only place a logical variable in the position of a list element, but not of the tail, since the tail
always has the type \lstinline{$\alpha$ list}, fixed in the definition of constructor \lstinline{Cons}. In order to create
a full-fledged logical representation, we first have to abstract the type into a fully-polymorphic functor:
\begin{lstlisting}
type ($\alpha$, $\beta$) $\mathcal L$ = Nil | Cons of $\alpha$ * $\beta$
\end{lstlisting}
Now, the original type can be expressed as
\begin{lstlisting}
type $\alpha$ list = ($\alpha$, $\alpha$ list) $\mathcal L$
\end{lstlisting}
\noindent and its logical representation~--- as
\begin{lstlisting}
type $\alpha$ list$^o$ = $[$($[\alpha]$, $\alpha$ list$^o$) $\mathcal L]$
\end{lstlisting}
Moreover, with the aid of conventional functor-specific mapping function
\begin{lstlisting}
val fmap$_{\mathcal L}$ : ($\alpha$ -> $\alpha^\prime$) -> ($\beta$ -> $\beta^\prime$) -> ($\alpha$, $\beta$) $\mathcal L$ -> ($\alpha^\prime$, $\beta^\prime$) $\mathcal L$
\end{lstlisting}
\noindent both the injection and the projection functions can be implemented:
\begin{lstlisting}[mathescape=true]
let rec $\uparrow_{\mbox{\texttt{list}}}$ l = $\uparrow_{\forall}$(fmap$_{\mathcal L}$ ($\uparrow_\forall$) $\uparrow_{\mbox{\texttt{list}}}$ l)
let rec $\downarrow_{\mbox{\texttt{list}}}$ l = fmap$_{\mathcal L}$ ($\downarrow_\forall$) $\downarrow_{\mbox{\texttt{list}}}$ ($\downarrow_\forall$ l)
\end{lstlisting}
As functor-specific mapping functions can be easily written or, better, derived automatically using a number of existing frameworks for
generic programming for OCaml, one can easily provide injection/projection pair for user-defined data types.
We now can address the problem of variable identification during polymorphic unification. As we do not know the types, we cannot discriminate logical
variables by their tags only and, thus, cannot simply use pattern matching. In our implementation we perform a variable test
as follows:
\begin{itemize}
\item in an environment, we additionally keep some unique boxed value~--- the \emph{anchor}~--- created by \lstinline{run} at the moment of initial
state generation; the anchor is inherited unchanged in all derived environments during the search session;
\item we change the logic type definition into
\begin{lstlisting}
type $[\alpha]$ = Var of int * anchor | Value of $\alpha$
\end{lstlisting}
\noindent making it possible to save in each variable the anchor, inherited from the environment, in which the variable was created;
\item inside the unification, in order to check if we are dealing with a variable, we test the conjunction of the following properties:
\begin{enumerate}
\item the scrutinee is boxed;
\item the scrutinee's tag and layout correspond to those for variables (i.e. the values, created with the constructor \lstinline{Var} of
type \lstinline{[$\alpha$]});
\item the scrutinee's anchor and the current environment's anchor have equal addresses.
\end{enumerate}
\end{itemize}
Taking into account that the state type is abstract at the user level, we guarantee that only those variables which were
created during the current run session would pass the test, since the pointer to the anchor is unique among all pointers to a boxed value
and could not be disclosed anywhere but in the variable-creation primitive.
The only thing to describe now is the implementation of the reification stage. The reification is represented by the following
function:
\begin{lstlisting}
val reify : state -> $[\alpha]$ -> $[\alpha]$
\end{lstlisting}
This function takes a state and a logic value and recursively substitutes all logic variables in that value w.r.t.
the substitution in the state until no occurrences of bound variables are left. Since in our implementation the type of a substitution is
not polymorphic, \lstinline{reify} is also implemented in an unsafe manner. However, it is easy to see that \lstinline{reify}
does not produce ill-typed terms. Indeed, all original types of variables are preserved in a substitution; unification does not
change unified terms, so all terms bound in a substitution are well-typed. Hence, \lstinline{reify} always substitutes
some subterms in a well-typed term with other terms of the corresponding types, which preserves the well-typedness.
In addition to performing substitutions, \lstinline{reify} also reifies disequality constraints. Constraint reification
attaches to each free variable in a reified term a list of reified terms, describing the disequality constraints for that
free variable. Note, disequality can be established only for equally-typed terms, which justifies the type-safety of reification.
Note also, additional care has to be taken to avoid infinite looping, since reification of answers and constraints are
mutually recursive, and the reification of a variable can be potentially invoked from itself due to a chain of disequality
constraints. In the following example
\begin{lstlisting}
let foo q =
fresh (r s)
(q === $\uparrow_{\forall}$ (Some r)) &&&
(r =/= s) &&&
(s =/= r)
\end{lstlisting}
\noindent the answer for the variable $q$ will contain a disequality constraint for the variable $r$; the reification of $r$ will in turn
lead to the reification of its own constraint, this time the variable $s$; finally, the reification of $s$ will again invoke the
reification of $r$, etc.
After the reification, the content of a logical value can be inspected via the following function:
\begin{lstlisting}
val destruct : $[\alpha]$ -> [`Var of int * $[\alpha]$ list | `Value of $\alpha$]
\end{lstlisting}
Constructor \lstinline{`Var} corresponds to a free variable with unique integer identifier and a list of terms,
representing all disequality constraints for this variable.
\subsection{Tagless Logical Values and Type Bookkeeping}
The solution presented in the previous subsection suffers from the following deficiency: in order to perform unification,
we inject terms into the logic domain, making them as twice as large. As a result, this implementation loses to the original one in
terms of performance in many important applications, which compromises the very idea of using OCaml as a host language.
Here we develop an advanced version, which eliminates this penalty. As a first step, let's try to eliminate the tagging with
a drastic measure:
\begin{lstlisting}
type $[\alpha]$ = $\alpha$
\end{lstlisting}
What consequences would this have? Of course, we would not be able to create logical variables in a conventional way. However,
we still could have a separate type of variables
\begin{lstlisting}
type var = Var of int * anchor
\end{lstlisting}
\noindent and use \emph{the same} variable test procedure. As the type $[\alpha]$ is abstract, this modification does not change the interface.
As we reuse the variable test, polymorphic unification can continue to work \emph{almost} correctly. The problem is that
now it can introduce the occurrences of free logic variables in non-logical, tagless, data structures. These free logic variables
do not get in the way of unification itself (since it can handle them properly, thanks to the variable test), but they can not
be disclosed to the outer world as is.
Our idea is to use this generally unsound representation for all internal actions, and perform tagging only during the reification
stage. However, this scenario raises the following question: what would the type of \lstinline{reify} be? It can not be simply
\begin{lstlisting}
val reify : state -> $[\alpha]$ -> $[\alpha]$
\end{lstlisting}
anymore since $[\alpha]$ now equals $\alpha$. We \emph{want}, however, it be something like
\begin{lstlisting}
val reify : state -> $[\alpha]$ -> $(\mbox{``tagged'' } [\alpha])$
\end{lstlisting}
If $\alpha$ is not a parametric type, we can simply test if the value is a variable, and if yes, tag it with the constructor \lstinline{Var};
we tag it with \lstinline{Value} otherwise, and we're done. This trick, however, would not work for parametric types. Consider, for example,
the reification of a value of type \lstinline{$[[$int$]$ list$]$}. The (hypothetical) approach being described would return a value of
type \lstinline{$(\mbox{``tagged'' }[[$int$]$ list$])$}, i.e. tagged only on the top level; we need to repeat the procedure
recursively. In other words, we need the following (meta) type for the reification primitive:
\begin{lstlisting}
val reify : state -> $[\alpha]$ -> $\mbox{(``tagged''} [\beta])$
\end{lstlisting}
\noindent where $\beta$ is the result of tagging $\alpha$.
These considerations can be boiled down to the following concrete implementation.
First, we roll back to the initial definition of $[\alpha]$~--- it will play the role of our ``tagged'' type.
We introduce a new, two-parameter type\footnote{In concrete syntax called ``$(\alpha,\;\beta)\;$\lstinline{injected}''.}
\begin{lstlisting}
type $\{\alpha,\;\beta\}$ = $\alpha$
\end{lstlisting}
Of course, this type is kept abstract at the end-user level. Informally speaking, the type $\{\alpha,\;\beta\}$ designates the
injection of a tagless type $\alpha$ into a tagged type $\beta$; the value itself is kept in the tagless form, but
the tagged type can be used during the reify stage as a constraint, which would allow us to reify a tagless
representation only to a feasible tagged one. In other words, we record the injection steps using the second
type parameter of the type ``\{,\}'', performing the bookkeeping on the type level rather than on the value level.
We introduce the following primitives for the type $\{\alpha,\;\beta\}$:
\begin{lstlisting}
val lift : $\alpha$ -> $\{\alpha,\;\alpha\}$
val inj : $\{\alpha,\;\beta\}$ -> $\{\alpha,\;[\beta]\}$
let lift x = x
let inj x = x
\end{lstlisting}
The function \lstinline{lift} puts a value into the ``bookkeeping injection'' domain for the first time, while
\lstinline{inj} plays the role of the injection itself. Their composition is analogous to what was
called ``$\uparrow_\forall$'' in the previous implementation:
\begin{lstlisting}
val $\uparrow_\forall$ : $\alpha$ -> $\{\alpha,\;[\alpha]\}$
let $\uparrow_\forall$ x = inj (lift x)
\end{lstlisting}
In order to deal with parametric types, we can again utilize generic programming. To handle the types with
one parameter, we introduce the following functor:
\begin{lstlisting}
module FMap (T : sig type $\alpha$ t val fmap : ($\alpha$ -> $\beta$) -> $\alpha$ t -> $\beta$ t end) :
sig
val distrib : $\{\alpha,\;\beta\}$ T.t -> $\{\alpha$ T.t, $\beta$ T.t$\}$
end =
struct
let distrib x = x
end
\end{lstlisting}
Note, that we do not use the function ``\lstinline{T.fmap}'' in the implementation; however, first, we need an inhabitant of the
corresponding type to make sure we are indeed dealing with a functor, and next, we actually will use it in the
implementation of type-specific reification, see below.
In order to handle two-, three-, etc. parameter types we need higher-kinded polymorphism, which is
not supported in a direct form in OCaml. So, unfortunately, we need to introduce separate
functors for the types with two-, three- etc. parameters; existing works on higher-kinded
polymorphism in OCaml~\cite{HKinded} require the similar scaffolding to be erected as a bootstrap step.
Given the functor(s) of the described shape, we can implement logic representations for
all type's constructors. For example, for standard type \lstinline{$\alpha$ option} with two constructors
\lstinline{None} and \lstinline{Some} the implementation looks like as follows:
\begin{lstlisting}
module FOption = FMap (struct
type $\alpha$ t = $\alpha$ option
let fmap = fmap$_{\mbox{\texttt{option}}}$
end)
val some : $\{\alpha, \beta\}$ -> $\{\alpha\mbox{\texttt{ option}},\;\beta\mbox{\texttt{ option}}\}$
val none : unit -> $\{\alpha\mbox{\texttt{ option}},\;\beta\mbox{\texttt{ option}}\}$
let some x = inj (FOption.distrib (Some x))
let none () = inj (FOption.distrib None)
\end{lstlisting}
In other words, we can in a very systematic manner define logic representations for all constructors
of types of interest. These representations can be used in the relational code, providing a well-bookkept
typing~--- for each logical type we would be able to reconstruct its original, tagless preimage.
With the new implementation, the types of basic goal constructors have to be adjusted:
\begin{lstlisting}
val (===) : $\{\alpha,\;[\beta]\}$ -> $\{\alpha,\;[\beta]\}$ -> goal
val (=/=) : $\{\alpha,\;[\beta]\}$ -> $\{\alpha,\;[\beta]\}$ -> goal
\end{lstlisting}
As always, we require both arguments of unification and disequality constraint to be of the same type; in addition
we require the injected part of the type to be logical.
During the reification stage the bindings for the top-level variables, reconstructed using the final
substitution, have to be properly tagged. This process is implemented in a datatype-generic manner as well:
first, we have reifiers for all primitive types:
\begin{lstlisting}
val reify$_{\mbox{\texttt{int}}}$ : helper -> $\{$int,$[$int$]\}$ -> $[$int$]$
val reify$_{\mbox{\texttt{string}}}$ : helper -> $\{$string,$[$string$]\}$ -> $[$string$]$
...
\end{lstlisting}
and, then, we add the reifier to the output signature in all \lstinline{FMap}-like functors:
\begin{lstlisting}
val reify: (helper -> $\{\alpha,\;\beta\}$ -> $\beta$) -> helper -> $\{\alpha$ T.t, $[\beta$ T.t$]$ as $\gamma\}$ -> $\gamma$
\end{lstlisting}
Note, since now \lstinline{reify} is a type-specific and, hence, constructed at the user-level, we refrain from passing
it a state (which is inaccessible on the user level). Instead, we wrap all state-specific functionality in
an abstract \lstinline{helper} data type, which encapsulates all state-dependent functionality needed for \lstinline{reify}
to work properly.
\section{Introduction}
\label{intro}
Relational programming~\cite{TRS} is an attractive technique, based on the idea
of constructing programs as relations. As a result, relational programs can be
``queried'' in various ``directions'', making it possible, for example, to simulate
reversed execution. Apart from being interesting from a purely theoretical standpoint,
this approach may have a practical value: some problems look much simpler
when considered as queries to some relational specification~\cite{WillThesis}. There are a
number of appealing examples confirming this observation: a type checker
for simply typed lambda calculus (and, at the same time, a type inferencer and solver
for the inhabitation problem), an interpreter (capable of generating ``quines''~---
programs producing themselves as a result)~\cite{Untagged}, list sorting (capable of
producing all permutations), etc.
Many logic programming languages, such as Prolog, Mercury~\cite{MercuryFirstPaper},
or Curry~\cite{CurryFirstPaper} to some extent
can be considered relational. We have chosen miniKanren\xspace\footnote{\url{http://minikanren.org}}
as a model language, because it was specifically designed as a relational DSL, embedded in Scheme/Racket.
Being rather a minimalistic language, which can be implemented with just a few data structures and
combinators~\cite{MicroKanren, MuKanrenNew}, miniKanren\xspace found its way into dozens of host languages, including Scala, Haskell and Standard ML.
The paradigm behind miniKanren\xspace can be described as ``lightweight logic programming''\footnote{An in-depth comparison of miniKanren\xspace
and Prolog can be found here: \url{http://minikanren.org/minikanren-and-prolog.html}}.
This paper addresses the problem of embedding miniKanren\xspace into OCaml\footnote{\url{http://ocaml.org}}~--- a statically-typed functional language with
a rich type system. A statically-typed implementation would bring us a number of benefits. First, as always,
we expect typing to provide a certain level of correctness guarantees, ruling out some pathological programs, which
otherwise would provide pathological results. In the context of relational programming, however, typing would additionally
help us to interpret the results of queries. Often an answer to a relational query contains a number of
free variables, which are supposed to take arbitrary values. In the typed case these variables become typed,
facilitating the understanding of the answers, especially those with multiple free variables. Next, a number of miniKanren\xspace
applications require additional constraints to be implemented. In the untyped setting, when everything can be anything,
some symbols or data structures tend to percolate into undesirable contexts~\cite{Untagged}. In order to prevent this from happening, some
auxiliary constraints (``\lstinline{absent$^o$}'', ``\lstinline{symbol$^o$}'', etc.) were introduced. These constraints play a role
of a weak dynamic type system, cutting undesirable answers out at runtime. Conversely, in a typed language this work can be
entrusted to the type checker (at the price of enforcing an end user to write properly typed specifications), not only improving the
performance of the system but also reducing the number of constraints which have to be implemented. Finally, it is rather natural
to expect better performance of a statically-typed implementation.
We present an implementation of a set of relational combinators and syntax extensions for
OCaml\footnote{The source code of our implementation is accessible from \url{https://github.com/dboulytchev/OCanren}.},
which, technically speaking, corresponds to $\mu$Kanren~\cite{MicroKanren} with disequality
constraints~\cite{CKanren}. The contribution of our work is as follows:
\begin{enumerate}
\item Our embedding allows an end user to enjoy strong static typing and type inference in relational
specifications; in particular, all type errors are reported at compile time and the types for
all logical variables are inferred from the context.
\item Our implementation is based on the \emph{polymorphic unification}, which, like the polymorphic comparison,
can be used for arbitrary types. The implementation of polymorphic unification uses unsafe features and
relies on the intrinsic knowledge of the runtime representation of values; we show, however, that this does not
compromise type safety.
\item We describe a uniform and scalable pattern for using types for relational programming, which
helps in converting typed data to and from the relational domain. With this pattern, only one
generic feature (``\lstinline{Functor}'') is needed, and thus virtually any generic
framework for OCaml can be used. Although being rather a pragmatic observation, this pattern, we
believe, would lead to a more regular and easy way to maintain relational specifications.
\item We provide a simplified way to integrate relational and functional code. Our approach utilizes
a well-known pattern~\cite{Unparsing, DoWeNeed} for variadic function implementation and makes it
possible to hide the reification of the answers phase from an end user.
\end{enumerate}
The rest of the paper is organized as follows: in the next section we provide a short overview of the related
works. Then we briefly introduce miniKanren\xspace in
its original form to establish some notions; we do not intend to describe the language in its full bloom (interested readers can
refer to~\cite{TRS}). In Section~\ref{sec:goals} we describe some basic constructs behind a miniKanren\xspace implementation, this time
in OCaml. In Section~\ref{sec:unification} we discuss polymorphic unification, and show that unification with
triangular substitution respects typing. Then we present our approach to handle user-defined types by injecting them
into the logic domain, and describe a convenient generic programming pattern, which can be used to implement the conversions from/to logic
domain. We also consider a simple approach and a more elaborate and efficient tagless variant (see Section~\ref{sec:injection}).
Section~\ref{sec:reification} describes top-level primitives and addresses the problem of relational and functional code integration.
Then, in Section~\ref{sec:examples} we present a set of relational examples, written with the aid of our
library. Section~\ref{sec:evaluation} contains the results of a performance evaluation and a comparison of our implementations
with existing implementation for Scheme. The final section concludes.
The authors would like to express a deep gratitude to the anonymous rewievers for their numerous constructive comments, Michael Ballantyne, Greg Rosenblatt,
and the other attendees of the miniKanren Google Hangouts, who helped the authors in understanding the subtleties of the original miniKanren
implementation, Daniel Friedman for his remarks on the preliminary version of the paper, and William Byrd for all his help and support, which cannot be
overestimated.
\section{Polymorphic Unification}
\label{sec:unification}
We consider it rather natural to employ polymorphic unification in a language already equipped
with polymorphic comparison~--- a convenient, but somewhat disputable\footnote{See, for example,
\url{https://blogs.janestreet.com/the-perils-of-polymorphic-compare}} feature. Like polymorphic comparison,
polymorphic unification performs a traversal of values, exploiting intrinsic knowledge of their runtime
representation. The undeniable benefits of this solution are, first, that in order to perform unification
for user types no ``boilerplate'' code is needed, and, second, that this approach seems to deliver the
most efficient implementation. On the other hand, all the pitfalls of polymorphic comparison are inherited as
well; in particular, polymorphic unification loops for cyclic data structures and does not work for functional
values. Since we generally do not expect any reasonable outcome in these cases, the only remaining problem is that
the compiler is incapable of providing any assistance in identifying and avoiding these cases. Another drawback is that
the implementation of polymorphic unification relies on the runtime representation of values and has to be fixed
every time the representation changes. Finally, as it is written in an unsafe manner using the \lstinline{Obj} interface,
it has to be carefully developed and tested.
An important difference between polymorphic comparison and unification is that the former only inspects its operands,
while the results of unification are recorded in a substitution (a mapping from logical variables to terms), which
later is used to reify answers. So, generally speaking, we have to show that no ill-typed
terms are constructed as a result. Overall, this property seems to be maintained vacuously, since the very
nature of (syntactic) unification is to detect whether some things can be considered equal. Nevertheless there are
different type systems and different unification implementations; in addition \emph{equal things} can be
\emph{differently typed}, so we provide here a correctness justification for a well-defined abstract case, and will
reuse this conclusion for various concrete cases.
First, we consider three alphabets:
$$
\begin{array}{rcl}
\tau,\dots&-&\mbox{types}\\
x^\tau,\dots&-&\mbox{typed logic variables}\\
C_k^{\tau_1\times\tau_2\times\dots\times\tau_k\to\tau} (k\ge 0),\dots&-&\mbox{typed constructors}
\end{array}
$$
The set of all well-formed typed terms is defined by mutual induction for all types:
$$
t^\tau=x^\tau\mid C_k^{\tau_1\times\tau_2\times\dots\times\tau_k\to\tau}(t^{\tau_1},\,t^{\tau_2},\,\dots,\,t^{\tau_k})
$$
For simplicity from now on we abbreviate the notation $C_k^{\tau_1\times\tau_2\times\dots\times\tau_k\to\tau}(t^{\tau_1},\,t^{\tau_2},\,\dots,\,t^{\tau_k})$ into
$C_k^\tau(t^{\tau_1},\,t^{\tau_2},\,\dots,\,t^{\tau_k})$, keeping in mind that for any concrete constructor and for all its occurrences
in arbitrary terms all its subterms in corresponding positions agree in types.
\begin{comment}
We need also to define the notion of a subterm $t^\tau[p]$ of a term $t^\tau$ at given position $p$:
$$
\begin{array}{rcl}
p=\epsilon\mid\{1, 2, 3,\dots\}\bullet p&-&\mbox{the set of positions}\\
t^\tau[\epsilon]=t^\tau&-&\mbox{base case}\\
C_k^\tau(t_1^{\tau_1},\,t_2^{\tau_2},\dots,\,t_k^{\tau_k})[i\bullet p]=t_i^{\tau_i}[p], 1\le i \le k&-&\mbox{inductive case}
\end{array}
$$
\end{comment}
In this formulation we do not consider any structure over the set of types besides type equality, and we assume all terms are explicitly
attributed to their types at runtime. We employ this property to implement a unification algorithm in regular OCaml, using some
representation for terms and types:
\begin{lstlisting}[mathescape=true]
val unify : term -> term -> subst option -> subst option
\end{lstlisting}
\noindent where ``\lstinline{term}'' stands for the type representing typed terms, and ``\lstinline{subst}'' stands for the type of
substitution (a partial mapping from logic variables to terms). Unification can fail (hence ``\lstinline{option}'' in the result type),
is performed in the context of existing substitution (hence ``\lstinline{subst}'' in the third argument) and can be
chained (hence ``\lstinline{option}'' in the third argument).
We use exactly the same unification algorithm with triangular substitution as in the reference implementation~\cite{MicroKanren}. We
omit here some not-so-important details (like ``occurs check''), which are kept in the actual implementaion, and refrain from discussing
the nature and properties of the algorithm
itself (an excellent description, including a certified correctness proof, can be found in~\cite{Kumar}).
The following snippet presents the implementation:
\begin{lstlisting}[mathescape=true,numbers=left,numberstyle=\small,stepnumber=1,numbersep=-5pt]
let rec unify $t_1^\tau$ $t_2^\tau$ $subst$ =
let rec walk $s$ $t^\tau$ =
match $t^\tau$ with
| $x^\tau$ when $x^\tau\in dom(s)$ -> $\;\;$walk $s$ $(s\;\;x^\tau)$
| _ -> $t^\tau$
in
match $subst$ with
| None -> None
| Some $s$ ->
match walk $s$ $t_1^\tau$, walk $s$ $t_2^\tau$ with
| $x_1^\tau$, $x_2^\tau$ when $x_1^\tau$ = $x_2^\tau$ -> $subst$
| $x_1^\tau$, $q_2^\tau$ -> Some ($s\;[x_1^\tau \gets q_2^\tau]$)
| $q_1^\tau$, $x_2^\tau$ -> Some ($s\;[x_2^\tau \gets q_1^\tau]$)
| $C^\tau(t_1^{\tau_1},\dots,t_k^{\tau_k})$, $C^\tau(p_1^{\tau_1},\dots,p_k^{\tau_k})$ ->
unify $t_k^{\tau_k}$ $p_k^{\tau_k}$(.. (unify $t_1^{\tau_1}$ $p_1^{\tau_1}$ $subst$)$..$)
| $\_$, $\_$ -> None
\end{lstlisting}
We remind the reader that all superscripts correspond to type attributes, which we consider here as
parts of values being manipulated. For example, line 1 means that we apply \lstinline{unify}
to terms $t_1$ and $t_2$, and expect their types to be equal $\tau$. We assume that
at the top level unification is always applied to some terms of the same type and that any
substitution can only be acquired from the empty one by a sequence of unifications.
We are going to show that under these assumptions all type attributes are superfluous~--- they
do not affect the execution of \lstinline{unify} and can be removed. Note that the only place where we
were incapable of providing an explicit type attribute was in line 4, where the result of
substitution application was returned. However, we can prove by induction that any substitution
respects the following property: if a substitution $s$ is defined for a variable $x^\tau$,
then $s\;\;x^\tau$ is attributed with the type $\tau$ (and, consequently, \lstinline{walk $s$ $t^\tau$} always
returns a term of type $\tau$).
Indeed, this property vacuously holds for the empty substitution. Let $s$ be some substitution, for which the
property holds. In line 11 we return an unchanged substitution; in line 10 we perform two calls~---
\lstinline{walk $s$ $t_1^\tau$} and \lstinline{walk $s$ $t_2^\tau$} and match their results. However,
by our induction hypothesis these results are again attributed to the type $\tau$, which justifies the
pattern matching. In line 11 we return the substitution unchanged, in lines 12 and 13 we extend the
existing substitution but preserve the property of interest. Finally, in line 15 we chain a few
applications of \lstinline{unify}; note that, again, all these calls are performed for terms of equal
types, starting from a substitution possessing the property of interest. A simple induction on the
chain length completes the proof.
So, type attributes are inessential~--- they are never analyzed and never restrict pattern matching; hence,
they can be erased completely.
We can notice now that for the representation of terms we can use OCaml's native runtime representation.
It can not be done, however, using regular OCaml~--- we have to utilize the low-level, unsafe interface \lstinline{Obj}.
Note also, we need some way to identify the occurrences of logical variables inside the terms (in the original miniKanren\xspace
implementation the ambiguity between variable and non-variable datum representation is resolved by a convention~--- a luxury
we cannot afford). We postpone the discussion on this subject until the next section.
We call our implementation \emph{polymorphic}, since at the top level it is defined as
\begin{lstlisting}
val unify : $\alpha$ -> $\alpha$ -> subst option -> subst option
\end{lstlisting}
The type of substitution is not polymorphic, which means that the compiler completely loses the track
of types of values stored in a substitution. These types are recovered later during the reification-of-answers phase (see Section~\ref{sec:reification}).
Outside the unification the compiler maintains typing, which means that all terms, subterms, and variables agree in their types
in all contexts.
\section{Reification and Top-Level Primitives}
\label{sec:reification}
In Section~\ref{sec:goals} we presented a top-level function \lstinline{run}, which
runs a goal and returns a stream of states. To acquire answers to the query,
represented by that goal, its free variables have to be reified in these states, and
we described the reification primitives in Section~\ref{sec:injection}. However,
the states keep answers in an untyped form, and the types of answers are
recovered solely on the basis of the types of variables being reified. So, the
type safety of the reification critically depends on the requirement to
reify each variable only in those states, which are descendants (w.r.t. the search tree)
of the state, in which that variable was created. In this section we describe a set of
top-level primitives, which enforce this requirement.
We provide a set of top-level combinators, which should be used to surround relational code
and perform reification in a transparent manner only in correct states.
We reimplement the top-level primitive \lstinline{run} to take three
arguments. The exact type of \lstinline{run} is rather complex and non-instructive,
so we prefer to describe the typical form of its application:
\begin{lstlisting}[mathescape=true]
run $\overline{n}$ (fun $l_1\dots l_n$ -> $\;\;G$) (fun $a_1\dots a_n$ -> $\;\;H$)
\end{lstlisting}
Here $\overline{n}$ stands for a \emph{numeral}, which describes the number of
parameters for two other arguments of \lstinline{run}, \mbox{$l_1\dots l_n$}~---
free logical variables, $G$~--- a goal (which can make use of \mbox{$l_1\dots l_n$}),
\mbox{$a_1\dots a_n$}~--- reified answers for \mbox{$l_1\dots l_n$}, respectively, and,
finally, $H$~--- a \emph{handler} (which can make use of \mbox{$a_1\dots a_n$}).
The types of \mbox{$l_1\dots l_n$} are inferred from $G$ and always have a form
\begin{lstlisting}
$\{\alpha,\;[\beta]\}$
\end{lstlisting}
\noindent since the types of variables can be constrained only in unification or disequality constraints.
The types of \mbox{$a_1\dots a_n$} are inferred from the types of \mbox{$l_1\dots l_n$} and
have the form
\begin{lstlisting}
$(\alpha,\;\beta)$ reified stream
\end{lstlisting}
\noindent where the type \lstinline{reified}, in turn, is
\begin{lstlisting}
type ($\alpha$, $\beta$) reified = $<\;$prj : $\alpha$; reify : (helper -> $\{\alpha,\;\beta\}$ -> $\beta$) -> $\beta>$
\end{lstlisting}
Two methods of this type can be used to perform two different styles of reification: first, a value without
free variables can be returned as is (using the method \lstinline{prj} which checks that in the value of
interest no free variables occur, and raises an exception otherwise). If the value contains some free
variables, it has to be properly injected into the logic domain~--- this is what \lstinline{reify} stands
for. It takes as an argument a type-specific tagging function, constructed using generic
primitives described in the previous section.
In other words a user-defined handler takes streams of reified answers for all variables supplied to the top-level
goal. All streams $a_i$ contain coherent elements, so they all have the same length and $n$-th elements of all
streams correspond to the $n$-th answer, produced by the goal $G$.
There are a few predefined numerals for one, two, etc. arguments (called, traditionally,
\lstinline{q}, \lstinline{qr}, \lstinline{qrs} etc.), and a successor function, which
can be applied to existing numeral to increment the number of expected arguments. The
implementation technique generally follows~\cite{Unparsing, DoWeNeed}.
Thus, the search and reification are tightly coupled; it is simply impossible to perform the reification
for arbitrarily-taken state and variable. This solution both guarantees the type safety and frees an end
user from the necessity to call reification primitives manually.
\section{Related Works}
\label{sec:relworks}
There is a predictable difficulty in implementing miniKanren\xspace for a strongly typed language.
Designed in the metaprogramming-friendly and dynamically typed realm of Scheme/Racket, the original
miniKanren\xspace implementation pays very little attention to what has a significant importance in (specifically)
ML or Haskell. In particular, one of the capstone constructs of miniKanren\xspace~--- unification~--- has to work for
different data structures, which may have types different beyond parametricity.
There are a few ways to overcome this problem. The first one is simply to follow the untyped paradigm and
provide unification for some concrete type, rich enough to represent any reasonable data structures.
Some Haskell miniKanren\xspace libraries\footnote{\url{https://github.com/JaimieMurdock/HK}, \url{https://github.com/rntz/ukanren}}
as well as the previous OCaml implementation\footnote{\url{https://github.com/lightyang/minikanren-ocaml}} take this approach.
As a result, the original implementation can be retold with all its elegance; the relational specifications, however,
become weakly typed. A similar approach was taken in early works on embedding Prolog into Haskell~\cite{PrologInHaskell}.
Another approach is to utilize \emph{ad hoc} polymorphism and provide a type-specific unification for each ``interesting'' type.
Some miniKanren\xspace implementations, such as Molog\footnote{\url{https://github.com/acfoltzer/Molog}} and
MiniKanrenT\footnote{\url{https://github.com/jvranish/MiniKanrenT}}, both for Haskell, can be mentioned as examples.
While preserving strong typing, this approach requires a lot of ``boilerplate''
code to be written, so some automation --- for example, using
Template Haskell~\cite{SheardTMH}~---
is desirable. In~\cite{TypedLogicalVariables} a separate type class was introduced to both perform the unification
and detect free logical variables in end-user data structures. The requirement for end user to provide a way to represent
logical variables in custom data structures looks superfluous for us since these logical variables would require proper
handling in the rest of the code outside the logical programming subsystem.
There is, actually, another potential approach, but we do not know if anybody has tried
it: implementing unification for a generic representation of types as sum-of-products and fixpoints of
functors~\cite{InstantGenerics, ALaCarte}. Thus, unification would work for any type for which a representation
is provided. We believe that implementing this representation would require less boilerplate code to be written.
As follows from this exposition, a typed embedding of miniKanren\xspace in OCaml can be done with
a combination of datatype-generic programming~\cite{DGP} and \emph{ad hoc} polymorphism. There are
a number of generic frameworks for OCaml (for example,~\cite{Deriving}). On the other hand, the support
for \emph{ad hoc} polymorphism in OCaml is weak; there is nothing comparable in power to Haskell
type classes, and even though sometimes the object-oriented layer of the language can be used to mimic
desirable behavior, the result, as a rule, is far from satisfactory. Existing proposals for \emph{ad hoc} polymorphism (for example,
modular implicits~\cite{Implicits}) require patching the compiler, which we want to avoid. Therefore, we
take a different approach, implementing polymorphic unification once and for all logical types~--- a purely \emph{ad hoc}
approach, since the features which would provide a less \emph{ad hoc} solution are not yet well integrated into the language. To deal
with user-defined types in the relational subsystem, we propose to use their logical representations (see Section~\ref{sec:injection}),
which free an end user from the burden of maintaining logical variables, and we use generic programming to build conversions from and to logical
representations in a systematic manner.
|
1,116,691,497,786 | arxiv | \section{Introduction}
The NUclei of GAlaxies (NUGA) project \citep[][]{garcia03} is an IRAM Plateau
de Bure Interferometer (PdBI) survey of nearby active galaxies
to map the distribution and dynamics of molecular gas in the inner 1\,kpc at
high spatial resolution ($\sim\,0\,\farcs5 - 1^{\prime\prime}$,
corresponding to $\sim50-100\,{\rm pc}$), and to study the mechanisms for
gas fueling of low-luminosity active galactic nuclei (AGNs).
Most galaxies possess central supermassive black holes (SMBHs), and gas accretion
onto these black holes is the phenomenon usually invoked to explain nuclear
activity in galaxies. However, even if most galaxies host black
holes, the existence of nuclear activity is far from universal. It is not
clear whether the main limiting factor is the global gas mass available for
fueling the AGN or the mechanisms for efficiently removing the angular
momentum of the gas.
The main and non-trivial problem linked to the fueling of AGN is the
removal of angular momentum from the disk gas \citep[e.g.,][and
references therein]{jogee06}, a process which
can be accomplished through non-asymmetric perturbations.
These can be perturbations of external origin, such as galaxy collisions,
mergers, and mass accretion \citep{heckman86}, or of internal origin due to
density waves, such as spirals or bars, and their gravity torques
\citep[e.g.][]{sakamoto,combes01}.
In addition to primary bars, fueling processes can be associated with more
localized phenomena, such as nested nuclear bars \citep[e.g.][]{friedli},
lopsidedness or $m = 1$ perturbations \citep[e.g.][]{shu,garcia00}, or warped
nuclear disks \citep[e.g.][]{schinnerer00a,schinnerer00b}.
Since molecular gas is the predominant phase of the interstellar medium (ISM)
in the inner kiloparsec of spiral galaxies, the study of its morphology and
dynamics represents an optimal possibility for investigating AGN fueling
mechanisms and their link with circumnuclear star formation. In order to
do this, high-resolution maps of molecular gas are needed. Previous
CO single dish \citep[][]{heckman89,young,braine93,casoli96,vila98}
and interferometric \citep[][]{sakamoto,regan01,helfer03} surveys have mapped
the gas in galaxies with relatively low spatial resolution ($4^{\prime\prime}
-7^{\prime\prime}$). Moreover, the majority of these surveys only included
small numbers of AGN in their samples. This paper, dedicated to the galaxy
NGC\,3147, is the tenth of a series where results obtained for the galaxies
in the NUGA sample are described on a case-by-case basis.
\begin{table}
\caption[]{Fundamental parameters for NGC\,3147.}
\begin{center}
\begin{tabular}{lll}
\hline
\hline
Parameter & Value & References$^{\mathrm{a}}$ \\
\hline
RA (J2000) & 10$^h$16$^m$53.6$^s$ & (1) \\
DEC (J2000) & 73$^{\circ}$24$^{\prime}$03$^{\prime\prime}$ & (1) \\
$V_{\rm hel}$ & 2813 km\,s$^{-1}$ & (1) \\
RC3 Type & SA(rs)bc & (1) \\
T Hubble Type & 3.88 & (2) \\
Inclination & 29.5$^{\circ}$ & (2) \\
Position Angle & 150$^{\circ}$ & (2) \\
Distance & 40.9\,Mpc ($1^{\prime\prime} = 198\,{\rm pc}$) & (1) \\
$L_{B}$ & $4.3 \times 10^{10}\,L_{\odot}$ & (3) \\
$M_{\rm H\,I}$ & $8.3 \times 10^{8}\,M_{\odot}$ & (3) \\
$M_{\rm dust}$(60 and 100\,$\mu$m)& $1.7 \times 10^{6}\,M_{\odot}$ & (3) \\
$L_{\rm FIR}$ & $3.4 \times 10^{10}\,L_{\odot}$ & (4) \\
\hline
\hline
\end{tabular}
\label{table1}
\end{center}
\begin{list}{}{}
\item[$^{\mathrm{a}}$] (1) NASA/IPAC Extragalactic Database (NED);
(2) Lyon Extragalactic Database (LEDA); (3) \citet{bettoni}; (4) {\it IRAS} Catalog.
\end{list}
\end{table}
NGC\,3147 ($D = 40.9\,{\rm Mpc}$, $1^{\prime\prime} = 198\,{\rm pc}$,
$H_0 = 75\,{\rm km\,s^{-1}\,Mpc}^{-1}$)
is an isolated \citep{bettoni} Seyfert 2 galaxy \citep{ho} of Hubble
morphological type SA(rs)bc. Both \textit{ROSAT} \citep{roberts} and
\textit{Chandra} \citep{terashima03} observations have shown that this galaxy
possesses a pointlike nuclear X-ray source,
which is presumed to be the location of the
AGN. The 0.3--10\,keV \textit{Chandra} image reveals a
bright, compact source surrounded by very faint, soft, and diffuse emission. The
2--10\,keV core is clearly detected and confined to a region of
$2^{\prime\prime}$ in radius.
Interferometric observations of the continuum emission at cm wavelengths
with MERLIN and the VLBA have also shown a pointlike nonthermal continuum
source at the position of the nucleus of NGC\,3147,
coincident with the X-ray source \citep[][]{ulvestad,krips06,krips07a}.
Very recent optical and X-ray observations of the
nuclear region of this galaxy suggest
that NGC\,3147 is the first ``true'' Seyfert 2 in the sense that it
intrinsically lacks a broad-line region \citep{bianchi08}.
Table \ref{table1} reports the fundamental characteristics of
NGC\,3147.
The structure of this paper is as follows. In Sect. 2, we describe our new
observations of NGC\,3147 and the literature data with which we compare them.
In Sects. 3 and 4, we present the observational results, both single dish
and interferometric, describing the principal properties of the molecular gas
including its morphology, its kinematics, and its
excitation. A comparison between the CO observations and those obtained at
other wavelengths is given in Sect. 5. In Sect. 6, we describe the
computation of the gravity torques exerted on the gas by a weak stellar bar
that we have identified using a near-infrared (NIR) image. In Sect. 7, we discuss
our principal observational and theoretical results, which are summarized
in Sect. 8.
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth,angle=-90]{9545fig1n.ps}
\caption{$^{12}$CO(1-0) velocity channel maps observed with the IRAM PdBI+30\,m
in the nucleus of NGC\,3147, with a spatial resolution of 1\farcs9
$\times$ 1\farcs6 (HPBW). The center of the maps, given in Table
\ref{table1}, is $\alpha_{2000}$ = 10$^h$16$^m$53.6$^s$, $\delta_{2000}$ =
73$^{\circ}$24$^{\prime}$03$^{\prime\prime}$. Velocity channels range from
$\Delta V = -230\,{\rm km\,s^{-1}}$ to $+220\,{\rm km\,s^{-1}}$ in steps of
$10\,{\rm km\,s^{-1}}$ relative to $V_{\rm hel} = 2813\,{\rm km\,s^{-1}}$.
The contours begin at $5\,{\rm mJy\,beam^{-1}}$, their spacing is $5\,{\rm
mJy\,beam^{-1}}$, and their maximum is $50\,{\rm mJy\,beam^{-1}}$.}
\label{channel}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth,angle=-90]{9545fig2n.ps}
\caption{Same as Fig. \ref{channel} but for the $^{12}$CO(2-1) line,
with a spatial resolution of 1\farcs6 $\times$ 1\farcs4.
The velocity range is the same as for Fig. \ref{channel}.
The contours begin at $10\,{\rm mJy\,beam^{-1}}$, their spacing is $5\,{\rm
mJy\,beam^{-1}}$, and their maximum is $45\,{\rm mJy\,beam^{-1}}$.}
\label{channel21}
\end{figure*}
\section{Observations}
\subsection{IRAM interferometric CO observations}
We observed the $J = 1-0$ and $J = 2-1$ lines of ${}^{12}$CO in NGC\,3147
using the IRAM PdBI in October 2004, with the array deployed in its ABCD
configurations. The six 15\,m antennae were equipped with dual-band SIS
receivers yielding SSB receiver temperatures between 40 and 50\,K at both
frequencies. The precipitable water vapor ranged from 4 to 10\,mm (i.e.,
giving opacities of $\sim 0.2-0.3$), resulting in system temperatures of
approximately $200-300\,{\rm K}$ on average. The spectral correlators were
centered at 114.197\,GHz ($2793.75\,{\rm km\,s^{-1}}$) and 228.390\,GHz
($2793.27\,{\rm km\,s^{-1}}$), respectively. The coordinates of the PdBI
phase tracking center are given in Table \ref{table1} and correspond to the
(6\,cm VLBA) nuclear radio position of \citet{nagar02}.
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth,angle=0,bb=47 145 590 680]{9545fig3n.ps}
\caption{Radial brightness profiles are shown in the
top panel, with $J$ given as a
dotted line (blue), and IRAC $3.6\,{\rm \mu}m$ as a solid
one (red).
The vertical dashed line corresponds to 200\,pc.
The lower panels display the radial runs of ellipticity, $\epsilon$,
position angle $\theta$, and cos($4\theta$) terms in the ellipse
fitting residuals.
\label{fig:ellipse}
}
\end{figure}
Data cubes with 256 $\times$ 256 pixels
(0\,\farcs46\,${\rm pixel}^{-1}$ for $^{12}$CO(1-0) and
0\,\farcs23\,${\rm pixel}^{-1}$ for $^{12}$CO(2-1)) were created
over a velocity interval of -240 km\,s$^{-1}$ to +240 km\,s$^{-1}$
in bins of 10 km\,s$^{-1}$.
The images were reconstructed
using the standard IRAM/GILDAS\footnote{http://www.iram.fr/IRAMFR/GILDAS/}
software \citep{guilloteau} and restored with Gaussian beams of dimensions
1\farcs8 $\times$ 1\farcs6 (PA = $62^{\circ}$) at 115\,GHz
and 1\farcs4 $\times$ 1\farcs2 (PA = $61^{\circ}$) at
230\,GHz. In the cleaned maps, the rms levels are $2\,{\rm mJy\,beam^{-1}}$
and $4\,{\rm mJy\,beam^{-1}}$ for the $^{12}$CO(1-0) and $^{12}$CO(2-1) lines, respectively
(at a velocity resolution of $10\,{\rm km\,s^{-1}}$). The conversion
factors between intensity and brightness temperature are $32\,{\rm
K\,(Jy\,beam^{-1})^{-1}}$ at 115\,GHz and $14\,{\rm K\,(Jy\,beam^{-1})^{-1}}$
at 230\,GHz.
A continuum point source is detected at both 3\,mm and 1.3\,mm.
The continuum flux density is $5.2 \pm 0.2\,{\rm mJy}$ at 3\,mm
and $2.2 \pm 0.4\,{\rm mJy}$ at 1.3\,mm. Both values
are consistent with the synchrotron source detected at centimeter
wavelengths for a power-law spectrum with a slope of $-1$.
All maps presented in the paper are continuum-subtracted.
\subsection{IRAM single dish CO and HCN observations}
\label{singleo}
We performed IRAM 30\,m telescope observations in a $5 \times 5$
raster pattern with $7^{\prime\prime}$ spacing in
July 2002 and in June 2004. We used 4 SIS receivers to observe simultaneously
at the frequencies of the $^{12}$CO(1-0) (115\,GHz), the $^{12}$CO(2-1) (230\,GHz), and the
HCN(1-0) (89\,GHz) lines. The half power beam widths are
$22^{\prime\prime}$ for $^{12}$CO(1-0), $12^{\prime\prime}$ for $^{12}$CO(2-1), and
$29^{\prime\prime}$ for HCN(1-0). The typical system temperatures
were $\sim250 \,{\rm K}$ at 115\,GHz, $\sim350-700\,{\rm K}$ at
230\,GHz, and $\sim120$\,K at 89\,GHz. The line intensity scale throughout
this paper is expressed in units of $T_{\rm mb}$, the beam-averaged radiation
temperature. The value of $T_{\rm mb}$ is related to $T^{*}_{A}$, the
equivalent antenna temperature (corrected for rear spillover and ohmic
losses) reported above the atmosphere, by $\eta=T^{*}_{A}/T_{\rm mb}$ where
$\eta$ is the telescope main-beam efficiency. $\eta$ is 0.79 at 115\,GHz,
0.57 at 230\,GHz, and 0.82 at 89\,GHz. All observations were performed in
``wobbler-switching'' mode, with a minimum phase time for spectral line
observations of 2\,s and a maximum beam throw of $240^{\prime\prime}$. The
pointing accuracy was $\sim3^{\prime\prime}$ rms.
Single dish observations were used to compute short-spacings and
complete the interferometric measurements \citep[e.g.][]{combes04}. In
particular, short-spacing visibilities are computed from a map built by
interpolation of the 30\,m beam and multiplied by the PdBI primary beam.
We combined 30\,m and PdBI data, using the SHORT-SPACE task in the GILDAS
software. To find the best compromise between good angular resolution and
complete restoration of the missing extended flux, we varied the weights
attached to the 30\,m and PdBI data.
After writing the combined datasets to visibility tables,
converting to maps using standard data reduction procedures,
and deconvolving using the Clark algorithm, we obtained maps with
angular resolutions of 1\farcs88 $\times$ 1\farcs63
at PA 61.5$^{\circ}$ for the $^{12}$CO(1-0) map and
1\farcs61 $\times$ 1\farcs4 at PA 65.4$^{\circ}$ for
the $^{12}$CO(2-1) map.
The weights were adjusted in order to obtain the same mean
weights in the single-dish data as in the interferometric data in the $uv$
range of $1.25\,D/\lambda$ to $2.5\,D/\lambda$ ($D = 15$\,m). All figures
presented in this paper are made with short-spacing-corrected data.
Figs. \ref{channel} and \ref{channel21} display the channel maps
for the $^{12}$CO(1-0) and $^{12}$CO(2-1) lines, respectively.
All maps are centered on the position of Table \ref{table1}, and the
dynamical center coincides with this center, which is also the position
of the AGN (the radio continuum source).
Figs. \ref{co21-30m} and \ref{n3147-hcn} display the single dish data,
for the $^{12}$CO(1-0), $^{12}$CO(2-1), and HCN(1-0) lines. The two CO lines have been
mapped on a $5 \times 5$ grid with $7^{\prime\prime}$ spacings, while
the HCN(1-0) line has been mapped on a $3 \times 3$ grid with
$7^{\prime\prime}$ spacings and the nine HCN(1-0) spectra have been
averaged to improve the signal-to-noise.
To estimate the flux filtered out by the interferometric observations, we
have computed the flux measured by \citet{young} with the FCRAO 14\,m
telescope (HPBW = $45^{\prime\prime}$). They found a $^{12}$CO(1-0) intensity
towards the center of $I({\rm CO}) = 5.75\,{\rm K\,km\,s^{-1}}$ (in the
$T^{*}_{A}$ scale), which corresponds to an integrated flux of $S({\rm CO}) =
242\,{\rm Jy\,km\,s^{-1}}$ if we adopt a conversion factor of $42\,{\rm
Jy\,K^{-1}}$.
In the same region, we measure $130\,{\rm
Jy\,km\,s^{-1}}$ in $^{12}$CO(1-0) with the PdBI alone, and $290 \,{\rm
Jy\,km\,s^{-1}}$ with the IRAM 30\,m in a beam of $22^{\prime\prime}$.
With short-spacings added, we estimate the total flux to be
$295\,{\rm Jy\,km\,s^{-1}}$ in $^{12}$CO(1-0), which is compatible with the
FCRAO value within the uncertainties.
In $^{12}$CO(2-1), the PdBI alone indicates
$30\,{\rm Jy\,km\,s^{-1}}$ with a large region with negative flux
levels. With the IRAM 30\,m in $^{12}$CO(2-1) we measure a total
flux of $140\,{\rm Jy\,km\,s^{-1}}$ in a beam of 12$^{\prime\prime}$,
and when short-spacings are included we recover the value of
$151\,{\rm Jy\,km\,s^{-1}}$.
The implication is that much of the
$^{12}$CO(2--1) emission comes from smoothly distributed gas.
\begin{figure*}
\centering
\includegraphics[height=0.75\textwidth,angle=-90]{9545fg4tn.ps}
\label{co10-30m}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[height=0.75\textwidth,angle=-90]{9545fg4bn.ps}
\caption{Spectra maps of NGC\,3147 made with the IRAM 30\,m with
$7^{\prime\prime}$ spacing in $^{12}$CO(1-0) (top) and $^{12}$CO(2-1) (bottom). The
positions are offsets relative to the center of NGC\,3147, whose coordinates
are listed in Table \ref{table1}. Each spectrum has a velocity scale from
$-500$ to $500\,{\rm km\,s^{-1}}$, and an antenna temperature scale
($T^{*}_{A}$) from $-0.05$ to $0.25\,{\rm K}$ for $^{12}$CO(1-0) and to 0.20\,K
for $^{12}$CO(2-1).}
\label{co21-30m}
\end{figure*}
\subsection{Near-infrared observations}
For modeling the gravitational potential, we used NIR images
which have the advantage of being relatively free of dust extinction and more
representative of old stellar populations than blue or visible images.
The NIR images were obtained in December 1998 at the 3.6\,meter
Canada-France-Hawaii Telescope (CFHT), using the CFHT Adaptive Optics
Bonnette (AOB) and the KIR infrared camera.
The AOB, also called PUEO after the sharp-visioned Hawaiian owl,
delivers essentially diffraction-limited images at $H$ and $K'$
(0\,\farcs11 and 0\,\farcs14, respectively), and images
with FWHM $\sim0\,\farcs1$ at $J$ with guide stars as faint as $R=14$
\citep[see][]{rigaut98}. Here the Seyfert nucleus of the galaxy
was used for wavefront sensing.
The KIR 1024 $\times$ 1024 pixel HgCdTe array
has 0\,\farcs035/pixel, providing a field of view of
$36^{\prime\prime} \times 36^{\prime\prime}$.
The observations were obtained in excellent seeing conditions
($\sim0\,\farcs6$ in the $V$ band), with several images
taken in a dithering procedure to correct for camera defects.
The total on-source integration time was 8 minutes
each for $J$ and $K^{\prime}$, and the total observing time for
the two filters was 50 minutes including sky measurements and overheads.
A $J-K$ map was constructed by first subtracting sky
emission, as estimated from the outer regions of the images.
Then mean instrumental magnitude zero points were used to calculate a
$J-K$ image, after aligning them to a common center.
\subsection{Archival observations with \textit{Spitzer} and \textit{GALEX}\label{sec:archive}}
We acquired public images of NGC\,3147 at other wavelengths to compare our
molecular gas observations with other star formation tracers.
We used a far-ultraviolet (FUV) image from the \textit{GALEX} satellite,
whose band is centered at $\lambda_{\rm eff} = 1516\,{\rm \AA}$.
This image has been already used and studied in the context of
the \textit{GALEX} Nearby Galaxies Survey
\citep[NGS,][]{gildepaz04,bianchi03a,bianchi03b}, a project that spans a
large range of physical properties and morphological types of
nearby galaxies.
The image has been obtained with a total exposure
time of 1693\,s and covers a square
region on the sky of size $\sim 5500^{\prime\prime} \times
5500^{\prime\prime}$, i.e., much larger than the extent of the optical disk
of NGC\,3147, with 1\farcs5 pixels.
As the image was reduced with the \textit{GALEX} data pipeline,
it is already expressed in intensity units and sky-subtracted.
The total FUV
calibrated magnitude is $14.99 \pm 0.01$, corresponding to a FUV
flux density of $3669 \pm 17\,{\rm \mu Jy}$.
We also acquired
infrared (IR) images obtained with the IRAC camera on \textit{Spitzer},
available thanks to the project `A Mid-IR Hubble Atlas of Galaxies'
\citep[Principal Investigator: G. Fazio, see also][]{pahre04}.
These images range in wavelength from
$3.6\,{\rm \mu m}$ to $8\,{\rm \mu m}$, and were
reduced with the \textit{Spitzer} data pipeline
(version S14.0.0). As in the case of the {\it GALEX} image,
they cover a large sky area ($\sim1400^{\prime\prime} \times
700^{\prime\prime}$).
These images have been rotated to canonical
orientation (north up, east left) in order to allow correct superposition
on the molecular gas map.
The pixel size is 1\farcs20 in the final images.
The stellar component at 3.6\,$\mu$m and 4.5\,$\mu$m was
subtracted from the 8.0\,$\mu$m image
according to the prescriptions of
\citet{helou04} and \citet{pahre04}.
This procedure provides a ``dust-only'' image at
8\,$\mu$m, which we will discuss in Sect. \ref{sec:ring}.
\begin{figure}[!h]
\centering
\includegraphics[height=\columnwidth,angle=-90]{9545fig5n.ps}
\caption{HCN(1-0) spectrum towards the center of NGC\,3147, averaged over
the 9-point map made with the IRAM 30\,m with $7^{\prime\prime}$ spacing.}
\label{n3147-hcn}
\end{figure}
Using the $3.6\,{\rm \mu m}$ image,
we calculated radial surface-brightness profiles by fitting
ellipses (using the IRAF/STSDAS task \textit{ellipse}).
The center was fixed, but the ellipticity and the position angle
were allowed to vary.
This brightness profile is shown in Fig. \ref{fig:ellipse}, together with
the $J$-band radial profile obtained in a similar way from the
CFHT image.
The zero point for the IRAC image was taken from the
IRAC documentation at the \textit{Spitzer} Science Center
(http://ssc.spitzer.caltech.edu/irac/calib/).
No corrections were made to the ``standard'' IRAC photometric calibrations.
Also shown are the runs of ellipticity, ellipse position angle,
and cos($4\theta$) residuals of the ellipse fitting.
A local peak in the ellipticity profile at R\,$\simeq\,5^{\prime\prime}$
(1\,kpc) at constant position angle
(85$^\circ$)
corresponds to a weak bar/oval feature that will be discussed
in Sect. \ref{sec:bar}.
\section{Single dish results}
\label{singler}
The observations
performed with the A and B receivers of the IRAM 30\,m telescope in the two $^{12}$CO lines covered the
central $\sim$50$^{\prime\prime}$, corresponding to the central $\sim$10 kpc
(in diameter) of the galaxy (Fig. \ref{co21-30m}).
The 25 observed positions show that the central region of NGC~3147 presents extended molecular
emission in both $^{12}$CO(1-0) and $^{12}$CO(2-1) (Fig. \ref{co21-30m}). The maximum
detected $T_{mb}$ is 0.32 K in $^{12}$CO(1-0) and 0.31 K in $^{12}$CO(2-1), in the southeast region
corresponding to the offsets (10$^{\prime\prime}$,-10$^{\prime\prime}$)
and (5$^{\prime\prime}$,-10$^{\prime\prime}$) relative to the galaxy center.
The combination of observations of the $^{12}$CO(1-0) and $^{12}$CO(2-1) lines
allowed us to compute the line ratio,
$R_{21}=I_{21}/I_{10}=\int{T_{mb}(2-1)}$dv /$\int{T_{mb}(1-0)}$dv,
an indicator of physical properties of the molecular gas such as
excitation and optical depth.
After smoothing the $^{12}$CO(2-1) data to the $^{12}$CO(1-0) beam resolution
($22^{\prime\prime}$) and correcting for the different beam efficiencies, we found a mean
line ratio for the central region of NGC\,3147 of 0.8.
This value is consistent with the optically thick emission from molecular
gas expected for the central regions of spiral galaxies \citep{braine92}.
Assuming a CO-H$_{2}$ conversion factor $X = N(H_{2})/I_{CO} = 2.2 \times 10^{20}$ cm$^{-2}$
(K km s$^{-1}$)$^{-1}$ \citep{solomon}, the total H$_{2}$ mass derived from
these
single dish observations is
M$_{H_{2}}$ = 8.2 $\times$ 10$^9$ M$_{\odot}$, within a radius of 5 kpc.
The HCN(1-0) line has been observed for 9 positions with 7$^{\prime\prime}$ spacing
covering the central 43$^{\prime\prime}$, corresponding to the central $\sim$8.5 kpc
(in diameter). The HCN(1-0) emission has been detected most significantly
in the northwest part of the observed grid
corresponding to the offset (-7$^{\prime\prime}$,7$^{\prime\prime}$).
The HCN(1-0) average spectrum over the 3$\times$3 grid is plotted in Fig. \ref{n3147-hcn}.
The CO(1-0)/HCN(1-0) ratio is equal to 20 on average.
Because the observed region contains the AGN, we would expect enhanced
HCN emission there.
Hence, the observed CO(1-0)/HCN(1-0) ratio is unexpectedly high.
For instance in NGC 1097, this ratio is 3 in the nucleus
and 10 in the star-forming ring \citep{kohno}. NGC~3147 is more similar
to the starburst in NGC~6951 \citep[][]{krips07b},
where the ratio is 30 in the ring, and 2.5 in the nucleus.
To study the physics and distribution of the molecular gas in detail
we need to analyze both single dish and interferometric observations.
The analysis in the remainder of the paper will be performed on the
combined IRAM PdBI+30\,m datasets.
\begin{figure*}
\centering
\includegraphics[width=0.43\textwidth,angle=-90,bb=145 17 460 750]{9545fig6n.ps}
\caption{\textit{Left panel}: $^{12}$CO(1-0) integrated intensity
contours observed with the IRAM
PdBI+30\,m toward the center of NGC\,3147.
The cross marks the coordinates
of the center as given in Table \ref{table1}, with offsets in arcseconds.
The rms noise level is $\sigma = 0.15\,{\rm Jy\,beam^{-1}\,km\,s^{-1}}$.
The map, derived with 2$\sigma$ clipping, has not been corrected for
primary beam attenuation. Contour levels run from 4$\sigma$ to 17$\sigma$
with 2.5$\sigma$ spacing. In this map the full $\pm 240\,{\rm km\,s^{-1}}$
velocity range is used. The beam of 1\farcs9 $\times$ 1\farcs6
is plotted at lower left.
\textit{Right panel}: Same for $^{12}$CO(2-1).
The rms noise level is $\sigma = 0.2\,{\rm Jy\,beam^{-1}\,km\,s^{-1}}$.
Contour levels run from 5$\sigma$ to
18$\sigma$ with 2.5$\sigma$ spacing.
The beam of 1\farcs4 $\times$ 1\farcs2 is plotted at
lower left.}
\label{co10r}
\end{figure*}
\begin{figure*}
\centering
\hbox{
\includegraphics[width=0.34\textwidth,angle=-90]{9545fg7ln.ps}
\includegraphics[width=0.34\textwidth,angle=-90]{9545fg7rn.ps}
}
\caption{\textit{Left panel:}
Radial distribution (azimuthal average, deprojected to face-on
orientation) of the $^{12}$CO(1-0) integrated intensity shown in Fig. \ref{co10r}.
\textit{Right panel:} Same for $^{12}$CO(2-1).
}
\label{moment1021}
\end{figure*}
\section{Interferometric results: Molecular gas properties }
\subsection{Morphology and mass of the CO rings\label{sec:morph}}
The $^{12}$CO(1-0) and $^{12}$CO(2-1) integrated intensity distributions are
shown in Fig. \ref{co10r}.
The figure reveals a nuclear peak, a symmetric and
complete inner ring at a distance of about 10$^{\prime\prime}$
from the nucleus, and a larger and incomplete ring at
a distance of about 20$^{\prime\prime}$ from the nucleus (left panel).
In $^{12}$CO(2-1) the nuclear peak is stronger and more visible (right panel),
but there is no outer ring at a radius of 20$^{\prime\prime}$
because of the restricted field-of-view (FWHM of primary beam
of 22$^{\prime\prime}$).
The outer ring-like structure can also be interpreted as spiral arms,
more tightly wound in the inner regions than in the outer ones.
The radial distribution of the azimuthally averaged $^{12}$CO(1-0)
and $^{12}$CO(2-1) intensities
is shown in Fig. \ref{moment1021}, where there are clear dips
between the central peak and the inner ring and between the inner
ring and the outer spiral structure (partial outer ring).
Both $^{12}$CO(1-0) structures are clumpy, composed of individual giant
molecular cloud complexes,
each with a mass of a few 10$^{7}$ to 10$^{8}$ M$_{\odot}$. The total
$H_{2}$ mass derived from the interferometer maps,
assuming the same CO-to-H$_2$ conversion value used before, is
M$_{H_{2}}$ = 3.8 $\times$ 10$^9$ M$_{\odot}$.
The inner ring alone contributes more than half
of the total masses; M$_{H_{2}}$ = 2 $\times$ 10$^9$ M$_{\odot}$,
and the two parts of the outer spiral have similar mass,
the western part has a mass of M$_{H_{2}}$ = 9.5 $\times$ 10$^8$
M$_{\odot}$ and the eastern one
M$_{H_{2}}$ = 8.5 $\times$ 10$^8$ M$_{\odot}$.
The total M$_{H_{2}}$ (= 3.8 $\times$ 10$^9$ M$_{\odot}$)
we find is in good agreement with the mass
derived by \citet{young} for a position centered on
the galaxy nucleus, M$_{H_{2}}$ = 4.4 $\times$ 10$^9$ M$_{\odot}$.
Our mass estimate from single dish observations
presented in Sect. \ref{singler} is larger, since it corresponds
to a more extended region.
In any case, this molecular gas mass is very large, more than
three times larger than the most massive of
the other NUGA galaxies studied so far
\citep[NGC\,2782 and NGC\,4569, see][]{hunt,boone07}.
The comparison between the $^{12}$CO maps of the two transitions,
at the same resolution and with the same spatial frequency sampling,
gives insight about the excitation conditions
of the molecular gas locally, pixel by pixel.
Fig. \ref{co10-21} shows that when the $^{12}$CO(2-1) data are tapered
and convolved to the $^{12}$CO(1-0) resolution, the maxima of the $^{12}$CO(2-1) ring agree
quite well with the $^{12}$CO(1-0) peaks.
The ratio map is shown in Fig. \ref{ratio}
where the values range from 0.1 to 1.8.
The mean ratio is $\sim$0.7, consistent with the value found
with single dish observations
and also, in this case, consistent with the bulk of emission
being optically thick, as expected
in the nuclei of spiral galaxies.
Some regions, especially the center, reach a higher ratio ($\sim$1.3-1.4)
where the gas is warm and dense, while in the disk and spiral arms
it is colder and more diffuse.
The peak in the northwest region of the CO(2-1)/CO(1-0) ratio map
is coincident with the strongest HCN(1-0) emission corresponding
to the offset (-7$^{\prime\prime}$,7$^{\prime\prime}$).
\begin{figure}
\centering
\includegraphics[width=0.85\columnwidth,angle=-90]{9545fig8n.ps}
\caption{$^{12}$CO(1-0) contours as in Fig. \ref{co10r} (left panel) superposed on the
greyscale $^{12}$CO(2-1) map that has been tapered, convolved to the same resolution,
and corrected for primary beam attenuation, in units of ${\rm Jy\,beam^{-1}\,km\,s}^{-1}$.
The beam is plotted at lower left.}
\label{co10-21}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.85\columnwidth,angle=-90]{9545fig9n.ps}
\caption{Contours and greyscale of the CO(2-1)/CO(1-0) ratio map.
Contours run from 0.1 to 1.8 in steps of 0.1 in temperature units.}
\label{ratio}
\end{figure}
\subsection{Kinematics \label{sec:kinematics}}
The channel maps in Fig. \ref{channel} display overall a gross
regularity of the large scale kinematics,
following the expected spider diagram.
However, there are some local wiggles superposed on this pattern,
i.e., streaming motions.
Such streaming motions are also visible in
Fig. \ref{co-velo}, which shows the isovelocity curves (first-moment map) of the
$^{12}$CO(1-0) emission superposed on the $^{12}$CO(1-0) integrated intensity.
Some of the perturbations in the velocity field are
clearly coincident with the spiral arms, as expected from density wave theory.
\begin{figure}[ht]
\centering
\includegraphics[width=0.85\columnwidth,angle=-90]{9545fg10n.ps}
\caption{Overlay of the integrated $^{12}$CO(1-0) emission in grey scale,
same as Fig. \ref{co10r} (left panel), with the CO mean-velocity field in contours
spanning the range -200 km s$^{-1}$ to 200 km s$^{-1}$ in steps of
10 km s$^{-1}$. The velocities are referred
to V$_{hel}$ = 2813 km s$^{-1}$. Solid (dashed) lines are used for positive
(negative) velocities. The dotted line indicates the major axis (PA = 150$^\circ$).}
\label{co-velo}
\end{figure}
A position-velocity (p-v) cut along the major axis using a position angle of
150$^\circ$ is
presented in Fig. \ref{p-v}.
The major-axis p-v diagram shows approximately regular kinematics,
although there are clear signs of deviations from ``flatness'' at $\sim$15\arcsec
on either side of the nucleus of amplitude $\sim$40\,km\,s$^{-1}$.
The minor-axis p-v diagram is displayed in Fig. \ref{fig:minor}.
The wiggles are more conspicuous on the major axis in Fig. \ref{p-v}
than in Fig. \ref{fig:minor}.
Although the minor-axis velocities are
not completely symmetric about the nucleus, they hint that
the outer spiral arm/pseudo-ring at a radius of 20\arcsec\
is already outside corotation.
This is because the sign of the radial streaming motions
(visible on the minor-axis) is expected to change at corotation, while
the tangential streaming (visible on the major-axis) does not change
at corotation \citep[e.g.][]{canzian93}.
Thus, we conclude (although with substantial
uncertainty) that corotation occurs at $\sim$15\arcsec\ (3\,kpc),
consistent with the minor-axis p-v plot. This result will help constrain
the pattern speed of the weak bar/oval which we discuss in
Sect. \ref{sec:discussion}.
\subsection{Dynamical mass \label{sec:mass}}
The peak velocity of $\sim$200\,km\,s$^{-1}$
in the major-axis p-v diagram occurs
at a radius of $\sim$7-8\,\arcsec\ (1.5\,kpc).
This value, when corrected for disk
inclination (we adopt $i=29.5^\circ$, see Table \ref{table1}) by a
factor sin~$i$,
becomes 400\,km\,s$^{-1}$.
From the corrected peak velocity, we can estimate the
dynamical mass within this radius:
$M(R) = \alpha \times {\frac {R V^2(R)}{G}}$,
where $M(R)$ is in $M_\odot$, $R$ in kpc, and $V$ in km\,s$^{-1}$.
Assuming the most flattened disk-like distribution
($\alpha = 0.6$),
gives a dynamical mass $M_{\rm dyn}=3.3\times10^{10}$\,$M_\odot$
within a diameter of $\sim$3\,kpc.
Assuming a roughly flat rotation curve (see Sect. \ref{sec:discussion}),
this would imply a mass $M_{\rm dyn}=8.9\times10^{10}$\,$M_\odot$
within 4\,kpc (roughly the radius of the PdBI primary
beam for our $^{12}$CO(1-0) observations).
With a total molecular gas mass of $\sim4\times10^{9}$\,$M_\odot$
(see Sect. \ref{sec:morph}),
we would thus estimate a molecular mass fraction of $4-5$\%.
\section{Comparison with other wavelengths \label{sec:other}}
It is instructive to compare the distribution of the
molecular gas in the inner kpc of NGC~3147 with observations at
other wavelengths.
In particular, molecular clouds are thought to be the birth site of
future generations of stars, and thus should be strongly connected to other
emission tracing
star formation, such as far-ultraviolet and warm dust emission.
\subsection{The bar and the lopsided nucleus \label{sec:bar}}
The CO contours are superposed on the CFHT $J$-band image in Fig. \ref{JCO}.
There is a weak bar/oval feature at a position
angle PA = 85$^\circ$, just contained inside the inner CO
ring. Only the NIR reveals this bar, since
NGC~3147 is not classified optically as a barred galaxy. Its
late type (Sbc) corresponds to a non-dominating bulge,
and the bar feature is likely not diluted by this bulge.
The major axis orientation differs by 65$^\circ$ from the bar
PA, which minimizes confusion due to projection effects.
\citet{mh2001}
have obtained $JHK$ images of NGC\,3147 with the 2.2\,m telescope
of the Calar
Alto observatory using the MAGIC NICMOS3 camera, together with 40 other
unbarred galaxies with low inclinations. They
decomposed the images into bulge and disk components, and noticed that
the bulge in NGC~3147 appears at a very different position angle than the disk;
they concluded that the bulge must be triaxial, which was a rare
feature in their sample.
We have re-reduced their MAGIC $J$ and $K'$ images and the oval is present,
similar to the feature seen in our CFHT images, although with noisier
isophotes.
Our kinematic data suggest that
this bar may be the agent of the inflow that we discuss in
Sect. \ref{torq},
\begin{figure}[h]
\centering
\includegraphics[width=0.95\columnwidth]{9545fg11n.ps}
\caption{$^{12}$CO(1-0) position-velocity (p-v) diagram along the major
axis of NGC~3147 using the whole velocity range +/-240 km s$^{-1}$.
The coordinates of the center (0,0), the heliocentric velocity, and PA used are
given in Table \ref{table1}. Contour levels are from 3.2 to
$32\,{\rm mJy\,beam^{-1}}$ in steps of $3.2\,{\rm mJy\,beam^{-1}}$.}
\label{p-v}
\end{figure}
The ellipticity profile shown in Fig. \ref{fig:ellipse} (see Sect. \ref{sec:archive})
helps us estimate the length of the bar/oval distortion.
The ellipticity peaks at $\sim$5\arcsec\ (1\,kpc), but the bar
extends roughly to a radius of $\sim$7.5\arcsec\ (1.5\,kpc).
This is also seen in the $J$-band image (Fig. \ref{JCO}) where the bar is roughly
16\arcsec\ in diameter, and in the Fourier decomposition of the
stellar potential discussed in Sect. \ref{torq}.
We therefore adopt a bar radius of $\sim$7.5\arcsec\ or $\sim$1.5\,kpc,
although this estimate is subject to a rather large uncertainty, $\pm$30\%.
The $J-K$ image of NGC~3147
is shown in Fig. \ref{fig:jk} with the $^{12}$CO(2-1) integrated
intensity overlaid as contours (see Sect. \ref{sec:morph}).
Coincident with the nuclear CO emission,
there is a clearly-defined
non-axisymmetric structure of $\sim$700\,pc in size, which is bluer
in $J-K$ than the surrounding regions that are devoid of gas.
This central asymmetric ``blob'' or ``lopsided nucleus''
could be the destination of the gas inflow that we discuss in Sect. \ref{torq}.
\subsection{The inner ring and the outer spiral \label{sec:ring}}
Fig. \ref{onlygalex} displays the
FUV image obtained with the \textit{GALEX} satellite, described in
Sect. \ref{sec:archive}.
Inspection of Fig. \ref{onlygalex}
shows that NGC~3147 has a quite well-defined outer spiral
structure, outside the range of our CO observations.
There is also a compact nucleus with a central peak,
surrounded by a roughly axisymmetric ``plateau'' of emission
which breaks up into a pseudo ring at a radius of $\sim$20\arcsec.
Fig. \ref{galex-co} shows the superposition of \textit{GALEX} FUV
image with the $^{12}$CO(1-0) (left panel) and $^{12}$CO(2-1) (right)
integrated intensity contours.
The cross shown in Fig. \ref{onlygalex} is at
the center of our CO observations, and coincides with the peak of the FUV nuclear emission.
This agreement suggests that the FUV and CO emission are cospatial
in the nuclear region.
\begin{figure}[h]
\centering
\includegraphics[width=0.95\columnwidth]{9545fg12n.ps}
\caption{$^{12}$CO(1-0) position-velocity (p-v) diagram along the minor
axis of NGC~3147 using the same velocity range, center, PA, and
systemic velocity as for Fig. \ref{p-v}.
Contour levels are from 2.8 to $28\,{\rm mJy\,beam^{-1}}$ in steps of
$2.8\,{\rm mJy\,beam^{-1}}$.}
\label{fig:minor}
\end{figure}
\begin{figure*}
\includegraphics[width=0.43\textwidth,angle=-90]{9545f13cn.ps}
\caption{\textit{Left panel}: $^{12}$CO(1-0) contours
(0.6 to 2.6 by 0.35 Jy km s$^{-1}$ beam$^{-1}$)
superposed on the near-infrared $J$ image from the CFHT,
shown in logarithmic levels. The central 50$^{\prime\prime}$ are
shown. \textit{Right panel}: Same for $^{12}$CO(2-1) contours
(1.0 to 3.5 by 0.5 Jy km s$^{-1}$ beam$^{-1}$).
The central 30$^{\prime\prime}$ are
shown.
}
\label{JCO}
\end{figure*}
However, there is no clear ring in the FUV image.
The inner 10$^{\prime\prime}$-radius
CO ring is contained inside the circular region of the FUV emission, and the outer
CO spiral (20$^{\prime\prime}$-radius), at least its eastern part, has
some overlap with the FUV spiral, but there are clear shifts between the two distributions.
The $^{12}$CO(2-1) emission (essentially the inner ring) also appears to settle
inside the \textit{GALEX} FUV inner ``plateau'' (Fig. \ref{galex-co}).
The ``outer spiral'' in $^{12}$CO(1-0) falls in the interarm ``gap''
in the \textit{GALEX} image.
Offsets between FUV and CO (or FIR and HI) emissions are not
uncommon in grand design spiral galaxies, such as M\,100
\citep{rand95,sempere97} and M 51 \citep{calzetti05},
and may relate to star-formation efficiency and timescale variations
in response to a spiral density wave.
The star formation and all related tracers are often located
in different regions of the spiral arms: FUV emission is more prominent
at the outer edge of the arms, where the dust extinction is low, while
FIR and ${\rm H\alpha}$ emission is stronger at the inner edge.
However, in NGC 3147 we find CO in the middle
of the interarm region in the \textit{GALEX} image, not at the inner or
outer edge of the arm; this result is quite rare.
Unlike the FUV, the 8\,$\mu$m ``dust-only''
image described in Sect. \ref{sec:archive} shows an inner ring
apparently associated with molecular gas.
Fig. \ref{spitzer-8um} shows this image,
which clearly exhibits an inner-ring structure at about 10$^{\prime\prime}$ radius,
together with a central peak.
Such rings are frequently observed in barred galaxies with
\textit{Spitzer} \citep[e.g.][]{regan06}.
Fig. \ref{spitzer-co}
shows the dust-only 8\,$\mu$m image with the $^{12}$CO(1-0) (left panel) and $^{12}$CO(2-1)
(right) intensity contours superposed.
The ring visible at 8 $\mu$m corresponds to both the $^{12}$CO(1-0)
inner ring and the $^{12}$CO(2-1) one.
However, the outer $^{12}$CO(1-0) spiral is slightly inside the corresponding
structure in dust emission.
This may be a similar phenomenon to that seen in the \textit{GALEX} FUV
image, where molecular gas peaks in the interarm regions as described
above.
\begin{figure}
\centering
\includegraphics[width=0.87\columnwidth,angle=-90]{9545f14cn.ps}
\caption{The $J-K$ image from the CFHT with
$^{12}$CO(2-1) integrated intensity overlaid in contours
(1.0 to 3.5 by 0.5 Jy km s$^{-1}$ beam$^{-1}$).
North is up and east to the left.
}
\label{fig:jk}
\end{figure}
\begin{figure
\centering
\includegraphics[width=0.9\columnwidth,angle=-90]{9545f15cn.ps}
\caption{\textit{GALEX} FUV sky subtracted image of NGC~3147.
The central 120$^{\prime\prime}$ of the galaxy are
shown.
The box shows the central 50$^{\prime\prime}$,
field of view of our $^{12}$CO observations.
}
\label{onlygalex}
\end{figure}
We can use the 8\,$\mu$m dust-only image, together with
the CO emission itself, to define the size and width of the inner
gas (and dust) ring.
The ring is rather broad, extending over roughly 2--3\arcsec\ in radius,
or $\sim$500\,pc.
Measured to the middle of its width, the inner gas ring has a radius
of $\sim9-10$\arcsec\ (1.8--2\,kpc) , although it is slightly elongated, being
longer along the horizontal axis.
If instead we measure to the innermost portion of the ring, we
find a radius of $\sim7^{\prime\prime}-8^{\prime\prime}$ ($\sim$1.5\,kpc).
This uncertainty (20-30\%) will influence our discussion in Sect. \ref{sec:discussion},
where we will use the size of the ring in the context of the rotation
curve to help constrain the pattern speed and location of the resonances
that define the kinematics in NGC~3147.
\subsection{Star formation \label{sec:sf}}
The agreement, where present, between the molecular gas distribution and the
ultraviolet and infrared emission is consistent with on-going star formation
within the molecular clouds.
The star formation rate (SFR) can be computed from both
UV and FIR emission using the calibrations given by \citet{kennicutt98}
for a $0.1-100\,M_\odot$ Salpeter initial mass function:
\begin{eqnarray}
\mbox{SFR} \,\mbox{[M$_{\odot}$\,yr$^{-1}$]} = 1.4 \times 10^{-28}\,\mbox{L$_{UV}$}
\label{sfr1}
\end{eqnarray}
where L$_{UV}$ is the UV luminosity in
ergs s$^{-1}$ Hz$^{-1}$ over the wavelength range 1500-2800 $\AA$, and
\begin{eqnarray}
\mbox{SFR} \,\mbox{[M$_{\odot}$\,yr$^{-1}$]} = 4.5 \times 10^{-44}\, \mbox{L$_{FIR}$}
\label{sfr2}
\end{eqnarray}
where L$_{FIR}$, expressed in ergs s$^{-1}$, refers to the total infrared luminosity
integrated over the entire mid and far-IR spectrum
(8-1000 $\mu$m).
We first compute the UV luminosity from the \textit{GALEX}
map inside the FOV of 42$^{\prime\prime}$, to better separate the SFR in
the PdBI FOV from the total, and find a SFR $\sim$1\,M$_{\odot}$\,yr$^{-1}$.
\begin{figure*}
\centering
\includegraphics[width=0.43\textwidth,angle=-90]{9545f16cn.ps}
\caption{\textit{Left panel}: $^{12}$CO(1-0) contours
(0.6 to 2.6 by 0.35 Jy km s$^{-1}$ beam$^{-1}$) superposed on
the \textit{GALEX} FUV image of NGC~3147, shown in logarithmic
levels. The central 50$^{\prime\prime}$ are
shown. \textit{Right panel}: Same for $^{12}$CO(2-1) contours
(1.0 to 3.5 by 0.5 Jy km s$^{-1}$ beam$^{-1}$). The central
30$^{\prime\prime}$ are shown.}
\label{galex-co}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth,angle=-90]{9545f17cn.ps}
\caption{\textit{Spitzer/IRAC} ``dust-only'' image at 8\,$\mu$m of NGC~3147.
The central 120$^{\prime\prime}$ of the galaxy are shown.
The box shows the central 50$^{\prime\prime}$,
field of view of our $^{12}$CO observations.
}
\label{spitzer-8um}
\end{figure}
We then use the \textit{Spitzer} MIPS images available
from the archive to estimate
how much FIR flux falls within the FOV of the PdB primary beam
compared to the total integrated over the much larger \textit{IRAS} beams.
Rough measurements give a fraction of $\sim$12\% for 160\,$\mu$m and
$\sim$22\% for 24\,$\mu$m (the 70\,$\mu$m image is not available).
We then estimate the FIR luminosity by scaling the IRAS 100\,$\mu$m and 60\,$\mu$m fluxes
by 12\% and 17\%\footnote{17\% is a rough interpolation of the central concentration
we would expect at 60\,$\mu$m, given the more compact emission at 24\,$\mu$m
and the more extended emission at 160$\mu$m.}, respectively.
This gives 1.39\,Jy at 60\,$\mu$m and 3.55\,Jy at 100\,$\mu$m, which results in
an estimate for the SFR of $\sim$0.8\,M$_{\odot}$\,yr$^{-1}$,
similar to that obtained from \textit{GALEX}.
The compatibility between the two estimations of SFR (from UV and FIR) means that there is not
significant obscuration, which is probably due to the nearly face-on view of the galaxy.
\begin{figure*}
\centering
\includegraphics[width=0.43\textwidth,angle=-90]{9545f18cn.ps}
\caption{\textit{Left panel}: $^{12}$CO(1-0) contours
(0.6 to 2.6 by 0.35 Jy km s$^{-1}$ beam$^{-1}$) superposed on
the 8\,$\mu$m ``dust-only'' (\textit{Spitzer/IRAC}) image of
NGC~3147, shown in logarithmic levels. The central 50$^{\prime\prime}$ are
shown. \textit{Right panel}: Same for $^{12}$CO(2-1) contours (1.0 to 3.5 by
0.5 Jy km s$^{-1}$ beam$^{-1}$). The central 30$^{\prime\prime}$ are
shown.}
\label{spitzer-co}
\end{figure*}
\begin{figure}
\includegraphics[angle=-90,width=\columnwidth]{9545f19n.ps}
\caption{Strengths (Q$_1$, Q$_2$, and total Q$_T$) and phases ($\phi_1$ and $\phi_2$) of
the $m=1$ and $m=2$ Fourier
components of the stellar potential inside a field of
48$^{\prime\prime}$ in diameter (r $<$ 4.6 kpc).}
\label{pot3147}
\end{figure}
The SFR per unit area inferred from these values is
0.016\,M$_{\odot}$\,yr$^{-1}$\,kpc$^{-2}$,
high for spiral disks but similar to, although slightly lower than, typical
values for IR-selected circumnuclear starbursts \citep{kennicutt98}.
Moreover, given the gas mass we derived in Sect. \ref{sec:morph}
and the correlations shown in \citet{kennicutt98},
the global star-formation efficiency within the inner 8\,kpc is low, $\sim$1\% rather
than the more typical value of $\sim$10\%.
On the other hand, the molecular gas is concentrated into well-defined
structures; more than half the molecular mass is found in the inner ring
alone (see Sect. \ref{sec:morph}).
If we consider only the molecular gas in the inner ring, the gas surface
density is $\sim$320\,M$_{\odot}$\,pc$^{-2}$, roughly 4 times higher
than we obtain by averaging over the entire FOV.
Nevertheless, it is significantly lower than that in M\,51
spiral arms \citep{henry03}.
\section{Computation of the torques \label{torq}}
\subsection{Evaluation of the gravitational potential \label{grav-pot}}
As described in previous papers \citep[e.g.,][]{garcia05},
we assume that NIR images can give us the best approximation
for the total stellar mass distribution, being less affected than
optical images by
dust extinction or by stellar population biases. We here recall
the essential definitions and assumptions.
The $J$-band image was first deprojected according to the
angles PA = 150$^\circ$ and $i=29.5^\circ$. We have not deprojected
the bulge separately, since we do not know its actual flattening, but
because the galaxy is of late type (Sbc), it does not have
a large contribution. The image is
completed in the vertical
dimension by assuming an isothermal plane model with a constant scale height,
equal to $\sim$1/12th of the radial scale-length of the image. The potential is
then derived by a Fourier transform method, assuming a constant mass-to-light (M/L) ratio.
The M/L value is selected to reproduce the observed CO rotation curve.
The potential was also calculated from the lower-resolution
MAGIC images, which have a much larger field-of-view (162\arcsec).
The same results are obtained, and the potential is shown to be axisymmetric
at large radii.
Beyond a radius of 20\arcsec,
the mass density
is set to 0, thus suppressing any spurious $m=4$ terms;
this is sufficient to compute the potential over the PdB $^{12}$CO(1-0) primary beam.
For the non-axisymmetric part, the
potential $\Phi(R,\theta)$ is then decomposed in the different $\textit{m}$-modes:
$$
\Phi(R,\theta) = \Phi_0(R) + \sum_{m=1}^\infty \Phi_m(R) \cos (m \theta - \phi_m(R))
$$
\noindent
where $\Phi_m(R)$ and $\phi_m(R)$ represent the amplitude and phase of the $m$-mode.
The strength of the $m$-Fourier component, $Q_m(R)$ is defined as
$Q_m(R)=m \Phi_m / R | F_0(R) |$, i.e. by the ratio between tangential
and radial forces \citep[e.g.][]{combes81}.
The strength of the total non-axisymmetric perturbation is defined by:
$$
Q_T(R) = {F_T^{max}(R) \over F_0(R)}
$$
\noindent
where $F_T^{max}(R)$ represents the maximum amplitude of the tangential force and $F_{0}(R)$ is the mean
axisymmetric radial force.
Fig.~\ref{pot3147} displays these $Q$ values as a function of radius for NGC~3147.
A bar ($m=2$) can be seen clearly, with a peak radius of $\sim$ 1\,kpc (5\arcsec),
and a total extent of 1.5\,kpc (7.5\arcsec) in radius.
These dimensions are similar to those inferred from the
ellipticity distribution discussed in Sect. \ref{sec:bar}, and strengthen
our conclusion that this structure is ``bar-like''.
There is also an asymmetry in the $m=2$ profile towards
the center. The lopsidedness has a correspondence in the gas
morphology described in Sect. \ref{sec:morph} and the $J-K$ image
discussed in Sect. \ref{sec:archive}.
The bar is regular in phase, but its strength is relatively modest.
\begin{figure}
\includegraphics[angle=-90,width=\columnwidth,bb=80 80 530 610]{9545f20n.ps}
\caption{CO(1--0) contours overlaid on the gravitational torque
map (t(x,y)~$\times$~$\Sigma$(x,y), as defined in text) in the center of NGC~3147.
The torque map (grey/color scale) is
normalized to the maximum absolute value of the torques.
The derived torques change sign as expected in a {\it butterfly} diagram,
delineating four quadrants. The orientation of quadrants follow
the bar orientation in NGC~3147. In this deprojected picture,
the major axis of the galaxy is oriented parallel to the abscissa.
The line reproduces the mean orientation of the bar
(PA = 20$^\circ$ on the
deprojected image and 85$^\circ$ on the projected one).
}
\label{torq3147}
\end{figure}
\subsection{Evaluation of gravity torques}
After having calculated the forces per unit mass ($F_x$ and $F_y$) from the derivatives of $\Phi(R,\theta)$ on
each pixel, the torques per unit mass $t(x,y)$ can be computed by:
$$
t(x,y) = x~F_y -y~F_x
$$
The torque map is oriented according to the sense of rotation in the plane of the galaxy.
The combination of the torque map and the gas density $\Sigma$ map then
allows us to derive the net effect on the gas at each radius.
The gravitational torque map weighted by the gas surface density $t(x,y)\times
\Sigma(x,y)$, normalized to its maximum value, is shown in Figs.~\ref{torq3147}
and \ref{torq3147_2} for CO(1--0) and CO(2--1), respectively.
\begin{figure}
\includegraphics[angle=-90,width=\columnwidth,bb=80 80 530 610]{9545f21n.ps}
\caption{Same as Fig. \ref{torq3147} for the $^{12}$CO(2--1)
emission taken as a tracer of gas surface density.}
\label{torq3147_2}
\end{figure}
To estimate the radial gas flow induced by the torques, we first compute the torque
per unit mass averaged over azimuth, using $\Sigma(x,y)$ as the actual weighting function, i.e.:
$$
t(R) = \frac{\int_\theta \Sigma(x,y)\times(x~F_y -y~F_x)}{\int_\theta \Sigma(x,y)}
$$
By definition, $t(R)$ represents the time derivative of the specific angular momentum $L$ of the gas averaged
azimuthally, i.e., $t(R)$=$dL/dt~\vert_\theta$. To derive dimensionless quantities,
we normalize this variation of angular momentum per unit time
to the angular momentum at this radius and to the rotation period.
We then estimate the efficiency of the gas flow as
the average fraction of the gas specific angular momentum transferred in one rotation
($T_{rot}$) by the stellar potential, as a function of radius, i.e., by the function $\Delta L/L$ defined as:
$$
{\Delta L\over L}=\left.{dL\over dt}~\right\vert_\theta\times \left.{1\over L}~\right\vert_\theta\times
T_{rot}={t(R)\over L_\theta}\times T_{rot}
$$
\noindent
where $L_\theta$ is assumed to be well represented by its axisymmetric
estimate, i.e., $L_\theta=R\times v_{rot}$.
The $\Delta L/L$ curves for NGC~3147 derived from the CO(1--0) and
the CO(2--1) data are displayed in
Figs.~\ref{gastor3147}.
\begin{figure*}
\includegraphics[angle=-90,width=\textwidth]{9545f22n.ps}
\caption{The torque, or more precisely the fraction of the angular momentum transferred
from/to the gas in one rotation --$dL/L$-- is plotted for $^{12}$CO(1-0) (left) and
$^{12}$CO(2-1) (right).}
\label{gastor3147}
\end{figure*}
The gravitational torque maps in Fig.~\ref{torq3147} show
that the derived torques change sign following a characteristic 2D
{\it butterfly} pattern. The CO contours reveal that for the material in the inner ring
(radius 2\,kpc, 10\arcsec), part of the gas is trailing the bar,
while part of the gas is leading it. The observed gas distribution is representative
of the time spent by a molecular cloud on a typical orbit at this
location. The azimuthal average of the torques shown in Fig. \ref{gastor3147}
suggests that the torques are predominantly negative
inside a radius of $\sim$2\,kpc, while they become zero or positive outside.
If the near side is the northeast side, the rotation sense
of the galaxy is counterclockwise, and the whole spiral
structure is trailing with respect to the rotation.
In the spiral structure winding into the outer $^{12}$CO(1-0) ring,
the torques almost cancel out on average but are still predominantly positive.
Inside the inner ring (r $<$ 2 kpc), however, the dominating torques are negative,
although weak in absolute value.
The $^{12}$CO(2-1) emission is more clumpy, and our map suffers from lower sensitivity;
there is relatively more emission in the central part.
Although the emission is spread over three quadrants in the center, the negative
torques are still predominant there (Fig.~\ref{torq3147_2}).
The negative torques there are due to the central
asymmetric CO distribution, mainly the nuclear component,
at radii inside 3-4\arcsec. The component is resolved in $^{12}$CO(2-1),
and is not aligned with the nuclear bar, but rather shifted to the leading
side; therefore, negative torques dominate.
\section{Discussion \label{sec:discussion}}
We find that the two molecular structures (inner ring and outer spiral),
in addition to a central peak of $^{12}$CO(2-1) emission, dominate the
CO map of NGC~3147's inner region.
The inner ring coincides with the ring in the dust-only 8\,$\mu$m image,
and the lopsided central peak falls at the same position as
the asymmetric nuclear structure
in the $J-K$ map.
These features are not present in the stars (e.g., $J$ band, IRAC 3.6\,$\mu$m,
and \textit{GALEX}), but are rather features of the ISM.
On the other hand, the outer spiral also appears in the stars as shown by the
\textit{GALEX} image; however, there is an offset between
them, as the CO emission falls in the interarm region relative to
the outer spiral (see Fig. \ref{galex-co}).
A probable explanation is that the CO outer spiral is not a permanent structure but is transient.
The gas participates in a spiral wave and forms stars, and then
the newly formed stars decouple from the gas component.
The young stars heat only the dust present at their location, while the bulk
of the gas has moved to another density condensation.
The inward migration of the dense gas with respect to the initial resonance
has for been discussed by, e.g., \citet{regan03}.
This phenomenon, because it is very transient, is statistically rarely encountered.
In this scenario, the CO clouds would not trace stars of the same age as those
that have heated the dust observable at 8\,$\mu$m or of those whose emission
is detected in the FUV, but rather they will be the birth site of future stellar generations.
For the torques computation, we interpret the
results by identifying the resonances with the bar.
Let us note that CO resonant rings have already been observed
in weakly barred galaxies, where the bar is inconspicuous in the optical
\citep[e.g., NGC~5005, NGC~7217:][]{sakamoto00,combes04}.
The rotation curve derived from the CO kinematics
is rather peaked in the center, and corresponds well
to the gravitational model obtained from the NIR
image. Both are fitted by an axisymmetric mass
model, with a rotation curve as shown in Fig.~ \ref{vcur},
which allows us to derive the characteristic dynamical frequencies.
If the stellar bar ends at or just inside its ultra-harmonic resonance
(UHR), as is canonical for bar dynamics, this means
that the pattern speed of the bar would be roughly
$\Omega_p\,=\,120-130$ km s$^{-1}$ kpc$^{-1}$, corresponding to a bar
with one or two inner Lindblad resonances, as is common.
The presence of these ILRs is not certain, however, given
the possible non-circular motions in the bar.
The inner CO ring would then correspond to the UHR
(r\,=\,2\,kpc, 10\arcsec) just inside corotation, which we estimated to
be at a radius of $\sim$15\arcsec\ (3\,kpc) in Sect. \ref{sec:kinematics}.
The measurements of the inner ring radius and bar length suggest that
the bar/oval ends roughly at the onset of the inner ring. However,
the considerable uncertainties of these measurements and the finite
width of the ring lead to an uncertainty in the pattern speed $\Omega_p$
of the bar.
The outer CO spiral corresponds to the spiral
structure emerging from the bar, beyond its
corotation.
The pitch angle of these spiral arms is quite small,
as is expected for a weak bar.
In this scenario, the gas in the inner CO ring is inside
corotation; due to the negative torques there, the gas
loses angular momentum and is therefore
flowing inwards.
Due to the weak bar, the rate of inflow
is relatively low, with a time-scale
of about 5 rotations (Fig. \ref{gastor3147}).
\begin{figure}
\includegraphics[angle=-90,width=\columnwidth]{9545f23n.ps}
\caption{Model rotation curve and derived frequencies $\Omega$,
$\Omega-\kappa/2$, and $\Omega+\kappa/2$, for NGC~3147. The star symbols
are the data derived from the CO kinematics, deprojected
with an inclination of $i$ = 29.5$^\circ$.
The horizontal line corresponds to
$\Omega_p\,=\,125$ km s$^{-1}$ kpc$^{-1}$.}
\label{vcur}
\end{figure}
The amount of gas flowing inwards is consistent with
what might be needed to fuel the Seyfert~2 nucleus.
The gas surface density in the central kiloparsec is
about 10~M$_{\odot}$\,pc$^{-2}$;
if the gas at 1\,kpc from the center falls in within 5 dynamical time-scales
(of $\sim$20 Myr),
then the feeding rate is about 0.5 M$_\odot$\,yr$^{-1}$.
Assuming a mass-to-energy conversion efficiency
of $\epsilon\sim10$\% ($L = dM/dt c^2 \epsilon$), then
a luminosity of order 3 $\times$10$^{45}$ erg s$^{-1}$ is accounted
for, more than enough for this low-luminosity AGN.
The spatial resolution of the CO maps is not
yet sufficient to determine more precisely
whether the gas is actually feeding the AGN, but we
can at least conclude that we are observing the gas inflow
into the $\sim$100--200 pc scale circumnuclear region.
\section{Conclusions}
\label{sec:Conclusions}
The molecular gas in the Seyfert 2 galaxy NGC~3147 has been
mapped with high resolution (1\farcs9 $\times$ 1\farcs6 Gaussian beam
for the $^{12}$CO(1-0) line) inside a radius of 25$^{\prime\prime}$ ($\sim$5 kpc).
The CO emission shows a central peak (mainly in $^{12}$CO(2-1))
and a ring at distance of 2\,kpc.
In $^{12}$CO(1-0) also an outer spiral at distance of
4\,kpc is detected.
The observed CO has a mean line intensity
ratio $I_{21}/ I_{10}$ $\sim$0.7, consistent with the optically
thick emission expected in the nuclei of spiral galaxies,
and broadly regular kinematics with some evidence for local
non-circular motions.
Comparing the molecular gas distribution with tracers of
star formation, we find that central emission and the inner
CO ring coincide well with the FUV (\textit{GALEX}) and 8\,$\mu$m
(\textit{Spitzer}) emission, while the outer CO spiral structure/ring is
instead located
within an interarm interval
detected by \textit{GALEX} and \textit{Spitzer}.
We interpret this partial disagreement as due to the transient
nature of the outer CO ring.
It would be a structure
in re-condensation that traces a future stellar generation, but
is decoupled from those that have heated the dust observable at
8\,$\mu$m or from whose emission is detected in the FUV.
With a NIR image obtained with the Canada-France-Hawaii Telescope,
we have identified the presence of a weak bar in NGC~3147,
a galaxy classified as non-barred in the optical.
This stellar bar acting on the gas produces gravity torques.
A portion of the gas present
in the inner $^{12}$CO(1-0) ring is trailing the bar, while part
of the gas leads it. The gravity torques are negative inside
a radius of $\sim$2\,kpc, while they become zero or positive outside.
In the outer $^{12}$CO(1-0) spiral the torques almost cancel
out on average, but are positive. In the more clumpy inner $^{12}$CO(2-1) ring,
the predominant torques are negative.
We interpret these results by identifying the resonances with the bar.
The inner CO ring would correspond to the ultra-harmonic resonance (r
= 2 kpc) just inside corotation, and the outer CO spiral
to the spiral structure emerging from the bar, beyond
its corotation, which we estimate is located at $\sim$3\,kpc.
In the inner CO ring, the gas is inside corotation due to the negative
torques there; it loses angular momentum and is flowing inwards.
NGC~3147 is not the first case in the NUGA sample where inflowing gas has been found:
NGC 2782 \citep{hunt}, and NGC 6574 \citep{lindt}
also show this feature. However, only NGC 2782 has a strong
and significant inflow, due to a nuclear bar
embedded in the primary one and to gas aligned with the nuclear bar.
The amount of inflowing gas in NGC~3147, quantified
from the gravity torques computation, is modest but could still be sufficient to feed the
Seyfert 2 nucleus at a rate of about 0.5 M$_\odot$\,yr$^{-1}$.
Although higher spatial resolution is needed to determine whether the
gas is actually feeding the AGN, we are observing the gas inflow to the
$\sim$100--200 pc central region.
\begin{acknowledgements}
The authors would like to thank the anonymous referee,
whose comments have been very instructive
and useful for improving the original version of the paper.
We thank the scientific and technical staff at IRAM for their
work in making our 30\,m and PdBI observations possible.
V. Casasola is pleased to acknowledge the hospitality and
stimulating environment provided by the Observatoire de
Paris-LERMA, where part of the work on this paper was done
during her stay in Paris, thanks to the EARA agreement.
In this work, we have made use of \textit{MAST/GALEX}
images and the \textit{Spitzer} archive.
\end{acknowledgements}
|
1,116,691,497,787 | arxiv | \section{Introduction}
It is well known that, in local quantum field theory, one encounters
divergences which arise from taking products of field operators at the same
space-time point. As a result, these products do not have a well-defined
meaning. Quite some time ago, Dirac~\cite{Dirac34}
suggested point splitting as a remedy
for this difficulty: instead of taking all the field operators at the
space-time point $x$, a fixed four-vector $\epsilon$ is introduced so that only
field operators with different arguments $(x,x \pm \epsilon, \ldots)$ appear in
their products. As long as $\epsilon$ is taken to be different from 0, the
products of field operators are well defined and the theory can be expected to
be free of divergences, \mbox{i.e.}, to be regularized. At the end of the
calculation, the limit $\epsilon \rightarrow 0$ is taken, in order to recover
the original theory.
{\it A prioiri}, there are many ways in which such a point splitting procedure
can be implemented. For gauge theories, however, one must ensure that the
introduction
of this new parameter $\epsilon$ preserves the invariance under gauge
transformations. It thus appears reasonable that one first attempts to
construct gauge transformations involving products of field operators taken at
different space-time points, which we shall call generalized gauge
transformations. Once such generalized gauge transformations are found, one can
then attempt to construct an action, which is invariant under these generalized
gauge
transformations.
In this paper, I shall show how these ideas can be put to work for the \(U(1)\)
gauge symmetry, although several aspects can be carried over to the general
Yang-Mills case~\cite{Gastmans96,Gastmans98}.
\section{Framework}
The standard infinitesimal gauge transformations, $\delta_\Lambda$, for the
photon field \(A_\mu(x)\) and the Dirac field $\psi(x)$ in the Abelian
\(U(1)\) case take the form
\begin{eqnarray}
\delta_\Lambda \, A_\mu(x)& = &- \partial_\mu \, \Lambda(x) \, ,
\nonumber\\
\delta_\Lambda \, \psi(x) &= &- \imath \, e \, \Lambda(x) \, \psi(x)
\, , \label{eq2}\\
\delta_\Lambda \, \bar\psi(x) &=& \imath \, e \, \Lambda(x) \, \bar\psi(x)
\, ,
\nonumber \end{eqnarray}
where $\Lambda(x)$ is the gauge parameter.
The gauge transformations of the fields satisfy the \(U(1)\) group
property: the commutator of two subsequent gauge transformations vanishes.
In what follows, I shall require that the generalized gauge transformations
preserve this Abelian character, \mbox{i.e.}, two such generalized transformations
should commute. This requirement imposes strong restrictions on the form such
transformations
can take. Nevertheless, it was found that such infinitesimal transformations
can be constructed and that, for gauge transformations on the fermion fields,
they take the form of an infinite series in powers of the coupling constant
$e$:
\begin{equation}
\delta_\Lambda \, \psi(x) = \sum\limits_{n = 1}^{\infty}
\, e^n \, \delta_\Lambda^{(n)} \, \psi(x) \, .
\end{equation}
This is to be contrasted with the standard \(U(1)\) case
without point splitting, where the {\it finite} gauge transformations on the
fermion field are of infinite order in the coupling constant, the {\it
infinitesimal} ones in Eqs.\,(\ref{eq2}) being only of first order in $e$.
The action $A$, invariant under these infinitesimal generalized gauge
transformations,
is also an infinite series in the coupling constant $e$:
\begin{equation}
A = \int d^4x \, {\cal L}(x) = \sum_{n = 0}^{\infty}\, e^n \, \int d^4x \,
{\cal L}^{(n)}(x)
\,,
\end{equation}
where ${\cal L}(x)$ is the Lagrangian density and ${\cal L}^{(n)}(x)$ its expansion in powers
of $e$. The action being an infinite series in $e$
leads to new peculiar Feynman rules. Besides the one-photon vertex, there
are in this
generalized theory also two-, three-, four-, $\ldots$ photon vertices.
In Sections~3 and 4, I shall present two explicit examples of generalized gauge
transformations.
\section{First example}
Perhaps the simplest Ansatz one can imagine for generalized gauge
transformations is
\begin{eqnarray}
\delta_\Lambda \, A_\mu(x) &=& - \partial_\mu \, \Lambda(x) \, ,
\nonumber \\
\delta_\Lambda^{(1)} \, \psi(x) &=& - \imath
\, \Lambda(x + \epsilon) \, \psi(x + 2 \epsilon) \,,
\end{eqnarray}
where $\epsilon$ is the point splitting four-vector as discussed in the
introduction.
To satisfy the Abelian group property, one can take, \mbox{e.g.},
\begin{equation}
\delta_\Lambda^{(2)} \, \psi(x) = - \frac{1}{2} \,
[\Lambda(x + \epsilon) + \Lambda(x + 3 \epsilon)]
\, \psi(x +4\epsilon) \int_{x + \epsilon}^{x + 3 \epsilon}
dy^{\alpha} \, A_\alpha(y),
\end{equation}
and so on for the higher order terms.
When $\epsilon \rightarrow 0$, then $\delta_\Lambda^{(1)} \, \psi(x)$
reduces to the standard gauge transformation~$[$Eqs.\,(\ref{eq2})$]$, and
$\delta_\Lambda^{(2)}\, \psi(x) \rightarrow 0$.
The general proof that the higher order terms
\(\delta_\Lambda^{(n)} \, \psi(x)\) indeed exist, can be found in
Ref.\,\cite{Gastmans94}.
The expansion of the corresponding invariant action then yields the following
lowest order results for the Lagrangian density:
\begin{eqnarray}
{\cal L}^{(0)} &=& \bar\psi(x) \, (\imath \,
\gamma^\mu \,
\partial_\mu - m) \, \psi(x) - \frac{1}{4} \, F_{\mu \nu}(x) \,
F^{\mu \nu}(x) \,,
\\
{\cal L}^{(1)} &=& \bar\psi(x - \epsilon) \, \gamma^\mu \,
\psi(x + \epsilon) \, A_\mu(x)\,,
\\
{\cal L}^{(2)} &=& \frac{1}{2} \,
\bar\psi(x - 2 \epsilon)
\, \gamma^\mu \, \psi(x + 2 \epsilon)
\nonumber \\
&& \times \bigg[[ A_\mu(x - \epsilon) + A_\mu(x + \epsilon)]
\int_{x - \epsilon}^{x + \epsilon} dy^\alpha A_\alpha(y)
\nonumber \\
&& + [\int_{x - \infty}^{x - \epsilon} dy^\alpha
\, A_\alpha(y)
+ \int_{x - \infty}^{x + \epsilon} dy^\alpha \, A_\alpha(y)]
\, \int_{x - \epsilon}^{x + \epsilon} dy^\beta \, F_{\mu
\beta}(y) \bigg]\,.
\end{eqnarray}
It is now obvious that through the incorporation of point splitting in the gauge
transformation, one obtains a nonlocal Lagrangian density. It is also
non-Hermitian.
If $\epsilon \rightarrow 0$ then ${\cal L}^{(1)}$ reduces to the standard QED
interaction Lagrangian density, and ${\cal L}^{(n)} \rightarrow 0$ for $n \geq 2$.\\
The appearance of infinite line integrals is an unattractive feature of this
approach, which will be remedied in Section~4.
\section{Second example}
In this example, the separation between the different space-time points is still
characterized by a fixed four-vector $\epsilon$, but for the
construction
of the infinitesimal generalized gauge transformations, one takes
an average over the separation in $ \epsilon$ using a weight function
$\rho(\eta)$. The generalized gauge
transformations of the fermion fields are again infinite series, the first order
term being
\begin{eqnarray}
\delta_\Lambda^{(1)} \, \psi(x) &=& - \imath \int_{- \infty}^{+ \infty}
\rho(\alpha) \, d\alpha
\int_{- \infty}^{+ \infty} \rho(\beta) \, d\beta \int_{- \infty}^{+ \infty}
\rho(\gamma) \, d\gamma
\nonumber \\
&&\times \, \Lambda(x + (\alpha + \gamma)\epsilon) \, \psi(x + (\beta +
\gamma)\epsilon) \, . \label{eq11}
\end{eqnarray}
To avoid infinite line integrals, the weight function $\rho(\eta)$ is taken
to be real and even. It is also normalized
\begin{equation}
\int_{-\infty}^{+ \infty} d\eta \, \rho(\eta) = 1 \, ,
\end{equation}
which guarantees that $\delta_\Lambda^{(1)} \, \psi(x)$ reduces to the
expression in Eqs.\,(\ref{eq2}) when $\epsilon \rightarrow 0$. Finally, it must
obey the convolution property
\begin{equation}
\int_{- \infty}^{+ \infty} d\eta \, \rho(\eta) \, \rho(\xi
- \eta) = \rho(\xi) \,.
\end{equation}
An example of a function satisfying these four conditions is
\begin{equation} \rho(\eta) = \frac{1}{\pi}\frac{\sin (\eta)}{\eta} \,.
\label{eq14}
\end{equation}
In momentum space, the generalized gauge
transformations [Eq.\,(\ref{eq11})] become~:
\begin{eqnarray}
\delta_\Lambda^{(1)} \, \psi(k) &=& - \imath \int d^4k_1
\int d^4k_2
\, \delta^{(4)}(k - k_1 - k_2) \, \Lambda(k_1) \, \psi(k_2) \, \tilde{\rho}(k_1
\cdot
\epsilon) \nonumber \\
&& \times \, \tilde{\rho}(k_2 \cdot \epsilon) \, \tilde{\rho}((k_1 + k_2) \cdot
\epsilon) \, , \label{eq15}
\end{eqnarray}
with $\tilde\rho(\omega)$ the Fourier transform of $\rho(\eta)$,
$\Lambda(k)$ and $\psi(k)$ being the Fourier transforms of $\Lambda(x)$
and $\psi(x)$ respectively. The
properties of $\rho(\eta)$ translate into the following properties of
$\tilde\rho(\omega)$:
(i) $\tilde\rho(\omega)$ is real and even;
(ii) $\tilde\rho(0) = 1$;
(iii) $\tilde{\rho}^2(\omega) = \tilde{\rho}(\omega)$,
implying that $\tilde\rho(\omega) = 0$ or $1$.
For the example in Eq.\,(\ref{eq14}), one has that
\begin{eqnarray}
\tilde\rho(\omega) =
\left\{
\begin{array}{ll}
1, &{\rm if} \: \: |\omega| < 1\\
0, &{\rm if} \: \: |\omega| \geq 1 \, .
\end{array}
\right.
\label{eq17}
\end{eqnarray}
To satisfy the Abelian group property of Section 2, one can take, e.g.,
\begin{eqnarray}
\delta_\Lambda^{(2)} \, \psi(k) &=& - \imath \int d^4k_1
\int d^4k_2 \int d^4k_3
\, \delta^{(4)}(k - k_1 - k_2 - k_3) \nonumber \\
&&\times \, \frac{A(k_1) \cdot \epsilon}{k_1 \cdot \epsilon} \, \Lambda(k_2) \,
\psi(k_3) \, [1 - \tilde{\rho}((k_2 + k_3) \cdot \epsilon)] \nonumber \\
&&\times \, \tilde{\rho}(k_1 \cdot \epsilon) \,
\tilde{\rho}(k_2 \cdot \epsilon)\, \tilde{\rho}(k_3 \cdot \epsilon)\,
\tilde{\rho}((k_1 + k_2 + k_3) \cdot \epsilon) \, . \label{eq18}
\end{eqnarray}
and so on for the higher order terms \cite{Olivie97}.
In spite of the appearance of $k_1 \cdot \epsilon$ in the denominator in
Eq.\,(\ref{eq18}), the
expression for $\delta_\Lambda^{(2)} \, \psi(k)$ is free of singularities for
$k_1 \cdot \epsilon \rightarrow 0$. To see this, it suffices to observe that,
for $k_1 \cdot \epsilon \rightarrow 0$, $\tilde\rho((k_1 + k_2 + k_3) \cdot
\epsilon) \rightarrow \tilde\rho((k_2 + k_3)\cdot \epsilon)$ and that $[1 -
\tilde\rho(\omega)] \, \tilde\rho(\omega) = 0$ for all $\omega$.
Through this example, the relation between point splitting and the mitigation of
ultraviolet divergences becomes apparent. If the function $\tilde{\rho}(\omega)$ is
taken as in Eq.\,(\ref{eq17}), then, in Eq.\,(\ref{eq15}) for
$\delta_\Lambda^{(1)} \, \psi(k)$, the functions
$\tilde\rho(k_1 \cdot \epsilon)$, $\tilde\rho(k_2 \cdot \epsilon)$ and
$\tilde\rho((k_1 + k_2) \cdot \epsilon)$ cut off the high momentum
components in the direction of $\epsilon$ of
$\Lambda(k_1)$, $\psi(k_2)$ and $\delta_\Lambda^{(1)} \, \psi(k)$ respectively.
A similar property holds for the second order term in Eq.\,(\ref{eq18}),
and it is also valid for the higher order terms.
These generalized gauge transformations reduce to the standard
gauge transformations in the limit $\epsilon \rightarrow 0$.
Indeed, the first order term $\delta_\Lambda^{(1)} \,
\psi(k)$ [Eq.\,(\ref{eq15})] reduces to
\begin{eqnarray}
\delta_\Lambda^{(1)} \, \psi(k) &=& - \imath \int d^4k_1
\int d^4k_2
\, \delta^{(4)}(k - k_1 - k_2) \, \Lambda(k_1) \, \psi(k_2) \,, \label{eq40}
\end{eqnarray}
because $\tilde{\rho}(0) = 1$.
Eq.\,(\ref{eq40}) is exactly the Fourier transform of the standard infinitesimal gauge
transformation of Eq.\,(\ref{eq2}). Furhermore, when $\epsilon \rightarrow 0$,
the second order term $\delta_\Lambda^{(2)} \, \psi(k)
\rightarrow 0$ in Eq.\,(\ref{eq18}), and similarly for the higher order terms
$\delta_\Lambda^{(n)} \, \psi(k)$ for $n > 2$.
Again, one can construct a generalized action, invariant under
these generalized gauge transformations, the details of which are given in
Ref.\,\cite{Olivie97}.
The advantage of the approach in this second example is that it can also be
generalized to the case of Yang-Mills theories
\cite{Gastmans96,Gastmans98}. In those cases, one finds that it is absolutely necessary to average in the generalized
gauge transformations over the different arguments for the field operators.
One is still faced with a shortcoming in this approach: the momentum cut off of
the functions only occures in the direction of the four-vector $\epsilon$, and,
as a result, one cannot expect that all ultraviolet divergences are regulated.
\section{Conclusions}
We have shown that it is possible to construct generalized gauge transformations
for which the fields and the gauge parameters are taken at different space-time
points. The separation between the different space-time points is characterized
by a fixed four-vector $\epsilon$.
Two examples of generalized gauge transformations and invariant actions are
presented here.
It appears that the generalized infinitesimal gauge
transformations of the fermion fields and the invariant actions are infinite
series in the coupling constant $e$.
In the limit $\epsilon \rightarrow 0$, they reduce to the standard
expressions one encounters in QED.
In the second example, the conditions which the weight function has to satisfy
lead to the introduction of a cut off for the large momentum
components of the fields in the direction of $\epsilon$. We are thus
led to consider generalized gauge transformations which
only act on the small momentum components of the fields.
\section*{Acknowledgements}
I would like to thank Professors Raymond Gastmans and Tai Tsun Wu for sharing
their insights in this work with me.
|
1,116,691,497,788 | arxiv | \section{1. Introduction}
There is an increasing literature on the application of Structural VARMA
(SVARMA) models for the analysis of economic data which tries to solve the
identification problem of these models by incorporating information from the
distribution of non-Gaussian structural shocks. This information recovered
from the data can substitute, at least in part, the restrictions provided by
economic theory on the impulse response functions (IRF) of endogenous
variables to given shocks. Furthermore, there are many examples where it is
not possible to discard non-fundamental solutions to a wide class of dynamic
macroeconomic models with features affecting the flow of information used by agents to make
decisions, see e.g. the surveys in Alessi, Barigozzi and Capasso (2011) and
Gouri\'{e}roux, Monfort and Renne (2019). In parallel, the identification of
noncausal structural VAR models have been also investigated, see e.g. Lanne
and Saikkonen (2013).
The identification analysis of SVARMA models has to account for both static
(Structural) and dynamic (VARMA) aspects. The dynamic problem
is related to the location of the determinantal roots of the VAR and VMA
polynomials that lead to causal/noncausal or invertible/noninvertible
solutions, respectively, guaranteeing that model errors are unpredictable,
not just a serially uncorrelated white noise sequence. The static
identification relates to the choice of the particular rotation of the
reduced form errors that delivers the true vector of structural shocks with
proper economic interpretation. Therefore, these shocks must additionally
satisfy some mutual independence condition strengthening the uncorrelation
achieved by any square root transformation of the covariance matrix of the
reduced form errors.
Under Gaussianity, SVARMA identification is not possible in absence of
further restrictions provided by economic theory because uncorrelation is
equivalent to independence and therefore all the infinite sequences obtained
by different versions of the lag polynomials obtained by flipping roots
and/or rotating the different shocks through Blaschke orthogonal matrices
(see e.g. Lippi and Reichlin, 1994) would be admissible. However, under
non-Gaussianity and the independence component assumption (ICA) of the
structural shocks it is known that static rotations can be identified up to
permutation and sign, i.e. up to labeling of the shocks obtained, if at most
one of the innovations components is Gaussian (see Comon (1994), Hyv\"{a}%
rinen, Zhang, Shimizu and Hoyer (2010)), while a condition on higher order
cumulants and moments of serially
independent errors guarantees the dynamic identification (see Chan and Ho (2004), Chan, Ho and Tong (2006), Gouri%
\'{e}roux et al. (2019)). However, these results do not lead to specific
methods for designing parameter estimates and inference rules that rely on
the identifying assumptions and are easy to interpret and check for
particular models. Instead, most of available methods are based instead on
moment estimates for which local rank conditions are assumed after a basic
order condition is guaranteed or on (Pseudo) ML procedures which have to be
further justified. This is precisely the aim of this paper, to provide neat
inference methods exploiting efficiently global identification conditions
based on a minimal finite number of moments of the marginal and joint
distributions of the sequence of structural non-Gaussian errors.
Typically, non-Gaussianity is exploited for identification of dynamic models
through conditions on higher order cumulants (Gouri\'{e}roux et al., 2019;
Lanne and Luoto, 2019) or spectral densities (Lii and Rosenblatt, 1982;
Kumon, 1992), but it has also been imposed through particular probability
distribution assumptions on the shocks (Lanne and L\"{u}tkepohl, 2010) or
with conditional (Normandin and Phaneuf, 2004) and unconditional
heteroskedasticity conditions (Rigobon, 2003; Lanne and L\"{u}tkepohl,
2008), possibly with Markov switching dynamics (Lanne, L\"{u}tkepohl and
Maciejowska, 2010; L\"{u}tkepohl and Net\^{s}unajev, 2017). Then, estimation
is performed using ML or approximate versions of it (e.g. Lii and
Rosenblatt, 1992, 1996, for ARMA; Gouri\'{e}roux, Monfort and Renne, 2017,
and Lanne, Meitz and Saikkonen, 2017, for SVAR; Gouri\'{e}roux et al., 2019,
for SVARMA models) or non-Gaussian criteria like LAD or ranks (Breidt, Davis
and Trindade, 2001, and Andrews, Davis and Breidt, 2007, in the univariate
case). Methods based on higher order moments have been also developed, first
for the univariate case in the frequency and time domains (Lii and
Rosenblatt, 1982, Gospodinov and Ng, 2015, respectively). For multivariate
models, Gouri\'{e}roux et al. (2019) proposed a semiparametric 2-step
method, where first the VAR parameters are estimated using a 2SLS approach
under causality, and then the VMA parameters are estimated using moment
conditions on linear combinations of the residuals or using a PMLE
approximation to some prespecified non-Gaussian distribution. The
restrictions on moments of order 2, 3 and 4 are derived from ICA to improve
efficiency and raise the chances that they provide sufficient information to
guarantee usual local identification rank conditions. A similar approach is
pursued in Lanne and Luoto (2019) to achieve local identification of a SVAR
model by imposing a certain set of co-kurtosis conditions.
In this paper we study the problem of SVARMA identification extending the
frequency domain approach of Velasco and Lobato (2018), henceforth VL, to
the multivariate and structural case. VL showed that identification of a
possible noncausal or noninvertible ARMA model can be achieved by checking
that higher order spectral densities are sensitive to the location of the
roots of the lags polynomials unlike the usual second order spectral
density, i.e. they can achieve phase identification as noted in Lii and Rosenblatt
(1992). They also investigated model estimation using a minimum distance
criterion between the higher order periodograms and the parametric
specification of the corresponding higher order spectral densities of the
ARMA model that accounts efficiently for all moment conditions of a given
order at all lags. This approach provides a comprehensive method for dealing
with the problem of the location of roots of the lag polynomials and the
characterization of the non-Gaussian information through higher order
spectral densities to develop robust and efficient estimates, see also
Lobato and Velasco (2018).
To extend these ideas to the SVARMA setting, we first develop a new
representation of higher order spectral densities arrays for linear vector
processes and show that our identifying frequency domain criterion can
indeed discriminate processes that have observationally equivalent linear
covariance dynamics, but whose different IRFs are reflected on their higher
order dynamics. We are able to reproduce the previous dynamic and static
identification results found in the literature for non-Gaussian vector
models assuming only ICA and serial independence up to a given order (third
and/or fourth) and providing some extensions when some non-zero (i.e.
non-Gaussian) cumulant condition is violated
or when no version of ICA holds but we impose a rank condition on the
innovations third order cumulant array. These results rely on a simple
non-singularity condition on the transfer function of the VARMA system so
that our criterion can evaluate all versions of the model up to a Blaschke
factor and the value of higher order cumulants of structural errors.
This nonparametric global identification provides a constructive method for
designing minimum distance parameter estimates in the frequency domain which
can exploit efficiently all information contained in the dynamics of moments
of order 2, 3 and 4 without distributional assumptions or factorizations of
the matrix lag polynomials to deal with the simultaneous presence of roots
inside and outside the unit circle. Despite SVARMA identification up to a
signed permutation is enough for IRF and decomposition variance analyses
(given that a particular labeling can be attached to each shock), to obtain
standard asymptotic results for our parameter estimates we fix a unique
identified version of the model using a particular ordering and sign
structure on the innovations. These restrictions could be replaced by
alternative statistical conditions or economic information, which then would
become overidentification restrictions that could be tested in our
framework. We also develop bootstrap approximations for the asymptotic
distribution of parameter estimates and for the computation of efficient
estimates exploiting all moment conditions available. The finite sample
properties of a numerical algorithm to implement these identification and
inference methods are explored with real and simulated data.
The rest of the paper is organized as follows. Section~2 sets the
identification problem and introduces the main concepts and tools. Section~3
provides the basic identification results. Section~4 deals with parameter
identification and Section~5 with parameter minimum distance estimation.
Section~6 analyzes GMM efficient estimates exploiting information from
moments of several orders and bootstrap approximations to the distribution
of estimates. Section~7 presents the numerical methods and the simulation
experiment. Section~8 reanalyses Blanchard and Quah (1989) identification of
a bivariate system for US GNP growth and unemployment. A series of
appendices include additional discussion of concepts used in the paper,
together with proofs and auxiliary results.\bigskip
\section{2. Identification problem and assumptions}
We consider the SVARMA$\left( p,q\right) $ system
\begin{equation*}
\Phi \left( L\right) Y_{t}=\mu +\Theta \left( L\right) \mathbf{\varepsilon }%
_{t},
\end{equation*}%
where the $d$-vector $\mathbf{\varepsilon }_{t}$ behaves as an independent
identically distributed (\textit{iid}) sequence up to a finite number $k$ of
moments, $k\geq 3,$ with zero mean and covariance matrix $\mathbf{I}_{d},$
the $d$-dimensional identity matrix, but with components not necessarily
mutually independent.
The vector $\mu $ is an unknown level parameter and the lag polynomials with
matrix coefficients $\Phi \left( L\right) =\mathbf{I}_{p}+\Phi _{1}L+\cdots
+\Phi _{p}L^{p}$ and $\Theta \left( L\right) =\Theta _{0}+\Theta
_{1}L+\cdots +\Theta _{q}L^{q}$, $\Theta _{0}$ nonsingular, satisfy det$%
\left( \Phi \left( z\right) \right) $det$\left( \Theta \left( z\right)
\right) \neq 0$ for $\left \vert z\right \vert =1$. These conditions
guarantee the existence of a stationary solution for $Y_{t}.$ Note that we
allow the roots of the determinants of $\Theta \left( z\right) $ or $\Phi
\left( z\right) $ to be inside or outside the unit circle so that the
expansions of $\Psi \left( z\right) :=\Phi ^{-1}\left( z\right) \Theta
\left( z\right) $ and $\Psi ^{-1}\left( z\right) $ could include powers of $%
z $ and $z^{-1}$ simultaneously, accounting for noncausal or noninvertible
systems.
To investigate the identification problems on the location of the roots of
the matrix polynomials $\Phi \left( z\right) \ $and$\ \Theta \left( z\right)
$ and on the components of $\mathbf{\varepsilon }_{t}$ determined by $\Theta
_{0}$, we use the device of Blaschke matrices (BM) that are generalized
orthogonal matrices. Following Lippi and Reichlin (1994), and denoting by $%
\ast $ simultaneous transposition and complex conjugation, a $d\times d$
matrix $A\left( z\right) $ is a BM if \newline
\textbf{1}. $A\left( z\right) $ has no poles of modulus smaller or equal to
unity and
\textbf{2}. $A\left( z\right) ^{-1}=A^{\ast }\left( z^{-1}\right) ,$ i.e. $%
A\left( z\right) A^{\ast }\left( z^{-1}\right) =\mathbf{I}_{d}.$ \newline
Further, for any BM, there exists an integer $r$ and complex numbers $a_{j},$
$j=1,\ldots ,r,$ $\left \vert a_{j}\right \vert <1,$ such that%
\begin{equation}
A\left( z\right) =K_{0}R\left( a_{1},z\right) K_{1}R\left( a_{2},z\right)
K_{2}\cdots K_{r-1}R\left( a_{r},z\right) K_{r}, \label{BM}
\end{equation}%
where $K_{j}$ are orthogonal matrices\footnote{%
Notice that in Theorem~1 of Lippi and Reichlin (1995) it is fixed that $%
K_{0}=\mathbf{I}_{d},$ but in general $K_{0}$ needs to be different from
identity to complete their proof as can be seen for the BM $A\left( z\right)
=diag(1,g_{a}\left( z\right) )$, }, $K_{j}K_{j}^{\prime }=\mathbf{I}_{d},$ and
\begin{equation*}
R\left( a,z\right) =\left(
\begin{array}{cc}
g_{a}\left( z\right) & 0 \\
0 & \mathbf{I}_{d-1}%
\end{array}%
\right) ,\ \ \ g_{a}\left( z\right) =\frac{z-a}{1-a^{\ast }z},
\end{equation*}%
see also the discussion in Hannan (1970, pp.~65-67).
For any BM $A\left( z\right) ,$ we can write\ \
\begin{equation}
Y_{t}=\Psi \left( L\right) A\left( L\right) \mathbf{u}_{t},\ \ \ \
\label{u}
\end{equation}%
where $\mathbf{u}_{t}=A\left( L\right) ^{-1}\mathbf{\varepsilon }_{t}\ $is a
serially uncorrelated all-pass process though not independent, and, because
its spectral density matrix is constant, $f_{\mathbf{u}}\left( \lambda
\right) =\left( 2\pi \right) ^{-1}A^{-1}\left( e^{-i\lambda }\right)
A^{-1\ast }\left( e^{i\lambda }\right) =\left( 2\pi \right) ^{-1}\mathbf{I}%
_{d},$ we conclude that the spectral density implied by the representation ($%
\ref{u}$) for any BM $A\left( z\right) $ and any $\mathbf{u}_{t}$ is always
the same,%
\begin{equation*}
f\left( \lambda \right) =\Psi \left( e^{-i\lambda }\right) A\left(
e^{-i\lambda }\right) f_{\mathbf{u}}\left( \lambda \right) A^{\ast }\left(
e^{i\lambda }\right) \Psi ^{\ast }\left( e^{i\lambda }\right) =\frac{1}{2\pi
}\Psi \left( e^{-i\lambda }\right) \Psi ^{\ast }\left( e^{i\lambda }\right) .
\end{equation*}%
The same conclusion arises if $A$ is the inverse of a BM, and in particular
when in representation $\left( \ref{BM}\right) $ it holds that $%
1/a_{j}^{\ast }$ equals an actual root of det$\left( \Psi \left( z\right)
\right) ,$ irrespectively of being inside or outside the complex unit
circle, in a process of flipping the roots of det$\left( \Psi \left(
z\right) \right) $.
These facts imply at once that using only second order information we can
not identify the location of these roots with respect to the unit circle,
and, even with knowledge of $p$ and $q,$ there are infinite VARMA
representations with the same second order properties but different IRF $%
\Psi \left( L\right) A\left( L\right) $ and error sequence $A(L)^{-1}\mathbf{%
\varepsilon }_{t}$. These alternative IRFs and errors are associated to
invertible and noninvertible representations when $q>0$ and to causal and
noncausal representations when $q=0$ as in this case $\left( \Psi \left(
L\right) A\left( L\right) \right) ^{-1}=A^{-1}\left( L\right) \Theta
_{0}^{-1}\Phi \left( L\right) $ and the (inverse of the) roots of $%
A^{-1}\left( L\right) $ can match those of $\Phi \left( L\right) $ in our
generalized setup. Traditional estimation methods based on Gaussian PML,
like Whittle approximation, only consider causal and invertible
representations, but still have to deal with the static problem that arises
for $A\left( L\right) $ constant.
The static identification problem refers to the well known lack of
identification of standard Structural VAR(MA) models with respect to
orthogonal rotations $\mathbf{u }_{t}=K \mathbf{\varepsilon }_{t}$ of the
structural errors in absence of further identifying assumptions on the IRF
provided by economic theory and/or further model structure (see e.g.
Rubio-Ramirez, Waggoner and Zha, 2010, for equality restrictions, and
Granziera, Moon and Schorfheide, 2018, for sign restrictions). However, it
is possible to consider this static problem within the same framework by
allowing BM which are constant and equal to an orthogonal matrix. Then, when
dynamics are known or sufficient conditions for their identification are
imposed (and e.g. causality and invertibility), we can identify \textit{%
statistically} the structural shocks in SVARMA models by higher order moment
conditions implied by ICA under non-Gaussianity without further restrictions.
To consider all these situations when trying to identify a SVARMA model we
extend the concept of Blaschke Matrix (BM) to any matrix $A\left( z\right) $
that satisfies the orthogonality condition 2. and\newline
\textbf{1}$^{\ast }$. $A\left( z\right) $ has no poles of modulus equal to
unity, but could have some with modulus larger or smaller than unity.
\newline
Then, in the representation $\left( \ref{BM}\right) $ for a BM $A\left(
z\right) $ we allow for $\left \vert a_{j}\right \vert >1$ as well as $%
\left
\vert a_{j}\right \vert <1$, so that there exists an integer $%
r=0,1,\ldots $, complex numbers $a_{i},$ $j=1,\ldots ,r$ and a $\eta >0$
such that
\begin{equation*}
\min_{j}\left \vert \left \vert a_{j}\right \vert -1\right \vert \geq \eta
>0,
\end{equation*}%
where the case $r=0$ is interpreted as $A\left( z\right) =K_{0}$ being a
constant (in $z$) orthogonal matrix. Since we do not restrict the $a_{j}$ so
that $1/a_{j}^{\ast }$ matches a root of $\Psi \left( z\right) ,$
considering any BM $A\left( z\right) $ we can deal with both basic and
non-basic representations of VARMA models in the sense of Lippi and Reichlin
(1994).
To solve the problem that second order dynamics cannot identify the phase of
$\Psi $, we resort to higher order moments as proposed by Lii and Rosenblatt
(1992). In Appendix~A we develop a compact representation of the spectral
density $f_{\mathbf{a},k}$ of $\left( Y_{t,\mathbf{a}\left( 1\right)
},\ldots ,Y_{t,\mathbf{a}\left( k\right) }\right) $ for any order $%
k=2,3,\ldots ,$ and $k$-tuple $\mathbf{a}=\left( \mathbf{a}\left( 1\right)
,\ldots ,\mathbf{a}\left( k\right) \right) ,$ where $Y_{t}$ follows a linear
model with IRF $\Psi (L)$ and innovations $\mathbf{\varepsilon }_{t}$ which
are \emph{iid} up to moments of order $k.$ The $k$-th order cumulants of the
vector $\mathbf{\varepsilon }_{t}$ can be characterized by the $d^{2}\times
d^{k-2}$ matrix v$\mathbf{\kappa }_{k}^{0}$,
\begin{equation*}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}^{0}:=\left[ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{\kappa }%
_{\cdot \cdot 1\cdots 1}\right) \ \ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{\kappa }_{\cdot
\cdot 2\cdots 1}\right) \ \ \cdots \ \ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{\kappa }%
_{\cdot \cdot d\cdots d}\right) \right] ,
\end{equation*}%
where $\mathbf{\kappa }_{\cdot \cdot j\left( 3\right) \cdots j\left(
k\right) }$ is the $d\times d$ matrix with typical $\left( j\left( 1\right)
,j\left( 2\right) \right) $ element equal to the $k$-th order joint cumulant
cum$\left( \mathbf{\varepsilon }_{t,j\left( 1\right) },\mathbf{\varepsilon }%
_{t,j\left( 2\right) },\mathbf{\varepsilon }_{t,j\left( 3\right) },\ldots ,%
\mathbf{\varepsilon }_{t,j\left( k\right) }\right) ,$ $j\left( h\right) \in
\left \{ 1,\ldots ,d\right \} .$ Then we find that%
\begin{equation*}
f_{\mathbf{a},k}(\boldsymbol{\lambda })=\frac{1}{\left( 2\pi \right) ^{k-1}}%
\Psi _{\mathbf{a}}^{\otimes k}\left( \mathbf{\lambda }\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}%
\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}^{0}\right) ,
\end{equation*}%
where for $\boldsymbol{\lambda }=(\lambda _{1},\ldots ,\lambda _{k-1})$ we
define%
\begin{equation*}
\Psi _{\mathbf{a}}^{\otimes k}\left( \mathbf{\lambda }\right) :=\Psi _{%
\mathbf{a}\left( k\right) }\left( e^{i\left( \lambda _{1}+\cdots +\lambda
_{k-1}\right) }\right) \otimes \Psi _{\mathbf{a}\left( k-1\right) }\left(
e^{-i\lambda _{k-1}}\right) \otimes \cdots \otimes \Psi _{\mathbf{a}\left(
2\right) }\left( e^{-i\lambda _{2}}\right) \otimes \Psi _{\mathbf{a}\left(
1\right) }\left( e^{-i\lambda _{1}}\right)
\end{equation*}%
for the usual Kronecker product $\otimes $ on the rows $\Psi _{\mathbf{a}%
\left( j\right) }$ of $\Psi $. This representation produces the usual
spectral density for $k=2$ because
\begin{equation*}
f_{(\mathbf{a}(1),\mathbf{a}(2)),2}(\lambda )=\frac{1}{2\pi }\Psi _{\mathbf{a%
}(2)}(e^{i\lambda })\otimes \Psi _{\mathbf{a}(1)}(e^{-i\lambda })\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}%
\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{2}^{0}\right) =\frac{1}{2\pi }\Psi _{%
\mathbf{a}(1)}(e^{-i\lambda })\Psi _{\mathbf{a}(2)}^{\prime }(e^{i\lambda }),
\end{equation*}%
as in this case$\ $v$\mathbf{\kappa }_{2}^{0}=\ $vec$(E[\mathbf{\varepsilon }%
_{t}\mathbf{\varepsilon }_{t}^{\prime }])=\ $vec$(\mathbf{I}_{d})$, where $%
\mathbf{I}_{d}$ is the covariance matrix of $\mathbf{\varepsilon }_{t}$
under the imposed normalization.
We now discuss the intuition on why higher order spectral densities with $%
k\geq 3$ can achieve dynamics identification unlike for $k=2.$ Thus,
\begin{equation*}
f_{\mathbf{a},3}(\boldsymbol{\lambda };A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3})=\frac{%
1}{\left( 2\pi \right) ^{2}}\Psi _{\mathbf{a}}^{\otimes 3}\left( \mathbf{%
\lambda }\right) A^{\otimes 3}\left( \mathbf{\lambda }\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}%
\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}\right)
\end{equation*}%
is the implied $k=3$ spectral density for any third order marginal cumulants
matrix v$\mathbf{\kappa }_{3}$ under the (wrong) assumption that $\mathbf{u}%
_{t}=A\left( L\right) ^{-1}\mathbf{\varepsilon }_{t}$ is an $iid\left( 0,%
\mathbf{I}_{d}\right) $ sequence in (\ref{u}) for any non-constant BM $%
A\left( z\right) $, as the true $\mathbf{\varepsilon }_{t}$, and not just
serially uncorrelated.
Under some identification (rank) assumptions $f_{\mathbf{a},3}(\boldsymbol{%
\lambda };A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3})$ does differ from the true density
$f_{\mathbf{a},3}(\boldsymbol{\lambda })=f_{\mathbf{a},k}(\boldsymbol{%
\lambda };\mathbf{I}_{d},$v$\mathbf{\kappa }_{3}^{0})$ for all choices of v$%
\mathbf{\kappa }_{3}$ because $A^{\otimes 3}\left( \mathbf{\lambda }\right)
\,$vec$\, \left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}\right) $ depends on $\mathbf{%
\lambda} $ in general, unlike $A^{\otimes 2}\left( \lambda \right) $vec$%
\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{2}\right) =\ $vec$\left( A\left(
e^{-i\lambda _{1}}\right) \mathbf{I}_{d}A^{\prime }\left( e^{i\lambda
_{1}}\right) \right) =$ vec$\left( \mathbf{I}_{d}\right) .$ In particular,
for $d=1,$ $A^{\otimes 3}\left( \mathbf{\lambda }\right) $ is the bispectral
density of an all-pass process, which is not constant unlike its second
order spectral density. Similar arguments apply for any higher spectral
density, so, following VL, we can set up an $L^{2}$ distance between $f_{%
\mathbf{a},k}(\boldsymbol{\lambda };A,$v$\mathbf{\kappa }_{k})\ $and $f_{%
\mathbf{a},k}(\boldsymbol{\lambda }),$ for all $k$-tuples $\mathbf{a}$ from $%
\{1,2,\ldots ,d\}$,
\begin{equation*}
\mathcal{L}_{k}^{0}\left( A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}\right) :=\sum_{%
\mathbf{a}}\int_{\Pi ^{k-1}}\left \vert f_{\mathbf{a},k}(\boldsymbol{\lambda
};A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k})-f_{\mathbf{a},k}(\boldsymbol{\lambda }%
)\right \vert ^{2}d\boldsymbol{\lambda },
\end{equation*}%
with known $\Psi ,$ but unknown location of the roots of $\Theta \left(
z\right) $ (or $\Phi \left( z\right) $ if $q=0$) expressed by the factor $A$%
, possibly flipping inside or outside some of these roots, adding additional
all-pass dynamics, or simply rotating elements of $\mathbf{\varepsilon }%
_{t}. $ Here $\Pi =\left[ -\pi ,\pi \right] $ and $\Pi ^{k-1}$ is the ($k-1)$%
-th cartesian product of $\Pi $\textbf{.} Obviously $\mathcal{L}_{k}\left(
\mathbf{I}_{d},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}^{0}\right) =0,$ but we have to
rule out the possibility that for some spectral factor $A\neq \mathbf{I}_{d}$
it is possible to choose some v$\mathbf{\kappa }_{k}$ such that $\mathcal{L}%
_{k}\left( A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}\right) =0$ for a given $k>2$, as
indeed it is possible for $k=2$ just setting v$\mathbf{\kappa }_{2}=\mathbf{I%
}_{d}.$
To illustrate this problem notice that we can write $\mathcal{L}%
_{k}^{0}\left( A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}\right) $ as%
\begin{equation*}
\int_{\Pi ^{k-1}}\left \{ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\! \left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{k}\right) ^{\prime }A^{\otimes k}\! \left( \mathbf{\lambda }\right) ^{\ast
}-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\! \left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}^{0}\right) ^{\prime
}\right \} \Upsilon _{k}^{0}\left( \mathbf{I}_{d},\boldsymbol{\lambda }%
\right) \left \{ A^{\otimes k}\left( \mathbf{\lambda }\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}%
\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}\right) -\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{%
\kappa }_{k}^{0}\right) \right \} d\boldsymbol{\lambda },
\end{equation*}%
where for any $d\times d$ spectral factor $A$ and $\boldsymbol{\lambda }\in
\Pi ^{k-1}$ we define for $k=2,3,\ldots $
\begin{equation*}
\Upsilon _{k}^{0}\left( A,\boldsymbol{\lambda }\right) :=A^{\otimes k}\left(
\mathbf{\lambda }\right) ^{\ast }\sum_{\mathbf{a}}\Psi _{\mathbf{a}%
}^{\otimes k}\left( \mathbf{\lambda }\right) ^{\ast }\Psi _{\mathbf{a}%
}^{\otimes k}\left( \mathbf{\lambda }\right) A^{\otimes k}\left( \mathbf{%
\lambda }\right) .
\end{equation*}%
Then, for identification of $\Psi $ under the assumption that $\Upsilon
_{k}^{0}\left( \mathbf{I}_{d},\boldsymbol{\lambda }\right) $ is full rank
for every $\boldsymbol{\lambda }$, we have to rule out the possibility that
for some BM $A\neq \mathbf{I}_{d}$, possibly constant, and some choice of v$%
\mathbf{\kappa }_{k}$
it holds that $A^{\otimes k}\left( \mathbf{\lambda }\right) $vec$\left(
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}\right) -$vec$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{k}^{0}\right) =0$ for all $\mathbf{\lambda }$ (except possibly in a set of
measure zero), implying that $\mathcal{L}_{k}^{0}\left( A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{%
\kappa }_{k}\right) =0.$
We now introduce rank conditions on $\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}^{0}$ to
reduce to a minimum the range of situations where identification is lost for
$k=3$ and $4$. For dynamic identification it is sufficient to use a rank
condition on $\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0}$, but when using $k=4$
moments, we need to impose ICA with nonzero marginal kurtosis coefficients
for all components of $\mathbf{\varepsilon }_{t}$, though no further
conditions on higher order moments are required in contrast to Chan et al.
(2006) and Gouri\'{e}roux et al. (2019).
For static identification we also impose ICA of order $k$ among the
components of $\mathbf{\varepsilon }_{t}$ but we allow for at most one of
the components to have zero marginal cumulants, being possibly Gaussian,
resembling the result of Comon (1994). We also relax the serial \emph{idd}
assumption on $\mathbf{\varepsilon }_{t} $ to equal distribution and
independence up to moments of order $k$, which is sufficient for
stationarity of $Y_{t}$ of order $k$ and to specify the corresponding higher
order spectral densities without further conditions on probability
distributions or conditional moments. Denote by $\lambda _{\min }\left(
M\right) $ the minimum eigenvalue of a matrix $M$ and let $\mathbf{\alpha }%
_{k}^{0}=\left( \mathbf{\alpha }_{k1}^{0},\ldots ,\mathbf{\alpha }%
_{kd}^{0}\right) ^{\prime }$ be the true vector of marginal cumulants of
order $k$ of $\mathbf{\varepsilon }_{t}.$ \bigskip
\noindent \textbf{Assumption~1}$\left( k\right) $: The $\mathbf{\varepsilon }%
_{t}$ are stationary and serially independent up to $k$ moments$,$ $%
E\left
\Vert \mathbf{\varepsilon }_{t}\right \Vert ^{k}<\infty ,$ and are
standardized with zero mean and $\mathbf{I}_{d}$ covariance matrix. \bigskip
\noindent \textbf{Assumption~2}$\left( 3\right) $. Rank$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}%
\mathbf{\kappa }_{3}^{0}\right) =d.$\bigskip
\noindent \textbf{Assumption~3}$\left( k\right) $. The components of $%
\mathbf{\varepsilon }_{t}\ $are independent up to $k$ moments and
\begin{itemize}
\item For $k=3$ all marginal skewness coefficients are nonzero, $\mathbf{%
\alpha }_{3j}^{0}=\mathbf{\kappa }_{jjj}^{0}\neq 0,$ $j=1,\ldots ,d.$
\item For $k=4$ all marginal kurtosis coefficients are nonzero, $\mathbf{%
\alpha }_{4j}^{0}=\mathbf{\kappa }_{jjjj}^{0}\neq 0,$ $j=1,\ldots ,d.$%
\bigskip
\end{itemize}
\noindent \textbf{Assumption~4}: For some $\eta >0,$%
\begin{equation*}
\inf_{\left \vert z\right \vert =1}\lambda _{\min }\left( \Psi \left(
z\right) \right) \geq \eta >0.
\end{equation*}
\smallskip
Assumption~4 determines the full rank of the dynamic system excluding unit
roots on the AR and MA lag polynomials, so $\Upsilon _{k}^{0}\left( \mathbf{I%
}_{d},\mathbf{\lambda }\right) =\left( \Psi ^{\otimes k}\left( \mathbf{%
\lambda }\right) \right) ^{\ast }\Psi ^{\otimes k}\left( \mathbf{\lambda }%
\right) $ is positive definite as well as $\Upsilon _{k}^{0}\left( A,%
\boldsymbol{\lambda }\right) \ $for all $\mathbf{\lambda }$ and any BM $A,$
because $\Psi \left( z\right) $ has rank $d$ on the complex unit circle.
However, note that despite that under Assumption~4 $\Upsilon _{k}^{0}\left(
A,\boldsymbol{\lambda }\right) >0$ for all $\mathbf{\lambda }$ and any BM $%
A, $ it is not sufficient for dynamics identification and we need to make
sure that v$\mathbf{\kappa }_{k}^{0}$ is rich enough through Assumptions~2$%
\left( 3\right) $ or~3$\left( k\right) $.
In particular, Assumption~2$\left( 3\right) $ is equivalent to the linear
independence of the third order cumulant matrices $\left \{ \mathbf{\kappa }%
_{\cdot \cdot j}^{0}\right \} _{j=1}^{d}$ assumption of Chen, Choi and
Escanciano (2018) used to investigate the consistency of a fundamentalness
test by showing that the Wold (invertible) innovations of a nonfundamental
VARMA model cannot be a martingale difference sequence despite being white
noise.
The ICA of order $k$ among the elements of $\mathbf{\ \varepsilon }_{t}$
contained in Assumption~3$(k)$ implies that all joint higher order cumulants
up to order $k$ are zero, i.e. $\mathbf{\kappa }_{abc}^{0}=0$ when $a,b,c$
are not all equal, so Assumption~3$\left( 3\right) $ implies Assumption~2$%
\left( 3\right) $ but imposes further structure on the multivariate skewness
of the vector $\mathbf{\varepsilon }_{t}$ given by
\begin{equation}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0}=\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}%
}\left( \mathbf{\alpha }_{3}^{0}\right) :=\left( \mathbf{\alpha } _{31}^{0}
\mathbf{e}_{1}^{\otimes 2} ,\ \mathbf{\alpha } _{32}^{0} \mathbf{e}%
_{2}^{\otimes 2},\ \ldots ,\ \mathbf{\alpha } _{3d}^{0} \mathbf{e}%
_{d}^{\otimes 2} \right) =\sum_{j=1}^{d} \mathbf{\alpha} _{3j}^{0}\mathbf{e}%
_{j}^{\otimes 2}\mathbf{e}_{j}^{\prime } , \label{vk3}
\end{equation}%
where $\mathbf{\alpha }_{3}^{0}=\left( \mathbf{\alpha } _{31}^{0},\mathbf{%
\alpha } _{32}^{0},\ldots ,\mathbf{\alpha } _{3d}^{0}\right) ^{\prime }$ are
the marginal skewness coefficients of $\mathbf{\varepsilon }_{t}$ and $%
\mathbf{e}_{j}$ is the $j$-th column of $\mathbf{I}_{d}$. Therefore, under
Assumption~3(3), all $\mathbf{\kappa }_{\cdot \cdot j}^{0}$, $j=1,\ldots ,d,$
are $d\times d$ matrices of zeros with a unique nonzero element $\mathbf{%
\alpha }_{3j}^{0}=\mathbf{\kappa } _{jjj}^{0}$ in position $\left(
j,j\right) $. Note that orthogonal rotations $\mathbf{\eta }_{t}=K\mathbf{%
\varepsilon }_{t}$ have the same identity covariance matrix of $\mathbf{%
\varepsilon }_{t},$ but their components are not longer independent if $%
K\neq P_{d},$ a signed permutation matrix of dimension $d$, because e.g. v$%
\mathbf{\kappa }_{3}^{\eta }=K^{\otimes 2}\ $v$\mathbf{\kappa }_{3}^{0}\
K^{\prime }$ has not the same structure $\left( \ref{vk3}\right) $ of v$%
\mathbf{\kappa }_{3}^{0},$ despite it maintains its rank, see Appendix~A for
details.
By contrast, Assumption~3$\left( 4\right) $ implies that v$\mathbf{\kappa }%
_{4}^{0}$ satisfies
\begin{equation}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}^{0}=\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}%
}\left( \mathbf{\alpha }_{4}^{0}\right) :=\sum_{j=1}^{d}\mathbf{\alpha }%
_{4j}^{0}\mathbf{e}_{j}^{\otimes 2}\mathbf{e}_{j}^{\otimes 2\prime }
\label{vk4}
\end{equation}%
for the kurtosis coefficients $\mathbf{\alpha }_4^0$ and has only rank $d$,
because, despite $\mathbf{\kappa }_{\cdot \cdot jj}^{0}$ are all matrices of
zeros with a unique nonzero element in position $\left( j,j\right) $ given
by the marginal kurtosis coefficient $\mathbf{\alpha }_{4j}^{0}=\mathbf{%
\kappa }_{jjjj}^{0}$, $j=1,\ldots ,d,$ we have that $\mathbf{\kappa }_{\cdot
\cdot hj}^{0}=\mathbf{0},$ $h\neq j,$ so v$\mathbf{\kappa }_{4}^{0}$ has at
most $d$ columns different from zero. Note that Assumption~A.6 in Gouri\'{e}%
roux et al. (2019) similarly needs that each component of $\varepsilon _{t}$
has a nonzero cumulant of order $k,$ $k\geq 3,$ but it further requires that
components of $\mathbf{\varepsilon }_{t}$ are full independent with a finite
moment of order $s,$ where $s$ is an even integer greater than $k.$
Finally, it is possible to obtain static identifying results when there is
at most one Gaussian structural shock as in Comon (1994) result, or a
non-Gaussian one with zero higher order cumulant of order $k$, as formalized
in the next weaker version of Assumption~$3\left( k\right) .$\bigskip
\noindent \textbf{Assumption~3}$^{\ast }\left( k\right) $. The components of
$\mathbf{\varepsilon }_{t}\ $are independent up to $k$ moments and
\begin{itemize}
\item For $k=3$ the marginal skewness coefficients are nonzero, $\mathbf{%
\alpha }_{3j}^{0}=\mathbf{\kappa }_{jjj}^{0}\neq 0,$ for all $j=1,\ldots ,d$
but at most one index.
\item For $k=4$ the marginal kurtosis coefficients are nonzero, $\mathbf{%
\alpha }_{4j}^{0}=\mathbf{\kappa }_{jjjj}^{0}\neq 0,$ for all $j=1,\ldots ,d$
but at most one index.\bigskip
\end{itemize}
\section{3. Nonparametric Identification}
In this section we discuss general identification results for SVARMA models
based on the spectral loss functions $\mathcal{L}_{k}^{0}$ under the
assumption of known dynamics up to a Blaschke factor $A$ and the
corresponding cumulants of structural shocks. We consider first dynamic
identification using nonconstant BM, while we later move to the static
components identification using constant BM.\bigskip
\begin{theorem}
\label{1A} Under Assumptions 1$\left( k\right) ,$ 2$\left( 3\right) $ and 4,
for any nonconstant BM $A\left( z\right) $, there exists an $\epsilon >0$
such that,
\begin{equation*}
\inf_{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}}\mathcal{L}_{3}^{0}\left( A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}%
\mathbf{\kappa }_{3}\right) \geq \epsilon >0.
\end{equation*}
\end{theorem}
\bigskip
All proofs are contained in Appendix~B, while auxiliary results are included
in Appendices~C and D. As when $d=1$ in VL, Theorem~\ref{1A} implies that
there is no way of choosing v$\mathbf{\kappa }_{3}$ such that third order
dynamics can be replicated after introducing a Blaschke factor inverting any
root of $\Psi $ (as can be done for $k=2).$ The conditions of Theorem~\ref%
{1A} allow for general nonconstant $A\left( z\right) $ which have an
infinite expansion in positive and/or negative powers of $z$ under a rank
condition on v$\mathbf{\kappa }_{3}^{0}.$ \bigskip
\begin{theorem}
\label{1ANEW} Under Assumptions 1$\left( k\right) ,$ 3$\left( k\right) $, $%
k=3$ or$\ 4$, and 4, for any nonconstant BM $A\left( z\right) $, there
exists an $\epsilon >0$ such that,
\begin{equation*}
\inf_{\mathbf{\alpha }}\mathcal{L}_{k}^{0}\left( A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{k}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left( \mathbf{\alpha }\right) \right) \geq \epsilon >0.
\end{equation*}
\end{theorem}
\bigskip
Theorem~\ref{1ANEW} relies on the particular structure of v$\mathbf{\kappa }%
_{k}^{0}$ imposed by Assumption~$3\left( k\right) .$ In fact, for $k=3,$
this is just a particular case of Theorem~\ref{1A}, since rank$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v%
}\mathbf{\kappa }_{k}^{0}\right) =d$ under Assumption~3$\left( k\right) ,$ $%
k=3,4$. However, the argument of Theorem~\ref{1ANEW} can not be extended
under a generic rank condition on v$\mathbf{\kappa }_{4}^{0}$ to cover $%
\mathcal{L}_{4}^{0}$ in Theorem~\ref{1A} without further structure, because
for any v$\mathbf{\kappa }_{4}^{0}$ it holds rank$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{%
\kappa }_{4}^{0}\right) \leq d\left( d+1\right) /2<d^{2}=\ $rank$\left(
A\right) ^{2}$ for $d>1.$
For a signed permutation matrix $P_{d}$ of dimension $d$ with all elements
equal to zero but a single term equal to $+1$ or $-1$ in each column and
row, let $P_{d}^{+}$ be equal to $P_{d}$ but taking the absolute value of
all its elements. Then $P_{d}\mathbf{\alpha }$ and $P_{d}^{+}\mathbf{\alpha }
$ are (signed) permutations of the elements of the vector $\mathbf{\alpha }.$
\bigskip
\begin{theorem}
\label{1B} Under Assumptions 1$\left( k\right) ,$ 3$\left( k\right) $, $k=3\
$or $4,$ and 4, for any constant BM $K$ different from a signed permutation,
i.e. $K \neq P_{d}$, there exists an $\epsilon >0$ such that%
\begin{equation*}
\inf_{\mathbf{\alpha }}\mathcal{L}_{k}^{0}\left( K,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{k}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left( \mathbf{\alpha }\right) \right) \geq \epsilon >0.
\end{equation*}
\end{theorem}
\bigskip
Under marginal independence of order $k$, which implies the co-kurtosis
conditions of Lanne and Luoto (2019), Theorem~\ref{1B} shows that for any
orthogonal matrix $K$ different from $\mathbf{I}_{d}$ and any signed
permutation matrix $P_{d}$, it is not possible to find any $\mathbf{\alpha }$
so that $\mathcal{L}_{k}^{0}\left( K,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}%
}\left( \mathbf{\alpha }\right) \right) =0.$ This provides identification of
the components of $\mathbf{\varepsilon }_{t}$ up to signed permutations
because for any $P_{d}$ and any v$\mathbf{\kappa }_{k}^{0}=\ $v$\mathbf{%
\kappa }_{k}^{IC}\left( \mathbf{\alpha }_{k}^{0}\right) $ we could select $%
\mathbf{\alpha }_{3}=P_{d}^{\prime }\mathbf{\alpha }_{3}^{0}$ or $\mathbf{%
\alpha }_{4}=P_{d}^{+\prime }\mathbf{\alpha }_{4}^{0}$ to make $\mathcal{L}%
_{3}^{0}\left( P_{d},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left(
P_{d}^{\prime }\mathbf{\alpha }_{3}^{0}\right) \right) =\mathcal{L}%
_{4}^{0}\left( P_{d},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left(
P_{d}^{+\prime }\mathbf{\alpha }_{4}^{0}\right) \right) =0.$
The results of Theorems~\ref{1ANEW} and~\ref{1B} can be combined in the
following result that identifies SVARMA models under the assumption of $k$%
-order independence of innovation components. Denote by $\left \Vert
M\right
\Vert =trace\left( M^{\ast }M\right) ^{1/2}$ the Frobenious norm of
a matrix $M$ and by $\left \Vert M\right \Vert _{L^{2}}=\int_{\left \vert
z\right
\vert =1}\left \Vert M\left( z\right) \right \Vert ^{2}dz$ the $%
L^{2} $ norm of $\Vert M\left( z\right) \Vert $ over the unit circle.
\begin{theorem}
\label{1A1B} Under Assumptions 1$\left( k\right) ,$ 3$\left( k\right) ,$ $%
k=3\ $or $4,$ and 4, for all $\nu >0$ there exists an $\epsilon >0$ such that%
\begin{equation*}
\inf_{A,\mathbf{\alpha },P_{d}\mathbf{:}\left \Vert A-P_{d}\right \Vert
_{L^{2}}+\left \Vert \mathbf{\alpha }-P_{d}^{\prime }\mathbf{\alpha }%
_{k}^{0}\right \Vert \geq \nu >0}\mathcal{L}_{k}^{0}\left( A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{%
\kappa }_{k}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left( \mathbf{\alpha }\right) \right) \geq \epsilon
>0.
\end{equation*}%
where $A$ is any BM and$\ P_{d}$ is any signed permutation matrix, and $%
P_{d}^{\prime }\mathbf{\alpha }_{k}^{0}$ has to be replaced by $%
P_{d}^{+\prime }\mathbf{\alpha }_{k}^{0}$ for $k=4.$\bigskip
\end{theorem}
Note that under Assumption~3$(k)$ we can deal simultaneously with both
dynamics and marginal identification for both $k=3$ and$~4$, providing
identification up to signed permutations of the specific components of $%
\mathbf{\varepsilon }_{t}$ under mutual independence of order $k$.
We now explore the possibility of relaxing Assumption$~3 $ by allowing for
some marginal cumulants to be zero in the static identification of Theorem~%
\ref{1B} and also for the dynamics identification of Theorems~\ref{1A},~\ref%
{1ANEW} or~\ref{1A1B}, but only under conditions which guarantee that every
single component of $\mathbf{\varepsilon }_{t}$ is non-Gaussian. Then, for
static identification robustness we explore, first, the situation when just
one of the marginal cumulants is zero for a given $k$ and, second, when some
further marginal skewness coefficients are zero, but the corresponding
kurtosis coefficients are not, or vice versa, so they mutually compensate
for the lack of identification of some particular component, both mechanisms
indicating that at most one Gaussian component in $\mathbf{\varepsilon }_{t}$
can be allowed as in Comon (1994).
\bigskip
\begin{corollary}
\label{L0}
Under Assumptions 1$\left( k\right) ,$ 3*$\left( k\right) $, $k=3\ $or $4,$
and 4, the conclusions of Theorem~\ref{1B} hold.
\bigskip
\end{corollary}
Then, when a single higher order cumulant of a given order $k$ is zero only
signed permutation matrices are not discarded, but this is not true if more
than one marginal element of $\mathbf{\alpha }_{k}^{0}$ is zero. The next
result investigates the case when possibly more than one marginal cumulant
of the same order $k=3$ or $k=4$ is zero, but the corresponding marginal
cumulants of the other order are nonzero, allowing for at most one component
to have simultaneously zero skewness and zero kurtosis for static
identification, while the others need to have at least one coefficient
different from zero. As before, for dynamics identification we need to
guarantee that all components are non-Gaussian up to order four. To avoid
the potential lack of identification provided by a single set of cumulants
of a given order, we need to consider a robustified loss function involving
both third and fourth moments simultaneously.\bigskip
\begin{corollary}
\label{L1} Under Assumptions 1$\left( k\right) ,$ 3$\left( k\right) $, $k=3\
$and $4,$ and 4, with index sets $\mathcal{I}_{3}$ and $\mathcal{I}_{4}$,
subsets of $\left \{ 1,2,\ldots ,d\right \} ,$ such that $\mathbf{\alpha }%
_{3j}^{0}=0$ for $j\in \mathcal{I}_{3}$, $\mathbf{\alpha }_{4j}^{0}=0$ for $%
j\in \mathcal{I}_{4},$ and \#$\left \{ \mathcal{I}_{3}\cap \mathcal{I}%
_{4}\right \} \leq 1$, for any constant BM $A$ different from a signed
permutation, i.e. $A\neq P_{d}$, there exists an $\epsilon >0$ such that
\begin{equation}
\inf_{\mathbf{\alpha }_{3}}\mathcal{L}_{3}^{0}\left( A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{%
\kappa }_{3}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left( \mathbf{\alpha }_{3}\right) \right) +\inf_{%
\mathbf{\alpha }_{4}}\mathcal{L}_{4}^{0}\left( A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{4}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left( \mathbf{\alpha }_{4}\right) \right) \geq \epsilon >0,
\label{L34}
\end{equation}%
while if \#$\left \{ \mathcal{I}_{3}\cap \mathcal{I}_{4}\right \} =0$, then (%
\ref{L34}) holds for all non constant BM $A.$
\end{corollary}
\bigskip
\section{4. Parameter Identification}
We assume that the observed $d$-dimensional SVARMA$\left( p,q\right) $
process $Y_t$ admits the following parameterization
\begin{equation}
\Phi _{\mathbf{\theta }_{0}}\left( L\right) Y_{t}=\mu +\Theta _{\mathbf{%
\theta }_{0}}\left( L\right) \mathbf{\varepsilon }_{t},\ \ \ \mathbf{%
\varepsilon }_{t}\sim iid_{k}\left( \mathbf{0},\mathbf{I}_{d},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}%
\mathbf{\kappa }_{k}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left( \mathbf{\alpha }_{k}^{0}\right) ,k\in
\mathcal{K}\right) \label{Rep}
\end{equation}%
where the index set $\mathcal{K}\subseteq \left \{ 3,4\right \} $ is non
empty, the lag polynomials
\begin{eqnarray*}
\Phi _{\mathbf{\theta }}\left( L\right) &:=&\mathbf{I}_{d}+\Phi _{1}\left(
\mathbf{\theta }\right) L+\cdots +\Phi _{p}\left( \mathbf{\theta }\right)
L^{p} \\
\Theta _{\mathbf{\theta }}\left( L\right) &:=&\Theta _{0}\left( \mathbf{%
\theta }\right) +\Theta _{1}\left( \mathbf{\theta }\right) L+\cdots +\Theta
_{q}\left( \mathbf{\theta }\right) L^{q}
\end{eqnarray*}%
depend on a $m$-dimensional parameter $\mathbf{\theta }\in \mathcal{S}%
\subset \mathbb{R}^{m}$ and $iid_{k}$ means that Assumption~1$\left(
k\right) $ holds imposing serial $iid$-ness up to $k$ moments. The
parameterization v$\mathbf{\kappa }_{k}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left( \mathbf{\alpha }%
_{k}\right) $ given in (\ref{vk3}) and (\ref{vk4}) impose the independence
component condition of Assumption~3$\left( k\right) $ on the arrays of $k=3$
and/or $k=4$ order cumulants of the standardized error sequence $\mathbf{%
\varepsilon }_{t}$ with vectors $\mathbf{\alpha }_{k}\ \mathbf{\in \ }%
\mathcal{D}_{k}\subset \mathbb{R}^{d}$ of marginal skewness $\left(
k=3\right) $ and kurtosis coefficients $\left( k=4\right) $. Then $\mathbf{%
\theta }_{0}$ and $\mathbf{\alpha }_{0}$ denote the true value of the
parameters and, while the level $\mu $ could be estimated by OLS or GLS
based on estimates of $\mathbf{\theta }$ as usual, since our methods are
invariant to $\mu $ we do not discuss this further.
The $k$-th order spectral density parametric model for each index $\mathbf{a}%
=\left( \mathbf{a}(1),\ldots ,\mathbf{a}(k)\right) $ of components of $Y_{t}$
with representation $\left( \ref{Rep}\right) $ is given for $k=3,4,$ by
\begin{equation*}
f_{\mathbf{a},k}(\boldsymbol{\lambda };\mathbf{\theta },\mathbf{\alpha }) :
=\left( \Phi _{\mathbf{\theta }}^{-1}\Theta _{\mathbf{\theta }}\right) _{%
\mathbf{a}}^{\otimes k}\left( \mathbf{\lambda }\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left(
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left( \mathbf{\alpha }\right)
\right) =\mathbf{\Psi }\left( \lambda ;\mathbf{\theta }\right) _{\mathbf{a}%
}^{\otimes k}\mathbf{S}_{k}\mathbf{\alpha }
\end{equation*}%
where $\mathbf{S}_{k}:=\left( \mathbf{e}_{1}^{\otimes k},\mathbf{e}%
_{2}^{\otimes k},\ldots ,\mathbf{e}_{d}^{\otimes k}\right) $ is a rank $d$
selection matrix and $\mathbf{\Psi }\left( \lambda ;\mathbf{\theta }\right)
:=\Phi _{\mathbf{\theta }}^{-1}\left( e^{-i\lambda }\right) \Theta _{\mathbf{%
\theta }}\left( e^{-i\lambda }\right) $. For $k=2$ we replace $\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}%
\mathbf{\kappa }_{2}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left( \mathbf{\alpha }\right) $ by $\mathbf{%
I}_{d}$ in $f_{\mathbf{a},2}(\lambda ;\mathbf{\theta })$, to impose
normalization and uncorrelation of the components of $\mathbf{\varepsilon }%
_{t}$.
We assume that the parameterization $\left( \ref{Rep}\right) $ satisfies the
following conditions. \bigskip
\noindent \textbf{Assumption~5} \label{Ass4}
\noindent \textbf{5.1.} For all $\mathbf{\theta }\in \mathcal{S},$ det$%
\left( \Phi _{\mathbf{\theta }}\left( z\right) \right) $det$\left( \Theta _{%
\mathbf{\theta }}\left( z\right) \right) \neq 0$ for $\left \vert
z\right
\vert =1$ and $\Theta _{0}\left( \mathbf{\theta }\right) $ is
nonsingular.
\noindent \textbf{5.2.} For all $\mathbf{\theta }\neq \mathbf{\theta }_{0},$
$\ \Phi _{\mathbf{\theta }}^{-1}\left( z\right) \Theta _{\mathbf{\theta }%
}\left( z\right) \neq \Phi _{\mathbf{\theta }_{0}}^{-1}\left( z\right)
\Theta _{\mathbf{\theta }_{0}}\left( z\right) $ in a subset of positive
measure of $\left \{ z\in \mathbb{C}:\left \vert z\right \vert =1\right \} .$
\noindent \textbf{5.3.} $\mathbf{\theta }_{0}\in \mathcal{S}\ $and $\mathcal{%
S}$ is compact.
\noindent \textbf{5.4.} $\Phi _{i}\left( \mathbf{\theta }\right) ,$ $%
i=0,\ldots ,p,\ $and $\Theta _{i}\left( \mathbf{\theta }\right) ,$ $%
i=0,\ldots ,q,$ are continuously differentiable for $\mathbf{\theta }\in
\mathcal{S}$.\bigskip
Assumption$~5.1$ imposes Assumption~4 for each parameterized model, and with
the $iid$ condition of order $k$ on the sequence $\mathbf{\varepsilon }_{t}$
in Assumption~1$\left( k\right) $ guarantee that $Y_{t}$ with representation
$\left( \ref{Rep}\right) $ is $k$-stationary. The identifiability conditions
in Assumption~5.2 are satisfied when the parameter space $\mathcal{S}$ is
sufficiently constrained (cf. Boubacar Mainassara and Francq, 2011) as for
restricted versions of causal and invertible VARMA models (e.g. echelon or
final equations forms) that guarantee that $\Phi _{\mathbf{\theta }}\left(
L\right) \ $and $\Theta _{\mathbf{\theta }}\left( L\right) $ are left
coprime and that the unique unimodular common left divisor of $\Phi _{%
\mathbf{\theta }}\left( L\right) $ and $\Theta _{\mathbf{\theta }}\left(
L\right) $ is the identity matrix, see e.g. Section~12.1 in L\"{u}tkepohl
(2005). Assumption~5.3 is a standard parameter space restriction and,
together with Assumption~5.4, allows for uniformity arguments.
Note that parameterizations covering both invertible and noninvertible (or
causal and noncausal) solutions are allowed by Assumption~5.2, which
identifies uniquely the parametric transfer and impulse response functions.
However, it is possible that $\Phi _{\mathbf{\theta }}^{-1}\left( z\right)
\Theta _{\mathbf{\theta }}\left( z\right) A(z)=\Phi _{\mathbf{\theta }%
_{0}}^{-1}\left( z\right) \Theta _{\mathbf{\theta }_{0}}\left( z\right) $
for almost all $z,\ \left \vert z\right \vert =1,\ $some $\mathbf{\theta }%
\neq \mathbf{\theta }_{0}$ and some BM $A\left( z\right) \neq \mathbf{I}%
_{d}, $ and therefore $f_{\mathbf{a},2}\left( \lambda ;\mathbf{\theta }%
_{0}\right) $ can not identify $\Phi _{\mathbf{\theta }_{0}}^{-1}\left(
z\right) \Theta _{\mathbf{\theta }_{0}}\left( z\right) .$ However,
Assumption~5.2 is not sufficient either to identify $\Phi _{\mathbf{\theta }%
}^{-1}\left( z\right) \Theta _{\mathbf{\theta }}\left( z\right) $ uniquely
from $f_{\mathbf{a},k}(\boldsymbol{\lambda };\mathbf{\theta }_{0},\mathbf{%
\alpha }_{k}^{0})$ for $k=3$ or 4, without further restrictions to discard
signed permutations $A\left( z\right) =P_{d}\neq \mathbf{I}_{d}$ since
Assumption~3$\left( k\right) $ already prevents for all nonconstant $A\left(
z\right) .$ With this aim, we introduce the following assumptions, which fix
the signs of the components of $\mathbf{\varepsilon }_{t}$ and impose an
ordering by either imposing a given structure on $\Theta _{0}\left( \mathbf{%
\theta }_{0}\right) $ (6A), or by restricting the allowed set of values for
the marginal third or fourth order cumulants $\mathbf{\alpha }_{k}^{0}$ (6B$%
(k)$), or by directly excluding any signed permutations among the columns of
the transfer function (6C). \bigskip
\noindent \textbf{Assumption~6A}. The diagonal elements of $\Theta
_{0}\left( \mathbf{\theta }_{0}\right) $ are all positive and the elements
of $Y_{t}$ are ordered so that there is no signed permutation $P_{d}$ such
that the absolute value of the product of the diagonal elements of $\Theta
_{0}\left( \mathbf{\theta }_{0}\right) P_{d}$ is equal or larger than that
of $\Theta _{0}\left( \mathbf{\theta }_{0}\right) .$\bigskip
The restriction on the diagonal elements of $\Theta _{0}\left( \mathbf{%
\theta }_{0}\right) ,$ cf. Pham and Garat (1997) and Lanne and Luoto (2019),
fixes the signs of $\mathbf{\varepsilon }_{t}$ so that a positive increment
in a component of $\mathbf{\varepsilon }_{t}$ corresponds to a positive
increment in the element of $Y_{t}$ associated to this shock. This
restriction can be described as a sign restriction (at lag $0)$ on the IRF
of each endogenous variable with respect to the corresponding error term,
giving a unique interpretation of the IRF (as it is automatically imposed
when using Cholesky identification). Further, it is imposed a unique
permutation that maximizes the absolute value of the product of the diagonal
elements of $\Theta _{0}\left( \mathbf{\theta }_{0}\right) P_{d}$ for $P_{d}=%
\mathbf{I}_d.$ Alternative ordering schemes based on the elements of $\Theta
_{0}$ are possible, see e.g. Lanne et al. (2017), as well as schemes that
use information for the ordering from the vectors $\mathbf{\alpha }_{3}^{0}$
and $\mathbf{\alpha }_{4}^{0},$ as the following novel condition.\bigskip
\noindent \textbf{Assumption~6B}$\left( k\right) $. The diagonal elements of
$\Theta _{0}\left( \mathbf{\theta }_{0}\right) $ are all positive and it
holds for $k\in \mathcal{K}$,
\begin{itemize}
\item[\textbf{6B}$(3)$] \noindent For $k=3:$ $\mathbf{\alpha }_{3}^{0}\in
\mathcal{D}_{3}$ compact where
\begin{equation*}
\mathcal{D}_{3}\subseteq \left \{ \mathbf{\alpha }=\left( \alpha _{1},\ldots
,\alpha _{d}\right) ^{\prime }\in \mathbb{R}^{d}:-\infty <\alpha _{1}<\alpha
_{2}<\cdots <\alpha _{d}<\infty , \ \alpha _{j}\neq 0\right \} ,
\end{equation*}%
and there is no signed permutation $P_{d}\neq \mathbf{I}_{d}$ such that the
diagonal elements of $\Theta _{0}\left( \mathbf{\theta }_{0}\right) P_{d}$
are positive and $P_{d}^{\prime }\mathbf{\alpha }_{3}^{0}\in \mathcal{D}%
_{3}. $
\item[\textbf{6B}$(4)$] \noindent For $k=4:$ $\mathbf{\alpha }_{4}^{0}\in
\mathcal{D}_{4}$ compact where
\begin{equation*}
\mathcal{D}_{4}\subseteq \left \{ \mathbf{\alpha }=\left( \alpha _{1},\ldots
,\alpha _{d}\right) ^{\prime }\in \mathbb{R}^{d}:-2\leq \alpha _{1}<\alpha
_{2}<\cdots <\alpha _{d}<\infty ,\ \alpha _{j}\neq 0\right \} .
\end{equation*}
\end{itemize}
The sign of the diagonal elements of $\Theta _{0}\left( \mathbf{\theta }%
_{0}\right) $ also determines the sign of the elements of $\mathbf{\alpha }%
_{3}^{0}\in \mathcal{D}_{3},$ but not that of $\mathbf{\alpha }_{4}^{0}\in
\mathcal{D}_{4},$ so any signed permutation $P_{d},$ $P_{d}^{+}\neq \mathbf{I%
}_{d},$ would alter the ordering of the elements of $\mathbf{\alpha }%
_{4}^{0} $, but it might not that of $\mathbf{\alpha }_{3}^{0}$ depending on
the values of $\Theta _{0}\left( \mathbf{\theta }_{0}\right) $ and the shift
of signs. Then, Assumption~6B($3$) imposes explicitly a unique sign-ordering
associated with an increasing sequence of asymmetry coefficients to remove
these situations. While Assumption~6B imposes the non-zero cumulant part of
Assumption~3$\left( k\right) $, it further requires some further knowledge
on the marginal distributions of the structural errors, e.g. related to the
asymmetry sign or the tail behaviour, that allows to order the skewness or
kurtosis coefficients of the components of $\mathbf{\varepsilon }_{t}$.
Assumptions~6A and 6B fix the ordering of the columns of $\Theta _{0}\left(
\mathbf{\theta }_{0}\right) $ or of the cumulant coefficients, respectively,
to exclude permutations in the columns of the IRF (cf. Theorem~3) if $%
\mathcal{S}$ and the parameterization $\Phi _{\mathbf{\theta }}^{-1}\left(
z\right) \Theta _{\mathbf{\theta }}\left( z\right) $ allow for rotations
(i.e. constant Blaschke factors) different from $\mathbf{I}_{d}$, where
rotations different from $P_{d}$ are ruled out by $\mathcal{L}_{k}^{0},$ $%
k=3,4.$ If $\mathcal{S}$ restricts to only invertible and causal models,
then it would be possible to allow for a component of $\mathbf{\varepsilon }%
_{t}$ to be symmetric (and possibly Gaussian), i.e. $\mathbf{\alpha }%
_{3j}^{0}=0$ for a single $j\in \left \{ 1,\ldots ,d\right \} ,$ if second
order moments (to identify dynamics) were used in conjunction with third
order ones (to identify rotations), cf. Corollary~1. If in this case we
include information from $k=4$ simultaneously to $k=2$ and $k=3,$ we could
also allow for a component with zero skewness and kurtosis (and possibly
Gaussian), while the other ones should display at least one non-Gaussian
feature in their third and fourth order moments, i.e. $\left \vert \mathbf{%
\alpha }_{3j}^{0}\right \vert +\left \vert \mathbf{\alpha }%
_{4j}^{0}\right
\vert =0$ for at most one $j\in \left \{ 1,\ldots
,d\right
\} .$
In the next assumption, the parameterization and $\mathcal{S}$ are further
restricted not to allow that $\Phi _{\mathbf{\theta }}^{-1}\left( z\right)
\Theta _{\mathbf{\theta }}\left( z\right) =\Phi _{\mathbf{\theta }%
_{0}}^{-1}\left( z\right) \Theta _{\mathbf{\theta }_{0}}\left( z\right)
P_{d} $ for any $\mathbf{\theta }\neq \mathbf{\theta }_{0}$ and any constant
signed permutation matrix $P_{d}$ because some identifying restrictions have
been already imposed, such as a recursive system (Choleski) assumption with
a known ordering, so that Assumption~3$\left( k\right) $ becomes sufficient
for identification$,$ cf. Theorem~\ref{1A1B}. \bigskip
\noindent \textbf{Assumption~6C}. \label{Ass4NoRotC)}For all $\mathbf{\theta
}\neq \mathbf{\theta }_{0},$ $\mathbf{\theta }\in \mathcal{S}$, $\ \Phi _{%
\mathbf{\theta }}^{-1}\left( z\right) \Theta _{\mathbf{\theta }}\left(
z\right) \neq \Phi _{\mathbf{\theta }_{0}}^{-1}\left( z\right) \Theta _{%
\mathbf{\theta }_{0}}\left( z\right) P_{d}$ for any signed permutation
matrix $P_{d}$ in a subset of positive measure of $\left \{ z\in \mathbb{C}%
:\left \vert z\right \vert =1\right \} $.\bigskip
Define the $L^{2}$ distance between $f_{\mathbf{a},k}(\boldsymbol{\lambda };%
\mathbf{\theta },\mathbf{\alpha })\ $and $f_{\mathbf{a},k}(\boldsymbol{%
\lambda })=f_{\mathbf{a},k}(\boldsymbol{\lambda };\mathbf{\theta }_{0},%
\mathbf{\alpha }_{k}^{0})$ for $k=3,4$ and all possible indices $\mathbf{a}$%
,
\begin{equation*}
\mathcal{L}_{k}\left( \mathbf{\theta },\mathbf{\alpha }\right) :=\sum_{%
\mathbf{a}}\int_{\Pi ^{k-1}}\left \vert f_{\mathbf{a},k}(\boldsymbol{\lambda
};\mathbf{\theta },\mathbf{\alpha })-f_{\mathbf{a},k}(\boldsymbol{\lambda }%
)\right \vert ^{2}d\boldsymbol{\lambda },
\end{equation*}%
in terms of the parameters $\mathbf{\theta }$ and the marginal cumulants $%
\mathbf{\alpha }$ of order $k,$ and define%
\begin{equation*}
\mathcal{L}_{2}\left( \mathbf{\theta }\right) :=\sum_{\mathbf{a}}\int_{\Pi
}\left \vert f_{\mathbf{a},2}(\lambda ;\mathbf{\theta })-f_{\mathbf{a}%
,2}(\lambda )\right \vert ^{2}d\lambda,
\end{equation*}%
where $f_{\mathbf{a},2}(\lambda ;\mathbf{\theta })$ is the parametric model
for the second order spectral density $f_{\mathbf{a},2}(\lambda )$ for all
pairs $\mathbf{a}=\left( \mathbf{a}(1),\mathbf{a}(2)\right) ,$ which only
depends on $\mathbf{\theta }$ because of the normalization $\mathbb{V}\left(
\mathbf{\varepsilon }_{t}\right) =\mathbf{I}_{d}$ under $\left( \ref{Rep}%
\right) .$ The loss function $\mathcal{L}_{2}$ is only able to identify $%
\Phi _{\mathbf{\theta }}^{-1}\left( z\right) \Theta _{\mathbf{\theta }%
}\left( z\right) $ up to a Blaschke factor so it cannot identify $\mathbf{%
\theta }$ under Assumptions~3 and~5 in absence of more restrictions.
We now show that the conclusions of Theorem~\ref{1A1B} extend to these
parametric loss functions under any of the versions of Assumption~6 (A, B or
C) with the corresponding restrictions on the parameter space. Thus, let $%
\mathcal{S}^{+}\subseteq \mathcal{S}$ be a compact set where we impose the
restriction that the diagonal elements of $\Theta _{0}\left( \mathbf{\theta }%
\right) $ are positive, while in the compact $\mathcal{S}^{\max }\subseteq
\mathcal{S}^{+}$ we further impose that $\Pi _{j=1}^{d}\Theta _{0,jj}\left(
\mathbf{\theta }\right) >\max_{\mathbf{\theta }^{\prime }\in \mathcal{S}%
^{\max }}\Pi _{j=1}^{d}\Theta _{0,jj}\left( \mathbf{\theta }^{\prime
}\right) $ where $\mathbf{\theta }^{\prime }$ is any parameter vector which
describes a permutation of the columns of $\Theta _{\mathbf{\theta }}\left(
L\right) ,$ i.e. $\Theta _{\mathbf{\theta }^{\prime }}\left( L\right)
=\Theta _{\mathbf{\theta }}\left( L\right) P_{d}$ for some signed
permutation matrix $P_{d}.$ \bigskip
\begin{theorem}
\label{Th4} Under Assumptions 1$\left( k\right) ,$ 3$\left( k\right) ,$ 5
and 6$\left( k\right) $, $k\in \mathcal{K},$ for any $\nu >0,$ there exists
an $\epsilon >0$ such that,%
\begin{equation*}
\inf_{\mathbf{\theta }\in \mathcal{S},\mathbf{\alpha }:\left \Vert \mathbf{%
\theta }-\mathbf{\theta }_{0}\right \Vert +\left \Vert \mathbf{\alpha }-%
\mathbf{\alpha }_{k}^0\right \Vert \geq \nu >0}\mathcal{L}_{2}\left( \mathbf{%
\theta }\right) +\mathcal{L}_{k}\left( \mathbf{\theta },\mathbf{\alpha }%
\right) \geq \epsilon >0,
\end{equation*}%
where it is further imposed $\mathbf{\theta }\in \mathcal{S}^{\max }$ under
Assumption~6A and $\mathbf{\theta }\in \mathcal{S}^{+}\ $and $\mathbf{\alpha
}\in \mathcal{D}_{k}$ under Assumption~6B$\left( k\right) $.
\end{theorem}
This result provides identification of the dynamics and scaling parameters $%
\mathbf{\theta }\ $and the marginal cumulant vector $\mathbf{\alpha}$ in
absence of knowledge on the possible noninvertibility (or noncausality) of $%
Y_{t}.$ Following the comments to Theorem~\ref{1A1B} and despite
Assumption~5.2 allows that for some $\mathbf{\theta \neq \theta }_{0}$ and
some BM $A,$ $\Phi _{\mathbf{\theta }}^{-1}\left( z\right) \Theta _{\mathbf{%
\theta }}\left( z\right) =\Phi _{\mathbf{\theta }_{0}}^{-1}\left( z\right)
\Theta _{\mathbf{\theta }_{0}}\left( z\right) A(z)$ for almost all $z$ in
the unit circle so that $\mathcal{L}_{2}\left( \mathbf{\theta }\right) =0,$ $%
\mathcal{L}_{k}$ is not minimized unless $A(z)=P_{d}$ and $\mathbf{\alpha }%
_{3}=P_{d}^{\prime }\mathbf{\alpha }_{3}^{0}\ $or $\mathbf{\alpha }%
_{4}=P_{d}^{+\prime }\mathbf{\alpha }_{4}^{0}$. However, Assumption~6
imposes a unique ordering (and sign) to discard any signed permutations of
the columns of $\Phi _{\mathbf{\theta }_{0}}^{-1}\left( z\right) \Theta _{%
\mathbf{\theta }_{0}}\left( z\right) $ that would generate the same (second
and $k$-order) dynamics of $Y_{t}$ based on a signed permuted version of $%
\mathbf{\varepsilon }_{t}.$ Further, similarly as in VL, $\mathcal{L}_{3}$
and $\mathcal{L}_{4}$ could be considered jointly. Thus, if the target
function includes contributions from $\mathcal{L}_{3}$ and $\mathcal{L}_{4},$
then both $\mathbf{\alpha }_{3}$ and $\mathbf{\alpha }_{4}$ are identified
by the ordering of the columns of $\Theta _{0}\left( \mathbf{\theta }%
_{0}\right) $ (Assumption~6A) or by the ordering information from a single
higher order cumulant (Assumption~6B$\left( k\right) )$.
By contrast with the univariate analysis of VL, in the multivariate case it
is not possible to identify jointly $\mathbf{\theta }$ and $\mathbf{\alpha }%
_{k}$ from only $\mathcal{L}_{k}$ without $\mathcal{L}_{2},$ because the MA
matrix polynomial $\Theta _{\mathbf{\theta }}\left( z\right) $ incorporates
the scaling in $\Theta _{0}\left( \mathbf{\theta }\right) $ as we set $\mathbb{V}
\left( \mathbf{\varepsilon }_{t}\right) =\mathbf{I}_{d},$ so that $\mathbf{%
\alpha }_{k}$ are skewness and kurtosis \textit{coefficients}. In
alternative parameterizations, e.g. when setting $\Theta _{0}\left( \mathbf{%
\theta }\right) =\mathbf{I}_{d}$ for all $\mathbf{\theta }$ and%
\begin{equation*}
\Phi _{\mathbf{\theta }}\left( L\right) Y_{t}=\Theta _{\mathbf{\theta }%
}\left( L\right) \Omega \mathbf{\varepsilon }_{t},\ \ \ \mathbf{\varepsilon }%
_{t}\sim iid_{k}\left( \mathbf{0},\mathbf{I}_{d},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{k}^{IC}\left( \mathbf{\alpha }_{k}^{0}\right) ,k\in \mathcal{K}\right),
\end{equation*}%
where $\Omega $ is a non-singular matrix parameterized independently of $%
\mathbf{\theta },$ the dynamics parameters $\mathbf{\theta }$ and the
marginal cumulants $\mathbf{\alpha }_{k}$ could be identified jointly by a
single $\mathcal{L}_{k},$ $k=3\ $or $4,$ for a given scaling rotation $%
\Omega ,$ which has still to be identified together with $\mathcal{L}_{2}.$%
\bigskip
\section{5. Parameter Minimum Distance Estimation}
Given a time series of $Y_{t}$, $t=1,\ldots ,T$, we define sample analogs of
the loss functions $\mathcal{L}_{k}\left( \mathbf{\theta },\mathbf{\alpha }%
\right) $ for $k=3,4$ and $\mathcal{L}_{2}\left( \mathbf{\theta }\right) $
as in VL (2018) and Brillinger (1985),
\begin{equation*}
\mathcal{L}_{k,T}\left( \mathbf{\theta },\mathbf{\alpha }\right) :=\frac{%
\left( 2\pi \right) ^{2k-2}}{T^{k-1}} \sum_{\mathbf{a}}\sum_{\mathbf{\lambda
}_{\mathbf{j}}}\left \vert f_{\mathbf{a},k}(\mathbf{\lambda }_{\mathbf{j}};%
\mathbf{\theta },\mathbf{\alpha })-I_{\mathbf{a},k}(\mathbf{\lambda }_{%
\mathbf{j}})\right \vert ^{2},
\end{equation*}%
replacing the true spectral densities by the sample higher order
periodograms $I_{\mathbf{a},k}$,
\begin{equation*}
I_{\mathbf{a},k}\left( \lambda _{1},\ldots ,\lambda _{k-1}\right) :=\frac{1}{%
\left( 2\pi \right) ^{k-1}T}w_{T,\mathbf{a}(1)}\left( \lambda _{1}\right)
\cdots w_{T,\mathbf{a}(k-1)}\left( \lambda _{k-1}\right) w_{T,\mathbf{a}%
(k)}\left( -\lambda _{1}\cdots -\lambda _{k-1}\right),
\end{equation*}%
where $w_{T}\left( \lambda \right) =\sum_{t=1}^{T}e^{-i\lambda t}Y_{t}$ is
the discrete Fourier transform (DFT) of $Y_{t}.$ In $\mathcal{L}_{k,T},$ the
summation in $\mathbf{\lambda }_{\mathbf{j}}=(\lambda _{j_{1}},\ldots
,\lambda _{j_{k-1}}) $ for Fourier frequencies $\lambda _{j_n}=2\pi j_n/T$
runs for all $j_{n}=1,\ldots ,T-1,$ $n=1,\ldots ,k-1,$ excluding $%
j_{a}+j_{b}=0\func{mod}(T),$ $a\neq b, $ and $j_{a}+j_{b}+j_{c}=0\func{mod}%
(T),$ all $a$, $b$ and $c$ different, for sample mean correction. Similarly,%
\begin{equation*}
\mathcal{L}_{2,T}\left( \mathbf{\theta }\right) :=\frac{\left( 2\pi \right)
^{2}}{T}\sum_{\mathbf{a}}\sum_{j=1}^{T-1}\left \vert f_{\mathbf{a}%
,2}(\lambda _{j};\mathbf{\theta })-I_{\mathbf{a},2}(\lambda _{j})\right
\vert ^{2}
\end{equation*}%
for the usual periodogram $I_{\mathbf{a},2}$ and spectral density $f_{%
\mathbf{a},2}.$
The $d^{k}$-vector containing all $f_{\mathbf{a},k}$ spectral densities can
be written under Assumptions~1$\left( k\right) $ and~3$\left( k\right) $ as $%
\left( 2\pi \right) ^{1-k}\mathbf{\Psi }_{k}\left( \mathbf{\lambda ;\theta }%
\right) \mathbf{S}_{k}\mathbf{\alpha },\ $with$\ \mathbf{\Psi }_{k}\left(
\mathbf{\lambda ;\theta }\right) =\mathbf{\Psi }^{\otimes k}\left( \mathbf{%
\lambda ;\theta }\right) ,$ so that for $k=3,4$%
\begin{equation*}
\mathcal{L}_{k,T}\left( \mathbf{\theta },\mathbf{\alpha }\right) =\frac{1}{%
T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\left( \mathbf{\Psi }_{k}\left(
\mathbf{\lambda }_{\mathbf{j}}\mathbf{;\theta }\right) \mathbf{S}_{k}\mathbf{%
\alpha }-\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}})\right) ^{\ast
}\left( \mathbf{\Psi }_{k}\left( \mathbf{\lambda ;\theta }\right) \mathbf{S}%
_{k}\mathbf{\alpha }-\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}})\right) ,
\end{equation*}%
where $\mathbb{I}_{k}(\mathbf{\lambda })$ is the $d^{k}\times 1$ vector
stacking all ($k=3$ order) biperiodograms and ($k=4$ order) triperiodograms
of $Y_{t},$ normalized by $\left( 2\pi \right) ^{k-1},$
\begin{equation*}
\mathbb{I}_{k}(\mathbf{\lambda }) : =\frac{1}{T}w_{T}\left( -\lambda _{1}-
\cdots -\lambda _{k-1}\right)\otimes \cdots \otimes w_{T}\left( \lambda
_{2}\right) \otimes w_{T}\left( \lambda _{1}\right) .
\end{equation*}%
Correspondingly,%
\begin{equation*}
\mathcal{L}_{2,T}\left( \mathbf{\theta }\right) =\frac{1}{T}%
\sum_{j=1}^{T-1}\left( \mathbf{\Psi }_{2}\left( \lambda _{j}\mathbf{;\theta }%
\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}_{d}\right) -\mathbb{I}_{2}(\lambda
_{j})\right) ^{\ast }\left( \mathbf{\Psi }_{2}\left( \lambda _{j}\mathbf{%
;\theta }\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}_{d}\right) -\mathbb{I}%
_{2}(\lambda _{j})\right),
\end{equation*}%
where $\mathbb{I}_{2}\left( \lambda \right) =\frac{1}{T}w_{T}\left( -\lambda
\right) \otimes w_{T}\left( \lambda \right) =2\pi $vec$\left( I_{YY}\left(
\lambda \right) \right) $ and $I_{YY}\left( \lambda \right) =\left( 2\pi
T\right) ^{-1}w_{T}\left( \lambda \right) w_{T}^{\ast }\left( \lambda
\right) =\left \{ I_{(a,b),2}\right \} _{a,b=1,\ldots ,d}$ is the usual
periodogram matrix of $Y_{t}.$
We set the following minimum distance parameter estimates for weights $w_k$,
\begin{equation*}
\left( \mathbf{\hat{\theta}}_{w,T},\ \mathbf{\hat{\alpha}}_{k,T},\ k\in
\mathcal{K}\right) :=\arg \min_{\mathbf{\theta }\in \mathcal{S}, \mathbf{%
\alpha }_{k},k\in \mathcal{K}}\mathcal{L}_{2,T}\left( \mathbf{\theta }%
\right) +\sum_{k\in \mathcal{K}}w_{k}\mathcal{L}_{k,T}\left( \mathbf{\theta }%
,\mathbf{\alpha }_{k}\right) ,
\end{equation*}%
where min$_{k\in \mathcal{K}}w_{k}>0$ and we further restrict to $\mathbf{%
\theta }\in \mathcal{S}^{\max }$ under Assumption~6A and to $\mathbf{\theta }%
\in \mathcal{S}^{+}\ $and $\mathbf{\alpha }_{k} \in \mathcal{D}_{k}$ under
Assumption~6B$\left( k\right) ,$ $k\in \mathcal{K}$. The main purpose of
combining loss functions $\mathcal{L}_{k,T}$ involving cumulants of
different orders $k=3,4,$ is robustness to lack of identification due to
failure of the nonzero cumulant condition of Assumption~3$\left( k\right) $
for a single $k$, as efficiency gains are possible though difficult to
characterize, even in the univariate case, see Lobato and Velasco (2018).
Additionally, we always need to include $\mathcal{L}_{2,T}$ in our loss
function for scaling identification.
Solving the first order conditions for $\mathbf{\alpha }_{k},$ $k\in
\mathcal{K}$, using
\begin{equation*}
\frac{\partial }{\partial \mathbf{\alpha }}\mathcal{L}_{k,T}\left( \mathbf{%
\theta ,\alpha }\right) =\frac{2}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}%
}}\func{Re}\left \{ \mathbf{S}_{k}^{\prime }\mathbf{\Psi }_{k}^{\ast }(%
\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })\left( \mathbf{\Psi }_{k}(%
\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })\mathbf{S}_{k}\mathbf{\alpha
}-\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}})\right) \right \} ,
\end{equation*}%
we can obtain the (unrestricted) estimate of $\mathbf{\alpha }_{k}$ for a
given $\mathbf{\theta }$
\begin{equation*}
\mathbf{\hat{\alpha}}_{k,T}\left( \mathbf{\theta }\right): =\left( \sum_{%
\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{ \mathbf{S}_{k}^{\prime }%
\mathbf{\Psi }_{k}^{\ast }(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })%
\mathbf{\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })\mathbf{S}%
_{k}\right \} \right) ^{-1}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}%
\left \{ \mathbf{S}_{k}^{\prime }\mathbf{\Psi }_{k}^{\ast }(\mathbf{\lambda }%
_{\mathbf{j}};\mathbf{\theta })\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}%
})\right \} ,
\end{equation*}%
and concentrate out $\mathbf{\alpha }_{k}$ in $\mathcal{L}_{k,T}\left(
\mathbf{\theta },\mathbf{\alpha }_{k}\right) $ so that
\begin{equation*}
\mathbf{\hat{\theta}}_{w,T}=\arg \min_{\mathbf{\theta }\in \mathcal{S}}%
\mathcal{L}_{w,T}\left( \mathbf{\theta }\right) ,
\end{equation*}%
or restricted to $\mathbf{\theta }\in \mathcal{S}^{\max }$ under
Assumption~6A, where%
\begin{equation*}
\mathcal{L}_{w,T}\left( \mathbf{\theta }\right) :=\mathcal{L}_{2,T}\left(
\mathbf{\theta }\right) +\sum_{k\in \mathcal{K}}w_{k}\mathcal{\hat{L}}%
_{k,T}\left( \mathbf{\theta }\right) ,\ \ \mathcal{\hat{L}}_{k,T}\left(
\mathbf{\theta }\right) :=\mathcal{L}_{k,T}\left( \mathbf{\theta },\mathbf{%
\hat{\alpha}}_{k,T}\left( \mathbf{\theta }\right) \right),
\end{equation*}%
and $\mathbf{\hat{\alpha}}_{k,T}=\mathbf{\hat{\alpha}}_{k,T}\left( \mathbf{%
\hat{\theta}}_{w,T}\right) .$ However, under Assumption~6B$\left( k\right) $
there is no guarantee that for $\mathbf{\theta }\in \mathcal{S}^{+}\ $it
holds that $\mathbf{\hat{\alpha}}_{k,T}\left( \mathbf{\theta }\right) \in
\mathcal{D}_{k},$ $k\in \mathcal{K}$, and optimization should be done
simultaneously for $\left( \mathbf{\theta },\ \mathbf{\alpha }_{k}, \ k\in
\mathcal{K}\right)$ in $\mathcal{S}^{+}\times \prod_{ \mathcal{K}}\mathcal{D}%
_{k}$.
Then, consistency of estimates under our set of identifying conditions is
achieved exploiting that periodogram averages estimate consistently
integrals of the true spectral densities.\bigskip
\begin{theorem}
\label{Th5} Under Assumptions 1$\left( 2k\right) ,$ 3$\left( k\right) ,$ 5, 6%
$\left( k\right) $, $k\in \mathcal{K},$ as $T\rightarrow \infty ,$
\begin{equation*}
\left( \mathbf{\hat{\theta}}_{w,T},\ \mathbf{\hat{\alpha}}_{k,T},\ k\in
\mathcal{K}\right) \rightarrow _{p}\left( \mathbf{\theta }_{0}, \ \mathbf{\
\alpha }_{k}^{0},\ k\in \mathcal{K}\right) .
\end{equation*}
\end{theorem}
Note that the result for $\mathbf{\ \hat{\alpha}}_{k,T}$ also holds for $k
\notin \mathcal{K} $ from the consistency of $\mathbf{\hat{\theta}}_{w,T}$
when Assumptions~1$\left( 2k \right) $ and 3$\left( k \right)$ hold for both
$k=3,4$, but only one set of cumulants needs to satisfy the nonzero
conditions to guarantee identification. Further, independence and equal
distribution of order $2k$ in Assumption~1$\left( k\right) $ are used to
facilitate the asymptotic analysis despite are not necessary for
identification.\bigskip
\textbf{Asymptotic Distribution}\bigskip
Now we consider optimal weighting of higher order periodograms replacing $%
\mathcal{L}_{k,T}\left( \mathbf{\theta },\mathbf{\alpha }\right) $ by a
weighted loss function
\begin{equation*}
\mathcal{L}_{k,T}^{EFF}\left( \mathbf{\theta },\mathbf{\alpha }\right) :=%
\frac{1}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\left( \mathbf{\Psi }%
_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })\mathbf{S}_{k}\mathbf{%
\alpha }-\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}})\right) ^{\ast }%
\mathbf{W}_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\tilde{\theta}}%
_{T})\left( \mathbf{\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta
})\mathbf{S}_{k}\mathbf{\alpha }-\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}%
})\right) ,
\end{equation*}%
where $\mathbf{W}_{k}(\mathbf{\lambda };\mathbf{\tilde{\theta}}_{T}):=\left(
\mathbf{\Psi }_{k}(\mathbf{\lambda };\mathbf{\tilde{\theta}}_{T})\mathbf{%
\Psi }_{k}^{\ast }(\mathbf{\lambda };\mathbf{\tilde{\theta}}_{T})\right)
^{-1}$ gives weights inversely proportional to the modulus of the higher
order transfer function $\mathbf{\Psi }_{k}(\mathbf{\lambda };\mathbf{\theta
})$ evaluated at some preliminary estimate $\mathbf{\tilde{\theta}}%
_{T}\rightarrow _{p}\mathbf{\theta }_{0}.$ This weighting does not require
structural identification since $\mathbf{W}_{k}$ produces the same weighting
when $\mathbf{\Psi }(e^{-i\lambda };\mathbf{\tilde{\theta}}_{T})$ is
replaced by $\mathbf{\Psi }(e^{-i\lambda };\mathbf{\tilde{\theta}}%
_{T})A\left( e^{-i\lambda }\right) $ for a BM $A\left( z\right) $, because%
\begin{equation*}
\left( \mathbf{\Psi }(e^{-i\lambda };\mathbf{\theta })A(e^{-i\lambda
})\right) ^{\otimes k}=\mathbf{\Psi }(e^{-i\lambda };\mathbf{\theta }%
)^{\otimes k}A(e^{-i \lambda })^{\otimes k}=\mathbf{\Psi }_{k}(\mathbf{%
\lambda };\mathbf{\theta })A^{\otimes k}(\mathbf{\lambda }).
\end{equation*}%
Further, instead of using the solution from the first order condition for $%
\mathbf{\alpha }$, we propose to use instead the simpler estimate%
\begin{equation*}
\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }\right) :=\frac{1}{%
T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{ \mathbf{S}%
_{k}^{\prime }\mathbf{\Psi }_{k}^{-1}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{%
\theta })\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}})\right \} ,
\end{equation*}%
exploiting that $\mathbf{S}_{k}^{\prime }\mathbf{\Psi }_{k}^{\ast }(\mathbf{%
\lambda };\mathbf{\theta }_{0})\mathbf{W}_{k}(\mathbf{\lambda };\mathbf{%
\theta }_{0})\mathbf{\Psi }_{k}(\mathbf{\lambda };\mathbf{\theta }_{0})%
\mathbf{S}_{k}=\mathbf{S}_{k}^{\prime }\mathbf{S}_{k}=\mathbf{I}_{d},$
similar to the proposal in VL (2018), and replace $\mathcal{L}_{k,T}^{EFF}$
by the pseudo-profile loss function $\mathcal{\hat{L}}_{k,T}^{\dag }\left(
\mathbf{\theta }\right) :=\mathcal{L}_{k,T}^{EFF}\left( \mathbf{\theta ,\hat{%
\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }\right) \right) .$
The consistency of parameter estimates%
\begin{equation*}
\mathbf{\hat{\theta}}_{w,T}^{\dag }\ := \ \arg \min_{\mathbf{\theta }\in
\mathcal{S}} \mathcal{\hat{L}}_{w,T}^{\dag } \left( \mathbf{\theta } \right)
, \ \ \mathcal{\hat{L}}_{w,T}^{\dag } \left( \mathbf{\theta } \right) :=
\mathcal{L}_{2,T}\left( \mathbf{\theta }\right) +\sum_{k\in \mathcal{K}}w_{k}%
\mathcal{\hat{L}}_{k,T}^{\dag }\left( \mathbf{\theta }\right) ,\
\end{equation*}%
restricted to $\mathbf{\theta }\in \mathcal{S}^{\max }$ under Assumption~6A,
can be deduced by the same arguments as for $\mathbf{\hat{\theta}}_{w,T}$.
We focus on estimates $\mathbf{\hat{\theta}}_{w,T}^{\dag }$ based on $%
\mathcal{\hat{L}}_{w,T}^{\dag }$ because its analysis is simpler due to the
efficient weighting scheme and a more straightforward estimation effect from
$\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }\right) ,$ but it
is immediate to show the asymptotic equivalence with estimates based on
minimizing $\mathcal{L}_{k,T}^{EFF}\left( \mathbf{\theta },\mathbf{\alpha }%
\right) $ for $\mathbf{\theta }\in \mathcal{S}^{+}\ $and $\mathbf{\alpha }%
\in \mathcal{D}_{k},$ $k\in \mathcal{K}$, under Assumption~6B$\left(
k\right) $, as well as that of $\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left(
\mathbf{\theta }\right) $ with $\mathbf{\hat{\alpha}}_{k,T}^{EFF}\left(
\mathbf{\theta }\right) .$
To investigate the asymptotic distribution of parameter estimates we need
further restrictions on the parameterization and a local identification
condition. Define
\begin{equation*}
\mathbf{H}_{k}\left( \mathbf{\theta }\right) :=\left( 2\pi \right)
^{1-k}\int_{\Pi ^{k-1}}\func{Re}\left \{ \mathbf{B}_{k}^{\ast }\left(
\mathbf{\lambda };\mathbf{\theta }\right) \mathbf{B}_{k}\left( \mathbf{%
\lambda };\mathbf{\theta }\right) \right \} d\mathbf{\lambda }
\end{equation*}%
where for $k=3,4$ we set%
\begin{equation*}
\mathbf{B}_{k}\left( \mathbf{\lambda };\mathbf{\theta }\right) :=\mathbf{%
\Psi }_{k}^{-1}\left( \mathbf{\lambda };\mathbf{\theta }\right) \mathbf{\dot{%
\Psi}}_{k}(\mathbf{\lambda };\mathbf{\theta })-\mathbf{S}_{k}\mathbf{S}%
_{k}^{\prime }\mathbf{\bar{\Lambda}}_{k}\left( \mathbf{\theta }\right)
=\sum_{j=1}^{k}\mathbf{B}_{k,j}\left( \lambda _{j};\mathbf{\theta }\right)
,\ \ \
\end{equation*}%
with $\mathbf{\bar{\Lambda}}_{k}\left( \mathbf{\theta }\right) :=\left( 2\pi
\right) ^{1-k}\int_{\Pi ^{k-1}}\func{Re}\left \{ \mathbf{\Psi }%
_{k}^{-1}\left( \mathbf{\lambda };\mathbf{\theta }\right) \mathbf{\dot{\Psi}}%
_{k}(\mathbf{\lambda };\mathbf{\theta })\right \} d\mathbf{\lambda }$, and
$\mathbf{B}_{2}\left( \lambda ;\mathbf{\theta }\right) :=\mathbf{\Psi }%
_{2}^{-1}\left( \lambda ;\mathbf{\theta }\right) \mathbf{\dot{\Psi}}%
_{2}(\lambda ;\mathbf{\theta })$\\ $=\mathbf{B}_{2,1}\left( \lambda ;\mathbf{%
\theta }\right) +\mathbf{B}_{2,2}\left( -\lambda ;\mathbf{\theta }\right) ,$
with%
\begin{equation*}
\mathbf{\dot{\Psi}}_{k}(\mathbf{\lambda };\mathbf{\theta }):=\left( \mathbf{%
\dot{\Psi}}_{k}^{\left( 1\right) }(\mathbf{\lambda };\mathbf{\theta }%
),\ldots ,\mathbf{\dot{\Psi}}_{k}^{\left( m\right) }(\mathbf{\lambda };%
\mathbf{\theta })\right) ,\ \ \ \mathbf{\dot{\Psi}}_{k}^{\left( \ell \right)
}(\mathbf{\lambda };\mathbf{\theta }):=\frac{\partial }{\partial \mathbf{%
\theta }_{\ell }}\mathbf{\Psi }_{k}(\mathbf{\lambda };\mathbf{\theta }),
\end{equation*}%
where $\mathbf{B}_{k,j}\left( \lambda _{j};\mathbf{\theta }\right) ,$ $%
j=1,\ldots ,k,$ are obtained at once by the $k$-fold multiplicative
structure of $\mathbf{B}_{k}.$ Introduce for $\mathbf{\alpha }=\left \{
\mathbf{\alpha }_{k}\right \} _{k\in \mathcal{K}}$%
\begin{equation*}
\mathbf{\Sigma }\left( \mathbf{\theta ,\alpha }\right) :=\left( \mathbf{I}%
_{m}\otimes \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}_{d}\right) \right) ^{\prime }\mathbf{%
H}_{2}\left( \mathbf{\theta }\right) \left( \mathbf{I}_{m}\otimes \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}%
\left( \mathbf{I}_{d}\right) \right) +\sum_{k\in \mathcal{K}} w_{k}\left(
\mathbf{I}_{m}\otimes \mathbf{S}_{k}\mathbf{\alpha }_{k}\right) ^{\prime }%
\mathbf{H}_{k}\left( \mathbf{\theta }\right) \left( \mathbf{I}_{m}\otimes
\mathbf{S}_{k}\mathbf{\alpha }_{k}\right)
\end{equation*}%
and the following assumption which imposes a rank condition on $\mathbf{%
\Sigma }\left( \mathbf{\theta }_{0},\mathbf{\alpha }_{0}\right) $ and
reinforces the smoothness conditions of the parameterization.\bigskip
\noindent \textbf{Assumption~7}.
\noindent \textbf{7.1.}\emph{\ Let }$\Phi _{i}\left( \mathbf{\theta }\right)
,$\emph{\ }$i=0,\ldots ,p,$ \emph{and\ }$\Theta _{i}\left( \mathbf{\theta }%
\right) ,$\emph{\ }$i=0,\ldots ,q,$\emph{\ have continuous third order
derivatives for all }$\theta \in S.$
\noindent \textbf{7.2.} $\mathbf{\theta }_{0}\in Int\left( \mathcal{S}^{\max
}\right) .$
\noindent \textbf{7.3. }$\mathbf{\Sigma }\left( \mathbf{\theta }_{0},\mathbf{%
\alpha }_{0}\right) >0.$\bigskip
Assumptions 7.1 and 7.2 are standard for the analysis of asymptotic
properties of extremum estimates, while sufficient conditions for the local
identification Assumption~7.3 are the full rank of individual Hessian
matrices, $\mathbf{H}_{k}\left( \mathbf{\theta }_{0}\right) >0$, for at
least one $k=2,3,4$ with $w_{k}>0$ and Assumption~3($k$) for $k\in \mathcal{K%
}.$ $\mathbf{H}_{2}$ is similar to the Hessian of PMLE estimates under
Gaussianity (and causality and invertibility), like Whittle estimates, which
only use second order information, noting that here $\mathbf{H}_{2}$
includes also the scaling parameters in $\mathbf{\Theta }_{0}$. This
indicates that usual methods are sufficient for local identification and
local asymptotic inference, but non-Gaussian information is key to achieve
global identification and potential efficiency improvements. Further, the
centering terms in $\mathbf{\bar{\Lambda}}_{k}\left( \mathbf{\theta }\right)
$ reflect the higher order cumulant estimation effect, with the ones
corresponding to non scaling parameters being identically zero for causal
and invertible processes.
To simplify the asymptotic variance of $\mathbf{\hat{\theta}}_{w,T}^{\dag }$
we could strengthen Assumption~3$(k),$ $k\in \mathcal{K}$, to ICA of order $%
k=8$, as we do in Appendix~D to obtain explicit formulae for the variance of
the following vectors of powers of $\mathbf{\varepsilon }_{t}$ and $\mathbf{%
\varepsilon }_{r}$ for $t\neq r,$
\begin{eqnarray*}
\mathbf{\varepsilon }_{t,r}^{\left[ 2\right] } &:=&\left[
\begin{array}{c}
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r} \\
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}%
\end{array}%
\right] \\
\mathbf{\varepsilon }_{t,r}^{\left[ 3\right] } &:=&\left[
\begin{array}{c}
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t} \\
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}%
\end{array}%
\right] -\sum_{a=1}^{d}\left[
\begin{array}{c}
\mathbf{e}_{a}\otimes \mathbf{e}_{a}\otimes \mathbf{\varepsilon }_{r} \\
\mathbf{e}_{a}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{e}_{a} \\
\mathbf{\varepsilon }_{r}\otimes \mathbf{e}_{a}\otimes \mathbf{e}_{a}%
\end{array}%
\right] ,\ \ \\
\mathbf{\varepsilon }_{t,r}^{\left[ 4\right] } &:=&\left[
\begin{array}{c}
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t} \\
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}%
\end{array}%
\right] -\sum_{a=1}^{d}\mathbf{\alpha }_{3a}^{0}\left[
\begin{array}{c}
\mathbf{e}_{a}\otimes \mathbf{e}_{a}\otimes \mathbf{e}_{a}\otimes \mathbf{%
\varepsilon }_{r} \\
\mathbf{e}_{a}\otimes \mathbf{e}_{a}\otimes \mathbf{\varepsilon }_{r}\otimes
\mathbf{e}_{a} \\
\mathbf{e}_{a}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{e}_{a}\otimes
\mathbf{e}_{a} \\
\mathbf{\varepsilon }_{r}\otimes \mathbf{e}_{a}\otimes \mathbf{e}_{a}\otimes
\mathbf{e}_{a}%
\end{array}%
\right] .
\end{eqnarray*}%
However, there is no need of nonzero assumptions on marginal cumulants for
order larger than max$_{\mathcal{K}}k$, apart of existence of $2k$ moments
for $k\in \mathcal{K}$, as, e.g., it is straightforward to justify estimates
with $w_{4}=0$ based on up most $k=3$ information with only six bounded
moments.
Define $\mathbf{C}_{k}\left( 0\right) :=\left( 2\pi \right) ^{1-k}\int_{\Pi
^{k-1}}\mathbf{B}_{k}^{\ast }\left( \mathbf{\lambda };\mathbf{\theta }%
_{0}\right) d\mathbf{\lambda }$ for $k=2,3,4$ and the row block matrices for
$j=\pm 1,\pm 2,\ldots ,$
\begin{equation*}
\mathbf{\mathbf{C}}_{k}\left( j\right) :=\left \{ \left( 2\pi \right)
^{1-k}\int_{\Pi ^{k-1}}\mathbf{B}_{k,a}^{\ast }\left( \mathbf{\lambda };%
\mathbf{\theta }_{0}\right) e^{-ij\lambda _{a}}d\mathbf{\lambda }\right \}
_{a=1,\ldots ,k},
\end{equation*}%
and%
\begin{equation*}
\mathbf{\Omega }\left( \mathbf{\theta }_{0};\mathbf{C}\right) :=\left \{
\QATOP{{}}{{}}\mathbf{\Omega }_{ab}\left( \mathbf{\theta }_{0};\mathbf{C}%
\right) \QATOP{{}}{{}}\right \} _{a,b=2,3,4},
\end{equation*}%
where $\mathbf{\Omega }_{ab}\left( \mathbf{\theta }_{0};\mathbf{C}\right) :=%
\mathbf{\Phi }_{ab}^{0}\left( \mathbf{\theta }_{0};\mathbf{C}\right) +%
\mathbf{\Phi }_{ab}\left( \mathbf{\theta }_{0};\mathbf{C}\right) +\mathbf{%
\Phi }_{ab}^{\dag }\left( \mathbf{\theta }_{0};\mathbf{C}\right) $ with
\begin{eqnarray*}
\mathbf{\Phi }_{ab}^{0}\left( \mathbf{\theta }_{0};\mathbf{C}\right) &:= &%
\mathbf{C}_{a}\left( 0\right) \mathbb{C}\left[ \mathbf{\varepsilon }%
_{t}^{\otimes a},\mathbf{\varepsilon }_{t}^{\otimes b}\right] \mathbf{C}%
_{b}^{\prime }\left( 0\right) \\
\mathbf{\Phi }_{ab}\left( \mathbf{\theta }_{0};\mathbf{C}\right) &:=
&\sum_{j=-\infty ,\neq 0}^{\infty }\mathbf{C}_{a}\left( j\right) \mathbb{C}%
\left[ \mathbf{\varepsilon }_{t,r}^{\left[ a\right] },\mathbf{\varepsilon }%
_{r,t}^{\left[ b\right] }\right] \mathbf{C}_{b}^{\prime }\left( j\right) \\
\mathbf{\Phi }_{ab}^{\dag }\left( \mathbf{\theta }_{0};\mathbf{C}\right) &:=
&\sum_{j=-\infty ,\neq 0}^{\infty }\mathbf{C}_{a}\left( -j\right) \mathbb{C}%
\left[ \mathbf{\varepsilon }_{t,r}^{\left[ a\right] },\mathbf{\varepsilon }%
_{r,t}^{\left[ b\right] }\right] \mathbf{C}_{b}^{\prime }\left( j\right)
\end{eqnarray*}%
and variance-covariance matrices not depending on $t$ or $r$ by
stationarity of order $2k,$ $k\in \mathcal{K}$. These definitions provide
multivariate generalizations of the score variance expressions developed in
VL for averages of higher order periodograms accounting for possibly
nonfundamental solutions and allow a compact presentation of the asymptotic
distribution of $\mathbf{\hat{\theta}}_{w,T}^{\dag }$ in next theorem. See
Appendix~D for explicit expressions for each $k$.
\begin{theorem}
\label{Th6} Under Assumptions 1$\left( 2k\right) ,$ 3$\left( k\right) ,$ 5, 6%
$\left( k\right) ,$ 7, $k\in \mathcal{K},$ $\min_{\mathcal{K}}w_{k}>0,$ as $%
T\rightarrow \infty ,$
\begin{equation*}
\sqrt{T}\left( \mathbf{\hat{\theta}}_{w,T}^{\dag }-\mathbf{\theta }%
_{0}\right) \rightarrow _{d}N_{m}\left( 0\mathbf{,\Sigma }^{-1}\left(
\mathbf{\theta }_{0},\mathbf{\alpha }_{0}\right) \mathbf{\delta }\left(
\mathbf{\alpha }_{0}\right) \mathbf{\Omega }\left( \mathbf{\theta }_{0};%
\mathbf{B}\right) \mathbf{\delta }^{\prime }\left( \mathbf{\alpha }%
_{0}\right) \mathbf{\Sigma }^{-1}\left( \mathbf{\theta }_{0},\mathbf{\alpha }%
_{0}\right) \right) ,
\end{equation*}%
where%
\begin{equation*}
\mathbf{\delta }\left( \mathbf{\alpha }_{0}\right) :=\left[ \ \ \left(
\mathbf{I}_{m}\otimes \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}_{d}\right) \right)
^{\prime }\ \ \left \vert \ \ w_{3}\left( \mathbf{I}_{m}\otimes \mathbf{S}%
_{3}\mathbf{\alpha }_{3}^{0}\right) ^{\prime }\ \ \right \vert \ \
w_{4}\left( \mathbf{I}_{m}\otimes \mathbf{S}_{4}\mathbf{\alpha }%
_{4}^{0}\right) ^{\prime }\ \ \right] .
\end{equation*}
\end{theorem}
As in VL, the terms $\mathbf{\Phi }_{ab}^{\dag }$ are only different from
zero for noncausal or noninvertible models for which $\mathbf{B}_{k}\left(
\lambda ;\mathbf{\theta }\right) $ have a representation with terms in $%
e^{-ij\lambda }$ for $j=-1,-2,\ldots .$ Note also that $\mathbf{\Phi }%
_{22}^{0}$ incorporates the scaling estimation effect, which is treated
separately in other parameterizations, as in the univariate model in VL and,
partially, in the one discussed in Appendix~E, because, e.g. for causal and
invertible models, only the contributions from elements of $\mathbf{\theta }$
affecting $\Theta _{0}\left( \mathbf{\theta }\right) $ are different from
zero in $\mathbf{C}_{2}\left( 0\right) $. In the same line, the form of $\mathbf{\Phi }_{ab}^{0}$, $a,b>2$, accounts for the estimation of higher order cumulants under the ICA restriction of Assumption~3$(k)$, see Appendix~D for details.
We next present the asymptotic distribution of $\mathbf{\hat{\alpha}}%
_{k,T}^{\dag }\left( \mathbf{\hat{\theta}}_{w,T}^{\dag }\right) $ for a
particular $k\in \left \{ 3,4\right \} $ without assumptions on $\mathbf{%
\alpha }_{k}^{0},$ which could contain many zeros, obtaining an easy test of
overidentification. The only requisite is that $\mathbf{\hat{\theta}}%
_{w,T}^{\dag }$ is consistent and asymptotic normal with identification
provided by, possibly, a different set of cumulants. We introduce the
following vector of fourth powers of errors which include all permutations
of pairs, where $E_{t}$ stands for expectation conditional on $\mathbf{%
\varepsilon }_{t}$, $t\neq r,$ which is relevant for the variance of the
sample kurtosis vector coefficients $\mathbf{\hat{\alpha}}_{4,T}^{\dag
}\left( \mathbf{\theta }_{0}\right) ,$%
\begin{equation*}
\mathbf{\varepsilon }_{t}^{\left[ 4\right] }:=E_{t}\left[
\begin{array}{c}
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{r} \\
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r} \\
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t} \\
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}%
\end{array}%
\right] ,
\end{equation*}%
where, e.g., $\mathbf{\varepsilon }_{t,1}^{\left[ 4\right] }=\sum_{a,b=1}^{d}%
\mathbf{\varepsilon }_{t,a}^{2}\left( \mathbf{e}_{a}\otimes \mathbf{e}%
_{a}\otimes \mathbf{e}_{b}\otimes \mathbf{e}_{b}\right) $ given that $E\left[
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{r}\right]
=\sum_{b=1}^{d}E\left[ \mathbf{\varepsilon }_{r,b}^{2}\right] \left( \mathbf{%
e}_{b}\otimes \mathbf{e}_{b}\right) $\\ $=\sum_{b=1}^{d}\left( \mathbf{e}%
_{b}\otimes \mathbf{e}_{b}\right) .$ Define $\mathbf{\Delta }_{k}:=\mathbf{1}%
_{3}^{\prime }\otimes \mathbf{S}_{k}^{\prime }$ and $\mathbf{\Omega }\left(
\mathbf{\theta }_{0};\mathbf{D}_{k}\right) $ as $\mathbf{\Omega }\left(
\mathbf{\theta }_{0};\mathbf{C}\right) $ with $\mathbf{C}_{h}$ replaced by $%
\mathbf{D}_{k,h},$ $h=2,3,4,$ where for $j=\pm 1,\pm 2,\ldots ,$%
\begin{equation*}
\mathbf{D}_{k,h}\left( j\right) :=-w_{h}\mathbf{\bar{\Lambda}}_{k}\left(
\mathbf{\theta }_{0}\right) \left( \mathbf{I}_{m}\otimes \mathbf{S}_{k}%
\mathbf{\alpha }_{k}^{0}\right) \mathbf{\Sigma }^{-1}\left( \mathbf{\theta }%
_{0},\mathbf{\alpha }_{0}\right) \left( \mathbf{I}_{m}\otimes \mathbf{S}_{h}%
\mathbf{\alpha }_{h}^{0}\right) ^{\prime }\mathbf{\mathbf{C}}_{h}\left(
j\right)
\end{equation*}%
with $\mathbf{S}_{2}\mathbf{\alpha }_{2}^{0}=\ $vec$\left( \mathbf{I}%
_{d}\right) $ and $w_{2}=1,$ and for $j=0,$%
\begin{equation*}
\mathbf{D}_{k,h}\left( 0\right) :=\mathbf{I}_{d^{k}}1_{\left \{ k=h\right \}
}-w_{h}\mathbf{\bar{\Lambda}}_{k}\left( \mathbf{\theta }_{0}\right) \left(
\mathbf{I}_{m}\otimes \mathbf{S}_{k}\mathbf{\alpha }_{k}^{0}\right) \mathbf{%
\Sigma }^{-1}\left( \mathbf{\theta }_{0},\mathbf{\alpha }_{0}\right) \left(
\mathbf{I}_{m}\otimes \mathbf{S}_{h}\mathbf{\alpha }_{h}^{0}\right) ^{\prime
}\mathbf{\mathbf{C}}_{h}\left( 0\right) .
\end{equation*}%
Define also $\mathbf{\bar{\Delta}}_{4}:=\mathbf{1}_{4}^{\prime }\otimes
\mathbf{S}_{4}^{\prime }\ $and the block matrix $\mathbf{\bar{\Phi}}%
^{0}\left( \mathbf{\theta }_{0};\mathbf{D}_{4}\right) :=\left \{ \mathbf{%
\Phi }_{ab}^{0}\left( \mathbf{\theta }_{0};\mathbf{D}_{4}\right) \right \}
_{a,b=2,3,4,5}$ being equal to $\mathbf{\Phi }^{0}$ adding an extra row and
column$\ $given by, $b=2,3,4,$%
\begin{eqnarray*}
\mathbf{\Phi }_{5b}^{0}\left( \mathbf{\theta }_{0};\mathbf{D}_{4}\right) &:=
&\left( \mathbf{1}_{6}^{\prime }\otimes \mathbf{I}_{d^{4}}\right) \mathbb{C}%
\left[ \mathbf{\varepsilon }_{t}^{\left[ 4\right] },\mathbf{\varepsilon }%
_{t}^{\otimes b}\right] \mathbf{D}_{k,b}^{\prime }\left( 0\right) \ = \
\mathbf{\Phi }_{b5}^{0}\left( \mathbf{\theta }_{0};\mathbf{D}_{4}\right)
^{\prime } \\
\mathbf{\Phi }_{55}^{0}\left( \mathbf{\theta }_{0};\mathbf{D}_{4}\right) &:=
&\left( \mathbf{1}_{6}^{\prime }\otimes \mathbf{I}_{d^{4}}\right) \mathbb{V}%
\left[ \mathbf{\varepsilon }_{t}^{\left[ 4\right] }\right] \left( \mathbf{1}%
_{6}^{\prime }\otimes \mathbf{I}_{d^{4}}\right) ^{\prime }.
\end{eqnarray*}
\begin{theorem}
\label{Th7} Under Assumptions 1$\left( 2h\right) ,$ $3\left( h\right) ,$ 5, 6%
$\left( h\right) ,$ 7, $h\in \mathcal{K}\cup \{k\},$ $\min_{\mathcal{K}%
}w_{h}>0,$ as $T\rightarrow \infty ,$ for $k=3$,
\begin{equation*}
T^{1/2}\left( \mathbf{\hat{\alpha}}_{3,T}^{\dag }\left( \mathbf{\hat{\theta}}%
_{w,T}^{\dag }\right) -\mathbf{\alpha }_{3}^{0}\right) \rightarrow
_{d}N_{d}\left( 0,\mathbf{\Delta }_{3}\mathbf{\Omega }\left( \mathbf{\theta }%
_{0};\mathbf{D}_{3}\right) \mathbf{\Delta }_{3}^{\prime }\right)
\end{equation*}%
and for $k=4$,
\begin{equation*}
T^{1/2}\left( \mathbf{\hat{\alpha}}_{4,T}^{\dag }\left( \mathbf{\hat{\theta}}%
_{w,T}^{\dag }\right) -\mathbf{\alpha }_{4}^{0}\right) \rightarrow
_{d}N_{d}\left( 0,\mathbf{\bar{\Delta}}_{4}\mathbf{\bar{\Phi}}^{0}\left(
\mathbf{\theta }_{0};\mathbf{D}_{4}\right) \mathbf{\bar{\Delta}}_{4}^{\prime
}+\mathbf{\Delta }_{4}\left( \mathbf{\Phi }\left( \mathbf{\theta }_{0};%
\mathbf{D}_{4}\right) +\mathbf{\Phi }^{\dag }\left( \mathbf{\theta }_{0};%
\mathbf{D}_{4}\right) \right) \mathbf{\Delta }_{4}^{\prime }\right) .
\end{equation*}
\end{theorem}
\section{6. Parameter GMM Estimation and Bootstrap Approximations}
In this section we propose estimates that exploit efficiently all the
information used by the minimum distance estimates of the previous section
by minimizing simultaneously the score functions of $\mathcal{L}_{k,T}\left(
\mathbf{\theta },\mathbf{\alpha }\right) $ for $k=2,3,4$ without need to
specify $w.$ Denote as in VL the gradient vector of the concentrated loss
functions for all $k=2,3,4$ by%
\begin{equation*}
\mathbb{S}_{T}\left( \mathbf{\theta }\right) : =\left(
\begin{array}{c}
\frac{\partial }{\partial \theta }\mathcal{L}_{2,T}(\mathbf{\theta }) \\
\frac{\partial }{\partial \theta }\mathcal{\hat{L}}_{3,T}^{\dag }(\mathbf{%
\theta }) \\
\frac{\partial }{\partial \theta }\mathcal{\hat{L}}_{4,T}^{\dag }(\mathbf{%
\theta })%
\end{array}%
\right) ,
\end{equation*}%
and denote the asymptotic variance of $T^{1/2}\mathbb{S}_{T}\left( \mathbf{%
\theta }_{0}\right) $ by $\mathcal{V}$ depending on $\mathbf{\delta }$ and $%
\mathbf{\Omega }$ from Theorem~7, so we can consider the objective function
\begin{equation}
\mathcal{Q}_{T}\left( \mathbf{\theta }\right) =\mathbb{S}_{T}\left( \mathbf{%
\theta }\right) ^{\prime }\mathcal{\hat{V}}_{T}^{-}\, \mathbb{S}_{T}\left(
\mathbf{\theta }\right) , \label{gmmof}
\end{equation}%
to optimally weight the information on $\mathbf{\theta }_{0}$ contained in
the score vector $\mathbb{S}_{T}\left( \mathbf{\theta }\right) .$ Here $%
\mathcal{\hat{V}}_{T}^{-}$ is a consistent estimator of the matrix $\mathcal{%
V}^{-},$ which is a reflexive generalized inverse of $\mathcal{V}$, and
hence, satisfies $\mathcal{VV}^{-}\mathcal{V}=\mathcal{V}$ and $\mathcal{V}%
^{-}\mathcal{VV}^{-}=\mathcal{V}^{-}$ as the Moore-Penrose inverse. We
employ generalized inverses to take account of cases where the asymptotic
variance of $\mathbb{S}_{T}\left( \mathbf{\theta }\right) $ is default rank
when identification rank conditions in Assumption~3$\left( k\right) $ fail
for some $k=3,4$. Hence, the proposed efficient estimator of $\mathbf{\theta
}_{0}$ is a Newton-Raphson step using (\ref{gmmof}),
\begin{equation}
\mathbf{\hat{\theta}}_{GMM,T}:=\mathbf{\tilde{\theta}}_{T}-\left( \mathbb{H}%
_{T}\left( \mathbf{\tilde{\theta}}_{T}\right) ^{\prime }\mathcal{\hat{V}}%
_{T}^{-}\, \mathbb{H}_{T}\left( \mathbf{\tilde{\theta}}_{T}\right) \right)
^{-1}\mathbb{H}_{T}\left( \mathbf{\tilde{\theta}}_{T}\right) ^{\prime }%
\mathcal{\hat{V}}_{T}^{-}\, \mathbb{S}_{T}\left( \mathbf{\tilde{\theta}}%
_{T}\right) , \label{theestimator}
\end{equation}%
where $\mathbb{H}_{T}\left( \mathbf{\theta }\right) :=\left( \partial
/\partial \mathbf{\theta }^{\prime }\right) \mathbb{S}_{T}\left( \mathbf{%
\theta }\right) ,$ and the initial estimate $\tilde{\mathbf{\theta }}_{T}$
satisfying
\begin{equation}
\mathbf{\tilde{\theta}}_{T}-\mathbf{\theta }_{0}=O_{p}\left( T^{-1/2}\right)
, \label{condis}
\end{equation}%
could be $\mathbf{\hat{\theta}}_{w,T}^{\dag }$ or any other PML or GMM
estimate obtained under appropriate identifying conditions. Given (\ref%
{condis}), the consistency of $\mathbf{\hat{\theta}}_{GMM,T}$ is trivial and
the next theorem states its asymptotic distribution defining the column
block matrix with with $\mathbf{S}_{2}\mathbf{\alpha }_{2}^{0}=\ $vec$\left(
\mathbf{I}_{d}\right) .$
\begin{eqnarray*}
\mathbb{H} &:=&p\lim_{T\rightarrow \infty }\mathbb{H}_{T}\left( \mathbf{%
\tilde{\theta}}_{T}\right) =p\lim_{T\rightarrow \infty }\frac{\partial }{%
\partial \mathbf{\theta }^{\prime }}\mathbb{S}_{T}\left( \mathbf{\theta }%
_{0}\right) \\
&=&\left \{ \left( \mathbf{I}_{m}\otimes \mathbf{S}_{k}\mathbf{\alpha }%
_{k}^{0}\right) ^{\prime }\mathbf{H}_{k}\left( \mathbf{\theta }_{0}\right)
\left( \mathbf{I}_{m}\otimes \mathbf{S}_{k}\mathbf{\alpha }_{k}^{0}\right)
\right \} _{k=2,3,4}
\end{eqnarray*}
\begin{theorem}
\label{THGMM} Under Assumptions 1$\left( 2k\right) ,$ 3$(k),$ $k\in \left \{
3,4\right \} $, 5, 7, and $\mathcal{\hat{V}}_{T}^{-}\rightarrow _{p}\mathcal{%
V}^{-}$ as $T\rightarrow \infty ,$%
\begin{equation*}
\sqrt{T}(\mathbf{\hat{\theta}}_{GMM,T}-\mathbf{\theta }_{0})\rightarrow
_{d}N\left( 0,(\mathbb{H}^{\prime }\mathcal{V}^{-}\mathbb{H})^{-1}\right) .
\end{equation*}
\end{theorem}
Note that Theorem~\ref{THGMM} holds irrespective of zero values in $\mathbf{%
\alpha }_{k}^{0}$ for $k=3$ and $4,$ so no need of this part of Assumption~3(%
$k$) for both $k=3,4$ as far as (\ref{condis}) holds, and though estimation
procedures to obtain candidates for $\mathbf{\tilde{\theta}}_{T}$ can rely
on some form of non-Gaussianity, they might use only second-order
information complemented by economic identification restrictions. For the
same reason, Assumption~6 is not needed under (\ref{condis}).
To perform inference avoiding possibly imprecise estimation of $\mathbf{%
\Omega }$ or $\mathcal{V}$ it is possible to use a simple parametric
bootstrap based on resampling from the empirical distribution of model
residuals. However this procedure would require model simulation and
re-estimation in each resample, which could be costly for large high
dimensional models. Alternatively, we could resample the following
linearization of the estimates in terms of the higher order periodograms of
true errors $\mathbf{\varepsilon }_{t},$
\begin{equation*}
\mathbf{\hat{\theta}}_{w,T}^{\dag }-\mathbf{\theta }_{0}=\mathbf{\Sigma }%
^{-1}\left( \mathbf{\theta }_{0},\mathbf{\alpha }_{0}\right) \sum _{k\in
\mathcal{K} \cup 2}\frac{w_{k}}{T^{k-1}}\left( \mathbf{I}_{m}\otimes \mathbf{%
S}_{k}\mathbf{\alpha }_{k}^{0}\right) ^{\prime }\sum_{\mathbf{\lambda }_{%
\mathbf{j}}}\func{Re}\left \{ \mathbf{B}_{k}^{\ast }\left( \mathbf{\lambda }%
_{\mathbf{j}};\mathbf{\theta }_{0}\right) \mathbb{I}_{k}^{\mathbf{%
\varepsilon }}(\mathbf{\lambda }_{\mathbf{j}})\right \} +o_{p}\left(
T^{-1/2}\right) ,
\end{equation*}%
where for $k=2$ we set $\mathbf{S}_{2}\mathbf{\alpha }_{2}^{0}=$ vec$\left(
\mathbf{I}_{d}\right) $, $w_{2}=1,$ and replace $\mathbb{I}_{2}^{\mathbf{%
\varepsilon }}(\mathbf{\lambda }_{\mathbf{j}})$ by $\mathbb{I}_{2}^{\mathbf{%
\varepsilon }}(\mathbf{\lambda }_{\mathbf{j}})-$vec$\left( \mathbf{I}%
_{2}\right) $. Resampled versions of the estimates are obtained replacing by
$\left( \mathbf{\theta }_{0},\mathbf{\alpha }_{k}^{0}, k\in \mathcal{K}
\right) $ by $\left( \mathbf{\hat{\theta}}_{w,T}^{\dag },\mathbf{\hat{\alpha}%
}_{k,T}^{\dag }, k\in \mathcal{K} \right) $ and $\mathbb{I}_{k}^{\mathbf{%
\varepsilon }}$ by the periodograms $\mathbb{I}_{k}^{\mathbf{\hat{\varepsilon%
}}^{\star }}$ of resampled residuals $\mathbf{\hat{\varepsilon}}_{t}^{\star
} $ from the empirical distribution of residuals $\mathbf{\hat{\varepsilon}}%
_{t}=\mathbf{\varepsilon }_{t}\left( \mathbf{\hat{\theta}}_{w,T}^{\dag
}\right) ,$ $t=1,\ldots ,T,$ properly standardized,
\begin{equation*}
\mathbf{\hat{\theta}}_{w,T}^{\star }:=\mathbf{\hat{\theta}}_{w,T}^{\dag }+%
\mathbf{\Sigma }^{-1}\left( \mathbf{\hat{\theta}}_{w,T}^{\dag },\mathbf{\hat{%
\alpha}}_{T}^{\dag }\right) \sum _{k\in \mathcal{K} \cup 2}\frac{w_{k}}{%
T^{k-1}}\left( \mathbf{I}_{m}\otimes \mathbf{S}_{k}\mathbf{\hat{\alpha}}%
_{k,T}^{\dag }\right) ^{\prime }\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re%
}\left \{ \mathbf{B}_{k}^{\ast }\left( \mathbf{\lambda }_{\mathbf{j}};%
\mathbf{\hat{\theta}}_{w,T}^{\dag }\right) \mathbb{I}_{k}^{\mathbf{\hat{%
\varepsilon}}^{\star }}(\mathbf{\lambda }_{\mathbf{j}})\right \} ,
\end{equation*}%
which requires only one computation of $\mathbf{\Sigma }\left( \mathbf{\hat{%
\theta}}_{w,T}^{\dag },\mathbf{\hat{\alpha}}_{T}^{\dag }\right) $ and $%
\mathbf{B}_{k}\left( \mathbf{\lambda },\mathbf{\hat{\theta}}_{w,T}^{\dag
}\right) ,$ but not simulation or parameter re-estimation in each resample.
Interestingly, residuals $\mathbf{\hat{\varepsilon}}_{t}$ are obtained
directly in the frequency domain after inversion of the residual DFT $%
\mathbf{\Psi }(\lambda ;\mathbf{\hat{\theta}}_{w,T}^{\dag
})^{-1}w_{T}(\lambda ),$ without need to care about noninvertible or
noncausal roots or imposing ad-hoc factorizations. Resampling of residuals $%
\mathbf{\hat{\varepsilon}}_{t}$ and the FFT can be implemented efficiently
to obtain periodograms of resampled residuals $\mathbb{I}_{k}^{\mathbf{\hat{%
\varepsilon}}^{\star }}$ for any $k,$ as well as set $\mathbf{\hat{\alpha}}%
_{kT}^{\star }=\mathbf{\hat{\alpha}}_{kT}^{\dag }\left( \mathbf{\hat{\theta}}%
_{w,T}^{\star }\right) $ evaluated at the periodogram of data simulated from
resampled $\mathbf{\hat{\varepsilon}}_{t}^{\star }$ to approximate the
distribution of cumulant estimates.
Similar resampling methods to construct $\mathbf{\hat{\theta}}_{GMM,T}$ (and
approximate its finite sample distribution) can be implemented from (\ref%
{theestimator}) in the same way as for $\mathbf{\hat{\theta}}_{w,T}^{\dag }$%
, since the elements of $\mathbb{S}_{T}$ and $\mathbb{H}_{T}$ are the same
as those involved in the linearization of $\mathbf{\hat{\theta}}_{w,T}^{\dag
},$ but pooled instead of aggregated. In particular, it is straightforward
to estimate $\mathcal{V}$ by the sample covariance of bootstrap versions of $%
\mathbb{S}_{T}$ evaluated at a preliminary inefficient $\mathbf{\hat{\theta}}%
_{w,T}^{\dag }.$ Then, it is possible to construct bootstrap standard
errors, significance tests on parameter values and overidentification tests
based on redundant cumulants or restrictions on functions of the IRF.\bigskip
\section{7. Simulations}
In this section we consider several Monte Carlo experiments to check the
finite sample performance of our identification and estimation procedures.
We simulate bivariate $\left( d=2\right) $ SVARMA$\left( p,q\right) $
systems with $\left( p,q\right) =\left( 1,0\right) $ and $\left( 1,1\right)
, $ $\Theta \left( L\right) =\left( I-B_{1}L\right) \Omega $ and
\begin{equation*}
\Phi _{1}=\left(
\begin{array}{cc}
0.9 & 0 \\
-0.4 & 0.7%
\end{array}%
\right) ,\ B_{1}=\left(
\begin{array}{cc}
-\zeta _{1}^{\rho _{1}} & 0 \\
0 & -\zeta _{2}^{\rho _{2}}%
\end{array}%
\right) ,\ \ \Omega =\left(
\begin{array}{cc}
10 & 4 \\
-2 & 5%
\end{array}%
\right).
\end{equation*}%
Then, $\Theta _{0}=\Omega $ satisfies Assumption~6A and $\Theta
_{1}=-B_{1}\Omega $ with $\left \vert \zeta _{j}\right \vert <1$ and $\rho
_{j}=\pm 1,$ so the two roots of $\det \left( \Theta \left( z\right) \right)
$ are $-\zeta _{j}^{-\rho _{j}},$ $j=1,2$. We consider all configurations of
invertible and non-invertible roots (including mixed cases) when $p=0.$ We
also simulate the SVARMA$\left( 0,1\right) $ model considered by Gouri\'{e}%
roux et al. (2019) with mixed roots as for our VARMA$\left( 1,1\right) $
model. We set $\mathbf{\theta }_{0}=\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \Phi _{1}\right)
^{\prime },\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \Omega \right) ^{\prime },\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left(
B_{1}\right) ^{\prime }\right) ^{\prime }$ for $p=1$ and $\mathbf{\theta }%
_{0}=\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \Omega \right) ^{\prime },\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left(
B_{1}\right) ^{\prime }\right) ^{\prime }$ for $p=0$ and simulate two sample
sizes $T=100,200$ with 1000 and 500 replications, respectively.
We consider three sets of innovations. The first type are mutually
independent standardized $\chi _{n_{j}}^{2}$ variates, $(n_{1},n_{2})=(6,1)$,
designed to satisfy\ Assumption~6B$\left( k\right) $ for both $k=3$ or $k=4$
together with the value of $\Theta _{0}$ with $\mathbf{\alpha }%
_{3}^{0}=(1.155,2.828)^{\prime }$ and $\mathbf{\alpha }_{4}^{0}=(2,12)^{%
\prime }$. We also consider two further shock vectors, one composed of two
standardized $t_{n_{j}}$ variates, $(n_{1},n_{2})=(6,5)$ with $\mathbf{%
\alpha }_{3}^{0}=(0,0)^{\prime }$ and $\mathbf{\alpha }_{4}^{0}=(3,6)^{%
\prime }$ and one with a mixed normal and a $t_{6}$ variable as in Gouri\'{e}%
roux et al. (2019) with $\mathbf{\alpha }_{3}^{0}=(1,0)^{\prime }$ and $%
\mathbf{\alpha }_{4}^{0}=(6,3)^{\prime }.$ Note that these last two
distributions do not satisfy Assumption~3$\left( k\right) $ for $k=3$
because at least one component is symmetric, so (dynamic) identification
only relies on information from $k=4$ order cumulants. Also note that
Assumption~1$(2k)$ does not hold for these $t$ distributions, as they have 5
finite moments at most.
To identify both the correct rotation of the innovations and the location of
the MA lag polynomial roots we implement the following algorithm which
obtains different estimates in each step:
\begin{enumerate}
\item $\mathbf{\hat{\theta}}_{2,T}$: Causal and invertible reduced form
VARMA estimation using $\mathcal{L}_{2,T}$ loss function with Whittle
initial estimates to define the spectral weighting imposing uncorrelation
among components of $\mathcal{\varepsilon} _{t}$. If $p>0,$ preliminary IV
estimation of VAR parameters is performed as in Gouri\'{e}roux et al. (2019)
under the (correct) assumption of causality.
\item $\mathbf{\hat{\theta}}_{w,T}^\dag$: Minimum distance higher order
spectral estimation with unrestricted lag polynomial root location.
\begin{enumerate}
\item[2.1] Computation of all $2^{dq}=4$ MA basic representations of $%
\mathbf{\hat{\theta}}_{2,T}$ obtained by combinations of possibly inverted
MA roots using the procedure of Baggio and Ferrante (2019).
\item[2.2] Initial approximation of the rotation of the residuals for each
representation closest to component independence using a reconstruction ICA
algorithm as implemented in MATLAB RICA function.
\item[2.3] Higher order spectral estimation imposing independence of order $%
k=3$ or $k=3$ and $4$ in the components of $\mathbf{\varepsilon }_{t}$ by
minimizing $\mathcal{L}^\dag_{w,T}\left( \mathbf{\theta }\right) $ with $%
w\in \left \{ \left( 1,0\right) ,\left( 1,1\right) \right \} $ using the
rotated parameters from 2.2 of each specific root-model configurations as
initial estimates.
\item[2.4] Global minimum: choose the root configuration that minimizes $%
\mathcal{L}^\dag_{w,T}\left( \mathbf{\theta }\right) .$
\item[2.5] Permutation of components: choose the signed permutation of error
components to match Assumption~6A among all $2^{d}d!=8$ possible ones.
\end{enumerate}
\item $\mathbf{\hat{\theta}}_{GMM,T}$: Local GMM estimation based on
preliminary estimates $\mathbf{\hat{\theta}}_{w,T}^\dag $ from Step 2 and
bootstrap estimated variance of the score.\bigskip
\end{enumerate}
For Step 2.1 we also tried to minimize a penalized version of $\mathcal{L}%
_{2,T}$ that forces each specific root location configuration, with similar
results in most situations in terms of root identification, but less
efficient estimates. Alternatively, we also imposed Assumption~6B$\left(
k\right) $ in Step~2.5, but since it was not possible to get estimates of $%
\Theta _{0}$ with the right configuration of the implied cumulant estimates
in some replications, we do not report its results here. For comparison
purposes, we also report for invertible models the outcomes for an
infeasible version of the estimates $\mathbf{\hat{\theta}}_{2,T}$ obtained
in Step 1 (and after a RICA rotation to enforce component independence as in
Step 2.2) by finding the signed permuted version that minimizes the $%
\mathcal{L}^{2}$ distance to the true parameters. All $\mathcal{L}_{k,T}
\left( \mathbf{\theta }\right) $ objective functions minimized in the
different steps use the same causal and invertible weighting based on
Whittle estimates obtained in Step $1$ (as this is invariant to rotations of
innovations and flipping of polynomial roots). For all methods investigated
we also obtain estimates of $\mathbf{\alpha }_{k}$ and we report bias and
Root MSE across simulations of all estimates (and average absolute bias and
MSE for all elements of the vector $\mathbf{\theta }$).
\setlength{\tabcolsep}{3.3pt}
\renewcommand*{\arraystretch}{.8}
\begin{table}[h]
\centering
\noindent \textbf{Table 1. }Percentage of correct
identification of the location of the MA roots by $\mathcal{L}_{k,T}.$
\begin{tabular}{lllllcccccc}
\hline \hline
& & & & & \multicolumn{4}{c}{SVARMA$(0,1)$} & & SVARMA$(1,1)$ \\
\cline{6-9}\cline{11-11}
& Innovation & $T$ & $k$ & & Invert. & Mixed & Non-Inv. & Mix-GMR & & Mixed
\\ \hline \hline
& $\left( \chi _{6}^{2},\chi _{1}^{2}\right) $ & $100$ & $\mathcal{L}_{3}$ &
& \ \ \ 92.3\ \ \ \ & 83.3 & 92.0 & 66.7 & & 61.7 \\
& & & $\mathcal{L}_{3}+\mathcal{L}_{4}$ & & 91.4 & 79.5 & 92.1 & 67.3 &
& 60.6 \\
& & $200$ & $\mathcal{L}_{3}$ & & 99.0 & 99.0 & 99.6 & 76.6 & & 70.0 \\
& & & $\mathcal{L}_{3}+\mathcal{L}_{4}$ & & 89.2 & 86.8 & 88.4 & 76.4 &
& 71.4 \\ \hline
& $(MN,t_{6})$ & $100$ & $\mathcal{L}_{3}$ & & 78.0 & 75.3 & 77.3 & 36.8 &
& 61.7 \\
& & & $\mathcal{L}_{3}+\mathcal{L}_{4}$ & & 78.3 & 76.1 & 76.3 & 30.0 &
& 59.8 \\
& & $200$ & $\mathcal{L}_{3}$ & & 89.4 & 69.8 & 86.6 & 35.8 & & 72.0 \\
& & & $\mathcal{L}_{3}+\mathcal{L}_{4}$ & & 89.4 & 76.6 & 87.4 & 37.6 &
& 73.5 \\ \hline
& $(t_{6},t_{5})$ & $100$ & $\mathcal{L}_{3}$ & & 52.2 & 51.0 & 48.3 & 43.7
& & 39.3 \\
& & & $\mathcal{L}_{3}+\mathcal{L}_{4}$ & & 54.0 & 52.0 & 50.4 & 42.9 &
& 39.3 \\
& & $200$ & $\mathcal{L}_{3}$ & & 55.4 & 40.6 & 56.2 & 46.6 & & 42.0 \\
& & & $\mathcal{L}_{3}+\mathcal{L}_{4}$ & & 62.8 & 54.0 & 59.4 & 47.4 &
& 41.2 \\ \hline \hline
\end{tabular}
\caption{ \footnotesize
\noindent
Identification of SVARMA$\left( p, 1 \right)$ models
by minimizing $w_3 \mathcal{L}_{3,T} +w_4 \mathcal{L}_{4,T}$, $p\in \{0,1\}$, $d= 2 ,$
$\zeta _{1}= \zeta _{2}= 0.5 ,$ $\rho_{1}= 1 , \rho _{2}= 1 $. Mix-GMR is the bivariate SVARMA$(0,1)$ model simulated in Gouri\'{e}roux et al. (2019) with
$\mathbf{\theta} =(0,1,1,0.5,-0.5,1,0,-2)'$ and
mixed MA roots $(0.5,2)$. Innovations are $\left( \chi _{6}^{2},\chi
_{1}^{2}\right), (MN(2.12,1.41^2;-0.24,0.58^2;0.1,0.9),t_6),$ and $(t_6,t_5)$.\
$\mathcal{L}_{3}$ and $\mathcal{L}_{3}+\mathcal{L}_{4}$
identification use $w=(1,0)$ and $w=(1,1)$, respectively.}
\end{table}
\bigskip
In Table 1 we report the percentage of simulations that identified the right
location of roots for all the parameter configurations for the $p=0$ case
(just mixed case for Gouri\'{e}roux et al. (2019) and SVARMA$\left(
1,1\right) $ models), innovations, sample sizes and weights $w$ in the loss
functions. From the results for chi squared innovations we find that
skewness provides a very precise information for identification in
multivariate systems as was found for univariate models in VL, while the
additional use of kurtosis information does not help much for the sample
sizes considered and, even for the smallest ones, can introduce further
noise. For mixed normal and $t$ shocks our identification procedure is still
able to gain substantial information on the location of roots when
Assumption~3$\left( 3\right) $ fails for one component, but obtains quite
little when it fails for all components as for bivariate $t$ distributions
(about 50\% of right locations for each of the three configurations). When
adding information from kurtosis, i.e. $k=4$ cumulants, in these $k=3$
identification-failure situations, the results get better in almost all
cases, but with modest improvements in general given the small sample sizes
considered. Across all setups simulated, we found that mixed models with MA
roots both inside and outside the unit circle are more difficult to identify
than pure invertible or non-invertible systems, though only marginally in
same situations.
The results for the SVARMA$\left( 0,1\right) $ model simulated in Gouri\'{e}%
roux et al. (2019) also report significant differences for each set of
innovations, with kurtosis being relatively helpful for improving the
performance both for symmetric and asymmetric distributions. We observe a
similar pattern for SVARMA$\left( 1,1\right) $ models, but in general
identification of the MA polynomial roots is more complicated in the
presence of autorregressive dynamics. Still, kurtosis information seems more
valuable when skewness is default rank, but, when all series are perfectly
symmetric, identification results are poorer and only improve very slowly
with sample size.
\begin{table}[h!]
\renewcommand*{\arraystretch}{.75}
\centering
\noindent \textbf{Table 2.} Estimation of invertible SVARMA$\left( 0,1\right) .$ $T=100$
\begin{tabular}{rrrrrrrrr}
\hline \hline
& \multicolumn{8}{c}{Bias} \\ \cline{2-9}
& \multicolumn{2}{c}{$\left( \chi _{6}^{2},\chi _{1}^{2}\right) $} & &
\multicolumn{2}{c}{$(MN,t_{6})$} & & \multicolumn{2}{c}{$(t_{6},t_{5})$} \\
\cline{2-3}\cline{5-6}\cline{8-9}
$\mathbf{\theta }_{0}$ & $\mathbf{\hat{\theta}}_{2,T}$ & $\mathbf{\hat{\theta%
}}_{w,T}$ & & $\mathbf{\hat{\theta}}_{2,T}$ & $\mathbf{\hat{\theta}}_{w,T}$
& & $\mathbf{\hat{\theta}}_{2,T}$ & $\mathbf{\hat{\theta}}_{w,T}$ \\ \hline
10 & -0.612 & -0.299 & & -0.197 & -0.087 & & -0.304 & -0.648 \\
-2 & 0.075 & 0.046 & & -0.012 & 0.055 & & -0.075 & 0.542 \\
4 & -0.162 & -0.083 & & -0.078 & -0.180 & & -0.061 & -1.464 \\
5 & -0.193 & -0.049 & & -0.151 & -0.099 & & -0.171 & -0.284 \\
-0.5 & -0.016 & -0.005 & & -0.013 & -0.005 & & -0.017 & 0.005 \\
0 & -0.003 & -0.001 & & -0.002 & 0.001 & & 0.002 & 0.010 \\
0 & -0.006 & -0.002 & & -0.003 & 0.004 & & -0.005 & -0.016 \\
-0.5 & -0.018 & -0.001 & & -0.019 & -0.012 & & -0.011 & 0.016 \\
AVE & 0.134 & 0.061 & & 0.060 & 0.055 & & 0.080 & 0.373 \\
$\mathbf{\alpha }_{3}^{0}$ & $\mathbf{\hat{\alpha}}_{32,T}$ & $\mathbf{\hat{%
\alpha}}_{3w,T}$ & $\mathbf{\alpha }_{3}^{0}$ & $\mathbf{\hat{\alpha}}%
_{32,T} $ & $\mathbf{\hat{\alpha}}_{3w,T}$ & $\mathbf{\alpha }_{3}^{0}$ & $%
\mathbf{\hat{\alpha}}_{32,T}$ & $\mathbf{\hat{\alpha}}_{3w,T}$ \\ \hline
1.155 & -0.141 & -0.128 & 2 & -0.129 & -0.174 & 0 & -0.050 & -0.008 \\
2.828 & -0.503 & -0.420 & 0 & -0.011 & -0.036 & 0 & -0.008 & -0.038 \\
$\mathbf{\alpha }_{4}^{0}$ & $\mathbf{\hat{\alpha}}_{42,T}$ & $\mathbf{\hat{%
\alpha}}_{4w,T}$ & $\mathbf{\alpha }_{4}^{0}$ & $\mathbf{\hat{\alpha}}%
_{42,T} $ & $\mathbf{\hat{\alpha}}_{4w,T}$ & $\mathbf{\alpha }_{4}^{0}$ & $%
\mathbf{\hat{\alpha}}_{42,T}$ & $\mathbf{\hat{\alpha}}_{4w,T}$ \\ \hline
2 & -0.481 & -0.568 & 6 & -0.413 & -0.600 & 3 & -0.411 & 0.449 \\
12 & -4.286 & -4.073 & 3 & -1.233 & -1.144 & 6 & -2.704 & -1.554 \\
\hline \hline
& \multicolumn{8}{c}{RMSE} \\ \cline{2-9}
& \multicolumn{2}{c}{$\left( \chi _{6}^{2},\chi _{1}^{2}\right) $} & &
\multicolumn{2}{c}{$(MN,t_{6})$} & & \multicolumn{2}{c}{$(t_{6},t_{5})$} \\
\cline{2-3}\cline{5-6}\cline{8-9}
$\mathbf{\theta }_{0}$ & $\mathbf{\hat{\theta}}_{2,T}$ & $\mathbf{\hat{\theta%
}}_{w,T}$ & & $\mathbf{\hat{\theta}}_{2,T}$ & $\mathbf{\hat{\theta}}_{w,T}$
& & $\mathbf{\hat{\theta}}_{2,T}$ & $\mathbf{\hat{\theta}}_{w,T}$ \\ \hline
10 & 1.498 & 1.046 & & 1.720 & 1.607 & & 1.598 & 2.027 \\
-2 & 0.983 & 0.470 & & 0.785 & 0.921 & & 1.030 & 1.651 \\
4 & 2.233 & 1.340 & & 1.402 & 1.694 & & 2.140 & 3.740 \\
5 & 1.146 & 1.009 & & 0.647 & 0.593 & & 0.865 & 1.178 \\
-0.5 & 0.146 & 0.095 & & 0.124 & 0.089 & & 0.127 & 0.200 \\
0 & 0.059 & 0.051 & & 0.052 & 0.047 & & 0.057 & 0.088 \\
0 & 0.227 & 0.188 & & 0.225 & 0.199 & & 0.247 & 0.442 \\
-0.5 & 0.167 & 0.100 & & 0.125 & 0.101 & & 0.120 & 0.184 \\
AVE & 0.808 & 0.537 & & 0.635 & 0.656 & & 0.773 & 1.189 \\
$\mathbf{\alpha }_{3}^{0}$ & $\mathbf{\hat{\alpha}}_{32,T}$ & $\mathbf{\hat{%
\alpha}}_{3w,T}$ & $\mathbf{\alpha }_{3}^{0}$ & $\mathbf{\hat{\alpha}}%
_{32,T} $ & $\mathbf{\hat{\alpha}}_{3w,T}$ & $\mathbf{\alpha }_{3}^{0}$ & $%
\mathbf{\hat{\alpha}}_{32,T}$ & $\mathbf{\hat{\alpha}}_{3w,T}$ \\ \hline
1.155 & 0.419 & 0.394 & 2 & 0.483 & 0.510 & 0 & 0.797 & 0.887 \\
2.828 & 0.941 & 0.846 & 0 & 0.612 & 0.636 & 0 & 0.887 & 0.893 \\
$\mathbf{\alpha }_{4}^{0}$ & $\mathbf{\hat{\alpha}}_{42,T}$ & $\mathbf{\hat{%
\alpha}}_{4w,T}$ & $\mathbf{\alpha }_{4}^{0}$ & $\mathbf{\hat{\alpha}}%
_{42,T} $ & $\mathbf{\hat{\alpha}}_{4w,T}$ & $\mathbf{\alpha }_{4}^{0}$ & $%
\mathbf{\hat{\alpha}}_{42,T}$ & $\mathbf{\hat{\alpha}}_{4w,T}$ \\ \hline
2 & 2.137 & 2.116 & 6 & 2.851 & 2.815 & 3 & 3.556 & 3.219 \\
12 & 7.502 & 7.288 & 3 & 2.866 & 2.892 & 6 & 5.653 & 3.399 \\ \hline \hline
\end{tabular}
\caption{ \footnotesize
\noindent Estimation of SVARMA$\left( 0, 1 \right)$ models by GMM, $d= 2 ,$ $\zeta _{1}= \zeta _{2}= 0.5 ,$ $%
\rho_{1}= 1 , \rho _{2}= 1 $. Innovations $\left( \chi _{6}^{2},\chi
_{1}^{2}\right), (MN(2.12,1.41^2;-0.24,0.58^2;0.1,0.9),t_6),$ and $(t_6,t_5)$.\ $\mathcal{L}_{2}+\mathcal{L}_{3}+\mathcal{L}_{4}$ identification and estimation, $w=(1,1)$. AVE is the average of the absolute value of the column.}
\end{table}
In Tables 2-5 we report the results on the performance of parameter
estimates only for the replications which correctly identified the MA root
location to make the comparisons meaningful. We can check from the output in
Table 2 for SVARMA$\left( 0,1\right) $ invertible models that the behaviour
of parameter estimates $\mathbf{\hat{\theta}}_{GMM,T}$ compares well in
terms of bias and RMSE with the unfeasible estimate $\mathbf{\hat{\theta}}%
_{2,T},$ which uses information on the true value of the parameters. Among
the different distributions, the case of the symmetric $t$ distribution
seems to be the most complicated, both in terms of bias and variability,
given that its heavy tails violate the moment condition in Theorem~6, while
the $\chi ^{2}$ distribution is the most informative given its strong
skewness. The results for non-invertible models (mixed and non-invertible
cases) in Table~3 are similar but in general estimates are less precise
comparing, for instance, the estimation of $\Omega $ contained in the first
four components of $\mathbf{\theta }.$
\begin{table}[h!]
\renewcommand*{\arraystretch}{.75}
\centering
\noindent \textbf{Table 3.} Estimation of non-invertible SVARMA$\left( 0,1\right) .$ $T=100$
\begin{tabular}{rrrrrrrrrrrr}
\hline \hline
& \multicolumn{11}{c}{Bias} \\ \cline{2-12}
& \multicolumn{3}{c}{$\left( \chi _{6}^{2},\chi _{1}^{2}\right) $} & &
\multicolumn{3}{c}{$(MN,t_{6})$} & & \multicolumn{3}{c}{$(t_{6},t_{5})$} \\
\cline{2-4}\cline{6-8}\cline{10-12}
$\mathbf{\theta }_{0}$ & & Mixed & P.inv & & & Mixed & P.inv & & & Mixed
& P.inv \\ \hline
10 & & -0.407 & -0.145 & & & -0.832 & 0.098 & & & -1.686 & 0.094 \\
-2 & & 0.072 & 0.115 & & & 0.553 & -0.018 & & & 1.105 & 0.972 \\
4 & & -0.208 & -0.124 & & & -1.242 & -0.046 & & & -2.576 & -1.869 \\
5 & & 0.141 & 0.031 & & & 0.021 & 0.054 & & & 1.175 & 0.049 \\
-0.5/-2 & & -0.046 & -0.034 & & & -0.049 & -0.039 & & & -0.413 & -0.003
\\
0 & & 0.017 & 0.018 & & & 0.056 & -0.014 & & & -0.008 & -0.004 \\
0 & & 0.012 & 0.039 & & & 0.061 & 0.017 & & & -0.016 & -0.009 \\
-2 & & 0.017 & -0.053 & & & 0.050 & -0.008 & & & 0.409 & -0.021 \\
AVE & & 0.115 & 0.070 & & & 0.358 & 0.037 & & & 0.923 & 0.378 \\
& $\mathbf{\alpha }_{3}^{0}$ & & & & $\mathbf{\alpha }_{3}^{0}$ & & &
& $\mathbf{\alpha }_{3}^{0}$ & & \\ \hline
& 1.155 & -0.120 & -0.103 & & 2 & -0.095 & -0.175 & & 0 & -0.047 & -0.030
\\
& 2.828 & -0.394 & -0.435 & & 0 & -0.118 & -0.047 & & 0 & 0.031 & 0.007 \\
& $\mathbf{\alpha }_{4}^{0}$ & & & & $\mathbf{\alpha }_{4}^{0}$ & & &
& $\mathbf{\alpha }_{4}^{0}$ & & \\ \hline
& 2 & -0.491 & -0.340 & & 6 & 0.480 & -0.741 & & 3 & -1.392 & -1.080 \\
& 12 & -3.722 & -4.060 & & 3 & 0.436 & -1.157 & & 6 & -3.934 & -3.296 \\
\hline \hline
& \multicolumn{11}{c}{RMSE} \\ \cline{2-12}
& \multicolumn{3}{c}{$\left( \chi _{6}^{2},\chi _{1}^{2}\right) $} & &
\multicolumn{3}{c}{$(MN,t_{6})$} & & \multicolumn{3}{c}{$(t_{6},t_{5})$} \\
\cline{2-4}\cline{6-8}\cline{10-12}
$\mathbf{\theta }_{0}$ & & Mixed & P.inv & & & Mixed & P.inv & & & Mixed
& P.inv \\ \hline
10 & & 1.204 & 2.095 & & & 2.132 & 2.501 & & & 2.678 & 2.225 \\
-2 & & 1.559 & 1.218 & & & 2.034 & 1.434 & & & 3.589 & 2.576 \\
4 & & 1.675 & 2.752 & & & 3.829 & 2.882 & & & 4.952 & 5.020 \\
5 & & 1.504 & 1.449 & & & 1.840 & 1.161 & & & 2.331 & 1.411 \\
-0.5/-2 & & 0.138 & 0.430 & & & 0.257 & 0.406 & & & 0.657 & 0.389 \\
0 & & 0.285 & 0.224 & & & 1.013 & 0.188 & & & 0.439 & 0.223 \\
0 & & 0.281 & 0.959 & & & 0.580 & 0.980 & & & 0.473 & 0.759 \\
-2 & & 0.522 & 0.498 & & & 4.381 & 0.452 & & & 0.789 & 0.417 \\
AVE & & 0.896 & 1.203 & & & 2.008 & 1.250 & & & 1.989 & 1.628 \\
& $\mathbf{\alpha }_{3}^{0}$ & & & & $\mathbf{\alpha }_{3}^{0}$ & & &
& $\mathbf{\alpha }_{3}^{0}$ & & \\ \hline
& 1.155 & 0.414 & 0.415 & & 2 & 0.764 & 1.178 & & 0 & 0.718 & 0.711 \\
& 2.828 & 0.948 & 0.908 & & 0 & 0.937 & 0.396 & & 0 & 0.731 & 0.753 \\
& $\mathbf{\alpha }_{4}^{0}$ & & & & $\mathbf{\alpha }_{4}^{0}$ & & &
& $\mathbf{\alpha }_{4}^{0}$ & & \\ \hline
& 2 & 2.189 & 2.318 & & 6 & 2.059 & 6.607 & & 3 & 2.711 & 2.488 \\
& 12 & 7.606 & 7.409 & & 3 & 3.186 & 3.365 & & 6 & 4.657 & 4.051 \\
\hline \hline
\end{tabular}
\caption{\footnotesize
\noindent Estimation of SVARMA$\left( 0, 1 \right)$ models by GMM, $d= 2 ,$ $\zeta _{1}= \zeta _{2}= 0.5 ,$ $\rho_{1}= 1(\RIfM@\expandafter\text@\else\expandafter\mbox\fi{Mixed})/-1(\RIfM@\expandafter\text@\else\expandafter\mbox\fi{P.inv})$, $ \rho _{2}= -1 $. Innovations $\left( \chi _{6}^{2},\chi
_{1}^{2}\right), (MN(2.12,1.41^2;-0.24,0.58^2;0.1,0.9),t_6),$ and $(t_6,t_5)$.\ $\mathcal{L}_{2}+\mathcal{L}_{3}+\mathcal{L}_{4}$ identification and estimation, $w=(1,1)$. AVE is the average of the absolute value of the column.}
\end{table}
\bigskip
Summary results on estimation of higher order cumulants are severely
affected by some extreme replications, and, in general, confirm the usual
intuition that estimation of higher order moments is quite difficult for
small and moderate sample sizes and heavy tail distributions, even more
complicated for the kurtosis than for the skewness. Cumulant estimates are
biased towards zero, i.e. underestimate the non-Gaussianity, the bias
growing with the magnitude of the cumulant, and work similarly when based on
the unfeasible estimates $\mathbf{\hat{\theta}}_{2,T}.$
Table 4 reports the results for sample size $T=200$ and the same set-up, but
only for chi square innovations, confirming that models with non-invertible
roots tend to report more imprecise estimates and that estimation exploiting
higher order moments information in the invertible case can outperform
estimates using only second order moments.
The simulations on estimation of the SVARMA$\left( 1,1\right) $ mixed-root
model described in Table~5 for both $T=100$ and $200$ and $\chi ^{2}$
innovations using either $w=\left(1,0\right) $ and $\left( 1,1\right) $ to select different sets
of moments, confirm that kurtosis does not contribute much on top of
skewness in terms of bias and variability of estimates, in parallel with its
reduced identifying information. Cumulant estimation also becomes more
difficult with model complexity, though it improves substantially with the
sample size. \bigskip
\begin{table}[h!]
\centering
\renewcommand*{\arraystretch}{.75}
\noindent \noindent \textbf{Table 4.} Estimation of $\left( \chi _{6}^{2},\chi _{1}^{2}\right) $ SVARMA$\left( 0,1\right) .$ $T=200$
\begin{tabular}{rrrrrrrrr}
\hline \hline
\multicolumn{9}{c}{Bias} \\ \hline
\multicolumn{3}{c}{Invertible} & & \multicolumn{2}{c}{Mixed} & &
\multicolumn{2}{c}{Non-invert.} \\ \cline{1-3}\cline{5-6}\cline{8-9}
$\mathbf{\theta }_{0}$ & $\mathbf{\hat{\theta}}_{2,T}$ & $\mathbf{\hat{\theta%
}}_{w,T}$ & & $\mathbf{\theta }_{0}$ & $\mathbf{\hat{\theta}}_{w,T}$ & & $%
\mathbf{\theta }_{0}$ & $\mathbf{\hat{\theta}}_{w,T}$ \\ \hline
10 & -0.107 & -0.041 & & 10 & -0.237 & & 10 & 0.087 \\
-2 & 0.153 & 0.047 & & -2 & -0.080 & & -2 & -0.037 \\
4 & -0.324 & -0.076 & & 4 & 0.007 & & 4 & 0.095 \\
5 & -0.069 & -0.043 & & 5 & -0.073 & & 5 & -0.069 \\
-0.5 & -0.003 & -0.004 & & -0.5 & -0.009 & & -2 & 0.006 \\
0 & -0.001 & -0.001 & & 0 & -0.021 & & 0 & -0.007 \\
0 & -0.027 & -0.027 & & 0 & 0.001 & & 0 & -0.027 \\
-0.5 & 0.005 & 0.004 & & -2 & -0.019 & & -2 & -0.035 \\
AVE & 0.086 & 0.030 & & & 0.056 & & & 0.045 \\
$\mathbf{\alpha }_{3}^{0}$ & $\mathbf{\hat{\alpha}}_{32,T}$ & $\mathbf{\hat{%
\alpha}}_{3w,T}$ & & & $\mathbf{\hat{\alpha}}_{3w,T}$ & & & $\mathbf{%
\hat{\alpha}}_{3w,T}$ \\ \hline
1.155 & -0.054 & -0.051 & & & -0.010 & & & -0.035 \\
2.828 & -0.258 & -0.215 & & & -0.060 & & & -0.115 \\
$\mathbf{\alpha }_{4}^{0}$ & $\mathbf{\hat{\alpha}}_{42,T}$ & $\mathbf{\hat{%
\alpha}}_{4w,T}$ & & & $\mathbf{\hat{\alpha}}_{4w,T}$ & & & $\mathbf{%
\hat{\alpha}}_{4w,T}$ \\ \hline
2 & -0.210 & -0.275 & & & -0.213 & & & -0.267 \\
12 & -2.501 & -2.308 & & & -1.171 & & & -1.454 \\ \hline \hline
\multicolumn{9}{c}{RMSE} \\ \hline
\multicolumn{3}{c}{Invertible} & & \multicolumn{2}{c}{Mixed} & &
\multicolumn{2}{c}{Non-invert.} \\ \cline{1-3}\cline{5-6}\cline{8-9}
$\mathbf{\theta }_{0}$ & $\mathbf{\hat{\theta}}_{2,T}$ & $\mathbf{\hat{\theta%
}}_{w,T}$ & & $\mathbf{\theta }_{0}$ & $\mathbf{\hat{\theta}}_{w,T}$ & & $%
\mathbf{\theta }_{0}$ & $\mathbf{\hat{\theta}}_{w,T}$ \\ \hline
10 & 0.984 & 0.741 & & 10 & 0.737 & & 10 & 1.700 \\
-2 & 0.789 & 0.314 & & -2 & 0.773 & & -2 & 0.895 \\
4 & 1.780 & 1.000 & & 4 & 1.092 & & 4 & 1.752 \\
5 & 0.742 & 0.704 & & 5 & 1.067 & & 5 & 1.078 \\
-0.5 & 0.064 & 0.065 & & -0.5 & 0.080 & & -2 & 0.312 \\
0 & 0.028 & 0.029 & & 0 & 0.136 & & 0 & 0.162 \\
0 & 0.121 & 0.121 & & 0 & 0.157 & & 0 & 0.591 \\
-0.5 & 0.063 & 0.064 & & -2 & 0.297 & & -2 & 0.304 \\
AVE & 0.571 & 0.380 & & & 0.542 & & & 0.849 \\
$\mathbf{\alpha }_{3}^{0}$ & $\mathbf{\hat{\alpha}}_{32,T}$ & $\mathbf{\hat{%
\alpha}}_{3w,T}$ & & & $\mathbf{\hat{\alpha}}_{3w,T}$ & & & $\mathbf{%
\hat{\alpha}}_{3w,T}$ \\ \hline
1.155 & 0.335 & 0.308 & & & 0.284 & & & 0.273 \\
2.828 & 0.729 & 0.679 & & & 0.827 & & & 0.814 \\
$\mathbf{\alpha }_{4}^{0}$ & $\mathbf{\hat{\alpha}}_{42,T}$ & $\mathbf{\hat{%
\alpha}}_{4w,T}$ & & & $\mathbf{\hat{\alpha}}_{4w,T}$ & & & $\mathbf{%
\hat{\alpha}}_{4w,T}$ \\ \hline
2 & 2.019 & 1.933 & & & 1.750 & & & 1.681 \\
12 & 6.961 & 6.978 & & & 9.409 & & & 9.148 \\ \hline \hline
\end{tabular}
\caption{\footnotesize
\noindent Estimation of SVARMA$\left( 0, 1 \right)$ models by GMM, $d= 2 ,$ $\zeta _{1}= \zeta _{2}= 0.5 ,$ $%
\rho_{1}= \pm 1 , \rho _{2}= \pm 1 $. Innovations $\left( \chi _{6}^{2},\chi
_{1}^{2}\right).$\ $\mathcal{L}_{2}+\mathcal{L}_{3}+\mathcal{L}_{4}$ identification and estimation, $w=(1,1)$. AVE is the average of the absolute value of the column.}
\end{table}
\begin{table}[h!]
\centering
\renewcommand*{\arraystretch}{.58}
\noindent \textbf{Table 5.} Estimation of $\left( \chi
_{6}^{2},\chi _{1}^{2}\right) $-SVARMA$\left( 1,1\right) .$ Mixed Roots.
\begin{tabular}{rrcrrrrcrr}
\hline \hline
& \multicolumn{9}{c}{Bias} \\ \hline
& \multicolumn{4}{c}{$T=100$} & & \multicolumn{4}{c}{$T=200$} \\
\cline{2-5}\cline{7-10}
$\mathbf{\theta }_{0}$& & $\mathcal{L}_{3}$ & & \multicolumn{1}{c}{$\mathcal{L}_{3}+\mathcal{L}%
_{4}$} & & & $\mathcal{L}_{3}$ & & \multicolumn{1}{c}{$\mathcal{L}_{3}+%
\mathcal{L}_{4}$} \\ \hline
0.9 & & \multicolumn{1}{r}{-0.045} & & -0.002 & & & \multicolumn{1}{r}{
-0.026} & & -0.027 \\
-0.4 & & \multicolumn{1}{r}{0.155} & & 0.162 & & & \multicolumn{1}{r}{
0.170} & & 0.156 \\
0 & & \multicolumn{1}{r}{0.028} & & 0.039 & & & \multicolumn{1}{r}{-0.007
} & & -0.007 \\
0.7 & & \multicolumn{1}{r}{-0.041} & & -0.052 & & & \multicolumn{1}{r}{
-0.011} & & -0.017 \\
10 & & \multicolumn{1}{r}{-1.100} & & -1.178 & & & \multicolumn{1}{r}{
-0.384} & & -0.700 \\
-2 & & \multicolumn{1}{r}{0.200} & & 0.159 & & & \multicolumn{1}{r}{0.051
} & & 0.283 \\
4 & & \multicolumn{1}{r}{-1.773} & & -1.853 & & & \multicolumn{1}{r}{
-0.704} & & -1.010 \\
5 & & \multicolumn{1}{r}{0.401} & & 0.823 & & & \multicolumn{1}{r}{-0.037
} & & 0.224 \\
-0.5 & & \multicolumn{1}{r}{-0.159} & & -0.264 & & & \multicolumn{1}{r}{
-0.104} & & -0.149 \\
0 & & \multicolumn{1}{r}{-0.087} & & -0.035 & & & \multicolumn{1}{r}{
-0.055} & & -0.029 \\
0 & & \multicolumn{1}{r}{-0.008} & & 0.032 & & & \multicolumn{1}{r}{0.008
} & & 0.044 \\
-2 & & \multicolumn{1}{r}{-0.079} & & 0.203 & & & \multicolumn{1}{r}{
-0.038} & & -0.010 \\
AVE & & \multicolumn{1}{r}{0.340} & & 0.400 & & & \multicolumn{1}{r}{
0.133} & & 0.221 \\
$\mathbf{\alpha }_{3}^{0}$ & & \multicolumn{1}{r}{$\mathbf{\hat{\alpha}}%
_{3w,T}$} & & $\mathbf{\hat{\alpha}}_{3w,T}$ & & & \multicolumn{1}{r}{$%
\mathbf{\hat{\alpha}}_{3w,T}$} & & $\mathbf{\hat{\alpha}}_{3w,T}$ \\ \hline
1.155 & & \multicolumn{1}{r}{-0.192} & & -0.265 & & & \multicolumn{1}{r}{
-0.014} & & -0.162 \\
2.828 & & \multicolumn{1}{r}{-1.005} & & -1.227 & & & \multicolumn{1}{r}{
-0.511} & & -0.934 \\
$\mathbf{\alpha }_{4}^{0}$ & & \multicolumn{1}{r}{$\mathbf{\hat{\alpha}}%
_{4w,T}$} & & $\mathbf{\hat{\alpha}}_{4w,T}$ & & & \multicolumn{1}{r}{$%
\mathbf{\hat{\alpha}}_{4w,T}$} & & $\mathbf{\hat{\alpha}}_{4w,T}$ \\ \hline
2 & & \multicolumn{1}{r}{0.277} & & 0.060 & & & \multicolumn{1}{r}{0.788}
& & -0.239 \\
12 & & \multicolumn{1}{r}{-10.415} & & -7.540 & & & \multicolumn{1}{r}{
-2.112} & & -6.346 \\ \hline \hline
& \multicolumn{9}{c}{RMSE} \\ \hline
& \multicolumn{4}{c}{$T=100$} & & \multicolumn{4}{c}{$T=200$} \\
\cline{2-5}\cline{7-10}
$\mathbf{\theta }_{0}$ & & $\mathcal{L}_{3}$ & & \multicolumn{1}{c}{$\mathcal{L}_{3}+\mathcal{L}%
_{4}$} & & & $\mathcal{L}_{3}$ & & \multicolumn{1}{c}{$\mathcal{L}_{3}+%
\mathcal{L}_{4}$} \\ \hline
0.9 & & \multicolumn{1}{r}{0.117} & & 0.420 & & & \multicolumn{1}{r}{
0.071} & & 0.118 \\
-0.4 & & \multicolumn{1}{r}{0.172} & & 0.213 & & & \multicolumn{1}{r}{
0.174} & & 0.165 \\
0 & & \multicolumn{1}{r}{0.179} & & 0.217 & & & \multicolumn{1}{r}{0.104}
& & 0.161 \\
0.7 & & \multicolumn{1}{r}{0.113} & & 0.128 & & & \multicolumn{1}{r}{
0.054} & & 0.109 \\
10 & & \multicolumn{1}{r}{2.378} & & 2.790 & & & \multicolumn{1}{r}{1.355
} & & 1.644 \\
-2 & & \multicolumn{1}{r}{2.942} & & 2.986 & & & \multicolumn{1}{r}{2.398
} & & 2.457 \\
4 & & \multicolumn{1}{r}{4.549} & & 4.816 & & & \multicolumn{1}{r}{2.744}
& & 3.204 \\
5 & & \multicolumn{1}{r}{2.210} & & 2.462 & & & \multicolumn{1}{r}{1.534}
& & 1.857 \\
-0.5 & & \multicolumn{1}{r}{0.856} & & 0.837 & & & \multicolumn{1}{r}{
0.259} & & 0.361 \\
0 & & \multicolumn{1}{r}{0.951} & & 0.977 & & & \multicolumn{1}{r}{0.439}
& & 0.412 \\
0 & & \multicolumn{1}{r}{0.853} & & 0.977 & & & \multicolumn{1}{r}{0.471}
& & 0.382 \\
-2 & & \multicolumn{1}{r}{1.071} & & 1.475 & & & \multicolumn{1}{r}{0.542
} & & 0.635 \\
AVE & & \multicolumn{1}{r}{1.366} & & 1.525 & & & \multicolumn{1}{r}{
0.845} & & 0.959 \\
$\mathbf{\alpha }_{3}^{0}$ & & \multicolumn{1}{r}{$\mathbf{\hat{\alpha}}%
_{3w,T}$} & & $\mathbf{\hat{\alpha}}_{3w,T}$ & & & \multicolumn{1}{r}{$%
\mathbf{\hat{\alpha}}_{3w,T}$} & & $\mathbf{\hat{\alpha}}_{3w,T}$ \\ \hline
1.155 & & \multicolumn{1}{r}{0.809} & & 0.936 & & & \multicolumn{1}{r}{
0.747} & & 0.422 \\
2.828 & & \multicolumn{1}{r}{1.838} & & 1.799 & & & \multicolumn{1}{r}{
1.392} & & 1.338 \\
$\mathbf{\alpha }_{4}^{0}$ & & \multicolumn{1}{r}{$\mathbf{\hat{\alpha}}%
_{4w,T}$} & & $\mathbf{\hat{\alpha}}_{4w,T}$ & & & \multicolumn{1}{r}{$%
\mathbf{\hat{\alpha}}_{4w,T}$} & & $\mathbf{\hat{\alpha}}_{4w,T}$ \\ \hline
2 & & \multicolumn{1}{r}{7.205} & & 2.674 & & & \multicolumn{1}{r}{6.950}
& & 2.007 \\
12 & & \multicolumn{1}{r}{66.500} & & 8.112 & & & \multicolumn{1}{r}{
9.998} & & 6.859 \\ \hline
\% root loc. & & \multicolumn{1}{r}{61.7} & & 60.7 & & &
\multicolumn{1}{r}{70.2} & & 67.0 \\ \hline \hline
\end{tabular}
\caption{\footnotesize
\noindent Estimation of SVARMA$\left( 1, 1 \right)$ models by GMM, $d= 2 ,$ $\zeta _{1}= \zeta _{2}= 0.5 ,$ $%
\rho_{1}= 1 , \rho _{2}= -1 $. Innovations $\left( \chi _{6}^{2},\chi
_{1}^{2}\right).$\ $\mathcal{L}_{2}+\mathcal{L}_{3}$ and $\mathcal{L}_{2}+\mathcal{L}_{3}+\mathcal{L}_{4}$ identification and estimation, $w=(1,0)$ and $w=(1,1)$, respectively. Last line provides the percentage of simulated paths with correct MA roots location. AVE is the average of the absolute value of the column. $T=100,200$.}
\end{table}
\section{8. Empirical analysis}
We apply our identification and estimation methods to Blanchard and Quah
(1989) bivariate system for the US real GNP growth and unemployment rate
after linear detrending. They fit a SVAR with 8 lags (1948Q2-1987Q4, $T=159)$
and use for identification a long-run restriction by which the demand shock
has no long-run effect on real GNP in the same way as both supply and demand
shocks have no long-run effect on unemployment. Lippi and Reichlin (1994)
alternatively propose that these long VAR dynamics could be better
approximated by a VARMA model and explore the properties of the different
versions of the IRF obtained by inverting the MA roots of the fundamental
VARMA$\left( 1,1\right) $ representation deduced from the fitted VAR$\left(
8\right) $ parameterization. Gouri\'{e}roux et al. (2019) fit SVARMA$\left(
p,1\right) $ models to the same dataset for $p\in \left \{ 1,\ldots
,6\right
\} $ by PMLE using GMM initial estimates based on IV estimation of
the AR parameters. The shocks are assumed mixed Gaussian distributed and $%
p=4 $ is the order chosen by a combination of model selection criteria and
correlation diagnostics, finding a representation with a mixed
invertible/noninvertible MA roots.
We fit with our methods a series of simple SVARMA$\left( p,q\right) $ models
to the original dataset to investigate the possible nonfundamentalness of
the dynamics of the system using our higher order cumulant identification.
We follow the same MA polynomial parameterization and the same procedure as
in the Monte Carlo simulations and base the preliminary choice of the MA
roots configuration on minimization of $\mathcal{L}_{3,T}^\dag+\mathcal{L}%
_{4,T}^\dag$, $w=(1,1)$, among all the versions of the model obtained by MA
root flipping of Whittle estimates using Baggio and Ferrante (2019)
algorithm imposing causality. Then, a local of minimization of $\mathcal{L}%
_{w,T}^\dag=\mathcal{L}_{2,T}+\mathcal{L}_{3,T}^\dag+\mathcal{L}_{4,T}^\dag$
is performed to find $\mathbf{\hat{\theta}}_{w,T}^\dag$ and finally GMM
estimates $\mathbf{\hat{\theta}}_{GMM,T}$ are obtained with $\mathcal{V}$
estimated by bootstrap using the linear representation of the score. Each $%
\mathcal{L}^\dag _{k,T}$ is normalized by the number of spectral densities
of order $k$ in the system, $d^{k},$ and by $T^{k-2},$ so that their values
are close to one and can be compared easily across $k$ and model orders,
though the larger $k, $ the larger the relative contribution from the sample
variation of the $k $-order periodogram and the smaller the contribution of
the particular parameter value at which the loss function is evaluated.
We report in Table 6 the values of $\mathcal{L}_{2,T}$, $\mathcal{L}%
_{3,T}^\dag$, $\mathcal{L}_{4,T}^\dag$ and $\mathcal{L}_{w,T}^\dag$
evaluated at the final estimates with weighting defined by initial Whittle
estimates (and the value $\mathcal{L}_{2,T}^{0}$ obtained by minimizing only
$\mathcal{L}_{2,T}$ as a benchmark for best linear fit), the modulus of the
MA roots of the estimated parameterization and the estimates of $\mathbf{%
\alpha} _{3}$ and $\mathbf{\alpha} _{4},$ using the scheme of Assumption~6A
for component identification, which imposes positive diagonal values of $%
\Theta _{0}$ and maximizes their product. We also report the estimates of $%
\Omega $ for the instantaneous impact of the shocks on the endogenous
variables. We use a similar identification strategy as Gouri\'{e}roux et al.
(2019) to label error components as transitory or demand and permanent or
supply shocks, facilitating an easy comparison to previous analysis.
The model which appears to best fit the data is the SVARMA$\left( 1,1\right)
, $ reporting the smallest value for the overall loss function $\mathcal{L}%
_{w,T}^\dag$ after joint optimization and an efficient GMM Newton-Raphson
step, just third best for $\mathcal{L}_{2,T},$ outperformed by the much
larger SVARMA$\left( 4,1\right) $ and VAR$\left( 8\right) $ models, which
did also a better job when fitting only second order dynamics attending to $%
\mathcal{L}_{2,T}^{0}$. Note that this comparison is made against the
initial Whittle estimates found only using second moments for the same order
$\left( p,q\right) $, so additional local optimization of $\mathcal{L}%
_{3,T}^\dag$ and $\mathcal{L}_{4,T}^\dag$ to match higher order dynamics is
made by penalizing the second order goodness-of-fit at the same that imposes
ICA and the correct location of MA roots. For the same reason, and despite
the model nesting, larger models do not necessarily provide better fit in
finite samples attending to initial-estimates weighted $\mathcal{L}%
_{k,T}^\dag$ loss functions.
The SVARMA$\left( 1,1\right) $ model provides an invertible solution, but
some more complex models present non-invertible dynamics. Attending to
higher cumulants, invertible solutions indicate that both shocks have
moderate skewness, possibly of different signs. The SVARMA$\left( 1,1\right)
$ identifies the first shock as the transitory with negative skewness, while
the second shock would be the permanent one with much larger positive
asymmetry, so in both cases negative news (decreasing GNP growth and
increasing unemployment) tend to be more extreme than positive ones.
Further, typically one shock displays large positive kurtosis (the permanent
one for the SVARMA$\left( 1,1\right) )$, and there is no conclusive evidence
about the kurtosis of the other one, given the large bootstrap standard
errors which make most estimates not significatively different from zero. At
least for the SVARMA$\left( 1,1\right) $ model, these estimation results
would confirm model dynamics identification using jointly third and fourth
cumulants by Corollary~2, while none of them in isolation would be
sufficient.
\begin{table}[t]
\renewcommand*{\arraystretch}{.8}
\setlength{\tabcolsep}{2.8pt}
\textbf{Table 6.} US real GNP growth
and unemployment rate. GMM SVARMA$(p,q)$ estimates.
\begin{tabular}{lrrrrrrrrr}
\hline \hline
$(p,q)$ & $(1,0)\ $ & $(4,0)\ $ & $(8,0)\ $ & $(0,1)\ $ & $(1,1)\ $ & $%
(2,1)\ $ & $(4,1)\ $ & $\left( 1,2\right) \ $ & $\left( 2,2\right) \ $ \\
\hline \hline
$\mathcal{L}_{2,T}^{0}$ & $0.9608$ & $0.9991$ & $0.7694$ & $2.6118$ & $%
0.9136 $ & $0.9342$ & $0.8928$ & $0.9395$ & $0.9371$ \\
$\mathcal{L}_{2,T}$ & $0.9608$ & $1.0393$ & $0.8325$ & $2.6118$ & $0.9148$ &
$0.9329$ & $0.8885$ & $0.9395$ & $0.9364$ \\
$\mathcal{\hat{L}}_{3,T}^\dag$ & $1.0258$ & $1.0345$ & $1.0343$ & $1.3832$ & $%
1.0284$ & $1.0349$ & $1.0335$ & $1.0285$ & $1.0297$ \\
$\mathcal{\hat{L}}_{4,T}^\dag$ & $1.1066$ & $1.0784$ & $1.2122$ & $1.8540$ & $%
1.0116$ & $1.0693$ & $1.1070$ & $1.0578$ & $1.0622$ \\
$\mathcal{\hat{L}}_{w,T}^\dag$ & $3.0932$ & $3.1522$ & $3.0790$ & $5.8490$ & $%
2.9548$ & $3.0370$ & $3.0290$ & $3.0258$ & $3.0284$ \\ \hline
$\left \vert \RIfM@\expandafter\text@\else\expandafter\mbox\fi{MA\ roots}\right \vert $ & $-$ & $-$ & $-$ & $3.1072$ & $%
8.7673$ & $0.7873$ & $1.0002$ & $0.25,0.51$ & $1.13,1.32$ \\
& $-$ & $-$ & $-$ & $3.1072$ & $8.7673$ & $5.2772$ & $1.0002$ & $0.51,2.81$
& $1.32,2.99$ \\
& $-$ & $-$ & $-$ & $(Inv.)$ & $(Inv.)$ & $(Mix.)$ & $(Inv.)$ & $(Mix.)$ & $%
(Inv.)$ \\ \hline
$\hat{\mathbf{\alpha }}_{3}$ & $-0.4474$ & $-0.5422$ & $-1.3776$ & $%
-0.4611 $ & $-0.5260 $ & $-0.3023^{{}}$ & $-0.2722$ & $%
-0.8550 $ & $-0.2959$ \\
& $1.4965$ & $1.5745$ & $-0.3435$ & $0.5114 $ & $1.2839$ & $1.4849$ &
$1.9261$ & $-1.2217$ & $1.4230$ \\
$\hat{\mathbf{\alpha }}_{4}$ & $-1.6447$ & $0.1856$ & $6.3477 $ & $%
0.4471 $ & $-0.2134$ & $-1.0077$ & $-2.6953$ & $-0.0563$ & $-0.9281$
\\
& $5.4016$ & $8.1305 $ & $-1.8098$ & $1.1451 $ & $6.8220$ & $7.1990 $ & $-1.8966$ & $7.9476 $ & $7.0659 $ \\
\hline
$\hat{\Omega}_{1,1}$ & $\underset{\left( .103\right) }{0.8648}$ & $\underset{%
\left( .190\right) }{0.7465}$ & $\underset{\left( .067\right) }{0.4192}$ & $%
\underset{\left( .072\right) }{1.4697}$ & $\underset{\left( .088\right) }{%
0.8077}$ & $\underset{\left( .091\right) }{0.9184}$ & $\underset{\left(
.026\right) }{0.7116}$ & $\underset{\left( .305\right) }{0.2338}$ & $%
\underset{\left( .287\right) }{0.8178}$ \\
$\hat{\Omega}_{2,1}$ & $\underset{\left( .068\right) }{-0.2131}$ & $\underset%
{\left( .088\right) }{-0.1993}$ & $\underset{\left( .070\right) }{-0.0356}$
& $\underset{\left( .040\right) }{-0.6311}$ & $\underset{\left( .136\right) }%
{-0.2504}$ & $\underset{\left( .122\right) }{-0.2046}$ & $\underset{\left(
.027\right) }{-0.2568}$ & $\underset{\left( .194\right) }{-0.0364}$ & $%
\underset{\left( .173\right) }{-0.2042}$ \\
$\hat{\Omega}_{1,2}$ & $\underset{\left( .308\right) }{0.0886}$ & $\underset{%
\left( .202\right) }{0.1438}$ & $\underset{\left( .074\right) }{0.0379}$ & $%
\underset{\left( .194\right) }{0.3036}$ & $\underset{\left( .619\right) }{%
0.2445}$ & $\underset{\left( .187\right) }{-0.0575}$ & $\underset{\left(
.022\right) }{0.2805}$ & $\underset{\left( .394\right) }{-0.0704}$ & $%
\underset{\left( .261\right) }{0.1052}$ \\
$\hat{\Omega}_{2,2}$ & $\underset{\left( .111\right) }{0.1910}$ & $\underset{%
\left( .097\right) }{0.1688}$ & $\underset{\left( .073\right) }{0.0565}$ & $%
\underset{\left( .208\right) }{0.1961}$ & $\underset{\left( .140\right) }{%
0.1468}$ & $\underset{\left( .156\right) }{0.2301}$ & $\underset{\left(
.057\right) }{0.1094}$ & $\underset{\left( .170\right) }{0.0701}$ & $%
\underset{\left( .192\right) }{0.1825}$ \\ \hline \hline
\end{tabular}
\caption{\footnotesize SVARMA$(p,q)$ model fitting for US real GNP growth and unemployment rate after linear detrending, 1948Q2-1987Q4, $T=159$. GMM estimates $\hat{\mathbf{\theta}}_{GMM,T}$ in (\ref{theestimator}) obtained with $\tilde{\mathbf{\theta}}_T= \hat{\mathbf{\theta}}_{w,T}$ with $w=(1,1)$ and $\hat{\mathbb{V}}_T^-$ obtained by inverting the sample variance of the joint score $\mathbb{S}_T$ evaluated at $\hat{\mathbf{\theta}}_{w,T}$ for 400 draws on the empirical distribution of residuals $\hat{\mathbf{\varepsilon}}_t$. Component identification uses Assumption 6A.}
\end{table}
The estimates of the lag zero impact matrix $\Omega $ of our preferred
model, as for other models with only invertible dynamics in Table 6,
indicate a much higher impact of the first (transitory) shock on output
compared to the impacts of the second (permanent) shock on both output and
employment, while the transitory shock has a negative short run effect on
unemployment. This last feature is preserved in models which identify
non-invertible components, but the sign of the instantaneous impact of the
permanent shock on output is reversed.
Finally, we provide in Figure 1 the plots of the IRF's identified by our
estimation methods for the SVARMA $\left( 1,1\right) $ model together with
those reported by Blanchard and Quah (1989) and Gouri\'{e}roux et al.
(2019). We can observe that the effect of the supply (or permanent) shock on
both endogenous variables is very close to the results of Blanchard and Quah
for the SVAR(8) model, though there is a slight delay in the maximum effect
and a quite more persistent effect on output. For the demand (transitory)
shock, the shape of both IRF's and the timing of maximal effects are almost
the same as for the SVAR model, but now the long run effect on output is not
restricted to zero and the IRF displays a very slow rate of decay but close
to the horizontal axis. The bootstrap tests of the long-run identification
restriction used by Blanchard and Quah (1989), based on the distance at lags
40 and 100 of the IRF of GNP growth on this transitory shock, can not reject
the hypothesis of zero long run effect, confirming the plausibility of this
identification strategy. On the other hand, the IRF obtained by Gouri\'{e}%
roux et al. (2019) with noninvertible dynamics retain some of the previous
properties for the unemployment response, but GNP growth behaves quite
differently in the long run after either type of structural shocks hitting
the system. \bigskip
\begin{center}
{\normalsize \includegraphics[width=1.05%
\textwidth]{IRFfinal.pdf}
\captionof{figure}{IRF for US GNP
growth and unemployment based on our GMM estimates (\ref{theestimator}) for the SVARMA$\left(1,1\right) $ model, BQ: Blanchard and Quah (1989) SVAR$\left( 8\right) $ and
GMR: Gouri\'{e}roux et al. (2019) SVARMA$\left( 4,1\right) $ model.} }
\end{center}
\section{9. Conclusions}
In this paper we have showed how to achieve identification of non-Gaussian
SVARMA models using basic restrictions on higher order moments under serial
and component independence conditions of finite order $k=3$ or 4 on the
structural shocks sequence. We use an identification criterion in the
frequency domain that leads to easy to check global and local identification
conditions and permits the design of consistent and asymptotically normal
parameter estimates which exploit all dynamic and static information in
second, third and fourth order moments. These results provide consistent
estimation of IRFs without need to specify the fundamentalness of the system
and can be combined with different sources of information to proper label
the structural shocks of the model or to test relevant hypothesis and
overidentification conditions.\bigskip
\section{\protect \small Appendix A: Higher order cumulants and spectral
densities}
{\small \noindent \textbf{Cumulants of linear combinations. }The third order
cumulant matrices $\mathbf{\kappa }_{\cdot \cdot j}$,\ $j=1,\ldots ,d,$ of a
random vector $\mathbf{\varepsilon }_{t}$ can be updated easily under linear
transformations for a $d\times d$ matrix $K,$ where $K$ is orthogonal, $%
KK^{\prime }=\mathbf{I}_{d}, $ to maintain the covariance structure of $%
\mathbf{\varepsilon }_{t},$%
\begin{equation*}
\eta _{t}=K \mathbf{\varepsilon} _{t}
\end{equation*}%
so that it holds%
\begin{equation*}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{\eta }\right) =K_{1}^{\otimes
3}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0}\right)
\end{equation*}%
or, alternatively,
\begin{equation*}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{\eta }=K^{\otimes 2}\ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{3}^{0}\ K^{\prime },
\end{equation*}%
and in general, for any $k\geq 2$, see Jammalamadaka, Rao and Terdik (2006),
\begin{equation*}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}^{\eta }\right) =K^{\otimes k}%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}^{0}\right) =\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left(
K^{\otimes 2}\ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}^{0}\ \left( K^{\otimes
(k-2)}\right) ^{\prime }\right) ,
\end{equation*}%
with $K^{\otimes 0}=\mathbf{I}_{d}$ using that vec$\left( ABC\right) =\left(
C^{\prime }\otimes A\right) $vec$\left( B\right) $ for conformable matrices $%
A,B,C.$ Note however that, for $k\geq 4,$ v$\mathbf{\kappa }_{k}^{0}$
contains many repeated columns, since for instance, for $k=4,$ vec$\left(
\mathbf{\kappa }_{\cdot \cdot jh}\right) =\ $vec$\left( \mathbf{\kappa }%
_{\cdot \cdot hj}\right) ,$ so in this particular case rank$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}%
\mathbf{\kappa }_{k}^{0}\right) \leq d(d+1)/2.\ $\newline
}
{\small \noindent \textbf{Higher order spectral densities.} Consider a $d$%
-dimensional stationary stochastic process $\{Y_{t}\}_{t\in Z}$ with $%
E[\left \Vert Y_{t}\right \Vert ^{k}]<\infty ,$ some $k\geq 3,$ and call $%
\mu =E[Y_{t}].$ Define the autocovariance of order $j$ as
\begin{equation*}
\Gamma \left( j\right) =Cov[Y_{t},Y_{t-j}]=E[(Y_{t}-\mu )(Y_{t-j}-\mu
)^{\prime }],\qquad \RIfM@\expandafter\text@\else\expandafter\mbox\fi{for }j=0,\pm 1,\ldots ,
\end{equation*}%
and the spectral density matrix, $f(\lambda ),$ is defined implicitly as%
\begin{equation*}
\Gamma \left( j\right) =\int_{-\pi }^{\pi }f(\lambda )\exp (-ij\lambda
)d\lambda .
\end{equation*}%
The autocovariance sequence and the spectral density are measures of the
dependence of the stochastic process based on second moments, hence they are
the objects of interest of usual time series analysis. The dependence
contained in higher order moments can also be described by the cumulants
which are defined in terms of higher order moments as%
\begin{equation*}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{cum}\left( Y_{t\left( 1\right) ,\mathbf{a}\left( 1\right) },\ldots
,Y_{t\left( k\right) ,\mathbf{a}\left( k\right) }\right) =\sum
(-1)^{p-1}(p-1)!E(\Pi _{j\in v_{1}}Y_{t\left( j\right) ,\mathbf{a}\left(
j\right) })\cdots E(\Pi _{j\in v_{p}}Y_{t\left( j\right) ,\mathbf{a}\left(
j\right) }),\ \ k=1,2,\ldots ,
\end{equation*}%
where $v_{1},\ldots ,v_{p}$ is a partition of $(1,2,\ldots ,k),$ and the sum
runs over all these partitions, $\mathbf{a}\left( j\right) \in \left \{
1,\ldots ,d\right \} ,$ $t\left( j\right) =0,\pm 1,\ldots ,$ for $j=1.\ldots
,k,$ see Brillinger (1975) or Rosenblatt (1985, p.~34). Hence, the first and
second cumulants are the mean and the variance, respectively. }
{\small We also define the $k$-th order cumulant spectral density $%
k=2,3,\ldots ,$ for \textbf{$a$}$=\left( \mathbf{a}\left( 1\right) ,\ldots ,%
\mathbf{a}\left( k\right) \right) $ which is the Fourier transform of the $k$%
-th order cumulants for elements $\left( \mathbf{a}\left( 1\right) ,\ldots ,%
\mathbf{a}\left( k\right) \right) $ of the vector $Y_{t},$ $f_{\mathbf{a},k}(%
\boldsymbol{\lambda })=f_{\mathbf{a}\left( 1\right) ,\ldots ,\mathbf{a}%
\left( k\right) }(\lambda _{1},\ldots ,\lambda _{k-1}),$ as
\begin{equation}
f_{\mathbf{a},k}(\boldsymbol{\lambda })=\left( 2\pi \right)
^{1-k}\sum_{j_{1},\ldots ,j_{k-1}=-\infty }^{\infty }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{cum}(Y_{\mathbf{a}%
(1),t},Y_{\mathbf{a}(2),t+j_{1}},\ldots ,Y_{\mathbf{a}(k),t+j_{k-1}})\exp
\left( -\sum_{s=1}^{k-1}ij_{s}\lambda _{s}\right) , \notag
\end{equation}%
introducing for simplicity, when there is no ambiguity, the notation $%
\boldsymbol{\lambda }=(\lambda _{1},\ldots ,\lambda _{k-1})$. Existence of $%
f_{\mathbf{a},k}(\boldsymbol{\lambda })$ can be guaranteed by summability
conditions on cumulants, as those implied by a linear process condition.
Note that the elements of the usual spectral density (matrix) are recovered
for $k=2$, while, in general, $f_{\mathbf{a},k}$ can be complex valued for
any $k\geq 2$, except the diagonal elements of the usual spectral density
matrix $f=f_{2}$, i.e. $f_{a,a}\left( \lambda \right) ,$ $a=1,\ldots ,d,$
which are always real valued.$\ $\newline
}
{\small \noindent \textbf{VARMA higher order spectral densities}. The second
order spectral density matrix of $Y_{t}$ is given by%
\begin{eqnarray*}
f\left( \lambda \right) &=&\frac{1}{2\pi }\Phi ^{-1}\left( e^{-i\lambda
}\right) \Theta \left( e^{-i\lambda }\right) \left( \Phi ^{-1}\left(
e^{-i\lambda }\right) \Theta \left( e^{-i\lambda }\right) \right) ^{\ast } \\
&=&\frac{1}{2\pi }\Psi \left( e^{-i\lambda }\right) \Psi ^{\ast }\left(
e^{i\lambda }\right) ,
\end{eqnarray*}%
where $^{\ast }$ denotes complex conjugation and transposition, and in
particular for any two components \textbf{$a$}$\mathbf{=}\left( \mathbf{a}%
\left( 1\right) ,\mathbf{a}\left( 2\right) \right) $ of $Y_{t},$ its
spectral density $f_{\mathbf{a}}\left( \lambda \right) =f_{\left( a\left(
1\right) ,a\left( 2\right) \right) }\left( \lambda \right) $ satisfies
\begin{eqnarray*}
f_{\mathbf{a}}\left( \lambda \right) &=&\frac{1}{2\pi }\Psi _{\mathbf{a}%
\left( 1\right) }\left( e^{-i\lambda }\right) \Psi _{\mathbf{a}\left(
2\right) }^{\prime }\left( e^{i\lambda }\right) \\
&=&\frac{1}{2\pi }\sum_{h,j=1}^{d}\Psi _{\mathbf{a}\left( 1\right) ,h}\left(
e^{-i\lambda }\right) \Psi _{\mathbf{a}\left( 2\right) ,j}\left( e^{i\lambda
}\right)
\end{eqnarray*}%
where $\Psi _{j}\left( z\right) =\left( \Psi _{j,1}\left( z\right) ,\ldots
,\Psi _{j,d}\left( z\right) \right) $ is the $j$-th row of $\Psi \left(
z\right) ,$ because the $j$-th element of the vector $Y_{t}$ is obtained as $%
Y_{t,j}=\Psi _{j}\left( L\right) \mathbf{\varepsilon }_{t}.$ }
{\small For any triplet \textbf{$a$}$\mathbf{=}\left( \mathbf{a}\left(
1\right) ,\mathbf{a}\left( 2\right) ,\mathbf{a}\left( 3\right) \right) $,
the third order spectral density of the 3-dimensional vector $\left( Y_{t,%
\mathbf{a}(1)},Y_{t,\mathbf{a}(2)},Y_{t,\mathbf{a}(3)}\right) ,$ $f_{\mathbf{%
a},3}(\boldsymbol{\lambda })=f_{\left( \mathbf{a}\left( 1\right) ,\mathbf{a}%
\left( 2\right) ,\mathbf{a}\left( 3\right) \right) }(\lambda _{1},\lambda
_{2}),$ is given by%
\begin{equation*}
f_{\mathbf{a},3}(\boldsymbol{\lambda })=\left( 2\pi \right) ^{-2}\sum_{%
\mathbf{j}=1}^{d}\Psi _{\mathbf{a}\left( 1\right) ,j(1)}\left( e^{-i\lambda
_{1}}\right) \Psi _{\mathbf{a}\left( 2\right) ,j(2)}\left( e^{-i\lambda
_{2}}\right) \Psi _{\mathbf{a}\left( 3\right) ,j(3)}\left( e^{i\left(
\lambda _{1}+\lambda _{2}\right) }\right) \mathbf{\kappa}
_{j(1),j(2),j(3)}^{0}
\end{equation*}%
where $\mathbf{\kappa }_{\mathbf{j}}^{0}=\ $cum$\left( \mathbf{\varepsilon }%
_{t,j(1)},\mathbf{\varepsilon }_{t,j(2)},\mathbf{\varepsilon }%
_{t,j(3)}\right) $, $\mathbf{j}=\left( j(1),j(2),j(3)\right) ,$ is the joint
third order cumulants of the innovations indexed by \textbf{$a$}. This third
order spectral density can be we written more compactly as%
\begin{equation*}
f_{\mathbf{a},3}(\boldsymbol{\lambda })=\left( 2\pi \right) ^{-2}\Psi _{%
\mathbf{a}}^{\otimes 3}\left( \mathbf{\lambda }\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left(
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0}\right) ,
\end{equation*}%
where, using Kronecker product $\otimes ,$%
\begin{equation*}
\Psi _{\mathbf{a}}^{\otimes 3}\left( \mathbf{\lambda }\right) =\Psi _{%
\mathbf{a}\left( 3\right) }\left( e^{i\left( \lambda _{1}+\lambda
_{2}\right) }\right) \otimes \Psi _{\mathbf{a}\left( 2\right) }\left(
e^{-i\lambda _{2}}\right) \otimes \Psi _{\mathbf{a}\left( 1\right) }\left(
e^{-i\lambda _{1}}\right)
\end{equation*}%
and, denoting $\mathbf{\kappa }_{\cdot \cdot j}=E\left[ \mathbf{\varepsilon }%
_{t}\mathbf{\varepsilon }_{t}^{\prime }\mathbf{\varepsilon }_{t,j}\right] ,\
j=1,\ldots ,d,$%
\begin{equation*}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0}=\left[ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{\kappa }%
_{\cdot \cdot 1}\right) \ \ \cdots \ \ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{\kappa }%
_{\cdot \cdot d}\right) \right]
\end{equation*}%
is a $d^{2}\times d$ matrix, where vec is the usual operator stacking all
the columns of a matrix in a single column vector, and the indexes in the $%
d^{3}$-dimensional vector vec$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0}\right)
=\left \{ \mathbf{\kappa }_{hj\ell }\right \} $ run first from left to right
from 1 to $d.$ }
{\small This notation extends readily to any $k$-order spectral density, $%
k=2,3,\ldots ,$%
\begin{equation*}
f_{\mathbf{a},k}(\boldsymbol{\lambda })=\frac{1}{\left( 2\pi \right) ^{k-1}}%
\Psi _{\mathbf{a}}^{\otimes k}\left( \mathbf{\lambda }\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}%
\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}^{0}\right) ,
\end{equation*}%
where for $\boldsymbol{\lambda }=(\lambda _{1},\ldots ,\lambda _{k-1})$ we
define%
\begin{equation*}
\Psi _{\mathbf{a}}^{\otimes k}\left( \mathbf{\lambda }\right) =\Psi _{%
\mathbf{a}\left( k\right) }\left( e^{i\left( \lambda _{1}+\cdots +\lambda
_{k-1}\right) }\right) \otimes \Psi _{\mathbf{a}\left( k-1\right) }\left(
e^{-i\lambda _{k-1}}\right) \otimes \cdots \otimes \Psi _{\mathbf{a}\left(
2\right) }\left( e^{-i\lambda _{2}}\right) \otimes \Psi _{\mathbf{a}\left(
1\right) }\left( e^{-i\lambda _{1}}\right)
\end{equation*}%
and the $d^{2}\times d^{k-2}$ matrix v$\mathbf{\kappa }_{k}^{0}$ satisfies%
\begin{equation*}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}^{0}=\left[ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{\kappa }%
_{\cdot \cdot 1\cdots 1}\right) \ \ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{\kappa }_{\cdot
\cdot 2\cdots 1}\right) \ \ \cdots \ \ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{\kappa }%
_{\cdot \cdot d\cdots d}\right) \right]
\end{equation*}%
and $\mathbf{\kappa }_{\cdot \cdot j\left( 3\right) \cdots j\left( k\right)
} $ is a $d\times d$ matrix with typical $\left( j\left( 1\right) ,j\left(
2\right) \right) $ element equal to the $k$-th order joint cumulant cum$%
\left( \mathbf{\varepsilon }_{t,j\left( 1\right) },\mathbf{\varepsilon }%
_{t,j\left( 2\right) },\mathbf{\varepsilon }_{t,j\left( 3\right) },\ldots ,%
\mathbf{\varepsilon }_{t,j\left( k\right) }\right) ,$ $j\left( h\right) \in
\left \{ 1,\ldots ,d\right \} ,$ $h=1,\ldots ,k.$ \bigskip }
\section{\protect \small Appendix B: Proofs of Results}
{\small \noindent \textbf{Proof of Theorem~\ref{1A}. } }
{\small \noindent \textbf{Proof for }$k=3.$\textbf{\ }The integrand of $%
\mathcal{L}_{3}^{0}\left( A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}\right) ,$
satisfies, uniformly for $\mathbf{\lambda }\in \Pi ^{2},$%
\begin{eqnarray*}
&&\left \{ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\! \left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}\right) ^{\prime
}A^{\otimes 3}\! \left( \mathbf{\lambda }\right) ^{\ast }-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\!
\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0}\right) ^{\prime }\right \} \Upsilon
_{3}^{0}\left( \mathbf{I}_{d},\mathbf{\lambda }\right) \left \{ A^{\otimes
3}\left( \mathbf{\lambda }\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{3}\right) -\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0}\right) \right
\} \\
&\geq &\inf_{\mathbf{\lambda }}\lambda _{\min }\left( \Upsilon
_{3}^{0}\left( \mathbf{I}_{d},\boldsymbol{\lambda }\right) \right)
\inf_{\lambda }\left \vert \lambda _{\min }\left( A\left( \lambda \right)
\otimes \mathbf{I}_{d^{2}}\right) \right \vert ^{2}\left \Vert \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}%
\left( \left( A^{\otimes 2}\left( \lambda _{2},\lambda _{1}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}%
\mathbf{\kappa }_{3}-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0}A\left( \lambda
_{1}+\lambda _{2}\right) \right) \right) \right \Vert ^{2} \\
&\geq &\eta ^{6}\left \Vert \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( A^{\otimes 2}\left( \lambda
_{2},\lambda _{1}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa
}_{3}^{0}A\left( \lambda _{1}+\lambda _{2}\right) \right) \right \Vert ^{2}
\end{eqnarray*}%
because of Assumption~4, $A\left( z\right) $ is a BM (with unitary
eigenvalues for $\left \vert z\right \vert =1)$ and
\begin{eqnarray*}
A^{\otimes 3}\left( \mathbf{\lambda }\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}%
\mathbf{\kappa }_{3}\right) -\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{3}^{0}\right) &=&\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( A^{\otimes 2}\left( \lambda _{2},\lambda
_{1}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}A\left( -\lambda _{1}-\lambda
_{2}\right) ^{\prime }-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0}\right) \\
&=&\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \left( A^{\otimes 2}\left( \lambda _{2},\lambda
_{1}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{3}^{0}A^{\ast }\left( \lambda _{1}+\lambda _{2}\right) \right) A\left(
-\lambda _{1}-\lambda _{2}\right) ^{\prime }\right) \\
&=&\left( A\left( -\lambda _{1}-\lambda _{2}\right) \otimes \mathbf{I}%
_{d^{2}}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( A^{\otimes 2}\left( \lambda _{2},\lambda
_{1}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{3}^{0}A^{\ast }\left( \lambda _{1}+\lambda _{2}\right) \right)
\end{eqnarray*}%
denoting $A^{\otimes 2}\left( \lambda _{2},\lambda _{1}\right) =A\left(
e^{-i\lambda _{2}}\right) \otimes A\left( e^{-i\lambda _{1}}\right) .$ }
{\small Then, under the assumption that $A\left( z\right) $ is a
non-constant BM (even if $a_{n}={a}_{m}^{\star-1}$ for some $n\neq m$ and
therefore $g_{a_{n}}\left( z\right) g_{a_{m}}\left( z\right) =1$), and
because v$\mathbf{\kappa }_{3}^{0}$ is full rank $d,$ there is at least one
pair $\left( j,h\right) ,$ $j\in \left \{ 1,\ldots ,d^{2}\right \} ,$ $h\in
\left
\{ 1,\ldots ,d\right \} ,$ such that $\xi _{3}\left( \mathbf{\lambda }%
\right) :=\left \{ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0}A^{\ast }\left( \lambda
_{1}+\lambda _{2}\right) \right \} _{j,h}$ depends on $\mathbf{\lambda }$
through $\lambda _{1}+\lambda _{2}$ in a linear combination of products of $%
g_{a_{i}}\left( \lambda _{1}+\lambda _{2}\right) $ functions, $i=1,\ldots
,r. $\footnote{%
Note that in the representation $\left( \ref{BM}\right) $ of a GBM we can
allow for $a_{n}={a}_{m}^{\star-1}$ for some $n\neq m$, despite this implies
that $g_{a}\left( z\right) g_{{a}^{\star-1}}\left( z\right) =1$ for a real $%
a. $ Therefore, if $A\left( z\right) $ is assumed not constant in $z$, a
full cancellation of all roots is not allowed and the effect of $A$ is
shifting these flipped roots in different components of $\Psi $ when $d>1$,
avoiding cases like $A\left( z\right) =K_{0}R\left( {a}_{n}^{\star-1},z%
\right) R\left( a_{n},z\right) K_{0}^{\prime }=\mathbf{I}_{d},$ which leave
unchanged the dynamics as any constant $A$.}
Then, for such pair $\left( j,h\right) $ depending on the form of $A$
and v$\mathbf{\kappa }_{3}^{0},$
\begin{equation*}
\left \Vert \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( A^{\otimes 2}\left( \lambda _{2},\lambda
_{1}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{3}^{0}A^{\ast }\left( \lambda _{1}+\lambda _{2}\right) \right) \right
\Vert \geq \left \vert \zeta _{3}\left( \mathbf{\lambda }\right) -\xi
_{3}\left( \mathbf{\lambda }\right) \right \vert ,
\end{equation*}%
where $\zeta _{3}\left( \mathbf{\lambda }\right) :=\left \{ A^{\otimes
2}\left( \lambda _{2},\lambda _{1}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{3}\right
\} _{j,h}$ is a linear combination of products of
\begin{equation*}
1,g_{a_{i}}\left( \lambda _{1}\right) ,g_{a_{i}}\left( \lambda _{2}\right)
,\ \ \ i=1,\ldots ,r,
\end{equation*}%
possibly zero or constant even if v$\mathbf{\kappa }_{3}$ is full rank.
Therefore%
\begin{equation*}
\mathcal{L}_{3}^{0}\left( A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}\right) \geq \eta
^{6}\int_{\Pi ^{2}}\left \vert \zeta _{3}\left( \mathbf{\lambda }\right)
-\xi _{3}\left( \mathbf{\lambda }\right) \right \vert ^{2}d\mathbf{\lambda }%
\geq \epsilon >0
\end{equation*}%
for some $\epsilon $ not depending on v$\mathbf{\kappa }_{3},$ but depending
on $\eta ,$ v$\mathbf{\kappa }_{3}^{0}$ and $A$, because the functions $%
\zeta _{3}\left( \mathbf{\lambda }\right) \ $and $\xi _{3}\left( \mathbf{%
\lambda }\right) $ differ a.e. for any choice of v$\mathbf{\kappa }_{3}$
because $\xi _{3}\left( \mathbf{\lambda }\right) $ has an infinite expansion
on powers of $\exp \left( i\left( \lambda _{1}+\lambda _{2}\right) \right) $
which depends on $\mathbf{\lambda }$ only through $\lambda _{1}+\lambda _{2}$
(and can not be factorized in separated functions of $\lambda _{1}\ $and $%
\lambda _{2}$) while $\zeta _{3}\left( \mathbf{\lambda }\right) $ depends on
$\mathbf{\lambda }$ only through products of functions with infinite
representation on exponential functions of a single $\lambda _{i},\ i=1,2.$
Note that for $r=1$ it holds that $A\left( z\right) =K_{0}R\left(
a_{1},z\right) K_{1},$ $\left \vert a_{1}\right \vert \neq 1,$ $a_{1}$ real,
and we can chose $\left( j,h\right) $ such that for $\widetilde{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}%
\mathbf{\kappa }}_{3}^{0}=\ $v$\mathbf{\kappa }_{3}^{0}K_{0}$ and for some $%
c_{i}=c_{i}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0},K_{0},K_{1}\right) \in
\mathbb{R},$ $i=1,2,$ with $c_{1}\neq 0,$%
\begin{equation*}
\xi _{3}\left( \mathbf{\lambda }\right) =\left \{ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{3}^{0}A\left( \lambda _{1}+\lambda _{2}\right) \right \} _{j,h}=\left \{
\widetilde{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }}_{3}^{0}R\left( a_{1},z\right)
K_{1}\right \} _{j,h}=c_{0}+c_{1}g_{a_{1}}\left( \lambda _{1}+\lambda
_{2}\right)
\end{equation*}%
and for some constants $d_{i,\ell }=d_{i,\ell }\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa
}_{3},K_{0},K_{1}\right) \in \mathbb{R},$%
\begin{equation*}
\zeta _{3}\left( \mathbf{\lambda }\right) =\left(
d_{1,0}+d_{1,1}g_{a_{1}}\left( \lambda _{1}\right) \right) \left(
d_{2,0}+d_{2,1}g_{a_{1}}\left( \lambda _{2}\right) \right) ,
\end{equation*}%
so that for all $c_{0},c_{1}\neq 0$ and $d_{i,\ell },$ the function $\zeta
_{3}\left( \mathbf{\lambda }\right) -\xi _{3}\left( \mathbf{\lambda }\right)
\neq 0$ a.e. for any $a_{1},$ $\left \vert a_{1}\right \vert \neq 1.$%
In particular, if $r=d=1,$ then $\left( j,h\right) =\left(
1,1\right) $ and $\xi _{3}\left( \mathbf{\lambda }\right) =\kappa
_{3}^{0}g_{a_{1}}\left( \lambda _{1}+\lambda _{2}\right) ,$ $\zeta
_{3}\left( \mathbf{\lambda }\right) =\kappa _{3}g_{a_{1}}\left( \lambda
_{1}\right) g_{a_{1}}\left( \lambda _{2}\right) ,$ with $\kappa _{3}^{0}\neq
0,$ so that
\begin{eqnarray*}
\int_{\Pi ^{2}}\left \vert \zeta _{3}\left( \mathbf{\lambda }\right) -\xi
_{3}\left( \mathbf{\lambda }\right) \right \vert ^{2}d\mathbf{\lambda } &%
\mathbf{=}&\int_{\Pi ^{2}}\left \vert g_{a_{1}}\left( \lambda _{1}+\lambda
_{2}\right) \right \vert ^{2}\left \vert \kappa _{3}g_{a_{1}}\left( \lambda
_{1}\right) g_{a_{1}}\left( \lambda _{2}\right) g_{a_{1}}\left( -\lambda
_{1}-\lambda _{2}\right) -\kappa _{3}^{0}\right \vert ^{2}d\mathbf{\lambda }
\\
&=&\left( 2\pi \right) ^{2}\int_{\Pi ^{2}}\left \vert f_{3}\left( \mathbf{%
\lambda };\kappa _{3},a_{1}\right) -\left( 2\pi \right) ^{-2}\kappa
_{3}^{0}\right \vert ^{2}d\mathbf{\lambda >}0
\end{eqnarray*}%
because $\left \vert g_{a_{1}}\left( \lambda _{1}+\lambda _{2}\right)
\right
\vert ^{2}=1$ and where $f_{3}\left( \mathbf{\lambda };\kappa
_{3},a_{1}\right) $ is the bispectrum of an all-pass ARMA$\left( 1,1\right) $
model where $a_{1}$ is the root of the MA$\left( 1\right) $ polynomial (and $%
{a}_{1}^{\star-1}$ that of the AR$\left( 1\right) $ one$)$ and the third
order cumulant is equal to $\kappa _{3},$ whose $\mathcal{L}^{2}$ distance
to the nonzero constant function $\left( 2\pi \right) ^{-2}\kappa _{3}^{0}$
(the bispectrum of an independent series) is positive for all $\kappa
_{3}\in \mathbb{R}$, as showed by VL for any $r.$ $\Box $\bigskip }
{\small \noindent \textbf{Proof of Theorem~\ref{1ANEW}. } }
{\small \noindent \textbf{Proof for }$k=3.$\textbf{\ }It follows from
Theorem~\ref{1A}, because Assumption~3 implies Assumption~2 for $k=3.$ Note
in particular, that under Assumption~$3\left( k=3\right) $ it is easy to
check that v$\mathbf{\kappa }_{3}^{0}A^{\ast }\left( \lambda _{1}+\lambda
_{2}\right) $ contains only $d$ non-zero rows, $1,$ $d+2,\ldots ,d^{2},$
equal to the rows of $A^{\ast }\left( \lambda _{1}+\lambda _{2}\right) $ in
the same order multiplied by the corresponding marginal cumulant of $\mathbf{%
\kappa }_{3}^{0}$, so there is always at least one element $\left \{ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}%
\mathbf{\kappa }_{3}^{0}A^{\ast }\left( \lambda _{1}+\lambda _{2}\right)
\right \} _{j,h}$ depending on $\lambda _{1}+\lambda _{2}$ through linear
combinations of products of $g_{a_{i}^{\ast }}\left( \lambda _{1}+\lambda
_{2}\right) $ functions, $i=1,\ldots ,r$.\bigskip }
{\small \noindent \textbf{Proof for }$k=4.$ Arguing in a similar way as in
the proof of Theorem~\ref{1A}, denoting $\lambda _{4}:=\lambda _{1}+\lambda
_{2}+\lambda _{3}$ and recalling $A^{\otimes 2}\left( \lambda _{a},\lambda
_{b}\right) =A\left( e^{-i\lambda _{a}}\right) \otimes A\left( e^{-i\lambda
_{b}}\right) ,$ we find that
\begin{eqnarray*}
\left \Vert A^{\otimes 4}\left( \mathbf{\lambda }\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left(
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}\right) -\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa
}_{4}^{0}\right) \right \Vert &=&\left \Vert \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( A^{\otimes
2}\left( \lambda _{2},\lambda _{1}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{4}A^{\otimes 2}\left( -\lambda _{4},\lambda _{3}\right) ^{\prime }-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}%
\mathbf{\kappa }_{4}^{0}\right) \right \Vert \\
&=&\left \Vert \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \left( A^{\otimes 2}\left( \lambda
_{2},\lambda _{1}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa
}_{4}^{0}A^{\otimes 2\ast }\left( \lambda _{4},-\lambda _{3}\right) \right)
A^{\otimes 2}\left( -\lambda _{4},\lambda _{3}\right) ^{\prime }\right)
\right \Vert \\
&\geq &\left \Vert \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( A^{\otimes 2}\left( \lambda _{2},\lambda
_{1}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{4}^{0}A^{\otimes 2\ast }\left( \lambda _{4},-\lambda _{3}\right) \right)
\right \Vert .
\end{eqnarray*}
}
{\small Under Assumption~$3\left( k=4\right) $ and with the Kronecker
structure of $A^{\otimes 2\ast }\left( \lambda _{4},-\lambda _{3}\right) ,$
it is easy to check that v$\mathbf{\kappa }_{4}^{0}A^{\otimes 2\ast }\left(
\lambda _{4},-\lambda _{3}\right) $ contains only $d$ non-zero rows, $1,$ $%
d+2,\ldots ,d^{2},$ each containing all cross-products between the elements
of the same row of $A^{\ast }\left( \lambda _{4}\right) $ and of $A^{\ast
}\left( -\lambda _{3}\right) ,$ $A_{j}^{\ast }\left( e^{-i\lambda
_{4}}\right) \otimes A_{j}^{\ast }\left( e^{i\lambda _{3}}\right) ,$ $%
j=1,\ldots ,d$ (multiplied by the corresponding $j$-th element of $\mathbf{%
\kappa }_{4}^{0}$) so there is always at least one element $\left \{ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}%
\mathbf{\kappa }_{4}^{0}A^{\otimes 2\ast }\left( \lambda _{4},-\lambda
_{3}\right) \right \} _{j,h}:=\xi _{4}\left( \mathbf{\lambda }\right) ,$
say, depending on $\left( \lambda _{3},\lambda _{4}\right) $ in the form of
a product of two functions of $\lambda _{3}$ and $\lambda _{4}$,
respectively, each with infinite series expansion in powers of $\lambda _{i}$%
. Then, following the same argument as for $k=3$ in Theorem~\ref{1A},
defining $\zeta _{4}\left( \mathbf{\lambda }\right) :=\left \{ A^{\otimes
2}\left( \lambda _{2},\lambda _{1}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{4}\right
\} _{j,h}, $ we can show that
\begin{equation*}
\mathcal{L}_{4}^{0}\left( A,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}\right) \geq \eta
^{8}\int_{\Pi ^{3}}\left \vert \zeta _{4}\left( \mathbf{\lambda }\right)
-\xi _{4}\left( \mathbf{\lambda }\right) \right \vert ^{2}d\mathbf{\lambda }%
\geq \epsilon >0
\end{equation*}%
and the theorem follows.\ $\Box $\bigskip }
{\small \noindent \textbf{Proof of Theorem~\ref{1B}.} }
{\small \noindent \textbf{Proof for} $k=3.$ For $A=K$ orthogonal$,$ $%
KK^{\prime }=\mathbf{I}_{d},$ we need to show that $K=P_{d}$ and v$\mathbf{%
\kappa }_{3}=\ $v$\mathbf{\kappa }_{3}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left( P_{d}^{\prime }%
\mathbf{\alpha }_{3}^{0}\right) $ for signed permutation matrices $P_{d}$
are the only solutions that make true vec$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{3}^{0}\right) =K^{\otimes 3}\ $vec$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{3}\right) $ or equivalently v$\mathbf{\kappa }_{3}^{0}=K^{\otimes 2}\ $v$%
\mathbf{\kappa }_{3}\ K^{\prime },$ or
\begin{equation}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0}\ K=K^{\otimes 2}\ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{3}. \label{star}
\end{equation}%
Take any $d\geq 2$ and%
\begin{equation*}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0}:=\ v\mathbf{\kappa }_{3}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left(
\mathbf{\alpha }_{3}^{0}\right) =\left( \alpha _{1}^{0}\mathbf{e}%
_{1}^{\otimes 2},\ \ \alpha _{2}^{0}\mathbf{e}_{2}^{\otimes 2},\ldots
,\alpha _{d}^{0}\mathbf{e}_{d}^{\otimes 2}\right) ,\ \ K=\left \{
K_{ab}\right \} _{a,b=1}^{d}
\end{equation*}%
where the constants $\alpha _{j}^{0}$ satisfy $\alpha _{1}^{0}\alpha
_{2}^{0}\cdots \alpha _{d}^{0}\neq 0$ and $\mathbf{e}_{1}=\left( 1,0,\ldots
,0\right) ^{\prime }$ and so on are the unitary vectors of dimension $d$,
and we impose the same restriction to v$\mathbf{\kappa }_{3},$ i.e.
\begin{equation*}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}=\left( \alpha _{1}\mathbf{e}_{1}^{\otimes 2},\
\ \alpha _{2}\mathbf{e}_{2}^{\otimes 2},\ldots ,\alpha _{d}\mathbf{e}%
_{d}^{\otimes 2}\right) ,
\end{equation*}%
with $\alpha _{j}$ satisfying $\alpha _{1}\alpha _{2}\cdots \alpha _{d}\neq
0 $ and by orthogonality%
\begin{eqnarray}
K_{a1}^{2}+K_{a2}^{2}+\cdots +K_{ad}^{2} &=&1,\ \ a=1,\ldots ,d \label{O1.a}
\\
K_{a1}K_{b1}+K_{a2}K_{b2}+\cdots +K_{ad}K_{bd} &=&0,\ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ \ }%
b<a=2,\ldots ,d \label{O2.ab}
\end{eqnarray}%
so that also $K_{ab}^{2}\leq 1.$ }
{\small Then we have that (\ref{star}) is equivalent to%
\begin{equation*}
\left(
\begin{array}{cccc}
\alpha _{1}^{0}K_{11} & \alpha _{1}^{0}K_{12} & \cdots & \alpha
_{1}^{0}K_{1d} \\
0 & 0 & & 0 \\
\vdots & \vdots & & \vdots \\
0 & 0 & & 0 \\
\alpha _{2}^{0}K_{21} & \alpha _{2}^{0}K_{22} & \cdots & \alpha
_{2}^{0}K_{2d} \\
0 & 0 & & 0 \\
\vdots & \vdots & & \vdots \\
& & & \\
0 & 0 & & 0 \\
\alpha _{d}^{0}K_{d1} & \alpha _{d}^{0}K_{d2} & \cdots & \alpha
_{d}^{0}K_{dd}%
\end{array}%
\right) =\left(
\begin{array}{cccc}
\alpha _{1}K_{11}^{2} & \alpha _{2}K_{12}^{2} & \cdots & \alpha
_{d}K_{1d}^{2} \\
\vdots & \vdots & \cdots & \vdots \\
\alpha _{1}K_{11}K_{d1} & \alpha _{2}K_{12}K_{d2} & \cdots & \alpha
_{d}K_{1d}K_{dd} \\
\alpha _{1}K_{21}K_{11} & \alpha _{2}K_{22}K_{12} & & \alpha
_{d}K_{2d}K_{1d} \\
\alpha _{1}K_{21}^{2} & \alpha _{2}K_{22}^{2} & & \alpha _{d}K_{2d}^{2} \\
\vdots & \vdots & & \vdots \\
\alpha _{1}K_{21}K_{d1} & \alpha _{2}K_{22}K_{d2} & & \alpha
_{d}K_{2d}K_{dd} \\
\alpha _{1}K_{31}K_{11} & \alpha _{2}K_{32}K_{12} & & \alpha
_{d}K_{3d}K_{1d} \\
\vdots & \vdots & & \vdots \\
\alpha _{1}K_{d1}^{2} & \alpha _{2}K_{d2}^{2} & \cdots & \alpha
_{d}K_{dd}^{2}%
\end{array}%
\right)
\end{equation*}%
so that the restrictions imposed for all $a,b=1,\ldots ,d$ are%
\begin{equation}
\alpha _{a}^{0}K_{ab}=\alpha _{b}K_{ab}^{2}\ \ \Rightarrow \ \ K_{ab}=0\ \
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{or\ \ }=\alpha _{a}^{0}/\alpha _{b}\ast 1\left \{ 0<\left \vert \alpha
_{a}^{0}/\alpha _{b}\right \vert \leq 1\right \} . \label{V1.ab}
\end{equation}%
Note also that it must hold that $\alpha _{b}\neq 0$ for all $b,$ as
otherwise the $b$-th column of $K$ would be exactly zero.
We note the following consequences of these restrictions on $K$: }
\begin{enumerate}
\item {\small For $a=b,$ $K_{aa}=0$ or $=\alpha _{a}^{0}/\alpha _{a}$ with $%
0<\left
\vert \alpha _{a}^{0}/\alpha _{a}\right \vert \leq 1$ by (\ref{V1.ab}%
). }
\item {\small If $K_{aa}=\pm 1$ $\Rightarrow \ $ $\alpha _{a}=\pm \alpha
_{a}^{0},$ with$\ \ K_{ab}=0,b\neq a,$ by (\ref{O1.a}) ($\alpha _{b}$ no
restricted), i.e. the $a$-th row of $K$ is, up to sign, the unitary vector $%
\mathbf{e}_{a}.$ }
\item {\small If $K_{ab}=\pm 1\ \Rightarrow \ \ \alpha _{a}^{0}/\alpha
_{b}=\pm 1\ (a\neq b),\ $with$\ \ K_{aa}=0$ and $K_{aj}=0$ for all $j\neq b$
by (\ref{O1.a}). }
\item {\small If $K_{aa}\neq \pm 1$ $\ \ \Rightarrow \ \ \exists b\neq a,\ \
$s.t. $K_{ab}\neq 0$ by (\ref{O1.a}).\bigskip }
\end{enumerate}
{\small Then, to show that the only solutions to the equations (\ref{star})
are matrices $K$ which are signed permutations between row $b$ and row $a$
with signs given by $K_{ab}=\alpha _{a}^{0}/\alpha _{b}=\pm 1,$ $K_{aj}=0,$ $%
j\neq b,$ we have to show that it is not possible to select for any $%
a=1,\ldots ,d$ a set of indexes $\mathcal{I}_{a}=\left \{ j_{a,1},\ldots
,j_{a,p\left( a\right) }\right \} ,$ $\# \mathcal{I}_{a}=p\left( a\right)
\geq 2,$ $j_{a,\ell }\in \left \{ 1,\ldots ,d\right \} $ and values $%
K_{a\ell }\neq 0$ for $\ell \in \mathcal{I}_{a}$ that satisfy%
\begin{eqnarray}
\sum_{\ell \in \mathcal{I}_{a}}K_{a\ell }^{2} &=&\sum_{\ell \in \mathcal{I}%
_{a}}\left( \frac{\alpha _{a}^{0}}{\alpha _{\ell }}\right) ^{2}=1,\ \ \
\label{C1} \\
\sum_{\ell \in \mathcal{I}_{a}\cap \mathcal{I}_{b}}K_{a\ell }K_{b\ell }
&=&\sum_{\ell \in \mathcal{I}_{a}\cap \mathcal{I}_{b}}\frac{\alpha _{a}^{0}}{%
\alpha _{\ell }}\frac{\alpha _{b}^{0}}{\alpha _{\ell }}=0,\ \ \ b\neq a,
\label{C2}
\end{eqnarray}%
i.e. the normalization and orthogonalization conditions of $K,$ (\ref{O1.a}%
)-(\ref{O2.ab}). }
{\small Then note that condition (\ref{C1}) excludes permutation matrices
because \#$\mathcal{I}_{a}=p\left( a\right) \geq 2$ ($p\left( b\right) \geq
1,$ $b\neq a $) and that condition (\ref{C2}) implies that for all $a\neq b$%
\begin{equation*}
0=\sum_{\ell \in \mathcal{I}_{a}\cap \mathcal{I}_{b}}\frac{\alpha _{a}^{0}}{%
\alpha _{\ell }}\frac{\alpha _{b}^{0}}{\alpha _{\ell }}=\alpha
_{a}^{0}\alpha _{b}^{0}\sum_{\ell \in \mathcal{I}_{a}\cap \mathcal{I}_{b}}%
\frac{1}{\alpha _{\ell }^{2}}
\end{equation*}%
which, given that $\alpha _{\ell }\neq 0$ for all $\ell \in \mathcal{I}%
_{a}\cap \mathcal{I}_{b},$ is only feasible if $\mathcal{I}_{a}\cap \mathcal{%
I}_{b}=\emptyset $ for all $a\neq b,$ but since $p\left( a\right) \geq 2$,
even if $p\left( b\right) =1$, $b\neq a$, there must be some $b$ for which $%
\mathcal{I}_{a}\cap \mathcal{I}_{b}\not=\emptyset $ and therefore we
conclude that it is not possible to make such selection of indexes to
construct $K$ with elements different from $\pm 1.$\bigskip }
{\small \noindent \textbf{Proof for }$k=4.\ $Consider $d\geq 2,$ denote $%
\beta ^{0}=\mathbf{\alpha }_{4}^{0}$ and%
\begin{equation*}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}^{0}:=\ v\mathbf{\kappa }_{4}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left(
\mathbf{\alpha }_{4}^{0}\right) =\left( \beta _{1}^{0}\mathbf{e}_{1}\ 0\
\cdots \ 0\ \beta _{2}^{0}\mathbf{e}_{d+2}\ 0\ \cdots \ 0\ \beta _{d}^{0}%
\mathbf{e}_{d^{2}}\right) ,\ \
\end{equation*}%
where the constants $\beta _{j}^{0}$ satisfy $\beta _{1}^{0}\beta
_{2}^{0}\cdots \beta _{d}^{0}\neq 0$, v$\mathbf{\kappa }_{4}=\ $v$\mathbf{%
\kappa }_{4}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left( \mathbf{\beta }\right) $ has the same
structure for some constants $\beta _{j}$ satisfying $\beta _{1}\beta
_{2}\cdots \beta _{d}\neq 0$ and $K$ satisfies the normalization and
orthogonality conditions (\ref{O1.a}) and (\ref{O2.ab}). }
{\small Then we have that vec$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}^{0}\right)
=K^{\otimes 4}\ $vec$\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}\right) $ is
equivalent to v$\mathbf{\kappa }_{4}^{0}\ K^{\otimes 2}=K^{\otimes 2}\ $v$%
\mathbf{\kappa }_{4},$ or to%
\begin{equation*}
\left(
\begin{array}{cccccccc}
\beta _{1}^{0}K_{11}^{2} & \beta _{1}^{0}K_{11}K_{12} & \cdots & \beta
_{1}^{0}K_{11}K_{1d} & \beta _{1}^{0}K_{12}K_{11} & \beta _{1}^{0}K_{12}^{2}
& \cdots & \beta _{1}^{0}K_{1d}^{2} \\
0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots \\
0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 \\
\beta _{2}^{0}K_{21}^{2} & \beta _{2}^{0}K_{21}K_{22} & \cdots & \beta
_{2}^{0}K_{21}K_{1d} & \beta _{2}^{0}K_{22}K_{21} & \beta _{2}^{0}K_{22}^{2}
& \cdots & \beta _{2}^{0}K_{2d}^{2} \\
0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 \\
\vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots \\
0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 \\
\beta _{d}^{0}K_{d1}^{2} & \beta _{d}^{0}K_{d1}K_{d2} & \cdots & \beta
_{d}^{0}K_{d1}K_{dd} & \beta _{d}^{0}K_{d2}K_{d1} & \beta _{d}^{0}K_{d2}^{2}
& \cdots & \beta _{d}^{0}K_{dd}^{2}%
\end{array}%
\right)
\end{equation*}%
\begin{equation*}
=\left(
\begin{array}{ccccccccc}
\beta _{1}K_{11}^{2} & 0 & \cdots & 0 & \beta _{2}K_{12}^{2} & 0 & \cdots & 0
& \beta _{d}K_{1d}^{2} \\
\beta _{1}K_{11}K_{21} & 0 & \cdots & 0 & \beta _{2}K_{12}K_{22} & 0 & \cdots
& 0 & \beta _{d}K_{1d}K_{2d} \\
\vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots & \vdots \\
\beta _{1}K_{11}K_{d1} & 0 & \cdots & 0 & \beta _{2}K_{12}K_{d2} & 0 & \cdots
& 0 & \beta _{d}K_{1d}K_{dd} \\
\beta _{1}K_{21}K_{11} & 0 & \cdots & 0 & \beta _{2}K_{22}K_{12} & 0 & \cdots
& 0 & \beta _{d}K_{2d}K_{1d} \\
\beta _{1}K_{21}^{2} & 0 & \cdots & 0 & \beta _{2}K_{22}^{2} & 0 & \cdots & 0
& \beta _{d}K_{2d}^{2} \\
\vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots & \vdots \\
\beta _{1}K_{21}K_{d1} & 0 & \cdots & 0 & \beta _{2}K_{22}K_{d2} & 0 & \cdots
& 0 & \beta _{d}K_{2d}K_{dd} \\
\beta _{1}K_{31}K_{11} & 0 & \cdots & 0 & \beta _{2}K_{32}K_{12} & 0 & \cdots
& 0 & \beta _{d}K_{3d}K_{1d} \\
\vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots & \vdots \\
\beta _{1}K_{d1}^{2} & 0 & \cdots & 0 & \beta _{2}K_{d2}^{2} & 0 & \cdots & 0
& \beta _{d}K_{dd}^{2}%
\end{array}%
\right)
\end{equation*}%
so for each pair $\left( a,b\right) $%
\begin{equation*}
K_{ab}=0\ \ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{or\ \ }\beta _{b}=\beta _{a}^{0}\neq 0,
\end{equation*}%
and%
\begin{equation*}
K_{ab}K_{ac}=0,\ b\neq c,
\end{equation*}%
i.e. in each row $a$ there must be one single non-zero element, say $%
K_{a\ell }$, equal to $\pm 1$ to fulfill the orthogonality and normalization
restrictions, so all the restrictions can only hold when $K$ is a
permutation matrix $P_{d}$ and the assignment $\beta _{\ell }=\beta _{a}^{0}$
is done attending to the location of these nonzero elements $K_{a\ell }=\pm
1,$ i.e. $\mathbf{\beta }=P_{d}^{+\prime }\mathbf{\beta .}$ \ }
{\small This provides identification of the components of $\mathbf{%
\varepsilon }_{t}$ up to signed permutations because for any $A\left(
z\right) =P_{d}$ and any v$\mathbf{\kappa }_{k}^{0}=\ $v$\mathbf{\kappa }%
_{k}^{IC}\left( \mathbf{\alpha }_{k}^{0}\right) $ we could select v$\mathbf{%
\kappa }_{3}=\ $v$\mathbf{\kappa }_{3}^{IC}\left( P_{d}^{\prime }\mathbf{%
\alpha }_{3}^{0}\right) $ or v$\mathbf{\kappa }_{4}=\ $v$\mathbf{\kappa }%
_{4}^{IC}\left( P_{d}^{+\prime }\mathbf{\alpha }_{4}^{0}\right) $ so that $%
\mathcal{L}_{3}^{0}\left( P_{d},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}%
}\left( P_{d}^{\prime }\mathbf{\alpha }_{3}^{0}\right) \right) =\mathcal{L}%
_{4}^{0}\left( P_{d},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left(
P_{d}^{+\prime }\mathbf{\alpha }_{4}^{0}\right) \right) =0\ $because that
for these choices of v$\mathbf{\kappa }_{k},$
\begin{equation*}
P_{d}^{\otimes 3}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}\right) =\RIfM@\expandafter\text@\else\expandafter\mbox\fi{%
vec}\left( P_{d}^{\otimes 2}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{IC}\left(
P_{d}^{\prime }\mathbf{\alpha }_{3}^{0}\right) P_{d}^{\prime }\right) =\RIfM@\expandafter\text@\else\expandafter\mbox\fi{%
vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{IC}\left( \mathbf{\alpha }%
_{3}^{0}\right) \right) =\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{3}^{0}\right)
\end{equation*}%
and%
\begin{equation*}
P_{d}^{\otimes 4}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}\right) =\RIfM@\expandafter\text@\else\expandafter\mbox\fi{%
vec}\left( P_{d}^{\otimes 2}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}^{IC}\left(
P_{d}^{+\prime }\mathbf{\alpha }_{4}^{0}\right) P_{d}^{\otimes 2\prime
}\right) =\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}^{IC}\left( \mathbf{%
\alpha }_{4}^{0}\right) \right) =\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{4}^{0}\right) ,
\end{equation*}%
reflecting that kurtosis is identified independently of the sign of $\mathbf{%
\varepsilon }_{t},$ but skewness is not, because $\alpha _{3,j}^{0}=\mathbf{e%
}_{j}^{\prime }\mathbf{\alpha }_{3}^{0}$ and from (\ref{vk3}) it follows that%
\begin{equation*}
P_{d}^{\otimes 2}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{IC}\left( P_{d}^{\prime }%
\mathbf{\alpha }_{3}^{0}\right) P_{d}^{\prime }=\sum_{j=1}^{d}\mathbf{e}%
_{j}^{\prime }\left( P_{d}^{\prime }\mathbf{\alpha }_{3}^{0}\right) \left(
P_{d}\mathbf{e}_{j}\right) ^{\otimes 2}\left( P_{d}\mathbf{e}_{j}\right)
^{\prime }=\sum_{j=1}^{d}\alpha _{3j}^{0}\mathbf{e}_{j}^{\otimes 2}\mathbf{e}%
_{j}^{\prime }=\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{IC}\left( \mathbf{\alpha }%
_{3}^{0}\right) .
\end{equation*}%
$\Box $\bigskip }
{\small \noindent \textbf{Proof of Theorem~\ref{1A1B}}. The results follows
from combining Theorems~\ref{1A} and~\ref{1ANEW} for non-constant and
constant $A\left( z\right) ,$ respectively. $\Box $\bigskip }
{\small \noindent \textbf{Proof of Corollary \ref{L0}. }For $k=3,$
proceeding as in the proof of Theorem~\ref{1B}, and assuming w.l.o.g. $%
\alpha _{1}^{0}=0,$ the set of assumptions imposed by (\ref{V1.ab}) only
affect now to rows $a=2,\ldots ,d,$ of \ $K,$ while for $a=1\ $the
implication is
\begin{equation}
0=\alpha _{b}K_{1b}^{2}\ \ \Rightarrow \ \ K_{1b}=0\ \ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{and/or}\ \
\alpha _{b}=0\ \ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{(any }K_{1b}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{).} \label{V1.ab*}
\end{equation}%
Therefore, for $a=2,\ldots ,d,$ it is only possible to set $K_{ab}=\alpha
_{a}^{0}/\alpha _{b}\in \left[ -1,1\right] /\left \{ 0\right \} $ when $%
\alpha _{b}\neq 0$ (which implies $K_{1b}=0$ by (\ref{V1.ab*})), otherwise,
if $\alpha _{b}=0,$ then $K_{ab}=0$ for all $a>1.$ Hence, at most there can
be one $b$ such that $\alpha _{b}=0,$ otherwise there would be more than one
row in $K$ with a single nonzero element in the first column (and $K$ would
be no full rank). On the other hand, there must be at least one $\alpha
_{b}=0$ because otherwise the first row of $K$ would be zero. Then, for this
unique $b,$ $K_{1b}=\pm 1,$ $K_{1\ell }=0,$ $\ell \neq b,$ $K_{ab}=0,$ $%
a\neq 1,$ while for the rest of rows and columns of $K$ we can apply the
same argument as in the proof of Theorem~\ref{1B} to show that it must be
also a sign-permutation. }
{\small When there are two or more values of $a$ such that $\alpha
_{a}^{0}=0,$ e.g. $\alpha _{1}^{0}=\alpha _{2}^{0}=0,$ then it is possible
to set up to two values $\alpha _{b}=0,$ for $b=1,2,$ say, where the top
left corner of $K$ can be chosen freely as any orthogonal $2\times 2$
matrix, with the off diagonal blocks of $K$ being zero, while the bottom
right corner should remain of permutation type. }
{\small For $k=4$ and $\beta _{1}^{0}=0,$ we find again that $K_{1b}=0$
and/or $\beta _{b}=0$ (any $K_{1b}$), so the same reasoning as for $k=3,$
applies, since we can only set up one $\beta _{b}=0,$ and the first row (and
column) of $K$ will only contain a single nonzero element equal to $\pm 1,$
the rest of $K$ being also a permutation matrix orthogonal to this first
row.\bigskip }
{\small \noindent \textbf{Proof of Corollary \ref{L1}. }We only consider the
case $\mathcal{I}_{3}\cap \mathcal{I}_{4}=\varnothing ,$ i.e. \#$\left \{
\mathcal{I}_{3}\cap \mathcal{I}_{4}\right \} =0,$ the case with $\alpha
_{j}^{0}=\beta _{j}^{0}=0$ for a single $j$ can be dealt with using the
arguments in Corollary~\ref{L0} to show static identification. }
{\small Proceeding as in the proof of Theorem~\ref{1B}, the restrictions
given by v$\mathbf{\kappa }_{3}^{0}\ K=K^{\otimes 2}\ $v$\mathbf{\kappa }%
_{3}\ $for $k=3\ $in the typical case when all $\alpha _{j}^{0}\neq 0,$ $%
j=1,\ldots ,m,$ but $\alpha _{m+1}^{0}=\cdots =\alpha _{d}=0,$ so that
\begin{equation*}
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{3}^{0}\ K=\left(
\begin{array}{cccc}
\alpha _{1}^{0}K_{11} & \alpha _{1}^{0}K_{12} & \cdots & \alpha
_{1}^{0}K_{1d} \\
0 & 0 & & 0 \\
\vdots & \vdots & & \vdots \\
0 & 0 & & 0 \\
\alpha _{2}^{0}K_{21} & \alpha _{2}^{0}K_{22} & \cdots & \alpha
_{2}^{0}K_{2d} \\
0 & 0 & & 0 \\
\vdots & \vdots & & \vdots \\
0 & 0 & \cdots & 0 \\
\alpha _{m}^{0}K_{m1} & \alpha _{m}^{0}K_{m1} & & \alpha _{m}^{0}K_{21} \\
0 & 0 & \cdots & 0 \\
\vdots & & & \vdots \\
0 & 0 & \cdots & 0%
\end{array}%
\right)
\end{equation*}%
are
\begin{equation*}
K_{ab}=0\ \ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{or\ }=\alpha _{a}^{0}/\alpha _{b}\ \ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{for\ \ }%
a=1,\ldots ,m,
\end{equation*}%
while the rows $j=m+1,\ldots ,d$ of $K$ are essentially unrestricted when $%
d-m\geq 2,$ cf. proof of Corollary~\ref{L0}. This implies by the same
argument as in Theorem~\ref{1B} that these restrictions can only be
satisfied for $K$ whose top $m$ rows are of permutation type if we show that
the remaining rows have to be also of permutation type. }
{\small The restrictions v$\mathbf{\kappa }_{4}^{0}\ K^{\otimes
2}=K^{\otimes 2}\ $v$\mathbf{\kappa }_{4}$ for $k=4$, under independence and
$\beta _{j}^{0}\neq 0,$ $j=m+1,\ldots ,d$ and $\beta _{1}^{0}=\cdots =\beta
_{m}^{0}=0,$ where now%
\begin{equation*}
v\mathbf{\kappa }_{4}^{0}\ K^{\otimes 2}=\left(
\begin{array}{cccc}
0 & 0 & \cdots & 0 \\
0 & 0 & \cdots & 0 \\
\vdots & \vdots & & \vdots \\
0 & 0 & \cdots & 0 \\
\beta _{m+1}^{0}K_{m+11}^{2} & \beta _{m+1}^{0}K_{m+11}K_{m+12} & \cdots &
\beta _{m+1}^{0}K_{m+1d}^{2} \\
\vdots & \vdots & & \vdots \\
\beta _{d}^{0}K_{d1}^{2} & \beta _{d}^{0}K_{d1}K_{d2} & \cdots & \beta
_{d}^{0}K_{dd}^{2}%
\end{array}%
\right) ,
\end{equation*}%
are, following as in Theorem~\ref{1B},%
\begin{eqnarray*}
K_{ab} &=&0\ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{or otherwise }\beta _{b}=\beta _{a}^{0}=0,\ \ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{for }%
a=1,\ldots ,m \\
K_{ab} &=&0\ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{or otherwise }\beta _{b}=\beta _{a}^{0}\neq 0,\ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{for
}a=m+1,\ldots ,d,
\end{eqnarray*}%
and still%
\begin{equation*}
K_{ab}K_{ac}=0,\ \ b\neq c,\ \ \ a=m+1,\ldots ,d,
\end{equation*}%
i.e. each of the last $d-m$ rows must contain one single non-zero element,
equal to $\pm 1$ to fulfill the orthogonality and normalization
restrictions. Then the restrictions for $k=3$ imply that the top $m$ rows
have to be also of signed permutation type, and therefore, considering both
types of restrictions together, we obtain that they can only hold
simultaneously for $K$ equal to a signed permutation $P_{d},$ where the
non-zero and zero values of $\alpha _{j}^{0}$ and $\beta _{j}^{0}$ in v$%
\mathbf{\kappa }_{3}^{0}$ and v$\mathbf{\kappa }_{4}^{0}$ (and signs for $%
\alpha _{j}^{0}$) are permuted in the same order by $\mathbf{\alpha }%
=P_{d}^{\prime }\mathbf{\alpha }^{0}$ and $\mathbf{\beta }=P_{d}^{+\prime }%
\mathbf{\beta }^{0}.$ }
{\small To show dynamic identification when $\mathcal{I}_{3}\cap \mathcal{I}%
_{4}=\varnothing ,$ we take in the proof of Theorem~\ref{1ANEW} the simplest
case in which $\alpha _{41}^{0}=0,$ but all order marginal cumulants of
order $k=4$ are nonzero. In this case, the argument for identification fails
because there are BM $A(z)$ for which the only row of $A^{\otimes 2\ast
}\left( \lambda _{4},-\lambda _{3}\right) $ which depends on $\left( \lambda
_{3},\lambda _{4}\right) $ is the first one, so that $\xi _{4}(\mathbf{%
\lambda })$ can not be constructed and it would be possible to choose $\RIfM@\expandafter\text@\else\expandafter\mbox\fi{%
v}\mathbf{\kappa }_{4}$ to achieve $\left \Vert \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( A^{\otimes
2}\left( \lambda _{2},\lambda _{1}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{4}-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{%
v}\mathbf{\kappa }_{4}^{0}A^{\otimes 2\ast }\left( \lambda _{4},-\lambda
_{3}\right) \right) \right \Vert =0$ a.e.. However, if at the same time $%
\alpha _{31}^{0}\neq 0$, we can take $j=1$ in the argument of the proof of
Theorem~\ref{1ANEW} to define $\xi _{3}(\lambda _{1}+\lambda _{2})$ for an
appropriate $h$ so that it has an infinite expansion in $\lambda
_{1}+\lambda _{2}$ that can not be matched by any function $\zeta
_{3}(\lambda _{1},\lambda _{2})$ for any choice of $\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }%
_{3}$ and such BM $A\left( z\right) .$ The same reasoning applies in the
other direction and when more than one marginal cumulant is zero for any
order $k$ as far as the corresponding cumulants of the other order are
different from zero. $\square \bigskip $ }
{\small \noindent \textbf{Proof of Theorem~\ref{Th4}}. By Theorem~3 of Lippi
and Reichlin (1994), all second-order equivalent representations of $%
Y_{t}=\Phi _{\mathbf{\theta }_{0}}^{-1}\left( z\right) \Theta _{\mathbf{%
\theta }_{0}}\left( z\right) \mathbf{\varepsilon }_{t},$ $\mathbf{%
\varepsilon }_{t}=iid\left( \mathbf{0},\mathbf{I}_{d},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa
}_{k}^{IC}\left( \mathbf{\alpha }_{k}^{0}\right) \right) ,$ giving $\mathcal{%
L}_{2}\left( \mathbf{\theta }\right) =0$ for some $\mathbf{\theta \neq
\theta }_{0}\mathbf{,\ \theta }\in \mathcal{S}$, involve a matrix polynomial
$\Phi _{\mathbf{\theta }}^{-1}\left( z\right) \Theta _{\mathbf{\theta }%
}\left( z\right) =\Phi _{\mathbf{\theta }_{0}}^{-1}\left( z\right) \Theta _{%
\mathbf{\theta }_{0}}\left( z\right) K$ up to an orthogonal matrix $K$ when $%
\Theta _{\mathbf{\theta }}\left( z\right) $ has the same roots as $\Theta _{%
\mathbf{\theta }_{0}}\left( z\right) ,$ while by Theorem~2 of Lippi and
Reichlin (1994) the transfer functions differ by a (non constant) BM $%
A\left( z\right) $ when some of the roots of $\Theta _{\mathbf{\theta }%
_{0}}\left( z\right) $ are flipped, $\Phi _{\mathbf{\theta }}^{-1}\left(
z\right) \Theta _{\mathbf{\theta }}\left( z\right) =\Phi _{\mathbf{\theta }%
_{0}}^{-1}\left( z\right) \Theta _{\mathbf{\theta }_{0}}\left( z\right)
A\left( z\right) .$ }
{\small Then the proof follows from our Theorem~\ref{1A1B} which shows that $%
\mathcal{L}_{k}\left( \mathbf{\theta ,\alpha }_{k}\right) =0$ could only
hold if such $A$ is equal to a signed permutation $P_{d}$ for $k=3$ or $k=4$
and $\mathbf{\alpha }_{3}=P_{d}^{\prime }\mathbf{\alpha }_{3}^{0}$ or $%
\mathbf{\alpha }_{4}=P_{d}^{+\prime }\mathbf{\alpha }_{4}^{0}$ (with $%
P_{d}^{+}$ being equal to $P_{d}$ with positive entries), while Assumption~6
discards all such signed permutations except the identity one by fixing an
ordering and a sign pattern, and Assumption~5.2 excludes all values $\mathbf{%
\theta }\neq \mathbf{\theta }_{0}$ such that $\Phi _{\mathbf{\theta }%
}^{-1}\left( z\right) \Theta _{\mathbf{\theta }}\left( z\right) =\Phi _{%
\mathbf{\theta }_{0}}^{-1}\left( z\right) \Theta _{\mathbf{\theta }%
_{0}}\left( z\right) A\left( z\right) $ a.e. for $A\left( z\right) =\mathbf{I%
}_{d}.$ $\Box $\bigskip }
{\small \noindent \textbf{Proof of Theorem~\ref{Th5}}. As in VL (2018) we
can show uniformly for $\mathbf{\theta \in \ }\mathcal{S}$ and for $k=3\ $or
$4$ that%
\begin{equation*}
\mathbf{\hat{\alpha}}_{k,T}\left( \mathbf{\theta }\right) \rightarrow _{p}%
\mathbf{\alpha }_{k}\left( \mathbf{\theta }\right)
\end{equation*}%
where
\begin{equation*}
\mathbf{\alpha }_{k}\left( \mathbf{\theta }\right) :=\left( \int_{\Pi ^{k-1}}%
\func{Re}\left \{ \mathbf{S}_{k}^{\prime }\mathbf{\Psi }_{k}^{\ast }(\mathbf{%
\lambda };\mathbf{\theta })\mathbf{\Psi }_{k}(\mathbf{\lambda };\mathbf{%
\theta })\mathbf{S}_{k}\right \} d\mathbf{\lambda }\right) ^{-1}\int_{\Pi
^{k-1}}\func{Re}\left \{ \mathbf{S}_{k}^{\prime }\mathbf{\Psi }_{k}^{\ast }(%
\mathbf{\lambda };\mathbf{\theta })\mathbf{\Psi }_{k}(\mathbf{\lambda };%
\mathbf{\theta }_{0})\mathbf{S}_{k}\right \} d\mathbf{\lambda \ \mathbf{%
\alpha }}_{k}^{0}
\end{equation*}%
satisfies $\mathbf{\alpha }_{k}\left( \mathbf{\theta }_{0}\right) =\mathbf{%
\alpha }_{k}^{0},$ $\mathcal{\hat{L}}_{k,T}\left( \mathbf{\theta }\right) $
converges uniformly to
\begin{equation*}
\mathcal{\hat{L}}_{k}\left( \mathbf{\theta }\right) =\left( 2\pi \right)
^{1-k}\int_{\Pi ^{k-1}}\left( \mathbf{\Psi }_{k}\left( \mathbf{\lambda
;\theta }\right) \mathbf{S}_{k}\mathbf{\alpha }_{k}\left( \mathbf{\theta }%
\right) -\mathbf{\Psi }_{k}\left( \mathbf{\lambda ;\theta }_{0}\right)
\mathbf{S}_{k}\mathbf{\alpha }_{k}^{0}\right) ^{\ast }\left( \mathbf{\Psi }%
_{k}\left( \mathbf{\lambda ;\theta }\right) \mathbf{S}_{k}\mathbf{\alpha }%
_{k}\left( \mathbf{\theta }\right) -\mathbf{\Psi }_{k}\left( \mathbf{\lambda
;\theta }_{0}\right) \mathbf{S}_{k}\mathbf{\alpha }_{k}^{0}\right) d\mathbf{%
\lambda }
\end{equation*}%
while $\mathcal{L}_{2,T}\left( \mathbf{\theta }\right) $ converges uniformly
to%
\begin{equation*}
\mathcal{L}_{2}\left( \mathbf{\theta }\right) =\left( 2\pi \right)
^{-1}\int_{\Pi }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}_{d}\right) ^{\prime }\left(
\mathbf{\Psi }_{2}\left( \lambda \mathbf{;\theta }\right) -\mathbf{\Psi }%
_{2}\left( \lambda \mathbf{;\theta }_{0}\right) \right) ^{\ast }\left(
\mathbf{\Psi }_{2}\left( \lambda \mathbf{;\theta }\right) -\mathbf{\Psi }%
_{2}\left( \lambda \mathbf{;\theta }_{0}\right) \right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left(
\mathbf{I}_{d}\right) d\lambda
\end{equation*}%
which is minimized for $\mathbf{\theta =\theta }_{0}$ and for all $\mathbf{%
\theta \neq \theta }_{0}$ such that $\mathbf{\Psi }\left( e^{-i\lambda }%
\mathbf{;\theta }\right) =\mathbf{\Psi }\left( e^{-i\lambda }\mathbf{;\theta
}_{0}\right) A(e^{-i\lambda })\ $where the BM factor satisfies $%
A(e^{-i\lambda })\neq \mathbf{I}_{d}$ in a set of positive measure by
Assumption~5.2, so that $\mathbf{\Psi }_{2}\left( \lambda \mathbf{;\theta }%
\right) =\mathbf{\Psi }_{2}\left( \lambda \mathbf{;\theta }_{0}\right) $
a.e. and $\mathcal{L}_{2}\left( \mathbf{\theta }\right) =0.$ }
{\small Then for those $\mathbf{\theta \neq \theta }_{0}$ for which $%
\mathcal{L}_{2}\left( \mathbf{\theta }\right) =0$, \textbf{$\Psi $}$_{k}(%
\mathbf{\lambda };\mathbf{\theta })\mathbf{S}_{k}\mathbf{\alpha }_{k}\left(
\mathbf{\theta }\right) =\mathbf{\Psi }_{k}(\mathbf{\lambda };\mathbf{\theta
}_{0})A^{\otimes k}(\mathbf{\lambda })\mathbf{S}_{k}\mathbf{\alpha }%
_{k}\left( \mathbf{\theta }\right) ,$ and therefore for $\mathcal{\hat{L}}%
_{k}\left( \mathbf{\theta }\right) =0$ to be true given Assumption~5.2, it
must hold that $A^{\otimes k}(\mathbf{\lambda })\mathbf{S}_{k}\mathbf{\alpha
}_{k}\left( \mathbf{\theta }\right) =\mathbf{S}_{k}\mathbf{\alpha }_{k}^{0}$
a.e., which by Theorem~3 can only hold when $A(e^{-i\lambda })$ is a signed
permutation matrix $P_{d}$ a.e.. }
{\small Therefore, under Assumption~6A, all $\mathbf{\theta \neq \theta }%
_{0} $ such that $\mathbf{\Psi }\left( e^{-i\lambda }\mathbf{;\theta }%
\right) =\mathbf{\Psi }\left( e^{-i\lambda }\mathbf{;\theta }_{0}\right)
P_{d}$ with $P_{d}\neq \mathbf{I}_{d}$ are discarded as $\mathbf{\theta \not
\in }\mathcal{S}^{\max }$ because the product of the absolute value of the
diagonal elements of $\Theta _{0}\left( \mathbf{\theta }_{0}\right) $ is a
unique maximum up to permutations, and the consistency of $\mathbf{\hat{%
\theta}}_{k,T}$ follows by the standard argument. Alternatively,
Assumption~6C directly discard that for any $\mathbf{\theta \neq \theta }%
_{0} $ in $\mathcal{S}$ there exists a signed permutation $P_{d}$ so that $%
\mathbf{\Psi }\left( e^{-i\lambda }\mathbf{;\theta }\right) =\mathbf{\Psi }%
\left( e^{-i\lambda }\mathbf{;\theta }_{0}\right) P_{d}$ a.e. exists, so it
must hold that $\mathcal{\hat{L}}_{k}\left( \mathbf{\theta }\right) >0$. }
{\small Further, Lemma~\ref{LemmaA} shows that if $\mathbf{\Psi }\left(
e^{-i\lambda }\mathbf{;\theta }\right) =\mathbf{\Psi }\left( e^{-i\lambda }%
\mathbf{;\theta }_{0}\right) P_{d}$ a.e., then $\mathbf{\alpha }_{3}\left(
\mathbf{\theta }\right) =P_{d}^{\prime }\mathbf{\alpha }_{3}^{0}\ $and $%
\mathbf{\alpha }_{4}\left( \mathbf{\theta }\right) =P_{d}^{+\prime }\mathbf{%
\alpha }_{4}^{0}$ (where $P_{d}^{+}$ is equal to $P_{d}$ with positive
entries)$,$ and therefore $\mathcal{L}_{k}\left( \mathbf{\theta ,\alpha }%
_{k}\left( \mathbf{\theta }\right) \right) =0$.\ But for such $\mathbf{%
\theta }\neq \mathbf{\theta }_{0},$ even if $\mathbf{\theta \in }\mathcal{S}%
^{+},$ it holds $\mathbf{\alpha }_{k}\left( \mathbf{\theta }\right) \neq
\mathbf{\alpha }_{k}^{0}$ under Assumption~6B$\left( k\right) $ because $%
P_{d}\neq I_{d}$ and $P_{d}^{+}\neq I_{d}$ for $k=3$ and $4,$ respectively,
by Assumption~5.2, so that for such $\mathbf{\theta ,}$ $\mathbf{\alpha }%
_{k}\left( \mathbf{\theta }\right) \not \in \mathcal{D}_{k}$ and min$_{%
\mathbf{\alpha }\in \mathcal{D}_{k}}\mathcal{L}_{k}\left( \mathbf{\theta
,\alpha }\right) >0\ $because of the compactness of $\mathcal{D}_{k},$ and
therefore $\mathcal{L}_{2}\left( \mathbf{\theta }\right) +\mathcal{L}%
_{k}\left( \mathbf{\theta ,\alpha }\right) $ is uniquely miminized at $%
\left( \mathbf{\theta ,\alpha }\right) =\left( \mathbf{\theta }_{0},\mathbf{%
\alpha }_{0}\right) \ $in $\mathcal{S}^{+}\times \mathcal{D}_{k}.$ }
{\small Notice that the case where $P_{d}\neq \mathbf{I}_{d}$ is a pure
sign-flipping diagonal matrix so that $P_{d}^{+}=\mathbf{I}_{d}$ and $%
\mathbf{\Psi }_{4}(\mathbf{\lambda };\mathbf{\theta })=\mathbf{\Psi }_{4}(%
\mathbf{\lambda };\mathbf{\theta }_{0})$ is excluded by Assumption~6B$\left(
4\right) $ by imposing all the diagonal elements of $\mathbf{\Theta }%
_{0}\left( \mathbf{\theta }_{0}\right) $ being strictly positive, which is
not true for $\mathbf{\Psi }(e^{-i\lambda };\mathbf{\theta }_{0})P_{d}$ and
such diagonal $P_{d}.$ However, Assumption~6B$\left( 3\right) $ needs to
explicit prevent alternative orderings in $\mathbf{\alpha }_{3}$ due to sign
changes in $P_{d}^{\prime }\mathbf{\alpha }_{3}^{0}.$ }
{\small Finally, the consistency of $\mathbf{\hat{\alpha}}_{k,T}\left(
\mathbf{\hat{\theta}}_{k,T}\right) \ $follows from the consistency of $%
\mathbf{\hat{\theta}}_{k,T}$ using similar methods as in VL and
Assumption~5.4.$\ \ \Box $\bigskip }
{\small \noindent \textbf{Proof of Theorem~\ref{Th6}}. The score of the
concentrated loss function wrt to each component of $\mathbf{\theta }$ is
given by, $k=3,4,$%
\begin{eqnarray*}
\frac{\partial }{\partial \mathbf{\theta }_{\ell }}\mathcal{\hat{L}}%
_{k,T}^{\dag }\left( \mathbf{\theta }\right) &=&\frac{2}{T^{k-1}}\sum_{%
\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{ \left( \mathbf{\Psi }%
_{k}\left( \mathbf{\lambda _{\mathbf{j}};\theta }\right) \mathbf{S}_{k}%
\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }\right) -\mathbb{I}%
_{k}(\mathbf{\lambda }_{\mathbf{j}})\right) ^{\ast }\mathbf{W}_{k}(\mathbf{%
\lambda }_{\mathbf{j}};\mathbf{\tilde{\theta}}_{T})\mathbf{\dot{\Psi}}%
_{k}^{\left( \ell \right) }(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })%
\mathbf{S}_{k}\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }%
\right) \right \} \\
&+&\frac{2}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{
\left( \mathbf{\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })%
\mathbf{S}_{k}\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }%
\right) -\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}})\right) ^{\ast }%
\mathbf{W}_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\tilde{\theta}}_{T})%
\mathbf{\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })\mathbf{S}%
_{k}\frac{\partial }{\partial \mathbf{\theta }_{\ell }}\mathbf{\hat{\alpha}}%
_{k,T}^{\dag }\left( \mathbf{\theta }\right) \right \}
\end{eqnarray*}%
where the estimation effect of $\mathbf{\alpha }_{k}$ is similar to VL,%
\begin{equation*}
\frac{\partial }{\partial \mathbf{\theta }_{\ell }}\mathbf{\hat{\alpha}}%
_{k,T}^{\dag }\left( \mathbf{\theta }\right) =-\frac{1}{T^{k-1}}\sum_{%
\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{ \mathbf{S}_{k}^{\prime }%
\mathbf{\Psi }_{k}^{-1}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })%
\mathbf{\dot{\Psi}}_{k}^{\left( \ell \right) }(\mathbf{\lambda }_{\mathbf{j}%
};\mathbf{\theta })\mathbf{\Psi }_{k}^{-1}(\mathbf{\lambda }_{\mathbf{j}};%
\mathbf{\theta })\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}})\right \} ,
\end{equation*}%
which shows that estimation of $\mathbf{\theta }$ is not independent of
estimation of $\mathbf{\alpha }_{k}$ unlike with second order methods as we
can check that
\begin{equation*}
\frac{\partial }{\partial \mathbf{\theta }_{\ell }}\mathbf{\hat{\alpha}}%
_{k,T}^{\dag }\left( \mathbf{\theta }_{0}\right) \rightarrow _{p}-\mathbf{S}%
_{k}^{\prime }\mathbf{\bar{\Lambda}}_{k}^{\left( \ell \right) }\left(
\mathbf{\theta }_{0}\right) \mathbf{S}_{k}\mathbf{\alpha }_{0},
\end{equation*}%
which is also the limit of $\frac{\partial }{\partial \mathbf{\theta }_{\ell
}}\mathbf{\hat{\alpha}}_{k,T}^{EFF}\left( \mathbf{\theta }_{0}\right) ,$ so $%
\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }_{0}\right) $ and $%
\mathbf{\hat{\alpha}}_{k,T}^{EFF}\left( \mathbf{\theta }_{0}\right) $ share
the same asymptotic distribution because $\frac{\partial }{\partial \mathbf{%
\tilde{\theta}}_{T,\ell }}\mathbf{\hat{\alpha}}_{k,T}^{EFF}\left( \mathbf{%
\theta }_{0}\right) \rightarrow _{p}0.$ }
{\small Similarly for $k=2,$ we have%
\begin{equation*}
\frac{\partial }{\partial \mathbf{\theta }_{\ell }}\mathcal{L}_{2,T}\left(
\mathbf{\theta }\right) =\frac{2}{T}\sum_{\lambda _{j}}\func{Re}\left \{
\left( \mathbf{\Psi }_{2}\left( \lambda _{j}\mathbf{;\theta }\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{%
vec}\left( \mathbf{I}_{d}\right) -\mathbb{I}_{2}(\lambda _{j})\right) ^{\ast
}\mathbf{W}_{2}(\lambda _{j};\mathbf{\tilde{\theta}}_{T})\mathbf{\dot{\Psi}}%
_{2}^{\left( \ell \right) }(\lambda _{j};\mathbf{\theta )}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left(
\mathbf{I}_{d}\right) \right \} .
\end{equation*}
}
{\small Then, using that $\mathbf{S}_{k}\mathbf{S}_{k}^{\prime }T^{1-k}\sum_{%
\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{ \mathbf{B}_{k}\left(
\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0}\right) \right \} =%
\mathbf{S}_{k}\mathbf{S}_{k}^{\prime }\left( 2\pi \right) ^{1-k}\int \func{Re%
}\left \{ \mathbf{B}_{k}\left( \mathbf{\lambda };\mathbf{\theta }_{0}\right)
\right \} d\mathbf{\lambda }+O\left( T^{-1}\right) =O\left( T^{-1}\right)
\mathbf{,\ }$because $\mathbf{S}_{k}\mathbf{S}_{k}^{\prime }\left( 2\pi
\right) ^{1-k}\int_{\Pi ^{k-1}}\func{Re}\left \{ \mathbf{B}_{k}\left(
\mathbf{\lambda };\mathbf{\theta }_{0}\right) \right \} d\mathbf{\lambda }=0$
as $\mathbf{S}_{k}^{\prime }\mathbf{S}_{k}=\mathbf{I}_{d},$ up to $%
o_{p}\left( 1\right) $ terms,
\begin{eqnarray*}
&&T^{1/2}\mathbf{\Sigma }\left( \mathbf{\theta }_{0},\mathbf{\alpha }%
_{0}\right) \left( \mathbf{\hat{\theta}}_{w,T}^{\dag }-\mathbf{\theta }%
_{0}\right) \\
&=&-\sum_{k=3}^{4}w_{k}\frac{T^{1/2}}{T^{k-1}}\left( \mathbf{I}_{m}\otimes
\mathbf{\alpha }_{k}^{0\prime }\mathbf{S}_{k}^{\prime }\right) \sum_{\mathbf{%
\lambda }_{\mathbf{j}}}\func{Re}\left \{ \mathbf{B}_{k}^{\ast }\left(
\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0}\right) \mathbf{\Psi }%
_{k}^{-1}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0})\left( \mathbf{%
\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0})\mathbf{S}_{k}%
\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }_{0}\right) -%
\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}})\right) \right \} \\
&&-\frac{1}{T^{1/2}}\left( \mathbf{I}_{m}\otimes \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}%
_{d}\right) ^{\prime }\right) \sum_{\lambda _{j}}\func{Re}\left \{ \mathbf{B}%
_{2}^{\ast }\left( \lambda _{j}\mathbf{;\theta }_{0}\right) \mathbf{\Psi }%
_{2}^{-1}(\lambda _{j};\mathbf{\theta }_{0})\left( \mathbf{\Psi }_{2}\left(
\lambda _{j}\mathbf{;\theta }_{0}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}%
_{d}\right) -\mathbb{I}_{2}(\lambda _{j})\right) \right \} \\
&=&\sum_{k=3}^{4}w_{k}\frac{T^{1/2}}{T^{k-1}}\left( \mathbf{I}_{m}\otimes
\mathbf{\alpha }_{k}^{0\prime }\mathbf{S}_{k}^{\prime }\right) \sum_{\mathbf{%
\lambda }_{\mathbf{j}}}\func{Re}\left \{ \mathbf{B}_{k}^{\ast }\left(
\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0}\right) \left \{ \mathbf{%
\Psi }_{k}^{-1}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0})\mathbb{I%
}_{k}(\mathbf{\lambda }_{\mathbf{j}})-\mathbf{S}_{k}\mathbf{\hat{\alpha}}%
_{k,T}^{\dag }\left( \mathbf{\theta }_{0}\right) \right \} \right \} \\
&&+\frac{1}{T^{1/2}}\left( \mathbf{I}_{m}\otimes \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}%
_{d}\right) ^{\prime }\right) \sum_{\lambda _{j}}\func{Re}\left \{ \mathbf{B}%
_{2}^{\ast }\left( \lambda _{j}\mathbf{;\theta }_{0}\right) \left( \mathbf{%
\Psi }_{2}^{-1}(\lambda _{j};\mathbf{\theta }_{0})\mathbb{I}_{2}(\lambda
_{j})-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}_{d}\right) \right) \right \}
\end{eqnarray*}
\begin{eqnarray*}
&=&\sum_{k=3}^{4}w_{k}\frac{T^{1/2}}{T^{k-1}}\left( \mathbf{I}_{m}\otimes
\mathbf{\alpha }_{k}^{0\prime }\mathbf{S}_{k}^{\prime }\right) \sum_{\mathbf{%
\lambda }_{\mathbf{j}}}\func{Re}\left \{ \mathbf{B}_{k}^{\ast }\left(
\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0}\right) \left \{ \mathbb{I%
}_{k}^{\mathbf{\varepsilon }}(\mathbf{\lambda }_{\mathbf{j}})-\mathbf{S}_{k}%
\mathbf{S}_{k}^{\prime }\frac{1}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}%
}}\mathbb{I}_{k}^{\mathbf{\varepsilon }}(\mathbf{\lambda }_{\mathbf{j}%
})\right \} \right \} \\
&&+\frac{1}{T^{1/2}}\left( \mathbf{I}_{m}\otimes \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}%
_{d}\right) ^{\prime }\right) \sum_{\lambda _{j}}\func{Re}\left \{ \mathbf{B}%
_{2}^{\ast }\left( \lambda _{j}\mathbf{;\theta }_{0}\right) \left( \mathbb{I}%
_{2}^{\mathbf{\varepsilon }}(\lambda _{j})-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}%
_{d}\right) \right) \right \} \\
&=&\sum_{k=3}^{4}w_{k}\frac{T^{1/2}}{T^{k-1}}\left( \mathbf{I}_{m}\otimes
\mathbf{\alpha }_{k}^{0\prime }\mathbf{S}_{k}^{\prime }\right) \sum_{\mathbf{%
\lambda }_{\mathbf{j}}}\func{Re}\left \{ \mathbf{B}_{k}^{\ast }\left(
\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0}\right) \left \{ \mathbb{I%
}_{k}^{\mathbf{\varepsilon }}(\mathbf{\lambda }_{\mathbf{j}})-E\left[
\mathbb{I}_{k}^{\mathbf{\varepsilon }}(\mathbf{\lambda }_{\mathbf{j}})\right]
\right \} \right \} \\
&&+\frac{1}{T^{1/2}}\left( \mathbf{I}_{m}\otimes \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}%
_{d}\right) ^{\prime }\right) \sum_{\lambda _{j}}\func{Re}\left \{ \mathbf{B}%
_{2}^{\ast }\left( \lambda _{j}\mathbf{;\theta }_{0}\right) \left( \mathbb{I}%
_{2}^{\mathbf{\varepsilon }}(\lambda _{j})-E\left[ \mathbb{I}_{2}^{\mathbf{%
\varepsilon }}(\lambda _{j})\right] \right) \right \} \\
& \rightarrow _{d} &N_{m}\left( 0,\mathbf{\delta }\left( \mathbf{\alpha }%
_{0}\right) \mathbf{\Omega }\left( \mathbf{\theta }_{0}\right) \mathbf{%
\delta }\left( \mathbf{\alpha }_{0}\right) ^{\prime }\right) ,
\end{eqnarray*}%
applying the CLT in Appendix~D to weighted sums of $\mathbb{I}_{2}^{\mathbf{%
\varepsilon }}(\lambda _{j})$ and $\mathbb{I}_{k}^{\mathbf{\varepsilon }}(%
\mathbf{\lambda }_{\mathbf{j}})$ as in VL, because for $k=3,4,$ $%
T^{1-k}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{ \mathbf{B}%
_{k}^{\ast }\left( \mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }%
_{0}\right) E\left[ \mathbb{I}_{k}^{\mathbf{\varepsilon }}(\mathbf{\lambda }%
_{\mathbf{j}})\right] \right \} =T^{1-k}\sum_{\mathbf{\lambda }_{\mathbf{j}}}%
\func{Re}\left \{ \mathbf{B}_{k}^{\ast }\left( \mathbf{\lambda }_{\mathbf{j}%
};\mathbf{\theta }_{0}\right) \right \} \mathbf{S}_{k}\mathbf{\alpha }_{0}=%
\mathbf{\bar{\Lambda}}_{k}^{\prime }\left( \mathbf{\theta }\right) \mathbf{S}%
_{k}\mathbf{\alpha }_{0}-\mathbf{\bar{\Lambda}}_{k}^{\prime }\left( \mathbf{%
\theta }\right) \mathbf{S}_{k}\mathbf{S}_{k}^{\prime }\mathbf{S}_{k}\mathbf{%
\alpha }_{0}+O\left( T^{-1}\right) =O\left( T^{-1}\right) $ as $\mathbf{S}%
_{k}\mathbf{S}_{k}^{\prime }\mathbf{S}_{k}=\mathbf{S}_{k}.$ The proof is
completed using Lemma~\ref{Hessian} for the convergence of the Hessian. $%
\mathbf{\ \ \Box }$\bigskip }
{\small \noindent \textbf{Proof of Theorem~\ref{Th7}}. We can write%
\begin{equation*}
\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\hat{\theta}}_{k,T}^{\dag
}\right) -\mathbf{\alpha }_{k}^{0}=\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left(
\mathbf{\hat{\theta}}_{w,T}^{\dag }\right) -\mathbf{\hat{\alpha}}%
_{k,T}^{\dag }\left( \mathbf{\theta }_{0}\right) +\mathbf{\hat{\alpha}}%
_{k,T}^{\dag }\left( \mathbf{\theta }_{0}\right) -\mathbf{\alpha }_{k}^{0}
\end{equation*}%
where%
\begin{equation*}
\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\hat{\theta}}_{k,T}^{\dag
}\right) -\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }%
_{0}\right) =\frac{\partial }{\partial \mathbf{\theta }^{\prime }}\mathbf{%
\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }_{T}\right) \left( \mathbf{%
\hat{\theta}}_{w,T}^{\dag }-\mathbf{\theta }_{0}\right)
\end{equation*}%
for some $\mathbf{\theta }_{T}\rightarrow _{p}\mathbf{\theta }_{0}$ and%
\begin{eqnarray*}
T^{1/2}\left( \mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }%
_{0}\right) -\mathbf{\alpha }_{k}^{0}\right) &=&\frac{T^{1/2}}{T^{k-1}}\sum_{%
\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{ \mathbf{S}_{k}^{\prime }%
\mathbf{\Psi }_{k}^{-1}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0})%
\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}})-\mathbf{\alpha }%
_{k}^{0}\right \} \\
&=&\frac{T^{1/2}}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\mathbf{S}%
_{k}^{\prime }\func{Re}\left \{ \mathbb{I}_{k}^{\mathbf{\varepsilon }}(%
\mathbf{\lambda }_{\mathbf{j}})-E\left[ \mathbb{I}_{k}^{\mathbf{\varepsilon }%
}(\mathbf{\lambda }_{\mathbf{j}})\right] \right \} +o_{p}\left( 1\right) .
\end{eqnarray*}%
Then%
\begin{eqnarray*}
\frac{\partial }{\partial \mathbf{\theta }_{\ell }}\mathbf{\hat{\alpha}}%
_{k,T}^{\dag }\left( \mathbf{\theta }_{T}\right) &=&-\frac{1}{T^{k-1}}\sum_{%
\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{ \mathbf{S}_{k}^{\prime }%
\mathbf{\Psi }_{k}^{-1}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{T})%
\mathbf{\dot{\Psi}}_{k}^{\left( \ell \right) }(\mathbf{\lambda }_{\mathbf{j}%
};\mathbf{\theta }_{T})\mathbf{\Psi }_{k}^{-1}(\mathbf{\lambda }_{\mathbf{j}%
};\mathbf{\theta }_{T})\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}})\right
\} \\
& \rightarrow _{p} &-\frac{1}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}%
\func{Re}\left \{ \mathbf{S}_{k}^{\prime }\mathbf{\Psi }_{k}^{-1}(\mathbf{%
\lambda }_{\mathbf{j}};\mathbf{\theta }_{0})\mathbf{\dot{\Psi}}_{k}^{\left(
\ell \right) }(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0})\mathbf{%
\Psi }_{k}^{-1}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0})\mathbb{I%
}_{k}(\mathbf{\lambda }_{\mathbf{j}})\right \} \\
& \rightarrow _{p} &-\frac{1}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}%
\func{Re}\left \{ \mathbf{S}_{k}^{\prime }\mathbf{\Psi }_{k}^{-1}(\mathbf{%
\lambda }_{\mathbf{j}};\mathbf{\theta }_{0})\mathbf{\dot{\Psi}}_{k}^{\left(
\ell \right) }(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0})\mathbb{I}%
_{k}^{\mathbf{\varepsilon }}(\mathbf{\lambda }_{\mathbf{j}})\right \} \\
& \rightarrow _{p} &-\mathbf{S}_{k}^{\prime }\mathbf{\bar{\Lambda}}%
_{k}^{\left( \ell \right) }\left( \mathbf{\theta }_{0}\right) \mathbf{S}_{k}%
\mathbf{\alpha }_{k}^{0}
\end{eqnarray*}%
so that%
\begin{equation*}
\frac{\partial }{\partial \mathbf{\theta }^{\prime }}\mathbf{\hat{\alpha}}%
_{k,T}^{\dag }\left( \mathbf{\theta }_{T}\right) \rightarrow _{p}-\mathbf{S}%
_{k}^{\prime }\mathbf{\bar{\Lambda}}_{k}\left( \mathbf{\theta }_{0}\right)
\left( \mathbf{I}_{m}\otimes \mathbf{S}_{k}\mathbf{\alpha }_{k}^{0}\right) .
\end{equation*}%
Then, pooling all results, with $ \mathbf{\Sigma }_0 := \mathbf{\Sigma }\left( \mathbf{\theta }_{0},%
\mathbf{\alpha }_{k}^0 \right), $
\begin{eqnarray*}
&&T^{1/2}\left( \mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\hat{\theta%
}}_{w,T}^{\dag }\right) -\mathbf{\alpha }_{k}^{0}\right) \\
&=&-T^{1/2}\mathbf{S}_{k}^{\prime }\mathbf{\bar{\Lambda}}_{k}\left( \mathbf{%
\theta }_{0}\right) \left( \mathbf{I}_{m}\otimes \mathbf{S}_{k}\mathbf{%
\alpha }_{k}^{0}\right) \mathbf{\Sigma }^{-1}_0 \sum_{h=3}^{4}\frac{w_{h}}{T^{h-1}}\left(
\mathbf{I}_{m}\otimes \mathbf{\alpha }_{h}^{0\prime }\mathbf{S}_{hk}^{\prime
}\right) \sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{ \mathbf{B}%
_{h}^{\ast }\left( \mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }%
_{0}\right) \left \{ \mathbb{I}_{h}^{\mathbf{\varepsilon }}(\mathbf{\lambda }%
_{\mathbf{j}})-E\left[ \mathbb{I}_{h}^{\mathbf{\varepsilon }}(\mathbf{%
\lambda }_{\mathbf{j}})\right] \right \} \right \} \\
&&-\frac{1}{T^{1/2}}\mathbf{S}_{k}^{\prime }\mathbf{\bar{\Lambda}}_{k}\left(
\mathbf{\theta }_{0}\right) \left( \mathbf{I}_{m}\otimes \mathbf{S}_{k}%
\mathbf{\alpha }_{k}^{0}\right) \mathbf{\Sigma }^{-1}_0 \left( \mathbf{I}_{m}\otimes \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}%
\left( \mathbf{I}_{d}\right) \right) ^{\prime }\sum_{\lambda _{j}}\func{Re}%
\left \{ \mathbf{B}_{2}^{\ast }\left( \lambda _{j}\mathbf{;\theta }%
_{0}\right) \left( \mathbb{I}_{2}^{\mathbf{\varepsilon }}(\lambda _{j})-E%
\left[ \mathbb{I}_{2}^{\mathbf{\varepsilon }}(\lambda _{j})\right] \right)
\right \} \\
&&+\frac{T^{1/2}}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\mathbf{S}%
_{k}^{\prime }\func{Re}\left \{ \mathbb{I}_{k}^{\mathbf{\varepsilon }}(%
\mathbf{\lambda }_{\mathbf{j}})-E\left[ \mathbb{I}_{k}^{\mathbf{\varepsilon }%
}(\mathbf{\lambda }_{\mathbf{j}})\right] \right \} +o_{p}\left( 1\right) \\
&=&T^{1/2}\sum_{h=3}^{4}\frac{1}{T^{h-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}%
}}\func{Re}\left \{ \mathbf{S}_{k}^{\prime }\mathbf{D}_{k,h}^{\ast }\left(
\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0}\right) \left \{ \mathbb{I%
}_{h}^{\mathbf{\varepsilon }}(\mathbf{\lambda }_{\mathbf{j}})-E\left[
\mathbb{I}_{h}^{\mathbf{\varepsilon }}(\mathbf{\lambda }_{\mathbf{j}})\right]
\right \} \right \} \\
&&+\frac{1}{T^{1/2}}\sum_{\lambda _{j}}\func{Re}\left \{ \mathbf{S}%
_{k}^{\prime }\mathbf{D}_{k,2}^{\ast }\left( \lambda _{j}\mathbf{;\theta }%
_{0}\right) \left( \mathbb{I}_{2}^{\mathbf{\varepsilon }}(\lambda _{j})-E%
\left[ \mathbb{I}_{2}^{\mathbf{\varepsilon }}(\lambda _{j})\right] \right)
\right \} +o_{p}\left( 1\right)
\end{eqnarray*}%
and the result follows as Theorem~\ref{Th6} and Appendix~D for $k=2,3,$
while for $k=4$ we have to consider the extra term $\mathbf{\eta }_{t}$
coming for the decomposition of%
\begin{eqnarray*}
\frac{1}{T^{3}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\mathbb{I}_{4}^{\mathbf{%
\varepsilon }}(\mathbf{\lambda }_{\mathbf{j}}) &=&\frac{1}{T}\sum_{t=1}^{T}%
\mathbf{\varepsilon }_{t}^{\otimes 4}-\frac{1}{T^{2}}\sum_{t=1}^{T}%
\sum_{r=1}^{T}\left[
\begin{array}{c}
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{r} \\
+\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r} \\
+\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}%
\end{array}%
\right] +O_{p}\left( T^{-1}\right) \\
&=&\frac{1}{T}\sum_{t=1}^{T}\mathbf{\varepsilon }_{t}^{\otimes 4}-\frac{1}{T}%
\sum_{t=1}^{T}\left( \mathbf{1}_{6}^{\prime }\otimes \mathbf{I}%
_{d^{4}}\right) \mathbf{\eta }_{t}+E_{4}^{\left( a\right) }+E_{4}^{\left(
b\right) }+E_{4}^{\left( c\right) }+O_{p}\left( T^{-1}\right)
\end{eqnarray*}%
because the sum $\mathbf{\lambda }_{\mathbf{j}}$ does not include the terms $%
j_{a}=0\func{mod}T$, e.g., for $E_{4}^{\left( a\right) }:=E\left[ \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{r}\right] =E\left[ \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\right] \otimes E\left[
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{r}\right]
=\sum_{ab}\left( \mathbf{e}_{a}\otimes \mathbf{e}_{a}\otimes \mathbf{e}%
_{b}\otimes \mathbf{e}_{b}\right) :=E_{2}\otimes E_{2},$ say, for $t\neq r,$
we find that
\begin{eqnarray*}
&&\frac{1}{T^{2}}\sum_{t=1}^{T}\sum_{r=1}^{T}\left( \left( \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\pm E_{2}\right) \otimes
\left( \mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{r}\pm
E_{2}\right) \right) \\
&=&\frac{1}{T}\sum_{t=1}^{T}\left( \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}-E_{2}\right) \otimes \frac{1}{T}\sum_{r=1}^{T}\left(
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{r}-E_{2}\right)
+E_{4}^{\left( a\right) } \\
&&+\frac{1}{T^{2}}\sum_{t=1}^{T}\sum_{r=1}^{T}\left( \left( \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}-E_{2}\right) \otimes
E_{2}\right) +\frac{1}{T^{2}}\sum_{t=1}^{T}\sum_{r=1}^{T}\left( E_{2}\otimes
\left( \mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }%
_{r}-E_{2}\right) \right) \\
&=&\frac{1}{T^{2}}\sum_{t=1}^{T}\sum_{r=1}^{T}\left( \mathbf{\varepsilon }%
_{t}\otimes \mathbf{\varepsilon }_{t}\otimes E_{2}\right) +\frac{1}{T^{2}}%
\sum_{t=1}^{T}\sum_{r=1}^{T}\left( E_{2}\otimes \mathbf{\varepsilon }%
_{r}\otimes \mathbf{\varepsilon }_{r}\right) -E_{4}^{\left( a\right)
}+O_{p}\left( T^{-1}\right)
\end{eqnarray*}%
where $\left( \mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }%
_{t}\otimes E_{2}\right) =E_{r}\left[ \mathbf{\varepsilon }_{t}\otimes
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{r}\right] ,$ and we can proceed similarly for $E_{4}^{\left(
b\right) }:=E\left[ \mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }%
_{r}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}\right]
$ and $E_{4}^{\left( c\right) }:=E\left[ \mathbf{\varepsilon }_{t}\otimes
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t}\right] ,$ where these constant terms cancel with the
expectation of $\mathbb{I}_{h}^{\mathbf{\varepsilon }}(\mathbf{\lambda }_{%
\mathbf{j}})\ $and do not contribute to the variance of $\ \mathbf{\hat{%
\alpha}}_{4,T}^{\dag }.\ \ \Box $\bigskip }
\section{\protect \small Appendix C: Auxiliary results}
\begin{lemma}
{\small \label{LemmaA} Under the conditions of Theorem~\ref{Th5}, for any
signed permutation matrix $P_{d}$, if $\mathbf{\Psi }\left( e^{-i\lambda }%
\mathbf{;\theta }\right) =\mathbf{\Psi }\left( e^{-i\lambda }\mathbf{;\theta
}_{0}\right) P_{d}$ a.e., then $\mathbf{\alpha }_{3}\left( \mathbf{\theta }%
\right) =P_{d}^{\prime }\mathbf{\alpha }_{3}^{0}\ $and $\mathbf{\alpha }%
_{4}\left( \mathbf{\theta }\right) =P_{d}^{+\prime }\mathbf{\alpha }_{4}^{0}$%
.\bigskip }
\end{lemma}
{\small \noindent \textbf{Proof of Lemma~\ref{LemmaA}}. We can show using $%
P_{d}P_{d}^{\prime }=\mathbf{I}_{d}$ and $P_{d}^{+}P_{d}^{+\prime }=\mathbf{I%
}_{d}$
\begin{eqnarray*}
\mathbf{\alpha }_{k}\left( \mathbf{\theta }\right) &=&\left( \int_{\Pi
^{k-1}}\func{Re}\left \{ \mathbf{S}_{k}^{\prime }P_{d}^{\otimes k\prime }%
\mathbf{\Psi }_{k}^{\ast }(\mathbf{\lambda };\mathbf{\theta }_{0})\mathbf{%
\Psi }_{k}(\mathbf{\lambda };\mathbf{\theta }_{0})P_{d}^{\otimes k}\mathbf{S}%
_{k}\right \} d\mathbf{\lambda }\right) ^{-1} \\
&& \times \int_{\Pi ^{k-1}}\func{Re}\left \{ \mathbf{S}_{k}^{\prime
}P_{d}^{\otimes k\prime }\mathbf{\Psi }_{k}^{\ast }(\mathbf{\lambda };%
\mathbf{\theta }_{0})\mathbf{\Psi }_{k}(\mathbf{\lambda };\mathbf{\theta }%
_{0})\mathbf{S}_{k}\right \} d\mathbf{\lambda \ }P_{d}P_{d}^{\prime }\mathbf{%
\alpha }_{k}^{0}
\end{eqnarray*}
is equal to $P_{d}^{\prime }\mathbf{\alpha }_{k}^{0}$ or $P_{d}^{+\prime }%
\mathbf{\alpha }_{k}^{0}$ iff%
\begin{equation*}
P_{d}^{\otimes 3}\mathbf{S}_{3}=\mathbf{S}_{3}P_{d}\ \ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{or\ \ }P_{4}%
\mathbf{S}_{4}=\mathbf{S}_{4}P_{d}^{+},\ \ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{respectively.}
\end{equation*}%
Then writing $P_{d}=\left( s_{j_{1}}\mathbf{e}_{j_{1}},\ldots ,s_{j_{d}}%
\mathbf{e}_{j_{d}}\right) $ and $P_{d}=\left( \mathbf{e}_{j_{1}},\ldots ,%
\mathbf{e}_{j_{d}}\right) $ for $j_{a}\in \left \{ 1,\ldots ,d\right \} ,$ $%
j_{a}\neq j_{b}$ for $a\neq b,$ and $s_{j_{a}}=\pm 1,$ we notice that%
\begin{eqnarray*}
\mathbf{S}_{k}P_{d} &=&\left( \mathbf{e}_{1}^{\otimes k},\ldots ,\mathbf{e}%
_{d}^{\otimes k}\right) \left( s_{j_{1}}\mathbf{e}_{j_{1}},\ldots ,s_{j_{d}}%
\mathbf{e}_{j_{d}}\right) =\left( s_{j_{1}}\mathbf{e}_{j_{1}}^{\otimes
k},\ldots ,s_{j_{d}}\mathbf{e}_{j_{d}}^{\otimes k}\right) \\
\mathbf{S}_{k}P_{d}^{+} &=&\left( \mathbf{e}_{1}^{\otimes k},\ldots ,\mathbf{%
e}_{d}^{\otimes k}\right) \left( \mathbf{e}_{j_{1}},\ldots ,\mathbf{e}%
_{j_{d}}\right) =\left( \mathbf{e}_{j_{1}}^{\otimes k},\ldots ,\mathbf{e}%
_{j_{d}}^{\otimes k}\right)
\end{eqnarray*}%
is a reordering of the columns of $\mathbf{S}_{k}$ with the appropriate
sign, while%
\begin{equation*}
P_{d}^{\otimes k}\mathbf{S}_{k}=P_{d}^{\otimes k}\left( \mathbf{e}%
_{1}^{\otimes k},\ldots ,\mathbf{e}_{d}^{\otimes k}\right) =\left[ \left(
P_{d}\mathbf{e}_{1}\right) ^{\otimes k},\ldots ,\left( P_{d}\mathbf{e}%
_{d}\right) ^{\otimes k}\right] =\left[ s_{j_{1}}^{k}\mathbf{e}%
_{j_{1}}^{\otimes k},\ldots ,s_{j_{d}}^{k}\mathbf{e}_{j_{d}}^{\otimes k}%
\right] ,
\end{equation*}%
so that $P_{d}^{\otimes 3}\mathbf{S}_{3}=\left[ s_{j_{1}}\mathbf{e}%
_{j_{1}}^{\otimes 3},\ldots ,s_{j_{d}}\mathbf{e}_{j_{d}}^{\otimes 3}\right] =%
\mathbf{S}_{3}P_{d}$ and $P_{d}^{\otimes 4}\mathbf{S}_{4}=\left[ \mathbf{e}%
_{j_{1}}^{\otimes 4},\ldots ,\mathbf{e}_{j_{d}}^{\otimes 4}\right] =\mathbf{S%
}_{4}P_{d}^{+}.\ \ \Box $\bigskip }
\begin{lemma}
{\small \label{Hessian} Under the Assumptions of Theorem~\ref{Th6}, for $%
\tilde{\mathbf{\theta}}_{T}\rightarrow _{p}\mathbf{\theta }_{0},$ $k=3,4,$
\begin{eqnarray*}
\frac{\partial ^{2}}{\partial \mathbf{\theta }\partial \mathbf{\theta }%
^{\prime }}\mathcal{\hat{L}}_{k,T}^{\dag }\left( \mathbf{\theta }_{T}\right)
& \rightarrow _{p} &\left( \mathbf{I}_{m}\otimes \mathbf{S}_{k}\mathbf{%
\alpha }_{k}^{0}\right) ^{\prime }\mathbf{H}_{k}\left( \mathbf{\theta }%
_{0}\right) \left( \mathbf{I}_{m}\otimes \mathbf{S}_{k}\mathbf{\alpha }%
_{k}^{0}\right) \\
\frac{\partial ^{2}}{\partial \mathbf{\theta }\partial \mathbf{\theta }%
^{\prime }}\mathcal{L}_{2,T}\left( \mathbf{\theta }_{T}\right)
&\rightarrow_{p} &\left( \mathbf{I}_{m}\otimes \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}%
_{d}\right) \right) ^{\prime }\mathbf{H}_{2}\left( \mathbf{\theta }%
_{0}\right) \left( \mathbf{I}_{m}\otimes \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}%
_{d}\right) \right) .
\end{eqnarray*}
}
\end{lemma}
{\small \noindent \textbf{Proof of Lemma~\ref{Hessian}}. We give the proof
for $k=3,4,$ the case for $k=2$ is much simpler,%
\begin{eqnarray*}
&&\frac{\partial ^{2}}{\partial \mathbf{\theta }_{\ell }\partial \mathbf{%
\theta }_{p}}\mathcal{\hat{L}}_{k,T}^{\dag }\left( \mathbf{\theta }\right) \\
&=&\frac{2}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{
\left( \mathbf{\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })%
\mathbf{S}_{k}\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }%
\right) -\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}})\right) ^{\ast }%
\mathbf{W}_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\tilde{\theta}}_{T})%
\mathbf{\dot{\Psi}}_{k}^{\left( \ell ,p\right) }(\mathbf{\lambda }_{\mathbf{j%
}};\mathbf{\theta })\mathbf{S}_{k}\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left(
\mathbf{\theta }\right) \right \} \\
&&+\frac{2}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{
\left( \mathbf{\dot{\Psi}}_{k}^{\left( p\right) }(\mathbf{\lambda }_{\mathbf{%
j}};\mathbf{\theta })\mathbf{S}_{k}\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left(
\mathbf{\theta }\right) \right) ^{\ast }\mathbf{W}_{k}(\mathbf{\lambda }_{%
\mathbf{j}};\mathbf{\tilde{\theta}}_{T})\mathbf{\dot{\Psi}}_{k}^{\left( \ell
\right) }(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })\mathbf{S}_{k}%
\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }\right) \right \}
\\
&&+\frac{2}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{
\left( \mathbf{\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })%
\mathbf{S}_{k}\frac{\partial }{\partial \mathbf{\theta }_{p}}\mathbf{\hat{%
\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }\right) \right) ^{\ast }\mathbf{%
W}_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\tilde{\theta}}_{T})\mathbf{%
\dot{\Psi}}_{k}^{\left( \ell \right) }(\mathbf{\lambda }_{\mathbf{j}};%
\mathbf{\theta })\mathbf{S}_{k}\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left(
\mathbf{\theta }\right) \right \} \\
&&+\frac{2}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{
\left( \mathbf{\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })%
\mathbf{S}_{k}\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }%
\right) -\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}})\right) ^{\ast }%
\mathbf{W}_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\tilde{\theta}}_{T})%
\mathbf{\dot{\Psi}}_{k}^{\left( \ell \right) }(\mathbf{\lambda }_{\mathbf{j}%
};\mathbf{\theta })\mathbf{S}_{k}\frac{\partial }{\partial \mathbf{\theta }%
_{p}}\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }\right)
\right \} \\
&&+\frac{2}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{
\left( \mathbf{\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })%
\mathbf{S}_{k}\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }%
\right) -\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}})\right) ^{\ast }%
\mathbf{W}_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\tilde{\theta}}_{T})%
\mathbf{\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })\mathbf{S}%
_{k}\frac{\partial ^{2}}{\partial \mathbf{\theta }_{\ell }\partial \mathbf{%
\theta }_{p}}\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }%
\right) \right \} \\
&&+\frac{2}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{
\left( \mathbf{\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })%
\mathbf{S}_{k}\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }%
\right) -\mathbb{I}_{k}(\mathbf{\lambda }_{\mathbf{j}})\right) ^{\ast }%
\mathbf{W}_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\tilde{\theta}}_{T})%
\mathbf{\dot{\Psi}}_{k}^{(p)}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }%
)\mathbf{S}_{k}\frac{\partial }{\partial \mathbf{\theta }_{\ell }}\mathbf{%
\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }\right) \right \} \\
&&+\frac{2}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{
\left( \mathbf{\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })%
\mathbf{S}_{k}\frac{\partial }{\partial \mathbf{\theta }_{p}}\mathbf{\hat{%
\alpha}}_{k,T}^{\dag }\left( \mathbf{\theta }\right) \right) ^{\ast }\mathbf{%
W}_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\tilde{\theta}}_{T})\mathbf{%
\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta })\mathbf{S}_{k}%
\frac{\partial }{\partial \mathbf{\theta }_{\ell }}\mathbf{\hat{\alpha}}%
_{k,T}^{\dag }\left( \mathbf{\theta }\right) \right \} \\
&&+\frac{2}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{
\left( \mathbf{\dot{\Psi}}_{k}^{\left( p\right) }(\mathbf{\lambda }_{\mathbf{%
j}};\mathbf{\theta })\mathbf{S}_{k}\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left(
\mathbf{\theta }\right) \right) ^{\ast }\mathbf{W}_{k}(\mathbf{\lambda }_{%
\mathbf{j}};\mathbf{\tilde{\theta}}_{T})\mathbf{\Psi }_{k}(\mathbf{\lambda }%
_{\mathbf{j}};\mathbf{\theta })\mathbf{S}_{k}\frac{\partial }{\partial
\mathbf{\theta }_{\ell }}\mathbf{\hat{\alpha}}_{k,T}^{\dag }\left( \mathbf{%
\theta }\right) \right \}
\end{eqnarray*}%
where the limits of lines 1, 4, 5 and 6 of the rhs do not contribute
asymptotically when evaluated at $\tilde{\mathbf{\theta }}_{T} \rightarrow_{p} \mathbf{\mathbf{\theta }}_{0}$ and the other ones converge to the
probability limit of%
\begin{eqnarray*}
&&\frac{\partial ^{2}}{\partial \mathbf{\theta }_{\ell }\partial \mathbf{%
\theta }_{p}}\mathcal{\hat{L}}_{k,T}^{\dag }\left( \mathbf{\theta }%
_{0}\right) \\
&=&\frac{2}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{
\left( \mathbf{\dot{\Psi}}_{k}^{(p)}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{%
\theta }_{0})\mathbf{S}_{k}\mathbf{\alpha }_{0}\right) ^{\ast }\mathbf{W}%
_{T}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0})\mathbf{\dot{\Psi}}%
_{k}^{\left( \ell \right) }(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }%
_{0})\mathbf{S}_{k}\mathbf{\alpha }_{0}\right \} \\
&&+\frac{2}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{
\left( \mathbf{\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }%
_{0})\mathbf{S}_{k}\left( -\mathbf{S}_{k}^{\prime }\mathbf{\bar{\Lambda}}%
_{k}^{\left( p\right) }\left( \mathbf{\theta }_{0}\right) \mathbf{S}_{k}%
\mathbf{\alpha }_{0}\right) \right) ^{\ast }\mathbf{W}_{k}(\mathbf{\lambda }%
_{\mathbf{j}};\mathbf{\theta }_{0})\mathbf{\dot{\Psi}}_{k}^{\left( \ell
\right) }(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0})\mathbf{S}_{k}%
\mathbf{\alpha }_{0}\right \} \\
&&+\frac{2}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{
\left( \mathbf{\Psi }_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }%
_{0})\mathbf{S}_{k}\left( -\mathbf{S}_{k}^{\prime }\mathbf{\bar{\Lambda}}%
_{k}^{\left( p\right) }\left( \mathbf{\theta }_{0}\right) \mathbf{S}_{k}%
\mathbf{\alpha }_{0}\right) \right) ^{\ast }\mathbf{W}_{k}(\mathbf{\lambda }%
_{\mathbf{j}};\mathbf{\theta }_{0})\mathbf{\Psi }_{k}(\mathbf{\lambda }_{%
\mathbf{j}};\mathbf{\theta }_{0})\mathbf{S}_{k}\left( -\mathbf{S}%
_{k}^{\prime }\mathbf{\bar{\Lambda}}_{k}^{\left( \ell \right) }\left(
\mathbf{\theta }_{0}\right) \mathbf{S}_{k}\mathbf{\alpha }_{0}\right) \right
\} \\
&&+\frac{2}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\func{Re}\left \{
\left( \mathbf{\dot{\Psi}}_{k}^{(p)}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{%
\theta }_{0})\mathbf{S}_{k}\mathbf{\alpha }_{0}\right) ^{\ast }\mathbf{W}%
_{k}(\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0})\mathbf{\Psi }_{k}(%
\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0})\mathbf{S}_{k}\left( -%
\mathbf{S}_{k}^{\prime }\mathbf{\bar{\Lambda}}_{k}^{\left( \ell \right)
}\left( \mathbf{\theta }_{0}\right) \mathbf{S}_{k}\mathbf{\alpha }%
_{0}\right) \right \} +o_{p}\left( 1\right) ,
\end{eqnarray*}
which converges to $2\mathbf{\alpha }_{k}^{0\prime }\mathbf{S}_{k}^{\prime
}\left \{ \mathbf{H}_{k}\left( \mathbf{\theta }_{0} \right) \right \}
_{\left( p,\ell \right) }\mathbf{S}_{k}\mathbf{\alpha }_{k}^0$ because%
\begin{eqnarray*}
&&\left( 2\pi \right) ^{k-1}\left \{ \mathbf{H}_{k}\left( \mathbf{\theta }%
\right) \right \} _{\left( p,\ell \right) } \\
&=&\int \func{Re}\left \{ \mathbf{\dot{\Psi}}_{k}^{(p)}(\mathbf{\lambda };%
\mathbf{\theta })^{\ast }\mathbf{W}_{k}(\mathbf{\lambda };\mathbf{\theta })%
\mathbf{\dot{\Psi}}_{k}^{\left( \ell \right) }(\mathbf{\lambda };\mathbf{%
\theta })\right \} d\mathbf{\lambda } \\
&&-\int \func{Re}\left \{ \mathbf{\bar{\Lambda}}_{k}^{\left( p\right)
}\left( \mathbf{\theta }\right) ^{\ast }\mathbf{S}_{k}\mathbf{S}_{k}^{\prime
}\mathbf{\Psi }_{k}^{\ast }(\mathbf{\lambda };\mathbf{\theta })\mathbf{W}%
_{k}(\mathbf{\lambda };\mathbf{\theta })\mathbf{\dot{\Psi}}_{k}^{\left( \ell
\right) }(\mathbf{\lambda };\mathbf{\theta })\right \} d\mathbf{\lambda } \\
&&+\int \func{Re}\left \{ \mathbf{\bar{\Lambda}}_{k}^{\left( p\right)
}\left( \mathbf{\theta }\right) ^{\ast }\mathbf{S}_{k}\mathbf{S}_{k}^{\prime
}\mathbf{\Psi }_{k}^{\ast }(\mathbf{\lambda };\mathbf{\theta })\mathbf{W}%
_{k}(\mathbf{\lambda };\mathbf{\theta })\mathbf{\Psi }_{k}(\mathbf{\lambda };%
\mathbf{\theta })\mathbf{S}_{k}\mathbf{S}_{k}^{\prime }\mathbf{\bar{\Lambda}}%
_{k}^{\left( \ell \right) }\left( \mathbf{\theta }\right) \right \} d\mathbf{%
\lambda } \\
&&-\int \func{Re}\left \{ \mathbf{\dot{\Psi}}_{k}^{(p)}(\mathbf{\lambda };%
\mathbf{\theta })^{\ast }\mathbf{W}_{k}(\mathbf{\lambda };\mathbf{\theta })%
\mathbf{\Psi }_{k}(\mathbf{\lambda };\mathbf{\theta })\mathbf{S}_{k}\mathbf{S%
}_{k}^{\prime }\mathbf{\bar{\Lambda}}_{k}^{\left( \ell \right) }\left(
\mathbf{\theta }\right) \right \} d\mathbf{\lambda } \\
&=&\int \func{Re}\left \{ \left( \mathbf{\dot{\Psi}}_{k}^{(p)}(\mathbf{%
\lambda };\mathbf{\theta })-\mathbf{\Psi }_{k}(\mathbf{\lambda };\mathbf{%
\theta })\mathbf{S}_{k}\mathbf{S}_{k}^{\prime }\mathbf{\bar{\Lambda}}%
_{k}^{\left( p\right) }\left( \mathbf{\theta }\right) \right) ^{\ast }%
\mathbf{W}_{k}(\mathbf{\lambda };\mathbf{\theta })\left( \mathbf{\dot{\Psi}}%
_{k}^{(\ell )}(\mathbf{\lambda };\mathbf{\theta })-\mathbf{\Psi }_{k}(%
\mathbf{\lambda };\mathbf{\theta })\mathbf{S}_{k}\mathbf{S}_{k}^{\prime }%
\mathbf{\bar{\Lambda}}_{k}^{\left( \ell \right) }\left( \mathbf{\theta }%
\right) \right) \right \} d\mathbf{\lambda } \\
&=&\int \func{Re}\left \{ \left( \mathbf{\Psi }_{k}^{-1}(\mathbf{\lambda };%
\mathbf{\theta })\mathbf{\dot{\Psi}}_{k}^{(p)}(\mathbf{\lambda };\mathbf{%
\theta })-\mathbf{S}_{k}\mathbf{S}_{k}^{\prime }\mathbf{\bar{\Lambda}}%
_{k}^{\left( p\right) }\left( \mathbf{\theta }\right) \right) ^{\ast }\left(
\mathbf{\Psi }_{k}^{-1}(\mathbf{\lambda };\mathbf{\theta })\mathbf{\dot{\Psi}%
}_{k}^{(\ell )}(\mathbf{\lambda };\mathbf{\theta })-\mathbf{S}_{k}\mathbf{S}%
_{k}^{\prime }\mathbf{\bar{\Lambda}}_{k}^{\left( \ell \right) }\left(
\mathbf{\theta }\right) \right) \right \} d\mathbf{\lambda .}
\end{eqnarray*}
$\Box $\bigskip}
\section{\protect \small Appendix D: Asymptotics of averages of higher order
periodograms}
{\small Define for $k=2,3,4,$ the average of periodograms of order }$k,$%
{\small
\begin{equation*}
\mathbf{Z}_{k,T}:=\frac{T^{1/2}}{T^{k-1}}\sum_{\mathbf{\lambda }_{\mathbf{j}%
}}\func{Re}\left \{ \mathbf{B}_{k}^{\ast }\left( \mathbf{\lambda }_{\mathbf{j%
}}\right) \left( \mathbb{I}_{k}^{\varepsilon }(\mathbf{\lambda }_{\mathbf{j}%
})-E\left[ \mathbb{I}_{k}^{\varepsilon }(\mathbf{\lambda }_{\mathbf{j}})%
\right] \right) \right \}
\end{equation*}%
where
\begin{equation*}
\mathbb{I}_{k}^{\mathbf{\varepsilon }}(\mathbf{\lambda })=\frac{1}{T}w_{T}^{%
\mathbf{\varepsilon }}\left( -\lambda _{1}-\cdots -\lambda _{k-1}\right)
\otimes w_{T}^{\mathbf{\varepsilon }}\left( \lambda _{k-1}\right) \otimes
\cdots \otimes w_{T}^{\mathbf{\varepsilon }}\left( \lambda _{1}\right) ,\ \
w_{T}^{\mathbf{\varepsilon }}\left( \lambda \right)
=\sum_{t=1}^{T}e^{-i\lambda t}\mathbf{\varepsilon }_{t}.
\end{equation*}%
is the $k$-periodogram of shocks satisfying Assumptions 1$(2k)$ and 3$(k)$.
Denote $\mathbf{\dot{\Lambda}}^{(\ell )}\left( \lambda ;\mathbf{%
\theta }\right) :=\mathbf{\Psi }^{-1}\left( e^{-i\lambda }\mathbf{;\theta }%
\right) \mathbf{\dot{\Psi}}^{(\ell )}\left( e^{-i\lambda }\mathbf{;\theta }%
\right) $ and $\mathbf{\dot{\Lambda}}\left( \lambda ;\mathbf{\theta }\right)
:=\left( \mathbf{\dot{\Lambda}}^{(1)}\left( \lambda ;\mathbf{\theta }\right)
,\ldots ,\mathbf{\dot{\Lambda}}^{(m)}\left( \lambda ;\mathbf{\theta }\right)
\right) ,$ and set for $j=0,\pm 1,\ldots ,$
\begin{equation*}
\mathbf{\mathbf{C}}\left( j\right) :=\left( 2\pi \right) ^{-1}\int_{-\pi
}^{\pi }\mathbf{\dot{\Lambda}}^{\ast }\left( \lambda ;\mathbf{\theta }%
_{0}\right) \exp \left( -ij\lambda \right) d\lambda
\end{equation*}%
so for $\mathbf{C}_{k}\left( 0\right) :=\left( 2\pi \right) ^{1-k}\int_{\Pi
^{k-1}}\mathbf{B}_{k}^{\ast }\left( \mathbf{\lambda };\mathbf{\theta }%
_{0}\right) d\mathbf{\lambda }$ for $k=2,3,4,$ we find that $\mathbf{C}%
_{2}\left( 0\right) =\mathbf{I}_{d}\otimes \mathbf{\mathbf{C}}\left(
0\right) +\mathbf{\mathbf{C}}\left( 0\right) \otimes \mathbf{I}_{d}$ while%
\begin{eqnarray*}
\mathbf{C}_{3}\left( 0\right) &=&\left[ \mathbf{I}_{d}\otimes \mathbf{I}%
_{d}\otimes \mathbf{\mathbf{C}}\left( 0\right) +\mathbf{I}_{d}\otimes
\mathbf{\mathbf{C}}\left( 0\right) \otimes \mathbf{I}_{d}+\mathbf{\mathbf{C}}%
\left( 0\right) \otimes \mathbf{I}_{d}\otimes \mathbf{I}_{d}\right] \left(
\mathbf{I}_{d^{3}}-\mathbf{S}_{3}\mathbf{S}_{3}^{\prime }\right) \\
\mathbf{C}_{4}\left( 0\right) &=&\left[ \mathbf{I}_{d}\otimes \mathbf{I}%
_{d}\otimes \mathbf{I}_{d}\otimes \mathbf{\mathbf{C}}\left( 0\right) +%
\mathbf{I}_{d}\otimes \mathbf{I}_{d}\otimes \mathbf{\mathbf{C}}\left(
0\right) \otimes \mathbf{I}_{d}+\cdots +\mathbf{\mathbf{C}}\left( 0\right)
\otimes \mathbf{I}_{d}\otimes \mathbf{I}_{d}\otimes \mathbf{I}_{d}\right]
\left( \mathbf{I}_{d^{4}}-\mathbf{S}_{4}\mathbf{S}_{4}^{\prime }\right) ,\ \
\ \
\end{eqnarray*}%
and for the block row matrices, $j\neq 0,$
\begin{equation*}
\mathbf{\mathbf{C}}_{k}\left( j\right) :=\left \{ \left( 2\pi \right)
^{1-k}\int_{\Pi ^{k-1}}\mathbf{B}_{k,a}^{\ast }\left( \mathbf{\lambda };%
\mathbf{\theta }_{0}\right) \exp \left( -ij\lambda _{a}\right) d\mathbf{%
\lambda }\right \} _{a=1,\ldots ,k}\mathbf{,}
\end{equation*}%
we find that%
\begin{eqnarray*}
\mathbf{C}_{2}\left( j\right) &=&\left[ \mathbf{I}_{d}\otimes \mathbf{%
\mathbf{C}}\left( j\right) \ \QATOP{{}}{{}}\ \mathbf{\mathbf{C}}\left(
j\right) \otimes \mathbf{I}_{d}\right] \\
\mathbf{C}_{3}\left( j\right) &=&\left[ \mathbf{I}_{d}\otimes \mathbf{I}%
_{d}\otimes \mathbf{\mathbf{C}}\left( j\right) \ \QATOP{{}}{{}}\ \mathbf{I}%
_{d}\otimes \mathbf{\mathbf{C}}\left( j\right) \otimes \mathbf{I}_{d}\
\QATOP{{}}{{}}\ \mathbf{\mathbf{C}}\left( j\right) \otimes \mathbf{I}%
_{d}\otimes \mathbf{I}_{d}\right] \\
\mathbf{C}_{4}\left( j\right) &=&\left[ \mathbf{I}_{d}\otimes \mathbf{I}%
_{d}\otimes \mathbf{I}_{d}\otimes \mathbf{\mathbf{C}}\left( j\right) \
\QATOP{{}}{{}}\ \mathbf{I}_{d}\otimes \mathbf{I}_{d}\otimes \mathbf{\mathbf{C%
}}\left( j\right) \otimes \mathbf{I}_{d}\QATOP{{}}{{}}\ \cdots \ \QATOP{{}}{%
{}}\mathbf{\mathbf{C}}\left( j\right) \otimes \mathbf{I}_{d}\otimes \mathbf{I%
}_{d}\otimes \mathbf{I}_{d}\right] .
\end{eqnarray*}%
The factors $\left( \mathbf{I}_{d^{k}}-\mathbf{S}_{k}\mathbf{S}_{k}^{\prime }\right) $that appear in $\mathbf{C}_{3}\left( 0\right) $, $\mathbf{C}_{4}\left(
0\right) $ and $\mathbf{\Phi }_{ab}^{0}$, $a$ or $b>2$, indicate that the ICA
restriction is imposed when estimating higher order cumulants, so not all
elements in the cumulant array v$\mathbf{\kappa}_k$ are estimated. In an
unrestricted estimation, based e.g. on Assumption~2(3), $\mathbf{S}_{k}$
would be replaced by $\mathbf{I}_{d^{k}}$ so the contributions from $\Phi
_{ab}^{0}\left( \mathbf{\theta }_{0};\mathbf{C}\right) $ for $a>2\ $or $b>2$
cancel out.
For $k=2,$
\begin{eqnarray*}
\mathbf{B}_{2,1}^{\left( \ell \right) }\left( \lambda ;\mathbf{\theta }%
\right) &=&\mathbf{B}_{2,1}^{\left( \ell \right) }\left( \lambda ;\mathbf{%
\theta }\right) =\mathbf{I}_{d}\otimes \left \{ \mathbf{\dot{\Lambda}}%
^{(\ell )}\left( \lambda \mathbf{;\theta }\right) \right \} \\
\mathbf{B}_{2,2}^{\left( \ell \right) }\left( \lambda ;\mathbf{\theta }%
\right) &=&\mathbf{B}_{2,2}^{\left( \ell \right) }\left( \lambda ;\mathbf{%
\theta }\right) =\left \{ \mathbf{\dot{\Lambda}}^{(\ell )}\left( \lambda
\mathbf{;\theta }\right) \right \} \otimes \mathbf{I}_{d}
\end{eqnarray*}%
because
$
\mathbf{\Psi }_{2}^{-1}(\lambda ;\mathbf{\theta })\mathbf{\dot{\Psi}}%
_{2}^{\left( \ell \right) }(\lambda ;\mathbf{\theta })=\left( \mathbf{I}%
_{d}\otimes \mathbf{\dot{\Lambda}}^{(\ell )}\left( \lambda \mathbf{;\theta }%
\right) \right) +\left( \mathbf{\dot{\Lambda}}^{(\ell )}\left( \lambda
\mathbf{;\theta }\right) \otimes \mathbf{I}_{d}\right) ,\
$
and
\begin{eqnarray*}
\mathbb{I}_{2}^{\mathbf{\varepsilon }}(\lambda ) &=&\frac{1}{T}w_{T}\left(
-\lambda \right) \otimes w_{T}\left( \lambda \right) =\frac{1}{T}%
\sum_{t,r=1}^{T}\exp \left( -i\left( t-r\right) \lambda \right) \left(
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}\right) \\
&=&\frac{1}{T}\sum_{t=1}^{T}\left( \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}\right) +\frac{1}{T}\sum_{t,r=1}^{T}%
\sum_{r=1}^{t-1}A_{t,r}^{\left[ 2\right] }\left( \lambda _{j}\right) \left[
\begin{array}{c}
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}%
\end{array}%
\right]
\end{eqnarray*}%
where
\begin{equation*}
A_{t,r}^{\left[ 2\right] }\left( \lambda _{j}\right) :=\left[ \exp \left \{
-i\left( t-r\right) \lambda _{j}\right \} \ \QATOP{{}}{{}}\ \ \exp \left \{
-i\left( r-t\right) \lambda _{j}\right \} \right] .
\end{equation*}%
Then
\begin{equation*}
\mathbf{Z}_{2,T}=\frac{T^{1/2}}{T}\sum_{\lambda _{j}}\func{Re}\left \{
\mathbf{B}_{2}^{\ast }(\lambda _{j})\left( \mathbb{I}_{2}^{\varepsilon
}(\lambda _{j})-E\left[ \mathbb{I}_{2}^{\varepsilon }(\lambda _{j})\right]
\right) \right \} =\sum_{t=1}^{T}Z_{2,t}
\end{equation*}%
where $Z_{2,t}$ is a martingale difference sequence (MDS)
\begin{eqnarray*}
Z_{2,t} &:= &\frac{1}{T^{1/2}}\mathbf{C}_{T}^{\left[ 2,0\right] }\left \{
\left( \mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\right) -%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}_{d}\right) \right \} \\
&&+\frac{1}{T^{1/2}}\sum_{r=1}^{t-1}\left[ \mathbf{C}_{T}^{\left[ 2,1\right]
}\left( r-t\right) \ \ \ \mathbf{C}_{T}^{\left[ 2,2\right] }\left(
r-t\right) \right] \left[ \QATOP{\mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r}}{\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}%
}\right] \\
&&+\frac{1}{T^{1/2}}\sum_{r=1}^{t-1}\left[ \mathbf{C}_{T}^{\left[ 2,1\right]
}\left( t-r\right) \ \ \ \ \mathbf{C}_{T}^{\left[ 2,2\right] }\left(
t-r\right) \right] \left[ \QATOP{\mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t}}{\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}%
}\right] \\
&=&\frac{1}{T^{1/2}}\mathbf{C}_{T}^{\left[ 2,0\right] }\left \{ \left(
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\right) -\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}%
\left( \mathbf{I}_{d}\right) \right \} +\frac{1}{T^{1/2}}\sum_{r=1}^{t-1}%
\mathbf{C}_{T}^{\left[ 2\right] }\left( r-t\right) \mathbf{\varepsilon }%
_{t,r}^{\left[ 2\right] }+\frac{1}{T^{1/2}}\sum_{r=1}^{t-1}\mathbf{C}_{T}^{%
\left[ 2\right] }\left( t-r\right) \mathbf{\varepsilon }_{r,t}^{\left[ 2%
\right] }
\end{eqnarray*}%
with%
\begin{equation*}
\mathbf{C}_{T}^{\left[ 2,0\right] }:=\frac{1}{T}\sum_{\lambda _{j}}\func{Re}%
\mathbf{B}_{2}^{\ast }\left( \lambda _{j};\mathbf{\theta }\right)
\rightarrow \mathbf{C}_{2}\left( 0\right) :=\frac{1}{2\pi }\int_{-\pi }^{\pi
}\mathbf{B}_{2}^{\ast }\left( \lambda ;\mathbf{\theta }\right) d\lambda ,
\end{equation*}%
which in general is different from zero since $\mathbf{B}_{2}$ is not
centered as the scaling is incorporated in $\mathbf{\Psi }$ and second order
cumulants (or covariances) are not estimated separately due to the
normalization assumption, and
\begin{eqnarray*}
\mathbf{C}_{T}^{\left[ 2\right] }\left( r-t\right) &:=&\left[ \mathbf{C}%
_{T}^{\left[ 2,1\right] }\left( r-t\right) \ \ \ \ \mathbf{C}_{T}^{\left[ 2,2%
\right] }\left( r-t\right) \right] \\
&:=&\left[ \frac{1}{T}\sum_{\lambda _{j}}\func{Re}\left \{ \mathbf{B}%
_{2,1}^{\ast }\left( \lambda _{j};\mathbf{\theta }\right) \exp \left(
i\left( t-r\right) \lambda _{j}\right) \right \} \ \ \ \frac{1}{T}%
\sum_{\lambda _{j}}\func{Re}\left \{ \mathbf{B}_{2,2}^{\ast }\left( \lambda
_{j};\mathbf{\theta }\right) \exp \left( -i\left( t-r\right) \lambda
_{j}\right) \right \} \right] \\
&=&\mathbf{\mathbf{C}}_{2}\left( r-t\right) +O\left( T^{-1}\right) ,
\end{eqnarray*}%
as $T\rightarrow \infty $ where%
\begin{eqnarray*}
\mathbf{C}_{2}\left( j\right) &:=&\left[ \mathbf{C}_{\left[ 2,1\right]
}\left( j\right) \ \ \ \mathbf{C}_{\left[ 2,2\right] }\left( j\right) \right]
:=\left[ \left( \mathbf{I}_{d}\otimes \mathbf{\mathbf{C}}\left( j\right)
\right) \ \ \ \left( \mathbf{\mathbf{C}}\left( j\right) \otimes \mathbf{I}%
_{d}\right) \right] \\
\mathbf{\mathbf{C}}\left( j\right) &:=&\frac{1}{2\pi }\int_{-\pi }^{\pi }%
\mathbf{\dot{\Lambda}}^{\ast }\left( \lambda \mathbf{;\theta }\right) \exp
\left( -ij\lambda \right) d\lambda .
\end{eqnarray*}%
Then we can write $\mathbf{Z}_{2,T}=\sum_{t=1}^{T}Z_{2,t}^{0}+o_{p}%
\left( 1\right) ,$ where%
\begin{equation*}
Z_{2,t}^{0}:=\frac{1}{T^{1/2}}\sum_{r=1}^{t-1}\left[ \mathbf{C}_{2}\left(
0\right) \ \ \ \mathbf{C}_{2}\left( r-t\right) \ \ \ \mathbf{C}_{2}\left(
t-r\right) \right] \left[
\begin{array}{c}
\frac{1}{t-1}\left( \mathbf{\varepsilon }_{t}^{\otimes 2}-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left(
\mathbf{I}_{d}\right) \right) \\
\mathbf{\varepsilon }_{t,r}^{\left[ 2\right] } \\
\mathbf{\varepsilon }_{r,t}^{\left[ 2\right] }%
\end{array}%
\right]
\end{equation*}%
where%
\begin{equation*}
\mathbf{\varepsilon }_{t,r}:=\left[
\begin{array}{c}
\frac{1}{t-1}\left( \mathbf{\varepsilon }_{t}^{\otimes 2}-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left(
\mathbf{I}_{d}\right) \right) \\
\mathbf{\varepsilon }_{t,r}^{\left[ 2\right] } \\
\mathbf{\varepsilon }_{r,t}^{\left[ 2\right] }%
\end{array}%
\right] :=\left[
\begin{array}{c}
\frac{1}{t-1}\left( \mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }%
_{t}-\RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}\left( \mathbf{I}_{d}\right) \right) \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r} \\
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}%
\end{array}%
\right]
\end{equation*}%
so that exploiting symmetries, $\mathbb{C}\left[ \mathbf{\varepsilon }%
_{t}^{\otimes 2},\mathbf{\varepsilon }_{r,t}^{\left[ 2\right] }\right] =%
\mathbb{C}\left[ \mathbf{\varepsilon }_{t}^{\otimes 2},\mathbf{\varepsilon }%
_{t,r}^{\left[ 2\right] }\right] =0,$ $\mathbb{V}\left[ \mathbf{Z}_{2,T}%
\right] =$}$\sum_{t=1}^{T}\mathbb{V}\left[ Z_{2,t}^{0}\right] ${\small \ can
be approximated by
\begin{eqnarray*}
&&\mathbf{C}_{2}\left( 0\right) V_{0}\mathbf{C}_{2}^{\prime }\left( 0\right)
\\
&&+\frac{1}{T}\sum_{t=1}^{T}\sum_{r=1}^{t-1}\left[ \mathbf{C}^{\left[ 2%
\right] }\left( r-t\right) \ \ \ \mathbf{C}^{\left[ 2\right] }\left(
t-r\right) \right] \left[
\begin{array}{cc}
\mathbb{V}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 2\right] }\right] &
\mathbb{C}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 2\right] },\mathbf{%
\varepsilon }_{r,t}^{\left[ 2\right] }\right] \\
\mathbb{C}\left[ \mathbf{\varepsilon }_{r,t}^{\left[ 2\right] },\mathbf{%
\varepsilon }_{t,r}^{\left[ 2\right] }\right] & \mathbb{V}\left[ \mathbf{%
\varepsilon }_{r,t}^{\left[ 2\right] }\right]%
\end{array}%
\right] \left[
\begin{array}{c}
\mathbf{C}^{\left[ 2\right] }\left( r-t\right) ^{\prime } \\
\mathbf{C}^{\left[ 2\right] }\left( t-r\right) ^{\prime }%
\end{array}%
\right] \\
&=&\mathbf{C}_{2}\left( 0\right) V_{0}\mathbf{C}_{2}^{\prime }\left(
0\right) +\frac{1}{T}\sum_{t=1}^{T}\sum_{r=1}^{t-1}\mathbf{C}^{\left[ 2%
\right] }\left( r-t\right) \mathbb{V}\left[ \mathbf{\varepsilon }_{t,r}^{%
\left[ 2\right] }\right] \mathbf{C}^{\left[ 2\right] }\left( r-t\right)
^{\prime }+\mathbf{C}^{\left[ 2\right] }\left( t-r\right) \mathbb{V}\left[
\mathbf{\varepsilon }_{r,t}^{\left[ 2\right] }\right] \mathbf{C}^{\left[ 2%
\right] }\left( t-r\right) ^{\prime } \\
&&+ \mathbf{C}^{\left[ 2\right] }\left( r-t\right) \mathbb{C}\left[ \mathbf{%
\varepsilon }_{t,r}^{\left[ 2\right] },\mathbf{\varepsilon }_{r,t}^{\left[ 2%
\right] }\right] \mathbf{C}^{\left[ 2\right] }\left( t-r\right) ^{\prime }+%
\mathbf{C}^{\left[ 2\right] }\left( t-r\right) \mathbb{C}\left[ \mathbf{%
\varepsilon }_{r,t}^{\left[ 2\right] },\mathbf{\varepsilon }_{t,r}^{\left[ 2%
\right] }\right] \mathbf{C}^{\left[ 2\right] }\left( r-t\right) ^{\prime } \\
&\rightarrow &\mathbf{C}_{2}\left( 0\right) V_{0}\mathbf{C}_{2}^{\prime
}\left( 0\right) +\sum_{j=-\infty }^{\infty }\mathbf{C}^{\left[ 2\right]
}\left( j\right) \mathbb{V}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 2%
\right] }\right] \mathbf{C}^{\left[ 2\right] }\left( j\right) ^{\prime
}+\sum_{j=-\infty }^{\infty }\mathbf{C}^{\left[ 2\right] }\left( -j\right)
\mathbb{C}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 2\right] },\mathbf{%
\varepsilon }_{r,t}^{\left[ 2\right] }\right] \mathbf{C}^{\left[ 2\right]
}\left( j\right) ^{\prime } \\
&=&\mathbf{\Phi }_{22}^{0}+\mathbf{\Phi }_{22}+\mathbf{\Phi }_{22}^{\dag }
\end{eqnarray*}%
where, under Assumption 3$(4)$, $V_{0}=V_{0}\left( \mathbf{\kappa }%
_{4}^{0}\right) =\mathbb{V}\left[ \mathbf{\varepsilon }_{t}^{\otimes 2}%
\right] =\sum_{a,b}\left( \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes
\mathbf{e}_{b}\mathbf{e}_{b}^{\prime }+\mathbf{e}_{a}\mathbf{e}_{b}^{\prime
}\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime }\right) +\sum_{a}\mathbf{%
\kappa }_{4a}^{0}\left( \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes
\mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\right) ,$
\begin{equation*}
\mathbb{V}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 2\right] }\right] =%
\mathbb{V}\left[ \mathbf{\varepsilon }_{r,t}^{\left[ 2\right] }\right]
=\left( \mathbf{I}_{d}\otimes \mathbf{I}_{d}\right) =\mathbf{I}_{d^{2}}
\end{equation*}%
and for $t\neq r,$%
\begin{equation*}
\mathbb{C}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 2\right] },\mathbf{%
\varepsilon }_{r,t}^{\left[ 2\right] }\right] =\sum_{a,b}\left( \mathbf{e}%
_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime
}\right) .
\end{equation*}%
For $k=3$ and $\mathbf{\bar{\Lambda}}^{(\ell )}\left( \mathbf{\theta
}\right) =\left( 2\pi \right) ^{-1}\int_{\Pi }\mathbf{\dot{\Lambda}}^{(\ell
)}\left( \lambda \mathbf{;\theta }\right) d\lambda $ we obtain%
\begin{eqnarray*}
\mathbf{B}_{3,1}^{\left( \ell \right) }\left( \mathbf{\lambda };\mathbf{%
\theta }\right) &=&\mathbf{B}_{3,1}^{\left( \ell \right) }\left( \lambda
_{1};\mathbf{\theta }\right) =\mathbf{I}_{d}\otimes \mathbf{I}_{d}\otimes
\left \{ \mathbf{\dot{\Lambda}}^{(\ell )}\left( \lambda _{1}\mathbf{;\theta }%
\right) -\mathbf{S}_{3}\mathbf{S}_{3}^{\prime }\mathbf{\bar{\Lambda}}^{(\ell
)}\left( \mathbf{\theta }\right) \right \} \\
\mathbf{B}_{3,2}^{\left( \ell \right) }\left( \mathbf{\lambda };\mathbf{%
\theta }\right) &=&\mathbf{B}_{3,2}^{\left( \ell \right) }\left( \lambda
_{2};\mathbf{\theta }\right) =\mathbf{I}_{d}\otimes \left \{ \mathbf{\dot{%
\Lambda}}^{(\ell )}\left( \lambda _{2}\mathbf{;\theta }\right) -\mathbf{S}%
_{3}\mathbf{S}_{3}^{\prime }\mathbf{\bar{\Lambda}}^{(\ell )}\left( \mathbf{%
\theta }\right) \right \} \otimes \mathbf{I}_{d} \\
\mathbf{B}_{3,2}^{\left( \ell \right) }\left( \mathbf{\lambda };\mathbf{%
\theta }\right) &=&\mathbf{B}_{3,1}^{\left( \ell \right) }\left( \lambda
_{3};\mathbf{\theta }\right) =\left \{ \mathbf{\dot{\Lambda}}^{(\ell
)}\left( -\lambda _{1}-\lambda _{2}\mathbf{;\theta }\right) -\mathbf{S}_{3}%
\mathbf{S}_{3}^{\prime }\mathbf{\bar{\Lambda}}^{(\ell )}\left( \mathbf{%
\theta }\right) \right \} \otimes \mathbf{I}_{d}\otimes \mathbf{I}_{d},
\end{eqnarray*}%
because
\begin{equation*}
\mathbf{\Psi }_{3}^{-1}(\mathbf{\lambda };\mathbf{\theta })\mathbf{\dot{\Psi}%
}_{3}^{\left( \ell \right) }(\mathbf{\lambda };\mathbf{\theta })=\left(
\mathbf{I}_{d}\otimes \mathbf{I}_{d}\otimes \mathbf{\dot{\Lambda}}^{(\ell
)}\left( \lambda _{1}\mathbf{;\theta }\right) \right) +\left( \mathbf{I}%
_{d}\otimes \mathbf{\dot{\Lambda}}^{(\ell )}\left( \lambda _{2}\mathbf{%
;\theta }\right) \otimes \mathbf{I}_{d}\right) +\left( \mathbf{\dot{\Lambda}}%
^{(\ell )}\left( -\lambda _{1}-\lambda _{2}\mathbf{;\theta }\right) \otimes
\mathbf{I}_{d}\otimes \mathbf{I}_{d}\right) ,\ \ \
\end{equation*}%
and the third order periodogram }$\mathbb{I}_{3}^{\mathbf{\varepsilon }}(%
\mathbf{\lambda })=T^{-1}w_{T}^{\mathbf{\varepsilon }}\left( -\lambda
_{1}-\lambda _{2}\right) \otimes w_{T}^{\mathbf{\varepsilon }}\left( \lambda
_{2}\right) \otimes w_{T}^{\mathbf{\varepsilon }}\left( \lambda _{1}\right) $
can be written as
\begin{eqnarray*}
&&\frac{1}{T}\sum_{t,r,s=1}^{T}\exp \left( is\left( \lambda _{1}+\lambda
_{2}\right) -ir\lambda _{2}-it\lambda _{1}\right) \left( \mathbf{\varepsilon
}_{s}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }%
_{t}\right) \\ \! \! \! \! \! \!
&=&\frac{1}{T}\sum_{t=1}^{T}\left( \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\right) +\frac{1}{T}%
\sum_{t=1}^{T}\sum_{r=1}^{t-1}A_{t,r}^{\left[ 3\right] }\left( \mathbf{%
\lambda }\right) \! \left[ \!
\begin{array}{c}
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t} \\
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}%
\end{array} \!
\right] \!+\frac{1}{T}\sum_{t=1}^{T}\sum_{r,s=1}^{t-1}G_{t,r,s}^{\left[ 3%
\right] }\left( \mathbf{\lambda }\right) \! \left[ \!
\begin{array}{c}
\mathbf{\varepsilon }_{s}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t} \\
\mathbf{\varepsilon }_{s}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{s}\otimes \mathbf{%
\varepsilon }_{r}%
\end{array}%
\! \right] \ \ \ \
\end{eqnarray*}%
where
\begin{eqnarray*}
A_{t,r}^{\left[ 3\right] }\left( \mathbf{\lambda }_{\mathbf{j}}\right) &:= &%
\left[ \exp \left \{ -i\left( \left( r-t\right) \lambda _{j_{1}}\right)
\right \} \QATOP{{}}{{}}\exp \left \{ -i\left( \left( r-t\right) \lambda
_{j_{2}}\right) \right \} \QATOP{{}}{{}}\exp \left \{ -i\left( \left(
t-r\right) \left( \lambda _{j_{1}}+\lambda _{j_{2}}\right) \right) \right \} %
\right] , \\
G_{t,r,s}^{\left[ 3\right] }\left( \mathbf{\lambda }_{\mathbf{j}}\right) &:=&
\left[
\begin{array}{c}
\exp \left \{ -i\left( \left( t-s\right) \lambda _{j_{1}}+\left( r-s\right)
\lambda _{j_{2}}\right) \right \} \\
\exp \left \{ -i\left( \left( r-s\right) \lambda _{j_{1}}+\left( t-s\right)
\lambda _{j_{2}}\right) \right \} \\
\exp \left \{ -i\left( \left( r-t\right) \lambda _{j_{1}}+\left( s-t\right)
\lambda _{j_{2}}\right) \right \}%
\end{array}%
\right] ^{\prime }.
\end{eqnarray*}%
Then
\begin{equation*}
\mathbf{Z}_{3,T}=\frac{T^{1/2}}{T^{2}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}%
\func{Re}\left \{ \mathbf{B}_{3}^{\ast }\left( \mathbf{\lambda }_{\mathbf{j}%
};\mathbf{\theta }_{0}\right) \left( \mathbb{I}_{3}^{\varepsilon }(\mathbf{%
\lambda }_{\mathbf{j}})-E\left[ \mathbb{I}_{3}^{\varepsilon }(\mathbf{%
\lambda }_{\mathbf{j}})\right] \right) \right \} +o_{p}\left( 1\right)
=\sum_{t=1}^{T}Z_{3,t}+o_{p}\left( 1\right)
\end{equation*}%
where, vec$\left( \mathbf{I}_{d^{3}}\right) =\sum_{a=1}^{d}\left( \mathbf{e}%
_{a}\otimes \mathbf{e}_{a}\otimes \mathbf{e}_{a}\right) ,$ $\mathbf{e}_{a}$
equal to the $a$-th column of $\mathbf{I}_{d},$
\begin{eqnarray*}
Z_{3,t}&:= &\frac{1}{T^{1/2}}\mathbf{C}_{T}^{\left[ 3,0\right] }\left \{
\left( \mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\otimes
\mathbf{\varepsilon }_{t}\right) -\sum_{a=1}^{d}\mathbf{\kappa }_{3,a}\left(
\mathbf{e}_{a}\otimes \mathbf{e}_{a}\otimes \mathbf{e}_{a}\right) \right \}
\\
&&+\frac{1}{T^{1/2}}\sum_{r=1}^{t-1}\left \{ \frac{1}{T^{2}}\sum_{\mathbf{%
\lambda }_{\mathbf{j}}}\mathbf{B}_{3}^{\ast }\left( \mathbf{\lambda }_{%
\mathbf{j}};\mathbf{\theta }_{0}\right) A_{t,r}^{\left[ 3\right] }\left(
\mathbf{\lambda }_{\mathbf{j}}\right) \right \} \left[
\begin{array}{c}
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t} \\
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}%
\end{array}%
\right] \\
&&+\frac{1}{T^{1/2}}\sum_{r,s=1}^{t-1}\left \{ \frac{1}{T^{2}}\sum_{\mathbf{%
\lambda }_{\mathbf{j}}}\mathbf{B}_{3}^{\ast }\left( \mathbf{\lambda }_{%
\mathbf{j}};\mathbf{\theta }_{0}\right) G_{t,r,s}^{\left[ 3\right] }\left(
\mathbf{\lambda }_{\mathbf{j}}\right) \right \} \left[
\begin{array}{c}
\mathbf{\varepsilon }_{s}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t} \\
\mathbf{\varepsilon }_{s}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{s}\otimes \mathbf{%
\varepsilon }_{r}%
\end{array}%
\right]
\end{eqnarray*}%
where%
\begin{equation*}
\mathbf{C}_{T}^{\left[ 3,0\right] }:=\frac{1}{T^{2}}\sum_{\mathbf{\lambda }_{%
\mathbf{j}}}\mathbf{B}_{3}^{\ast }\left( \mathbf{\lambda }_{\mathbf{j}};%
\mathbf{\theta }_{0}\right) =\mathbf{C}_{3}\left( 0\right) +O\left(
T^{-1}\right) ,
\end{equation*}%
for $\mathbf{C}_{3}\left( 0\right) :=\left( 2\pi \right) ^{-2}\int_{\Pi ^{2}}%
\mathbf{B}_{3}^{\ast }\left( \mathbf{\lambda };\mathbf{\theta }_{0}\right) d%
\mathbf{\lambda }=\mathbf{A}_{3}^{\prime }\left( \mathbf{\theta }_{0}\right)
\left( \mathbf{I}_{d^{3}}-\mathbf{S}_{3}\mathbf{S}_{3}^{\prime }\right) $,
while%
\begin{equation*}
\frac{1}{T^{2}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\mathbf{B}_{3}^{\ast
}\left( \mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0}\right) A_{t,r}^{%
\left[ 3\right] }\left( \mathbf{\lambda }_{\mathbf{j}}\right) =\mathbf{C}%
_{3}\left( r-t\right) +O(T^{-1})
\end{equation*}%
as $T\rightarrow \infty ,$ where%
\begin{eqnarray*}
\mathbf{C}_{3}\left( j\right) &:= &\left[ \mathbf{C}_{\left[ 3,1\right]
}\left( j\right) \ \ \mathbf{C}_{\left[ 3,2\right] }\left( j\right) \ \
\mathbf{C}_{\left[ 3,3\right] }\left( j\right) \right] \\
&:= &\left[ \left( \mathbf{I}_{d}\otimes \mathbf{I}_{d}\otimes \mathbf{%
\mathbf{C}}\left( j\right) \right) \QATOP{{}}{{}}\left( \mathbf{I}%
_{d}\otimes \mathbf{\mathbf{C}}\left( j\right) \otimes \mathbf{I}_{d}\right)
\QATOP{{}}{{}}\left( \mathbf{\mathbf{C}}\left( j\right) \otimes \mathbf{I}%
_{d}\otimes \mathbf{I}_{d}\right) \right] ,
\end{eqnarray*}%
and similarly, the $G_{t,r,s}^{\left[ 3\right] }$ terms only contribute when
$r=s$ with%
\begin{equation*}
\frac{1}{T^{2}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\mathbf{B}_{3}^{\ast
}\left( \mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0}\right)
G_{t,r,r}^{\left[ 3\right] }\left( \mathbf{\lambda }_{\mathbf{j}}\right) =%
\mathbf{C}_{3}\left( t-r\right) +O(T^{-1}).
\end{equation*}%
Then, using the same arguments as in VL, noting that the terms in $%
\mathbf{G}\ $for $r\neq s$ do not contribute, we can write
\begin{equation*}
\mathbf{Z}_{3,T}=\sum_{t=1}^{T}Z_{3,t}^{0}+o_{p}\left( 1\right)
\end{equation*}%
where $Z_{3,t}^{0}$ is a MDS
\begin{equation*}
Z_{3,t}^{0}=\frac{1}{T^{1/2}}\sum_{r=1}^{t-1}\left[ \mathbf{C}_{3}\left(
0\right) \ \ \mathbf{C}_{3}\left( r-t\right) \ \ \mathbf{C}_{3}\left(
t-r\right) \right] \left[
\begin{array}{c}
\frac{1}{t-1}\left( \mathbf{\varepsilon }_{t}^{\otimes 3}-E\left[ \mathbf{%
\varepsilon }_{t}^{\otimes 3}\right] \right) \\
\mathbf{\varepsilon }_{t,r}^{\left[ 3\right] } \\
\mathbf{\varepsilon }_{r,t}^{\left[ 3\right] }%
\end{array}%
\right]
\end{equation*}%
with $\mathbf{\varepsilon }_{t,r}^{\left[ 3\right] }$ defined in Section~5,
so that exploiting symmetries, $\mathbb{V}\left[ \mathbf{Z}_{3,T}\right] =
\sum_{t=1}^{T}\mathbb{V}\left[ Z_{3,t}^{0}\right] ${\small \ can be
approximated by%
\begin{eqnarray*}
&&\mathbf{C}_{3}\left( 0\right) \mathbb{V}\left[ \mathbf{\varepsilon }%
_{t}^{\otimes 3}\right] \mathbf{C}_{3}^{\prime }\left( 0\right) \\
&&+\frac{1}{T}\sum_{t=1}^{T}\sum_{r=1}^{t-1}\left[ \mathbf{C}^{\left[ 3%
\right] }\left( r-t\right) \ \ \mathbf{C}^{\left[ 3\right] }\left(
t-r\right) \right] \left[
\begin{array}{cc}
\mathbb{V}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 3\right] }\right] &
\mathbb{C}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 3\right] },\mathbf{%
\varepsilon }_{r,t}^{\left[ 3\right] }\right] \\
\mathbb{C}\left[ \mathbf{\varepsilon }_{r,t}^{\left[ 3\right] },\mathbf{%
\varepsilon }_{t,r}^{\left[ 3\right] }\right] & \mathbb{V}\left[ \mathbf{%
\varepsilon }_{r,t}^{\left[ 3\right] }\right]%
\end{array}%
\right] \left[
\begin{array}{c}
\mathbf{C}^{\left[ 3\right] }\left( r-t\right) ^{\prime } \\
\mathbf{C}^{\left[ 3\right] }\left( t-r\right) ^{\prime }%
\end{array}%
\right] \\
&=&\mathbf{C}_{3}\left( 0\right) \mathbb{V}\left[ \mathbf{\varepsilon }%
_{t}^{\otimes 3}\right] \mathbf{C}_{3}^{\prime }\left( 0\right) +\frac{1}{T}%
\sum_{t=1}^{T}\sum_{r=1}^{t-1}\mathbf{C}^{\left[ 3\right] }\left( r-t\right)
\mathbb{V}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 3\right] }\right]
\mathbf{C}^{\left[ 3\right] }\left( r-t\right) ^{\prime }+\mathbf{C}^{\left[
3\right] }\left( t-r\right) \mathbb{V}\left[ \mathbf{\varepsilon }%
_{r,t}^{\left[ 3\right] }\right] \mathbf{C}^{\left[ 3\right] }\left(
t-r\right) ^{\prime } \\
&&+\mathbf{C}^{\left[ 3\right] }\left( r-t\right) \mathbb{C}\left[ \mathbf{%
\varepsilon }_{t,r}^{\left[ 3\right] },\mathbf{\varepsilon }_{r,t}^{\left(
3\right) }\right] \mathbf{C}^{\left[ 3\right] }\left( t-r\right) ^{\prime }+%
\mathbf{C}^{\left[ 3\right] }\left( t-r\right) \mathbb{C}\left[ \mathbf{%
\varepsilon }_{r,t}^{\left[ 3\right] },\mathbf{\varepsilon }_{t,r}^{\left(
3\right) }\right] \mathbf{C}^{\left[ 3\right] }\left( r-t\right) ^{\prime }
\\
&\rightarrow &\mathbf{C}_{3}\left( 0\right) \mathbb{V}\left[ \mathbf{%
\varepsilon }_{t}^{\otimes 3}\right] \mathbf{C}_{3}^{\prime }\left( 0\right)
+\sum_{j=-\infty }^{\infty }\mathbf{C}^{\left[ 3\right] }\left( j\right)
\mathbb{V}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 3\right] }\right]
\mathbf{C}^{\left[ 3\right] }\left( j\right) ^{\prime }+\sum_{j=-\infty
}^{\infty }\mathbf{C}^{\left[ 3\right] }\left( -j\right) \mathbb{C}\left[
\mathbf{\varepsilon }_{t,r}^{\left[ 3\right] },\mathbf{\varepsilon }%
_{r,t}^{\left[ 3\right] }\right] \mathbf{C}^{\left[ 3\right] }\left(
j\right) ^{\prime } \\
&=&\mathbf{\Phi }_{33}^{0}+\mathbf{\Phi }_{33}+\mathbf{\Phi }_{33}^{\dag }=%
\mathbf{\Phi }_{33}^{0}+\mathbf{\Phi }_{33}\left( \mathbf{\kappa }%
_{4}^{0}\right) +\mathbf{\Phi }_{33}^{\dag }\left( \mathbf{\kappa }%
_{3}^{0}\right) ,\ \RIfM@\expandafter\text@\else\expandafter\mbox\fi{say,}
\end{eqnarray*}%
where $\mathbf{\Phi }_{33}^{0}:=\mathbf{C}_{3}\left( 0\right) \mathbb{V}%
\left[ \mathbf{\varepsilon }_{t}^{\otimes 3}\right] \mathbf{C}_{3}^{\prime
}\left( 0\right) \ $depends on $\mathbf{\kappa }_{2},\mathbf{\kappa }_{3}$
and $\mathbf{\kappa }_{4},$ but not on $\mathbf{\kappa }_{6}$ because the
right factor $\mathbf{I}_{d^{3}}-\mathbf{S}_{3}\mathbf{S}_{3}^{\prime }=%
\mathbf{I}_{d^{3}}-\sum_{a}\mathbf{e}_{a}^{\otimes 3}\mathbf{e}_{a}^{\otimes
3\prime }$ of $\mathbf{C}_{3}\left( 0\right) $ is orthogonal to the
contribution of $\mathbf{\kappa }_{6}$ to $\mathbb{V}\left[ \mathbf{%
\varepsilon }_{t}^{\otimes 3}\right] ,$ namely$\ \sum_{a}\mathbf{\kappa }%
_{6,a}\mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }=\sum_{a}\mathbf{%
\kappa }_{6,a}\mathbf{e}_{a}^{\otimes 3}\mathbf{e}_{a}^{\otimes 3\prime }$,
and $\mathbb{V}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 3\right] }\right] =%
\mathbb{V}\left[ \mathbf{\varepsilon }_{r,t}^{\left[ 3\right] }\right] ,$ $%
t\neq r,$ where under Assumption 3$(h)$, $h\in \{3,4\}$, $\mathbb{V}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 3\right] }\right]$ is equal to
\begin{eqnarray*}
&&\sum_{a,b,c}\left[
\begin{array}{ccc}
\left \{
\begin{array}{c}
\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{c}\mathbf{e}_{c}^{\prime } \\
+\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{c}\mathbf{e}_{c}^{\prime }%
\end{array}%
\right \} & \left \{
\begin{array}{c}
\mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{c}^{\prime }\otimes \mathbf{e}_{c}\mathbf{e}_{b}^{\prime } \\
+\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{c}^{\prime }\otimes \mathbf{e}_{c}\mathbf{e}_{a}^{\prime }%
\end{array}%
\right \} & \left \{
\begin{array}{c}
\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{c}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{c}^{\prime } \\
+\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{c}\mathbf{e}%
_{c}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime }%
\end{array}%
\right \} \\
& \left \{
\begin{array}{c}
\mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{c}\mathbf{e}_{c}^{\prime } \\
+\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{c}\mathbf{e}%
_{c}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime }%
\end{array}%
\right \} & \left \{
\begin{array}{c}
\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{c}\mathbf{e}_{c}^{\prime } \\
+\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{c}^{\prime }\otimes \mathbf{e}_{c}\mathbf{e}_{a}^{\prime }%
\end{array}%
\right \} \\
& & \left \{
\begin{array}{c}
\mathbf{e}_{c}\mathbf{e}_{c}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime } \\
+\mathbf{e}_{c}\mathbf{e}_{c}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime }%
\end{array}%
\right \}%
\end{array}%
\right] \\
&&+\sum_{a,b}\mathbf{\kappa }_{4,a}\left[
\begin{array}{ccc}
\mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{b}^{\prime } & \mathbf{e}_{a}%
\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime
}\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime } & \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e%
}_{b}\mathbf{e}_{a}^{\prime } \\
& \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime } & \mathbf{e}_{a}%
\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime
}\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime } \\
& & \mathbf{e}_{b}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }%
\end{array}%
\right]
\end{eqnarray*}%
and $\mathbb{C}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 3\right] },\mathbf{%
\varepsilon }_{r,t}^{\left[ 3\right] }\right] =\mathbb{C}\left[ \mathbf{%
\varepsilon }_{r,t}^{\left[ 3\right] },\mathbf{\varepsilon }_{t,r}^{\left(
3\right) }\right] ,$ with
\begin{equation*}
\mathbb{C}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 3\right] },\mathbf{%
\varepsilon }_{r,t}^{\left[ 3\right] }\right] =\sum_{a,b}\mathbf{\kappa }%
_{3,a}\mathbf{\kappa }_{3,b}\left[
\begin{array}{ccc}
\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{b}^{\prime } & \mathbf{e}_{a}%
\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime
}\otimes \mathbf{e}_{b}\mathbf{e}_{b}^{\prime } & \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e%
}_{b}\mathbf{e}_{b}^{\prime } \\
\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime } & \mathbf{e}_{a}%
\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime
}\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime } & \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e%
}_{a}\mathbf{e}_{b}^{\prime } \\
\mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{b}^{\prime } & \mathbf{e}_{a}%
\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{b}^{\prime
}\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime } & \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e%
}_{b}\mathbf{e}_{a}^{\prime }%
\end{array}%
\right] .
\end{equation*}%
For $k=4,$%
\begin{eqnarray*}
\mathbf{B}_{4,1}^{\left( \ell \right) }\left( \mathbf{\lambda };\mathbf{%
\theta }\right) &=&\mathbf{B}_{4,1}^{\left( \ell \right) }\left( \lambda
_{1};\mathbf{\theta }\right) =\mathbf{I}_{d}\otimes \mathbf{I}_{d}\otimes
\mathbf{I}_{d}\otimes \left \{ \mathbf{\dot{\Lambda}}^{(\ell )}\left(
e^{-i\lambda _{1}}\mathbf{;\theta }\right) -\mathbf{S}_{4}\mathbf{S}%
_{4}^{\prime }\mathbf{\bar{\Lambda}}^{(\ell )}\left( \mathbf{\theta }\right)
\right \} \\
\mathbf{B}_{4,2}^{\left( \ell \right) }\left( \mathbf{\lambda };\mathbf{%
\theta }\right) &=&\mathbf{B}_{4,2}^{\left( \ell \right) }\left( \lambda
_{2};\mathbf{\theta }\right) =\mathbf{I}_{d}\otimes \mathbf{I}_{d}\otimes
\left \{ \mathbf{\dot{\Lambda}}^{(\ell )}\left( e^{-i\lambda _{2}}\mathbf{%
;\theta }\right) -\mathbf{S}_{4}\mathbf{S}_{4}^{\prime }\mathbf{\bar{\Lambda}%
}^{(\ell )}\left( \mathbf{\theta }\right) \right \} \otimes \mathbf{I}_{d} \\
\mathbf{B}_{4,2}^{\left( \ell \right) }\left( \mathbf{\lambda };\mathbf{%
\theta }\right) &=&\mathbf{B}_{4,1}^{\left( \ell \right) }\left( \lambda
_{3};\mathbf{\theta }\right) =\mathbf{I}_{d}\otimes \left \{ \mathbf{\dot{%
\Lambda}}^{(\ell )}\left( e^{-i\lambda _{3}}\mathbf{;\theta }\right) -%
\mathbf{S}_{4}\mathbf{S}_{4}^{\prime }\mathbf{\bar{\Lambda}}^{(\ell )}\left(
\mathbf{\theta }\right) \right \} \otimes \mathbf{I}_{d}\otimes \mathbf{I}%
_{d} \\
\mathbf{B}_{4,2}^{\left( \ell \right) }\left( \mathbf{\lambda };\mathbf{%
\theta }\right) &=&\mathbf{B}_{4,1}^{\left( \ell \right) }\left( \lambda
_{4};\mathbf{\theta }\right) =\left \{ \mathbf{\dot{\Lambda}}^{(\ell
)}\left( e^{-i\lambda _{4}}\mathbf{;\theta }\right) -\mathbf{S}_{4}\mathbf{S}%
_{4}^{\prime }\mathbf{\bar{\Lambda}}^{(\ell )}\left( \mathbf{\theta }\right)
\right \} \otimes \mathbf{I}_{d}\otimes \mathbf{I}_{d}\otimes \mathbf{I}_{d},
\end{eqnarray*}%
because
\begin{eqnarray*}
\mathbf{\Psi }_{4}^{-1}(\mathbf{\lambda };\mathbf{\theta })\mathbf{\dot{\Psi}%
}_{4}^{\left( \ell \right) }(\mathbf{\lambda };\mathbf{\theta }) &=&\left(
\mathbf{I}_{d}\otimes \mathbf{I}_{d}\otimes \mathbf{I}_{d}\otimes \mathbf{%
\dot{\Lambda}}^{(\ell )}\left( \lambda _{1}\mathbf{;\theta }\right) \right)
+\left( \mathbf{I}_{d}\otimes \mathbf{I}_{d}\otimes \mathbf{\dot{\Lambda}}%
^{(\ell )}\left( \lambda _{2}\mathbf{;\theta }\right) \otimes \mathbf{I}%
_{d}\right) \\
&+&\left( \mathbf{I}_{d}\otimes \mathbf{\dot{\Lambda}}^{(\ell )}\left(
\lambda _{3}\mathbf{;\theta }\right) \otimes \mathbf{I}_{d}\otimes \mathbf{I}%
_{d}\right) +\left( \mathbf{\dot{\Lambda}}^{(\ell )}\left( -\lambda
_{1}-\lambda _{2}-\lambda _{3}\mathbf{;\theta }\right) \otimes \mathbf{I}%
_{d}\otimes \mathbf{I}_{d}\otimes \mathbf{I}_{d}\right) ,
\end{eqnarray*}%
and $\mathbb{I}_{4}^{\mathbf{\varepsilon }}(\mathbf{\lambda }) = T^{-1}
w_{T}^{\mathbf{\varepsilon }}\left( -\lambda _{1}-\lambda _{2}-\lambda
_{3}\right) \otimes w_{T}^{\mathbf{\varepsilon }}\left( \lambda _{3}\right)
\otimes w_{T}^{\mathbf{\varepsilon }}\left( \lambda _{2}\right) \otimes
w_{T}^{\mathbf{\varepsilon }}\left( \lambda _{1}\right) $ is equal to
\begin{eqnarray*}
&&\frac{1}{T}\sum_{t,r,s,u=1}^{T}\exp \left( iu\left( \lambda _{1}+\lambda
_{2}+\lambda _{3}\right) -is\lambda _{3}-ir\lambda _{2}-it\lambda
_{1}\right) \left( \mathbf{\varepsilon }_{u}\otimes \mathbf{\varepsilon }%
_{s}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}\right)
\end{eqnarray*}
or
\begin{eqnarray*}
&&\frac{1}{T}\sum_{t=1}^{T}\left( \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}\right) +\frac{1}{T}\sum_{t=1}^{T}\sum_{r=1}^{t-1}A_{t,r}^{%
\left[ 4\right] }\left( \mathbf{\lambda }_{\mathbf{j}}\right) \left[
\begin{array}{c}
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t} \\
\mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}%
\end{array}%
\right] \\
&+&\frac{1}{T}\sum_{t=1}^{T}\sum_{r,s=1}^{t-1}G_{t,r,s}^{\left[ 4\right]
}\left( \mathbf{\lambda }_{\mathbf{j}}\right) \left[
\begin{array}{c}
\mathbf{\varepsilon }_{s}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t} \\
\mathbf{\varepsilon }_{s}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{s}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t} \\
\mathbf{\varepsilon }_{s}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{s}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{s}\otimes \mathbf{\varepsilon }_{r}%
\end{array}%
\right] +\frac{1}{T}\sum_{t=1}^{T}\sum_{r,s,u=1}^{t-1}F_{t,r,s,u}^{\left[ 4%
\right] }\left( \mathbf{\lambda }_{\mathbf{j}}\right) \left[
\begin{array}{c}
\mathbf{\varepsilon }_{u}\otimes \mathbf{\varepsilon }_{s}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t} \\
\mathbf{\varepsilon }_{u}\otimes \mathbf{\varepsilon }_{s}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r} \\
\mathbf{\varepsilon }_{u}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{s} \\
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{s}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{u}%
\end{array}%
\right]
\end{eqnarray*}%
where%
\begin{equation*}
A_{t,r}^{\left[ 4\right] }\left( \mathbf{\lambda }_{\mathbf{j}}\right)
=\left \{ A_{t,r}^{\left[ 4,n\right] }\left( \mathbf{\lambda }_{\mathbf{j}%
}\right) \right \} :=\left[
\begin{array}{c}
\exp \left \{ -i\left( r-t\right) \lambda _{j_{1}}\right \} \\
\exp \left \{ -i\left( r-t\right) \lambda _{j_{2}}\right \} \\
\exp \left \{ -i\left( r-t\right) \lambda _{j_{3}}\right \} \\
\exp \left \{ -i\left( t-r\right) \left( \lambda _{j_{1}}+\lambda
_{j_{2}}+\lambda _{j_{3}}\right) \right \}%
\end{array}%
\right] ^{\prime }\
\end{equation*}%
\begin{equation*}
G_{t,r,s}^{\left[ 4\right] }\left( \mathbf{\lambda }_{\mathbf{j}}\right)
=\left \{ G_{t,r,s}^{\left[ 4,n\right] }\left( \mathbf{\lambda }_{\mathbf{j}%
}\right) \right \} :=\left[
\begin{array}{c}
\exp \left \{ -i\left( \left( t-s\right) \lambda _{j_{1}}+\left( t-s\right)
\lambda _{j_{2}}+\left( r-s\right) \lambda _{j_{3}}\right) \right \} \\
\exp \left \{ -i\left( \left( t-s\right) \lambda _{j_{1}}+\left( r-s\right)
\lambda _{j_{2}}+\left( t-s\right) \lambda _{j_{3}}\right) \right \} \\
\exp \left \{ -i\left( \left( r-t\right) \lambda _{j_{2}}+\left( s-t\right)
\lambda _{j_{3}}\right) \right \} \\
\exp \left \{ -i\left( \left( r-s\right) \lambda _{j_{1}}+\left( t-s\right)
\lambda _{j_{2}}+\left( t-s\right) \lambda _{j_{3}}\right) \right \} \\
\exp \left \{ -i\left( \left( r-t\right) \lambda _{j_{1}}+\left( s-t\right)
\lambda _{j_{3}}\right) \right \} \\
\exp \left \{ -i\left( \left( r-t\right) \lambda _{j_{1}}+\left( s-t\right)
\lambda _{j_{2}}\right) \right \}%
\end{array}%
\right] ^{\prime }
\end{equation*}%
\begin{equation*}
F_{t,r,s,u}^{\left[ 4\right] }\left( \mathbf{\lambda }_{\mathbf{j}}\right)
=\left \{ F_{t,r,s,u}^{\left[ 4,n\right] }\left( \mathbf{\lambda }_{\mathbf{j%
}}\right) \right \} :=\left[
\begin{array}{c}
\exp \left \{ -i\left( \left( t-u\right) \lambda _{j_{1}}+\left( r-u\right)
\lambda _{j_{2}}+\left( s-u\right) \lambda _{j_{3}}\right) \right \} \\
\exp \left \{ -i\left( \left( r-u\right) \lambda _{j_{1}}+\left( t-u\right)
\lambda _{j_{2}}+\left( s-u\right) \lambda _{j_{3}}\right) \right \} \\
\exp \left \{ -i\left( \left( s-u\right) \lambda _{j_{1}}+\left( r-u\right)
\lambda _{j_{2}}+\left( t-u\right) \lambda _{j_{3}}\right) \right \} \\
\exp \left \{ -i\left( \left( u-t\right) \lambda _{j_{1}}+\left( r-t\right)
\lambda _{j_{2}}+\left( s-t\right) \lambda _{j_{3}}\right) \right \}%
\end{array}%
\right] ^{\prime }.
\end{equation*}%
Then
\begin{equation*}
\mathbf{Z}_{4,T}=\frac{T^{1/2}}{T^{2}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}%
\func{Re}\left \{ \mathbf{B}_{4}^{\ast }\left( \mathbf{\lambda }_{\mathbf{j}%
}\right) \left( \mathbb{I}_{4}^{\varepsilon }(\mathbf{\lambda }_{\mathbf{j}%
})-E\left[ \mathbb{I}_{4}^{\varepsilon }(\mathbf{\lambda }_{\mathbf{j}})%
\right] \right) \right \} =\sum_{t=1}^{T}Z_{4,t} =\sum_{t=1}^{T} \sum_{n=0}^{3} Z^{(n)}_{4,t},
\end{equation*}%
where, noting that vec$\left( \mathbf{I}_{d^{4}}\right)
=\sum_{a=1}^{d}\left( \mathbf{e}_{a}\otimes \mathbf{e}_{a}\otimes \mathbf{e}%
_{a}\otimes \mathbf{e}_{a}\right) ,$ $\mathbf{e}_{a}$ equal to the $a$-th
column of $\mathbf{I}_{d},$}%
{\small
\begin{equation*}
Z_{4,t}^{\left( 0\right) }:=\frac{1}{T^{1/2}}\mathbf{C}_{T}^{\left[ 4,0%
\right] }\left \{ \left( \mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon
}_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }%
_{t}\right) -\sum_{a=1}^{d}\mathbf{\kappa }_{4,a}\left( \mathbf{e}%
_{a}\otimes \mathbf{e}_{a}\otimes \mathbf{e}_{a}\otimes \mathbf{e}%
_{a}\right) \right \}
\end{equation*}%
\begin{eqnarray*}
Z_{4,t}^{\left( 1\right) } &:=&\frac{1}{T^{1/2}}\sum_{r=1}^{t-1}\left \{
\mathbf{C}_{T}^{\left[ 4,1\right] }\left( t-r\right) \left( \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}\right) +\mathbf{C}_{T}^{%
\left[ 4,2\right] }\left( t-r\right) \left( \mathbf{\varepsilon }_{t}\otimes
\mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t}\right) \right \} \\
&&+\frac{1}{T^{1/2}}\sum_{r=1}^{t-1}\left \{ \mathbf{C}_{T}^{\left[ 4,3\right]
}\left( t-r\right) \left( \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{t}\right) +\mathbf{C}_{T}^{\left[ 4,4\right] }\left(
t-r\right) \left( \mathbf{\varepsilon }_{r}\otimes \mathbf{\varepsilon }%
_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }%
_{t}\right) \right \}
\end{eqnarray*}%
\begin{eqnarray*}
Z_{4,t}^{\left( 2\right) } &:=&\frac{1}{T^{1/2}}\sum_{r,s=1}^{t-1}\left \{
\mathbf{G}_{T}^{\left[ 4,1\right] }\left( t-r,t-s\right) \left( \mathbf{%
\varepsilon }_{s}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{t}\right) +\mathbf{G}_{T}^{%
\left[ 4,2\right] }\left( t-r,t-s\right) \left( \mathbf{\varepsilon }%
_{s}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }%
_{r}\otimes \mathbf{\varepsilon }_{t}\right) \right \} \\
&&+\frac{1}{T^{1/2}}\sum_{r,s=1}^{t-1}\left \{ \mathbf{G}_{T}^{\left[ 4,3%
\right] }\left( t-r,t-s\right) \left( \mathbf{\varepsilon }_{t}\otimes
\mathbf{\varepsilon }_{s}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{t}\right) +\mathbf{G}_{T}^{\left[ 4,4\right] }\left(
t-r,t-s\right) \left( \mathbf{\varepsilon }_{s}\otimes \mathbf{\varepsilon }%
_{t}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }%
_{r}\right) \right \} \\
&&+\frac{1}{T^{1/2}}\sum_{r,s=1}^{t-1}\left \{ \mathbf{G}_{T}^{\left[ 4,5%
\right] }\left( t-r,t-s\right) \left( \mathbf{\varepsilon }_{t}\otimes
\mathbf{\varepsilon }_{s}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r}\right) +\mathbf{G}_{T}^{\left[ 4,6\right] }\left(
t-r,t-s\right) \left( \mathbf{\varepsilon }_{t}\otimes \mathbf{\varepsilon }%
_{t}\otimes \mathbf{\varepsilon }_{s}\otimes \mathbf{\varepsilon }%
_{r}\right) \right \}
\end{eqnarray*}%
\begin{eqnarray*}
Z_{4,t}^{\left( 3\right) } &:=&\frac{1}{T^{1/2}}\sum_{r,s,u=1}^{t-1}\mathbf{F}%
_{T}^{\left[ 4,1\right] }\left( t-r,t-s,t-u\right) \left( \mathbf{%
\varepsilon }_{u}\otimes \mathbf{\varepsilon }_{s}\otimes \mathbf{%
\varepsilon }_{r}\otimes \mathbf{\varepsilon }_{t}\right) \\
&&+\frac{1}{T^{1/2}}\sum_{r,s,u=1}^{t-1}\mathbf{F}_{T}^{\left[ 4,2\right]
}\left( t-r,t-s,t-u\right) \left( \mathbf{\varepsilon }_{u}\otimes \mathbf{%
\varepsilon }_{s}\otimes \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{r}\right) \\
&&+\frac{1}{T^{1/2}}\sum_{r,s,u=1}^{t-1}\mathbf{F}_{T}^{\left[ 4,3\right]
}\left( t-r,t-s,t-u\right) \left( \mathbf{\varepsilon }_{u}\otimes \mathbf{%
\varepsilon }_{t}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{s}\right) \\
&&+\frac{1}{T^{1/2}}\sum_{r,s,u=1}^{t-1}\mathbf{F}_{T}^{\left[ 4,4\right]
}\left( t-r,t-s,t-u\right) \left( \mathbf{\varepsilon }_{t}\otimes \mathbf{%
\varepsilon }_{s}\otimes \mathbf{\varepsilon }_{r}\otimes \mathbf{%
\varepsilon }_{u}\right) .
\end{eqnarray*}%
Next,%
\begin{equation*}
\mathbf{C}_{T}^{\left[ 4,0\right] }:=\frac{1}{T^{3}}\sum_{\mathbf{\lambda }_{%
\mathbf{j}}}\mathbf{B}_{4}^{\ast }\left( \mathbf{\lambda }_{\mathbf{j}};%
\mathbf{\theta }_{0}\right) =\mathbf{C}_{4}\left( 0\right) +O\left(
T^{-1}\right) ,
\end{equation*}%
where $\mathbf{C}_{4}\left( 0\right) :=\left( 2\pi \right) ^{-3}\int_{\Pi
^{3}}\mathbf{B}_{4}^{\ast }\left( \mathbf{\lambda };\mathbf{\theta }%
_{0}\right) d\mathbf{\lambda =A}_{4}^{\prime }\left( \mathbf{\theta }%
_{0}\right) \left( \mathbf{I}_{d^{4}}-\mathbf{S}_{4}\mathbf{S}_{4}^{\prime
}\right) ,$ while the third block of terms does not contribute because for $%
t>r,s,$ as $T\rightarrow \infty ,$
\begin{eqnarray*}
\mathbf{C}_{T}^{\left[ 4,n\right] }\left( r-t\right) &=&\frac{1}{T^{3}}%
\sum_{\mathbf{\lambda }_{\mathbf{j}}}\mathbf{B}_{[4,n]}^{\ast }\left(
\mathbf{\lambda }_{\mathbf{j}\left( n\right) };\mathbf{\theta }_{0}\right)
A_{t,r}^{\left[ 4,n\right] }\left( \mathbf{\lambda }_{\mathbf{j}\left(
n\right) }\right) =\frac{1}{T}\sum_{j=1}^{T-1}\mathbf{B}_{[4,n]}^{\ast
}\left( \lambda _{j};\mathbf{\theta }_{0}\right) A_{t,r}^{\left[ 4,n\right]
}\left( \mathbf{\lambda }_{\mathbf{j}}\right) \\
&\rightarrow &\mathbf{\mathbf{C}}^{\left[ 4,n\right] }\left( r-t\right) :=%
\frac{1}{2\pi }\int_{-\pi }^{\pi }\mathbf{B}_{[4,n]}^{\ast }\left( \lambda ;%
\mathbf{\theta }_{0}\right) \exp \left( i\left( t-r\right) \lambda \right)
d\lambda
\end{eqnarray*}%
\begin{eqnarray*}
\mathbf{G}_{T}^{\left[ 4,n\right] }\left( r-t,s-t\right) &=&\frac{1}{T^{3}}%
\sum_{\mathbf{\lambda }_{\mathbf{j}}}\mathbf{B}_{[4,n]}^{\ast }\left(
\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0}\right) G_{t,r,s}^{\left[
4,n\right] }\left( \mathbf{\lambda }_{\mathbf{j}}\right) \rightarrow 0\ \ \\
\mathbf{F}_{T}^{\left[ 4,n\right] }\left( r-t,s-t,u-t\right) &=&\frac{1}{%
T^{3}}\sum_{\mathbf{\lambda }_{\mathbf{j}}}\mathbf{B}_{[4,n]}^{\ast }\left(
\mathbf{\lambda }_{\mathbf{j}};\mathbf{\theta }_{0}\right) F_{t,r,s,u}^{%
\left[ 4,n\right] }\left( \mathbf{\lambda }_{\mathbf{j}}\right) \rightarrow
\mathbf{\mathbf{C}}^{\left[ 4,n\right] }\left( r-t\right) 1_{\left \{
r=s=u\right \} }.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{eqnarray*}%
Then, proceeding as in VL, we can write%
\begin{equation*}
\mathbf{Z}_{4,T}=\sum_{t=1}^{T}Z_{4,t}^{0}+o_{p}\left( 1\right)
\end{equation*}%
where $Z_{4,t}^{0}$ is a MDS
\begin{equation*}
Z_{4,t}^{0}=\frac{1}{T^{1/2}}\sum_{r=1}^{t-1}\left[ \mathbf{C}_{4}\left(
0\right) \ \ \mathbf{C}_{4}\left( r-t\right) \ \ \mathbf{C}_{4}\left(
t-r\right) \right] \left[
\begin{array}{c}
\frac{1}{t-1}\left( \mathbf{\varepsilon }_{t}^{\otimes 4}-E\left[ \mathbf{%
\varepsilon }_{t}^{\otimes 4}\right] \right) \\
\mathbf{\varepsilon }_{t,r}^{\left[ 4\right] } \\
\mathbf{\varepsilon }_{r,t}^{\left[ 4\right] }%
\end{array}%
\right]
\end{equation*}%
with $\mathbf{C}_{4}\left( r-t\right) =\left[ \mathbf{C}_{\left[ 4,1\right]
}\left( r-t\right) \ \ \mathbf{C}_{\left[ 4,2\right] }\left( r-t\right) \ \
\mathbf{C}_{\left[ 4,3\right] }\left( r-t\right) \ \ \mathbf{C}_{\left[ 4,4%
\right] }\left( r-t\right) \right] $ and
$\mathbf{\varepsilon }_{t,r}^{\left[ 4\right] }$ defined in Section~5.
}
{\small Then, using that $\mathbb{V}\left[ \mathbf{\varepsilon }_{t,r}^{%
\left[ 4\right] }\right] =\mathbb{V}\left[ \mathbf{\varepsilon }_{r,t}^{%
\left[ 4\right] }\right] $ and $\mathbb{C}\left[ \mathbf{\varepsilon }%
_{t,r}^{\left[ 4\right] },\mathbf{\varepsilon }_{r,t}^{\left[ 4\right] }%
\right] =\mathbb{C}\left[ \mathbf{\varepsilon }_{r,t}^{\left[ 4\right] },%
\mathbf{\varepsilon }_{t,r}^{\left[ 4\right] }\right] $ and Assumption 3$(h)$%
, $h\in \{3,4,5,6\}$,
\begin{eqnarray*}
\mathbb{V}\left[ \mathbf{Z}_{4,T}\right] &\rightarrow &\mathbf{C}_{4}\left(
0\right) \mathbb{V}\left[ \mathbf{\varepsilon }_{t}^{\otimes 4}\right]
\mathbf{C}_{4}^{\prime }\left( 0\right) +\sum_{j=-\infty }^{\infty }\mathbf{C%
}_{4}\left( j\right) \mathbb{V}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 4%
\right] }\right] \mathbf{C}_{4}^{\prime }\left( j\right) +\sum_{j=-\infty
}^{\infty }\mathbf{C}_{4}\left( -j\right) \mathbb{C}\left[ \mathbf{%
\varepsilon }_{t,r}^{\left[ 4\right] },\mathbf{\varepsilon }_{r,t}^{\left[ 4%
\right] }\right] \mathbf{C}_{4}^{\prime }\left( j\right) \\
&=&\mathbf{\Phi }_{44}^{0}\left( \mathbf{\kappa }_{2}^{0},\mathbf{\kappa }%
_{3}^{0},\mathbf{\kappa }_{4}^{0},\mathbf{\kappa }_{5}^{0},\mathbf{\kappa }%
_{6}^{0}\right) +\mathbf{\Phi }_{44}\left( \mathbf{\mu }_{6}^{0},\mathbf{%
\kappa }_{3}^{0}\right) +\mathbf{\Phi }_{44}^{\dag }\left( \mathbf{\mu }%
_{4}^{0}\right) ,
\end{eqnarray*}%
where $\mathbf{\Phi }_{44}^{0}$ does not depend on $\mathbf{\kappa }_{8}^{0},
$ and $\mathbb{V}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 4\right] }\right]
$ is
\begin{eqnarray*}
&&\sum_{ab} \! \mathbf{\mu }_{6a}\left[ \!
\begin{array}{cccc}
\mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e%
}_{b}\mathbf{e}_{b}^{\prime } & \mathbf{e}_{a}\mathbf{e}_{a}^{\prime
}\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{%
e}_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime } & \cdots &
\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e%
}_{b}\mathbf{e}_{a}^{\prime } \\
& \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e%
}_{a}\mathbf{e}_{a}^{\prime } & & \vdots \\
& & \ddots & \\
& & & \mathbf{e}_{b}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e%
}_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{%
e}_{a}\mathbf{e}_{a}^{\prime }%
\end{array}%
\! \right] \\
&-&\!\! \sum_{abc} \! \mathbf{\kappa }_{3a}\mathbf{\kappa }_{3a} \! \left[ \!\!
\begin{array}{cccc}
\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e%
}_{c}\mathbf{e}_{c}^{\prime } & \mathbf{e}_{a}\mathbf{e}_{b}^{\prime
}\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{%
e}_{c}^{\prime }\otimes \mathbf{e}_{c}\mathbf{e}_{b}^{\prime } & \cdots &
\mathbf{e}_{a}\mathbf{e}_{c}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e%
}_{c}\mathbf{e}_{b}^{\prime } \\
& \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{c}\mathbf{e}_{c}^{\prime }\otimes \mathbf{e%
}_{a}\mathbf{e}_{b}^{\prime } & & \vdots \\
& & \ddots & \\
& & & \mathbf{e}_{c}\mathbf{e}_{c}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e%
}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{%
e}_{a}\mathbf{e}_{b}^{\prime }%
\end{array}%
\!\! \right] ,\ \ \ \ \
\end{eqnarray*}%
and $\mathbb{C}\left[ \mathbf{\varepsilon }_{t,r}^{\left[ 4\right] },\mathbf{%
\varepsilon }_{r,t}^{\left[ 4\right] }\right] $ is%
\begin{equation*}
\sum_{ab}\mathbf{\mu }_{4a}\mathbf{\mu }_{4b}\left[
\begin{array}{cccc}
\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e%
}_{b}\mathbf{e}_{a}^{\prime } & \mathbf{e}_{a}\mathbf{e}_{b}^{\prime
}\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{%
e}_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{b}^{\prime } & \cdots &
\mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e%
}_{b}\mathbf{e}_{b}^{\prime } \\
\vdots & \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}%
\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime
}\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime } & & \vdots \\
& & \ddots & \\
\mathbf{e}_{b}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e%
}_{a}\mathbf{e}_{a}^{\prime } & \cdots & & \mathbf{e}_{b}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e%
}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }%
\end{array}%
\right] .
\end{equation*}%
$\bigskip $ }
{\small \noindent \textbf{Covariance terms. }For $k=2,3,$ using that $%
\mathbb{C}\left( \mathbf{\varepsilon }_{t}^{\left( 0\right) },\mathbf{%
\varepsilon }_{t,r}^{\left[ 3\right] }\right) =0,$ we find that%
\begin{eqnarray*}
\mathbb{C}\left( \mathbf{Z}_{2,T},\mathbf{Z}_{3,T}\right) &\rightarrow &%
\mathbf{C}_{2}\left( 0\right) \mathbb{C}\left( \mathbf{\varepsilon }%
_{t}^{\otimes 3},\mathbf{\varepsilon }_{t}^{\otimes 3}\right) \mathbf{C}%
_{3}^{\prime }\left( 0\right) +\sum_{j=1}^{\infty }\left[
\begin{array}{c}
\mathbf{C}_{2}^{\prime }\left( -j\right) \\
\mathbf{C}_{2}^{\prime }\left( j\right)
\end{array}%
\right] ^{\prime }\mathbb{C}\left( \left[
\begin{array}{c}
\mathbf{\varepsilon }_{t,r}^{\left[ 2\right] } \\
\mathbf{\varepsilon }_{r,t}^{\left[ 2\right] }%
\end{array}%
\right] ,\left[
\begin{array}{c}
\mathbf{\varepsilon }_{t,r}^{\left[ 3\right] } \\
\mathbf{\varepsilon }_{r,t}^{\left[ 3\right] }%
\end{array}%
\right] \right) \left[
\begin{array}{c}
\mathbf{C}_{3}^{\prime }\left( -j\right) \\
\mathbf{C}_{3}^{\prime }\left( j\right)
\end{array}%
\right] \\
&=&\mathbf{C}_{2}\left( 0\right) \ \mathbb{C}\left( \mathbf{\varepsilon }%
_{t}^{\otimes 2},\mathbf{\varepsilon }_{t}^{\otimes 3}\right) \mathbf{C}%
_{3}^{\prime }\left( 0\right) +\sum_{j=-\infty ,\neq 0}^{\infty }\mathbf{C}%
_{2}\left( j\right) \ \mathbb{C}\left( \mathbf{\varepsilon }_{r,t}^{\left[ 2%
\right] },\mathbf{\varepsilon }_{r,t}^{\left[ 3\right] }\right) \mathbf{C}%
_{3}^{\prime }\left( j\right) \\
&&+\sum_{j=-\infty ,\neq 0}^{\infty }\mathbf{C}_{2}\left( j\right) \ \mathbb{%
C}\left( \mathbf{\varepsilon }_{r,t}^{\left[ 2\right] },\mathbf{\varepsilon }%
_{t,r}^{\left[ 3\right] }\right) \mathbf{C}_{3}^{\prime }\left( -j\right) \\
&=&\Phi _{2,3}^{0}+\Phi _{2,3}+\Phi _{2,3}^{\dag }
\end{eqnarray*}%
because $\mathbb{C}\left( \mathbf{\varepsilon }_{t}^{\otimes 2},\ \ \mathbf{%
\varepsilon }_{t,r}^{\left[ 3\right] }\right) =\mathbb{C}\left( \mathbf{%
\varepsilon }_{t}^{\otimes 2},\ \ \mathbf{\varepsilon }_{r,t}^{\left[ 3%
\right] }\right) =0$, $\mathbb{C}\left( \mathbf{\varepsilon }_{r,t}^{\left[ 3%
\right] },\ \ \mathbf{\varepsilon }_{t}^{\otimes 2}\right) =\mathbb{C}\left(
\mathbf{\varepsilon }_{t,r}^{\left[ 3\right] },\ \ \mathbf{\varepsilon }%
_{t}^{\otimes 2}\right) =0,$ and $\mathbb{C}\left( \mathbf{\varepsilon }%
_{t,r}^{\left[ 2\right] },\mathbf{\varepsilon }_{t,r}^{\left[ 3\right]
}\right) =\mathbb{C}\left( \mathbf{\varepsilon }_{r,t}^{\left[ 2\right] },%
\mathbf{\varepsilon }_{r,t}^{\left[ 3\right] }\right) $ with%
\begin{equation*}
\mathbb{C}\left( \mathbf{\varepsilon }_{t,r}^{\left[ 2\right] },\mathbf{%
\varepsilon }_{t,r}^{\left[ 3\right] }\right) =\sum_{a,b}\mathbf{\kappa }%
_{3,b}^{0}\left[
\begin{array}{ccc}
\mathbf{e}_{b}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}^{\prime } & \mathbf{e}_{b}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e%
}_{b}^{\prime } & \mathbf{e}_{b}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}%
_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}^{\prime } \\
\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}^{\prime } & \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e%
}_{b}^{\prime } & \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}%
_{b}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}^{\prime }%
\end{array}%
\right]
\end{equation*}%
and $\mathbb{C}\left( \mathbf{\varepsilon }_{t,r}^{\left[ 2\right] },\mathbf{%
\varepsilon }_{r,t}^{\left[ 3\right] }\right) =\mathbb{C}\left( \mathbf{%
\varepsilon }_{r,t}^{\left[ 2\right] },\mathbf{\varepsilon }_{t,r}^{\left[ 3%
\right] }\right) $ with%
\begin{equation*}
\mathbb{C}\left( \mathbf{\varepsilon }_{t,r}^{\left[ 2\right] },\mathbf{%
\varepsilon }_{r,t}^{\left[ 3\right] }\right) =\sum_{a,b}\mathbf{\kappa }%
_{3,b}^{0}\left[
\begin{array}{ccc}
\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}^{\prime } & \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e%
}_{b}^{\prime } & \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}%
_{b}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}^{\prime } \\
\mathbf{e}_{b}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}^{\prime } & \mathbf{e}_{b}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e%
}_{b}^{\prime } & \mathbf{e}_{b}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}%
_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}^{\prime }%
\end{array}%
\right] .
\end{equation*}%
For $k=2,4,$ using now that $\mathbb{C}\left( \mathbf{\varepsilon }%
_{t}^{\left( 0\right) },\mathbf{\varepsilon }_{t,r}^{\left[ 4\right]
}\right) =0,$%
\begin{eqnarray*}
\mathbb{C}\left( \mathbf{Z}_{2,T},\mathbf{Z}_{4,T}\right) &\rightarrow &%
\mathbf{C}_{2}\left( 0\right) \mathbb{C}\left( \mathbf{\varepsilon }%
_{t}^{\otimes 2},\mathbf{\varepsilon }_{t}^{\otimes 4}\right) \mathbf{C}%
_{4}^{\prime }\left( 0\right) +\sum_{j=1}^{\infty }\left[
\begin{array}{c}
\mathbf{C}_{2}^{\prime }\left( -j\right) \\
\mathbf{C}_{2}^{\prime }\left( j\right)
\end{array}%
\right] ^{\prime }\mathbb{C}\left( \left[
\begin{array}{c}
\mathbf{\varepsilon }_{t,r}^{\left[ 2\right] } \\
\mathbf{\varepsilon }_{r,t}^{\left[ 2\right] }%
\end{array}%
\right] ,\left[
\begin{array}{c}
\mathbf{\varepsilon }_{t,r}^{\left[ 4\right] } \\
\mathbf{\varepsilon }_{r,t}^{\left[ 4\right] }%
\end{array}%
\right] \right) \left[
\begin{array}{c}
\mathbf{C}_{4}^{\prime }\left( -j\right) \\
\mathbf{C}_{4}^{\prime }\left( j\right)
\end{array}%
\right] \\
&=&\Phi _{2,4}^{0}+\Phi _{2,4}+\Phi _{2,4}^{\dag }
\end{eqnarray*}%
because $\mathbb{C}\left( \mathbf{\varepsilon }_{t}^{\otimes 2},\ \ \mathbf{%
\varepsilon }_{t,r}^{\left[ 4\right] }\right) =\mathbb{C}\left( \mathbf{%
\varepsilon }_{t}^{\otimes 2},\ \ \mathbf{\varepsilon }_{r-t}^{\left[ 4%
\right] }\right) =0$, $\mathbb{C}\left( \mathbf{\varepsilon }_{r-t}^{\left[ 2%
\right] },\ \ \mathbf{\varepsilon }_{t}^{\otimes 4}\right) =\mathbb{C}\left(
\mathbf{\varepsilon }_{t,r}^{\left[ 2\right] },\ \ \mathbf{\varepsilon }%
_{t}^{\otimes 4}\right) =0,$ while
\begin{eqnarray*}
\mathbb{C}\left( \left[
\begin{array}{c}
\mathbf{\varepsilon }_{t,r}^{\left[ 2\right] } \\
\mathbf{\varepsilon }_{r,t}^{\left[ 2\right] }%
\end{array}%
\right] ,\ \ \mathbf{\varepsilon }_{t,r}^{\left[ 4\right] }\right) &=&%
\sum_{a,b}\mathbf{\mu }_{4,a}^{0}\left[
\begin{array}{cccc}
\mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{b}^{\prime
} & \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}^{\prime
} & \ddots & \\
\mathbf{e}_{b}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{b}^{\prime
} & \mathbf{e}_{b}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}^{\prime
} & & \ddots
\end{array}%
\right] ,
\end{eqnarray*}%
and
\begin{eqnarray*}
\mathbb{C}\left( \left[
\begin{array}{c}
\mathbf{\varepsilon }_{t,r}^{\left[ 2\right] } \\
\mathbf{\varepsilon }_{r,t}^{\left[ 2\right] }%
\end{array}%
\right] ,\ \ \mathbf{\varepsilon }_{r,t}^{\left[ 4\right] }\right) &=&%
\sum_{a,b}\mathbf{\mu }_{4,a}^{0}\left[
\begin{array}{cccc}
\mathbf{e}_{b}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{b}^{\prime
} & \mathbf{e}_{b}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}^{\prime
} & \ddots & \\
\mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{b}^{\prime
} & \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}^{\prime
} & & \ddots
\end{array}%
\right] .
\end{eqnarray*}%
Finally, for $k=3,4$.
\begin{eqnarray*}
\mathbb{C}\left( \mathbf{Z}_{3,T},\mathbf{Z}_{4,T}\right) &\rightarrow &%
\mathbf{C}_{3}\left( 0\right) \mathbb{C}\left( \mathbf{\varepsilon }%
_{t}^{\otimes 3},\mathbf{\varepsilon }_{t}^{\otimes 4}\right) \mathbf{C}%
_{4}^{\prime }\left( 0\right) +\sum_{j=0}^{\infty }\left[
\begin{array}{c}
\mathbf{C}_{3}^{\prime }\left( -j\right) \\
\mathbf{C}_{3}^{\prime }\left( j\right)
\end{array}%
\right] ^{\prime }\mathbb{C}\left( \left[
\begin{array}{c}
\mathbf{\varepsilon }_{t,r}^{\left[ 3\right] } \\
\mathbf{\varepsilon }_{r,t}^{\left[ 3\right] }%
\end{array}%
\right] ,\left[
\begin{array}{c}
\mathbf{\varepsilon }_{t,r}^{\left[ 4\right] } \\
\mathbf{\varepsilon }_{r,t}^{\left[ 4\right] }%
\end{array}%
\right] \right) \left[
\begin{array}{c}
\mathbf{C}_{4}\left( -j\right) ^{\prime } \\
\mathbf{C}_{4}\left( j\right) ^{\prime }%
\end{array}%
\right] \\
&=&\Phi _{3,4}^{0}+\Phi _{3,4}+\Phi _{3,4}^{\dag }
\end{eqnarray*}%
because $\mathbb{C}\left( \mathbf{\varepsilon }_{t}^{\otimes 3},\ \ \mathbf{%
\varepsilon }_{t,r}^{\left[ 4\right] }\right) =\mathbb{C}\left( \mathbf{%
\varepsilon }_{t}^{\otimes 3},\ \ \mathbf{\varepsilon }_{r,t}^{\left[ 4%
\right] }\right) =0$, $\mathbb{C}\left( \mathbf{\varepsilon }_{t,r}^{\left[ 3%
\right] },\ \ \mathbf{\varepsilon }_{t}^{\otimes 4}\right) =\mathbb{C}\left(
\mathbf{\varepsilon }_{r,t}^{\left[ 3\right] },\ \ \mathbf{\varepsilon }%
_{t}^{\otimes 4}\right) =0\ $and $\mathbb{C}\left( \mathbf{\varepsilon }%
_{t,r}^{\left[ 3\right] },\mathbf{\varepsilon }_{t,r}^{\left[ 4\right]
}\right) =\mathbb{C}\left( \mathbf{\varepsilon }_{r,t}^{\left[ 3\right] },%
\mathbf{\varepsilon }_{r,t}^{\left[ 4\right] }\right) $ is%
\begin{eqnarray*}
&&
\sum_{a,b}\mathbf{\mu }_{5a}^{0}\left(
\begin{array}{cccc}
\mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e%
}_{b}^{\prime } & \cdots & & \\
\vdots & \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{b}%
\mathbf{e}_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime
}\otimes \mathbf{e}_{a}^{\prime } & & \\
& & \mathbf{e}_{b}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime }\otimes \mathbf{e%
}_{a}^{\prime } & \ddots
\end{array}%
\right) ,
\end{eqnarray*}%
while $\mathbb{C}\left( \mathbf{\varepsilon }_{t,r}^{\left[ 3\right] },%
\mathbf{\varepsilon }_{r,t}^{\left[ 4\right] }\right) =\mathbb{C}\left(
\mathbf{\varepsilon }_{r,t}^{\left[ 3\right] },\mathbf{\varepsilon }_{t,r}^{%
\left[ 4\right] }\right) $ is
\begin{eqnarray*}
&&
\sum_{a,b}\mathbf{\mu }_{3a}^{0}\mathbf{\mu }_{4b}^{0}\left(
\begin{array}{cccc}
\mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{b}^{\prime }\otimes \mathbf{e}_{b}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e%
}_{a}^{\prime } & \cdots & & \\
\vdots & \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{b}%
\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{a}^{\prime
}\otimes \mathbf{e}_{b}^{\prime } & & \\
& & \mathbf{e}_{b}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}%
_{a}^{\prime }\otimes \mathbf{e}_{a}\mathbf{e}_{b}^{\prime }\otimes \mathbf{e%
}_{b}^{\prime } & \ddots
\end{array}%
\right) .
\end{eqnarray*}%
Then the joint CLT for averages of periodograms $\mathbf{Z}_{k,T}$
of orders $k=2,3,4$ follows as in VL using the Cramer-Wold device, noticing
that $\left( Z_{2,t}^{0\prime },Z_{3,t}^{0},Z_{4,t}^{0}\right) ^{\prime }$
is a martingale difference under independence of order $4$ with the given
asymptotic variance for independence of order $8,$ while the proof of the
convergence of conditional variances and Lindeberg Feller condition follows
as in the univariate case as they depend on the rate of decay of the norm of
the matrices $\mathbf{C}\left( j\right) $ of scores, which are similar to
the sequences $c_{j}$ in VL. \bigskip }
\section{\protect \small Appendix E: An alternative parameterization}
{\small We can formulate the $d$-dimensional VARMA$\left( p,q\right) $
process with no standardized errors by
\begin{equation}
\Phi _{\mathbf{\theta }}\left( L\right) Y_{t}=\mu + \bar{\Theta}_{\mathbf{%
\theta }}\left( L\right) \mathbf{\varepsilon }_{t},\ \ \ \mathbf{\varepsilon
}_{t}\sim iid_k\left( \mathbf{0},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}%
}\left( \mathbf{\alpha }_{k}\right) ,k=2,3,4\right) \label{Repnew}
\end{equation}%
where
\begin{equation*}
\bar{\Theta}_{\mathbf{\theta }}\left( L\right) =\bar{\Theta}_{0}\left(
\mathbf{\theta }\right) +\bar{\Theta}_{1}\left( \mathbf{\theta }\right)
L+\cdots +\bar{\Theta}_{q}\left( \mathbf{\theta }\right) L^{q}
\end{equation*}%
with diag$\left \{ \bar{\Theta}_{0}\left( \mathbf{\theta }\right) \right \}
=\left( 1,\ldots ,1\right) ^{\prime }$ for all $\mathbf{\theta }\in \mathcal{%
S}\subset \mathbb{R}^{m}$ imposing a normalization on top of the
positiveness of Assumptions~6A and B on the diagonal values of $\Theta _{0}$%
, and v$\mathbf{\kappa }_{k}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left( \mathbf{\alpha }_{k}\right) $
given in (\ref{vk3}) and (\ref{vk4}) for $k=3$ and 4, respectively, it is
also imposed for $k=2$ for a vector $\mathbf{\alpha }_{2}$ which now
includes the (positive) variance of all innovations, which are not
normalized to 1. Conversely, $\mathbf{\alpha }_{k}$ for $k=3,4,$ are now
interpreted as the level third and fourth cumulants (and not as skewness or
kurtosis coefficients), as $\bar{\Theta}_{\mathbf{\theta }}$ is now scaling
free. Likewise, $\mathbf{\theta }$ now, as $\bar{\Theta}_{\mathbf{\theta }},$
does not includes scale parameterization and the asymptotic properties of
spectral estimations are slightly simpler because of the symmetry for all $%
k=2,3,4$. Thus Assumption~3$\left( 2\right) $ implies uncorrelation, i.e.
independence of order $k=2$, so that the $k$-th order spectral density
parametric model for each time series combination $\mathbf{a}$ with
representation $\left( \ref{Repnew}\right) $ is given for all $k=2,3,4,$ by
\begin{equation*}
f_{\mathbf{a},k}(\boldsymbol{\lambda };\mathbf{\theta },\mathbf{\alpha }%
_{k})=\left( \Phi _{\mathbf{\theta }}^{-1}\bar{\Theta}_{\mathbf{\theta }%
}\right) _{\mathbf{a}}^{\otimes k}\left( \mathbf{\lambda }\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{vec}%
\left( \RIfM@\expandafter\text@\else\expandafter\mbox\fi{v}\mathbf{\kappa }_{k}^{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{IC}}\left( \mathbf{\alpha }%
_{k}\right) \right) =\left( \Phi _{\mathbf{\theta }}^{-1}\bar{\Theta}_{%
\mathbf{\theta }}\right) _{\mathbf{a}}^{\otimes k}\left( \mathbf{\lambda }%
\right) \mathbf{S}_{k}\mathbf{\alpha }_{k}\mathbf{.}
\end{equation*}
Following with the same arguments as in Section~5, we can show the
consistency of $\mathbf{\hat{\alpha}}_{2,T}\left( \mathbf{\theta }\right) $
and set the simpler estimate $\mathbf{\hat{\alpha}}_{2,T}^{\dag }\left(
\mathbf{\theta }\right) $ of the vector of variances, which is equivalent to
the usual prewhitened estimate in the frequency domain of the diagonal of
the variance covariance matrix of $\mathbf{\varepsilon }_{t},$
\begin{eqnarray*}
\mathbf{\hat{\alpha}}_{2,T}^{\dag }\left( \mathbf{\theta }\right) &:=&\frac{1%
}{T}\sum_{j=1}^{T-1}\func{Re}\left \{ \mathbf{S}_{2}^{\prime }\mathbf{\Psi }%
_{2}^{-1}(\lambda _{j};\mathbf{\theta })\mathbb{I}_{2}(\lambda _{j})\right \}
\\
&=&\mathbf{S}_{2}^{\prime }\frac{1}{T}\sum_{j=1}^{T-1}\func{Re}\left \{
\mathbf{\Psi }^{-1}(-\lambda _{j};\mathbf{\theta })w_{T}(-\lambda
_{j})\otimes \mathbf{\Psi }^{-1}(\lambda _{j};\mathbf{\theta })w_{T}(\lambda
_{j})\right \} \\
&=&\frac{1}{T}\sum_{j=1}^{T-1}\func{Re}\left \{ \mathbf{e}_{a}^{\prime }%
\mathbf{\Psi }^{-1}(-\lambda _{j};\mathbf{\theta })w_{T}(-\lambda
_{j})\otimes \mathbf{e}_{a}^{\prime }\mathbf{\Psi }^{-1}(\lambda _{j};%
\mathbf{\theta })w_{T}(\lambda _{j})\right \} _{a=1,\ldots ,d} \\
&=&\frac{1}{T}\sum_{j=1}^{T-1}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{diag}\left \{ \mathbf{\Psi }%
^{-1}(\lambda _{j};\mathbf{\theta })I_{T}\left( \lambda _{j}\right) \mathbf{%
\Psi }^{-1}(-\lambda _{j};\mathbf{\theta })^{\prime }\right \}
\end{eqnarray*}%
where $I_{T}\left( \lambda _{j}\right) =w_{T}(\lambda _{j})w_{T}^{\prime
}(-\lambda _{j})$ is the usual (second order) periodogram matrix. Then,
following as in Theorem~$\ref{Th5},$ we can show the consistency of $\mathbf{%
\hat{\alpha}}_{2,T}^{\dag },$ and in fact, the consistency of $\mathbf{\hat{%
\theta}}_{k,T}$ without using $\mathcal{L}_{2,T}$ in the aggregated loss
function, as scaling is now excluded from $\mathbf{\theta }$ as $\mathbf{%
\bar{\Theta}}$ is normalized. Then, the three sets of cumulants (no
coefficients) can be identified independently and $\mathbf{\theta }$ can be
identified using a unique $\mathcal{L}_{k,T},$ $k=3,4.$ }
{\small To describe the asymptotic distribution of estimates, we also update
for $k=2 $
\begin{equation*}
\mathbf{B}_{2}\left( \mathbf{\lambda };\mathbf{\theta }\right) :=\mathbf{%
\Psi }_{2}^{-1}\left( \mathbf{\lambda };\mathbf{\theta }\right) \mathbf{\dot{%
\Psi}}_{2}(\mathbf{\lambda };\mathbf{\theta })-\mathbf{S}_{2}\mathbf{S}%
_{2}^{\prime }\bar{\mathbf{\Lambda}}_{2}\left( \mathbf{\theta }\right) ,\ \
\ \bar{\mathbf{\Lambda}}_{2}\left( \mathbf{\theta }\right) :=\left( 2\pi
\right) ^{-1}\int_{\Pi }\func{Re}\left \{ \mathbf{\Psi }_{2}^{-1}\left(
\lambda ;\mathbf{\theta }\right) \mathbf{\dot{\Psi}}_{2}(\lambda ;\mathbf{%
\theta })\right \} d\lambda
\end{equation*}%
and redefine for $\mathbf{\alpha }=\left( \mathbf{\alpha }_{2}^{\prime }%
\mathbf{,\alpha }_{3}^{\prime }\mathbf{,\alpha }_{4}^{\prime }\right)
^{\prime }$%
\begin{equation*}
\mathbf{\Sigma }\left( \mathbf{\theta ,\alpha }\right) :=\sum _{k\in
\mathcal{K}}w_{k}\left( \mathbf{I}_{m}\otimes \mathbf{S}_{k}\mathbf{\alpha }%
_{k}\right) ^{\prime }\mathbf{H}_{k}\left( \mathbf{\theta }\right) \left(
\mathbf{I}_{m}\otimes \mathbf{S}_{k}\mathbf{\alpha }_{k}\right)
\end{equation*}%
where now $w_{2}\geq 0$ and $\mathbf{H}_{2}$ is updated with the new $%
\mathbf{B}_{2}.$ }
{\small Then, Theorem~$\ref{Th6}$ holds for the new parameterization under
the same regularity conditions with%
\begin{equation*}
\mathbf{\delta }\left( \mathbf{\alpha }_{0}\right) :=\left[ \ \ w_{2} \left(
\mathbf{I}_{m}\otimes \mathbf{S}_{2}\mathbf{\alpha }_{2}^{0} \right)
^{\prime }\ \ \left \vert \ \ w_{3}\left( \mathbf{I}_{m}\otimes \mathbf{S}%
_{3}\mathbf{\alpha }_{3}^{0}\right) ^{\prime }\ \ \right \vert \ \
w_{4}\left( \mathbf{I}_{m}\otimes \mathbf{S}_{4}\mathbf{\alpha }%
_{4}^{0}\right) ^{\prime }\ \ \right] .
\end{equation*}
and where now $\mathbf{C}_{2}\left( 0\right) =\left( \mathbf{I}_{d}\otimes
\mathbf{\mathbf{C}}\left( 0\right) +\mathbf{\mathbf{C}}\left( 0\right)
\otimes \mathbf{I}_{d}\right) \left( \mathbf{I}_{d^{2}}-\mathbf{S}_{2}%
\mathbf{S}_{2}^{\prime }\right) $ neatly incorporates the effect of the
scaling estimation with $\mathbf{\Phi }_{22}^{0}\left( \mathbf{\theta }_{0};%
\mathbf{C}\right) =\mathbf{C}_{2}\left( 0\right) \mathbb{V}\left[ \mathbf{%
\varepsilon }_{t}^{\otimes 2}\right] \mathbf{C}_{2}^{\prime }\left( 0\right)
$ not depending on fourth order cumulants for both fundamental or
non-fundamental models as in VL, unlike in the original parameterization, in
the same way as $\mathbf{\Phi }_{kk}^{0}\left( \mathbf{\theta }_{0};\mathbf{C%
}\right) $ does not depend on $\mathbf{\alpha }_{2k}^{0}$ for either
parametrization. }
{\small The properties of $\mathbf{\hat{\alpha}}_{2,T}^{\dag }\left( \mathbf{%
\hat{\theta}}_{w,T}^{\dag }\right) $ can be described in an extended version
of Theorem~$\ref{Th7},$ which covers $\mathbf{\hat{\alpha}}_{k,T}^{\dag }$
for all $k=2,3,4$ with $\mathbf{D}_{k,2}$ defined also by the general
formulation of $\mathbf{D}_{k,h},$ i.e.,%
\begin{equation*}
\mathbf{D}_{k,2}\left( 0\right) :=\mathbf{I}_{d^{k}}1_{\left \{ k=2\right \}
}-w_{2}\mathbf{\bar{\Lambda}}_{k}\left( \mathbf{\theta }_{0}\right) \left(
\mathbf{I}_{m}\otimes \mathbf{S}_{k}\mathbf{\alpha }_{k}^{0}\right) \mathbf{%
\Sigma }^{-1}\left( \mathbf{\theta }_{0},\mathbf{\alpha }_{0}\right) \left(
\mathbf{I}_{m}\otimes \mathbf{S}_{2}\mathbf{\alpha }_{2}^{0}\right) ^{\prime
}\mathbf{\mathbf{C}}_{2}\left( 0\right) .
\end{equation*}
}
\vspace{-.8cm}
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\end{document}
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\def\alpha{{\Greekmath 010B}}%
\def\beta{{\Greekmath 010C}}%
\def\gamma{{\Greekmath 010D}}%
\def\delta{{\Greekmath 010E}}%
\def\epsilon{{\Greekmath 010F}}%
\def\zeta{{\Greekmath 0110}}%
\def\eta{{\Greekmath 0111}}%
\def\theta{{\Greekmath 0112}}%
\def\iota{{\Greekmath 0113}}%
\def\kappa{{\Greekmath 0114}}%
\def\lambda{{\Greekmath 0115}}%
\def\mu{{\Greekmath 0116}}%
\def\nu{{\Greekmath 0117}}%
\def\xi{{\Greekmath 0118}}%
\def\pi{{\Greekmath 0119}}%
\def\rho{{\Greekmath 011A}}%
\def\sigma{{\Greekmath 011B}}%
\def\tau{{\Greekmath 011C}}%
\def\upsilon{{\Greekmath 011D}}%
\def\phi{{\Greekmath 011E}}%
\def\chi{{\Greekmath 011F}}%
\def\psi{{\Greekmath 0120}}%
\def\omega{{\Greekmath 0121}}%
\def\varepsilon{{\Greekmath 0122}}%
\def\vartheta{{\Greekmath 0123}}%
\def\varpi{{\Greekmath 0124}}%
\def\varrho{{\Greekmath 0125}}%
\def\varsigma{{\Greekmath 0126}}%
\def\varphi{{\Greekmath 0127}}%
\def{\Greekmath 0272}{{\Greekmath 0272}}
\def\FindBoldGroup{%
{\setbox0=\hbox{$\mathbf{x\global\edef\theboldgroup{\the\mathgroup}}$}}%
}
\def\Greekmath#1#2#3#4{%
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{\mbox{\boldmath$\textstyle\mathchar"#1#2#3#4$}}%
{\mbox{\boldmath$\scriptstyle\mathchar"#1#2#3#4$}}%
{\mbox{\boldmath$\scriptscriptstyle\mathchar"#1#2#3#4$}}%
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\mathchar"#1#2#3#
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\FindBoldGroup
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{\mbox{\boldmath$\scriptstyle\mathchar"#1#2#3#4$}}%
{\mbox{\boldmath$\scriptscriptstyle\mathchar"#1#2#3#4$}}%
\else
\mathchar"#1#2#3#
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\fi}
\newif\ifGreekBold \GreekBoldfalse
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\global\def\@taggnum{(#1)}%
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\def\QABOVE#1#2#3{{#2 \above#1 #3}}%
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\def\QDATOPD#1#2#3#4{{\displaystyle {#3 \atopwithdelims#1#2 #4}}}%
\def\QABOVED#1#2#3#4#5{{#4 \abovewithdelims#1#2#3 #5}}%
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\def\tiint{\mathop{\textstyle \iint }}%
\def\tiiint{\mathop{\textstyle \iiint }}%
\def\tiiiint{\mathop{\textstyle \iiiint }}%
\def\tidotsint{\mathop{\textstyle \idotsint }}%
\def\toint{\mathop{\textstyle \oint}}%
\def\tsum{\mathop{\textstyle \sum }}%
\def\tprod{\mathop{\textstyle \prod }}%
\def\tbigcap{\mathop{\textstyle \bigcap }}%
\def\tbigwedge{\mathop{\textstyle \bigwedge }}%
\def\tbigoplus{\mathop{\textstyle \bigoplus }}%
\def\tbigodot{\mathop{\textstyle \bigodot }}%
\def\tbigsqcup{\mathop{\textstyle \bigsqcup }}%
\def\tcoprod{\mathop{\textstyle \coprod }}%
\def\tbigcup{\mathop{\textstyle \bigcup }}%
\def\tbigvee{\mathop{\textstyle \bigvee }}%
\def\tbigotimes{\mathop{\textstyle \bigotimes }}%
\def\tbiguplus{\mathop{\textstyle \biguplus }}%
\def\dint{\mathop{\displaystyle \int}}%
\def\diint{\mathop{\displaystyle \iint}}%
\def\diiint{\mathop{\displaystyle \iiint}}%
\def\diiiint{\mathop{\displaystyle \iiiint }}%
\def\didotsint{\mathop{\displaystyle \idotsint }}%
\def\doint{\mathop{\displaystyle \oint}}%
\def\dsum{\mathop{\displaystyle \sum }}%
\def\dprod{\mathop{\displaystyle \prod }}%
\def\dbigcap{\mathop{\displaystyle \bigcap }}%
\def\dbigwedge{\mathop{\displaystyle \bigwedge }}%
\def\dbigoplus{\mathop{\displaystyle \bigoplus }}%
\def\dbigodot{\mathop{\displaystyle \bigodot }}%
\def\dbigsqcup{\mathop{\displaystyle \bigsqcup }}%
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\def\dbigcup{\mathop{\displaystyle \bigcup }}%
\def\dbigvee{\mathop{\displaystyle \bigvee }}%
\def\dbigotimes{\mathop{\displaystyle \bigotimes }}%
\def\dbiguplus{\mathop{\displaystyle \biguplus }}%
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\RequirePackage{amsmath}
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\def\makeatother\endinput{\makeatother\endinput}
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\@ifpackageloaded{amsmath}%
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\def\iiint{\DOTSI\intno@\thr@@\FN@\ints@}%
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\newif\iflimtoken@
\newif\iflimits@
\def\findlimits@{\limtoken@true\ifx\next\limits\limits@true
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\ifnum\intno@>\tw@\int\intkern@\fi
\ifnum\intno@>\thr@@\int\intkern@\fi
\int
\def\multintlimits@{\intop\ifnum\intno@=\z@\intdots@\else\intkern@\fi
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\mathchoice{\hskip.5em}{\hskip.4em}{\hskip.4em}{\hskip.4em}}%
\def\negintic@{\mathchoice
{\hskip-.5em}{\hskip-.4em}{\hskip-.4em}{\hskip-.4em}}%
\def\ints@@{\iflimtoken@
\def\ints@@@{\iflimits@\negintic@
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\else\multint@\nolimits\fi
\eat@
\else
\def\ints@@@{\iflimits@\negintic@
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\def\intkern@{\mathchoice{\!\!\!}{\!\!}{\!\!}{\!\!}}%
\def\plaincdots@{\mathinner{\cdotp\cdotp\cdotp}}%
\def\intdots@{\mathchoice{\plaincdots@}%
{{\cdotp}\mkern1.5mu{\cdotp}\mkern1.5mu{\cdotp}}%
{{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}%
{{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}}%
\def\RIfM@{\relax\protect\ifmmode}
\def\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\RIfM@\expandafter\RIfM@\expandafter\text@\else\expandafter\mbox\fi@\else\expandafter\mbox\fi}
\let\nfss@text\RIfM@\expandafter\text@\else\expandafter\mbox\fi
\def\RIfM@\expandafter\text@\else\expandafter\mbox\fi@#1{\mathchoice
{\textdef@\displaystyle\f@size{#1}}%
{\textdef@\textstyle\tf@size{\firstchoice@false #1}}%
{\textdef@\textstyle\sf@size{\firstchoice@false #1}}%
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\glb@settings}
\def\textdef@#1#2#3{\hbox{{%
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\let\f@size#2\selectfont
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\newif\iffirstchoice@
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\def\vspace@{\def\vspace##1{\crcr\noalign{\vskip##1\relax}}}%
\def\multilimits@{\bgroup\vspace@\Let@
\baselineskip\fontdimen10 \scriptfont\tw@
\advance\baselineskip\fontdimen12 \scriptfont\tw@
\lineskip\thr@@\fontdimen8 \scriptfont\thr@@
\lineskiplimit\lineskip
\vbox\bgroup\ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil\crcr}%
\def\Sb{_\multilimits@}%
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\def\Sp{^\multilimits@}%
\let\endSp\endSb
\newdimen\ex@
\[email protected]
\def\rightarrowfill@#1{$#1\m@th\mathord-\mkern-6mu\cleaders
\hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill
\mkern-6mu\mathord\rightarrow$}%
\def\leftarrowfill@#1{$#1\m@th\mathord\leftarrow\mkern-6mu\cleaders
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\def\leftrightarrowfill@#1{$#1\m@th\mathord\leftarrow
\mkern-6mu\cleaders
\hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill
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\def\overrightarrow{\mathpalette\overrightarrow@}%
\def\overrightarrow@#1#2{\vbox{\ialign{##\crcr\rightarrowfill@#1\crcr
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\let\overarrow\overrightarrow
\def\overleftarrow{\mathpalette\overleftarrow@}%
\def\overleftarrow@#1#2{\vbox{\ialign{##\crcr\leftarrowfill@#1\crcr
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\def\overleftrightarrow{\mathpalette\overleftrightarrow@}%
\def\overleftrightarrow@#1#2{\vbox{\ialign{##\crcr
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\noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\def\underrightarrow{\mathpalette\underrightarrow@}%
\def\underrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil
$\crcr\noalign{\nointerlineskip}\rightarrowfill@#1\crcr}}}%
\let\underarrow\underrightarrow
\def\underleftarrow{\mathpalette\underleftarrow@}%
\def\underleftarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil
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\def\underleftrightarrow{\mathpalette\underleftrightarrow@}%
\def\underleftrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th
\hfil#1#2\hfil$\crcr
\noalign{\nointerlineskip}\leftrightarrowfill@#1\crcr}}}%
\def\qopnamewl@#1{\mathop{\operator@font#1}\nlimits@}
\let\nlimits@\displaylimits
\def\setboxz@h{\setbox\z@\hbox}
\def\varlim@#1#2{\mathop{\vtop{\ialign{##\crcr
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\noalign{\nointerlineskip}#2#1\crcr
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\def\rightarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
$#1\copy\z@\mkern-6mu\cleaders
\hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
\mkern-6mu\mathord\rightarrow$}
\def\leftarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
$#1\mathord\leftarrow\mkern-6mu\cleaders
\hbox{$#1\mkern-2mu\copy\z@\mkern-2mu$}\hfill
\mkern-6mu\box\z@$}
\def\qopnamewl@{proj\,lim}{\qopnamewl@{proj\,lim}}
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\def\mathpalette\varliminf@{}@#1{\mathop{\underline{\vrule\@depth.2\ex@\@width\z@
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\def\stackunder#1#2{\mathrel{\mathop{#2}\limits_{#1}}}%
\begingroup \catcode `|=0 \catcode `[= 1
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|gdef|@alignverbatim#1\end{align}[#1|end[align]]
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|gdef|@gatherverbatim#1\end{gather}[#1|end[gather]]
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|gdef|@arraxverbatim#1\end{arrax}[#1|end[arrax]]
|gdef|@sarraxverbatim#1\end{arrax*}[#1|end[arrax*]]
|gdef|@tabulaxverbatim#1\end{tabulax}[#1|end[tabulax]]
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\def\align{\@verbatim \frenchspacing\@vobeyspaces \@alignverbatim
You are using the "align" environment in a style in which it is not defined.}
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\@namedef{align*}{\@verbatim\@salignverbatim
You are using the "align*" environment in a style in which it is not defined.}
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You are using the "alignat" environment in a style in which it is not defined.}
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\@namedef{alignat*}{\@verbatim\@salignatverbatim
You are using the "alignat*" environment in a style in which it is not defined.}
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You are using the "xalignat" environment in a style in which it is not defined.}
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\@namedef{xalignat*}{\@verbatim\@sxalignatverbatim
You are using the "xalignat*" environment in a style in which it is not defined.}
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You are using the "gather" environment in a style in which it is not defined.}
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\@namedef{gather*}{\@verbatim\@sgatherverbatim
You are using the "gather*" environment in a style in which it is not defined.}
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You are using the "multiline" environment in a style in which it is not defined.}
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\@namedef{multiline*}{\@verbatim\@smultilineverbatim
You are using the "multiline*" environment in a style in which it is not defined.}
\expandafter\let\csname endmultiline*\endcsname =\endtrivlist
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You are using a type of "array" construct that is only allowed in AmS-LaTeX.}
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You are using a type of "tabular" construct that is only allowed in AmS-LaTeX.}
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You are using a type of "array*" construct that is only allowed in AmS-LaTeX.}
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\@namedef{tabulax*}{\@verbatim\@stabulaxverbatim
You are using a type of "tabular*" construct that is only allowed in AmS-LaTeX.}
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\ifmmode\ifinner
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$\hfil
\displaywidth\linewidth\@taggnum\egroup \endtrivlist
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$\hfil
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\iftag@
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\eqno \hbox{\@taggnum}
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$$\global\@ignoretrue
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$$\global\@ignoretrue
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\newif\iftag@ \@ifnextchar*{\@tagstar}{\@tag}@false
\def\@ifnextchar*{\@TCItagstar}{\@TCItag}{\@ifnextchar*{\@TCItagstar}{\@TCItag}}
\def\@TCItag#1{%
\global\@ifnextchar*{\@tagstar}{\@tag}@true
\global\def\@taggnum{(#1)}%
\global\def\@currentlabel{#1}}
\def\@TCItagstar*#1{%
\global\@ifnextchar*{\@tagstar}{\@tag}@true
\global\def\@taggnum{#1}%
\global\def\@currentlabel{#1}}
\@ifundefined{tag}{
\def\@ifnextchar*{\@tagstar}{\@tag}{\@ifnextchar*{\@tagstar}{\@tag}}
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\global\@ifnextchar*{\@tagstar}{\@tag}@true
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}{}
\def\tfrac#1#2{{\textstyle {#1 \over #2}}}%
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\def\binom#1#2{{#1 \choose #2}}%
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\def\dbinom#1#2{{\displaystyle {#1 \choose #2}}}%
\makeatother
\endinput
|
1,116,691,497,789 | arxiv | \section{Introduction}\label{sec-intro}
Supermassive black holes (SMBH)s, with masses between $10^{6}$ and $10^{10}$ solar masses,
lie at the heart of nearly every galaxy~\citep[e.g.][]{kr95}. With a few notable exceptions, these SMBHs dwell within
stellar spheroids -- spiral bulges, ellipticals, and S0s -- and observationally the SMBH mass is highly correlated with properties of its host spherioid~\citep[e.g.][]{geb00, fer00, mcc13, gs15}. This seems to show that the evolution of the SMBH and its host are deeply tied, and innumerable observational and theoretical studies bear this out~\citep{she14, vol12, hop10, mic07, dim05}.
There is an emerging picture
that SMBH hosts all rotate to some degree.
In terms of spiral bulges, rotational support is common. The early-type stars in our own galactic center have a net counterrotation of 120 km/sec~\citep{gen96,sch09}.
The prototypical pseudobulges are rapidly-rotating, disky structures~\citep{Kormendy04}, and most classical bulges show some rotation as well~\citep{gad12}.
Low mass ellipticals and S0s are well-known to be disky and rotationally-supported~\citep[e.g.][]{Faber97, ben94, cap07}. However, rotation may be pervasive in all early-type galaxies; a volume-limited census of the nearest ~250 early-type galaxies ATLAS$^{\rm 3D}$ found that an astonishing 86 percent are fast rotators~\citep{ems11}, with less than 3 percent of the sample being described as non-rotating. One of these non-rotators was thought to be the giant cD galaxy M87~\citep{ems04, kra11}, and yet new IFU data reveal that even this canonical non-rotator has an unmistakable kinematically-distinct core~\citep{arn14, ems14}. In fact, nearly half of the ATLAS$^{\rm 3D}$ sample contains kinematically-decoupled cores~\citep{kra11} (see also~\citep{fra88, ben88, dav01, ems04} ), so it appears that SMBH hosts not only rotate, but that the rotational structure is quite complex.
It is easy, theoretically, to expect that rotation is practically ubiquitous in stellar spheroids. Gas-rich formation scenarios give rise
to rotation, as do dry, non-radial, major mergers~\citep{ben92, kho09, boi11, tsa15}. Indeed, a non-rotating galaxy seems to be a special, and rare,
class that may be created exclusively through a slew of gas-poor minor mergers~\citep{naa14}.
Since SMBH hosts clearly rotate, and since the link between the SMBH and its host is so well-established, any study of the dynamics with a SMBH host should consider rotation. This paper explores the evolution of binary SMBHs in a rotating, flattened stellar system, and its effect on the structure of the stellar host.
Binary SMBHs are expected to form within the galaxy core after a merger, and if the two
SMBHs coalesce, they are arguably the most powerful sources of gravitational radiation
in the Universe~\citep{Hughes03}. One problem that plagues SMBHs is how they merge together. The binary can eject stars via 3-body scattering, bringing the SMBHs ever closer. Once the ejected stars extract enough energy from the binary orbit to shrink the separation to roughly milli-parsec scales, gravitational radiation dominates, and the SMBHs coalesce~\citep{BBR80}. However, the problem is that analytical calculations and simulations of static, spherical galaxies show that the binary's orbital separation stalls before the SMBHs can plunge toward merger (see ~\citep{MM03} for a review). The root cause of this hang up is simply a lack of low-angular momentum stars capable of interacting with the binary via 3-body scattering. This theoretical bottleneck has become known as the infamous ``final parsec problem". Recent work by several teams has shown that SMBHs can readily coalesce in more realistically-shaped galaxy models~\citep{ber06,kh11,Preto11,gm12,kh13}, because the stellar orbits in triaxial and flattened galaxies can continually replenish the SMBH binary loss cone~\citep[e.g.][]{Yu02, MP2004, khb06, va14, li14}.
SMBH binary dynamics has been studied in spherical rotating systems, but the resolution of the simulations (fewer than 100000 particles) could not accurately track the evolution of the binary to the gravitational wave regime. Experiments suggest that we need of the order of 1 million particles to resolve the true evolution of the binary beyond the hard binary stage. These previous studies primarily looked at the evolution of binary eccentricity and inclination with respect to the host~\citep{ses11, ama10, wan14}, and discovered that the eccentricity of the binary
increased dramatically in counterrotating systems, while in corotating systems, the binary tends to circularize.
Here we study SMBH binary dynamics and evolution in a flat rotating galaxy model with b/a = 0.8, which is quite a bit less flattened than a realistic galactic nucleus. For this model, we found an $N$-independent SMBH binary evolution in~\citet{kh13}. In this study we explore the dependence of SMBH binary evolution on particle number for various amounts of rotation of surrounding cusp. We also study the energy, angular momentum and eccentricity evolution of the SMBH binary and estimate the coalescence times by scaling our model to various observed nearby galaxies.
The paper is organized in the following way: In Section~2 we describe the initial conditions and numerical methods for our direct $N$-body simulations. The results for the evolution of the SMBH binary in rotating axisymmetric galaxy models are explained in Section~3. Finally, Section~4 concludes, discussing caveats and future work.
\section{Initial conditions and numerical methods}\label{sec-model}
\subsection{The host galaxies and their SMBHs}
\begin{figure}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig1a.pdf}}
}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig1b.pdf}}
}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig1c.pdf}}
}
\caption[]{Evolution of our flat rotating model (A0) in isolation.
Top Panel: Stellar density profile at various times. $\gamma = 1.0$ is the reference theoretical profile, shown as a dashed line. Middle Panel: Evolution of intermediate to major (b/a) and minor to major (c/a) axis ratio as measured at the half mass radius. Bottom Panel: Ratio of rotational velocity to the 3-d velocity dispersion as a function of distance at various times. In isolation, the rotational support in the center decreases over time as the central SMBH increases the velocity dispersion.
} \label{stab}
\end{figure}
To isolate the effect of rotation, we used the Flat8 model in \citet{kh13} (hereafter KH13) to generate our rotating models; this allows us to directly compare the SMBH binary orbital evolution results here to those in the identical non-rotating model. For this flattening ratio, we found that the SMBH binary evolved into the gravitational wave regime, and the evolution did not depend on particle number for $N>500K$.
We introduced rotation by flipping the z component of the angular momentum, $L_z$, in the positive direction for a subset of particles which have negative $L_z$. Our fiducial model, the A series, flips every particle with negative $L_z$, while our B series only flips 50$\%$ of the negative $L_z$ particles, meaning that $75\%$ of the particles have a positive $L_z$.
Our galaxy models have an inner density slope, $\gamma$, of 1.0 and minor to major axis ratio of 0.8, measured at the half-mass radius. In model units, the total galaxy mass is 1 and the SMBH at the center of each model has a mass of 0.005. Because we adiabatically squeeze the model to generate its shape~\citep{khb01}, the system can and does change its shape with radius as it adiabatically adjusts to a live and changing SMBH-embedded potential. In addition, the scale radius slowly shrinks from 1.0 to 0.5; we could resize our model such that the scale radius is 1.0 again, but we instead choose to adopt physical units to reflect this increased central density. Discussion on scaling of our model to observed galaxies is given in section \ref{coal} and Table \ref{scale} scales the model to physical units.
To determine the stability of our rotating models, we ran them in isolation for 40 time units; this duration is about half the maximum evolution time of the SMBH binaries in this study. Figure \ref{stab} shows that both the density profile and half-mass axes ratio remain very stable for whole duration of the run. The kinematics are also fairly stable; inside the radius of influence, however, the system does become hotter, with $v/\sigma$ decreasing from ~0.6 to ~0.4 as the SMBH re-establishes its characteristic cusp in velocity dispersion. This decrease in rotational support affects only the innermost $\sim$ 5000 particles, but since it is also in the region of the model that exhibits a less flattened, but triaxial shape, it may well be that this region contains orbits that are less stable to rotation~\citep{dei11}. The orbit content of this model will be a subject for future study.
To explore the SMBH binary evolution in flat rotating galaxies, we introduce an equal mass secondary SMBH at a distance of 0.5 with 70 $\%$ of the galaxy's circular velocity at that initial separation. We investigated SMBH binary orbits that are corotating and counterrotating with the sense of the galaxy rotation. Table \ref{TableA} describes parameters of our SMBH binary study. Note that we also evolved the SMBH binary in non-rotating spherical and flattened galaxy models with the same density profile to facilitate the comparison.
\begin{table}
\caption{Rotating Axisymmetric SMBH Binary Parameters}
\centering
\begin{tabular}{c c c c c }
\hline
Run & $N$ & $\gamma$ & $c/a$ & $Rotation$\\
\hline
S0 & 1500k & $1.0$& $1.0$& none\\
S1 & 1000k & $1.0$& $1.0$& none\\
S2 & 500k & $1.0$& $1.0$& none\\
A0 & 1500k & $1.0$& $0.8$& corotating\\
A1 & 1000k & $1.0$& $0.8$& corotating\\
A2 & 800k & $1.0$& $0.8$& corotating\\
A3 & 500k & $1.0$& $0.8$& corotating\\
A4 & 250k & $1.0$& $0.8$& corotating\\[0.2ex]
B0 & 1500k & $1.0$& $0.8$& 0.75\% corotating\\
B1 & 1000k & $1.0$& $0.8$& 0.75\% corotating\\
B2 & 800k & $1.0$& $0.8$& 0.75\% corotating\\
B3 & 500k & $1.0$& $0.8$& 0.75\% corotating\\
B4 & 250k & $1.0$& $0.8$& 0.75\% corotating\\[0.2ex]
C0 & 1500k & $1.0$& $0.8$& counterrotating\\
C1 & 1000k & $1.0$& $0.8$& counterrotating\\
C2 & 800k & $1.0$& $0.8$& counterrotating\\
C3 & 500k & $1.0$& $0.8$& counterrotating\\
\hline
\end{tabular}\label{TableA}
\tablecomments{Column 1: Galaxy model. Column 2: Number of particles. Column 3: Central density slope $\gamma$.
Column 4: Axes ratio. Column 5: Sense of galaxy rotation with respect to the SMBH binary initial orbit.}
\end{table}
The numerical methods and hardware used for this work is described in section 2.2 of KH13.
\section{SMBH BINARY EVOLUTION IN FLAT ROTATING GALAXY MODELS}\label{Resutls}
Here we discuss the results of our numerical studies of SMBH binary evolution in flat rotating galaxy models. The top panel of Figure \ref{rot1} shows the evolution of inverse semi-major axis for the A models. We see that for N greater than 500K, the inverse semi major axis evolution is independent of N, unlike in \citet{va14}. To be conservative, we approach $N$ as high as 1.5 million, never used before in such a study. For reference, we also plot the 1/a evolution of a flat non-rotating model with 1 million from our previous study (KH13), as well as a 1.5 million particle run in spherical galaxy (S0) model with the same density profile as our rotating galaxy model. We can see that in rotating flat models, the SMBH binary evolves at a rate considerably faster than in mere flat galaxy models. We see $N$-independent evolution of the SMBH binary in flat rotating galaxy models for $N$ as large as 1.5 million. This points to a potential stellar dynamical solution to the final parsec problem within flattened galaxy models.
\begin{figure}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig2a.pdf}}
}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig2b.pdf}}
}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig2c.pdf}}
}
\caption[]{Evolution of the semimajor axis of the SMBH binary for the A (top) B (middle) and C (bottom) model suites (see table \ref{TableA}). For comparison, the grey lines represent non-rotating flattened (uppermost grey) and non-rotating spherical (bottommost grey) one million particle models. The opacity of the purple lines scale with the particle number -- the most transparent line has the fewest particles in the suite.
} \label{rot1}
\end{figure}
We also studied SMBH binary evolution in a galaxy with less dramatic bulk rotation; in this case, only 75 $\%$ particles corotate with the massive binary (models B in table \ref{TableA}). The middle panel of figure \ref{rot1} again shows that the model experiences $N$-independent evolution of 1/a. for $N \geq 800K$. However the binary coalesces at a slower pace when compared to models A for the same particle number. For example, the convergent A models pass 1/a$=1500$ at 40 time units, while the
convergent B models take until nearly 50 time units to pass this same point.
The bottom panel of figure \ref{rot1} shows the evolution in the semi-major axis for the SMBH binary in a counterrotating orbital orientation (models C). Here, the binary orbit shrinks at slightly different rates than the previous models at various time intervals, though not with a clear trend. Af around 50 time units, the slope of 1/a line seems very similar for $N>500K$, and again we notice rapid evolution of 1/a when compared to flat and spherical models. However we have doubts that we are capturing the SMBH binary evolution accurately, because there is a clear dependence on particle number at later stages. In section \ref{coal} we discuss why we believe that SMBH binary coalescence is achieved in our counterrotating models, despite the lack of convergence in the model suite.
For consistency between runs, we calculate the hardening rates $s$ for all our runs by fitting a straight line to the inverse semi major axis $s = \frac{d}{dt}(\frac{1}{a})$ in the interval 50-70 time units (fig \ref{hard}). Both corotating and counterrotating binaries (runs A \& C) have hardening rates of about $s \sim 28$ which is about $30 \%$ higher than mere flat models.
The B runs have slightly lower values of $s$ $\sim$ 25 when compared to (runs A \& C). Overall, we find that $s$ in both co- and counterrotating models is approximately 4 times higher than in spherical models with exactly the same density profile, for our best resolved runs with 1.5 million particles.
\begin{figure}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig3.pdf}}
}
\caption[]{
Hardening rates for all our numerical experiments. Grey points are for the spherical models (S); Purple represents models A; Green is for models B, and the red points are for models C.
} \label{hard}
\end{figure}
For a spherical, homogeneous, isothermal background, the hardening parameter $H$ is related to hardening rate $s$ through $H = {s\sigma}/{G\rho}$ -- for the full loss cone regime, $H$ $\approx$ $15$~\citep{Q96,ses06}. We calculate $H$ by substituting values of $\rho$ and $\sigma$ at the influence radius, defined as sphere around the SMBH binary containing twice the mass of the binary in stars. Figure \ref{h} shows that for runs A2, A1 \& A0 with $N \geq 800k$, where the evolution is independent of N, the value of $H$ remains constant around $11$. The value of H from these $N$-body simulations is within 70\% of scattering experiments. For models B, again we see a constant value of $H$ $\approx$ $10$ for runs with $N \geq 800k$. For spherical run S0 with greatest particle number $N$, $H$ $\simeq$ $1.9$ almost $8$ times smaller than what is predicted for a full loss cone in scattering experiments. We would like to point out that in our models, the background profile is not at all isothermal and there is also some ambiguity for where one should measure $\sigma$ and $\rho$ to make a fair comparison with scattering experiments. With this in mind, it is very encouraging that $H$ obtained from our study is well within a factor of two of idealized scattering experiments.
\begin{figure}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig4.pdf}}
}
\caption[]{
Hardening parameter $H$ for all our numerical experiments. Colors are as in figure~\ref{hard}.
} \label{h}
\end{figure}
Figure \ref{ec} shows the eccentricity evolution of SMBH binaries for all runs with $N \geq 1$ million. For corotating SMBH binaries, the eccentricities are consistently small ($e \sim 0.2$). For counter-rotating SMBH binaries, the eccentricity approaches $e \sim 1$ as soon as the binary forms. This is consistent with the findings of \citet{ses11}.
\begin{figure}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig5.pdf}}
}
\caption[]{
SMBH binaries eccentricity for models with $N \geq 1$ million. SMBH binaries in co-rotating, flat and spherical models have very small values of eccentricity $\sim$ 0.1 whereas eccentricity approaches unity in counter-rotating models. Here, the transparency
represents the particle number; opaque lines are the $N=1.5$ million runs, while the fainter lines show results for $N=1$ million
particles. Orange lines show the counterrotating model; Green lines represent models B; Purple lines show models A, and the grey
lines are for the spherical and non-rotating axisymmetric models.
} \label{ec}
\end{figure}
The dichotomy in eccentricity behavior is borne out in a difference in the evolution of the angular momentum loss. Figure \ref{ang1} shows the angular momentum evolution of SMBH binaries for our best resolved co- and counterrotating models.
\begin{figure}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig6a.pdf}}
}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig6b.pdf}}
}
\caption[]{
Angular momentum evolution for corotating (top) and counterrotating (bottom) SMBH binaries. The green line is Lz, while the pink and brown lines are Lx and Ly, respectively.
} \label{ang1}
\end{figure}
It is clear that in the counterrotating case, the angular momentum loss is much more rapid. As we see from figure \ref{rot1}, the inverse semi-major axis evolution (and hence energy loss) is very similar for both co- and counterrotating models. This faster loss of angular momentum translates into a rapid rise in eccentricity.
We also investigated the center of mass motion of the binary in models A and C. Figure \ref{corot} shows the position of the center of mass throughout the run.
\begin{figure}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig7.pdf}}
}
\caption[]{
Motion of the center of mass of the SMBH binary in model A0 (purple) and the non-rotating flattened model (grey). Fainter colors indicate earlier epochs. Note that for model A0, the binary center of mass
settles into a roughly circular orbit about the galactic center with a radius of $\sim 0.05$ in model units, roughly the SMBH radius of influence. In contrast, the center of mass of the SMBH binary in the non-rotating model undergoes simple Brownian motion about the galactic center.
} \label{corot}
\end{figure}
The trajectory of the SMBH binary center of mass is strongly effected by rotation. For non-rotating flattened models, the binary center of mass exhibits a small random walk characteristic of Brownian motion~\citep{cha02}. On the other hand, the center of mass in the corotating system settles into a corotation resonance at the radius of influence, following a roughly circular orbit of radius $R_{\rm infl}$ nearly in the x-y plane.
\begin{figure}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig8.pdf}}
}
\caption[]{
Motion of the SMBH binary center of mass in model C0 (purple) and the non-rotating flattened model (grey). The color transparency is as in figure~\ref{corot}. In our counterrotating model, the binary center of mass executes a random walk like the non-rotating model.
} \label{count-rot}
\end{figure}
The counterrotating case, on the other hand, shows no binary orbital coupling, and the SMBH binary center of mass undergoes a random walk very similar to the non-rotating case.
\section{SMBH Binary Coalescence} \label{coal}
We also estimate the SMBH binary evolution for each case after the end of our simulation. We choose three Virgo cluster galaxies as reference to physically scale our models: M87, NGC4472, NGC4486A. In each case, the mass scale is set by the observed SMBH mass. For length scale, we set the influence radius of the SMBH binary-embedded galaxy model to the size of the observed influence radius of reference galaxy model. NGC4472 may best represent our model density profile with its moderate central cusp, while the central core in M87 and the steep cusp of NGC4486A span the range of typical density cusps. Table \ref{scale} shows useful quantities for the physical scales in our models.
\begin{table}
\caption{Physical Scaling of our Models}
\centering
\begin{tabular}{c c c c c c c c }
\hline
Galaxy & $M_{\bullet} (M_\odot)$ & $r_\mathrm{h}$(pc) & $T (\mathrm{Myr})$ & $L(\mathrm{kpc})$ & $M (M_{\odot})$ \\
\hline
M87 & $3.6 \times 10^9 $& $460$& $3.92$& $3.68$& $7.2 \times 10^{11}$\\
NGC4472 & $5.94 \times 10^8 $& $130$& $1.45$& $1.04$& $1.2 \times 10^{11}$\\
N4486A & $1.3 \times 10^7 $& $31$& $1.14$& $0.25$& $2.6 \times 10^9$\\
\hline
\end{tabular}\label{scale}
\tablecomments{Columns from left to right; (1) Reference galaxy, (2) Observed SMBH mass, (3) Observed SMBH radius of influence, (4) time unit, (5) length unit, (6) mass unit}
\end{table}
Our technique for extrapolating the evolution of the SMBH binary beyond the endpoint of the simulation is explained in detail in section 4.3 of \citet{kh12b}. We choose the runs with highest particle number in each model (A0, B0, C0) for this extrapolation technique, and the evolution is shown in figure \ref{grav}. The top panel shows the SMBH binary evolution scaled to M87.
In the non-rotating case, the SMBH binary coalesces in roughly 1.5 Gyr for this physical scaling. For corotating models A0 and B0, coalescence times are roughly 1.3 and 1.1 Gyr. For the counterrotating run C0, we evolve the SMBH binary from $1/a = 1000$ (see figure \ref{rot1}) at a system time T = 27, and we only consider hardening by gravitational waves; this is because we are not certain that the scattering results converge when the binary orbit is smaller than this. The SMBH binary coalesces in a mere 100 Myr -- essentially immediately -- due to its near radial eccentricity. The middle panel of figure \ref{grav} shows the SMBH binary evolution for NGC4472. Here, the SMBH binaries coalesce approximately two times faster than in M87; a case in point: the corotating model A0, the SMBH binary coalesces in roughly 500 Myr. Finally, the bottom panel of \ref{grav} scales to NGC4486A, and in this case the SMBHs merge in model A0 in about 1.5 Gyr while the SMBH binaries coalesce in almost 2 Gyr within the non-rotating model. Neglecting stellar hardening, the SMBH binary in the counterrotating case coalesces in roughly 2 Gyr. However, if we assume that SMBH binary reaches an asymptotic hardening rate in C0, coalescence happens immediately after a hard binary forms. Clearly, SMBH binaries coalesce faster in rotating flattened models, but the mechanism behind the coalescence is very different depending on the sense of rotation. In corotating models, the rapid coalescence is due to higher hardening rates, but for counterrotating models high eccentricity drives the merger. Out of three representative galaxies, the SMBH binary coalescence time is shortest for NGC4472.
\begin{figure}
\bigskip
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig9a.pdf}}
}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig9b.pdf}}
}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig9c.pdf}}
}
\caption[]{
Complete orbital evolution of the SMBH binary from formation to coalescence for runs A0 (yellow), B0 (orange), C0 (blue) and Flat (gray) when our galaxy model is scaled to M87 (top panel), NGC4472 (middle panel) and NGC4486A (bottom panel). In the bottom panel, the very different counterrotating timescales either include (short timescale with fainter hue) or exclude (longer timescale with stronger hue) stellar hardening.}
\label{grav}
\end{figure}
We show the coalescence time for different runs and physical scaling in Table \ref{time}. Technically, the clock starts here when two SMBHs form a pair with a separation $\simeq$ 10 influence radii (presumably after a major galaxy merger), and ends with the
coalescence of the binary from gravitational radiation.
\begin{table}
\caption{Physical Scaling of our Models}
\centering
\begin{tabular}{c c c c c c c}
\hline
Run & $s$ & $H$ & $e$ & $T_{c,M87}$ & $T_{c,4472}$ & $T_{c,4486A}$\\
\hline
A0 & $27.7$& $10.87$& $0.17$& $1.17$& $0.57$& $1.52$\\
B0 & $23.90$& $9.20$& $0.06$& $1.27$& $0.62$& $1.67$\\
C0 & $23.6$& $8.66$& $0.99$& $0.11$& $0.04$& $2.04(0.14)$\\
\hline
\end{tabular}\label{time}
\tablecomments{Columns from left to right; (1) SMBH binary evolution run, (2) SMBH binary hardening rate, (3) Hardening parameter H, (4) SMBH binary eccentricity. (5) Coalescence time (in Gyr) when our model is scaled to M87, (6) NGC4472 (7) and NGC4486A.}
\end{table}
\section{Structure of the merger remnant}
In this section, we go over the imprint of the binary black hole merger on the galaxy remnant.
As expected, figure \ref{scour} shows that the density cusp is scoured out by 3-body scattering
as the black holes coalesce~\citep{gra04}. One puzzling result can be seen in figure \ref{sigma}, where the
final velocity dispersion spikes; this is in contradiction to the dip in velocity dispersion expected
from the stellar hardening phase~\citep{Mei13}, and fully consistent with the velocity dispersion cusp
expected in an equilibrium SMBH-embedded nucleus. Further study is needed, using simulations
with shorter snapshot output cadence, to help pinpoint the occurrence and longevity of this potential
kinematic signature of 3-body scattering.
\begin{figure}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig10.pdf}}
}
\caption[]{
This is the initial (green) and final (orange) density profile of model A, showing the clear mass deficit
out to $r\sim0.3$ of 1.3 times the binary SMBH mass.
} \label{scour}
\end{figure}
\begin{figure}
\centerline{
\resizebox{0.95\hsize}{!}{\includegraphics[angle=0]{fig11.pdf}}
}
\caption[]{
This is the initial (orange) and final (red) 3-dimensional velocity dispersion profile of model A, showing a sharp spike in the velocity dispersion at the end of the run.
}
\bigskip
\label{sigma}
\end{figure}
\section{Conclusion}
We investigated the effect of bulk rotation on SMBH binary coalescence in N-body-generated flattened galaxy models. Overall, we found that rotation drives the SMBH binary more efficiently through the 3-body scattering phase, resulting in coalescence timescales
that are between 3 to 30 times faster than the same non-rotating model for co- and counterrotating models, respectively.
The 3-body scattering phase removes roughly 1.3 times the binary SMBH mass, scouring the density cusp out to about 1.1 kiloparsec if scaled to the M87 core.
We found that when the SMBH binary and the galaxy are corotating, the eccentricity remains low at approximately 0.1, while
counter-rotation acts to pump the SMBH binary eccentricity up to nearly one during the inspiral phase. Such a high eccentricity
enhances the coalescence of the SMBH binary, as is seen in many previous studies~\citep[e.g.][]{ses10,kh12a}. Though we caution that the
eccentricity behavior is not convergent even for 1.5 million particles, we suspect that the eccentricity will remain high in the convergent regime; when the SMBH is counterrotating, the abundance of retrograde orbits can extract angular momentum from the binary very efficiently, and secular dynamical anti-friction~\citep{mad12} torques the orbit so that it bleeds angular momentum.
A systematic study is needed to gauge the degree of counterrotation and binary orbital plane alignment needed to pump the eccentricity into the nearly radial regime; if the binary eccentricity is very sensitive to minor degrees of counterrotation, then few mergers will linger in the 3-body scattering stage. For such high eccentricities, we should expect residual eccentricity to persist into the last few orbits in the gravitational wave regime; this will have profound implications for gravitational wave detection using waveform template matching.
\acknowledgments
We would like to thank Massimo Dotti for very helpful discussions during the early stages of this work. The simulations were conducted in part using the resources of the Advanced Computing Center for Research and Education at Vanderbilt University, Nashville, TN.
KHB also acknowledges support from the NSF Career award AST-0847696.
|
1,116,691,497,790 | arxiv | \section{Introduction}
The optimal transport (OT) problem asks to compute the minimum cost needed to send supplies to demands. It is formally described as the following linear optimization problem, where
\[ \min_{\mathbf{X} \in \mathbf{U}(\boldsymbol{r},\boldsymbol{c})} \sum_{i\in[n],j\in[m]} \mathbf{X}_{ij}\mathbf{Q}_{ij}, \hspace{0.3cm} \mathbf{U}(\boldsymbol{r}, \boldsymbol{c}) \stackrel{\mathrm{\scriptscriptstyle def}}{=} \left\{\mathbf{X} \in \mathbb{R}_{+}^{n \times m}: \mathbf{X} \mathbf{1}_{n}=\boldsymbol{r} \ \text {and} \ \mathbf{X}^\top \mathbf{1}_m =\boldsymbol{c}\right\}, \]
where $\mathbf{Q}$ is the given cost matrix, and $\boldsymbol{r} \in \mathbb{Z}^n$ and $\boldsymbol{c} \in \mathbb{Z}^m$ are the (integral) demand and supply vectors. OT has been widely used as an important subroutine in computer vision \cite{damodaran2018deepjdot,Kolkin_2019_CVPR}, natural language processing \cite{kusner2015word}, deep learning \cite{oh2020unpaired,zhang2021optimal}, clustering \cite{ho2017multilevel}, unsupervised learning \cite{arjovsky2017wasserstein}, and semi-supervised learning \cite{solomon2014wasserstein}.
In $2013$, Cuturi proposed solving regularized OT as a possible approach to computing approximate OT solutions \cite{Cut13}. This is achieved by adding an entropy regularizer to the OT problem:
\begin{align}
\label{equation:regularizedOT}
\min_{\mathbf{X} \in \mathbf{U}(\boldsymbol{r},\boldsymbol{c})} \sum_{i\in[n],j\in[m]} \mathbf{X}_{ij}\mathbf{Q}_{ij} + \eta^{-1} \sum_{i \in [n], j \in [m]} \mathbf{X}_{ij}(\log \mathbf{X}_{ij} - 1).
\end{align}
The optimal solution $\mathbf{X}$ of (\ref{equation:regularizedOT}) has the structure of $\mathbf{X} = \mathrm{diag}(\boldsymbol{a})\mathbf{K}\mathrm{diag}(\boldsymbol{b})$, where $\mathbf{K}$ is a constant matrix depending on the costs $\mathbf{Q}$ and regularization $\eta$, and $\mathrm{diag}(\boldsymbol{a}), \mathrm{diag}(\boldsymbol{b})$ are two diagonal matrices. The solution structure enables us to reduce the regularized OT problem to the matrix scaling problem, which asks to compute scalings $\mathrm{diag}(\boldsymbol{a})$ and $\mathrm{diag}(\boldsymbol{b})$ given $\mathbf{K}$ so that $\mathbf{X}$ has row sums $\boldsymbol{r}$ and column sums $\boldsymbol{c}$, corresponding to the requirement that $\mathbf{X} \in \mathbf{U}(\boldsymbol{r}, \boldsymbol{c})$. Therefore, we can use the classic \textsc{Sinkhorn} algorithm to compute the OT solution. The \textsc{Sinkhorn} algorithm alternates between rescaling $\boldsymbol{a}$ (corresponding to rows) and $\boldsymbol{b}$ (corresponding to columns) until $\mathbf{X}$ is close to having row sums $\boldsymbol{r}$ and column sums $\boldsymbol{c}$.
The convergence rate of the \textsc{Sinkhorn} algorithm has been extensively analyzed both theoretically and practically. After \cite{Cut13}, many scaling methods have been developed \cite{benamou2015iterative,genevay2016stochastic} by leveraging the idea of entropic regularization. While these methods have been efficient in practice, there were few theoretical guarantees. \cite{altschuler2017near} shows that with an appropriate choice of parameters, the normal \textsc{Sinkhorn} or \textsc{Greenkhorn} algorithm is in fact a near-linear time approximation algorithm.
For input data of $n$ dimensions, it takes $O(n^2 N^3 (\log n)\varepsilon^{-3})$ runtime
to give a solution within $OPT + \varepsilon$.
Here $N$ denotes the largest element in the cost matrix $\mathbf{Q}$,
and $OPT$ denotes the optimal OT solution.
Despite the substantial work dedicated to the optimal transport problem, there is still a significant gap between theoretical and practical understanding of optimal transport \cite{OTML19}.
A significant contributing factor is in the selection of $\eta$, the regularization parameter which trades off the desired accuracy and iteration complexity of the subroutine. This motivates our work to study the theoretical and practical effects of \emph{adaptively} choosing the regularization $\eta$ to achieve our desired accuracy $\varepsilon$.
\paragraph{Contribution.} We introduce the exponentially converging Sinkhorn algorithm (\textsc{ExpSinkhorn}), which adaptively chooses the regularization parameter $\eta$ depending on a desired accuracy $\epsilon$. The algorithm, which is presented in Algorithm \ref{algo:scaling}, doubles $\eta$ when the matrix $\mathbf{X}$ we are storing is close to being properly scaled. This contrasts with previous approaches that select $\eta$ manually \cite{OTML19}.
We show a theoretical bound on the convergence of our method. In contrast with previous analyses \cite{altschuler2017near}, our algorithm converges \emph{exponentially} to the optimal solution, in that the number of iterations depends on $\log(1/\varepsilon)$ instead of the larger $\varepsilon^{-O(1)}$. However, our iteration count does have a polynomial dependence on $n$ through $\boldsymbol{r}$, while \cite{altschuler2017near} does not. Exponential convergence implies that our algorithm performs nearly as well in high accuracy regimes as in low accuracy regimes, unlike the standard \textsc{Sinkhorn} algorithm. Our main theoretical guarantee is stated in the following theorem.
\begin{theorem}[Algorithmic result]
\label{theo:algo}
Consider the cost matrix $\mathbf{Q}$, the demand vector $\boldsymbol{r} \in \mathbb{Z}^n$, the supply vector $\boldsymbol{c} \in \mathbb{Z}^m$, and desired accuracy $\epsilon > 0$, let $\mu := \max\{\|\boldsymbol{r}\|_\infty, \|\boldsymbol{c}\|_\infty\}$, and \[ \mathbf{U}(\boldsymbol{r}, \boldsymbol{c}) \stackrel{\mathrm{\scriptscriptstyle def}}{=} \left\{\mathbf{X} \in \mathbb{R}_{+}^{n \times n}: \mathbf{X} \mathbf{1}_{n}=\boldsymbol{r} \quad \text { and } \quad \mathbf{X}^\top \mathbf{1}_m =\boldsymbol{c}\right\} \]
\[ OPT \stackrel{\mathrm{\scriptscriptstyle def}}{=} \min_{\mathbf{X} \in \mathbf{U}(\boldsymbol{r},\boldsymbol{c})}\sum_{i\in[n],j\in[m]} \mathbf{X}_{ij}\mathbf{Q}_{ij}. \]
Given $\mathbf{Q},\boldsymbol{r},\boldsymbol{c},\epsilon$ as inputs, Algorithm \ref{algo:scaling} gives a solution $\mathbf{X} \in \mathbf{U}(\boldsymbol{r},\boldsymbol{c})$ satisfying
\[\sum_{i\in[n],j\in[m]} \mathbf{X}_{ij}\mathbf{Q}_{ij} \leq OPT + \varepsilon,\]
in $O(\|\boldsymbol{r}\|_1^2 \log(n\mu) \log(\|\mathbf{Q}\|_\infty\|\boldsymbol{r}\|_1\varepsilon^{-1}))$ iterations of row/column scaling.
\end{theorem}
Empirical results (Figure \ref{fig1}, Table \ref{tab2}) also demonstrate the efficiency and stability of our new algorithm. We verify that our algorithm produces high-accuracy solutions, as reflected by the improvement of the $\epsilon$ dependence from $\mathrm{poly}(1/\epsilon)$ to $\log(1/\epsilon)$ in Theorem \ref{theo:algo}. Additionally, we also verify that our algorithm which adaptive selects $\eta$ indeed produces high accuracy solutions, without requiring us to manually tune any parameters. Finally, we demonstrate that the iteration count of our algorithm is competitive with the previous Sinkhorn algorithm of \cite{Cut13}, on both synthetic data and instances arising from OT-based image processing pipelines.
Our algorithm is also numerically robust in addition to being efficient:
repeatedly increasing $\eta$ avoids the \textit{nan} values
that result from taking numbers far from $1$ to large powers.
Such a property is not shared by previous Sinkhorn based algorithms.
\subsection{Related work}
\paragraph{Optimal transport}
Many combinatorial techniques have been introduced to compute the exact solution for certain kinds of OT problems. The Hungarian method invented by Kuhn \cite{kuhn1955hungarian} in $1955$ solves the assignment problem (equivalent to OT) in $O(n^3)$ time. In $1991$, Gabow and Tarjan gave an $O(n^{2.5} \log(nN))$ time cost/capacity scaling algorithm \cite{GR91} to solve OT, where $N$ is the largest element in the scaled cost matrix. Using cost/capacity scaling techniques, min-cost flow algorithms such as network simplex also provide exact algorithms for the optimal transport problem in $O(n^3 \log n \log(nN))$ time \cite{OTML19}. There are also studies on certain kinds of OT problems, such as geometric OT \cite{alvarez2020geometric} \cite{peyre2019computational}. Additionally, there has been significant recent theoretical work studying the runtime of solving mincost flow, which generalizes OT \cite{LS19,matching,mincost,maxflow}. These methods rely heavily on second order methods and primitives from graph theory.
\paragraph{Regularization} In machine learning, regularization is widely used to resolve various kinds of datasets' heterogeneity \cite{tian2022comprehensive,zhu2018machine,neyshabur2017implicit,goodfellow2016regularization}. Recently there are more works on developing adaptive regularization methods, including deep learning towards imbalanced data \cite{cao2020heteroskedastic}, neural networks learning \cite{NEURIPS2019_2281f5c8}. There are also studies on regularization hyperparameters selection \cite{luketina2016scalable,leung1999adaptive}. On the other hand, there are few studies on choosing proper regularization hyperparameters in solving the regularized OT problem, especially with respect to a high accuracy solution.
\subsection{Notation}
We use bold lowercase characters such as $\boldsymbol{a}$ to denote vectors. Specially, we use $\mathbf{1}$ or $\mathbf{1}_n$ to denote the all ones vector with proper length. We use bold capital letters (such as $\mathbf{Q}$) as matrices. Specially, we denote the matrix that we are rescaling as $\mathbf{X}$. We denote the inner product of two matrix as $\langle \cdot, \cdot \rangle$, so $\langle \mathbf{X}, \mathbf{Q} \rangle = \sum_{i\in[n],j\in[m]} \mathbf{X}_{ij}\mathbf{Q}_{ij}$.
We use the integral vectors $\boldsymbol{r} \in \mathbb{Z}^n$ and $\boldsymbol{c} \in \mathbb{Z}^m$ to denote the desired row and column sums. Note that the matrix $\mathbf{X}$ has row sums $\mathbf{X} \mathbf{1}$ and column sums $\mathbf{X}^\top \mathbf{1}$. We use $\alpha_i$ for $i \in [n]$ and $\beta_j$ for $j \in [m]$ as the dual variables in our matrix scaling algorithm. As above, $\eta$ is the regularization parameter.
\section{Matrix scaling with regularization scheduling}
\label{sec2}
We propose an algorithm \textsc{ExpSinkhorn} to solve the OT problem to high accuracy. The algorithm maintains a matrix $\mathbf{X}$ to be scaled and a regularization parameter $\eta$. It rescales the rows and columns iteratively for this fixed parameter $\eta$. When the rows and columns are close enough to scaled, the algorithm doubles $\eta$. We ultimately show that this algorithm converges in time depending logarithmically on $\varepsilon^{-1}$ (see Theorem \ref{theo:algo}), as opposed to the standard Sinkhorn algorithm requiring time depending polynomially on $\varepsilon^{-1}$ to converge \cite{Cut13,altschuler2017near}.
The analysis of our algorithm hinges on understanding the interaction between the $\ell_1$ error of the row/column scaling and a \emph{dual objective}. Formally, when the quantities \[ \|\mathbf{X}\mathbf{1}-\boldsymbol{r}\|_1 \enspace \text{ and } \enspace \|\mathbf{X}^\top\mathbf{1} - \boldsymbol{c}\|_{1} \]
are small, the algorithm doubles the regularization parameter $\eta$. We show that when they are large, then rescaling the rows or columns of $\mathbf{X}$ causes the \emph{dual objective} to significantly improve (see Lemma \ref{lemma:dualincrease}). We also prove that when the $\ell_1$ errors are small, the duality gap is small (see Lemma \ref{lemma:dualitygap}), which bounds the number of iterations (see Lemma \ref{lemma:itercount}).
We now formally present our matrix scaling algorithm that doubles $\eta$ over time to give a high accuracy solution to optimal transport.
\begin{algo}
\label{algo:scaling}
\textbf{Input:} A $n \times m$ cost matrix $\mathbf{Q}$.
\textbf{Output:} A $n \times m$ matrix $\mathbf{X}_{i j} \geq 0$ such that $\mathbf{X} \in \mathbf{U}(\boldsymbol{r},\boldsymbol{c})$ and $\langle \mathbf{X}, \mathbf{Q} \rangle \le OPT + \epsilon$, where
\[
OPT \stackrel{\mathrm{\scriptscriptstyle def}}{=} \min_{\mathbf{X}_{i j} \geq 0, \mathbf{X} \in \mathbf{U}(\boldsymbol{r}, \boldsymbol{c})} \langle \mathbf{X}, \mathbf{Q} \rangle. \]
\begin{algorithm}[h]
\caption{\textsc{ExpSinkhorn}$(\mathbf{Q}, \boldsymbol{r}, \boldsymbol{c}, \epsilon)$ - Algorithm for solving the optimal transport problem.}
$\mu \leftarrow \max\{\max_{i \in [n]} r_i, \max_{j\in[m]} c_j\}.$ \\
$\boldsymbol{r} \leftarrow \boldsymbol{r}/\mu, \boldsymbol{c} \leftarrow \boldsymbol{c}/\mu$ \Comment{Scale $\boldsymbol{r}, \boldsymbol{c}$ to have $\|\boldsymbol{r}\|_\infty \le 1, \|\boldsymbol{c}\|_\infty \le 1$.} \label{line:scaler} \\
$\eta \leftarrow 10\|\mathbf{Q}\|_{\infty}^{-1} \log(n\mu).$ \Comment{Starting value of $\eta$.} \\
$\alpha_i \leftarrow -\|\mathbf{Q}\|_\infty, \beta_j \leftarrow -\|\mathbf{Q}\|_\infty$ for $i\in[n]$, $j\in[m]$. \Comment{Dual variable initialization} \\
\While{$\eta \le 4\mu\varepsilon^{-1}\|\boldsymbol{r}\|_1\log(n\mu)$}{ \label{line:mainwhile}
$\mathbf{X}_{ij} \leftarrow \exp(\eta(\alpha_i+\beta_j-\mathbf{Q}_{ij})).$ \Comment{Initialize matrix to be scaled.} \\
\For{$k \ge 0$}{
\If{$k$ is even}{\label{line:secondwhile}
$\boldsymbol{a} \leftarrow \mathbf{X}\mathbf{1}.$ \Comment{Row sums} \\
\If{$\|\boldsymbol{a}-\boldsymbol{r}\|_1 \le 1/(2\mu)$\label{line:timetodouble}}{\label{line:rif}
$\eta \leftarrow 2\eta$ and return to line \ref{line:mainwhile}. \label{line:double} \\
}
$\mathbf{X}_{ij} \leftarrow (\boldsymbol{a}_i/\boldsymbol{r}_i)^{-1}\mathbf{X}_{ij}$ for $1 \le i \le n$, $1 \le j \le m$ \Comment{Row scaling} \label{line:rescale} \\
$\alpha_i \leftarrow \alpha_i - \eta^{-1}\log(\boldsymbol{a}_i/\boldsymbol{r}_i)$ for $1 \le i \le n$ \label{line:adjustr} \Comment{Row dual adjustment} \\
}\Else{
$\boldsymbol{b} \leftarrow \mathbf{X}^T\mathbf{1}.$ \Comment{Column sums} \\
\If{$\|\boldsymbol{b}-\boldsymbol{c}\|_1 \le 1/(2\mu)$ \label{line:timetodouble2}}{\label{line:cif}
$\eta \leftarrow 2\eta$ and return to line \ref{line:mainwhile}. \label{line:double2} \\
}
$\mathbf{X}_{ij} \leftarrow (\boldsymbol{b}_j/\boldsymbol{c}_j)^{-1}\mathbf{X}_{ij}$ for $1 \le i \le n$, $1 \le j \le m$ \label{line:rescale2} \Comment{Column scaling}\\
$\beta_j \leftarrow \beta_j - \eta^{-1}\log(\boldsymbol{b}_j/\boldsymbol{c}_j)$ for $1 \le j \le m$ \Comment{Row dual adjustment} \label{line:adjustc}
}
}
}
$\mathbf{X} \leftarrow \mu\mathbf{X}$ \Comment{Scale $\mathbf{X}$ back up.} \\
Repair the demands routed by $\mathbf{X}$ and return $\mathbf{X}$.
\end{algorithm}
\end{algo}
We will assume throughout this analysis that $\|\boldsymbol{r}\|_\infty, \|\boldsymbol{c}\|_\infty \le 1$, such that $\mu\boldsymbol{r}, \mu\boldsymbol{c} \in \mathbb{Z}^n$. This is because we scale $\boldsymbol{r}, \boldsymbol{c}$, which are originally in $\mathbb{Z}^n$, down by $\mu$ in line \ref{line:scaler} of Algorithm \ref{algo:scaling}.
The analysis is based on looking at the dual program of the optimal transport objective: \[ \max_{\alpha_i+\beta_j \le \mathbf{Q}_{ij} \forall i \in [n], j \in [m]} \sum_{i\in[n]} \boldsymbol{r}_i\alpha_i + \sum_{j\in[m]} \boldsymbol{c}_j\beta_j. \]
The value of this program is also $OPT$, the same as the value of the optimal transport objective $\min_{\mathbf{X} \ge 0, \mathbf{X} \in \mathbf{U}(\boldsymbol{r}, \boldsymbol{c})} \langle \mathbf{X}, \mathbf{Q} \rangle$ by linear programming duality.
Thus, as long as we can guarantee that the $\alpha_i, \beta_j$ parameters in Algorithm \ref{algo:scaling} always satisfy $\alpha_i + \beta_j \le \mathbf{Q}_{ij}$, then the dual potential $D := \sum_{i\in[n]} \boldsymbol{r}_i\alpha_i + \sum_{j\in[m]} \boldsymbol{c}_j\beta_j \le OPT$ at all times. We will show these by induction.
\begin{lemma}[Algorithm invariants]
\label{lemma:valid}
At all times during an execution of Algorithm \ref{algo:scaling}, we have that $\mathbf{X}_{ij} \le 1$ for all $i \in [n], j \in [m]$ and $\sum_{i, j} \mathbf{X}_{ij} \le \|\boldsymbol{r}\|_1$. Hence $\alpha_i + \beta_j \le \mathbf{Q}_{ij}$ at all times.
\end{lemma}
\begin{proof}
The ``hence'' part follows because $\mathbf{X}_{ij} = \exp(\eta(\alpha_i + \beta_j - \mathbf{Q}_{ij}))$, so if $\mathbf{X}_{ij} \le 1$ then $\alpha_i + \beta_j \le \mathbf{Q}_{ij}$. Thus, in this proof we focus on showing the claims about $\mathbf{X}$.
We will proceed by induction. We first check that all conditions hold at the start of the algorithm. For the initial choices of $\eta, \alpha_i, \beta_j$ we have that
\[ \mathbf{X}_{ij} = \exp(10\|\mathbf{Q}\|_\infty^{-1} \log n(-2\|\mathbf{Q}\|_\infty + \mathbf{Q}_{ij})) \le \exp(-10\log n) \le n^{-10}. \]
Hence, $\sum_{ij} \mathbf{X}_{ij} \le n^{-8} \le 1 \le \|\boldsymbol{r}\|_1$, and $\mathbf{X}_{ij} \le 1$ for all $i, j$.
Now, we check that the condition continues to hold after we double $\eta$ in line \ref{line:double} and \ref{line:double2}. Let $\mathbf{X}^{\mathrm{new}}$ be the new matrix after $\eta$ is doubled. Clearly, $\mathbf{X}^{\mathrm{new}}_{ij} = \mathbf{X}_{ij}^2 \le \mathbf{X}_{ij}$ because $\mathbf{X}_{ij} \le 1$ by induction. So $\sum_{ij} \mathbf{X}^{\mathrm{new}}_{ij} \le \sum_{ij} \mathbf{X}_{ij} \le \|\boldsymbol{r}\|_1$ by induction, and $\mathbf{X}^{\mathrm{new}}_{ij} = \mathbf{X}_{ij}^2 \le \mathbf{X}_{ij} \le 1$.
Finally, we check the conditions after a rescaling step in lines \ref{line:rescale}, \ref{line:rescale2}. By symmetry, we consider a row rescaling step in line \ref{line:rescale}. After such a step, we know that $\sum_{j \in [m]} \mathbf{X}_{ij} = \boldsymbol{r}_i$ for all $i \in [n]$. Because $\|\boldsymbol{r}\|_\infty \le 1$, we deduce that $\mathbf{X}_{ij} \le 1$ for all $i, j$ and $\sum_{ij} \mathbf{X}_{ij} \le \|\boldsymbol{r}\|_1$ as desired. The same argument applies to a column rescaling in line \ref{line:rescale2}, if we note that $\|\boldsymbol{c}\|_1 = \|\boldsymbol{r}\|_1$.
\end{proof}
Because the dual potentials $\alpha_i, \beta_j$ are feasible, we know that the dual potential is upper bounded.
\begin{corollary}[Dual potential upper bound]
\label{cor:valid}
During an execution of Algorithm \ref{algo:scaling}, $\alpha_i, \beta_j$ satisfy $D := \sum_{i\in[n]} \boldsymbol{r}_i\alpha_i + \sum_{j\in[m]} \boldsymbol{c}_j\beta_j \le OPT$ at all times.
\end{corollary}
\begin{proof}
By Lemma \ref{lemma:valid} we know that $\alpha_i + \beta_j \le \mathbf{Q}_{ij}$ at all times. As noted above, by linear programming duality \[ D \le \max_{\alpha_i + \beta_j \le \mathbf{Q}_{ij} \forall i\in[n],j\in[m]} \sum_{i\in[n]} \boldsymbol{r}_i\alpha_i + \sum_{j\in[m]} \boldsymbol{c}_j\beta_j = OPT. \qedhere \]
\end{proof}
The remainder of the analysis requires the following claims. First, we show that the duality gap $OPT - D$ is small when either $\|\boldsymbol{r}-\boldsymbol{a}\|_1 \le 1/(2\mu)$ or $\|\boldsymbol{c}-\boldsymbol{b}\|_1 \le 1/(2\mu)$ triggers, i.e. lines \ref{line:timetodouble} or \ref{line:timetodouble2}. When these do not hold, we show that a rescaling step in lines \ref{line:adjustr} or \ref{line:adjustc} causes $D$ to significantly increase. Finally, we will show how to round our approximately scaled solution $\mathbf{X}$ to a feasible point.
Towards this, we show the following useful helper lemma which intuitively shows that an approximately feasible $\mathbf{X}$ ``contains'' half of a truly feasible solution.
\begin{lemma}[Containing a feasible solution]
\label{lemma:hatmx}
Let $\boldsymbol{r}, \boldsymbol{c}$ be vectors with $\|\boldsymbol{r}\|_1, \|\boldsymbol{c}\|_1 \le 1$ and $\mu\boldsymbol{r}, \mu\boldsymbol{c} \in \mathbb{Z}^n$. If $\mathbf{X} \ge 0$ satisfies $\mathbf{X}\mathbf{1} = \boldsymbol{r}$ and $\|\mathbf{X}^\top\mathbf{1} - \boldsymbol{c}\|_1 \le 1/(2\mu)$, then there is a vector $\hat{\mathbf{X}} \in \mathbb{R}^{n \times m}$ with $0 \le \hat{\mathbf{X}}_{ij} \le \mathbf{X}_{ij}$ for all $i \in [n], j \in [m]$ and $\hat{\mathbf{X}}\mathbf{1} = \boldsymbol{r}/2$ and $\hat{\mathbf{X}}^\top\mathbf{1} = \boldsymbol{c}/2$.
Additionally, such an $\hat{\mathbf{X}}$ can be found by running any maximum flow algorithm.
\end{lemma}
Clearly we may swap the roles of $\boldsymbol{r}, \boldsymbol{c}$ above. We state only one case in Lemma \ref{lemma:hatmx} for brevity.
\begin{proof}
Let $\alpha \ge 0$ be maximal so that there exists a $0 \le \hat{\mathbf{X}} \le \mathbf{X}$ such that $\hat{\mathbf{X}}\mathbf{1}=\alpha\boldsymbol{r}$ and $\hat{\mathbf{X}}^\top\mathbf{1}=\alpha\boldsymbol{c}$. Let $\mathbf{Y}$ satisfy $\mathbf{Y}\mathbf{1}=\alpha\boldsymbol{r}$ and $\mathbf{Y}^\top\mathbf{1}=\alpha\boldsymbol{c}$. We wish to show that $\alpha\ge1/2$.
Assume $\alpha < 1/2$ for contradiction, and let $\mathbf{X}^{(1)} = \mathbf{X} - \mathbf{Y}$, so that $\mathbf{X}^{(1)}\mathbf{1} = (1-\alpha)\boldsymbol{r}$ and $\|(\mathbf{X}^{(1)})^\top\mathbf{1} - (1-\alpha)\boldsymbol{c}\|_1 \le 1/(2\mu)$. Multiplying the previous equations by $(1-\alpha)^{-1}\mu$ on both sides yields that
\begin{align}
\bar{\mathbf{X}}\mathbf{1} = \mu\boldsymbol{r} \enspace \text{ and } \enspace \left\|\bar{\mathbf{X}}^\top\mathbf{1} - \mu\boldsymbol{c}\right\|_1 \le \frac{1}{2(1-\alpha)} < 1, \label{eq:scaledup}
\end{align}
where $\bar{\mathbf{X}} := (1-\alpha)^{-1}\mu\mathbf{X}^{(1)}$. Using that both $\mu\boldsymbol{r}$ and $\mu\boldsymbol{c}$ are integral vectors, we will construct $0 \le \mathbf{Z} \le \bar{\mathbf{X}}$ and $\delta > 0$ such that $\mathbf{Z}\mathbf{1} = \delta\boldsymbol{r}$ and $\mathbf{Z}\mathbf{1} = \delta\boldsymbol{c}$, contradicting the maximality of $\alpha$.
Let $E$ be the support of $\bar{\mathbf{X}}$, i.e. $E := \left\{(i, j) : \bar{\mathbf{X}}_{ij} > 0 \right\}$. For a subset $S \subseteq [n]$, let $N(S) := \{t : \exists s \in S, (s, t) \in E\}$, i.e. the neighborhood of $S$. By Hall's marriage theorem (for weighted sources and sinks), the subset $E$ supports a flow between $\mu\boldsymbol{r}$ and $\mu\boldsymbol{c}$ as long as for all subsets $S \subseteq [n]$, we have that $\sum_{s \in S} (\mu\boldsymbol{r})_s \le \sum_{t \in N(S)} (\mu\boldsymbol{c})_t$. By the guarantee in (\ref{eq:scaledup}) we know that
\begin{align*}
\sum_{s \in S} (\mu\boldsymbol{r})_s &= \sum_{(s, t) \in E} \bar{\mathbf{X}}_{st} \le \sum_{t \in N(S)} \sum_{s \in [n]} \bar{\mathbf{X}}_{st} \\ &\le \sum_{t \in N(S)} (\mu\boldsymbol{c})_t - \|\bar{\mathbf{X}}^\top\mathbf{1} - \mu\boldsymbol{c}\|_1 < \sum_{t \in N(S)} (\mu\boldsymbol{c})_t - 1.
\end{align*}
Because $\sum_{s \in S} (\mu\boldsymbol{r})_s$ and $\sum_{t \in N(S)} (\mu\boldsymbol{c})_t$ are both integral quantities, the previous equation implies that $\sum_{s \in S} (\mu\boldsymbol{r})_s \le \sum_{t \in N(S)} (\mu\boldsymbol{c})_t$ as desired. This shows that there is some $0 \le \mathbf{Z} \le \bar{\mathbf{X}}$ and strictly positive $\delta > 0$ such that $\mathbf{Z}\mathbf{1} = \delta\boldsymbol{r}$ and $\mathbf{Z}\mathbf{1} = \delta\boldsymbol{c}$. This completes the proof.
\end{proof}
The above lemma lets us bound the duality gap right before we double $\eta$, i.e. when lines \ref{line:timetodouble}, \ref{line:timetodouble2} occur.
\begin{lemma}[Duality gap]
\label{lemma:dualitygap}
Let $D = \sum_{i \in [n]} \alpha_i\boldsymbol{r}_i + \sum_{j \in [m]} \beta_j\boldsymbol{c}_j$. During an execution of Algorithm \ref{algo:scaling} when lines \ref{line:timetodouble}, \ref{line:timetodouble2} occur, we have that $OPT - D \le 2\eta^{-1}\|\boldsymbol{r}\|_1 \log(n\mu)$.
\end{lemma}
\begin{proof}
We only handle the case where line \ref{line:timetodouble2} occurs, as the other case is symmetric (recall that $\|\boldsymbol{r}\|_1 = \|\boldsymbol{c}\|_1$). Hence $\mathbf{X}^\top\mathbf{1}=\boldsymbol{r}$.
By Jensen's inequality, we know that
\begin{align*}
\sum_{i\in[n],j\in[m]} \mathbf{X}_{ij} \log \mathbf{X}_{ij} &= -\sum_{i \in [n]} \boldsymbol{r}_i \sum_{j \in [m]} \frac{\mathbf{X}_{ij}}{\boldsymbol{r}_i} \log(1/\mathbf{X}_{ij}) \\ &\ge -\sum_{i \in [n]} \boldsymbol{r}_i \log(m/\boldsymbol{r}_i) \ge -\|\boldsymbol{r}\|_1 \log(n\mu),
\end{align*}
because $\boldsymbol{r}_i \ge \mu^{-1}$ for all $i$, because $\mu\boldsymbol{r} \in \mathbb{Z}^n$ by assumption. Let $\hat{\mathbf{X}}$ be as constructed in Lemma \ref{lemma:hatmx}. Because $\mathbf{X}_{ij} \le 1$ for all $i, j$ by Lemma \ref{lemma:valid} (so $\log \mathbf{X}_{ij} \le 0$), we can write
\begin{align*}
\sum_{i\in[n],j\in[m]} \mathbf{X}_{ij} \log \mathbf{X}_{ij} &\le \sum_{i\in[n],j\in[m]} \hat{\mathbf{X}}_{ij} \log \mathbf{X}_{ij} = \eta\sum_{i\in[n],j\in[m]} \hat{\mathbf{X}}_{ij}(\alpha_i + \beta_j - \mathbf{Q}_{ij}) \\
&= \eta(D/2 - \langle \hat{\mathbf{X}}, \mathbf{Q} \rangle) \le \eta(D/2 - OPT/2),
\end{align*}
where the final inequality follows because $\hat{\mathbf{X}}\mathbf{1} = \boldsymbol{r}/2$ and $\hat{\mathbf{X}}^\top\mathbf{1} = \boldsymbol{c}/2$, hence $\langle \hat{\mathbf{X}}, \mathbf{Q} \rangle \ge OPT/2$ by the minimality of OPT.
Combining the previous two expressions completes the proof.
\end{proof}
Now, we prove that if line \ref{line:timetodouble} and \ref{line:timetodouble2} do not occur, then the dual solution increases significantly.
\begin{lemma}[Dual increase]
\label{lemma:dualincrease}
Let $\boldsymbol{a} = \mathbf{X}\mathbf{1}$, and consider updating $\alpha$ as in line \ref{line:adjustr}. Then the dual $D := \sum_{i \in [n]} \alpha_i \boldsymbol{r}_i + \sum_{j \in [m]} \beta_j \boldsymbol{c}_j$ increases by at least \[ \eta^{-1}/10 \cdot \min\{\mu^{-1}, \|\boldsymbol{r}\|_1^{-1}\|\boldsymbol{a} - \boldsymbol{r}\|_1^2\}. \]
\end{lemma}
\begin{proof}
Note the following numerical bound: $-\log(1-t) \ge t + \min\{1/10, t^2/3\}$ for all $t < 1$.
By the formula in line \ref{line:adjustr}, the dual increases by
\begin{align*}
-\eta^{-1} \sum_{i \in [n]} \boldsymbol{r}_i \log(\boldsymbol{a}_i/\boldsymbol{r}_i) &= \eta^{-1} \sum_{i \in [n]} \boldsymbol{r}_i(-\log(1-(1-\boldsymbol{a}_i/\boldsymbol{r}_i))) \\ &\ge \eta^{-1} \sum_{i \in [n]} \boldsymbol{r}_i \cdot ((1-\boldsymbol{a}_i/\boldsymbol{r}_i) + \min\{(1-\boldsymbol{a}_i/\boldsymbol{r}_i)^2/3, 1/10)\} \\
&= \eta^{-1} \sum_{i \in [n]} \boldsymbol{r}_i \cdot \min\{(1-\boldsymbol{a}_i/\boldsymbol{r}_i)^2/3, 1/10\},
\end{align*}
because $\|\boldsymbol{a}\|_1 = \sum_{i,j} \mathbf{X}_{ij} \le \|\boldsymbol{r}\|_1$ by Lemma \ref{lemma:valid}. If any of the $\min$'s in the previous expression evaluate to $1/10$, then the expression is clearly at least $\eta^{-1}\boldsymbol{r}_i/10 \ge \eta^{-1}/10 \cdot \mu^{-1}$, because $\mu\boldsymbol{r}$ is integral. Otherwise, by the Cauchy-Schwarz inequality,
\begin{align*}
\eta^{-1} \sum_{i \in [n]} \boldsymbol{r}_i (1-\boldsymbol{a}_i/\boldsymbol{r}_i)^2/3 = \eta^{-1}/3 \cdot \sum_{i \in [n]} (\boldsymbol{a}_i - \boldsymbol{r}_i)^2/\boldsymbol{r}_i \ge \eta^{-1}/3 \cdot \frac{\|\boldsymbol{a}-\boldsymbol{r}\|_1^2}{\|\boldsymbol{r}\|_1},
\end{align*}
as desired. This completes the proof.
\end{proof}
We can now bound the total number of iterations of the algorithm.
\begin{lemma}[Iteration count]
\label{lemma:itercount}
For integral vectors $\boldsymbol{r}, \boldsymbol{c} \in \mathbb{Z}^n$, and $\mu := \max\{\|\boldsymbol{r}\|_\infty, \|\boldsymbol{c}\|_\infty\}$ an execution of Algorithm \ref{algo:scaling} uses at most $O(\|\boldsymbol{r}\|_1^2 \log(n\mu) \log(\varepsilon^{-1}\|\mathbf{Q}\|_\infty\mu))$ iterations.
\end{lemma}
\begin{proof}
After doubling $\eta$ the duality gap is at most $4\eta^{-1}\|\boldsymbol{r}\|_1 \log(n\mu)$ by Lemma \ref{lemma:dualitygap}. If $\|\boldsymbol{a} - \boldsymbol{r}\|_1 \ge 1/(2\mu)$, then the dual increase is at least $\eta^{-1}/10 \cdot \min\{\mu^{-1}, \|\boldsymbol{r}\|_1^{-1}\|\boldsymbol{a} - \boldsymbol{r}\|_1^2\} \ge 1/40 \cdot \eta^{-1}\|\boldsymbol{r}\|_1^{-1}\mu^{-2}$. Hence the number of iterations during a doubling phase is bounded by $\frac{4\eta^{-1}\|\boldsymbol{r}\|_1 \log(n\mu)}{1/40 \cdot \eta^{-1}\|\boldsymbol{r}\|_1^{-1}\mu^{-2}} = O((\mu\|\boldsymbol{r}\|_1)^2 \log(n\mu))$. Additionally, the total number of doubling phases is bounded by $\log((4\mu\varepsilon^{-1}\|\boldsymbol{r}\|_1\log(n\mu))/(10\|\mathbf{Q}\|_{\infty}^{-1} \log(n\mu)))$.
Thus, the lemma follows (recall that the $\boldsymbol{r}$ in the Lemma statement is really $\mu\boldsymbol{r}$ after scaling).
\end{proof}
Finally, we show how to recover a feasible solution from $\mathbf{X}$, and complete the proof of Theorem \ref{theo:algo}.
\begin{proof}[Proof of Theorem \ref{theo:algo}]
The iteration complexity bound follows from Lemma \ref{lemma:itercount}, so it suffices to explain how to round our final solution $\mathbf{X}$ to an accurate solution $\mathbf{Y}$.
To construct $\mathbf{Y}$, let $\hat{\mathbf{X}}$ be as in Lemma \ref{lemma:hatmx}, and let $\mathbf{Y} = 2\hat{\mathbf{X}}$. By definition, we know that $\mathbf{Y}\mathbf{1} = 2\hat{\mathbf{X}}\mathbf{1} = \boldsymbol{r}$, and similarly $\mathbf{Y}^\top\mathbf{1} = \boldsymbol{c}$. To bound the optimality gap of $\mathbf{Y}$, note by the equations in the proof of Lemma \ref{lemma:dualitygap} that $-\|\boldsymbol{r}\|_1 \log(n\mu) \le \eta(D/2 - \langle \hat{\mathbf{X}}, \mathbf{Q} \rangle)$, so \[ \langle \hat{\mathbf{X}}, \mathbf{Q} \rangle \le \eta^{-1}\|\boldsymbol{r}\|_1 \log(n\mu) + D/2 \le \eta^{-1}\|\boldsymbol{r}\|_1 \log(n\mu) + OPT/2, \] as $D \le OPT$ by Corollary \ref{cor:valid}.
Hence \[ \langle \mathbf{Y}, \mathbf{Q} \rangle = 2\langle \hat{\mathbf{X}}, \mathbf{Q}\rangle \le 2\eta^{-1}\|\boldsymbol{r}\|_1 \log(n \mu) + OPT \le OPT + \mu^{-1}\epsilon \] by the ending choice of $\eta$. Because Algorithm \ref{algo:scaling} scales everything down by $\mu$, the error in terms of the original objective is $\epsilon$, as desired. $\mathbf{Y}$ can be computed efficiently by calling maximum flow.
\end{proof}
\section{Empirical results}
\label{sec4}
In this section, we provide empirical results of our proposed algorithm \textsc{ExpSinkhorn} under different accuracy settings, and on different data sets.
\footnote{All the code is written in C++. Experiments are conducted on a Mac with 2.3 GHz 8-Core Intel Core i9, 16 GB 2667 MHz DDR4, Intel UHD Graphics 630 1536 MB.}
We also conduct experiments to compare
\textsc{ExpSinkhorn} and \textsc{Sinkhorn}~\cite{Cut13}.
\paragraph{Methodology} Due to the possible variance in algorithm implementation and experiment environment, instead of actual run-time, we use the number of iterations that the algorithm calls for scaling the whole matrix to measure the efficiency cost. Note that in our comparison experiment, both \textsc{ExpSinkhorn} and \textsc{Sinkhorn} are fit in such metric since they scale the entire matrix each time.
\paragraph{Dataset}\footnote{The majority of the datasets were generated by ourselves, with an exception of the data for MNIST, NLP, CIFAR, and CircleSquare, which were taken from the Github repository from https://github.com/sawlani/OPOT from \cite{OTML19}} We select three datasets from real-world scenarios:
\begin{itemize}
\item \textbf{NLP}: These distributions are created from disjoint portions of the novel \textit{The Count of Monte Cristo}. Each sample uses $900$ lines of text from the novel, which is tokenized with AllenNLP \cite{GGN+18}, and then embedded into $\mathbb{R}^{100}$ with 100-dimensional GloVe \cite{PSM14} word embeddings, creating $N$ points in $\mathbb{R}^{100}$, where $N$ is the number of tokens. The cost matrix uses Euclidean distances between the points.
\item \textbf{MNIST}: The MNIST \cite{LC10} dataset contains black and white images of size $28 \times 28$ pixels, thus giving a capacity distribution of length-784 ($28^2$). The cost matrix used in the dataset are Euclidean distances between the points. The supply and demand capacity vectors are appropriately normalized such that the supply and demand sum to the same.
\item \textbf{CIFAR}: The CIFAR \cite{Kri09} dataset contains color images of size $32 \times 32$ pixels. We represent the pixels as $(x, y, r, g, b)$ points in $\mathbb{R}^5$ and normalize each direction into $[0, 1]$. These are used as the supply/demand point sets. The cost matrix represents the Euclidean distance between points. The capacities are uniformly distributed among the points.
\end{itemize}
We also generate three kinds of synthetic datasets:
\begin{itemize}
\item \textbf{CircleSquare}: CircleSquare dataset describes a min cost matching problem between integral points on $\mathbb{R}^2$. The two point sets form a circle and a square that have the same center and the roughly the same diameter, with both the supply and demand capacities being uniformly distributed among the points. The dataset uses Euclidean distance as the transportation cost.
\item \textbf{Rand}: Rand uses the random images generating method described in \cite{AWR17} to form a dataset. Two random images are generated with a randomly positioned bright foreground and a dark background. These images are used as the initial and required distributions, and Manhattan distance is used as the transportation cost.
\item \textbf{RandPoints}: RandPoints describes a min cost matching problem between two sets of random integral points within a $10^6 \times 10^6$ area on 2-D plane, using Euclidean distance as the transportation cost. The two capacities are uniform generated.
\end{itemize}
We conduct all our experiments on the datasets from above. Here we give a table on the properties of the data used for our experiments.
\begin{table}[htbp]
\caption{Information of datasets in the experiment}
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Name & Dimension & S/D Range & Demand & Cost range & Exact Solution\\
\hline
\textbf{MNIST} & 64-169 & 0 - 13818 & 999920-999961 & 0 - 234 & $1.3 \times 10^7 - 4.3 \times 10^7$ \\
\hline
\textbf{NLP} & 1389-1788 & 0 - 7270 & 80210-89030 & 0 - 11577502 & $8.6 \times 10^{10} - 1.2 \times 10^{11}$ \\
\hline
\textbf{CIFAR} & 1024-1024 & 1 - 1 & 1024-1024 & 3 - 619 & $5.7 \times 10^4 - 1.3 \times 10^5$ \\
\hline
\textbf{CircleSquare} & 900-2500 & 1 - 1 & 900-2500 & 0 - 1579922 & $9.0 \times 10^5 - 9.6 \times 10^6$ \\
\hline
\textbf{RandPoints} & 100-900 & 1 - 1 & 100-900 & 171 - 1397290 & $1.1 \times 10^7 - 8.9 \times 10^7$ \\
\hline
\textbf{Rand} & 100-900 & 1 - 5033 & 247825-2193442 & 0 - 58 & $1.6 \times 10^5 - 3.4 \times 10^7$ \\
\hline
\textbf{Uniform} & 100-900 & 1 - 1 & 100-900 & 5 - 610 & $1.4 \times 10^6 - 1.7 \times 10^6$ \\
\hline
\end{tabular}
}
\label{tab1}
\end{table}
"Dimension" refers to the min and max dimensions over all the datasets of each type. We have included ranges of the min / max entries of the supply and demand, as well as the ranges of the total demand and the ranges of the exact solution. The exact solution was computed using the Lemon Network Simplex subroutine. \cite{DJK11}
\subsection{Accuracy and efficiency}
In this part, we study the relation between the accuracy goal and the iteration count,
giving a full picture of how \textsc{ExpSinkhorn} performs in different accuracy requirements. Our goal is to both empirically confirm
the logarithmic dependence on $\log(1/\varepsilon)$ predicted
from \ref{theo:algo},
and to demonstrate that such dependencies are effects
of $\eta$ scheduling.
We calculate the number of matrix scaling iterations of \textsc{ExpSinkhorn}, and we choose the appropriate $\varepsilon$ to get $x$\%-approximations for $x = 10, 3, 1, 0.3, 0.1$. This is done by letting $\varepsilon = \frac{x}{100} OPT$ where OPT is the optimal solution to the OT problem.
\textsc{ExpSinkhorn}, along with standard variants of approximate OT, includes a rounding step. Since the rounding step results in a computational overhead and a lower-quality solution, and since we only are concerned with the iteration count prior to rounding, we do not perform the rounding step in our implementation of the algorithm.
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[/pgf/number format/1000 sep={},/pgf/number format/fixed]
\begin{axis}[
xmode=log,
x dir=reverse,
ytick distance = 25000,
ymin=0,
log ticks with fixed point,
scaled ticks = false,
legend pos=north west,
y label/.style = {anchor=near ticklabel},
xlabel = {Accuracy Parameter},
ylabel = Iterations,
]
\addplot[color=blue,mark=none]
coordinates {
(0.100000, 4467) (0.030000, 8905) (0.010000, 14892) (0.003000, 17396) (0.001000, 17492)
};
\addplot[color=orange,mark=none]
coordinates {
(0.100000, 4016) (0.030000, 14036) (0.010000, 32146) (0.003000, 43203) (0.001000, 65315)
};
\addplot[color=ForestGreen,mark=none]
coordinates {
(0.100000, 28675) (0.030000, 53663) (0.010000, 72715) (0.003000, 144695) (0.001000, 207893)
};
\addplot[color=red,mark=none]
coordinates {
(0.100000, 12487) (0.030000, 19411) (0.010000, 22885) (0.003000, 29901) (0.001000, 37263)
};
\addplot[color=Periwinkle,mark=none]
coordinates {
(0.100000, 18083) (0.030000, 19949) (0.010000, 20672) (0.003000, 20708) (0.001000, 20730)
};
\addplot[color=Sepia,mark=none]
coordinates {
(0.100000, 23138) (0.030000, 31892) (0.010000, 40646) (0.003000, 45023) (0.001000, 53777)
};
\legend{CIFAR,MNIST,NLP,CS900,RP900,R100}
\end{axis}
\end{tikzpicture}
\captionof{figure}{\textsc{ExpSinkhorn} performance}
\label{fig1}
\end{figure}
From the pseudocode of \textsc{ExpSinkhorn}, we can see that reducing $\varepsilon$ makes the bound on $\eta$ large, and results in more iterations. This is consistent with the results in Figure \ref{fig1}, where the number of iterations increases as the accuracy parameter gets smaller. We include the exact number of iterations from the real-world datasets (MNIST, CIFAR, and NLP) in Table \ref{tab2}.
\begin{figure}[!ht]
\centering
\captionof{table}{Iterations on real-world data}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Name & 0.1 & 0.03 & 0.01 & 0.003 & 0.001 \\
\hline
\textbf{CIFAR1} & 4059 & 8774 & 9910 & 10162 & 10210 \\
\hline
\textbf{CIFAR2} & 2611 & 5735 & 9079 & 13342 & 14824 \\
\hline
\textbf{CIFAR3} & 3774 & 6088 & 11355 & 17035 & 19924 \\
\hline
\textbf{MNIST1} & 85979 & 147884 & 178836 & 240740 & 271692 \\
\hline
\textbf{MNIST2} & 33337 & 133375 & 249155 & 364935 & 422825 \\
\hline
\textbf{MNIST3} & 179351 & 361109 & 542867 & 724625 & 815504 \\
\hline
\textbf{NLP1} & 82140 & 171645 & 283843 & 608173 & 760374 \\
\hline
\textbf{NLP2} & 89513 & 133346 & 371964 & 805013 & 866408 \\
\hline
\end{tabular}
\label{tab2}
\end{figure}
\subsection{Comparison with \texorpdfstring{\cite{Cut13}}{cut13}}
We evaluate our algorithm against the \textsc{Sinkhorn} algorithm of \cite{Cut13}.\footnote{Implementation from https://github.com/sawlani/OPOT from \cite{OTML19}} As mentioned earlier, higher values of $\eta$ aid in achieving higher accuracy solutions, but result in a longer runtime. Because of this, the algorithm runs a binary search to find the smallest $\eta$ such that the approximation guarantee holds. We set the $\mathit{ETA\_INIT}$ parameter, which is the initial $\eta$ (and the midpoint of the search range from $[0, 2 * \mathit{ETA\_INIT}]$) in our experiments to $2000$. One exception is NLP, for which we set $\mathit{ETA\_INIT}$ to a smaler value (1000) to avoid \textit{nan} issues (later explained in detail in the analysis).
We run the \textsc{Sinkhorn} of \cite{Cut13} on the datasets CIFAR, R100 and RP900. We approximate solutions to $x\%$ for $x = 10, 1, 3, 1, 0.3, 0.1$. Here, CIFAR refers to cifar0 from Table \ref{tab2}.
We compare the iteration counts of \cite{Cut13} and the iteration counts of Algorithm \ref{algo:scaling}, where the iterations are counted as before; i.e. each iteration is a scaling of the matrix. The number of iterations of \cite{Cut13} is measured over the entire binary search that is used to compute $\eta$. We use the iteration count over the entire binary search, rather than for just the best $\eta$, since in a real-world setting, we do not know what an appropriate $\eta$ is beforehand, and given an OT, the amount of work done to find $\eta$ is an important cost to consider when searching for the appropriate $\eta$.
Figure \ref{fig2} are the plots comparing the iteration counts of \textsc{ExpSinkhorn} and \textsc{Sinkhorn}. In these plots, given our experiment configurations, \textsc{ExpSinkhorn} requires fewer iterations than Sinkhorn. This provides evidence that \textsc{ExpSinkhorn} is competitive with (and can outperform) existing algorithms in practice. \textsc{Sinkhorn}'s iterations increases greatly while accuracy gets higher while \textsc{ExpSinkhorn} gives a far more gradual increase, linearly to $\log(1/\varepsilon)$. This is consistent with our theoretical improvement.
There is one important key weakness that is inherent in the \textsc{Sinkhorn} algorithm, which is vulnerability to encountering \textit{nan} values. This is because there is a step where the matrix $K$ is equal to $e^{-\eta C}$. When $\eta$ is too large, many of the entries of $K$ will equal $0$, so scaling rows and columns results in \textit{nan} values. This is why we had to reduce the value of $\mathit{ETA\_INIT}$ to $1000$ for NLP. On the contrary, \textsc{ExpSinkhorn} does not run into this issue since $\eta$, which changes adaptively, starts out very small.
\begin{figure}[!ht]
\begin{minipage}[c]{0.32\textwidth}
\begin{tikzpicture}[scale=0.5,/pgf/number format/1000 sep={},/pgf/number format/fixed]
\begin{axis}[
xmode=log,
x dir=reverse,
ytick distance = 5000,
ymin=0,
log ticks with fixed point,
scaled ticks = false,
legend pos=north west,
y label/.style = {anchor=near ticklabel},
title = CIFAR,
xlabel = {Accuracy parameter},
ylabel = Iterations,
]
\addplot[color=blue,mark=none]
table {
0.100000 4467
0.050000 6010
0.020000 8905
0.010000 14892
0.005000 17077
0.002000 17396
0.001000 17492
};
\addplot[color=orange,mark=none]
table {
0.100000 19407
0.050000 21154
0.020000 24637
0.010000 27777
0.005000 33028
0.002000 37374
0.001000 50495
};
\legend{\textsc{ExpSinkhorn},\textsc{Sinkhorn}}
\end{axis}
\end{tikzpicture}
\end{minipage}\hfill
\begin{minipage}[c]{0.32\textwidth}
\begin{tikzpicture}[scale=0.5,/pgf/number format/1000 sep={},/pgf/number format/fixed]
\begin{axis}[
xmode=log,
x dir=reverse,
ytick distance = 50000,
ymin=0,
log ticks with fixed point,
scaled ticks = false,
legend pos=north west,
y label/.style = {anchor=near ticklabel},
title = R100,
xlabel = {Accuracy parameter},
ylabel = Iterations,
]
\addplot[color=blue,mark=none]
table {
0.100000 33776
0.050000 38858
0.020000 49022
0.010000 54104
0.005000 59186
0.002000 64268
0.001000 69350
};
\addplot[color=orange,mark=none]
table {
0.100000 85457
0.050000 86813
0.020000 87503
0.010000 88320
0.005000 88657
0.002000 568537
0.001000 568537
};
\legend{\textsc{ExpSinkhorn},\textsc{Sinkhorn}}
\end{axis}
\end{tikzpicture}
\end{minipage}\hfill
\begin{minipage}[c]{0.32\textwidth}
\begin{tikzpicture}[scale=0.5,/pgf/number format/1000 sep={},/pgf/number format/fixed]
\begin{axis}[
xmode=log,
x dir=reverse,
ytick distance = 20000,
ymin=0,
log ticks with fixed point,
scaled ticks = false,
legend pos=north west,
y label/.style = {anchor=near ticklabel},
title = RP900,
xlabel = {Accuracy parameter},
ylabel = Iterations,
]
\addplot[color=blue,mark=none]
table {
0.100000 18083
0.050000 19376
0.020000 20406
0.010000 20672
0.005000 20708
0.002000 20727
0.001000 20730
};
\addplot[color=orange,mark=none]
table {
0.100000 49035
0.050000 56745
0.020000 74094
0.010000 84819
0.005000 121202
0.002000 135917
0.001000 147492
};
\legend{\textsc{ExpSinkhorn},\textsc{Sinkhorn}}
\end{axis}
\end{tikzpicture}
\end{minipage}\hfill
\caption{Iterations of \textsc{ExpSinkhorn} and \textsc{Sinkhorn}}
\label{fig2}
\end{figure}
\bibliographystyle{alpha}
|
1,116,691,497,791 | arxiv | \section{Introduction}
In the recent years, there has been a growing interest regarding the non-commutative spaces of the Lie algebra type, {i.e. \/} spaces where coordinates can be represented as Lie algebra generators. These types of models have indeed appeared in the context of quantum gravity, as candidates for the flat semi-classical limit. In 3d, it has been actually shown that a $\mathfrak{su}(2)$ space-time correctly describes the flat semi-classical limit of Euclidian 3d quantum gravity \cite{etera-laurent}. Meanwhile, in 4d, different arguments have been proposed to argue that $\mathfrak{an}_3$, {i.e. \/} $\kappa$-Minkowski, is the right flat semi-classical limit \cite{dsr}.
These non-commutative spaces are flat spaces, that is it is possible to introduce a deformation of the Poincar\'e group, in particular of the translations, such that the non-commutativity is consistent with the Poincar\'e transformations (for a classification of the Poincar\'e group deformations see for example \cite{zak}). If the space-time is given in terms of a Lie algebra, momentum space (or \emph{Fourier space}) is in turn defined as a (non-abelian) Lie group and acquires some curvature.
Under the Fourier transform from the Lie algebra coordinate space to the Lie group momentum space, a scalar field theory defined over a non-commutative space-time of the Lie algebra type can therefore be seen as a scalar field theory defined on a \emph{group manifold}.
\medskip
This is a well-known theory in the context of quantum gravity, where one calls it a \emph{group field theory} (GFT). These GFT are used to generate spinfoam models \cite{GFT}. The fact that group field theories are at the same time a fundamental object in the spinfoam framework and a natural object to consider in non-commutative geometry framework is very interesting and certainly a sign of fruitful future interplays between these two different frameworks. Recently this connection was used to relate in a new and original way the different flat semi-classical space-time to some spinfoam models \cite{dsr-GFT}.
\medskip
In general momentum space can be seen as an homogenous space. For example in the usual case, one considers the Poincar\'e group $\ppp \sim \mathrm{SO}(n-1,1){\,\triangleright\hspace{-1.45mm}< \,} \mathbb{R}^n$, where the Lorentz group acts on $\mathbb{R}^n$ on the left. Momentum space can then be obtained as $\mm\sim \ppp/\mathrm{SO}(n-1,1)\sim \mathbb{R}^n$. The most natural generalization of this construction is to consider a pair of Lie groups $G,\, H$ which acts on each other in a consistent way to obtain a "double cross product group" $G{\triangleright\hspace{-0.5mm}\triangleleft} H$ \cite{majid}. A typical way to have such pair is to consider a group $\ggg$ which factorizes $\ggg\sim G\cdot H$. Momentum space is then $\mm\sim \ggg/H\sim G$. In this case the scalar field will be a representation of the quantum group introduced by Majid, the bicrossproduct quantum group \cite{majid}. The most well known example is given by the Iwasawa decomposition of $\mathrm{SO}(4,1)$ \cite{klymyk}, which encodes that $\mathrm{SO}(4,1)$ is factorizable $\mathrm{SO}(4,1)\sim \mathrm{AN}_3 \cdot \mathrm{SO}(3,1)$. Momentum space is then obtained as the homogenous space $\mm\sim\mathrm{AN}_3\sim \mathrm{SO}(4,1)/\mathrm{SO}(3,1)$, which has a group structure. The corresponding Lie algebra is $\mathfrak{an}_3$ and defines the $\kappa$-Minkowski space-time. We shall present in further details the use of the bicrossproduct construction in the context of GFT in \cite{bicross en prep}.
\medskip
An homogenous space does not have in general a group structure. By considering the Cartan decomposition of a group $G$ using the maximal subgroup $H$, the homogenous space $G/H$ is only a \emph{coset}. These types of momentum spaces are again interesting both from the non-commutative geometry and the spinfoam GFT points of view. Indeed, in 1947 Snyder introduced one of the first examples of non-commutative geometry \cite{snyder} which can be encoded in terms of coordinates as
$$X_\mu\sim\frac{1}{\kappa}J_{p\mu} \in \mathfrak{so}(p-1,1)/\mathfrak{so}(p-2,1), \qquad [X_\mu,X_\nu]=i\frac{1}{\kappa^2}J_{{\mu\nu}},$$
where $J_{\mu\nu}$ is an infinitesimal Lorentz transformation and $\kappa$ is the Planck scale. Momentum space is the de Sitter space $dS$ seen as the homogenous space obtained from the Cartan decomposition $dS\sim \mathrm{SO}(4,1)/\mathrm{SO}(3,1)$. There has been up to now very few attempts to identify the deformation of the Poincar\'e group associated to this non-commutative space and to construct a scalar field theory transforming under these deformed symmetries. On the other hand the GFT which generates spinfoam models for 4d quantum gravity are typically field theories defined on the product of coset spaces $G/H$, where for example $G=\mathrm{SO}(4)$ and $H=\mathrm{SO}(3)$ in the Euclidian case \cite{gft-4d}.
\medskip
We intend here to construct a scalar field theory defined on a coset seen as momentum space, and introduce a generalized Fourier transform to analyze the properties of the dual space-time (for different approaches regarding the definition of a scalar field theory in Snyder space-time see \cite{othersnyder}).
\medskip
We show first how using the coset structure we can define a momenta addition and a convolution product (both of them will be non-associative). These are the necessary ingredients to construct the scalar field action. We show then how one can introduce a generalized Fourier transform which will allow to define a non-commutative star product as the dual of the convolution product. We pinpoint that this star product is in fact a realization of Snyder's non-commutative space-time. Before ending with some concluding remarks, we discuss the notion of symmetries in this non-commutative space.
\section{Momentum space as a coset }
In this section we construct an action for a scalar field theory defined on the right coset $G/H$. In the following we work with the 3d hyperboloid ${\cal H}} \def\ii{{\cal I}} \def\jj{{\cal J}} \def\kk{{\cal K}} \def\lll{{\cal L}} \def\mm{{\cal M}} \def\nn{{\nonumber}_3=\mathrm{SO}(3,1)/\mathrm{SO}(3)=G/H$ as a guiding example, but all the steps described here can be extended to more general cosets. Any element $g\in G$ can be written as $g=ah$, with $a\in G/H$ and $h\in H$. We note $[da]$ the measure over the coset and $[dh]$ the Haar measure on $H$.
Momentum is identified as the coordinate system on the coset space. The 3d hyperboloid is defined as
\begin{equation}\label{hyperboloid}
{\cal H}} \def\ii{{\cal I}} \def\jj{{\cal J}} \def\kk{{\cal K}} \def\lll{{\cal L}} \def\mm{{\cal M}} \def\nn{{\nonumber}_3\sim \mathrm{SO}(3,1)/\mathrm{SO}(3)\sim \{v_\mu\in\mathbb{R}^4, \, v_0^2-v_i^2=1\}.
\end{equation}
We define the Snyder coordinates $P_i= \kappa \frac{v_i}{v_0}$ as parameterizing our 3-momentum space~\footnotemark. The $\mathrm{SO}(3,1)$-invariant measure on is ${\cal H}} \def\ii{{\cal I}} \def\jj{{\cal J}} \def\kk{{\cal K}} \def\lll{{\cal L}} \def\mm{{\cal M}} \def\nn{{\nonumber}_3$:
$$
[dP]
\,=\,
\kappa^3\,d^4v \,\delta(v_0^2-v_i^2-1)
\,=\,
d^3P\,\left(1- \frac{\vec P^2}{\kappa^2}\right)^{-1}.
$$
\footnotetext{{Of course different choices of parametrization are possible. For example, using $p_i= \kappa v_i$ \cite{snyder gili}, we would get a different deformed momentum addition:
$$
\vec p_1 \oplus \vec p_2\,=\,
\left(\gamma_2+ \frac{1}{1+\gamma_1}\frac{\vec p_1\cdot \vec p_2}{\kappa^2}\right)\vec p_1 +\,\vec p_2.
$$
The main difference between $\vec p$ and the Snyder momentum $\vec P$ is that $P$ is bounded by $\kappa$ while $p$ remains unbounded.}}
A coset element is given by a boost $a=e^{i \frac} \def\t{\textrm{Tr}} \def\MP{M_P\eta2 \vec b\cdot \vec K}\,\in\mathrm{SO}(3,1)$, where $K_i\equiv J_{0i}$ are the boost generators, $\eta$ the angle of the boost and the unitary vector $\vec b$ its direction. For the explicit calculations, we actually work in the spinorial representation of $\mathrm{SO}(3,1)$, given in term of $2\times 2$ group elements belonging to $\mathrm{SL}(2,\mathbb{C})$. Such a group element acts on the 3+1d-Minkowski space and boosts the unit time-like vector $(1,0,0,0)$ to a vector $v_\mu$:
$$
v\,=\, a\rhd\,(1,0,0,0)\,=\,(\cosh \eta,\sinh\eta\,\vec{b}).
$$
This defines a Snyder momentum expressed in terms of $\eta$ and $\vec b$:
\begin{equation} \label{snyder coordinates} \vec P= \frac{\kappa}{v_0}\,\vec{v}\,=\, \kappa\tanh\eta\, \vec b.\end{equation}
From this definition of the section, one can check that the measure $[da]$ on the coset is as expected $[da]=[dP]$.
The group product on $\mathrm{SO}(3,1)$ induces a product on the coset between elements $a_i\in G/H$, which we note $a_{12}\equiv a_1\cdot a_2$. In turn, this product induces a momenta addition:
\begin{equation}\label{notation product non assoc}
a_1a_2 =a_{12}h_{12}\, {\rightarrow} \, a_{12}=e^{i \frac} \def\t{\textrm{Tr}} \def\MP{M_P{\eta_1} 2\vec b_1\cdot \vec K} \cdot e^{i\frac} \def\t{\textrm{Tr}} \def\MP{M_P{\eta_2} 2\vec b_2\cdot \vec K}=e^{i \frac} \def\t{\textrm{Tr}} \def\MP{M_P{\eta_{12}}2(\vec b_1\oplus \vec b_2)\cdot \vec K} \quad a_i\in G/H, \, h_{12}\in H. \end{equation}
The $\mathrm{SO}(3)$-group element $h_{12}$ is uniquely defined and an explicit formula can be found in \cite{SR}. The relation \eqref{snyder coordinates} is then used together with \eqref{notation product non assoc} to define the addition in terms of the Snyder coordinates~:
\begin{equation}\label{non assoc add}
\vec P_1 \oplus \vec P_2= \frac{1}{1+\frac{\vec P_1\cdot \vec P_2}{\kappa^2}}\left(\left(1+ \frac{\gamma_1}{1+\gamma_1}\frac{\vec P_1\cdot \vec P_2}{\kappa^2}\right)\vec P_1 + \frac{1}{\gamma_1}\vec P_2\right), \quad \gamma_1= \frac{1}{\sqrt{1-\frac{\vec P^2_1}{\kappa^2}}},
\end{equation}
This addition is non-associative due to the coset structure and therefore the ordering in which we group the addition is important.
We would like to emphasize here the parallel with Special Relativity. Indeed in this context the hyperboloid ${\cal H}} \def\ii{{\cal I}} \def\jj{{\cal J}} \def\kk{{\cal K}} \def\lll{{\cal L}} \def\mm{{\cal M}} \def\nn{{\nonumber}_3$ is the space of 3d speeds $\vec \upsilon$. The embedding space $\mathbb{R}^4$ is the space of relativistic speeds $v_\mu$. The usual choice of coordinates on ${\cal H}} \def\ii{{\cal I}} \def\jj{{\cal J}} \def\kk{{\cal K}} \def\lll{{\cal L}} \def\mm{{\cal M}} \def\nn{{\nonumber}_3$ is given by $\vec \upsilon = c\frac{\vec v}{v_0}= c\tanh\eta \vec b$. By noting $v_0=\gamma$, we have equivalently $\vec v= \gamma \vec \upsilon/c$, the standard expression for the relativistic speed. Finally the addition of speeds is generated by the coset product in \eqref{notation product non assoc}, and one obtains explicitly this addition by considering \eqref{non assoc add} and replacing $\vec P_i$ by $\vec \upsilon_i$ and $\kappa$ by $c$ \cite{SR}.
To build an action, we consider the distributions $\phi(a)$ and $\delta(a)$ which are respectively a function (the real scalar field) on the coset seen as a distribution and the Dirac delta function. The $\delta$-distribution on the coset is inherited from the group structure:
$$
\delta_{{\cal H}} \def\ii{{\cal I}} \def\jj{{\cal J}} \def\kk{{\cal K}} \def\lll{{\cal L}} \def\mm{{\cal M}} \def\nn{{\nonumber}_3}(a)=\int [dh]\,\delta_{\mathrm{SO}(3,1)}(ah)=\delta^{(3)}(\vec{P}).
$$
In the following, we will drop the indices ${\cal H}} \def\ii{{\cal I}} \def\jj{{\cal J}} \def\kk{{\cal K}} \def\lll{{\cal L}} \def\mm{{\cal M}} \def\nn{{\nonumber}_3$ or $\mathrm{SO}(3,1)$.
We introduce the convolution product $\diamond$ using the product on the coset \eqref{notation product non assoc}.
\begin{eqnarray}\label{coset convolution}
\phi \diamond \psi (a)&=& \int [da_1][da_2][dh]\, \phi(a_1)\psi(a_2)\delta(a^{-1} a_1a_2h)\nn\\
&=& \int [da_1][da_2]\, \phi(a_1)\psi(a_2)\delta(a^{-1} \cdot a_{12}).
\end{eqnarray}
We insist on the fact that the $\delta$-distribution in the first line is on the $\mathrm{SO}(3,1)$-group, while the $\delta$-distribution of the second line is on the coset.
In particular, we evaluate the convolution at the identity $a=e$, which corresponds to a zero-momentum $\vec{P}=0$~:
\begin{eqnarray}\label{coset convolution1}
\phi \diamond \psi (e)&=& \int [da_1][da_2]\, \phi(a_1)\psi(a_2)\delta(a_{12})\nn\\
&=& \int [dP]^2\, \phi(P_1)\psi(P_2)\delta(P_1\oplus P_2).
\end{eqnarray}
This convolution product is also non-associative, we need to keep track of the grouping of the convolution products we take. Thanks to this convolution product, we can introduce our proposal for a $\phi^3$ type scalar field action defined on the homogenous momentum space ${\cal H}} \def\ii{{\cal I}} \def\jj{{\cal J}} \def\kk{{\cal K}} \def\lll{{\cal L}} \def\mm{{\cal M}} \def\nn{{\nonumber}_3$.
\begin{eqnarray}\label{action field dia}
S_\diamond(\phi)&=& \int [da]^2\, \phi(a_1)(\vec P_1^2(a)-m^2)\phi(a_2)\int[dh]\,\delta(a_1a_2h)+ \frac{\lambda}{3!} \phi\diamond(\phi\diamond\phi) (e)\nn\\
&=& \int [dP]^2\, \phi(P_1)(\vec P_1^2-m^2)\phi(P_2)\,\delta(P_1\oplus P_2) \nn\\
&&+ \frac{\lambda}{3!} \int [dP]^3\,\phi(P_1) \phi(P_2)\phi (P_3)\,\delta(P_1\oplus(P_2\oplus P_3)).
\end{eqnarray}
\section{Fourier transform and Snyder space-time}
We now construct a generalized Fourier in order to define the dual space-time. We first introduce the plane-wave $e^{iP\cdot x}=e^{iP(a)\cdot x}$ with $x_\mu\in \mathbb{R}^3$ and star product noted $\star$ between the plane-waves in order to represent the modified momenta addition \eqref{non assoc add}~:
\begin{eqnarray} \label{product planewave}
e^{iP_1\cdot x }\star e^{iP_2\cdot x }&\equiv& e^{i(P_1\oplus P_2)\cdot x }
\end{eqnarray}
We define the Fourier transform of a distribution $\phi$ as
\begin{equation}
\hat \phi(x)\equiv \int [da] \, e^{iP(a)\cdot x } \phi(P) ,
\end{equation}
The $\star$ product is the dual of the convolution product
\begin{eqnarray}
\int [da]\, e^{iP(a)\cdot x} (\psi\diamond \phi)(a) &=&\int [da_i]^2\, \psi(a_1)\phi(a_2)\int [da][dh]\, e^{iP(a)\cdot x} \delta(a^{-1} a_1a_2 h )\nn\\
&=& \hat \psi \star \hat \phi (x)
\end{eqnarray}
Since the convolution product is non-associative, the $\star$ product will also be non-associative. We explore the properties of this $\star$ product by considering the products between monomials \cite{snyder gili}.
\begin{eqnarray}\label{1st order monomial}
x_\mu \star x_\nu& =&x_\mu\,x_\nu \\
x_\mu \star (x_\nu\star x_\alpha)&=& -\frac{1}{\kappa^2}\delta_{\nu\alpha}x_\mu +x_\mu x_\nu x_\alpha\neq ( x_\mu \star x_\nu) \star x_\alpha. \label{2nd order monomial}
\end{eqnarray}
The $\star$ product encodes therefore some type of non-commutativity in a slightly different way than usual: the $\star$ product between two coordinate functions is commutative but the position operators still do not commute. To understand the non-commutative geometry structure behind this construction we introduce the operator positions which act by $\star$-multiplication by $x_\mu$, and calculate the commutator $[X_\mu,X_\nu]$ for example on the function $x_\alpha$ using \eqref{2nd order monomial}:
\begin{eqnarray}
[X_\mu,X_\nu]{\, \triangleright\, } x_\alpha&=& x_\mu\star (x_\nu\star x_\alpha)-x_\nu\star (x_\mu\star x_\alpha)\nn\\
&=& \frac{1}{\kappa^2}\left(\delta_{\mu\alpha}x_\nu -\delta_{\nu\alpha}x_\mu\right)= i\frac{1}{\kappa^2}J_{\mu\nu}{\, \triangleright\, } x_\alpha,
\end{eqnarray}
where $J_{\mu\nu}\in \mathfrak{so}(3)$ act in the usual way on the space-time. Using the Fourier transform, this calculation can be extended to an arbitrary function $f$ and we see therefore that
\begin{equation} \label{snyder bracket}
[X_\mu,X_\nu]= i\frac{1}{\kappa^2}J_{\mu\nu}.
\end{equation}
This commutator properly encodes the commutation relation of the Snyder coordinates \cite{snyder}. \emph{We have therefore constructed a realization of the (Euclidian) Snyder space-time in terms of a star product}.
The commutator of the \emph{classical} coordinates is zero following \eqref{1st order monomial}.
$$[x_\mu,x_\nu]_\star= x_\mu\star x_\nu- x_\nu\star x_\mu=0$$
This might seem puzzling if one has in mind the commutator \eqref{snyder bracket}. This however should not come as a surprise. Indeed when working with the star product, we work at the level of the classical (deformed) algebra of continuous functions ${\cal C}_\star$ over Minkowski space. The star product is therefore stable in ${\cal C}_\star$ and it is not possible to have a commutator of the type $[x_\mu,x_\nu]_\star\sim J_{\mu\nu}$ since $J_{\mu\nu}$ is not in ${\cal C}_\star$. However at the level of \emph{operators}, both $X_\mu\sim J_{0\mu}$ and $J_{\mu\nu}$ have a well defined action on ${\cal C}_\star$, so that the commutator $[X_\mu,X_\nu]\sim J_{\mu\nu}$ does make sense. To our knowledge there is no other example of non-commutative space where the commutator $[x_\mu,x_\nu]_\star$ is actually different than its operator representation $[X_\mu,X_\nu]$ where $X_\mu$ acts by $\star$-multiplication. The difference can be traced back to the non-associative structure inherent to the coset.
\section{Symmetries}
By construction the action of $H=\mathrm{SO}(3)$ on the coset is given by the adjoint action, which in terms of the Snyder coordinates, is simply the usual "Lorentz" action $$a{\rightarrow} hah^{-1} \,\Rightarrow\, [J_{ij},P_l]=\delta_{jl}P_i-\delta_{il}P_j, \quad J_{ij}\in \mathfrak{su}(2).$$ The measure $[da]= d^3P\, \left(1- \frac{\vec P^2}{\kappa^2} \right)^{-1}$ is clearly invariant under the adjoint action of $H$. The distributions $\phi,\delta$ are transformed in the standard way
$$\phi(a)\,{\rightarrow}\, \phi(hah^{-1}), \qquad\delta(a)\,{\rightarrow}\, \delta(hah^{-1}). $$
We note in particular that a function $f$ on the coset evaluated at the origin $e$ is then left invariant under the adjoint action of $H$. As a direct consequence, the convolution product $\phi\diamond\psi(e)$ will be invariant as well. With these properties in mind, it is straightforward to check that the action \eqref{action field dia} is invariant under the adjoint action of $\mathrm{SO}(3)$ ({i.e. \/} the "Lorentz transformations").
To introduce the notion of translations, we use space-time. We shall then realize these transformations in momentum space in order to check that the action has also some kind of translation invariance.
\begin{eqnarray}
{\hat \phi}(x+\epsilon)&=& \int [dP]\, \phi(P) e^{iP\cdot(x+\epsilon) }\\
({\hat \phi} \star {\hat \phi})(x+\epsilon)&=& \int [dP_i]^2\, e^{i(P_1\oplus P_2)\cdot (x+\epsilon) } \phi(P_1)\phi(P_2) \nn
\end{eqnarray}
We see therefore that in momentum space the translations act a phase multiplication. Moreover, when dealing with many fields, we use the $\star$ product between the plane-waves to define the transformation of the star product of fields.
\begin{eqnarray}
\phi(P)&{\rightarrow}& e^{iP(a)\cdot \epsilon} \phi(P)\\
\phi(P_1)\phi(P_2)& {\rightarrow}& e^{iP_{1}\cdot\epsilon} \star e^{iP_{2}\cdot\epsilon} \phi(P_1)\phi(P_2) =e^{i(P_{1}\oplus P_2)\cdot\epsilon} \phi(P_1)\phi(P_2) .
\end{eqnarray}
Thanks to the Dirac delta function, encoding the conservation of momentum, we see therefore that the action is invariant under the translations as well. For example we have
\begin{eqnarray}
\int [dP]^2\,\phi(P_1) \phi(P_2)\,e^{i(P_1\oplus P_2)\cdot \epsilon}\delta(P_1\oplus P_2)= \int [dP]^2\,\phi(P_1) \phi(P_2)\,\delta(P_1\oplus P_2).
\end{eqnarray}
The action \eqref{action field dia} is therefore invariant under a deformation of the Poincar\'e group. The Lorentz part is not deformed, indeed Snyder noticed already that the non-commutative structure is consistent with the Lorentz symmetries \cite{snyder}. On the other hand the translations sector is deformed in a consistent way with the non-commutative structure. We do not expect this type of deformation to be among the ones classified in \cite{zak}. Indeed, this deformation is non-(co)associative whereas the classification \cite{zak} looked at the deformations which preserved the (co)associativity. It would be interesting to see if the quantum group we are dealing with could be encoded in a type of quasi-Hopf algebra, that is if there is a 3-cocycle that would rule the lack of (co)associativity (cf for example \cite{majid}).
\section{Concluding remarks}
We have constructed an action for a scalar field with momentum space a coset. We used the properties of this coset to define a modified momenta addition. We then introduced a generalized Fourier transform which allowed to defined the dual non-commutative space-time, which show to be a realization of Snyder space-time. The key feature of the construction is the non-associativity of the different products we used, property which is traced back to the geometry of the coset.
A non-associative addition of momenta is something that is definitely physically difficult to understand from the usual field theory perspective. One can therefore try instead to introduce an associative convolution product on the coset. Using the group product from $G$ we define then a convolution product based on the group product on $G$. Starting with $H$-invariant fields on $G$, $\phi(g)=\phi(gh),\,\forall h\in H$, we can define their usual convolution product:
$$
\phi \circ\psi (g)=\int [dg_1][dg_2]\, \phi(g_1)\psi(g_2)\,\delta( g^{-1} g_1g_2 ).
$$
Since the resulting convolution is still $H$-invariant, this can be written as a convolution product on the coset $G/H$ directly~\footnotemark:
\begin{eqnarray}\label{G convolution}
\phi \circ\psi (a)&=& \int [da_1][da_2]\, \phi(a_1)\psi(a_2)\int [dh_1][dh_2]\,\delta( a^{-1} a_1 h_1 a_2 h_2 ) \nn\\
&=& \int [da_1][da_2]\, \phi(a_1)\psi(a_2)\int [dh_1]\,\delta(a^{-1}\cdot a_1 \cdot (h_1 a_2 h_1^{-1}) ).
\end{eqnarray}
\footnotetext{We could start with the equivalent definition $\phi \circ\psi (a)= \int [da_1][da_2]\, \phi(a_1)\psi(a_2)\int [dh_1][dh_2][dh]\,\delta( a_1 h_1 a_2 h_2 h^{-1} a^{-1})$ and use the right invariance of $[dh_2]$ to eliminate $h$. Nevertheless, this definition is divergent if $H$ is non-compact. We therefore work with \eqref{G convolution}.}
Expressed in term of the Snyder momentum, we get:
\begin{equation}
\phi \circ\psi (P)\,=\,
\int [dP]^2\, \phi(P_1)\psi(P_2)\int [dh]\,\delta((-P)\oplus (P_1\oplus h\rhd P_2)).
\end{equation}
First, we see that the integral over $h$ averages over the direction of the second momentum $\vec{P}_2$, thus the convolution loses all information about this direction and only remembers the modulus of $\vec{P}_2$. In other words, $\phi \circ\psi$ does not depend on the entire field $\psi(P)$ but only on its radial component $\widetilde{\psi}(P)=\int dh \psi(h\rhd P)$.
Moreover, this convolution product can not be expressed simply in term of a deformed addition of momenta. At the end of the day, we obtain a non-trivial distribution of the resulting (final) momentum $\vec P$ which not only depends on the initial momenta $P_1$ and $P_2$ but also on the arbitrary group rotation $h$. This seems to be the price in order to have an associative convolution product, and thus star-product, on the coset.
For these reasons, this associative product does not seem to be neither physically motivated, nor interesting from the usual quantum field theory point of view. As a conclusion it seems that the natural non-associative convolution product inherent to the coset is the most interesting structure to study from the quantum field theory approach. It provides us with a proper $star$-product representation of the non-commutative Snyder space-time. The next step will be to study the quantization of the (scalar) field theory on the Snyder space-time and to see whether a Fock space representation is possible or not. The present $\star$-product will likely be instrumental in this development.
\begin{theacknowledgments}
F.G. wants to thank the organizers of the conference, in particular J. Kowalski-Glikman, for their kind hospitality.
\end{theacknowledgments}
\bibliographystyle{aipproc}
|
1,116,691,497,792 | arxiv | \section{Introduction}
\label{sec:intro} Synchronization of oscillations is one of the
fundamental nonlinear phenomena in biology, physics, chemistry,
communication science, and many other branches of science and
engineering~\cite{book}. Recently, the dynamics of synchronization
of oscillators located at each vertex in complex networks has
attracted much attentions. That is because the small-world feature
of complex networks is closely related to their synchronizability.
By the small-world feature, we mean that the average separation
$\langle d \rangle$ between a pair of vertices scales at most
$\langle d \rangle \sim \log N$, where $N$ is the number of
vertices in the system. It was shown that a small-world network
model introduced by Watts and Strogatz~\cite{watts98} is more
synchronizable than regular lattice~\cite{pecora}. However, such
feature is not observed in scale-free (SF) networks. SF networks
are the networks that exhibit a power-law degree distribution
$P_d(k) \sim k^{-\lambda}$ and degree $k$ is the number of edges
connected to a given vertex~\cite{barabasi99}. In SF networks, the
heterogeneity in the degree distribution suppresses their
synchronizability~\cite{synch_SF1}. Thus, it was desired to
introduce a new dynamic model that prompts SF networks to be more
synchronizable.
The dynamics of synchronization is described by various forms of
coupled equations. A linearly coupled model is probably the
simplest one. In the model, $N$ oscillators are coupled when they
are connected via edges. Coupling constant is normally symmetric;
however, it is not necessarily symmetric to achieve a better
synchronizability. This case can happen in SF networks: It was
shown recently~\cite{motter05} that the synchronizability becomes
maximum when information flow diffuses and reaches a uniform
stationary state over the entire system. Here, the mapping from
synchronization dynamics to information flow can be naturally
introduced, because the linearly coupled equation is nothing but
the diffusion equation. It was shown that the uniform-stationary
state can be reached by introducing asymmetric and weighted
coupling strength between a pair of vertices or oscillators.
To be specific, the dynamic model with the asymmetric coupling
strength is written as
\begin{equation} \frac{d\phi_i}{dt}= f(\phi_i)
-\frac{J}{k_i^{1-\eta}}\sum_{j=1}^{N} a_{ij}\left(
h(\phi_i)-h(\phi_j)\right)\label{eqn:linear}
\end{equation}
for $i=1, \ldots, N$. Here, $\phi_i$ is the phase of an oscillator
located at vertex $i$, $f(\phi)$ describes the dynamics of an
individual oscillator, $J$ is the overall coupling strength. $k_i$
is the degree of vertex $i$, and $a_{ij}$ is an element of the
adjacent matrix, which is 1 if vertices $i$ and $j$ are connected
and 0 otherwise. $h(\phi_i)$ is the output function and take a
form of $h(\phi_i)=\phi_i$ for the linear case. It is noteworthy
that the coupling strength of Eq.~(\ref{eqn:linear}) is asymmetric
and weighted due to the factor $1/k_i^{1-\eta}$ unless $\eta=1$.
When $\eta>0$, vertices with large degree can influence other
vertices significantly on regulating phases due to their large
numbers of connections; on the other hand, when $\eta<0$, the
influence is reduced. It was found~\cite{motter05} that the system
is most synchronizable when $\eta=0$, irrespective of the value of
the degree exponent of a given SF network.
In this paper, we study the pattern of synchronization transition
for a modified Kuramoto model~\cite{kuramoto}, which is similarly
modified with the asymmetric and weighted coupling strength as
\begin{equation}
{\frac{d \phi_i}{dt}} = \omega_i - \frac {J} {k_i^{1-\eta}}
\sum_{j=1}^{N} a_{ij}\sin(\phi_i-\phi_j).\label{eqn:eom}
\end{equation}
The oscillators are located at each vertex $i=1,\dots,N$ of a SF
network with degree exponent $\lambda$. Here, $\omega_i$ is the
natural frequency of the $i$th oscillator selected from the
Gaussian distribution $g(\omega)=e^{-\omega^2/2}/\sqrt{2\pi}$. We
find that the modified Kuramoto dynamic model displays a very
complex and rich behaviors in the space of the two tunable
parameters ($\eta,\lambda$).
The synchronization transition from a desynchronized to a
synchronized state occurs at the critical point $J_c$. For small
$J \ll J_c$, the coupling strength is so weak that an individual
vertex maintains its own phase different from others; therefore,
the entire system is desynchronized. As the coupling strength $J$
increases, a cluster of vertices is more likely to be coupled, to
be in a common or almost the same phase, and thus forms a cluster
of synchrony. Size of such clusters becomes diverse as the
coupling strength $J$ increases. At the critical point $J_c$, the
system reaches a self-organized state, and the cluster-size
distribution follows a power law,
\begin{equation}
n(s)\sim s^{-\tau-1}\label{cluster} \end{equation} in the
thermodynamic limit. For $J \gg J_c$, the power-law behavior no
longer holds and the entire system is synchronized.
The order parameter of the synchronization transition is defined
as
\begin{equation}
\label{eqn:glovalop} r e^{i\theta} =
\frac{1}{N}\sum_{i=1}^N e^{i\phi_i}.
\end{equation}
In the synchronized state, the phases $\phi_i$ of each vertex are
narrowly distributed around an average phase $\theta$. The
amplitude $r$ of the order parameter has a finite value; on the
other hand, $r\approx 0$ in the desynchronized state. Thus, the
exponent $\beta$ associated with the order parameter is defined
via the relation, \begin{equation} r\sim \Delta^{\beta},
\end{equation} where $\Delta=(J-J_c)/J_c$. In finite-size systems,
the order parameter is described in terms of a scaling function as
\begin{equation} r \sim N^{-\beta/\mu}\psi(\Delta N^{1/\mu}).\end{equation}
In the recent works~\cite{synch_SF2,synch_SF5}, the nature of the
transitions and the finite-size scalings have been studied for the
case of $\eta=1$. In this work, we determine the order parameter
and the size distribution of synchronized clusters for general
$\eta$ using the mean-field approach and the generating function
technique. Moreover, we construct a finite-size scaling function
for the order parameter, and determine the exponent $\mu$. Even
for a simple extension of $\eta\ne 1$, we find that the obtained
result is very rich. There exist eight distinct transition
behaviors depending on the values of $\eta$ and $\lambda$.
Therefore, the result can be helpful in understanding diverse
dynamic phenomena arising on SF networks.
The paper is organized as follows: In Sec.~\ref{sec:meanfield}, we
first introduce and apply the mean-field approach to the dynamic
equation (\ref{eqn:eom}). We construct a self-consistent equation
for a local field and determine the order parameter. Next, the
critical point is determined and the behavior of the order
parameter near the critical point is obtained in
Sec.~\ref{sec:anal1}. The size distribution of synchronized
clusters and the largest cluster size at the critical point are
solved in Secs.~\ref{sec:anal2} and \ref{sec:fss}, respectively.
The finite-size scaling analysis for the order parameter is
performed and the results are checked numerically in
Sec.~\ref{sec:nu}. Summary and discussion follow in
Sec.~\ref{sec:conclusion}.
\section{order parameter equation}
\label{sec:meanfield}
In this section, we analyze the modified Kuramoto equation
(\ref{eqn:eom}) in the framework of the mean-field approach by
constructing a self-consistent equation for a local field. To
proceed, we define $\bar{r}_i$ and $\bar{\theta}_i$ as the
amplitude and phase of the local field at vertex $i$,
respectively, via
\begin{equation} \label{eqn:localop} \bar{r}_i
e^{i\bar{\theta}_i} = \frac {1} {k_i} \sum_{j=1}^N
a_{ij}e^{i\phi_j}.
\end{equation} Then,
Eq.~(\ref{eqn:eom}) is rewritten in terms of the local field as
\begin{equation}
\label{eqn:localemo} {\frac{d \phi_i}{dt}} =
\omega_i - J \bar{r}_i k_i^{\eta} \sin (\phi_i -
\bar{\theta_i}).
\end{equation}
Once the amplitude $\bar{r}_i$ and the phase $\bar{\theta}_i$
of the local field are determined, one can solve
Eq.~(\ref{eqn:localemo}) easily. The local field $\bar{r}_i$
is determined in a self-consistent manner.
We consider the probability density
$\rho_i^{(s)}(\phi|\omega)d\phi$ that the phase of an oscillator
$i$ with natural frequency $\omega$ lies between $\phi$ and
$\phi+d\phi$ in the steady state~\cite{strogatz_physica}. Using a
previous result \cite{strogatz_physica} that
$\rho_i^{(s)}(\phi|\omega)d\phi$ is inversely proportional to the
speed of $\phi$, one can obtain that \begin{equation} \rho_i^{\rm (s)}
(\phi|\omega) =
\begin{cases}
\delta \left[ \phi -\bar{\theta}_i- \sin^{-1} \left( \displaystyle{
\frac{\omega}{\omega_{*,i}}}\right)\right] &
{\rm if }\ |\omega|\leq \omega_{*,i}, \\
&\\
\displaystyle{ \frac{\sqrt{\omega^2 - \omega_{*,i}^2}}{ 2\pi | \omega -
\omega_{*,i} \sin (\phi-\bar{\theta}_i) |}} & \rm{otherwise},
\end{cases}
\label{eqn:rho} \end{equation} where $\omega_{*,i} = J
\bar{r}_i k_i^\eta$. This result implies that an oscillator
$i$ with natural frequency $\omega$ has its phase that is locked
at $\phi=\bar{\theta}_i+\sin^{-1}(\omega/\omega_{*,i})$ and
$d\phi_i/dt=0$ if $|\omega|\leq \omega_{*,i}$. Otherwise, its
phase drifts with a finite speed, $d\phi_i/dt \ne 0$. Next, we can
evaluate the order parameter using the stationary probability
density in Eq.~(\ref{eqn:rho}) as \begin{equation} r
e^{i\theta}=\frac{1}{N}\sum_i \int_{-\infty}^{\infty} d\omega
g(\omega) \int d\phi \rho_i^{(s)} (\phi|\omega) e^{i\phi}.
\label{eqn:glovalop-decomp} \end{equation}
Although $\bar{r}_i$, $\bar{\theta}_i$, and
$\rho_i^{\rm (s)}(\phi|\omega)$ can fluctuate over $i$ in the
steady state, we assume here that they depend only on degree
$k_i$. This is a mean field approximation. Keeping only the
degree-dependent fluctuations, one can obtain a self-consistent
equation for the local field through Eq.~(\ref{eqn:localop}) as
\begin{align}
\bar{r}(k) e^{i\bar{\theta}(k)} &= \sum_{k'=1}^{k_m} P(k'|k)\nonumber\\
&~~~\times
\int_{-\infty}^{\infty} d\omega g(\omega)
\int_0^{2\pi} d\phi \rho^{\rm (s)}(\phi|\omega,k') e^{i\phi},
\label{localfield}
\end{align}
where $\rho^{\rm (s)}(\phi| \omega,k)$ is given by the right
hand side of Eq.~(\ref{eqn:rho}) with
$\omega_*(k)=J\bar{r}(k) k^\eta$ replacing
$\omega_{*,i}$. $P(k^{\prime}|k)$ denotes the probability that a
neighboring vertex of a given vertex with degree $k$ has degree
$k^{\prime}$, and $k_m$ is the natural cutoff of degree. Here, we
consider only the case that the network is random and does not
have any type of degree-degree correlation; then,
$P(k^{\prime}|k)$ can be written as
$k^{\prime}P_d(k^{\prime})/\langle k\rangle$ with $\langle
k\rangle = \sum_k k P_d(k)$. After that, one can see that both
$\bar{r}(k)$ and $\bar{\theta}(k)$ are independent of degree
$k$, and therefore, we can drop the $k$-dependence in
$\bar{r}$ and $\bar{\theta}$ from now on.
The last integral of Eq.~(\ref{localfield}) is evaluated as
\begin{align}
&\int_{0}^{2\pi} d\phi \rho^{(s)}(\phi|\omega,k) e^{i\phi}\nonumber \\
&= e^{i\bar{\theta}}
\begin{cases}
i(\omega/\omega_*(k))-i\sqrt{(\omega/\omega_*(k))^2-1}, &(\omega > \omega_*(k)), \\
i(\omega/\omega_*(k)) + \sqrt{ 1-(\omega/\omega_*(k))^2}, &(|\omega|\leq \omega_*(k)), \\
i(\omega/\omega_*(k)) + i \sqrt{ (\omega/\omega_*(k))^2 - 1},
&(\omega < -\omega_*(k)).
\end{cases}
\end{align}
The remaining integration in Eq.(\ref{localfield}) for $\omega
> \omega_*(k)$ and $\omega < \omega_*(k)$ cancel out due to
the fact $g(\omega)=g(-\omega)$. As a result, only the oscillators
having the frequency within the range $|\omega|\leq \omega_*(k)$
contribute to the local field in Eq.~(\ref{eqn:localop}). Thus,
one obtains \begin{equation} \label{eqn:localopcontinum} \bar{r} =
\sum_{k=1}^{k_m} \frac{k P_d(k)}{ \langle k \rangle}
\int_{-\omega_*(k)}^{\omega_*(k)} d\omega g(\omega)
\sqrt{1-
\left(\displaystyle{\frac{\omega}{\omega_*(k)}}\right)^2},
\end{equation} which is the self-consistent equation for $\bar{r}$. Note
that $\bar{r}$ is contained in $\omega_*(k)=J{\bar
r}k^{\eta}$. After the local field is obtained, the order
parameter in Eqs.~(\ref{eqn:glovalop}) or
(\ref{eqn:glovalop-decomp}) is calculated as \begin{equation}
\label{eqn:glovalopcontinuum} r=
\sum_{k=1}^{k_m}P_d(k)\int_{-\omega_*(k)}^{\omega_*(k)} d\omega
g(\omega)
\sqrt{1-\left(\displaystyle{\frac{\omega}{\omega_*(k)}}\right)^2}.
\end{equation}
\section{Synchronization transition}
\label{sec:anal1} In this section, we solve the self-consistent
equation, Eq.~(\ref{eqn:localopcontinum}) explicitly, and then
investigate the behavior of the order parameter near the critical
point via Eq.~(\ref{eqn:glovalopcontinuum}). To proceed, we first
recall that the degree distribution is given in a closed form as
$P_d(k)=k^{-\lambda}/H_{k_m}^{\lambda}$ for $\lambda>2$, where
$H_{k_m}^\lambda$ is the generalized harmonic number, defined by
$H_m^q \equiv \sum_{k=1}^{m} k^{-q}$ and $k_m \sim
N^{1/(\lambda-1)}$. Substituting
$g(\omega)=e^{-\omega^2/2}/\sqrt{2\pi}$ to
Eq.~(\ref{eqn:localopcontinum}), one can derive the local field
$\bar{r}$ as
\begin{align}
\label{eqn:localfield-expand} \bar{r} & = \sum_{k=1}^{k_m}
\frac {k P_d(k)} {\langle k \rangle} \omega_*(k)
\int_{-\pi/2}^{\pi/2} d\phi \cos^2\phi \frac {1}{\sqrt{2 \pi}}
e^{-{[\omega_*(k) \sin\phi]^2}/{2}}
\nonumber\\
&= \sum_{n=0}^{\infty}\frac{ (n-1/2)!\,(-1)^n \,
H_{k_m}^{\lambda-\eta-2n \eta-1}}{n!\,(n+1)!\,2^{n+3/2}\,
H_{k_m}^{\lambda-1}} (J\bar{r})^{2n+1}
\nonumber\\
&\equiv \sum_{n=0}^\infty \bar{A}_n (J \bar{r})^{2n+1},
\end{align}
where we used the Taylor expansion of
$e^{-(\omega_*(k)\sin\phi)^2/2}$ and the integration
$\int_{-\pi/2}^{\pi/2} d\phi \cos^2\phi \sin^{2n}\phi = \pi^{1/2}
(n-1/2)! /\left(2(n+1)!\right)$. Similarly, the order parameter is
evaluated as
\begin{align}
r &= \sum_{n=0}^{\infty} \frac{ (n-1/2)! \,\,(-1)^n
H_{k_m}^{\lambda-\eta-2n \eta}} { 2^{n+3/2}\,\, n!\,(n+1)!\,\,
H_{k_m}^\lambda}
(J\bar{r})^{2n+1}\nonumber\\
&\equiv \sum_{n=0}^\infty A_n (J \bar{r})^{2n+1}.
\label{eqn:orderparameter-expand}
\end{align}
If $\eta\leq 0$, the generalized harmonic numbers in $\bar{A}_n$
and $A_n$ are finite, and then they can be represented in terms of
the Riemann zeta functions for all $n$ below and they are denoted
as $\bar{B}_n$ and $B_n$, respectively. That is,
\begin{equation}
\bar{A}_n\approx \bar{B}_n = \frac{ (n-1/2)! (-1)^n
\zeta(\lambda-\eta-2n \eta-1)} { 2^{n+3/2} n!(n+1)!
\zeta(\lambda-1)} \label{eqn:BBbar}
\end{equation}
and
\begin{equation}
{A}_n\approx {B}_n = \frac{ (n-1/2)! (-1)^n \zeta(\lambda-\eta-2n
\eta)} { 2^{n+3/2} n!(n+1)! \zeta(\lambda)}, \label{eqn:BB}
\end{equation}
respectively. Using these formulae, the local field and the order
parameter are determined by Eqs.~\sceqs.
On the other hand, it is remarkable that when $0 < q <1$, the
generalized harmonic number diverges as $H_m^q \simeq
(m+1)^{1-q}/(1-q)+\zeta(q)+{\cal O}(m^{-q})$ in the $m\to\infty$
limit, which is shown in Appendix. Then, Eqs.~\sceqs are divided
into analytic and singular parts as
\begin{equation}
\bar{r}=\sum_n \bar{B}_n (J
\bar{r})^{2n+1}+ \bar{C}(J \bar{r} k_m^\eta)
(J \bar{r})^{(\lambda-2)/\eta}
\label{eqn:localrs}
\end{equation} and
\begin{equation}
r=\sum_n B_n (J \bar{r})^{2n+1}+
{C}(J \bar{r} k_m^\eta)
(J \bar{r})^{(\lambda-1)/\eta},
\label{eqn:globalrs}
\end{equation}respectively, where the functions $\bar{C}(x)$ and
${C}(x)$ are defined in Appendix. In the $x\to\infty$ limit
corresponding to the thermodynamic limit, $\bar{C}(x)$ and $C(x)$
reduce to $\bar{C}_\infty$ and $C_\infty$, respectively, defined
as
\begin{align}
\bar{C}_\infty&=\displaystyle{
\frac{[(\lambda-2\eta-2)/2\eta]![(2-\lambda-\eta)/2\eta]!} {\eta
2^{(\lambda+4\eta-2)/2\eta}[(\lambda+\eta-2)/2\eta]! \zeta(\lambda-1)}},
\nonumber\\
{C}_\infty&=\displaystyle{
\frac{[(\lambda-2\eta-1)/2\eta]![(1-\lambda-\eta)/2\eta]!} {\eta
2^{(\lambda+4\eta-1)/2\eta}[(\lambda+\eta-1)/2\eta]!
\zeta(\lambda)}}. \label{eqn:c0}
\end{align}
Thus, the local field and the order parameter are written as
\begin{equation} \bar{r} = \sum_{n=0}^{\infty}
\bar{B}_n(J\bar{r})^{2n+1}
+ \bar{C}_\infty(J\bar{r})^{(\lambda-2)/\eta} +\ldots,
\label{eqn:local-pos-eta}
\end{equation}
and
\begin{align}
\label{eqn:orderparameter-pos-eta}
r= \sum_{n=0}^{\infty} B_n (J
\bar{r})^{2n+1} + C_\infty (J \bar{r})^{(\lambda-1)/\eta} + \ldots,
\end{align}
for $J\bar{r} k_m^\eta\gg 1$. We remark that the
singular terms appear only in the limit $J
\bar{r} k_m^\eta\to \infty$. For the case of
$J\bar{r} k_m^\eta\ll 1$, however, Eqs.~\sceqs
are valid.
Next, we determine the critical point. To proceed, we investigate
the behavior of the local field as a function of $J$,
which depends on the sign of $\eta$.
{\bf (i) In the case of $\eta\leq 0$,} $\bar{A}_n$ and $A_n$ are
finite. One can see from Eq.~(\ref{eqn:localfield-expand}) that
the local field is zero for $\bar{A}_0 J< 1$ and non-zero for
$\bar{A}_0 J> 1$. The order parameter behaves in the same manner
as that of the local field from
Eq.~(\ref{eqn:orderparameter-expand}). Thus, we obtain the
critical point as \begin{equation} J_c = \frac{1}{\bar{A}_0} =
\frac {2\sqrt{2}}{\sqrt{\pi}}
\frac {H_{k_m}^{\lambda-1}}{H_{k_m}^{\lambda-1-\eta}}.
\label{eqn:Jc} \end{equation} As $\lambda\to\infty$, the critical point $J_c$
approaches $2\sqrt{2}/\sqrt{\pi} \simeq 1.60$ in the limit
$N\to\infty$, which is consistent with that found in case of the
globally-coupled oscillators~\cite{kuramoto}.
When $J > J_c$, the local field
$\bar{r}$ and the order parameter $r$ are
non-zero. When $J$ is close to $J_c$,
\begin{equation} \bar{r} \simeq (|\bar{A}_1| J_c^3)^{-1/2}
\Delta^{1/2}\label{r10}\end{equation} and \begin{equation}r\simeq {A}_0
(|\bar{A}_1| J_c)^{-1/2} \Delta^{1/2},\end{equation} where
$\Delta= {(J-J_c)}/{J_c}$.
Thus, we obtain that $\beta=1/2$. Again, this result is consistent
with the one obtained from the globally-coupled
oscillators~\cite{kuramoto}.
{\bf (ii) In the case of $\eta>0$,} the singular terms in
Eqs.~(\ref{eqn:local-pos-eta}) and
(\ref{eqn:orderparameter-pos-eta}) can be crucial in determining
the critical point and the order parameter. Depending on relative
magnitude of $\lambda$ and $\eta$, we divide the case of $\eta
>0$ into four subcases:
\begin{itemize}
\item[(I)] When $0<\eta<(\lambda-2)/3$ (i.e., $\lambda > 3\eta+2$),
$\bar{r} \simeq \bar{B}_0 J \bar{r} +
\bar{B}_1 J^3 \bar{r}^3 + \cdots$ for small
$\bar{r}$ from Eq.~(\ref{eqn:local-pos-eta}). Then
$J_c$ and $\bar{r}$ behave as those for $\eta <
0$ presented in Eqs.~(\ref{eqn:Jc}) and (\ref{r10}). \item[(II)]
When $(\lambda-2)/3 <\eta<\lambda-2$ (i.e., $\eta+2 < \lambda <
3\eta+2$), the dominant contribution is made from the singular
term of Eq.~(\ref{eqn:local-pos-eta}). Then \begin{equation} \bar{r} \simeq
\bar{B}_0 J\bar{r}+\bar{C}_\infty
(J\bar{r})^{(\lambda-2)/\eta}+\ldots, \end{equation} leading
to \begin{equation} J_c\sim
1/\bar{B}_0=\frac{2\sqrt{2}}{\sqrt{\pi}}\frac{\zeta(\lambda-1)}{\zeta(\lambda-\eta-1)}
\end{equation} and \begin{equation} \bar{r}\sim r \sim
\Delta^{\eta/(\lambda-2-\eta)}. \end{equation}
\item[(III)] When $\lambda-2 < \eta < \lambda-1$ (i.e., $\eta+1
< \lambda < \eta+2$), the critical point in Eq.~(\ref{eqn:Jc}) for
finite $N$ behaves as \begin{equation} J_c \sim k_m^{-(\eta-\lambda+2)}\sim
N^{-(\eta-\lambda+2)/(\lambda-1)}.\end{equation} Thus, it approaches zero in
the thermodynamic limit. $\bar{r}$ is always positive unless
$J$ is zero as $\bar{r}\sim
J^{(\lambda-2)/(\eta-\lambda+2)}$ for small $J$ and
$r \sim J\bar{r} \sim
J^{\eta/(\eta-\lambda+2)}$.
\item[(IV)] When $\eta > \lambda-1$ (i.e., $\lambda < \eta+1$),
we obtain that $r \sim (J
\bar{r})^{(\lambda-1)/\eta}$. Using the result of
$\bar{r}$ obtained in (III), we obtain that \begin{equation}
r \sim
J^{(\lambda-1)/(\eta-\lambda+2)}. \end{equation}
\end{itemize}
We summarize the result as follows: When $\eta < \lambda-2$ (in
the (I) and (II) cases), the critical point $J_c$ is finite;
however, when $\eta > \lambda-2$ (in the (III) and (IV) cases),
$J_c=0$ in the thermodynamic limit $N\to \infty$. Thus, the
critical exponent $\beta$ associated with the order parameter is
defined through the relation, $r\sim \Delta^{\beta}$ ($r\sim
J^{\beta}$) for the former (latter) case. The exponent $\beta$ is
evaluated in each case as follows: \begin{equation} \beta =
\begin{cases}
1/2 & \text{in~(I),}\\
\eta/(\lambda-2-\eta) & \text{in~(II),}\\
\eta/(\eta-\lambda+2) & \text{in~(III),}\\
(\lambda-1)/(\eta-\lambda+2) & \text{in~(IV).}
\end{cases}
\label{eqn:beta}
\end{equation}
The result of the critical point is consistent with those of other
phase transition problems such as the percolation transition and
the epidemic spreading in SF networks. Moreover, the result for
the case of $\eta=1$ reduces to the previous result
\cite{synch_SF2,synch_SF5}. Moreover, the result $\beta=1/2$ for
$\eta=1$ and $\lambda > 5$ is reduced to the mean-field result in
regular lattice.
\section{Cluster formation of synchronized oscillators}
\label{sec:anal2}
In this section, we investigate in detail how the coupled
oscillator system develops its synchrony as the coupling strength
increases. To this end, we study the formation of clusters
comprising synchronized vertices as a function of the coupling
strength $J$. We use the generating function approach to derive
the cluster-size distribution.
\subsection{Cooperative versus background synchrony}
\label{sec:synchrony}
The order parameter averaged over the natural frequency
distribution $g(\omega)$ can be written as \begin{equation}
r=\frac{1}{N}\sqrt{\sum_{i}\langle \cos^2\phi_i
+\sin^2\phi_i \rangle+\sum_i \sum_{j \ne i}\langle
\cos(\phi_i-\phi_j)\rangle}\label{eqn:order-anal-J0} \end{equation} from
Eq.(\ref{eqn:glovalop}). Here, the brackets represent the average
over $g(\omega)$. For the case of $J=0$, each element
oscillates independently, so that $\langle
\cos(\phi_i-\phi_j)\rangle=0$ for $i \ne j$. Thus, the order
parameter is evaluated as \begin{equation} r_{J=0}
\sim \frac{1}{\sqrt{N}}. \label{eqn:order-anal-J0} \end{equation} As
$J$ increases, clusters comprising synchronized
oscillators are more likely to form. We here define a cluster as
a group of vertices (or oscillators) which are connected and in
the same coherent state: Two oscillators are coherent if its
time-average correlation function $C_{ij}$, defined as \begin{equation}
C_{ij}=\frac{1}{(t_1-t_0)} \sum_{t=t_0+1}^{t_1} \langle \cos
(\phi_i(t) - \phi_j(t)) \rangle,\end{equation} is larger than a preassigned
threshold value $C_{\rm th}$. We choose $C_{\rm th}$ value to
generate the cluster-size distribution in a power law form at the
critical point $J_c$. As such clusters form, the term of $\sum_i
\sum_{j \ne i}\langle \cos(\phi_i-\phi_j)\rangle$ becomes nonzero
and dominant. The order parameter is then evaluated as \begin{equation}
r \sim \frac{\sqrt{\sum_{\kappa} s^2_{\kappa}}}{N},
\label{eqn:order-anal-J-pos} \end{equation} where $\kappa$ is the index of
cluster and $s_{\kappa}$ is the size of cluster $\kappa$, i.e.,
the number of vertices within the cluster $\kappa$. Note that
$\sum_{\kappa} s_{\kappa}=N$, and Eq.~(\ref{eqn:order-anal-J-pos})
reduces to Eq.~(\ref{eqn:order-anal-J0}) when $J=0$
because each cluster size is 1. In the synchronized state when
$J \gg J_c$, the size of the largest
cluster, denoted as $S$, is of ${\cal O}(N)$, and thus the order
parameter is approximately given as \begin{equation} r \sim S/N.
\end{equation} Next, we study the cluster-size distribution and the size of
the largest cluster as a function of $J$.
The dynamics of cluster merging with increasing $J$
results in the change of the cluster-size distribution. Let $n(s)$
be the number of $s$-size clusters. Then $\sum_s sn(s)=N$. The
cluster-size distribution is defined as $n(s)/\sum_s n(s)$. For $J
< J_c$, the cluster-size distribution decays exponentially for
large $s$. However, it decays in the power law form
(\ref{cluster}) at the critical point $J=J_c$, and the associated
exponent $\tau$ depends on the parameters $\eta$ and $\lambda$. We
determine $\tau$ using the generating function method in the next
subsection. For $J
> J_c$, a giant cluster forms and the distribution of finite-size
clusters decays exponentially. The cluster-size distributions for
various values of $J$ are shown in Fig.~\ref{fig:cluster}.
\begin{figure*}
\resizebox*{!}{5cm}{\includegraphics{fig1.eps}} \caption { (Color
online) Cluster-size distributions $n(s)/\sum_s n(s)$ of
synchronized clusters for the networks generated with
$\lambda=4.0$, $\eta=0.0$, and $N=3000$ at $ J=0.7J_c $
(\textcolor{red}{$\bigcirc$}), $J=0.8J_c$
(\textcolor{blue}{$\square)$}, $J=0.9J_c$
(\textcolor{green}{$\triangle$}) for (a), $J=J_c$
(\textcolor{red}{$\bigcirc$}) for (b), $J=1.1J_c$
(\textcolor{red}{$\bigcirc$}), $J=1.2J_c$
(\textcolor{blue}{$\square$}), and $J=1.3J_c$
(\textcolor{green}{$\triangle$}) for (c), respectively. Solid
lines drawn for reference have a slope of $7/3$ for all. }
\label{fig:cluster}
\end{figure*}
\subsection{Generating function of the cluster-size distribution}
The probability that a vertex belongs to a cluster with size $s$ is
given by $sn(s)/N$, which is denoted as $p(s)$. Invoking the
percolation theory, $p(s)$ follows a power law with an exponential
cutoff, \begin{equation} p(s)\sim s^{-\tau}e^{-s/s_c}, \label{ps} \end{equation} where $s_c$
is the characteristic size, which depends on $J$ and system size
$N$. In the thermodynamic limit $N\to \infty$, $s_c$ diverges at
$J=J_c$. As in the percolation theory, the generating function
${\cal P}(z)\equiv\sum_s p(s)z^s$ is useful for studying structural
feature of the synchronized clusters, since its singular behavior is
related to the critical behavior of the synchronization transition.
(i) The order parameter $r\sim S/N$ can be obtained
from the relation $r \simeq \lim_{N\to\infty} [1-{\cal
P}(z_N^*)]$, where ${\cal P}(z_N^*) =\sum_{s<S} p(s)$, i.e., the
contribution by finite-size clusters. This can be achieved by
choosing $z_N^* \approx e^{-1/S_m}$, where $S_m$ is a cluster size
smaller than the largest cluster but larger the second largest
cluster. (ii) From Eq.~(\ref{ps}), one can find that ${\cal P}(z)$
diverges for $z > z_c=\lim_{s\to \infty} p(s)^{-1/s}$, i.e., $z_c
\approx e^{1/s_c}$. Thus, at $J=J_c$, ${\cal P}(z)\sim
(1-z)^{\tau-1}$ as $z\to z_c=1$ in the thermodynamic limit. Thus,
finding the singularity of ${\cal P}(z)$ enables one to obtain
$p(s)$.
For the purpose, we introduce another generating function
$\bar{\cal P}(z)$ as a partner of the local field ${\bar r}$. From
$\bar{\cal P}(z)$, one can define a probability ${\bar p}(s)$ via
the relation $\bar{\cal P}(z)\equiv \sum_s \bar{p}(s)z^s$, where
${\bar p}(s)$ is defined similarly to $p(s)$ as the probability
that a vertex belongs to a synchronized cluster of size $s$
composed of the vertex and $s-1$ neighboring vertices. For finite
$N$, the generating function $\bar{\cal P}(z)$ is analytic for
$|z|\leq 1$ and so is its inverse function $\bar{\cal P}^{-1}(z)$.
To investigate the singularity of $\bar{\cal P}(z)$ near $z=1$, we
consider the expansion of the inverse function $z=\bar{\cal
P}^{-1}(\omega)=1-\sum_{n\geq 1} b_n (1-\omega)^n$ around
$\omega=1$. The coefficient $b_n$ depends on $J$.
Using Eqs.~(\ref{eqn:localfield-expand}) and
(\ref{eqn:local-pos-eta}) and replacing $\bar{r}$ by
$1-\omega$, we can find that the generating function $\bar{\cal
P}(z)$ satisfies the self-consistent relations given below:
\begin{align}
z = &\bar{\cal P}(z) +
\sum_{n=0}^{\infty} \bar{B}_n[J(1-\bar{\cal P}(z))]^{2n+1}\nonumber\\
& + \bar{C}_\infty[J(1-\bar{\cal P}(z))]^{(\lambda-2)/\eta} +\ldots,
\label{eqn:pbar-small-s}
\end{align}
for $J (1-\bar{\cal P}(z)) k_m^\eta \gg 1$, and \begin{equation} z
=\bar{\cal P}(z)+\sum_{n=0}^{\infty}
\bar{A}_n[J(1-\bar{\cal P}(z))]^{2n+1},
\label{eqn:pbar-large-s}\end{equation} for $J (1-\bar{\cal
P}(z)) k_m^\eta \ll 1$. Similarly, ${\cal P}(z)$ is determined as
\begin{eqnarray}
z-{\cal P}(z)&=&\sum_{n=0}^{\infty} B_n[J(1-\bar{\cal
P}(z))]^{2n+1} \nonumber
\\&+&C_\infty[J(1-\bar{\cal
P}(z))]^{(\lambda-1)/\eta}+\ldots, \label{eqn:pandpbar-small-s}
\end{eqnarray}
for $J (1-\bar{\cal P}(z)) k_m^\eta \gg 1$, and \begin{equation}
z-{\cal P}(z)=\sum_{n=0}^{\infty} A_n[J(1-\bar{\cal
P}(z))]^{2n+1}, \label{eqn:pandpbar-large-s} \end{equation} for
$J (1-\bar{\cal P}(z)) k_m^\eta \ll 1$.
\subsection{Behavior of $p(s)$ at the critical point}
Here, we calculate the probability $p(s)$ to find a vertex in the
$s$-size cluster at the critical
point $J=J_c$ explicitly in each case defined in Sec.\ref{sec:anal1}.\\
{\bf In the case (I),} since $\eta < (\lambda-2)/3$, we obtain
that $z=1+\bar{B}_1 \left(J_c (1-\bar{\cal
P}(z))\right)^3 + \ldots$ to the leading orders by expanding
$z=\bar{\cal P}^{-1}(\omega)$ around $\omega=1$ in either
Eqs.~(\ref{eqn:pbar-small-s}) or (\ref{eqn:pbar-large-s}). Thus,
we obtain that $1-\bar{\cal P}(z)\sim (1-z)^{1/3}$, leading to \begin{equation}
\bar{p}(s)\sim s^{-4/3} \end{equation} for large $s$.
Using the obtained leading behaviors of $\bar{\cal P}(z)$ around
$z=1$ in Eqs.~(\ref{eqn:pandpbar-small-s}) and
(\ref{eqn:pandpbar-large-s}), one obtains the behavior of ${\cal
P}(z)$ around $z=1$ as $1-{\cal P}(z) \sim (1-z)^{1/3}$. Thus, \begin{equation}
p(s)\sim s^{-4/3}. \label{eqn:ps1} \end{equation}
{\bf In the case (II),} the singular term in
Eq.~(\ref{eqn:pbar-small-s}) is relevant. In this case, the
behavior of $\bar{\cal P}(z)$ for $z < z_c$ differs from that for
$z > z_c$. $z_c$ is determined by the criterion $J
(1-\bar{\cal P}(z_c))k_m^\eta \sim 1$. This case also happens for
the cases (III) and (IV).
In this case, the singular term
$\bar{C}_\infty[J_c(1-\bar{\cal
P}(z)))]^{(\lambda-2)/\eta}$ is dominant in
Eq.~(\ref{eqn:pbar-small-s}); therefore, it follows that
$1-\bar{\cal P}(z)\sim J_c^{-1}(1-z)^{\eta/(\lambda-2)}$ at
$J=J_c$, which is valid for $z\gg z_c$. From this result,
$\bar{p}(s)$ is obtained as \begin{equation} \bar{p}(s)\sim
J_c^{-1}s^{-(\eta+\lambda-2)/(\lambda-2)},\label{eqn:pbar<} \end{equation}
which is valid for $s\ll s_c$.
On the other hand, when $J(1-\bar{P}(z))k_m^\eta \ll 1$ so that
$|\bar{A}_{n+1}/\bar{A}_n| [J(1-\bar{P}(z))]^2\ll 1$, one can
obtain that $1-\bar{\cal P}(z)\sim J_c^{-1}
|\bar{A}_1|^{-1/3}(1-z)^{1/3}$ for $z \ll z_c$; therefore, \begin{equation}
\bar{p}(s)\sim J_c^{-1} k_m^{(\lambda-2)/3-\eta}
s^{-4/3} \label{eqn:pbar>} \end{equation} for large $s \gg s_c$. $s_c$ is
evaluated as follows: Substituting the result of $1-\bar{\cal
P}(z)$ in the criterion $J_c (1-\bar{\cal P}(z_c))
k_m^\eta \sim 1$ and using $(1-z_c) \sim s_c^{-1}$, one can obtain
system-size dependence of the characteristic size $s_c$ explicitly
as \begin{equation} s_c\sim k_m^{\lambda-2}\sim N^{(\lambda-2)/(\lambda-1)},
\label{eqn:s_c}\end{equation} which diverges as $N\to \infty$.
Together with Eqs.~(\ref{eqn:pbar<}) and (\ref{eqn:pbar>}), we
obtain that \begin{equation} \bar{p}(s)\sim
\begin{cases}
s^{-(\eta+\lambda-2)/(\lambda-2)} & (s\ll s_c),\\
k_m^{(\lambda-2)/3-\eta} s^{-4/3} & (s\gg s_c).
\end{cases}
\end{equation}
Next, using the result of $1-{\cal P}(z)\sim 1-\bar{\cal P}(z)$
obtained from both Eqs.~(\ref{eqn:pandpbar-small-s}) and
(\ref{eqn:pandpbar-large-s}), one can find that $p(s)$ behaves
similarly to $\bar{p}(s)$. That is, \begin{equation} p(s)\sim
\begin{cases}
s^{-(\eta+\lambda-2)/(\lambda-2)} & (s\ll s_c),\\
k_m^{(\lambda-2)/3-\eta} s^{-4/3} & (s\gg s_c).
\end{cases}
\label{eqn:ps2} \end{equation}
{\bf In the case (III),} the critical point $J_c$ is finite in
finite-size systems as being of order $J_c \sim
k_m^{\lambda-2-\eta}\sim N^{(\lambda-2-\eta)/(\lambda-1)}$.
Plugging the $N$-dependence into Eq.~(\ref{eqn:pbar>}) and the
expression, $1-\bar{\cal{P}}(z) \simeq 1-z +\bar{C}_\infty
[J_c(1-z)]^{(\lambda-2)/\eta}$ for $s\ll s_c$ from
Eq.~(\ref{eqn:pbar-small-s}), one obtains $\bar{p}(s)$ as follows:
\begin{equation} \bar{p}(s)\sim
\begin{cases}
k_m^{(\lambda-2-\eta)(\lambda-2)/\eta}
s^{-(\lambda-2+\eta)/\eta}& (s\ll s_c),\\
k_m^{-2(\lambda-2)/3} s^{-4/3} & (s\gg s_c).
\end{cases}
\end{equation}
Next, we derive $p(s)$. We find that the leading singular term in
${\cal P}(z)$ for the case $1-z\gg s_c^{-1}\sim k_m^{2-\lambda}$
shows up in two ways. Substituting $1-\bar{\cal P}(z)\approx 1-z +
\bar{C}_\infty [J_c (1-z)]^{(\lambda-2)/\eta}$ to
Eq.~(\ref{eqn:pandpbar-small-s}), we obtain that $1-{\cal
P}(z)\approx B_0 J_c (1-z)+B_0 J_c
\bar{C}_\infty [J_c
(1-z)]^{(\lambda-2)/\eta}+C_\infty [J_c
(1-z)]^{(\lambda-1)/\eta}+\cdots$. We compare the second with the
third terms in order of magnitude. Using the fact that $J_c \sim
k_m^{\lambda-2-\eta}$, we find that there exist two subcases for
$s \ll s_c$. The second term $B_0 J_c \bar{C}_\infty
[J_c (1-z)]^{(\lambda-2)/\eta}$ is more dominant than
the third term $C_\infty [J_c
(1-z)]^{(\lambda-1)/\eta}$ when $1-z\ll s_*^{-1}$ and vice versa.
Here, it is found that a new crossover size $s_*$ scales as \begin{equation}
s_*\sim k_m^{(\eta-1)(\eta-\lambda+2)}. \end{equation} From the behaviors of
${\cal P}(z)$ in the three different subcases, we obtain the
probability $p(s)$ as \begin{equation} p(s)\sim
\begin{cases}
k_m^{(\lambda-1)(\lambda-2-\eta)/\eta}
s^{-(\lambda-1+\eta)/\eta}& (s\ll s_*),\\
k_m^{(\lambda-2-\eta)(\lambda-2+\eta)/\eta}
s^{-(\lambda-2+\eta)/\eta}& (s_*\ll s\ll s_c),\\
k_m^{(\lambda-2)/3 - \eta} s^{-4/3} & (s\gg s_c).
\end{cases}
\label{eqn:ps3} \end{equation} One can notice that the subcase $s\ll s_*$
diminishes when $\eta\leq 1$, but it is extended as the parameter
$\eta$ increases.
{\bf In the case (IV),} the third term $C_\infty [J_c
(1-z)]^{(\lambda-1)/\eta}$ in $1-{\cal P}(z)$ in the case (III) is
always dominant when $1-z \gg s_c^{-1}$. Moreover, $A_0$ in
Eq.~(\ref{eqn:pandpbar-large-s}) diverges as $A_0\sim
k_m^{\eta-\lambda+1}$, which has to be considered in the relation,
$1-\bar{P}(z)\simeq A_i J_c (1-\bar{P}(z))$ for $1-z\ll s_c^{-1}$.
Consequently, $p(s)$ behaves as \begin{equation} p(s)\sim
\begin{cases}
k_m^{(\lambda-1)(\lambda-2-\eta)/\eta}
s^{-(\lambda-1+\eta)/\eta}& (s\ll s_c),\\
k_m^{(1-2\lambda)/3} s^{-4/3} & (s\gg s_c).
\end{cases}
\label{eqn:ps4}
\end{equation}
To substantiate the predictions of this section, we investigate
the asymptotic behavior of $p(s)$ in a numerical manner. The
static model introduced in~\cite{staticmodel} is used for
underlying network in our simulations. The network has $N=3000$
oscillators and its mean degree $\langle k \rangle$ is 4.0. The
values of $\lambda$ and $\eta$ are chosen as 4.0 and 0.0,
respectively. This pair belongs to the case (I). First, we
simulate the system at $J=J_c$ to determine $C_{\rm th}$ defined
in Sec.~\ref{sec:synchrony}. During the simulation, we assume a
large value of $C_{\rm th}$ and then collect the pairs of vertices
that the $C_{ij}$ of each pair is larger than the assumed $C_{\rm
th}$. After that, we determine clusters and obtain the
cluster-size distribution. We then adjust $C_{\rm th}$ by somewhat
decreasing or increasing it, and repeat these procedures until the
power-law distribution appears in the cluster-size distribution.
If the cluster-size distribution follows the power-law form of
$p(s) \sim s^{-\tau}$, the corresponding value of $C_{\rm th}$ is
considered as the threshold value $C_{\rm th}$. It is found
numerically that $C_{\rm th}\approx 0.7$, independent of the
system size $N$. In our simulations, we obtain $\tau+1 \approx
7/3$, which is close to the theoretical value in
Eq.~(\ref{eqn:ps1}), as shown in Fig.~\ref{fig:cluster}(b). We
also performed simulations for various $J < J_c$ and $> J_c$ as
shown in Figs.~\ref{fig:cluster}(a) and (c), respectively. The
power-law behaviors in the cluster-size distribution do not appear
in these cases.
\section{Largest cluster size and finite-size scaling}
\label{sec:fss}
\begin{figure}
\includegraphics[width=\columnwidth]{fig2.eps}
\caption{(Color online) Diagram in the space of $(\eta,\lambda)$
of eight different domains each corresponding to a distinct
synchronization transition. The transition nature of each domain
is listed in Table~\ref{table:exponents}.} \label{fig:regions}
\end{figure}
\begin{table*}
\caption{The probability to find a vertex in $s$-size cluster
$p(s)$, the largest cluster size $S$ at the critical point, and the
critical exponents $\beta$ and $\mu$ for the eight cases shown in
Fig.~\ref{fig:regions}.} \label{table:exponents}
\begin{tabular}{c|c|c|c|c}
\hline
domain & $p(s)$ & $S$ & $\beta$ & $\mu$\\
\hline\hline
(I) &
$s^{-4/3}$ &
$N^{3/4}$ &
$\displaystyle{\frac{1}{2}}$ &
$2$\\ \cline{2-5}
(IIa) &
\multirow{2}{5cm}{$\begin{cases}
s^{-(\eta+\lambda-2)/(\lambda-2)} & (s\ll s_c),\\
k_m^{(\lambda-2)/3 -\eta} s^{-4/3} & (s\gg s_c)
\end{cases}$} &
$N^{(4\lambda-5-3\eta)/[4(\lambda-1)]}$ &
\multirow{2}{1.5cm}{$\displaystyle{\frac{\eta}{\lambda-2-\eta}}$} &
$\displaystyle{\frac{\eta}{1+3\eta} \frac{4(\lambda-1)}{\lambda-2-\eta}}$
\\
\cline{3-3} \cline{5-5}
(IIb)&&
$N^{(\lambda-2)/(\lambda-2+\eta)}$ &&
$\displaystyle{\frac{\lambda-2+\eta}{\lambda-2-\eta}}$\\
\cline{2-5}
(IIIa1) &
\multirow{4}{7cm}{
$\begin{cases}
k_m^{(\lambda-1)(\lambda-2-\eta)/\eta}
s^{-(\lambda-1+\eta)/\eta}& (s\ll s_*)\\
k_m^{(\lambda-2-\eta)(\lambda-2+\eta)/\eta}
s^{-(\lambda-2+\eta)/\eta}& (s_*\ll s\ll s_c)\\
k_m^{(\lambda-2)/3 - \eta} s^{-4/3} & (s\gg s_c).
\end{cases}$}&
\multirow{2}{4cm}{$
N^{(4\lambda-5-3\eta)/[4(\lambda-1)]}$}&
\multirow{4}{1.5cm}{$\displaystyle{\frac{\eta}{\eta-\lambda+2}}$} &
$\displaystyle{\frac{\eta}{1+3\eta} \frac{4(\lambda-1)}{\eta-\lambda+2}}$ \\
\cline{5-5}
(IIIa2) &&&&
\multirow{3}{3cm}{$\displaystyle{\frac{2 \eta}{\eta-\lambda+2}}$}\\
\cline{3-3}
(IIIb)&&
$N^{\eta/(\lambda-2+\eta)+(\lambda-2-\eta)/(\lambda-1)}$&&\\
\cline{3-3}
(IIIc)&&
$N^{(\lambda-2)/(\lambda-1+\eta)}$&&\\
\cline{2-5}
(IV) &
$\begin{cases}
k_m^{(\lambda-1)(\lambda-2-\eta)/\eta}
s^{-(\lambda-1+\eta)/\eta}& (s\ll s_c)\\
k_m^{(1-2\lambda)/3} s^{-4/3} & (s\gg s_c).
\end{cases}$ &
$N^{(\lambda-2)/(\lambda-1+\eta)}$&
$\displaystyle{\frac{\lambda-1}{\eta-\lambda+2}}$ &
$\displaystyle{\frac{2(\lambda-1)}{\eta-\lambda+2}}$
\\ \hline
\end{tabular}
\end{table*}
In this section, we first investigate the $N$-dependence of the
largest cluster size at $J_c$. Next, based on this
result, we derive a finite-size scaling form for the order
parameter near $J_c$.
\subsection{The largest cluster size}
The largest cluster size $S$ can be obtain from the
relation~\cite{percolation}, \begin{equation} \sum_{s>S} p(s) \sim \frac{S}{N}.
\label{eqn:extreme} \end{equation}
{\bf In the case (I),} we use the result of Eq.~(\ref{eqn:ps1}),
and obtain simply that \begin{equation} S \sim N^{3/4}. \label{eqn:S1} \end{equation}
{\bf In the case (II),} $p(s)$ displays a crossover at $s_c$, and
thus the obtained value of the largest cluster size must satisfy
the self-consistency conditions. For instance, the largest cluster
size obtained by Eq.~(\ref{eqn:extreme}) for $s\ll s_c$ in
Eq.~(\ref{eqn:ps2}) has to be smaller than $s_c$. As a result, the
largest cluster size behaves differently in the two subcases $\eta
< 1$ and $\eta \ge 1$, which we denote (IIa) and (IIb),
respectively. In each subcase, we obtain that \begin{equation} S\sim
\begin{cases}
N^{(4\lambda-5-3\eta)/[4(\lambda-1)]} & \text{in (IIa),}\\
N^{(\lambda-2)/(\lambda-2+\eta)} & \text{in (IIb)}.
\end{cases}
\end{equation} The largest cluster size $S$ in (IIa) was determined from
$p(s)$ for $s\gg s_c$, and is indeed much larger than $s_c$,
whereas it in (IIb) was done from $p(s)$ for $s\ll s_c$.
{\bf In the case (III),} $p(s)$ exhibits three distinct power-law
behaviors. Thus, this case is divided into three subcases. They
are as follows: $\eta < 1$ (IIIa), $1 \le \eta \le
\sqrt{\lambda^2-3\lambda+3}$ (IIIb), and
$\eta>\sqrt{\lambda^2-3\lambda+3}$ (IIIc). The largest cluster
size in each subcase is given as \begin{equation} S\sim
\begin{cases}
N^{(4\lambda-5-3\eta)/[4(\lambda-1)]} & \text{in (IIIa),}\\
N^{\eta/(\lambda-2+\eta)+ (\lambda-2-\eta)/(\lambda-1)} & \text{in (IIIb),} \\
N^{(\lambda-2)/(\lambda-1+\eta)} & \text{in (IIIc).}
\end{cases}
\end{equation}
{\bf In the case (IV),} the largest cluster size is determined
simply by $p(s)$ for $s\ll s_c$ since the resulting largest
cluster size fulfils the criterion $S < s_c$ for $\eta>0$. Thus,
\begin{equation} S\sim N^{(\lambda-2)/(\lambda-1+\eta)}. \end{equation}
\subsection{Finite-size scaling}
Here, we evaluate the magnitude of the order parameter
$r_c$ at $J_c$ and establish the finite-size scaling
function. To proceed, we compare the magnitude of cooperative
synchrony $S/N$ with that of the background synchrony $\sim
N^{-1/2}$. The order parameter $r_c$ is defined as $\sim S/N$ if
$S/N\gg N^{-1/2}$, and $\sim N^{-1/2}$ otherwise. Under this
criterion, we obtain $r_c$ as $\sim S/N$ in the cases (I) and
(II), and $\sim N^{-1/2}$ in the cases (IIIb), (IIIc), and (IV).
The case (IIIa) is divided into two subcases, $2\lambda-3\eta-3
\ge ( < ) 0$. They are denoted as (IIIa1) and (IIIa2),
respectively. The order parameter $r_c$ behaves as $\sim S/N$ and
$\sim N^{-1/2}$ in (IIIa1) and (IIIa2), respectively.
By using that $r\sim \Delta^{\beta}$ and $N$-dependent behavior of
$r_c$ at $J_c$, we can construct a finite-size scaling form as \begin{equation}
r= N^{-\beta/\mu} \psi (
\Delta N^{1/\mu})
\label{eqn:fss12} \end{equation} for the cases (I) and (II), and \begin{equation}
r= N^{-\beta/\mu} \psi (
J N^{1/\mu})
\label{eqn:fss34} \end{equation} for the cases (III) and (IV), where \begin{equation}
\psi(x)\sim
\begin{cases}
\text{const} & \text{for}~~x\ll 1, \\
x^\beta & \text{for}~~x\gg 1.
\end{cases}
\end{equation} The critical exponent $\mu$ is determined by the relation
$r_c\sim N^{-\beta/\mu}$. The value of $\mu$ varies depending on
the cases determined by the magnitude of $\eta$ and $\lambda$.
We present the diagram in Fig.~\ref{fig:regions} comprising eight
distinct cases in the $(\eta,\lambda)$ plane. Each case in the
diagram corresponds to a distinct behavior of the critical
exponents $\beta$ and $\mu$, the cluster-size distribution, and
the largest cluster size. We summarize those features in
Table~\ref{table:exponents}.
\begin{table*}[t]
\caption{Numerical values of the parameters ($\eta$,\,$\lambda$)
we used for Figs.~\ref{fig:numericalresults}. $\beta_t$ and
$\mu_t$ are theoretical values for a given set of
($\eta$,\,$\lambda$) in the third column. $\beta_n$ and $\mu_n$
are numerical values to draw Figs~\ref{fig:numericalresults} for
each case. For (a)--(c), the theoretical and numerical values are
the same each other for both $\beta$ and $\mu$. However, they can
be different for (d)--(h). $r_t$ and $r_n$ are the order
parameters in scaling form formulated with the theoretical values
of $\beta_t$ and $\mu_t$, and the numerical values $\beta_n$ and
$\mu_n$, respectively.} \label{table:lambdanu}
\begin{tabular}{c|c|c|c|c|c|c|l|l}
\hline
Fig.~\ref{fig:numericalresults} & domain & ($\eta$,\,\,$\lambda$)& $\beta_t$ & $\beta_n$ & $\mu_t$ & $\mu_n$ &~~~~~~~~~~~~~$r_t$ & ~~~~~~~~~~~~~$r_n$ \\
\hline\hline
(a) & (I) & $\left(1/3,4\right)$ & $1/2$ & $1/2$ & $2$ & $2$ & ~~$N^{-1/4}\psi(\Delta N^{1/2})$ & ~~$N^{-1/4}\psi(\Delta N^{1/2})$ \\
(b) & (IIa) & $\left(5/6,4\right)$ & $5/7$ & $5/7$ &
~$49/120$~ &
~$49/120$~ & ~~$N^{-7/24} \psi(\Delta N^{49/120})$ &~~$N^{-7/24} \psi(\Delta N^{49/120})$ \\
(c) & (IIb) & $\left(4/3,4\right)$ & $2$ & $2$ & $5$ & $5$ & ~~$N^{-2/5}\psi(\Delta N^{1/5})$ & ~~$N^{-2/5}\psi(\Delta N^{1/5})$ \\
(d) & (IIIa1) & $~\left(1/3,13/6 \right)$ & $2$ & $20/7$ & $14/3$ & $20/3$ & ~~$N^{-3/7}\psi(J N^{3/14})$ & ~~$N^{-3/7}\psi(J N^{3/20})$ \\
(e) & (IIIa2)~& $~\left({2}/{3},{13}/{6}\right)$~& ~$4/3$~ & ~$50/23$~ & $8/3$ & $100/23$ & ~~$N^{-1/2}\psi(J N^{3/8})$ & ~~$N^{-1/2} \psi(J N^{23/100})$ \\
(f) & (IIIb) & $\left({5}/{2},4 \right)$ & $5$ & $5$ & $10$ & $10$ & ~~$N^{-1/2}\psi(J N^{1/10})$ & ~~$N^{-1/2}\psi(J N^{1/10})$ \\
(g) & (IIIc) & $\left({11}/{4},4 \right)$ & $11/3$ & $11/3$ & $22/3$ & $22/3$ & ~~$N^{-1/2}\psi(J N^{3/22})$ & ~~$N^{-1/2}\psi(J N^{3/22})$ \\
(h) & (IV) & $\left(4,4\right)$ & $3/2$ & $5$ & $3$ & $10$ & ~~$N^{-1/2}\psi(J N^{1/3})$ & ~~$N^{-1/2}\psi(J N^{1/10})$ \\
\hline
\end{tabular}
\end{table*}
\begin{figure*}
\resizebox*{!}{8cm}{\includegraphics{fig3.eps}} \caption {(Color
online) Finite-size scaling behaviors of the order parameter $r$.
Data are collected from the static network model with mean degree
$\langle k \rangle = 4$ and system sizes
$N=$400(\textcolor{red}{{\large{$\diamond$}}}),
800(\textcolor{green}{$\triangle$})
1600(\textcolor{blue}{$\bigcirc$}), and
3200(\textcolor{magenta}{$\square$}). Numerical values of the
tunable parameters ($\eta$,\,$\lambda$) and the critical exponents
are given in Table.~\ref{table:lambdanu} for each case. For the
critical point $J_c$, theoretical $J_c^{(t)}$ and numerical
$J_c^{(n)}$ values used in (a)--(c) are different as $(J_c^{(t)},
J_c^{(n)})=(0.92,1.32)$ (a), (0.37,0.50) (b), and (0.13,0.18)
(c).} \label{fig:numericalresults}
\end{figure*}
\section{Numerical simulations}
\label{sec:nu}
We perform direct numerical integration of Eq.~(\ref{eqn:eom}) to
confirm the analytic results. In particular, the finite-size
scaling behaviors in Eqs.~(\ref{eqn:fss12}) and (\ref{eqn:fss34})
are compared. For the purpose, we generate random SF networks
using the static model~\cite{staticmodel} with system sizes of
$N=400, 800, 1600$, and $3200$, mean degree of $\langle k \rangle
= 4.0$ and values of $\eta$ and $\lambda$ chosen from each domain
in Fig.~\ref{fig:regions}. Numerical values of $(\lambda,\eta)$ we
used are listed in Table~\ref{table:lambdanu}. For the numerical
integration, we apply the Heun's method~\cite{heun}. Time is
discretized by a unit step $\delta t=0.005$ and runs up to $t=1.2
\times 10^4$. Ensemble average is taken over ${\cal O}(10^2)\sim
{\cal O}(10^3)$ different configurations of natural frequencies
and network realizations, respectively.
Numerical results are presented in
Fig.~\ref{fig:numericalresults}. For each
Fig.~\ref{fig:numericalresults}(a)--(c), the critical point $J_c$
is finite. We find $J_c$ numerically to make the obtained
numerical data collapsed for different system size $N$ in the
scaling plot with theoretical values of $\beta$ and $\mu$.
Theoretical and numerically found values of $J_c$, denoted as
$J_c^{(t)}$ and $J_c^{(n)}$, respectively, are compared as
$(J_c^{(t)}, J_c^{(n)})=(0.92,1.32)$ for (a), (0.37,0.50) for (b),
and (0.13,0.18) for (c). They belong to the cases (I), (IIa), and
(IIb), respectively.
For Figs.3~(d)--(h), the critical point $J_c$ is zero. For the
cases of (d),(e), and (h), we find that numerical data do not
collapse well in the scaling plot of $rN^{\beta/\mu}$ versus $J
N^{1/\mu}$ with theoretical values of $\beta_t$ and $\mu_t$
tabulated in Table~I. Instead, we adjust numerical values of
$\beta_n$ and $\mu_n$ values empirically to make the obtained
numerical data collapsed. Those empirical values of $\beta_n$ and
$\mu_n$ are compared with the theoretical values as listed in
Table II. The discrepancy in (d) and (e) originates from the
presence of intrinsic degree-degree correlation in the static
model when the degree exponent $2 < \lambda < 3$, while the
theoretical values were obtained under the assumption that the
degree-degree correlation is absent.
\section{Conclusions and Discussion}
\label{sec:conclusion}
In this paper, we have investigated the nature of the
synchronization transition generated by $N$ limit-cycle oscillators
located at random SF networks with degree exponent $\lambda$. The
dynamics is given by a modified Kuramoto equation with the
asymmetric and degree-dependent weighted coupling strength in the
form of $J k_i^{1-\eta}$, where $k_i$ is the degree of vertex $i$.
Depending on the sign and magnitude of $\eta$, the influence of the
hub vertices on the dynamics can be moderated or amplified, and
determines the nature of the synchronization transition. Applying
the mean-field approach to the modified Kuramoto equation, we
derived the critical point, the size distribution of synchronized
clusters, and the largest cluster size at the critical point. The
critical exponents associated with the order parameter and the
finite-size scaling are determined in terms of the two tunable
parameters, ($\eta, \lambda$). All results are summarized in
Table.~\ref{table:exponents}. The parameter space of
($\eta,\lambda$) is divided into eight different domains, in each of
which the transition nature is distinct.
It would be interesting to notice that the critical exponents
$\beta$ and $\mu$ associated with the order parameter and the
finite-size scaling of the synchronization transition depend on
the parameter $\eta$ defined in the coupling strength. The result
is unusual from the perspective of the universality in the
critical phenomena in regular lattice where the details of the
couplings are mostly irrelevant unless they are long
ranged~\cite{fisher}. This result implies that structural features
of SF networks such as the degree distribution are not sufficient
to understand the dynamic process on such networks. Asymmetric
coupling in dynamics is a relevant perturbation in such networks,
because of the heterogeneity of the degree distribution. Such a
behavior was also observed in the sandpile model~\cite{sandpile}.
\section{Acknowledgement}
This work was supported by KRF Grant No.~R14-2002-059-010000-0 of
the ABRL program funded by the Korean government (MOEHRD) and the
Seoul R \& BD program.
|
1,116,691,497,793 | arxiv | \section{Introduction and summary of results}
\setcounter{equation}{0}
Recent years have seen growing efforts to extend the dictionary of holography beyond the original paradigm of the AdS/CFT correspondence, and generalize it to spacetimes with different asymptotics from those of AdS. In particular, there has been interest in extensions to quantum field theories that may not be relativistic in the UV, with Lifshitz theories providing a prime example \cite{Kachru:2008yh,Taylor:2008tg,Ross:2009ar,Ross:2011gu,Mann:2011hg,Baggio:2011cp,Griffin:2011xs,Griffin:2012qx,Chemissany:2014xsa}. These efforts are part of a broader program to enlarge the universality class of theories to which holographic techniques can be applied, and also to understand how symmetries and symmetry breaking mechanisms are encoded in a spacetime description. Much of this program has been motivated by the intriguing behavior of many novel, strongly correlated quantum phases of matter that typically lack a quasiparticle description, but are amenable to being probed with the tools of holography. Indeed, the latter can offer a new analytical window into the dynamics of such phases, and provide insight into their unconventional transport behavior. For a comprehensive review with a focus on condensed matter applications see \emph{e.g.}, \cite{Hartnoll:2016apf}.
Thus far, most holographic studies of transport have been framed within the context of systems
which break Lorentz invariance in the IR, but emanate from a relativistic conformal fixed point in the UV. Such constructions, which rely on geometries that approach AdS at the boundary, have proven fruitful for probing a rich variety of IR phases, whose behavior is captured by the near-horizon region of the geometry. However, non-relativistic UV fixed points and their hyperscaling violating generalizations are interesting in their own right, and furthermore broaden the class of renormalization group (RG) flows that can be modeled holographically, offering novel ways to make contact with condensed matter systems. In particular, in order to identify the physical observables in non-relativistic theories holographically it is necessary to place them at the UV of the RG flow, that is to consider bulk backgrounds that are asymptotically non-relativistic, and develop the corresponding holographic dictionary.
To this end, in this paper we examine a four dimensional Einstein-Maxwell-Axion-Dilaton theory that admits a class of non-relativistic geometries that are asymptotically Lifshitz and exhibit hyperscaling violation. They are parametrized by a dynamical critical exponent $z$ and a hyperscaling violating parameter $\theta$, which characterizes the anomalous scaling of the free energy of the system. The axionic fields included in the model break translational invariance along the boundary directions, thus providing a mechanism to dissipate momentum \cite{Karch:2007pd,Hartnoll:2009ns,Faulkner:2010zz,Faulkner:2013bna}, a crucial ingredient for a realistic description of materials with impurities and an underlying lattice structure. The theory we focus on involves two massless $U(1)$ gauge fields, one responsible for the Lifshitz-like nature of the background solutions, while the other is analogous to a standard Maxwell field in asymptotically-AdS charged black holes.
The electrical DC conductivity matrix for this model was computed originally
in \cite{Cremonini:2016avj}, using the near horizon analysis of \cite{Donos:2014cya}. Subsequently, \cite{Bhatnagar:2017twr} computed the full thermoelectric conductivity matrix in the presence of a background magnetic field, using the same near horizon analysis.\footnote{See also \cite{Ge:2016lyn,Chen:2017gsl}. Note, however, that there are a number of subtleties in their analysis, including choices of parameters which lead to violations of the null energy condition. Some of their results differ from ours.}
However, the near horizon analysis does not suffice to identify the conserved currents in the dual Lifshitz theory whose two-point functions determine the thermoelectric conductivities. Moreover, both the identification of the conserved currents in the Lifshitz theory and the corresponding conductivities depend on the boundary conditions for the bulk fields imposed at infinity. In particular, the conductivities obtained from the near horizon analysis correspond only to one specific boundary condition, out of an infinite family of possible boundary conditions.
In this paper we consider linearized fluctuations around a class of purely electric asymptotically hyperscaling violating Lifshitz backgrounds with axion charge that turn on all the modes necessary to obtain the full matrix of thermoelectric conductivities. Through a careful analysis of the asymptotic behavior of the solutions at the UV we determine the boundary counterterms necessary to renormalize the theory and identify the physical observables in the dual Lifshitz theory. One of our main observations is that the fluctuations necessary to compute the thermoelectric conductivities turn on a source for the energy flux, which is an irrelevant operator in the energy-momentum complex of the dual Lifshitz theory \cite{Ross:2011gu}. This leads to several subtleties in the holographic renormalization of the theory, which we address in detail. Moreover, we consider both Dirichlet and Neumann boundary conditions for the gauge field supporting the Lifshitz background and show that these boundary conditions result in a different set of thermoelectric conductivities for the second Maxwell field. In fact, there is an infinite set of possible boundary conditions, parameterized by $SL(2,\bb Z)$ \cite{Witten:2003ya}, all leading to a different set of conductivities. Only the conductivities corresponding to Dirichlet boundary conditions on the gauge field supporting the Lifshitz background match those obtained from the near horizon analysis, which has mostly been used in the literature. However, we show that boundary conditions at the UV provide a mechanism for obtaining different conductivities from the same bulk theory.
The conductivities we obtain from Dirichlet and Neumann boundary conditions have a rich but distinct behavior as a function of temperature and are sensitive to the various parameters characterizing the background solution. We perform a preliminary analysis of their temperature dependence by looking for regions in the parameter space where they exhibit approximate scaling behavior with the temperature. We identify several clean scaling regimes in the limit of large temperature, which in our setup probes the Lifshitz theory in the UV. Rather intriguingly, the only case we could identify that can potentially lead to a linear resistivity arises in the case of Dirichlet boundary conditions for the specific value $z=4/3$ of the dynamical exponent, which was also singled out in the field theory analysis of \cite{Hartnoll:2015sea}.
The paper is organized as follows. In section \ref{sec:model} we present the general class of models we are interested in, both in the Einstein and the so called `dual frame' \cite{Kanitscheider:2009as}, as well as the generic backgrounds around which we linearize the equations of motion. We consider backgrounds carrying only electric Maxwell charge and magnetic axion charge in order to incorporate momentum dissipation. A significant part of our analysis is carried out for generic backgrounds and applies to a wide range of UV asymptotics, including AdS$_4$ and hyperscaling violating Lifshitz with arbitrary dynamical exponents. A specific exact asymptotically hyperscaling violating Lifshitz black brane solution studied in \cite{Cremonini:2016avj} is reviewed in subsection \ref{example} and is the focus of most of our subsequent analysis. Section \ref{sec:linear} contains our study of the linearized fluctuation equations, which are collected in appendix \ref{sec:fluctuations}. In particular, we obtain the general solution of the fluctuation equations in the small frequency limit and with infalling boundary conditions on the horizon. Specializing to the exact background of subsection \ref{example}, we then obtain the asymptotic UV expansions of the linear fluctuations and determine the boundary terms required in order to renormalize the theory. Finally, we identify the physical observables in the dual Lifshitz theory for two different boundary conditions on the bulk gauge field supporting the Lifshitz background. All two-point functions captured by the linear fluctuations we consider are presented in section \ref{sec:conductivities}, and the corresponding DC conductivities are obtained after the identification of the thermal current in the dual Lifshitz theory. We conclude in section \ref{sec:conclusion}, where we also discuss a number of possible directions for future work. The radial Hamiltonian description of the bulk dynamics, which is used to renormalize the theory, is summarized in appendix \ref{ham}.
\section{Hyperscaling violating Lifshitz black branes with axion charge}
\label{sec:model}
\setcounter{equation}{0}
We are interested in asymptotically hyperscaling violating Lifshitz backgrounds supported by a massless gauge field and a running dilaton \cite{Taylor:2008tg}. Additionally, we turn on spatial profiles for a number of axion fields in order to break translation invariance \cite{Donos:2013eha,Andrade:2013gsa}, which ensures that the DC conductivities are finite.
\paragraph{Einstein frame} Specifically, the model we consider is described by the Einstein frame action
\bbepsilon\label{action-0}
S=\frac{1}{2\kappa^2}\int_{\cal M} d^{d_s+2}x\sqrt{-g}\Big(R-\alpha (\pa\phi)^2-\Sigma_{IJ}(\phi)F^I_{\mu\nu}F^{J\mu\nu}-Z(\phi)(\pa\chi^a)^2-V(\phi)\Big)+S\sbtx{GH},
\end{equation}
where $d_s$ denotes the number of spatial dimensions of the conformal boundary, $\kappa^2=8\pi G_{d_s+2}$ is the gravitational constant in $d_s+2$ dimensions, and $S\sbtx{GH}$ denotes the Gibbons-Hawking term
\bbepsilon
S\sbtx{GH}=\frac{1}{2\kappa^2}\int_{\pa{\cal M}} d^{d_s+1}x\sqrt{-\gamma}\;2K.
\end{equation}
We have allowed for a generic normalization of the dilaton kinetic term, corresponding to the positive definite parameter $\alpha$, for later convenience. The indices $a=1,\cdots, d_s$ run over all $d_s$ axion fields, while the indices $I,J$ run over a yet unspecified number of Abelian gauge fields. The summation convention is adopted for the indices $a$ and $I, J$, in addition to the spacetime indices $\mu, \nu$. Finally, the symmetric matrix $\Sigma_{IJ}(\phi)$ and the functions $Z(\phi)$ and $V(\phi)$ are going to be kept generic for most part of our analysis, except that $\Sigma_{IJ}(\phi)$ is required to be invertible with strictly positive eigenvalues.
The equations of motion following from the action \eqref{action-0} are
\bal\label{eoms-0}
&R_{\mu\nu}=Z(\phi)\pa_\mu\chi^a\pa_\nu\chi^a+\alpha\pa_\mu\phi\pa_\nu\phi+\frac{1}{d_s}V(\phi)g_{\mu\nu}+2\Sigma_{IJ}(\phi)\(F_{\mu\rho}^I F^J_\nu{}^\rho-\frac{1}{2d_s}F^I_{\rho\sigma}F^{J\rho\sigma}g_{\mu\nu}\),\nonumber\\
&\nabla^\mu(Z(\phi)\pa_\mu\chi^a)=0,\qquad 2\alpha\square\phi-V'(\phi)=\Sigma'_{IJ}(\phi)F^I_{\rho\sigma}F^{J\rho\sigma},\qquad \nabla^\mu(\Sigma_{IJ}(\phi)F^J_{\mu\nu})=0.
\eal
\paragraph{Dual frame} As for non-conformal branes \cite{Kanitscheider:2009as}, the holographic dictionary for asymptotically hyperscaling violating Lifshitz backgrounds is best understood in the so called `dual frame' \cite{Chemissany:2014xsa}, obtained by a Weyl rescaling of the Einstein frame metric as
\bbepsilon\label{dual-metric}
\lbar g_{\mu\nu}=e^{-2\xi\phi}g_{\mu\nu},
\end{equation}
where the parameter $\xi$ is proportional to the hyperscaling violating exponent $\theta$ (see eq.~\eqref{xi-theta}). In the dual frame the action \eqref{action-0} takes the form
\bbepsilon\label{action}
S_\xi=\frac{1}{2\kappa^2}\int_{\cal M} d^{d_s+2}x\sqrt{-\lbar g}\;e^{d_s\xi \phi}\Big(\lbar R
-\alpha_\xi (\lbar \pa\phi)^2-\Sigma^\xi_{IJ}(\phi) \lbar F^I_{\mu\nu}\lbar F^{J\mu\nu}-Z_\xi(\phi)(\lbar \pa\chi^a)^2-V_\xi(\phi)\Big)
+S\sbtx{GH}^\xi,
\end{equation}
where we have defined
\bbepsilon
\alpha_\xi=\alpha-d_s(d_s+1)\xi^2,\quad
\Sigma^\xi_{IJ}(\phi)=e^{-2\xi\phi}\Sigma_{IJ}(\phi),\quad
Z_\xi(\phi)=Z(\phi),\quad
V_\xi(\phi)=e^{2\xi\phi}V(\phi).
\end{equation}
Moreover, the Gibbons-Hawking term in the dual frame is
\bbepsilon
S\sbtx{GH}^\xi=\frac{1}{2\kappa^2}\int_{\pa{\cal M}} d^{d_s+2}x\sqrt{-\lbar\gamma}\;e^{d_s\xi\phi}2\lbar K.
\end{equation}
All quantities with an over-bar are constructed using the dual frame metric \eqref{dual-metric}. In the analysis that follows we will use both the Einstein and dual frame variables, depending on which frame is more convenient for different aspects of the analysis.
\subsection{Anisotropic black brane solutions}
Depending on the choice of the functions $Z(\phi)$, $\Sigma_{IJ}(\phi)$ and $V(\phi)$, the action \eqref{action-0} admits a wide range of planar solutions, including black holes and domain walls at finite charge density. Such solutions can be asymptotically AdS, conformal to AdS, Lifshitz, or hyperscaling violating Lifshitz. In this paper the functions of the dilaton in the action \eqref{action-0} will be chosen such that the equations of motion admit asymptotically hyperscaling violating Lifshitz backgrounds, with generic values of the Lifshitz and hyperscaling violating exponents $z$ and $\theta$.
In the Einstein frame we parameterize the planar background solutions we are interested in as
\bal\label{Bans}
&ds^2_B=dr^2+e^{2A(r)}\left(-f(r)dt^2+dx^adx^a\right), \nonumber\\
&A_B^I=a^I(r)dt, \qquad \phi_B=\phi_B(r), \qquad
\chi_B^a=p x^a,
\eal
where again the index $a=1,2,\ldots,d_s$, runs over all spatial dimensions along the conformal boundary and $p\neq 0$ is the isotropic axion charge \cite{Caldarelli:2016nni}. The fieldstrength of the Abelian gauge fields on the backgrounds \eqref{Bans} is given by
\bbepsilon\label{FB}
F_B^I=\tx dA_B^I=\dot{a}^I\;\tx dr\wedge \tx dt,
\end{equation}
where the dot denotes a derivative with respect to the canonical Einstein frame radial coordinate $r$. In the following we will occasionally use the alternative `domain wall' radial coordinate $u$, defined through the relation
\bbepsilon\label{def-u}
\pa_r=-\sqrt{f}\; e^{-A}\pa_u.
\end{equation}
In terms of the radial coordinate $u$ the Einstein frame background metric \eqref{Bans} takes the form
\bbepsilon\label{dw-metric}
ds^2=e^{2A(u)}\(\frac{du^2}{f(u)}-f(u)dt^2+dx^adx^a\).
\end{equation}
\paragraph{Einstein frame background field equations}
Inserting the ansatz \eqref{Bans} in the field equations \eqref{eoms-0} leads to the system of coupled equations
\begin{subequations}\label{BEOM}
\begin{align}
& d_s\dot A\Big((d_s+1)\dot A+\frac{\dot{f}}{f}\Big)-\alpha\dot\phi^2_B+V(\phi_B)+d_sp^2Z(\phi_B)e^{-2A}+2f^{-1}e^{-2A}\Sigma_{IJ}(\phi_B)\dot a^I\dot a^J=0,\\
&\ddot{A}+\dot{A}\Big((d_s+1)\dot{A}+\frac{\dot{f}}{2f}\Big)+p^2Z(\phi_B)e^{-2A}+\frac{1}{d_s}\Big(V(\phi_B)+2e^{-2A}f^{-1}\Sigma_{IJ}(\phi_B)\dot{a}^I\dot a^J\Big)=0, \\
&\ddot{f}+\dot{f}\Big((d_s+1)\dot{A}-\frac{\dot{f}}{2f}\Big)-2p^2fZ(\phi_B)e^{-2A}-4e^{-2A}\Sigma_{IJ}(\phi_B)\dot{a}^I\dot a^J=0, \\
& 2\alpha\ddot{\phi}_B+2\alpha\Big((d_s+1)\dot{A}+\frac{\dot{f}}{2f}\Big)\dot{\phi}_B-V'(\phi_B)-d_sp^2Z'(\phi_B)e^{-2A}+\frac{2}{f}e^{-2A}\Sigma'_{IJ}(\phi_B)\dot{a}^I\dot a^J=0,\\
&\pa_r\left(\Sigma_{IJ}(\phi_B) e^{(d_s-1)A}f^{-1/2}\dot{a}^J\right)=0.
\end{align}
\end{subequations}
The Maxwell equation can be integrated directly to obtain
\bbepsilon\label{charge}
\Sigma_{IJ}(\phi_B) e^{(d_s-1)A} f^{-1/2}\dot a^J=-q_I,
\end{equation}
where the integration constants $q_I$ are electric charges associated with the Abelian gauge fields $A^I_B$.
\subsection{Asymptotically hyperscaling violating Lifshitz backgrounds}
\label{hvLf-backgrounds}
The system of equations \eqref{BEOM} admits asymptotically hyperscaling violating Lifshitz solutions with generic dynamical exponent $z$ and hyperscaling violation parameter $\theta$ provided at least one of the gauge fields, here taken to be $a^1$, has a non-trivial profile, while, at least asymptotically \cite{Charmousis:2010zz,Huijse:2011ef,Iizuka:2011hg,Ogawa:2011bz,Shaghoulian:2011aa,Dong:2012se,Iizuka:2012iv,Gouteraux:2012yr,Gath:2012pg,Chemissany:2014xsa}
\bbepsilon\label{asym-potentials}
V(\phi)\sim-(d_s+z-\theta)(d_s+z-\theta-1)e^{\frac{2\theta}{d_s\mu}\phi},\qquad \Sigma_{11}(\phi)\sim \Sigma_o e^{\frac{2(d_s-1)\theta-2d_s^2}{d_s\mu}\phi},
\end{equation}
where $\Sigma_o$ is an arbitrary positive definite constant and
\bbepsilon
\mu^2=\frac{(d_s-\theta)(d_s z-d_s-\theta)}{d_s\alpha}.
\end{equation}
Recall that the parameter $\alpha$ corresponds to the normalization of the dilaton kinetic term in \eqref{action-0} and can be specified at will. Moreover, the electric charge of the gauge field supporting the hyperscaling violating Lifshitz background is related to the dynamical exponents as
\bbepsilon
q_1=\pm\sqrt{\frac{(z-1)(d_s+z-\theta)}{32\Sigma_o}}\,.
\end{equation}
The null energy condition (NEC) for this class of solutions requires that $z\geq1$ and $\theta\leq d_s+z$.
There exists another class of hyperscaling violating Lifshitz solutions of \eqref{BEOM} with vanishing gauge fields, but those solutions have a fixed hyperscaling violating exponent $\theta=d_s+z$ \cite{Gath:2012pg,Chemissany:2014xsa} and we will not consider them here further.
In terms of the radial coordinate $r$ in the ansatz \eqref{Bans}, the hyperscaling violating Lifshitz solutions of \eqref{BEOM} asymptotically take the form
\bal\label{Einstein-background-r}
&ds^2_B\sim dr^2-\(|\theta|r/d_s\)^{2-\frac{2d_sz}{\theta}}dt^2+\(|\theta|r/d_s\)^{2-\frac{2d_s}{\theta}}dx^adx^a,\nonumber\\
&a^1(r)\sim \frac{4{\rm sgn}(\theta)q_1}{d_s+z-\theta}\(|\theta|r/d_s\)^{-\frac{d_s(d_s+z-\theta)}{\theta}},\quad \phi_B(r)\sim -\frac{d_s\mu}{\theta} \log r,
\eal
while the asymptotic behavior of the second gauge field, $a^2(r)$, depends on the choice of the function $\Sigma_{22}(\phi)$ and it is assumed to contribute to the bulk stress tensor at subleading order relative to the gauge field $a^1(r)$ that supports the asymptotic solution. Notice that in these coordinates the UV is located at $r\to\infty$ for $\theta<0$ and at $r=0$ for $\theta>0$ \cite{Chemissany:2014xsa}. Moreover, the case $\theta=0$, corresponding to an asymptotically Lifshitz background, looks like a singular limit of the asymptotic solution \eqref{Einstein-background-r}.
A radial coordinate that is better suited for describing hyperscaling violating Lifshitz backgrounds can be defined as \cite{Chemissany:2014xsa}
\bbepsilon\label{def-bar-r}
\boxed{d\bar r=-{\rm sgn}(\theta)e^{\frac{\theta}{d_s\mu}\phi}dr,}
\end{equation}
which leads to the asymptotic relation
\bbepsilon\label{r-bar-r}
r\sim\frac{d_s}{|\theta|}e^{-\frac{\theta}{d_s}\bar r}.
\end{equation}
In terms of the radial coordinate $\bar r$ the asymptotic solution \eqref{Einstein-background-r} becomes
\bal\label{hvLf-bgnd-asymptotics}
&ds^2_B\sim e^{-\frac{2\theta\bar r}{d_s}}\Big(d\bar r^2-e^{2z\bar r}dt^2+e^{2\bar r}dx^adx^a\Big),\nonumber\\
&a^1(\bar r)\sim \frac{4{\rm sgn}(\theta)q_1}{d_s+z-\theta}\;e^{(d_s+z-\theta)\bar r},\quad \phi_B(\bar r)\sim\mu \bar r.
\eal
In this coordinate system the metric is manifestly conformally related to a Lifshitz geometry and the solution for $\theta=0$ can be smoothly obtained as a limiting case. Moreover, setting the parameter $\xi$ that defines the dual frame metric in \eqref{dual-metric} to
\bbepsilon\label{xi-theta}\boxed{
\xi=-\frac{\theta}{d_s\mu},}
\end{equation}
one can identify the radial coordinate $\bar r$ with the canonical radial coordinate in the dual frame, where the background \eqref{Einstein-background-r} becomes asymptotically Lifshitz, i.e. in the dual frame
\bbepsilon\label{dual-metric-background}
d\bar s^2_B\sim d\bar r^2-e^{2z\bar r}dt^2+e^{2\bar r}dx^adx^a.
\end{equation}
Another advantage of the radial coordinate $\bar r$ is that, as follows from the asymptotic relation \eqref{r-bar-r}, the UV is located at $\bar r\to\infty$ for any value of $\theta$.
\paragraph{Evading the curvature singularity for $\theta>0$} For $\theta<0$ the conformal factor in the Einstein frame metric \eqref{hvLf-bgnd-asymptotics} blows up as $\bar r\to \infty$, while it approaches zero for $\theta>0$. As a consequence, the Einstein frame metric possesses a well defined conformal boundary for $\theta<0$, but for $\theta>0$ there is a curvature singularity at $\bar r\to \infty$. In contrast, the dual frame metric \eqref{dual-metric-background} is independent of the hyperscaling exponent $\theta$ and, hence, possesses a well defined conformal boundary for all $\theta$. Since hyperscaling violating Lifshitz backgrounds necessarily involve a running dilaton that diverges at the UV, it was argued in \cite{Chemissany:2014xsa} that the presence of a metric curvature singularity is a frame-dependent property that can be avoided simply by going to the dual frame. Instead of discarding the case $\theta>0$, therefore, we postulate that the holographic dictionary for all values of $\theta$ should be constructed in the {\em dual frame}.
\subsection{An exact black brane solution}
\label{example}
An exact asymptotically hyperscaling violating black brane solution of \eqref{BEOM} with non-zero axion charge was presented in \cite{Cremonini:2016avj} (see also \cite{Ge:2016lyn,Bhatnagar:2017twr}) for the case $d_s=2$ and generic exponents $z$ and $\theta$, subject to a number of conditions that we will specify shortly. This solution exists when the expressions \eqref{asym-potentials} hold exactly, instead of merely asymptotically near the UV.
Specifically, the Lagrangian that admits the solutions found in \cite{Cremonini:2016avj} corresponds to the choice of functions
\bal\label{asym-potentials-exact}
&V(\phi)=-(2+z-\theta)(1+z-\theta)e^{\theta\phi/\mu},\qquad \Sigma_{11}(\phi)=\frac14 e^{(\theta-4)\phi/\mu},\nonumber\\
&\Sigma_{12}(\phi)=\Sigma_{21}(\phi)=0,\qquad \Sigma_{22}(\phi)=\frac14 e^{(2z-2-\theta)\phi/\mu},\qquad Z(\phi)=\frac12e^{\frac{\mu}{\theta-2}\phi},
\eal
with the normalization of the dilaton kinetic term in \eqref{action-0} chosen such that $\alpha=1/2$. The exact black brane solution discussed in \cite{Cremonini:2016avj,Bhatnagar:2017twr} then takes the form\footnote{Note the change of notation relative to \cite{Cremonini:2016avj}:
$r\to v$, $\theta\to -\theta$, $\gamma\to\mu$, $f\to{\cal F}$, $\alpha\to p$ and $Q_I\to 4{\rm sgn}(\theta) q_I$.}
\bal
\label{HVbackground}
&ds^2 = v^{-\theta} \left( - v^{2z} {\cal F}(v) dt^2 + \frac{dv^2}{{\cal F}(v) v^2} + v^2 d\vec{x}^2 \right),\quad \phi = \mu \log v, \quad \chi^a = p \, x^a,\nonumber\\
&a^1= \frac{4{\rm sgn}(\theta)q_1}{2+z-\theta}(v^{2+z-\theta}-v^{2+z-\theta}_h),\quad a^2 = \frac{4{\rm sgn}(\theta)q_2}{\theta-z}(v^{\theta-z}-v^{\theta-z}_h),
\eal
where the axion charge $p$ denotes the amount of momentum dissipation in the dual system and
\bbepsilon
q_1=\pm\sqrt{\frac{(2+z-\theta)(z-1)}{8}}\;.
\end{equation}
$v_h$ denotes the location of the horizon and corresponds to the largest real root of the equation ${\cal F}(v)=0$, where the blackening factor is given by
\bbepsilon\label{blackening-sol}
{\cal F}(v)=1+\frac{p^2}{(2-\theta)(z-2)v^{2z-\theta}}-\frac{m}{v^{2+z-\theta}}+\frac{8q_2^2}{(2-\theta)(z-\theta)v^{2(z+1-\theta)}}.
\end{equation}
Although this expression for the blackening factor holds provided $z\neq 2$, $\theta\neq 2$ and $\theta\neq z$, the solution can be extended to these cases by including certain logarithmic terms. For example, the blackening factor for the case $z=2$, $\theta\neq z$ is given in \cite{Cremonini:2016avj}. The radial coordinate $v$ in \eqref{HVbackground} is related to the radial coordinates, $r$, defined in \eqref{Bans}, $\bar r$, defined in \eqref{def-bar-r}, and $u$ defined in \eqref{def-u} through the identities
\bbepsilon\label{coords}\boxed{
dr=-{\rm sgn}(\theta)v^{-\theta/2}{\cal F}^{-1/2}(v)\frac{dv}{v},\qquad d\bar r={\cal F}^{-1/2}(v)\frac{dv}{v},\qquad du={\rm sgn}(\theta)v^{z-3}dv.}
\end{equation}
In particular, the coordinate $v$ is asymptotically related to the canonical radial coordinate in the dual frame as
$v\sim e^{\bar r}$ for all values of $\theta$. All of these coordinates are useful for different aspects of the subsequent analysis.
In order for the solution \eqref{HVbackground} to be asymptotically hyperscaling violating Lifshitz with exponents $z$ and $\theta$, the blackening factor ${\cal F}(v)$ must asymptote to 1 as $v\to\infty$. For $q_2\neq 0$ and $p\neq 0$, together with the NEC, this requires that $z\geq 1$ and $\theta<z+1$. Moreover, the axion charge $p$ is a magnetic charge that corresponds to a non-normalizable mode and should be kept fixed for all solutions of the theory, while the mass $m$ and the electric charge $q_2$ correspond to normalizable modes, i.e. to state-dependent parameters. We therefore demand that the term proportional to $p^2$ is asymptotically dominant relative to the terms involving $m$ and $q_2$, which further restricts the values of the dynamical exponents to
\bbepsilon\label{exponents}\boxed{
1\leq z <2,\qquad \theta<z.}
\end{equation}
Demanding that the exponents $z$ and $\theta$ satisfy \eqref{exponents} ensures that the terms in \eqref{blackening-sol} are increasingly asymptotically subleading in the order they are written, from left to right. Note that the upper bounds in these inequalities can be saturated, but we will not consider these cases here since the form of the blackening factor \eqref{blackening-sol} would acquire logarithmic terms, which must be analyzed separately. It is important to emphasize that the conditions \eqref{exponents} are {\em not} a generic requirement for hyperscaling violating backgrounds, but rather a specific feature of the solution \eqref{HVbackground}.
\section{Linear fluctuations and the holographic dictionary}
\label{sec:linear}
\setcounter{equation}{0}
In order to compute the conductivities for hyperscaling violating Lifshitz backgrounds of the form \eqref{Bans} it is necessary to not only solve the system of linearized field equations around such backgrounds, but also to correctly identify the holographic observables, i.e. the sources and the local operators, in the dual theory. As we will see in the subsequent analysis, identifying the holographic observables in such theories is far from trivial and requires a novel form of holographic renormalization.
In appendix \ref{sec:fluctuations} we derive the linearized equations for a consistent set of spatially homogeneous, time dependent fluctuations around a generic background of the form \eqref{Bans} for the special case $d_s=2$, i.e. four dimensional bulk. The system of equations we derive can be used to compute certain two-point functions at arbitrary frequency and in any background of the form \eqref{Bans}, including asymptotically AdS$_4$ backgrounds. However, in this section we focus exclusively on the zero frequency limit of the fluctuations around the hyperscaling violating Lifshitz background \eqref{HVbackground}.
Without loss of generality, we consider fluctuations that preserve the Einstein frame gauge
\bbepsilon
ds^2=dr^2+\gamma_{ij}(r,t)dx^idx^j,\qquad A_r^I=0,
\end{equation}
where $x^i=\{t,x^a\}$, and we parameterize the fluctuations as
\bbepsilon\label{fluctuations}
\gamma_{ij}=\gamma_{Bij}+h_{ij},\quad
A_i^I=A_{Bi}^I+\frak a_i^I, \quad
\phi=\phi_B+\varphi,\quad
\chi^a=\chi_B^a+\tau^a.
\end{equation}
Defining $S_i^j\equiv\gamma_B^{jk}h_{ki}$, the linearized equations allow one to consistently set $S_t^{t}=S_x^x=S_y^y=S_x^y=\varphi=\frak a_t=0$ and to consider only the non-zero components $\frak a_a^I=\frak a_a^I(r,t)$, $S_t^a=S_t^a(r,t)$, and $\tau^a(r,t)$. The resulting system of linear equations can be reduced to the coupled system of equations \eqref{fluctuation-eqs}, involving only the variables $\frak a^I_a$ and
\bbepsilon\label{Theta}
\Theta^a\equiv S^a_t-\frac{i\omega}{p} \tau^a,
\end{equation}
where $\omega$ is the frequency. Both $S^a_t$ and $\tau^a$ can then be determined from the solution for $\frak a^I_a$ and $\Theta^a$.
\subsection{Near horizon solutions and ingoing boundary conditions}
To compute the retarded two-point functions in a generic background of the form
\eqref{Bans} we need to impose ingoing boundary conditions at the black brane horizon. In the domain wall coordinates \eqref{dw-metric}, near the horizon the blackening factor $f$ takes the form
\bbepsilon
f(\rho)=4\pi T\rho+{\cal O}(\rho^2), \qquad \rho\equiv u_h-u,
\end{equation}
where $u_h$ is the smallest root of the equation $f(u)=0$ and $T$ is the Hawking temperature. To leading order in $\rho$, the system of linear equations \eqref{fluctuation-eqs} becomes
\begin{subequations}
\bal
&\Sigma_{IJ}(\phi_B(u_h))\[\rho\pa_\rho\(\rho\pa_\rho{\frak a}^J_a\)+\(\frac{\omega}{4\pi T}\)^2\frak a^J_a\]+2p^2\omega^{-2}q_IZ(\phi_B(u_h))\rho^2\pa_\rho \Theta^a=0,\\
&\rho\pa_\rho\(\rho\pa_\rho(e^{2A}\Theta^a)\)+ \(\frac{\omega}{4\pi T}\)^2e^{2A}\Theta^a-4q_I\rho\pa_\rho(\rho\frak a_a^I)=0.
\eal
\end{subequations}
It follows that ingoing solutions at the horizon, to leading order in the near horizon expansion, are of the form
\bbepsilon\label{ingoing}\boxed{
\frak a_a^{I\;(\tx{in})}\propto \rho^{-\frac{i\omega}{4\pi T}},\qquad \Theta^{a\;(\tx{in})}\propto e^{-2A}\rho^{-\frac{i\omega}{4\pi T}}.}
\end{equation}
\subsection{Small frequency expansions}
Our next goal is to solve the system of equations \eqref{fluctuation-eqs} for a generic background of the form \eqref{Bans} in a small frequency expansion, i.e.
\bbepsilon\label{small-freq-exp}
\frak a^I_a =\frak a_{a}^{I(0)}+\omega^2 \frak a_{a}^{I(2)}+{\cal O}(\omega^4),\qquad \Theta^a =\Theta^{a(0)}+\omega^2 \Theta^{a(2)}+{\cal O}(\omega^4).
\end{equation}
In order to compute the DC conductivities it suffices to determine the ${\cal O}(\omega^0)$ terms in these expansions only. However, as we will see, identifying the ${\cal O}(\omega^0)$ solution that corresponds to ingoing boundary conditions at the horizon requires to compute part of the ${\cal O}(\omega^2)$ terms too.
\paragraph{Solution at ${\cal O}(\omega^0)$} Inserting the small frequency expansions \eqref{small-freq-exp} in the system of linear equations \eqref{fluctuation-eqs} we find that the leading order terms satisfy
\begin{subequations}
\label{fluctuation-eqs-0}
\bal
&\pa_r\(\Sigma_{IJ}(\phi_B) f^{1/2}e^A\dot{\frak a}^{J(0)}_a-q_I\Theta^{a(0)}\)=0,\\
&\pa_r\(e^{3A}f^{-1/2}\dot\Theta^{a(0)}-4q_I\frak a^{I(0)}_a\)-2p^2Z(\phi_B) e^Af^{-1/2}\Theta^{a(0)}=0.
\eal
\end{subequations}
The first of these equations can be integrated immediately to obtain
\bbepsilon\label{1st-integral}
\Sigma_{IJ}(\phi_B) f^{1/2}e^A\dot{\frak a}^{J(0)}_a-q_I\Theta^{a(0)}=c_I^a(\omega),
\end{equation}
where $c_I^a(\omega)$ are arbitrary frequency dependent integration constants. Inserting this expression in the second equation in \eqref{fluctuation-eqs-0}, rearranging the resulting equation using the background equations \eqref{BEOM} and \eqref{charge}, and changing to the domain wall coordinates \eqref{dw-metric} we obtain
\bbepsilon
\pa_u\(e^{2A}f^{2}\pa_u(f^{-1}\Theta^{a(0)})-4c^a_I(\omega) a^{I}\)=0.
\end{equation}
Hence, the general solution for $\Theta^{a(0)}$ to leading order in the small frequency expansion is
\bbepsilon\label{theta-sol}\boxed{
\Theta^{a(0)}(u,\omega)=\Theta_{1}^{a}(\omega)f+\Theta_{2}^{a}(\omega)f\int_{u_*}^u \frac{du'}{e^{2A(u')}f(u')^{2}}+4c^a_I(\omega)f\int_{u_*}^u \frac{du'a^I(u')}{e^{2A(u')}f(u')^{2}},}
\end{equation}
where $\Theta_{1}^{a}(\omega)$, $\Theta_{2}^{a}(\omega)$ are arbitrary frequency dependent integration constants, and $u_*$ is an unspecified reference point that should be chosen such that the integrals are well defined. Notice that the solution \eqref{theta-sol} is valid for any background of the form \eqref{Bans} and so the choice of the reference point $u_*$ depends on the specific background. Inserting this solution for $\Theta^{a(0)}$ in \eqref{1st-integral} determines
\begin{align}
\label{a-sol}
\boxed{
\begin{aligned}
\frak a^{I(0)}_a(u,\omega)=&\;\frak a^I_{oa}(\omega)-\Theta_{1}^{a}(\omega)a^I(u)-q_J\Theta_{2}^{a}(\omega)\int_{u_*}^u du'\Sigma^{IJ}(\phi_B(u'))\int_{u_*}^{u'} \frac{du''}{e^{2A(u'')}f(u'')^{2}}\\
&-\int_{u_*}^u du'\Sigma^{IJ}(\phi_B(u')) f^{-1}(u')\(\delta^K_J+4q_Jf\int_{u_*}^{u'}\frac{du'' a^K(u'')}{e^{2A(u'')}f(u'')^{2}}\)c^a_K(\omega),
\end{aligned}
}
\end{align}
where $\frak a^I_{oa}(\omega)$ are yet another set of arbitrary frequency dependent integration constants. Finally, \eqref{tau} implies that, to leading order in $\omega$, $\tau^a$ is determined from the equation
\bbepsilon\label{tau-dot}
\dot\tau^{a(0)}=\frac{i\omega }{2pZ(\phi_B)f}\Big(4e^{-3A}f^{1/2}q_I\frak a^{I(0)}_a-\dot\Theta^{a(0)}\Big).
\end{equation}
Recall that the dot $\dot{}$ stands for a derivative with respect to the Einstein frame radial coordinate $r$ defined in \eqref{Bans}.
\paragraph{Ingoing boundary conditions at the horizon} The expressions \eqref{theta-sol} and \eqref{a-sol} correspond to the general solutions of the linearized equations \eqref{fluctuation-eqs} to leading order in the small frequency expansion and suffice for computing the DC conductivities. However, to do so we must first determine the relations among the integration constants imposed by ingoing boundary conditions at the horizon.
In a gauge where the background gauge fields vanish at the horizon, i.e. $a^I(\rho)={\cal O}(\rho)$ as $\rho\to 0$, the near horizon behavior of the solution \eqref{a-sol} is
\bbepsilon\label{NH0-2}
\frak a^{I(0)}_a(\rho,\omega)= \frak a^I_{oa}(\omega)+\Theta_{1}^{a}(\omega){\cal O}(\rho)+\frac{\Sigma^{IJ}(\phi_B(u_h))}{4\pi T}\(c^a_J(\omega)+\frac{q_J\Theta_{2}^{a}(\omega)}{4\pi T e^{2A(u_h)}}\)(\log\rho+{\cal O}(\rho^0)).
\end{equation}
Comparing this with the small $\rho$ expansion of the ingoing solution \eqref{ingoing} determines
that for ingoing solutions the integration constants in \eqref{a-sol} must satisfy
\bbepsilon\label{rel-1}
c_I(\omega)+\frac{q_I\Theta_{2}^{a}(\omega)}{4\pi T e^{2A(u_h)}}=-i\omega \Sigma_{IJ}(\phi_B(u_h))\frak a^J_{oa}(\omega).
\end{equation}
Similarly, the near horizon behavior of the solution \eqref{theta-sol} is
\bbepsilon\label{NH0-1}
\Theta^{a(0)}(\rho,\omega)= 4\pi T\Theta_{1}^{a}(\omega)\rho+\frac{\Theta_{2}^{a}(\omega)}{4\pi T e^{2A(u_h)}}(1+{\cal O}(\rho\log\rho))+c_I^a(\omega){\cal O}(\rho\log\rho).
\end{equation}
However, in order to compare this with the small $\rho$ expansion of the ingoing solution \eqref{ingoing} we need to determine $\Theta^{a}(\rho,\omega)$ up to ${\cal O}(\omega^2)$.
\paragraph{Solution at ${\cal O}(\omega^2)$} To determine $\Theta^{a}(\rho,\omega)$ to ${\cal O}(\omega^2)$ we insert the expansions \eqref{small-freq-exp} in \eqref{fluctuation-eqs}, which leads to the ${\cal O}(\omega^2)$ equations
\begin{subequations}
\label{fluctuation-eqs-2}
\bal
&\pa_r\(\Sigma_{IJ}(\phi_B) f^{1/2}e^A\dot{\frak a}^{J(2)}_a-q_I\Theta^{a(2)}\)\\
&\hskip0.5in+f^{-1/2}e^{-A}\(\Sigma_{IJ}(\phi_B)+\frac{2e^{-2A}q_Iq_J}{p^2Z(\phi_B)}\) \frak a^{J(0)}_a-\frac{q_I}{2p^2fZ(\phi_B)}\dot\Theta^{a(0)}=0,\nonumber\\\nonumber\\
&\pa_r\(e^{3A}f^{-1/2}\dot\Theta^{a(2)}-4q_I\frak a^{I(2)}_a\)-2p^2Z(\phi_B) e^Af^{-1/2}\Theta^{a(2)}\nonumber\\
&\hskip0.5in+\pa_r\(\frac{1}{2p^2f Z(\phi_B)}\(e^{3A}f^{-1/2}\dot\Theta^{a(0)}-4q_I\frak a^{I(0)}_a\)\)=0.
\eal
\end{subequations}
Without loss of generality, the homogeneous solution of these equations can be absorbed into the integration constants $\frak a^I_{oa}(\omega)$, $\Theta_{1}^{a}(\omega)$, $\Theta_{2}^{a}(\omega)$ and $c^a_I(\omega)$ of the ${\cal O}(\omega^0)$ solution \eqref{theta-sol} and \eqref{a-sol}. Hence, at ${\cal O}(\omega^2)$ we are only interested in the inhomogeneous solution of the system \eqref{fluctuation-eqs-2}.
The first equation in \eqref{fluctuation-eqs-2} determines
\bbepsilon
-\Sigma_{IJ}(\phi_B) f\pa_u{\frak a}^{J(2)}_a-q_I\Theta^{a(2)}=v_I^a,
\end{equation}
where
\bbepsilon
v_I^a=\int du f^{-1}\(\Sigma_{IJ}(\phi_B)+\frac{2e^{-2A}q_Iq_J}{p^2Z(\phi_B)}\)\frak a^{J(0)}_a+\frac{q_I}{2p^2}\int \frac{du}{fZ(\phi_B)}\pa_u\Theta^{a(0)}.
\end{equation}
Inserting this in the second equation in \eqref{fluctuation-eqs-2} gives
\bbepsilon\label{Theta-2-eq}
\pa_u\(e^{2A}f^{2}\pa_u(f^{-1}\Theta^{a(2)})\)
=-f\pa_u\(\frac{1}{2p^2f Z(\phi_B)}\(e^{2A}\pa_u\Theta^{a(0)}+4q_I\frak a^{I(0)}_a\)\)+4\pa_u a^Jv_J^a.
\end{equation}
The near horizon behavior of $\Theta^{a(2)}$ can be determined straightforwardly from this equation. Using the near horizon behavior of the ${\cal O}(\omega^0)$ solution that we obtained above, one can easily deduce that near the horizon $v_I^a={\cal O}(\log\rho)^2$. Expanding \eqref{Theta-2-eq} near the horizon then determines
\bbepsilon
\Theta^{a(2)}= \frac{1}{2p^2Z(\phi_B(u_h))4\pi T e^{2A(u_h)}}\(-4\pi Te^{2A(u_h)}\Theta_{1}^{a}(\omega)+4q_I\frak a^I_{oa}(\omega)\)\log\rho+{\cal O}(\log\rho)^2.
\end{equation}
Combining this with the near horizon expansion \eqref{NH0-1} of the ${\cal O}(\omega^0)$ solution and comparing with the small $\rho$ expansion of the ingoing solution \eqref{ingoing} leads to a second relation among the integration constants, namely
\bbepsilon\label{IR1}\boxed{
\Theta_{2}^{a}(\omega)=\frac{i\omega 4\pi T}{2p^2Z(\phi_B(u_h))}\(-4\pi Te^{2A(u_h)}\Theta_{1}^{a}(\omega)+4q_I\frak a^I_{oa}(\omega)\).
}
\end{equation}
Inserting this in the relation \eqref{rel-1} we obtained above gives
\bbepsilon\label{IR2}\boxed{
c^a_I(\omega)=-i\omega \(\Sigma_{IJ}(\phi_B(u_h))+\frac{2q_Iq_J}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\)\frak a^J_{oa}(\omega)+\frac{i\omega 4\pi T q_I}{2p^2Z(\phi_B(u_h))}\Theta_{1}^{a}(\omega).}
\end{equation}
These two relations allow us to express the integration constants $\Theta_{2}^{a}(\omega)$ and $c^a_I(\omega)$ in terms of the arbitrary constants $\Theta_{1}^{a}(\omega)$ and $\frak a^I_{oa}(\omega)$, which we will later identify with sources of certain dual operators.
The solutions \eqref{theta-sol} and \eqref{a-sol}, together with the identifications \eqref{IR1} and \eqref{IR2}, provide the general solution of the linear equations \eqref{fluctuation-eqs} to lowest order in the frequency and with ingoing boundary conditions at the horizon. These solutions apply to any background of the form \eqref{Bans} and can therefore be used to compute the DC conductivities in a large class of holographic theories with momentum dissipation. However, identifying the operators whose two-point functions these fluctuations compute is a very non-trivial question that requires a careful analysis of the asymptotic behavior of the solutions at the UV, which depends strongly on the specific choice of background.
\subsection{Asymptotic analysis at the UV}
In our analysis of the linear fluctuations so far we have not specified the background other than demanding that it takes the generic form \eqref{Bans}. The asymptotic analysis in the UV, however, and the holographic dictionary, are sensitive to the asymptotic form of the background. We are therefore compelled at this point to make a concrete choice. Ideally, one would like to have general expressions for the DC conductivities valid for any asymptotically hyperscaling violating Lifshitz background described in section \ref{hvLf-backgrounds}. However, even though they are not directly involved in supporting the asymptotic form of the background to leading order, additional matter fields and their asymptotic behavior do in general affect the form of the UV expansions of the fluctuations, and consequently the holographic dictionary. We already encountered an example of the effect extra matter fields can have on the asymptotics in section \ref{example}, where we saw that in order for the exact background \eqref{HVbackground} to have a well defined asymptotic behavior in the UV, the dynamical exponents must lie in the range \eqref{exponents}, which is more stringent than the conditions imposed by the null energy condition.
In the subsequent analysis, therefore, we consider fluctuations around the specific exact background \eqref{HVbackground}. In particular, we set the functions that specify the bulk Lagrangian as in \eqref{asym-potentials-exact} and we assume that the dynamical exponents satisfy \eqref{exponents}, although we exclude the relativistic case $z=1$, since it requires a separate analysis. Moreover, we use the radial coordinate $v$ for the asymptotic analysis, since this is the natural coordinate for the exact background \eqref{HVbackground} and also it is simply related asymptotically to the canonical radial coordinate $\bar r$ in the dual frame. Crucially, the UV is located at $v\to\infty$ independently of the value of the hyperscaling violation exponent $\theta$. In terms of $v$, the functions \eqref{asym-potentials-exact} evaluated on the background \eqref{HVbackground} take the form
\bbepsilon\label{model-functions-v}
\Sigma_{11}(\phi_B)=\frac14 v^{\theta-4},\qquad \Sigma_{22}(\phi_B)=\frac14 v^{2z-2-\theta},\qquad Z(\phi_B)=\frac12v^{-(2z-2-\theta)},
\end{equation}
while a comparison between \eqref{HVbackground} and the general background ansatz \eqref{Bans} allows us to read off
\bbepsilon
e^{2A}=v^{2-\theta},\qquad f=v^{2(z-1)}{\cal F}(v).
\end{equation}
To determine the UV asymptotic expansions of the linear fluctuations \eqref{theta-sol} and \eqref{a-sol} it is useful to introduce the functions
\bbepsilon
\Psi(u)=\int_{u_*}^u \frac{du'}{e^{2A(u')}f(u')^{2}},\qquad {\cal Y}^I(u)=\int_{u^*}^u \frac{du'a^I(u')}{e^{2A(u')}f(u')^{2}},
\end{equation}
which in terms of the radial coordinate $v$ become
\begin{subequations}
\label{aux}
\bal
\Psi(v)=&\;{\rm sgn}(\theta)\int^v dv\; v^{\theta-3z-1}{\cal F}^{-2}(v),\\
{\cal Y}^1(v)=&\;\frac{4{\rm sgn}(\theta)q_1}{2+z-\theta}\Big(-v_h^{2+z-\theta}\Psi(v)+{\rm sgn}(\theta)\int^v dv\; v^{-2z+1}{\cal F}^{-2}(v)\Big),\\
{\cal Y}^2(v)=&\;\frac{4{\rm sgn}(\theta)q_2}{\theta-z}\Big(-v_h^{\theta-z}\Psi(v)+{\rm sgn}(\theta)\int^v dv\; v^{2\theta-4z-1}{\cal F}^{-2}(v)\Big).
\eal
\end{subequations}
The small frequency solutions of the fluctuation equations, \eqref{theta-sol} and \eqref{a-sol}, can then be expressed in the $v$ coordinate in the compact form
\begin{subequations}
\label{sols-v}
\bal
\Theta^{a(0)}(v,\omega)=&\;v^{2(z-1)}{\cal F}(v)\Big(\Theta_{1}^{a}(\omega)+\Theta_{2}^{a}(\omega)\Psi(v)+4c^a_I(\omega){\cal Y}^I(v)\Big),\\
\rule{0cm}{1.cm}\frak a^{I(0)}_a(v,\omega)=&\;\frak a^I_{oa}(\omega)-\Theta_{1}^{a}(\omega)a^I(v)-{\rm sgn}(\theta)q_J\Theta_{2}^{a}(\omega)\int^v dv\; \Sigma^{IJ}v^{z-3}\Psi(v)\nonumber\\
&-{\rm sgn}(\theta)\int^v dv\;\Sigma^{IJ}v^{-(z+1)}\({\cal F}^{-1}(v)\delta^K_J+4q_J v^{2(z-1)}{\cal Y}^K(v)\)c^a_K(\omega).
\eal
\end{subequations}
These expressions, together with \eqref{aux}, allow us to straightforwardly determine the UV asymptotic expansions of the fluctuations.
\paragraph{UV expansions} Using the expression \eqref{blackening-sol} for the function ${\cal F}(v)$, as well as the allowed range \eqref{exponents} for the dynamical exponents, the integral expressions in \eqref{aux} determine that asymptotically
\begin{subequations}
\label{uv-exp-aux}
\bal
\Psi(v)=&\;{\rm sgn}(\theta)\Big(\frac{v^{\theta-3z}}{\theta-3z}+\cdots\Big),\\
\rule{0cm}{1.cm}{\cal Y}^1(v)=&\;\frac{-4q_1}{2+z-\theta}\Big(\frac{v^{-2(z-1)}}{2(z-1)}+\frac{2p^2v^{\theta+2-4z}}{(2-\theta)(z-2)(\theta-4z+2)}-\frac{(2m-v_h^{2+z-\theta})v^{\theta-3z}}{\theta-3z}+\cdots\Big),\\
\rule{0cm}{1.cm}{\cal Y}^2(v)=&\;\frac{4q_2}{\theta-z}\Big(-\frac{v_h^{\theta-z}v^{\theta-3z}}{\theta-3z}+\cdots\Big),
\eal
\end{subequations}
where the ellipses stand for subleading terms that are not required for the holographic calculation of the conductivities. Inserting these expansions in turn in \eqref{sols-v}, we obtain the UV expansions of the linear fluctuations in the small frequency limit, namely
\begin{subequations}
\label{uv-exp}
\bal
\Theta^{a(0)}(v,\omega)=&\;v^{2(z-1)}{\cal F}(v)\Bigg(\Theta_{1}^{a}(\omega)-\frac{8q_1c_1^a(\omega)v^{-2(z-1)}}{(z-1)(2+z-\theta)}-\frac{32p^2q_1c_1^a(\omega)v^{\theta+2-4z}}{(2-\theta)(z-2)(\theta-4z+2)(2+z-\theta)}\nonumber\\
&\hskip-1.7cm+\frac{v^{\theta-3z}}{\theta-3z}\Big(\frac{16q_1(2m-v_h^{2+z-\theta})c_1^a(\omega)}{2+z-\theta}+\frac{16q_2v_h^{\theta-z}c_2^a(\omega)}{z-\theta}+{\rm sgn}(\theta)\Theta_{2}^{a}(\omega)\Big)+\cdots\Bigg),\\
\rule{0cm}{1.cm}\frak a^{1(0)}_a(v,\omega)=&\;\frak a^1_{oa}(\omega)-\Theta_{1}^{a}(\omega)\frac{4{\rm sgn}(\theta)q_1(v^{2+z-\theta}-v^{2+z-\theta}_h)}{2+z-\theta}+\frac{2q_1\Theta_{2}^{a}(\omega)v^{-2(z-1)}}{(z-1)(\theta-3z)}\nonumber\\
&\hskip-1.7cm-\frac{32{\rm sgn}(\theta)q_1q_2v_h^{\theta-z}v^{-2(z-1)}}{(z-1)(\theta-z)(\theta-3z)}c_2^a(\omega)-4{\rm sgn}(\theta)c^a_1(\omega)\Bigg(\frac{p^2v^{4-3z}}{(z-2)(\theta-4z+2)(4-3z)}\nonumber\\
&\hskip-1.7cm-\frac{\Big((z+\theta-4)m-2(z-1)v_h^{2+z-\theta}\Big)v^{-2(z-1)}}{2(z-1)(\theta-3z)}\Bigg)+\cdots,\\
\rule{0cm}{1.cm}\frak a^{2(0)}_a(v,\omega)=&\;\frak a^2_{oa}(\omega)-\Theta_{1}^{a}(\omega)\frac{4{\rm sgn}(\theta)q_2(v^{\theta-z}-v^{\theta-z}_h)}{\theta-z}-\frac{2q_2\Theta_{2}^{a}(\omega)v^{2(\theta-2z)}}{(\theta-2z)(\theta-3z)}\nonumber\\
&\hskip-1.7cm-4{\rm sgn}(\theta)c^a_2(\omega)\Bigg(\frac{v^{2+\theta-3z}}{2+\theta-3z}-\frac{p^2v^{2\theta-5z+2}}{(2-\theta)(z-2)(2\theta-5z+2)}+\Big(m-\frac{16q_2^2v_h^{\theta-z}}{(\theta-z)(\theta-3z)}\Big)\frac{v^{2(\theta-2z)}}{2(\theta-2z)}\Bigg)\nonumber\\
&\hskip-1.7cm+\frac{64{\rm sgn}(\theta)q_1q_2}{(2+z-\theta)}c_1^a(\omega)\Bigg(\frac{v^{2+\theta-3z}}{2(z-1)(2+\theta-3z)}+\frac{2p^2v^{2\theta-5z+2}}{(2-\theta)(z-2)(\theta-4z+2)(2\theta-5z+2)}\nonumber\\
&\hskip-1.7cm-\frac{(2m-v_h^{2+z-\theta})v^{2(\theta-2z)}}{2(\theta-3z)(\theta-2z)}\Bigg)+\cdots.
\eal
\end{subequations}
Finally, from these expansions and \eqref{tau-dot} follows that the UV expansion of the axion fluctuations takes the form $\tau^{a(0)}(v,\omega)=\tau_o^a(\omega)+\cdots$, where $\tau_o^a(\omega)$ is an arbitrary integration constant.
These asymptotic expansions, together with the relations \eqref{IR1} and \eqref{IR2}, encode all information about the thermoelectric DC conductivities. However, in order to extract this information it is necessary to properly identify the local operators and their sources in the dual hyperscaling violating Lifshitz theory, which is the goal of the next subsection.
\subsection{Renormalized holographic observables and Ward identities}
\label{dictionary}
In order to construct the holographic dictionary one must determine a canonical set of symplectic data that parameterize the space of asymptotic solutions \cite{Papadimitriou:2010as}. This symplectic space is then holographically identified with the space of local sources and operators in the dual quantum field theory. As we pointed out in section \ref{hvLf-backgrounds}, a well defined space of asymptotic hyperscaling violating Lifshitz solutions exists for both positive and negative values of the hyperscaling violation exponent $\theta$ only in the dual frame, defined by the Weyl transformation \eqref{dual-metric}, where the parameter $\xi$ is related to $\theta$ as in eq.~\eqref{xi-theta}. Moreover, a natural basis of symplectic variables that can be used to parameterize the space of asymptotic solutions is provided by the set of generalized coordinates and canonical momenta in the radial Hamiltonian formulation of the bulk dynamics, summarized in appendix \ref{ham}.
Our first goal in this section, therefore, is to determine the asymptotic expansions of all canonical variables in the dual frame in terms of the modes in the solutions of the linear fluctuations equations. Since we do not consider fluctuations of the dilaton, the relation between the linear fluctuations of the fields in the Einstein and dual frames is straightforward. In particular, the asymptotic expansions of the linear fluctuations in \eqref{uv-exp} apply to both the Einstein and dual frames. The relation between the canonical momenta in the two frames is less trivial, but also straightforward. Evaluating the dual frame canonical momenta \eqref{momenta} in the gauge \eqref{dual-FG} and observing that the extrinsic curvature in the dual frame is related to that in the Einstein frame as
\bbepsilon
\lbar K_{ij}=\frac{1}{2\lbar N}\dot{\lbar\gamma}_{ij}=\frac12 e^{\xi\phi}\pa_r(e^{-2\xi\phi}\gamma_{ij})=e^{-\xi\phi}(K_{ij}-\xi\dot\phi \gamma_{ij}),
\end{equation}
we can express the dual frame canonical momenta in terms of Einstein frame variables as
\bal\label{momentum-map}
\lbar\pi^{ij}=&\;\frac{1}{2\kappa^2}\sqrt{-\gamma}\;e^{2\xi\phi}(K\gamma^{ij}-K^{ij}),\nonumber\\
\lbar \pi^{i}_I=&\;-\frac{2}{\kappa^2}\sqrt{-\gamma}\;\Sigma_{IJ}(\phi)
\gamma^{ij}F^J_{rj},\nonumber\\
\lbar\pi_\phi=&\;\frac{1}{\kappa^2}\sqrt{-\gamma}\;
(d_s\xi K-\alpha \dot\phi),\nonumber\\
\lbar\pi_{\chi a}=&\;-\frac{1}{\kappa^2}\sqrt{-\gamma}\;Z(\phi)\dot\chi^a.
\eal
Linearizing these expressions around the background \eqref{Bans} using the fluctuations \eqref{fluctuations} we find that the only non-zero linearized momenta in the dual frame are
\bal\label{lin-momenta}
\lbar\pi^{ta[1]}=&\;\frac{1}{4\kappa^2}e^{2\xi\phi_B}e^{-(d_s+1)A}f^{-1/2}\pa_r\big(e^{2d_s A}S^a_t\big),\nonumber\\
\lbar\pi^{a[1]}_I=&\;-\frac{2}{\kappa^2}e^{(d_s-1)A}f^{1/2}\;\Sigma_{IJ}(\phi_B)
\(\dot{\frak a}^J_{a}+f^{-1}\dot a^J S^a_t\),\nonumber\\
\lbar\pi^{[1]}_{\chi a}=&\;-\frac{1}{\kappa^2}e^{(d_s+1)A}f^{1/2}Z(\phi_B)\dot\tau^a,
\eal
where the superscript $[1]$ indicates that these expressions are linear in the fluctuations of the fields. Using the first coordinate transformation in \eqref{coords}, as well as equations \eqref{Theta}, \eqref{tau-dot} and \eqref{model-functions-v}, we can express these linearized momenta in terms of the background \eqref{HVbackground} and the small frequency fluctuations as
\bal
\lbar\pi^{ta[1]}=&\;-\frac{{\rm sgn}(\theta)}{4\kappa^2}v^{\theta-z-1}\pa_v\Big(v^{4-2\theta}\Big(\Theta^{a(0)}+\frac{i\omega}{p} \tau^{a(0)}\Big)\Big),\nonumber\\
\lbar\pi^{a[1]}_1=&\;\frac{{\rm sgn}(\theta)}{2\kappa^2}
\(v^{z+\theta-3}{\cal F}(v)\pa_v{\frak a}^{1(0)}_{a}+4{\rm sgn}(\theta)q_1\Big(\Theta^{a(0)}+\frac{i\omega}{p} \tau^{a(0)}\Big)\),\nonumber\\
\lbar\pi^{a[1]}_2=&\;\frac{{\rm sgn}(\theta)}{2\kappa^2}
\(v^{3z-1-\theta}{\cal F}(v)\pa_v{\frak a}^{2(0)}_{a}+4{\rm sgn}(\theta)q_2 \Big(\Theta^{a(0)}+\frac{i\omega}{p} \tau^{a(0)}\Big)\),\nonumber\\
\lbar\pi^{[1]}_{\chi a}=&\;\frac{i\omega}{2p\kappa^2}\Big(-{\rm sgn}(\theta)v^{5-z-\theta}\pa_v\Theta^{a(0)}-4q_I\frak a^{I(0)}_a\Big).
\eal
Finally, inserting the asymptotic expansions \eqref{uv-exp} for the linear fluctuations in these expressions we obtain the asymptotic expansions for the linearized momenta, namely
\bal\label{uv-exp-momenta}
\lbar\pi^{ta[1]}=&\;\frac{{\rm sgn}(\theta)}{4\kappa^2}\Big(-2(z+1-\theta)\Theta_1^a(\omega)v^{z-\theta}-\frac{p^2}{z-2}\Theta_1^a(\omega)v^{-z}+(z-\theta)m\Theta_1^a(\omega)v^{-2}\nonumber\\
&\hskip-0.9cm+\frac{2(2-\theta)}{q_1}\Big(c_1^a(\omega)-\frac{i\omega}{p}q_1\tau_o^a(\omega)\Big)v^{2-\theta-z}-\frac{(4-2z-\theta)p^2 }{(z-2)(\theta-4z+2)q_1}c_1^a(\omega)v^{2-3z}\nonumber\\
&\hskip-0.9cm-\Big(\frac{2-z-\theta}{\theta-3z}\Big)\Big[\frac{8(\theta+z-4)mq_1c_1^a(\omega)}{(z-1)(2+z-\theta)}-\frac{16q_1 v_h^{2+z-\theta}c_1^a(\omega)}{2+z-\theta}-\frac{16q_2v_h^{\theta-z}c_2^a(\omega)}{\theta-z}+{\rm sgn}(\theta)\Theta_{2}^{a}(\omega)\Big]v^{-2z}+\cdots\Big),\nonumber\\
\lbar\pi^{a[1]}_1=&\;-\frac{2}{\kappa^2}
\Big(c_1^a(\omega)-\frac{i\omega q_1}{p} \tau^{a}_o(\omega)+\cdots\Big),\nonumber\\
\lbar\pi^{a[1]}_2=&\;-\frac{2}{\kappa^2}
\Big(c_2^a(\omega)-\frac{i\omega q_2}{p} \tau^{a}_o(\omega)+\cdots\Big),\nonumber\\
\lbar\pi^{[1]}_{\chi a}=&\;\frac{i\omega{\rm sgn}(\theta)}{2p\kappa^2}\Bigg(\frac{p^2}{z-2}\Theta^a_1(\omega)v^{2-z}+\frac{p^2c_1^a(\omega)}{(4-3z)q_1}v^{4-3z}-4{\rm sgn}(\theta)q_1\frak a^1_{oa}(\omega)-4{\rm sgn}(\theta)q_2\frak a^2_{oa}(\omega)\nonumber\\
&+\Big((z+\theta-4)m-2(z-1)v_h^{2+z-\theta}-\frac{16q_2^2v_h^{\theta-z}}{\theta-z}\Big)\Theta^a_1(\omega)
+\cdots\Bigg).
\eal
The asymptotic expansions \eqref{uv-exp} and \eqref{uv-exp-momenta} provide the symplectic map between the canonical variables in the dual frame, i.e. the fluctuations $\Theta^{a(0)}$, $\frak a_a^{I(0)}$, $\tau^{a(0)}$ and their conjugate momenta, and the modes $\Theta_1^a(\omega)$, $\Theta_2^a(\omega)$, $\frak a_{oa}^I(\omega)$, $c_I^a(\omega)$ and $\tau_o^a(\omega)$ parameterizing the solutions of the linearized equations. The covariant fluctuations and their conjugate momenta provide the correct symplectic variables that should be identified with the local sources and operators in the dual theory, but they depend on the radial coordinate, while the sources and operators in the dual theory should be solely expressed in terms of the radially independent modes $\Theta_1^a(\omega)$, $\Theta_2^a(\omega)$, $\frak a_{oa}^I(\omega)$, $c_I^a(\omega)$ and $\tau_o^a(\omega)$ parameterizing the solutions of the linearized equations. The resolution to this problem is provided by a suitable canonical transformation that diagonalizes the symplectic map between the canonical variables and the modes parameterizing the solutions of the linearized equations. This procedure is a generalization of the method of holographic renormalization that can be applied to a wider class of backgrounds, beyond asymptotically AdS ones \cite{Papadimitriou:2010as}.
As we will see below, the fluctuations we have turned on contain a non-zero source, namely $\Theta_1^a(\omega)$, for the energy flux, which is an irrelevant operator in the dual Lifshitz theory \cite{Ross:2011gu}. As a result, the canonical transformation required to renormalize the asymptotically hyperscaling violating theory we consider here is qualitatively different from the usual canonical transformations that implement holographic renormalization in asymptotically AdS backgrounds or asymptotically Lifshitz backgrounds without a source for the energy flux. However, it shares some features with the renormalization of gauge fields in AdS$_2$ and AdS$_3$ \cite{An:2016fzu,Cvetic:2016eiv,Erdmenger:2016jjg}.
\paragraph{Holographic renormalization} In order to determine the canonical transformation that diagonalizes the symplectic map between the canonical variables and the modes parameterizing the solutions of the linearized equations, i.e. the asymptotic expansions \eqref{uv-exp} and \eqref{uv-exp-momenta}, it is necessary to decompose the induced metric $\lbar\gamma_{ij}$ in the dual frame in time and spatial components in order to account for the non-relativistic scaling. Following \cite{Chemissany:2014xsa} we parameterize $\lbar\gamma_{ij}$ as
\bbepsilon
\lbar\gamma_{ij}dx^idx^j=-(n^2-n_a n^a)dt^2+2n_adtdx^a+\sigma_{ab}dx^a dx^b,\qquad a,b=1,\ldots,d_s,
\end{equation}
in terms of the lapse function $n$, the shift function $n^a$ and the spatial metric $\sigma_{ab}$. These variables are all dynamical since they are components of the induced metric, in contrast to the non-dynamical fields $\lbar N$ and $\lbar N_i$ in the radial decomposition of the bulk metric discussed in appendix \ref{ham}.
The values of these variables in a generic Einstein frame background of the form \eqref{Bans} are
\beta
\sigma_{ab}^B=e^{-2\xi\phi_B}e^{2A}\delta_{ab},\qquad n_a^B=0,\qquad n_B=e^{-\xi\phi_B}e^{A}\sqrt{f}
\end{equation}
while the linear fluctuations \eqref{fluctuations} become
\beta
e^{-2\xi\phi_B}h_{tt}=-2n_B\delta n,\qquad e^{-2\xi\phi_B}h_{ta}=e^{-2\xi\phi_B}h_{at}=\delta n_a,\qquad e^{-2\xi\phi_B} h_{ab}=\delta\sigma_{ab}
\end{equation}
It follows that, for the components of the linear fluctuations we turn on,
\beta
S^a_t=\sigma_B^{ab}\delta n_b,\qquad S^t_a=-\frac{1}{n_B^2}\delta n_a,\qquad \delta n=0,\qquad \delta\sigma_{ab}=0
\end{equation}
Using these identities we can express the variational problem of the regularized on-shell action, within the space of fluctuations we consider, in terms of the non-relativistic variables as
\bal\label{var}
\delta S_{\rm reg}=&\;\int d^{d_s+1}x\;\Big(\lbar\pi^{ij}\delta\lbar\gamma_{ij}+\lbar\pi_{\chi\; a}\delta\chi^a+\lbar\pi^i_1\delta A_i^1+\lbar\pi^i_2\delta A_i^2\Big)\nonumber\\
=&\;\int d^{d_s+1}x\;\Big(2\lbar\pi^{ta}\delta n_a+\lbar\pi_{\chi\; a}\delta\chi^a+\lbar\pi^a_1\delta A_a^1+\lbar\pi^a_2\delta A_a^2\Big).
\eal
We now demonstrate that the canonical transformation required to renormalize the symplectic variables within the space of fluctuations we have turned on is generated by the boundary term
\bal\label{bt}
S_{\rm b}=&\;-\int d^{d_s+1}x\;A_a^1\lbar\pi_1^a\\
&\hskip-0.6cm+\int d^{d_s+1}x\;\Big(h_1(\phi)n_a\lbar\pi_1^a+h_2(\phi)\frac{\lbar\pi^a_1
\lbar\pi_{1a}}{\sqrt{-\lbar\gamma}}+\sqrt{-\lbar\gamma}\;h_3(\phi)n_a\pa_t\chi^a+h_4(\phi)\lbar\pi_{1a}\pa_t\chi^a+\sqrt{-\lbar\gamma}\;h_5(\phi)n_an^a\Big),\nonumber
\eal
where the functions $h_1(\phi)$, $h_2(\phi)$, $h_3(\phi)$, $h_4(\phi)$ and $h_5(\phi)$ are yet unspecified. Adding this boundary term to the regularized dual frame on-shell action $S_{\rm reg}$ and varying only the field components that get contributions from the fluctuations we turn on here, the variational principle \eqref{var} becomes
\bbepsilon\label{var-1}
\delta(S_{\rm reg}+S_{\rm b})=\int d^{d_s+1}x\;\Big(2\lbar\Pi^{ta}\delta n_a+\lbar\Pi_{\chi\; a}\delta\chi^a-\bb A_a^1\delta\lbar\pi^a_1+\lbar\pi^a_2\delta A_a^2\Big),
\end{equation}
where the canonical variables $\lbar\pi^{ta}$, $A_a^1$ and $\lbar\pi_{\chi a}$ are transformed according to
\bal\label{transformed-variables}
\lbar\pi^{ta}\to&\;\lbar\Pi^{ta}=\lbar\pi^{ta}+\frac12 h_1(\phi)\lbar\pi^a_1+\frac12\sqrt{-\lbar\gamma}\;h_3(\phi)\pa_t\chi^a+\sqrt{-\lbar\gamma}\;h_5(\phi)n^a,\nonumber\\
A_a^1\to&\;\bb A_a^1=A_a^1-h_1(\phi)n_a-2h_2(\phi)\frac{\lbar\pi_{1a}}{\sqrt{-\lbar\gamma}}-h_4(\phi)\pa_t\chi_a,\nonumber\\
\lbar\pi_{\chi a}\to&\;\lbar\Pi_{\chi a}=\lbar\pi_{\chi a}-\sqrt{-\lbar\gamma}\;\lbar D_t(h_3(\phi)n_a)-\lbar D_t(h_4(\phi)\lbar\pi_{1a}),
\eal
while $A_a^2$ and its conjugate momentum remain unchanged.
Several comments are in order here. Firstly, in writing \eqref{bt} we have only included terms that contribute to the renormalization of the linearized canonical variables in the low frequency limit. In particular, we have not specified the boundary terms required to renormalize the background values of the canonical variables, or terms that involve spatial derivatives or more than one time derivative. Such terms can be determined systematically, but we do not need them here. In fact, this is one of the advantages of formulating holographic renormalization in the language of canonical transformations: it can be carried out directly within the context one works in, without having to first carry out the general procedure and then specialize to the case of interest.
Secondly, the boundary term \eqref{bt} is gauge invariant and so the renormalized gauge field $\bb A_a^1$ transforms as the unrenormalized gauge potential $A_a^1$. Since the bulk action \eqref{action-0} is also gauge invariant, it follows that the dual theory possesses a $U(1)^2$ global symmetry associated with the bulk gauge fields $A_a^I$.
Thirdly, notice that the boundary term \eqref{bt} is the generating function of a canonical transformation that is qualitatively different from the standard canonical transformation required for holographically renormalizing asymptotically AdS backgrounds. In particular, the term in the first line of \eqref{bt} is a Legendre transform that changes the boundary condition imposed on the gauge field $A^1_a$ from Dirichlet to Neumann, while the second line in \eqref{bt} is a local function of the canonical momentum $\pi^a_1$, which is holographically identified with the source when Neumann boundary conditions are imposed on $A^1_a$. This type of canonical transformation is also required for renormalizing gauge fields in asymptotically AdS$_2$ or AdS$_3$ backgrounds \cite{An:2016fzu,Cvetic:2016eiv,Erdmenger:2016jjg}. As in those cases, the gauge field $A_a^1$ diverges asymptotically in the UV, as is evident from the asymptotic expansion in \eqref{uv-exp}. Recall that the standard counterterms implement a canonical transformation that renormalizes the canonical momenta, while leaving the induced fields unchanged \cite{Papadimitriou:2010as}. However, whenever the electric chemical potential is not the leading term in the asymptotic expansion of a gauge field, the canonical transformation required to render the variational problem well posed renormalizes the gauge potential instead of its conjugate momentum, as for example in \eqref{transformed-variables}. Although the variational principle \eqref{var-1} suggests that Neumann boundary conditions be imposed on the gauge field $A_a^1$, this is not necessarily the case. As we will see shortly, once the symplectic variables have been renormalized by means of the canonical transformation generated by \eqref{bt}, an extra {\em finite} boundary term can be added in order to change the boundary conditions on the gauge field back to Dirichlet, if so desired.
Finally, to compare the boundary terms \eqref{bt} with the more standard counterterms for asymptotically Lifshitz theories, e.g. in \cite{Ross:2009ar,Ross:2011gu,Chemissany:2014xsa}, it is necessary to determine the general asymptotic solutions without linearizing around a background, which we leave for future work. Nevertheless, it is important to keep in mind that there are some crucial differences between the system we consider here and typical asymptotically Lifshitz theories. Besides the hyperscaling violation that was considered also in \cite{Chemissany:2014xsa}, the presence of a non-zero axion charge and of the second gauge field affects the asymptotic UV expansions non-trivially, as can be seen from \eqref{uv-exp}. Moreover, in the present analysis we turn on a linear source for the energy flux, which is an irrelevant operator relative to the Lifshitz theory, but only consider massless gauge fields in the bulk.
In order to determine the functions $h_1(\phi)$, $h_2(\phi)$, $h_3(\phi)$, $h_4(\phi)$ and $h_5(\phi)$ and demonstrate that the canonical transformation generated by the boundary term \eqref{bt} renormalizes the symplectic variables, we need to evaluate the renormalized canonical variables \eqref{transformed-variables} asymptotically. Linearizing the expressions \eqref{transformed-variables} for the canonically transformed variables using the identities
\bbepsilon
\sqrt{-\lbar\gamma}=n_B\sqrt{\sigma}=v^2n_B=v^{2+z}{\cal F}^{1/2}(v),
\end{equation}
and
\bbepsilon
\delta n_a=\sigma_{ab}^B S^b_t=v^2\Big(\Theta^{a}+\frac{i\omega}{p} \tau^{a}\Big),
\end{equation}
we obtain the following expressions for the linearized and renormalized canonical variables:
\bal\label{transformed-variables-explicit}
\lbar\Pi^{at[1]}=&\;\lbar\pi^{at[1]}+\frac12 h_1(\phi)\lbar\pi^{a[1]}_1+\frac{i\omega}{2}h_3(\phi)v^{z+2}{\cal F}^{1/2}(v)\tau^a+h_5(\phi)v^{z+2}{\cal F}^{1/2}(v)\Big(\Theta^{a}+\frac{i\omega}{p} \tau^{a}\Big),\nonumber\\
\bb a_a^1=&\;\frak a_a^1-h_1(\phi)v^2\Big(\Theta^{a}+\frac{i\omega}{p} \tau^{a}\Big)-2h_2(\phi)v^{-z}{\cal F}^{-1/2}(v)\lbar\pi_{1}^{a[1]}-i\omega v^2h_4(\phi)\tau^a,\nonumber\\
\lbar\Pi_{\chi a}^{[1]}=&\;\lbar\pi_{\chi a}^{[1]}-i\omega v^{4+z}{\cal F}^{1/2}(v)h_3(\phi)\Big(\Theta^{a}+\frac{i\omega}{p} \tau^{a}\Big)-i\omega h_4(\phi)v^2\lbar\pi_{1}^{a[1]}.
\eal
Using the expression \eqref{blackening-sol} for the blackening factor of the background, ${\cal F}(v)$, and the UV expansions \eqref{uv-exp} and \eqref{uv-exp-momenta} for the induced fields and their conjugate momenta, we determine that the transformed variables \eqref{transformed-variables-explicit} are renormalized provided
\bal\label{h-expressions}
&h_1(\phi)=-\frac{4q_1{\rm sgn}(\theta)e^{(2-z-\theta)\phi/\mu}}{2+z-\theta}\Big(1-\frac{ p^2e^{(\theta-2z)\phi/\mu}}{(2-\theta)(z-2)}+m e^{-(2+z-\theta)\phi/\mu}\Big),\nonumber\\\nonumber\\
&h_2(\phi)=\left\{\begin{matrix}
\hskip-2.2cm\frac{\kappa^2{\rm sgn}(\theta)e^{(4-\theta)\phi/\mu}}{2+z-\theta}\Big(1-\Big(\frac12+\frac{2-\theta}{4-3z}\Big)\frac{ p^2e^{(\theta-2z)\phi/\mu}}{(2-\theta)(z-2)}\Big), & z\neq 4/3,\\&\\
\frac{\kappa^2{\rm sgn}(\theta)e^{(4-\theta)\phi/\mu}}{2+z-\theta}\Big(1+\Big(\frac{2-z-\theta}{2(2+z-\theta)}-(2-\theta)\phi/\mu\Big)\frac{ p^2e^{(\theta-2z)\phi/\mu}}{(2-\theta)(z-2)}\Big), & z=4/3,
\end{matrix}\right.\nonumber\\\nonumber\\
&h_3(\phi)=\frac{p\;{\rm sgn}(\theta)e^{-4z\phi/\mu}}{2(z-2)\kappa^2}\Big(1-\frac{(2-z)(z-4+\theta)me^{(z-2)\phi/\mu}}{p^2}\Big),\,\nonumber\\\nonumber\\
&h_4(\phi)=\left\{\begin{matrix}
\hskip-2.7cm\frac{p(z-1)\;{\rm sgn}(\theta)e^{(2-3z)\phi/\mu}}{2(z-2)(4-3z)q_1}, & z\neq 4/3,\\&\\
\frac{p(z-1)\;{\rm sgn}(\theta)e^{(2-3z)\phi/\mu}}{2(z-2)q_1}\Big(\phi/\mu-\frac{1}{4z-2-\theta}\Big), & z=4/3,
\end{matrix}\right.\nonumber\\\nonumber\\
&h_5(\phi)=\frac{(z+1-\theta){\rm sgn}(\theta)}{2\kappa^2}e^{-(2z+\theta)\phi/\mu}\Big(1-\frac{(3z+1-2\theta)p^2e^{(\theta-2z)\phi/\mu}}{2(2-\theta)(z-2)(z+1-\theta)}\nonumber\\
&\hskip2.4in+ \frac{(3+2z-2\theta)me^{-(2+z-\theta)\phi/\mu}}{2(z+1-\theta)}-\frac{12q_2^2e^{-2(z+1-\theta)\phi/\mu}}{(2-\theta)(z-\theta)}\Big).
\eal
Inserting these expressions for the functions $h_1(\phi)$, $h_2(\phi)$, $h_3(\phi)$, $h_4(\phi)$ and $h_5(\phi)$ in the UV asymptotic expansions of the transformed variables \eqref{transformed-variables-explicit} gives
\bal\label{ren-var}
\lbar\Pi^{at[1]}=&\;-\frac{1}{4\kappa^2}v^{-2z}\big(\Theta_{2}^{a}(\omega)+4\mu^Ic_I^a(\omega)+\cdots\big),\nonumber\\
\bb a_a^1=&\;\frak a^1_{oa}(\omega)-\mu^1\Theta_{1}^{a}(\omega)+\cdots,\nonumber\\
\lbar\Pi_{\chi a}^{[1]}=&\;-\frac{2i\omega}{p\kappa^2}q_I\bb a^I_{a}(\omega)+\cdots,
\eal
where again the ellipses stand for asymptotically subleading terms,
\bbepsilon\label{chemical-potentials
\mu^1\equiv-\frac{4{\rm sgn}(\theta)q_1v^{2+z-\theta}_h}{2+z-\theta},\qquad \mu^2\equiv-\frac{4{\rm sgn}(\theta)q_2v^{\theta-z}_h}{\theta-z}
\end{equation}
are the electric chemical potentials of the background \eqref{HVbackground}, and we have defined
\bbepsilon\label{a2}
\bb a_a^2\equiv\frak a^2_{oa}(\omega)-\mu^2\Theta_{1}^{a}(\omega),
\end{equation}
which is the leading order term in UV expansion of gauge field $a_a^2$ in \eqref{uv-exp}. From the variational principle \eqref{var-1} follows that these renormalized variables have the correct radial scaling to be the conjugates of respectively $n_a$, $\lbar\pi_1^a$ and $\chi^a$. Moreover, since they are related to the original symplectic variables by a canonical transformation they are automatically compatible with the symplectic structure of the theory and, therefore, can be directly identified with observables in the dual field theory.
The astute reader will have noticed that the scalar functions \eqref{h-expressions} that define the boundary counterterms surprisingly depend on the mass $m$ and the electric charge $q_2$ of the background, in addition to the axion charge $p$ and the electric charge $q_1$ that is fixed by the Lifshitz boundary conditions. As we have pointed out above, while $p$ and $q_1$ parameterize non-normalizable modes, i.e. boundary conditions, $m$ and $q_2$ correspond to normalizable modes and define a state in the dual field theory. The local counterterms that renormalize a UV complete theory should not depend on the state! However, as we will see shortly, the mode $\Theta_1^a$ in the linearized fluctuations corresponds to the source of the energy flux, which is an irrelevant operator in the dual Lifshitz theory \cite{Ross:2011gu}. It is well known that the counterterms required to renormalize a theory perturbed by a source of an irrelevant operator do in fact depend on the background one-point functions, i.e. on the state \cite{vanRees:2011fr}. For example, using \eqref{momentum-map} and \eqref{Bans} one can relate the function $h_5(\phi)$ to the background canonical momentum
\bbepsilon
\lbar\pi_{B\; b}^a=-n_B^2\sqrt{-\lbar\gamma}\;h_5(\phi)\delta^a_b+\cdots,
\end{equation}
which determines the VEV of the spatial stress tensor in the dual Lifshitz theory. A similar situation was encountered in the computation of two-point functions in a holographic Kondo model \cite{Erdmenger:2016jjg}.
Finally, notice that the functions $h_2(\phi)$ and $h_4(\phi)$ contain a logarithmic term, i.e. linear in $\phi$, when $z=4/3$. Such terms would normally signify a conformal anomaly, but in the presence of a background running dilaton the notion of a conformal anomaly is ambiguous without reference to a UV fixed point \cite{Chemissany:2014xsa,Taylor:2015glc}. Examples where a conformal anomaly in the presence of a running dilaton can be defined unambiguously include the D4-brane theory with its M5-brane UV completion \cite{Kanitscheider:2009as}, as well as the AdS$_2$ dilaton-gravity theory studied in \cite{Cvetic:2016eiv}, whose UV completions is provided by a two dimensional conformal field theory. However, in the present bottom-up context we are agnostic about the far UV completion of the theory and so we cannot unambiguously define a notion of a conformal anomaly. Nevertheless, it is interesting to observe that the logarithmic terms appear for the unique value of $z$ that, as we will see in section \ref{sec:conductivities}, potentially leads to a linear resistivity at high temperature.
Having identified the canonical transformation that renormalizes the symplectic variables, we can define different holographic duals by imposing different boundary conditions on the bulk fields through additional {\em finite} boundary terms. In the following we consider the two distinct theories obtained by imposing Dirichlet or Neumann boundary conditions on the spatial components $A_a^1$ of gauge field $A_i^1$, whose time component supports the Lifshitz background.\footnote{We stress that imposing Dirichlet or Neumann boundary conditions on the gauge fields in the bulk {\em is not} equivalent to changing thermodynamic ensemble, as is often stated. Changing boundary conditions in general changes the dual theory \cite{Witten:2003ya}. Moreover, a given theory can be studied in {\em any} thermodynamic ensemble.} Different boundary conditions for scalars and metric fluctuations in Lifshitz backgrounds have been studied in \cite{Andrade:2012xy,Andrade:2013wsa}.
\paragraph{Renormalized observables for Dirichlet boundary conditions} The variational problem \eqref{var-1} in terms of the renormalized canonical variables is well posed provided $\lbar\pi^a_1$ is kept fixed, which corresponds to Neumann boundary conditions on the
gauge field $A_a^1$. In order to impose Dirichlet boundary conditions on $A_a^1$ we need to add the {\em finite} term
\bbepsilon\label{D-term}
S_{D}=\int d^{d_s+1}x\;\bb A^1_a\lbar\pi^a_1,
\end{equation}
to the renormalized on-shell action and define
\bbepsilon\label{Sren}
S^D_{\rm ren}=\lim_{r\to\infty}\big(S_{\rm reg}+S_{\rm b}+S_{D}\big).
\end{equation}
From \eqref{var-1} follows that the variation of the renormalized action \eqref{Sren} takes the form
\bbepsilon\label{var-2}
\delta(S_{\rm reg}+S_b+S_{D})=\int d^{d_s+1}x\;\Big(2\lbar\Pi^{ta}\delta n_a+\lbar\Pi_{\chi\; a}\delta\chi^a+\lbar\pi^a_1\delta\bb A_a^1+\lbar\pi^a_2\delta A_a^2\Big),
\end{equation}
which is well posed provided $\bb A_a^1$ is kept fixed.
The renormalized variational principle \eqref{var-2} leads to the following identification of local operators and sources in the dual field theory:
\begin{table}[H]
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
& Operator & Source & Dimension\\\hline\hline
Energy flux & ${\cal E}^a=2\lim_{\lbar r\to\infty}e^{2z\bar r}\lbar\Pi^{ta}$ & $\Theta_1^a=\lim_{\lbar r\to\infty}e^{-2z\bar r}n_a$ & $d_s+2z-\theta-1$\\&&&\\
$U(1)$ currents & ${\cal J}^a_I=\lim_{\lbar r\to\infty}\lbar\pi^a_I$ & $\bb a_{oa}^I$ & $d_s+z-\theta-1$\\ &&&\\
Pseudoscalars & ${\cal X}_a=\lim_{\lbar r\to\infty}\lbar\Pi_{\chi a}$ & $\tau_o^a=\lim_{\lbar r\to\infty}\tau^a$ & $d_s+z-\theta$\\
\hline
\end{tabular}
\end{center}
\caption{Spectrum of operators corresponding to Dirichlet boundary conditions on all fields.}
\label{D-observables}
\end{table}
\vskip-0.3cm\hskip-0.6cm Recall that $\bar r$ is the canonical radial coordinate in the dual frame defined in \eqref{dual-metric-background}. Of course, these observables are only those turned on by the fluctuations we consider here. In particular, the energy density and the spatial stress tensor operators of the energy-momentum complex \cite{Ross:2011gu} are outside the space of fluctuations we turn on. Using the expressions \eqref{ren-var} for the renormalized variables in terms of the modes in the linear fluctuations we obtain
\bbepsilon\label{operators}\boxed{
{\cal E}^a=-\frac{1}{2\kappa^2}\big(\Theta_{2}^{a}+4\mu^Ic_I^a\big),\qquad
{\cal J}_I^a=-\frac{2}{\kappa^2}
\Big(c_I^a-\frac{i\omega q_I}{p} \tau^{a}_o\Big),\qquad
{\cal X}_a=-\frac{2i\omega}{p\kappa^2}q_I\bb a^I_{a}.}
\end{equation}
Notice that, as anticipated, the mode $\Theta_1^a$ corresponds to the source of the energy flux ${\cal E}^a$, which is an irrelevant operator for $z>1$.
\paragraph{Renormalized observables for Neumann boundary conditions} If the boundary term \eqref{D-term} is not added to the variational principle \eqref{var-1}, then Neumann boundary conditions must be imposed on the gauge field $A_a^1$, i.e. $\lbar\pi^a_1$ should be identified with the source of the dual operator and kept fixed. More accurately, since $\lbar\pi_1^i$ is constrained by the conservation equation $\pa_i\lbar\pi_1^i=0$ it cannot be directly identified with the arbitrary source of a local operator. The general (local) solution of the conservation equation, however, takes the form $\lbar\pi_1^i=\frac{2}{\kappa^2}\lbar\epsilon^{ijk}\pa_j\wt{\bb a}_{ok}$, for some arbitrary $\wt{\bb a}_{ok}$ that can be identified with the unconstrained source of the dual operator. This is a special case (the $S$-transformation) of the $SL(2,\bb Z)$ transformation on the dual field theory first discussed in \cite{Witten:2003ya}, or equivalently, particle-vortex duality \cite{Herzog:2007ij}. In the bulk it corresponds to electric-magnetic duality. In fact, Neumann boundary conditions on $A_a^1$ are equivalent to Dirichlet boundary conditions for the magnetic dual $\wt A_a^1$. Moreover, had we dualized the gauge field $A_a^1$ in the action \eqref{action-0} from the start, the boundary term \eqref{bt} would take the standard form of the local boundary counterterms and would involve the fieldstrength $\wt F^1_{ij}$ of the magnetic dual gauge field $\wt A_a^1$, in direct analogy to the electric and magnetic frame boundary counterterms computed in \cite{An:2016fzu}.
For the fluctuations we turn on the only non-trivial components of the canonical momentum $\lbar\pi_1^i$ are dualized according to
\bbepsilon
\lbar\pi_1^a=\frac{2i\omega}{\kappa^2}\varepsilon^{ab}\wt{\bb a}_{ob}^1,
\end{equation}
where $\varepsilon^{ab}$ is the Levi-Civita symbol in two dimensions. The spectrum of (dynamical) operators in the theory defined by the variational principle \eqref{var-1} therefore is:
\begin{table}[H]
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
& Operator & Source & Dimension\\\hline\hline
Energy flux & ${\cal E}^a=2\lim_{\lbar r\to\infty}e^{2z\bar r}\lbar\Pi^{ta}$ & $\Theta_1^a=\lim_{\lbar r\to\infty}e^{-2z\bar r}n_a$ & $d_s+2z-\theta-1$\\&&&\\
$\wt{U(1)}$ current & $\wt{\cal J}_1^a=\frac{2i\omega}{\kappa^2}\varepsilon^{ab}\bb a_{ob}^1$ & $\wt{\bb a}_{oa}^1=\frac{\kappa^2}{2i\omega}\varepsilon_{ab}\lim_{\lbar r\to\infty}\lbar\pi^b_1$ & $z+1\,\,\,$ ($d_s=2$)\\ &&&\\
$U(1)$ current & ${\cal J}^a_2=\lim_{\lbar r\to\infty}\lbar\pi^a_2$ & $\bb a_{oa}^2$ & $d_s+z-\theta-1$\\ &&&\\
Pseudoscalars & ${\cal X}_a=\lim_{\lbar r\to\infty}\lbar\Pi_{\chi a}$ & $\tau_o^a=\lim_{\lbar r\to\infty}\tau^a$ & $d_s+z-\theta$\\
\hline
\end{tabular}
\end{center}
\caption{Spectrum of operators corresponding to Neumann boundary conditions on $\bb A^1_a$.}
\label{N-observables}
\end{table}
\vskip-0.3cm\hskip-0.6cm
Note that both operators $\wt{\cal J}_1^a$ and ${\cal J}_1^a$ exist in the Dirichlet theory, as well as the Neumann one. However, $\wt{\cal J}_1^a$ is topological in the Dirichlet theory, while ${\cal J}_1^a$ is topological in the Neumann theory. In tables \ref{D-observables} and \ref{N-observables} we list only the dynamical operators, respectively in the Dirichlet and Neumann theories. We should also point out that the particle-vortex dual currents $\wt{\cal J}_1^a$ and ${\cal J}_1^a$ have the same dimension if and only if $\theta=0$.
The exchange of the $U(1)$ currents ${\cal J}_1^a$ and $\wt {\cal J}_1^a$ under the change of boundary conditions on the bulk gauge field $A^1$ is effectively a non-perturbative definition of the particle-vortex duality transformation in the dual Lifshitz theory, that applies even in the absence of a Lagrangian description. When the theory admits a Lagrangian description the particle-vortex duality transformation can be constructed explicitly, at least for certain values of the dynamical exponent $z$. For $z=2$ this transformation is reviewed e.g. in appendix A of \cite{Grandi:2012nu}.
\paragraph{Ward identities} The Ward identities in the dual theory are directly related to the first class constraints \eqref{constraints} in the radial Hamiltonian formulation of the theory reviewed in appendix \ref{ham}. The shift symmetry of the axions gives rise to an additional {\em global} Ward identity, which implies that the axion momentum $\lbar\pi_{\chi a}$ is a total derivative \cite{Caldarelli:2016nni}.
Within the space of linear fluctuations we consider in this paper the only non-trivial Ward identity comes from the diffeomorphism constraint
\bbepsilon\label{WI-constraint}
-2\lbar D_j\lbar\pi^{j}_i+\lbar F^{I}_{ij}\lbar\pi^j_I+\lbar\pi_\phi\pa_i\phi+\lbar\pi_{\chi a}\pa_i\chi^a=0.
\end{equation}
Linearizing this constraint around a background of the form \eqref{Bans} leads to a single non-trivial condition, namely\footnote{This constraint was also studied in \cite{Andrade:2016tbr} in the context of Lifshitz theories with momentum relaxation. However, the bulk gauge field supporting the Lifshitz asymptotics was massive in that case, which is presumably the reason why the corresponding Ward identity in terms of renormalized variables differs from our \eqref{WID-ren}.}
\bbepsilon\label{WID}\boxed{
-2\pa_t\lbar\pi^{t[1]}_a+\frac{2}{\kappa^2}q_I\pa_t\frak a^I_a+p\;\lbar\pi^{[1]}_{\chi a}=0.}
\end{equation}
From the expressions \eqref{lin-momenta} for the linearized momenta we easily see that this constraint is precisely the fluctuation equation \eqref{eqn:Stzdot-fourier-new} in appendix \ref{sec:fluctuations}.
Using the identities
\bal
\lbar\pi^{a[1]}_t=&\;-n_B^2\lbar\pi^{ta[1]}+\delta n_b\lbar\pi_B^{ba}=-v^{2z}{\cal F}(v)\lbar\pi^{ta[1]}+\delta n_b\lbar\pi_B^{ba},\nonumber\\
\lbar\pi^{t[1]}_a=&\;\sigma^B_{ab}\lbar\pi^{bt[1]}+\delta n_a\lbar\pi_B^{tt}=v^2\lbar\pi^{at[1]}+\delta n_a\lbar\pi_B^{tt},
\eal
and the canonical transformation \eqref{transformed-variables-explicit}, the constraint \eqref{WID} can be expressed in terms of the renormalized variables \eqref{ren-var}. This leads to the Ward identity
\bbepsilon\label{WID-ren}\boxed{
{\cal X}_a=-\frac{2i\omega}{p\kappa^2}q_I\bb a^I_{a}(\omega),}
\end{equation}
which we already obtained in \eqref{operators}. We therefore see that the correlation functions of the scalar operator dual to the axion fields are entirely determined by the diffeomorphism symmetry.
\section{Thermoelectric Lifshitz DC conductivities}
\label{sec:conductivities}
\setcounter{equation}{0}
Combining the general solution of the fluctuation equations and the identification of the renormalized physical observables in the previous section, we are now in a position to evaluate the renormalized two-point functions for both Dirichlet and Neumann boundary conditions on the gauge field $\bb A_a^1$. As we shall see, the DC conductivities in the Dirichlet and Neumann theories are closely related, but are distinct. This highlights the fact that the boundary conditions at the UV play a crucial role in identifying the physical observables and in computing the conductivities, which cannot be determined in general solely from a near horizon analysis.
\subsection{Two-point functions for Dirichlet boundary conditions}
In terms of the renormalized gauge field modes $\bb a_a^I$, specified in \eqref{ren-var} and \eqref{a2}, the expressions \eqref{IR1} and \eqref{IR2} following from imposing ingoing boundary conditions on the horizon become
\bal\label{explicit-ops}
\Theta_{2}^{a}(\omega)
=&\;\frac{i\omega 8\pi T}{p^2Z(\phi_B(u_h))}\Big(q_I \bb a^I_{oa}(\omega)+\big(q_I\mu^I-\pi Te^{2A(u_h)}\big)\Theta_{1}^{a}(\omega)\Big),\nonumber\\
c^a_I(\omega)=&\;-i\omega\(\Sigma_{IJ}(\phi_B(u_h))+\frac{2q_Iq_J}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\)\bb a^J_{oa}(\omega)\nonumber\\
&-i\omega\Bigg(\frac{2q_I\big(q_J\mu^J-\pi Te^{2A(u_h)}\big)}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}+\Sigma_{IJ}(\phi_B(u_h))\mu^J\Bigg)\Theta_{1}^{a}(\omega).
\eal
Inserting these in the renormalized operators of the Dirichlet theory in \eqref{operators} gives
\bal
{\cal E}^a=&\;\frac{2i\omega}{\kappa^2}\Bigg(\frac{2\big(q_I\mu^I-\pi T e^{2A(u_h)}\big)^2}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}+\Sigma_{IJ}(\phi_B(u_h))\mu^I\mu^J\Bigg)\Theta_{1}^{a}(\omega)\nonumber\\
&+\frac{2i\omega}{\kappa^2}\Bigg(\frac{2q_J\big(q_I\mu^I-\pi T e^{2A(u_h)}\big)}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}+\Sigma_{IJ}(\phi_B(u_h))\mu^I\Bigg)\bb a^J_{oa}(\omega),\nonumber\\
{\cal J}^a_I=&\;\frac{2i\omega}{\kappa^2}\Bigg(\frac{2q_I\big(q_J\mu^J-\pi Te^{2A(u_h)}\big)}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}+\Sigma_{IJ}(\phi_B(u_h))\mu^J\Bigg)\Theta_{1}^{a}(\omega)\nonumber\\&+\frac{2i\omega}{\kappa^2}\(\Sigma_{IJ}(\phi_B(u_h))+\frac{2q_Iq_J}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\)\bb a^J_{oa}(\omega)+\frac{2i\omega q_I}{p\kappa^2} \tau^{a}_o(\omega),\nonumber\\
{\cal X}_a=&\;-\frac{2i\omega}{p\kappa^2}q_I\bb a^I_{a},
\eal
from which we read off the two-point functions
\begin{align}
\hskip-0.7cm
\boxed{
\begin{aligned}
\<{\cal J}_I^a(-\omega){\cal J}_J^b(\omega)\>=&\;\frac{2i\omega}{\kappa^2}\(\Sigma_{IJ}(\phi_B(u_h))+\frac{2q_Iq_J}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\)\delta^{ab},\\
\<{\cal E}^a(-\omega){\cal J}_I^b(\omega)\>=&\;\<{\cal J}_I^a(-\omega){\cal E}^b(\omega)\>
=\frac{2i\omega}{\kappa^2}\Bigg(\frac{2q_I\big(q_J\mu^J-\pi T e^{2A(u_h)}\big)}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}+\Sigma_{IJ}(\phi_B(u_h))\mu^J\Bigg)\delta^{ab},\\
\<{\cal E}^a(-\omega){\cal E}^b(\omega)\>=&\;\frac{2i\omega}{\kappa^2}\Bigg(\frac{2\big(q_I\mu^I-\pi T e^{2A(u_h)}\big)^2}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}+\Sigma_{IJ}(\phi_B(u_h))\mu^I\mu^J\Bigg)\delta^{ab},\\
\<{\cal J}_I^a(-\omega){\cal X}^b(\omega)\>=&\;-\<{\cal X}^a(-\omega){\cal J}_I^b(\omega)\>=-\frac{2i\omega q_I}{p\kappa^2}\delta^{ab}.
\end{aligned}
}
\end{align}
All other two-point functions are identically zero.
These correlation functions may be simplified by introducing the {\em Lifshitz heat current}\footnote{The heat current can also be defined through the Ward identity obtained from the time component of the first class constraint \eqref{WI-constraint}. This Ward identity involves the energy density ${\cal E}$ and takes the form $\pa_t{\cal E}+\pa_a{\cal Q}_D^a=0$ (see e.g. \cite{Hartnoll:2015sea}). However, within the space of linear fluctuations we consider here this identity is satisfied trivially.}
\bbepsilon\label{HC-D}\boxed{
{\cal Q}^a_D\equiv {\cal E}^a-\mu^J{\cal J}^a_J,}
\end{equation}
where $\mu^I$ are the electric chemical potentials of the background \eqref{HVbackground} and are given explicitly in \eqref{chemical-potentials}. Notice that in the relativistic limit it reduces to the standard heat current ${\cal T}^{ta}-\mu^J{\cal J}^a_J$, in terms of the relativistic stress tensor ${\cal T}^{ij}$. However, for $z>1$, the heat current is an irrelevant operator, as is the energy flux ${\cal E}^a$. Indeed, the scaling dimension of the heat current \eqref{HC-D} is the one expected for a hyperscaling violating Lifshitz theory \cite{Hartnoll:2015sea}. From expressions \eqref{operators} for the renormalized operators in the Dirichlet theory follows that the heat current takes the form
\bbepsilon
{\cal Q}^a_D=-\frac{1}{2\kappa^2}\Theta_{2}^{a}-\frac{2i\omega q_I\mu^I}{p\kappa^2}\tau^{a}_o,
\end{equation}
and so its correlation functions simplify to
\bal
\<{\cal Q}_D^a(-\omega){\cal J}_I^b(\omega)\>=&\;-\frac{2i\omega}{\kappa^2}\Bigg(\frac{2q_I\big(\pi T e^{2A(u_h)}\big)}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\Bigg)\delta^{ab},\nonumber\\ \<{\cal Q}_D^a(-\omega){\cal Q}_D^b(\omega)\>=&\;\frac{2i\omega}{\kappa^2}\Bigg(\frac{2\big(\pi T e^{2A(u_h)}\big)^2}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\Bigg)\delta^{ab}.
\eal
From these two-point functions we can read off the thermoelectric DC conductivity matrix
\bbepsilon\hskip-0.3cm
\bbsigma_D^{\rm DC}=\left(\begin{matrix}
T\lbar\bbkappa & T\bbalpha_J \\
T\bbalpha_I & \bbsigma_{IJ}
\end{matrix}\right)=\lim_{\omega\to 0}\frac{1}{i\omega}\left(\begin{matrix}
\<{\cal Q}_D^a(-\omega){\cal Q}_D^b(\omega)\> & \<{\cal Q}_D^a(-\omega){\cal J}_J^b(\omega)\> \\&\\
\<{\cal J}_I^a(-\omega){\cal Q}_D^b(\omega)\> & \<{\cal J}_I^a(-\omega){\cal J}_J^b(\omega)\>
\end{matrix}\right),
\end{equation}
namely
\begin{align}
\label{DC-conductivities-D}\boxed{\boxed{
\begin{aligned}
\lbar\bbkappa^{ab}=&\;\frac{\pi sT}{\kappa^2p^2Z(\phi_B(u_h))}\delta^{ab},\\
\bbalpha^{ab}_I=&\;-\frac{4q_I\lbar\bbkappa^{ab}}{sT},\\
\bbsigma^{ab}_{IJ}=&\;\frac{2}{\kappa^2}\Sigma_{IJ}(\phi_B(u_h))\delta^{ab}+\frac{16q_Iq_J\lbar\bbkappa^{ab}}{s^2T},
\end{aligned}
}}
\end{align}
where $s=4\pi e^{2A(u_h)}$ is the entropy density. These conductivities are consistent with the results of \cite{Donos:2014cya,Cremonini:2016avj,Bhatnagar:2017twr}, which were obtained exclusively from a near horizon analysis without identifying the corresponding conserved currents in the dual Lifshitz theory. Notice that the two $U(1)$ currents present in the theory defined by Dirichlet boundary conditions are on the same footing and so the full matrix $\bbsigma_{IJ}^{ab}$ should be identified as a matrix of electric conductivities.
\subsection{Two-point functions for Neumann boundary conditions}
In order to compute the two-point functions in the theory defined by Neumann boundary conditions we need to invert the expression for the current ${\cal J}_1^a$ in \eqref{explicit-ops} to express $\bb a_{oa}^1$ in terms of the sources in the Neumann theory, namely
\bal
\bb a^1_{oa}(\omega)=&\;\(\Sigma_{11}(\phi_B(u_h))+\frac{2q_1^2}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\)^{-1}\Bigg[-\frac{2q_1q_2}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\bb a^2_{oa}(\omega)\nonumber\\&-\Bigg(\frac{2q_1\big(q_J\mu^J-\pi Te^{2A(u_h)}\big)}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}+\Sigma_{11}(\phi_B(u_h))\mu^1\Bigg)\Theta_{1}^{a}(\omega)+\(\varepsilon^{ab}\wt{\bb a}^1_b-\frac{q_1}{p} \tau^{a}_o(\omega)\)\Bigg].
\eal
Using this relation in the remaining expressions for the operators in \eqref{explicit-ops} we determine that the non-zero two-point functions in the Neumann case are
\begin{align}
\label{N-2-pt-functions}
\hskip-0.7cm
\boxed{
\begin{aligned}
\<\wt{\cal J}^a_1(-\omega)\wt{\cal J}^b_1(\omega)\>=&\;\frac{2i\omega}{\kappa^2}{\cal K}\delta^{ab},\\
\<\wt{\cal J}^a_1(-\omega){\cal J}_2^b(\omega)\>=&\;\<{\cal J}_2^b(-\omega)\wt{\cal J}^a_1(\omega)\>=\frac{2i\omega}{\kappa^2}\(\frac{2q_1q_2{\cal K}}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\)\varepsilon^{ab},\\
\<{\cal J}_2^a(-\omega){\cal J}_2^b(\omega)\>=&\;\frac{2i\omega}{\kappa^2}\(\Sigma_{22}(\phi_B(u_h))+\frac{2q_2^2{\cal N}}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\)\delta^{ab},\\
\<\wt{\cal J}^a_1(-\omega){\cal E}^b(\omega)\>=&\;\<{\cal E}^b(-\omega)\wt{\cal J}^a_1(\omega)\>
=\frac{2i\omega}{\kappa^2}\(\mu^1+\frac{2q_1{\cal K}\big(q_2\mu^2-\pi Te^{2A(u_h)}\big)}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\)\varepsilon^{ab},\\
\<{\cal E}^a(-\omega){\cal J}_2^b(\omega)\>=&\;\<{\cal J}_2^a(-\omega){\cal E}^b(\omega)\>
=\frac{2i\omega}{\kappa^2}\(\frac{2q_2{\cal N}\big(q_2\mu^2-\pi T e^{2A(u_h)}\big)}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}+\Sigma_{22}(\phi_B(u_h))\mu^2\)\delta^{ab},\\
\<{\cal E}^a(-\omega){\cal E}^b(\omega)\>=&\;\frac{2i\omega}{\kappa^2}\(\frac{2{\cal N}\big(q_2\mu^2-\pi T e^{2A(u_h)}\big)^2}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}+\Sigma_{22}(\phi_B(u_h))(\mu^2)^2\)\delta^{ab},\\
\<\wt{\cal J}^a_1(-\omega){\cal X}^b(\omega)\>=&\;-\<{\cal X}^b(-\omega)\wt{\cal J}^a_1(\omega)\>=-\frac{2i\omega q_1}{p\kappa^2}{\cal K}\varepsilon^{ab},\\
\<{\cal J}_2^a(-\omega){\cal X}^b(\omega)\>=&\;-\<{\cal X}^a(-\omega){\cal J}_2^b(\omega)\>=-\frac{2i\omega q_2{\cal N}}{p\kappa^2}\delta^{ab},\\
\<{\cal X}^a(-\omega){\cal X}^b(\omega)\>=&\;\frac{2i\omega q_1^2}{p^2\kappa^2}{\cal K},
\end{aligned}
}
\end{align}
where we have defined
\bal\label{factors}
{\cal K}\equiv&\;\(\Sigma_{11}(\phi_B(u_h))+\frac{2q_1^2}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\)^{-1},\nonumber\\ {\cal N}\equiv&\;\Sigma_{11}(\phi_B(u_h)){\cal K}=\(1+\frac{2q_1^2\Sigma^{-1}_{11}(\phi_B(u_h))}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\)^{-1}.
\eal
Notice that ${\cal N}$ takes values in the interval $0<{\cal N}<1$.
Introducing the Lifshitz heat current for the Neumann theory\footnote{This form of the heat current in the case of Neumann boundary conditions on $A_a^1$ again follows from the time component of the Ward identity obtained from the constraint \eqref{WI-constraint}. The reason why $\mu^1$ does not enter in the heat current for Neumann boundary conditions is that $q_1$ corresponds a to magnetic charge in terms of the dual gauge field $\wt A_a^1$, and $\mu^1$ to a magnetic chemical potential. }
\bbepsilon\boxed{
{\cal Q}_N^a\equiv{\cal E}^b-\mu^2{\cal J}_2^b,}
\end{equation}
the two-point functions involving the energy flux and the electric current ${\cal J}_2^a$ take the simpler form
\bal
\<{\cal Q}_N^a(-\omega){\cal J}_2^b(\omega)\>=&\;-\frac{2i\omega}{\kappa^2}\Bigg(\frac{2q_2{\cal N}\big(\pi T e^{2A(u_h)}\big)}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\Bigg)\delta^{ab},\nonumber\\ \<{\cal Q}_N^a(-\omega){\cal Q}_N^b(\omega)\>=&\;\frac{2i\omega}{\kappa^2}\Bigg(\frac{2{\cal N}\big(\pi T e^{2A(u_h)}\big)^2}{p^2Z(\phi_B(u_h))e^{2A(u_h)}}\Bigg)\delta^{ab}.
\eal
Reading off the components of the thermoelectric DC conductivity matrix
\bbepsilon\hskip-0.3cm
\bbsigma_N^{\rm DC}=\left(\begin{matrix}
T\lbar\bbkappa_N & T\bbalpha_N \\
T\bbalpha_N & \bbsigma_N
\end{matrix}\right)=\lim_{\omega\to 0}\frac{1}{i\omega}\left(\begin{matrix}
\<{\cal Q}_N^a(-\omega){\cal Q}_N^b(\omega)\> & \<{\cal Q}_N^a(-\omega){\cal J}_2^b(\omega)\> \\&\\
\<{\cal J}_2^a(-\omega){\cal Q}_N^b(\omega)\> & \<{\cal J}_2^a(-\omega){\cal J}_2^b(\omega)\>
\end{matrix}\right),
\end{equation}
we obtain
\begin{align}
\label{DC-conductivities-N}\boxed{\boxed{
\begin{aligned}
\lbar\bbkappa^{ab}_N=&\;\frac{\pi sT{\cal N}}{\kappa^2p^2Z(\phi_B(u_h))}\delta^{ab},\\
\bbalpha_N^{ab}=&\;-\frac{4q_2\lbar\bbkappa_N^{ab}}{sT},\\
\bbsigma_N^{ab}=&\;\frac{2}{\kappa^2}\Sigma_{22}(\phi_B(u_h))\delta^{ab}+\frac{16q_2^2\lbar\bbkappa_N^{ab}}{s^2T}.
\end{aligned}
}}
\end{align}
Note that the thermal conductivities \eqref{DC-conductivities-N} and \eqref{DC-conductivities-D} satisfy the simple relation $\lbar\bbkappa/\bbalpha = - T s/(4 q)$, as in \cite{Donos:2014cya}. Moreover, the conductivities \eqref{DC-conductivities-N} are related to those of the Dirichlet theory in \eqref{DC-conductivities-D} by an effective rescaling of the function $Z$ according to $Z(\phi_B(u_h))\to Z(\phi_B(u_h))/{\cal N}$, where the factor ${\cal N}$ is defined in \eqref{factors}. Nevertheless, the conductivities \eqref{DC-conductivities-N} are not identical with those obtained from the near horizon analysis, which as we have seen above correspond to imposing Dirichlet boundary conditions on $\bb A_a^1$. The boundary conditions at the UV, therefore, crucially affect the computation of the conductivities. The thermoelectric conductivities \eqref{DC-conductivities-N} can alternatively be obtained by dualizing the gauge field $A_i^1$ in the action \eqref{action-0} from the very beginning and imposing Dirichlet boundary conditions on all fields. Presumably, the near horizon analysis of \cite{Donos:2014cya} in the theory with the dual gauge field $\wt A_i^1$ would produce the same result.
A notable property of the conductivities \eqref{DC-conductivities-N} is that, as their counterparts for Dirichlet boundary conditions in \eqref{DC-conductivities-D}, they obey the
Wiedemann-Franz law at weak momentum dissipation, i.e. $\lbar {\bbkappa}/\bbsigma\propto T$, at least for low temperatures.
Another interesting feature of both electric conductivities \eqref{DC-conductivities-D} and \eqref{DC-conductivities-N} is that they satisfy the lower bound found in \cite{Grozdanov:2015qia}, which in our theory takes the form\footnote{We are grateful to Sa\v{s}o Grozdanov for bringing this bound to our attention. It is unclear whether such a bound exits for the off-diagonal conductivities $\bbsigma^{ab}_{12}$ in the presence of two $U(1)$ gauge fields as in \eqref{DC-conductivities-D}, since \cite{Grozdanov:2015qia} considers only one gauge field.}
\bbepsilon\label{bound}
\bbsigma^{ab}_{11}\geq \frac{2}{\kappa^2}\Sigma_{11}(\phi_B(u_h))\delta^{ab},\qquad \bbsigma^{ab}_{22},\,\,\bbsigma^{ab}_N\geq \frac{2}{\kappa^2}\Sigma_{22}(\phi_B(u_h))\delta^{ab}.
\end{equation}
This is expected for the conductivities $\bbsigma^{ab}_{11}$ and $\bbsigma^{ab}_{22}$ obtained by imposing Dirichlet boundary conditions on $A_a^1$, since they agree with the near horizon result of \cite{Donos:2014cya}, on which the bound of \cite{Grozdanov:2015qia} relies. More interesting, and reassuring, is the fact that the Neumann conductivity $\bbsigma^{ab}_N$ also satisfies the same bound. A possible explanation for this is that the conductivities \eqref{DC-conductivities-N} can be obtained from a near horizon analysis after dualizing the gauge field $A^1_\mu$ in the original action \eqref{action-0}.
A lower bound on the thermal conductivity was also obtained in \cite{Grozdanov:2015djs}. However, in contrast to the bound of \cite{Grozdanov:2015qia} on the electric conductivity, the bound on the thermal conductivity assumes a relativistic conformal UV fixed point, and so it is not directly applicable to the class of theories we study here. Nevertheless, we can see traces of that bound in non-relativistic theories as well. The temperature of the black brane solution \eqref{HVbackground} is given below in eq.~\eqref{Texp}. For $\theta\leq 0$, positivity of this temperature requires that\footnote{For $\theta>0$ the inequality \eqref{entropy-bound} ceases to hold and so it is unclear whether a lower bound on the thermal conductivity exists in that case.}
\bbepsilon\label{entropy-bound}
\frac{\pi s}{\kappa^2p^2Z(\phi_B(u_h))}>\frac{16\pi^2 v_h^{-\theta}}{2\kappa^2(2-\theta)(2+z-\theta)},
\end{equation}
which translates to the thermal conductivity bounds
\bbepsilon\label{bound-thermal}
\frac{\lbar\bbkappa^{ab}}{T}>\frac{16\pi^2 v_h^{-\theta}}{2\kappa^2(2-\theta)(2+z-\theta)}\delta^{ab},\qquad \frac{\lbar\bbkappa^{ab}_N}{T}>\frac{16\pi^2 v_h^{-\theta}}{2\kappa^2(2-\theta)(2+z-\theta)}{\cal N}\delta^{ab},
\end{equation}
respectively for Dirichlet and Neumann boundary conditions on $A_a^1$. These expressions suggest that the thermal conductivity for Dirichlet boundary conditions in non-relativistic theories is still bounded from below, at least for $\theta\leq 0$, but the one obtained from Neumann boundary conditions is less constrained, since the factor ${\cal N}$ can be made arbitrarily small, e.g. by taking $p^2$ to be small.
Although the relativistic limit $z\to 1$ does not generically commute with the asymptotic analysis in the UV, naively setting $z=1$ and $\theta=0$ one finds that the bounds \eqref{bound-thermal} on the Dirichlet and Neumann conductivities coincide, since for $z=1$, $q_1=0$ and so ${\cal N}=1$, and take the simple form
\bbepsilon\label{bound-thermal-relativisitc}
\frac{\lbar\bbkappa^{ab}}{T},\,\,\,\frac{\lbar\bbkappa^{ab}_N}{T} >\frac{8\pi^2}{3}\frac{\delta^{ab}}{2\kappa^2}.
\end{equation}
This lower bound is twice the one found in \cite{Grozdanov:2015djs}, but this is not surprising given that the relativistic conductivities cannot in general be obtained simply by taking the $z\to 1$ limit of the non-relativistic ones. Nevertheless, the result \eqref{bound-thermal-relativisitc} of the naive relativistic limit of the non-relativistic bounds \eqref{bound-thermal} suggests that a lower bound on the thermal conductivity analogous to the one in \cite{Grozdanov:2015djs} exists for non-relativistic theories as well. It would be very interesting to see how universal these bounds are in non-relativistic theories.
Finally, the two-point functions in \eqref{N-2-pt-functions} involving the particle-vortex dual current $\wt {\cal J}_1^a$ give rise to the following thermoelectric conductivities:
\bal\label{magnetic-conductivities}
&\;\lim_{\omega\to 0}\frac{1}{i\omega}\<\wt{\cal J}_1^a(-\omega){\cal Q}_N^b(\omega)\>=\frac{2}{\kappa^2}\(\mu^1-\frac{2\pi q_1{\cal K} T}{p^2Z(\phi_B(u_h))}\)\varepsilon^{ab},\nonumber\\
&\;\lim_{\omega\to 0}\frac{1}{i\omega}\<\wt{\cal J}_1^a(-\omega){\cal J}_2^b(\omega)\>=\frac{2}{\kappa^2}\(\frac{8\pi q_1q_2{\cal K}}{p^2sZ(\phi_B(u_h))}\)\varepsilon^{ab},\nonumber\\
&\;\lim_{\omega\to 0}\frac{1}{i\omega}\<\wt{\cal J}_1^a(-\omega)\wt{\cal J}_1^b(\omega)\>=\frac{2}{\kappa^2}{\cal K}\delta^{ab}.
\eal
In particular, the $\wt{\cal J}_1\wt{\cal J}_1$ conductivity is the inverse of the ${\cal J}_1{\cal J}_1$ one obtained from the Dirichlet theory in \eqref{DC-conductivities-D}, in agreement with particle-vortex duality
\cite{Herzog:2007ij}. Note that the charge $q_1$ is a background magnetic charge in the Neumann theory, and $\mu^1$ is a magnetic chemical potential. Indeed, the thermoelectric conductivities \eqref{magnetic-conductivities} have precisely the expected structure for a background that is purely electric for one gauge field and purely magnetic for the other. This can be checked qualitatively by comparing with the conductivities obtained in \cite{Bhatnagar:2017twr}, where a background magnetic field is turned on for $A_i^2$. Replacing $B^{\rm there}\to q_1$, $q_1^{\rm there}\to q_2$, $q_2^{\rm there}\to 0$, where ``there'' refers to quantities defined in \cite{Bhatnagar:2017twr}, eq.~(49) in \cite{Bhatnagar:2017twr} is analogous to the $\wt{\cal J}_1\wt{\cal J}_1$ conductivity in \eqref{magnetic-conductivities}, eq.~(46) in \cite{Bhatnagar:2017twr} is analogous to the $\wt{\cal J}_1{\cal J}_2$ conductivity in \eqref{magnetic-conductivities}, and eq.~(57) in \cite{Bhatnagar:2017twr} is analogous to the $\wt{\cal J}_1{\cal Q}$ conductivity in \eqref{magnetic-conductivities}. This suggests that the thermoelectric DC conductivities on the dyonic background of \cite{Bhatnagar:2017twr} can alternatively be obtained by considering mixed boundary conditions for the gauge field $A_a^2$, but in the purely electric background \eqref{HVbackground}.
\subsection{Temperature dependence and scaling regimes}
Now that we have obtained the general form of the thermoelectric conductivities for Dirichlet and Neumann boundary conditions, we would like to apply these results to the analytical black brane solutions (\ref{HVbackground}) we considered in subsection \ref{example}, and examine the associated temperature dependence.
We are particularly interested in the differences between the possible scaling regimes for the two sets of
boundary conditions.
We remind the reader that the electrical part of the conductivity matrix for the model of subsection \ref{example} was already studied in \cite{Cremonini:2016avj}, and therefore we will see some overlap with that discussion.
However, while the analysis of \cite{Cremonini:2016avj}
provided some preliminary insight into the conductive behavior of the system,
it only discussed the electric conductivities corresponding to Dirichlet boundary conditions on $\bb A_a^1$ without fully identifying the physical observables in the dual Lifshitz theory, and was therefore incomplete. Moreover, the analysis of \cite{Cremonini:2016avj} considered only the case $\theta<0$ ($\theta>0$ in the conventions of \cite{Cremonini:2016avj}). Here we revisit those results in light of the analysis of section \ref{sec:linear} and expand on it by examining the full thermoelectric matrix for both Dirichlet and Neumann boundary conditions on $\bb A_a^1$, and for any $\theta<z$, both positive and negative.
The temperature $T = -\frac{{\rm sgn}(\theta)}{4\pi} v_h^{z+1} {\cal F}^\prime(v_h) $ of the black brane solutions (\ref{HVbackground}) is given by
\bbepsilon
\label{Texp}\boxed{
T = -\frac{{\rm sgn}(\theta)}{4\pi}\Big((z+2-\theta) v_h^z - \frac{8q_2^2}{2-\theta}v_h^{2\theta-z-2} - \frac{p^2}{2-\theta} v_h^{\theta-z}\Big) \, .}
\end{equation}
Note that for the range $1<z<2$, $\theta<z$ we are considering, the first term inside the parenthesis is always positive and the remaining two negative, independently of the sign of $\theta$.
To identify analytically relatively simple scaling regimes we would like to inspect two different limiting cases, high and low temperatures. These regimes will be sensitive to the sign of $\theta$,
as we show next:
\begin{itemize}
\item $\theta<0$ case:\\
{\bf Large temperatures:} When the hyperscaling violating parameter is negative, the high temperature
limit is defined by $ q_2^2 << v_h^{2z-2\theta+2}$ and $ p^2 << v_h^{2z-\theta}$, and (\ref{Texp}) can be well approximated by
\bbepsilon
\label{largeT}
T \sim \frac{z+2-\theta}{4\pi} v_h^z \sim q_1^2 v_h^z\, .
\end{equation}
This simple expression will facilitate the identification of clean scaling regimes.
{\bf Small temperatures:} On the other hand, the low temperature limit of the theory can be obtained in three ways, either by working with
\bbepsilon
\label{lowT1}
p^2 v_h^{\theta-z} << q_2^2 v_h^{2\theta-z-2} \lesssim v_h^z \, ,
\end{equation}
or alternatively with
\bbepsilon
\label{lowT2}
q_2^2 v_h^{2\theta-z-2} << p^2 v_h^{\theta-z} \lesssim v_h^z \, ,
\end{equation}
or finally by taking the two negative terms in (\ref{Texp}) to be comparable to each other,
\bbepsilon
\label{lowT3}
q_2^2 v_h^{2\theta-z-2} + p^2 v_h^{\theta-z} \lesssim v_h^z \, .
\end{equation}
These conditions ensure that the temperature is small and positive.
Unfortunately in these cases we don't have a simple analytical expression for $v_h$ as a function of temperature. For generic values of the scaling exponents expression (\ref{Texp}) must be inverted numerically. Still, these regimes will allow us to highlight some of the key differences in the transport behavior associated with Neumann and Dirichlet boundary conditions.
\item
$\theta>0$ case:\\
{\bf Large temperatures:} When the hyperscaling violation parameter is positive
the high temperature limit
corresponds to small values of the horizon radius, and in particular one has either
\bbepsilon
\label{poshighT1}
T \sim q_2^2 v_h^{2\theta-z-2} \quad \text{when} \quad q_2^2 v_h^{\theta-2} >> p^2 >> q_1^2 v_h^{2z-\theta} \, ,
\end{equation}
or alternatively
\bbepsilon
\label{poshighT2}
T \sim p^2 v_h^{\theta-z} \quad \text{when} \quad p^2 >> q_2^2 v_h^{\theta-2} >> q_1^2 v_h^{2z-\theta} \, .
\end{equation}
The case in which the two positive terms in (\ref{Texp}) are of comparable magnitude is also possible, but would lead to similar scalings for $T$ as a function of horizon radius.
{\bf Small temperatures:} Once again, there are several ways to work in the low temperature regime,
depending on the hierarchy of scales in the theory.
One can take
\bbepsilon
\label{poslowT1}
q_2^2 v_h^{\theta-2} << q_1^2 v_h^{2z-\theta} \lesssim p^2,
\end{equation}
or alternatively
\bbepsilon
\label{poslowT2}
p^2 << q_1^2 v_h^{2z-\theta} \lesssim q_2^2 v_h^{\theta-2},
\end{equation}
or finally the two positive terms in (\ref{Texp}) can be of comparable strength, so that
\bbepsilon
\label{poslowT3}
q_1^2 v_h^{2z-\theta} \lesssim q_2^2 v_h^{\theta-2} + p^2 \, .
\end{equation}
\end{itemize}
We can now ask whether in some of these simple regions of parameter space
we can identify clean scaling regimes for the thermoelectric conductivities.
\subsubsection{Dirichlet case}
Starting from the expressions for the thermoelectric conductivities for Dirichlet boundary conditions
\eqref{DC-conductivities-D}, and evaluating them on the black brane background (\ref{HVbackground}), we obtain
\bal
\label{DC-conductivities-D-background}
\lbar\bbkappa^{ab}=&\;\frac{8 \pi^2}{\kappa^2p^2} \, v_h^{2z-2\theta}\, T \; \delta^{ab},\nonumber\\
\bbalpha^{ab}_I=&\;-\frac{8\pi q_I}{\kappa^2p^2} \, v_h^{2z-\theta-2} \; \delta^{ab} ,\nonumber\\
\bbsigma^{ab}_{11}=&\;\frac{1}{2\kappa^2} \Big(v_h^{\theta-4}+ 16 \frac{q_1^2}{p^2} v_h^{2z-4} \Big) \delta^{ab},\nonumber\\
\bbsigma^{ab}_{12}=&\;\frac{8}{2\kappa^2} \frac{q_1 q_2}{p^2} v_h^{2z-4} \delta^{ab} \, ,\nonumber\\
\bbsigma^{ab}_{22}=&\;\frac{1}{2\kappa^2} \Big(v_h^{2z-2-\theta}+ 16 \frac{q_2^2}{p^2} v_h^{2z-4}\Big) \delta^{ab},
\eal
whose temperature dependence can then be extracted by expressing the horizon radius $v_h$ as a function of $T$.
This is easy to do at large temperatures, but more challenging analytically at small and intermediate values of $T$.
\begin{itemize}
\item $\theta<0$ case:\\
{\bf Large temperatures:} In the regime where (\ref{largeT}) holds they behave as
%
\bal
\label{DC-conductivities-D-temperature-highT}
\lbar\bbkappa^{ab} \sim &\;\frac{8 \pi^2}{\kappa^2p^2} \, T^{\frac{3z-2\theta}{z}} \; \delta^{ab},\nonumber\\
\bbalpha^{ab}_I \sim &\;-\frac{8\pi q_I}{\kappa^2p^2} \, T^{\frac{2z-\theta-2}{z}} \; \delta^{ab} ,\nonumber\\
\bbsigma^{ab}_{11} \sim &\;\frac{8}{\kappa^2} \frac{q_1^2}{p^2} T^{\frac{2z-4}{z}} \delta^{ab} ,\nonumber\\
\bbsigma^{ab}_{12} \sim &\;\frac{8}{2\kappa^2} \frac{q_1 q_2}{p^2} T^{\frac{2z-4}{z}} \delta^{ab} \, ,\nonumber\\
\bbsigma^{ab}_{22} \sim &\;\frac{1}{2\kappa^2} \Big( T^{\frac{2z-2-\theta}{z}}+ 16 \frac{q_2^2}{p^2} T^{\frac{2z-4}{z}} \Big) \delta^{ab} \; .
\eal
Note that the hierarchy between the two terms in $\bbsigma_{22}$ depends on the magnitude of $q_2, p$ relative to $v_h$. In particular,
\bbepsilon
\label{negcase1}
\bbsigma_{22} \sim \frac{1}{2\kappa^2}T^{\frac{2z-2-\theta}{z}} \quad \text{when} \quad q_2^2 << p^2 v_h^{2-\theta},
\end{equation}
while instead
\bbepsilon
\label{negcase2}
\bbsigma_{22} \sim \frac{8}{\kappa^2} \frac{q_2^2}{p^2} T^{\frac{2z-4}{z}} \quad \text{when} \quad p^2 v_h^{2-\theta} << q_2^2 << v_h^{2z-2\theta+2} \, .
\end{equation}
Since $\theta<0$ and $1<z<2$, note that $\bbsigma_{22}$ in (\ref{negcase1}) can only scale with positive powers of $T$.
The case described by (\ref{negcase2}) is more interesting, since it corresponds to each component of
$\bbsigma_{IJ}$ scaling in the same way $\sim T^{\frac{2z-4}{z}}$ as a function of temperature.
In particular, we see that the electric conductivities scale as $\sim 1/T$ when $z=4/3$, which would be interesting for systems that may exhibit a
linear temperature dependence for the resistivity, $\rho \sim T$. Recall that the $z=4/3$ case was singled out
by the analysis of \cite{Hartnoll:2015sea}.
{\bf Small temperatures:}
At low temperature it is challenging to obtain analytic expressions for the horizon radius as a function of temperature, which are needed to isolate easily recognizable scaling regimes in temperature.
Nonetheless, we can still examine how the thermoelectric matrix scales as a function of $v_h$,
in the three limits (\ref{lowT1}), (\ref{lowT2}) and (\ref{lowT3}), to highlight how it differs from the analogous expressions obtained assuming Neumann boundary conditions.
Notice that the two components $\lbar\bbkappa^{ab}$ and $\bbalpha^{ab}$ in (\ref{DC-conductivities-D-background}) cannot be simplified any further. Thus,
we focus on the electric conductivities $\bbsigma^{ab}_{IJ}$.
When (\ref{lowT1}) holds we find that
\bal
\label{DC-conductivities-D-background-lowT1}
\bbsigma^{ab}_{11}=&\;\frac{8}{\kappa^2} \frac{q_1^2}{p^2} v_h^{2z-4} \delta^{ab} \, , \qquad \bbsigma^{ab}_{12}= \frac{1}{2} \frac{q_2}{q_1}\, \bbsigma^{ab}_{11}\, , \qquad \bbsigma^{ab}_{22}= \frac{q_2^2}{q_1^2} \, \bbsigma^{ab}_{11} \, .
\eal
Alternatively, in the low temperature range (\ref{lowT2}) we find that the only component of (\ref{DC-conductivities-D-background}) which can be simplified is $\bbsigma^{ab}_{22}$ and takes the form
\bbepsilon
\bbsigma^{ab}_{22}= \frac{1}{2\kappa^2} v_h^{2z-2-\theta} \delta^{ab} \, .
\end{equation}
Finally, when (\ref{lowT3}) holds we have
\bbepsilon
\bbsigma^{ab}_{22} = \frac{1}{2\kappa^2 p^2} v_h^{4z-2\theta -2} \delta^{ab} \, ,
\end{equation}
with the remaining components of the thermoelectric matrix as in (\ref{DC-conductivities-D-background}).
Thus, we stress that one obtains different low temperature behaviors for the electric conductivities depending on the hierarchy between the different scales in the system,
with a variety of scaling regimes possible. While a more extensive analysis would have to be
done numerically, these simple estimates provide insight into the richness of this system.
\item $\theta>0$ case:\\
{\bf Large temperatures:} In the regime (\ref{poshighT1}) the thermoelectric conductivities
scale as
\bal
\label{DC-conductivities-D-temperature-highTpositive}
\lbar\bbkappa^{ab} \sim &\;\frac{8 \pi^2 q_2^2}{\kappa^2p^2} \, \left(\frac{T}{q_2^2}\right)^{\frac{z-2}{2\theta-z-2}} \; \delta^{ab},\nonumber\\
\bbalpha^{ab}_I \sim &\;-\frac{8\pi q_I}{\kappa^2p^2} \, \left(\frac{T}{q_2^2}\right)^{\frac{2z-\theta-2}{2\theta-z-2}} \; \delta^{ab} ,\nonumber\\
\bbsigma^{ab}_{11} \sim &\;\frac{1}{2\kappa^2} \left(\frac{T}{q_2^2}\right)^{\frac{\theta-4}{2\theta-z-2}}
\delta^{ab}, \quad
\bbsigma^{ab}_{12} \sim
\frac{4}{\kappa^2} \frac{q_1 q_2}{p^2} \left(\frac{T}{q_2^2}\right)^{\frac{2z-4}{2\theta-z-2}}
\delta^{ab}, \quad
\bbsigma^{ab}_{22} \sim \frac{2 q_2}{q_1} \, \bbsigma^{ab}_{12} \, .
\eal
It is easy to check that for these ranges of parameters none of the components of the electric matrix
$\sigma_{IJ}^{ab}$ can scale as $1/T$.
In the regime (\ref{poshighT2}) on the other hand we have
\bal
\label{DC-conductivities-D-temperature-highTpositive2}
\lbar\bbkappa^{ab} \sim &\;\frac{8 \pi^2 p^2}{\kappa^2} \, \frac{1}{T} \; \delta^{ab},\nonumber\\
\bbalpha^{ab}_I \sim &\;-\frac{8\pi q_I}{\kappa^2p^2} \, \left(\frac{T}{p^2}\right)^{\frac{2z-\theta-2}{\theta-z}} \; \delta^{ab} ,\nonumber\\
\bbsigma^{ab}_{11} \sim &\;\frac{1}{2\kappa^2} \left(\frac{T}{p^2}\right)^{\frac{\theta-4}{\theta-z}}
\delta^{ab}, \quad
\bbsigma^{ab}_{12} \sim
\frac{4}{\kappa^2} \frac{q_1 q_2}{p^2} \left(\frac{T}{p^2}\right)^{\frac{2z-4}{\theta-z}}
\delta^{ab}, \quad
\bbsigma^{ab}_{22} \sim \frac{1}{2\kappa^2} \left(\frac{T}{p^2}\right)^{\frac{2z-2-\theta}{\theta-z}}
\delta^{ab} \, .
\eal
As for the case above, none of the electric conductivities here can scale as $1/T$, for our range of $z$ and
$\theta$.
{\bf Small temperatures:}
In the low temperature regime, the only case for which the thermoelectric matrix simplifies significantly
corresponds to the case (\ref{poslowT2}). One can then show that the electrical conductivities scale as they do in (\ref{DC-conductivities-D-background-lowT1}), which we recall
corresponds to $\theta<0$.
\end{itemize}
\subsubsection{Neumann case}
Recall that the thermoelectric conductivities in the case of Neumann boundary conditions are modified by a factor of
\bbepsilon
{\cal N} = \Big(1+ 16 \frac{q_1^2}{p^2} v_h^{2z-\theta} \Big)^{-1} \, ,
\end{equation}
as compared to those associated with Dirichlet boundary conditions.
Thus, after evaluating on the background we have
\bal
\label{DC-conductivities-N-background}
\lbar\bbkappa^{ab}_N=&\;\frac{8 \pi^2}{\kappa^2p^2} \, v_h^{2z-2\theta}\, T \, \Big( 1+ 16 \frac{q_1^2}{p^2} v_h^{2z-\theta} \Big)^{-1} \; \delta^{ab},\nonumber\\
\bbalpha^{ab}_N=&\;-\frac{8\pi q_2}{\kappa^2p^2} \, v_h^{2z-\theta-2} \, \Big( 1+ 16 \frac{q_1^2}{p^2} v_h^{2z-\theta} \Big)^{-1} \; \delta^{ab} ,\nonumber\\
\bbsigma^{ab}_{N}=&\;\frac{1}{2\kappa^2} \Big[ v_h^{2z-2-\theta}+ 16 \frac{q_2^2}{p^2} v_h^{2z-4} \Big( 1+ 16 \frac{q_1^2}{p^2} v_h^{2z-\theta} \Big)^{-1} \Big]\delta^{ab}.
\eal
For generic values of $v_h$ these differ significantly from their Dirichlet counterparts.
Again, we consider separately positive and negative values of the hyperscaling violating exponent:
\begin{itemize}
\item $\theta<0$ case:\\
{\bf Large temperatures:}
In the regime (\ref{largeT}) it can be easily shown that ${\cal N}$ scales like
\bbepsilon
\label{cnlimit}
{\cal N} \sim \frac{p^2}{16 q_1^2} v_h^{\theta-2z} \sim \frac{p^2}{16 q_1^2} T^{\frac{\theta-2z}{z}} \, ,
\end{equation}
and therefore the leading behavior of the components of the thermoelectric matrix is
\bal
\label{DC-conductivities-N-background-temperature-highT}
\lbar\bbkappa^{ab}_N \sim &\; \frac{ \pi^2}{2 \kappa^2 q_1^2} \, T^{\frac{z-\theta}{z}}\delta^{ab}, \nonumber\\
\bbalpha^{ab}_N\sim&\;-\frac{ \pi q_2}{2 \kappa^2 q_1^2} \, T^{-\frac{2}{z}} \; \delta^{ab} ,\nonumber\\
\bbsigma^{ab}_{N}\sim&\;\frac{1}{2\kappa^2} T^{\frac{2z-2-\theta}{z}} \;\delta^{ab} .
\eal
Notice that while the temperature behavior of $ \lbar\bbkappa^{ab}_N $ and $ \bbalpha^{ab}_N$ is markedly different from their Dirichlet counterparts
$ \lbar\bbkappa^{ab} $ and $ \bbalpha^{ab}_2$ in (\ref{DC-conductivities-D-temperature-highT}),
the electric conductivity $\bbsigma^{ab}_{N} $ in the high temperature limit agrees with the first term in
$\bbsigma_{22}^{ab}$. Thus, the two scale in the same way with temperature in the regime $ q_2^2 << p^2 v_h^{2-\theta}$, as
seen in (\ref{negcase1}). On the other hand, in the regime defined by (\ref{negcase2}) the behavior of
$\bbsigma^{ab}_{N} $ differs from that of $\bbsigma^{ab}_{22} $.
Finally, notice that for the range $1<z<2$, $\theta<0$ of scaling exponents we are working with in this regime,
$\bbsigma_N$ can only scale as a positive power of $T$. As a result, a resistivity which is linear in temperature (or even quadratic) is not allowed in this case, at least not to leading order. This compels us to examine
the low temperature regime.
{\bf Small temperatures:} In the low temperature regime (\ref{lowT1}) the quantity ${\cal N}$ is still given by (\ref{cnlimit}), and we find the following simple scalings as a function of horizon radius,
\bal
\label{DC-conductivities-N-background-lowT1}
\lbar\bbkappa^{ab}_N=&\;\frac{ \pi^2}{2\kappa^2 q_1^2} \, v_h^{-\theta}\, T \; \delta^{ab},\nonumber\\
\bbalpha^{ab}_N=&\;-\frac{\pi q_2}{2\kappa^2 q_1^2} \, v_h^{-2} \; \delta^{ab} ,\nonumber\\
\bbsigma^{ab}_{N}=&\;\frac{1}{2\kappa^2} v_h^{2z-2-\theta}\delta^{ab}.
\eal
While the first two coefficients differ substantially from the analogous expressions in (\ref{DC-conductivities-D-background}), the electrical
conductivity $\bbsigma^{ab}_{N}$ is identical to the first term in $\bbsigma^{ab}_{22}$.
On the other hand, in the low temperature ranges (\ref{lowT2}) and (\ref{lowT3}) the quantity ${\cal N}$ cannot be simplified further, and thus we have to resort to
using the full expression (\ref{DC-conductivities-N-background}), whose structure is highly non-trivial.
Identifying clean scaling regimes analytically for the electrical conductivity in this case would be challenging
and one should resort to numerics.
\item $\theta>0$ case:
{\bf Large temperatures:}
In the high temperature regimes (\ref{poshighT1}) and (\ref{poshighT2}) the rescaling factor ${\cal N} \sim 1$ and therefore
the thermoelectric matrix agrees with the corresponding expressions (\ref{DC-conductivities-D-temperature-highTpositive} and (\ref{DC-conductivities-D-temperature-highTpositive2}) for Dirichlet boundary conditions, if we identify $\lbar\bbkappa^{ab}_N = \lbar\bbkappa^{ab}, \bbalpha^{ab}_N=\bbalpha^{ab}_2, \bbsigma^{ab}_{N} = \bbsigma^{ab}_{22}$.
{\bf Small temperatures:}
At low temperature the thermoelectric matrix simplifies significantly only when
(\ref{poslowT2}) is satisfied, for which we have
\bbepsilon
{\cal N} \sim \frac{p^2}{16 q_1^2} v_h^{\theta-2z} \, .
\end{equation}
In this case we find that (\ref{DC-conductivities-N-background-lowT1}) still describes the
thermoelectric matrix, which we stress once more is very different from its Dirichlet counterpart.
\end{itemize}
To summarize, at large temperatures the difference between the two sets of boundary conditions is most apparent for the case $\theta<0$.
When $\theta>0$ and $T$ is large, we find that ${\cal N} \sim 1$ and thus the components of the thermoelectric matrix for Neumann boundary conditions scale in the same way as the corresponding Dirichlet components.
For a negative value of the hyperscaling violating exponent, on the other hand, the two sets of boundary
conditions yield very different results.
The only partial agreement is for the special case of (\ref{negcase1}), in which the Neumann electric conductivity
$ \bbsigma_{N}$ behaves in the same way with temperature as its Dirichlet counterpart $\bbsigma_{22}$.
As mentioned earlier, because of the complexity of the background solution we work with, the only regime in which we can
analytically estimate the temperature scaling of the thermoelectric matrix is that of large temperatures.
As we already mentioned, an interesting phenomenological question is whether one can obtain a robust explanation for the linear scaling of the resistivity $\rho \sim T$ for the strange metal phase.
In this respect a particularly interesting case is described by the regime (\ref{negcase2}) for $\theta<0$ and
Dirichlet boundary conditions.
In this case all the components of the electrical DC conductivity $\bbsigma_{IJ}$ scale in the same way $\sim T^{(2z-4)/z}$ and in
particular for $z=4/3$ scale as $1/T$, leading to a linear resistivity.
As it turns out, in our analysis this is the only case which could in principle support a linear resistivity,
in the high temperature regime (independently of choices of boundary conditions).
This is intriguing because the special value $z=4/3$ was the one seemingly needed by the scaling arguments and field theoretic analysis of \cite{Hartnoll:2015sea}.
To better understand the significance of this point we would like to generalize this analysis to include a magnetic field, and extend the computation of observables to the Hall angle and the magnetoresistance. This would also complement the recent analysis of \cite{Cremonini:2017qwq,Blauvelt:2017koq}, in which the special values $z=4/3$, $\theta =0$ singled out by \cite{Hartnoll:2015sea} were associated with a minimal DBI holographic model which could be used to reproduce, in the probe limit, the scalings of the resistivity $\rho \sim T$ and
Hall angle $\cot \Theta_H \sim T^2$ of the cuprates \cite{Blauvelt:2017koq}.
Finally, at arbitrary and low temperatures the expressions for the thermoelectric conductivities in the Neumann and Dirichlet cases generically differ from each other -- for both positive and negative values of $\theta$ -- and a more extensive analysis of their temperature
behavior would have to be done numerically.
\section{Concluding remarks}
\label{sec:conclusion}
\setcounter{equation}{0}
In order to holographically identify the conserved currents responsible for the thermoelectric conductivities in a hyperscaling violating Lifshitz theory, it is necessary to place it at the UV, i.e. consider bulk solutions that are asymptotically locally hyperscaling violating Lifshitz, and to construct the physical observables through holographic renormalization. In this paper we have carried out this procedure for linearized fluctuations of the theory \eqref{action-0} around the family of hyperscaling violating Lifshitz backgrounds \eqref{HVbackground}. This analysis was considerably more involved compared to that for relativistic theories because the heat current involves the energy flux, which is an irrelevant operator in the dual Lifshitz theory when $z>1$. In particular, we showed that the boundary counterterms required to renormalize the theory in the presence of a source for the energy flux involve the radial canonical momentum conjugate to the gauge field supporting the Lifshitz asymptotics, in close resemblance to the renormalization of gauge fields in asymptotically AdS$_2$ and AdS$_3$ backgrounds \cite{Cvetic:2016eiv,Erdmenger:2016jjg}. However, if the gauge field supporting the Lifshitz asymptotics in the action \eqref{action-0} is dualized to its magnetic dual, the boundary counterterms would assume a more standard form that does not involve the canonical momenta \cite{An:2016fzu}.
An additional complication in the identification of the dual operators responsible for the thermoelectric conductivities is related to the choice of possible boundary conditions on the bulk fields. It is well known that the field theory dual of a given gravitational theory in the bulk is fully specified only once boundary conditions at infinity are imposed. In particular, the spectrum of local operators and their correlation functions are sensitive to the boundary conditions. Holographic conductivities are no exception. In this paper we have explicitly demonstrated the dependence of physical observables on the boundary conditions by computing analytically the thermoelectric DC conductivity matrix in a hyperscaling violating Lifshitz theory with two different boundary conditions on one of the two Maxwell fields present. Only Dirichlet boundary conditions lead to conductivities that agree with the result of the near horizon analysis \cite{Donos:2014cya}, while Neumann boundary conditions produce a different set of DC conductivities. However, we expect that a near horizon analysis in the theory obtained by dualizing the Maxwell field supporting the Lifshitz background should reproduce the thermoelectric conductivities obtained by imposing Neumann boundary conditions in the original theory. It would be interesting to confirm this explicitly.
Although the thermoelectric conductivities we computed have a rich behavior as functions of the temperature, depending on the various parameters characterizing the background solution, we carried out a preliminary analysis by looking for parameter regions where the DC conductivities exhibit approximate scaling behavior with the temperature. We identified several clean scaling regimes in the limit of large temperature, which in our setup probes the Lifshitz theory in the UV. The thermoelectric conductivities obtained from Dirichlet and Neumann boundary conditions on the Maxwell field supporting the Lifshitz background scale generically differently with temperature. Rather intriguingly, the only case we could identify that can potentially lead to a linear resistivity arises in the case of Dirichlet boundary conditions and the specific value of the dynamical exponent $z=4/3$, which was also singled out in the field theory analysis of \cite{Hartnoll:2015sea}.
The work presented in this paper can be extended in several promising directions. An interesting generalization is to compute the optical (AC) thermoelectric conductivities for the same model and background we consider here. This should be relatively straightforward since the linearized equations we have obtained in appendix \ref{sec:fluctuations} hold for arbitrary frequency, but solving them for general frequency would require extensive numerical analysis. In particular, one can ask what kinds of clean scaling laws can be supported in the intermediate frequency regime, along the lines of the analysis done in \cite{Bhattacharya:2014dea}.
Another obvious extension of the current analysis is to include a background magnetic field, as was done for the model we study here in \cite{Bhatnagar:2017twr}. A much simpler way to obtain the thermoelectric DC conductivities in a dyonic background, however, is to solve the linearized fluctuation equations for a purely electric background as we have done in this paper, and simply impose mixed boundary conditions on the gauge field $A_a^2$. Both procedures are equivalent and place the dual field theory on a dyonic background. In particular, the presence of a background magnetic field will facilitate the evaluation of the Hall angle and the magnetoresistance in the dual Lifshitz theory. These in turn will permit a more detailed comparison of the holographic DC conductivities with experimental observations for the cuprates.
Yet another potential direction to explore is turning on a mass for the bulk gauge field supporting the Lifshitz asymptotics so that the dual Lifshitz theory is characterized by an additional vector exponent \cite{Gouteraux:2012yr}. This again would allow for a more direct comparison with the field theory results of \cite{Hartnoll:2015sea}. Finally, it would be very interesting to revisit part of the asymptotic analysis of \cite{Chemissany:2014xsa} for generic asymptotically locally hyperscaling violating Lifshitz theories in order to include a non-zero source for the energy flux and consider the effect of different boundary conditions on the bulk gauge field. We hope to address some of these questions in the near future.
\section*{Acknowledgments}
S.C. would like to thank Blaise Gouteraux, Richard Davison and Li Li for valuable discussions.
This research is supported in part by the DOE Grant Award DE-SC0013528, (M.C.), the Fay R. and Eugene L. Langberg Endowed Chair (M.C.) and the Slovenian Research Agency (ARRS) (M.C.).
The work of S.C. is supported in part by the National Science Foundation grant PHY-1620169.
M.C. thanks CERN, and I.P. thanks CERN and the University of Pennsylvania, for the hospitality during the completion of this work.
|
1,116,691,497,794 | arxiv | \section{Introduction}
\label{sec:intro}
Instabilities due to magnetic buoyancy are driven by stratified horizontal magnetic fields in compressible plasmas \citep{Newcomb_1961}, and may thus occur in a range of astrophysical settings --- for example, in stars, in accretion disks and in the interstellar medium \citep[see][]{Parker_1979, Choudhuri_1998}. Of particular recent interest is the idea that such instabilities are responsible for the break-up and escape of predominantly toroidal magnetic field from the solar tachocline \citep[see][]{Hughes_2007}, producing the stitches of field that eventually appear at the surface as active regions.
One of the earliest studies of magnetic buoyancy instability was by \cite{Gilman_1970}, who considered the linear instability of a magnetohydrostatic atmosphere with a vertically stratified, horizontal magnetic field. Motivated by the extreme parameter values that pertain astrophysically, Gilman considered the case of an inviscid, perfectly conducting gas in which the thermal relaxation is sufficiently rapid that the temperature can be specified for all times. He anticipated that, under these assumptions, the fastest growing perturbations would be infinitesimally narrow in the horizontal direction perpendicular to the imposed horizontal magnetic field. Interestingly, this approach leads to a `dispersion relation' relating the (possibly complex) frequency to the horizontal wavenumber, but one in which the coefficients are dependent on the vertical coordinate $z$. Thus one may formally associate a different dispersion relation with each height. Mathematically, however, the problem can be posed as a two-point boundary value problem with well-defined (constant) eigenvalues. The aim of this paper is to clarify the connection between these two ostensibly rather different approaches.
The analysis, although new to this problem and, as far as we are aware, to the study of magnetohydrodynamic instabilities, may be regarded as essentially a Rayleigh-Schr\"odinger perturbation analysis exploiting the large transverse horizontal wavenumber. In geophysical fluid dynamics, inertial instabilities of a rotating, stratified flow with arbitrary horizontal cross-stream shear have been analyzed using this technique by \cite{Griffiths_2008}, who considered perturbations that are highly localized in the horizontal cross-stream direction. The alternative to this boundary layer (or internal layer) type approach is a WKB analysis, as used in magnetohydrodynamics by \citet{TP_1996} and \citet{Ogilvie_1998}, who considered highly localized (in radius) magnetic instabilities in accretion disks. Each method has its strengths: on the one hand, WKB analysis is more general; on the other, the boundary layer approach is mathematically simpler and more physically appealing.
The paper is organized as follows. The governing equations describing the magnetic buoyancy instability of a layer of gas with a vertically stratified horizontal magnetic field are set out in Section~\ref{sec:Math_form}, together with the formulation of the problem both as a two-point boundary value problem and as one yielding a `depth-dependent dispersion relation' with growth rate $\sigma(z)$. For simplicity we first restrict attention to two-dimensional (interchange) perturbations, for which the magnetic field remains unidirectional. In Section~\ref{sec:example}, in order to elucidate the key aspects of the analysis, we revisit the problem of the quantum harmonic oscillator, which shares an important common feature with the problem of magnetic buoyancy instability at high wavenumber. In Section~\ref{sec:MBE}, we apply a high wavenumber asymptotic analysis to the governing MHD equation, considering in detail two distinct cases, depending on whether $\sigma(z)$ is maximized strictly within the layer of gas (where $\sigma$ is locally quadratic in $z$) or whether the maximum occurs at the boundary (with $\sigma$ locally linear in $z$). As a result of this analysis we are able to reconcile the idea of a depth-dependent dispersion relation with the solutions of the two-point boundary eigenvalue problem. In Section~\ref{sec:TE} we consider the instability from the different standpoint of an initial value problem, calculating how an initial perturbation evolves with time, and demonstrating the relation of this solution to those derived in Section~\ref{sec:MBE}. In Section~\ref{sec:3D} we show how the high wavenumber analysis carries over to the case of fully three-dimensional perturbations. The concluding discussion is contained in Section~\ref{sec:Disc}.
\section{Mathematical formulation}
\label{sec:Math_form}
Following \cite{Gilman_1970}, we consider magnetic buoyancy instabilities under the assumptions that the gas is isothermal, inviscid and electrically perfectly conducting. The main argument of the paper can be advanced even when the system is rather simple, so here we choose to ignore the effects of rotation. By adopting the layer depth $d$, the free fall time $\sqrt{d/g}$ and the free fall velocity $\sqrt{gd}$ as units of length, time and velocity respectively (where $g$ is the acceleration of gravity, assumed constant), the equations of motion, induction and mass conservation, together with the perfect gas law, take the following dimensionless form:
\begin{equation}
\rho \left( \frac{\partial\mbox{\boldmath$u$}}{\partial t} + \left(\mbox{\boldmath$u$}\cdot\nabla\right) \mbox{\boldmath$u$} \right) = -\mathcal{P}\nabla p - \rho\hat{\mathbf{e}}_{z} + \Lambda\left(\nabla\times\mbox{\boldmath$B$}\right)\times\mbox{\boldmath$B$} ,
\label{eq:NS}
\end{equation}
\begin{equation}
\frac{\partial\mbox{\boldmath$B$}}{\partial t} + \left(\mbox{\boldmath$u$}\cdot\nabla\right)\mbox{\boldmath$B$} = \left(\mbox{\boldmath$B$}\cdot\nabla\right)\mbox{\boldmath$u$} - \mbox{\boldmath$B$} \left(\nabla\cdot\mbox{\boldmath$u$}\right) ,
\label{eq:IND}
\end{equation}
\begin{equation}
\nabla \cdot \mbox{\boldmath$B$} = 0 ,
\label{eq:divB}
\end{equation}
\begin{equation}
\frac{\partial\rho}{\partial t} + \nabla\cdot\left(\rho\mbox{\boldmath$u$}\right)=0 \, ,
\label{eq:MASS}
\end{equation}
\begin{equation}
p=\alpha\rho,
\label{eq:EQSTATE}
\end{equation}
where
\begin{equation}
\mathcal{P} = \frac{p_s}{\rho_s gd}, \quad \alpha = \frac{\rho_s RT_0}{p_s},
\quad
\Lambda=\frac{B_s^2}{\mu_0\rho_s gd}=\beta^{-1} \mathcal{P}.
\label{eq:param_def}
\end{equation}
Here $T_0$ is the constant temperature of the system, $R$ is the gas constant, $\mu_0$ is the permeability of free space, $p_{s}$, $\rho_{s}$ and $B_{s}$ are representative values of the pressure, density and magnetic field respectively. The plasma $\beta$, here defined by
\begin{equation}
\beta = \frac{\mu_0 p_s}{B_s^2} ,
\label{eq:plasma_beta}
\end{equation}
represents the ratio of the gas pressure to twice the magnetic pressure. Additionally, we define the non-dimensional isothermal speed of sound $\mathcal{U}_S=\sqrt{RT_0/gd}$, the non-dimensional Alfv\'en speed $\mathcal{U}_A=B_s/\sqrt{\mu_0\rho_s gd}$ and $\mathcal{U}_p=\sqrt{p_s/\rho_s gd}$. The following relations then hold between the various parameters:
\begin{equation}
\mathcal{P}=\mathcal{U}_{p}^2,\qquad\Lambda=\mathcal{U}_{A}^2,\qquad\alpha=\frac{\mathcal{U}_{S}^2}{\mathcal{U}_{p}^2}.\label{eq:Param_relations}\end{equation}
We now consider a stationary equilibrium layer of gas in the (dimensionless) region $0 \le z \le 1$ with a horizontal, depth-dependent magnetic field,
\begin{equation}
\mbox{\boldmath$B$} = {\bar B} \left( z \right) \mbox{\boldmath$e$}_x.
\label{eq:BS_field}
\end{equation}
The pressure and density of the static basic state are determined by the two coupled equations
\begin{equation}
\mathcal{P} \frac{\mathrm{d} \bar{p}}{\mathrm{d} z} = -\frac{\Lambda}{2} \frac{\mathrm{d} {\bar B}^2}{\mathrm{d} z} - {\bar \rho} ,
\label{eq:isotherm_BS_eq_1}
\end{equation}
\begin{equation}
\bar{p} = \alpha {\bar \rho} ,
\label{eq:isotherm_BS_eq_2}
\end{equation}
from which the density distribution is given by,
\begin{align}
\nonumber
{\bar \rho}(z) = &{\bar \rho}(0) \mathrm{e}^{-z/\calP \alpha} -
\frac{\Lambda}{2 \calP \alpha} \left( {\bar B}^2(z) - {\bar B}^2(0) \mathrm{e}^{-z/\calP \alpha} \right) \\
& + \frac{\Lambda}{2 \calP^2 \alpha^2} \mathrm{e}^{-z/\calP \alpha} \int_0^z \mathrm{e}^{-z/\calP \alpha} {\bar B}^2(z) \mathrm{d} z .
\label{eq:BS_rho}
\end{align}
In order to retain as much simplicity as possible, we shall first assume that the perturbations to the basic state are two-dimensional, varying in the directions perpendicular to the equilibrium magnetic field (\textit{interchange modes}) (three-dimensional perturbations are considered in Section~\ref{sec:3D}). Thus we adopt perturbations of the following form:
\begin{subequations}
\begin{equation}
\mbox{\boldmath$u$} = (0, v(z), w(z)) \mathrm{e}^{\sigma t+ i ky} ,
\label{eq:pert_form_intro_u}
\end{equation}
\begin{equation}
\mbox{\boldmath$b$} = (b_x(z), 0, 0)\mathrm{e}^{\sigma t+ i ky},
\label{eq:pert_form_intro_b}
\end{equation}
\begin{equation}
p = \tilde{p} (z) \, \mathrm{e}^{\sigma t+i ky}\,, \qquad \rho = \tilde{\rho} (z) \, \mathrm{e}^{\sigma t+ i ky}.
\label{eq:pert_form_intro_p_rho}
\end{equation}
\end{subequations}
Introducing these into equations~(\ref{eq:NS}) -- (\ref{eq:EQSTATE}) (equation~(\ref{eq:divB}) is trivially satisfied), with the basic state given by expressions~(\ref{eq:BS_field}), (\ref{eq:isotherm_BS_eq_2}) and (\ref{eq:BS_rho}), and then linearizing, leads to the following set of equations:
\begin{equation}
\sigma{\bar \rho} v=-i k\left(\mathcal{P}\tilde{p} + \Lambda {\bar B} b_x \right) ,
\label{eq:pert_v}
\end{equation}
\begin{equation}
\sigma{\bar \rho} w = -\frac{\mathrm{d}}{\mathrm{d} z} \left( \mathcal{P} \tilde{p} + \Lambda {\bar B} b_x \right) -\tilde{\rho},
\label{eq:pert_w}
\end{equation}
\begin{equation}
\sigma b_x=-i k {\bar B} v - \frac{\mathrm{d}}{\mathrm{d} z}\left( {\bar B} w \right),
\label{eq:pert_b}
\end{equation}
\begin{equation}
\sigma \tilde{\rho} = - i k {\bar \rho} v - \frac{\mathrm{d}}{\mathrm{d} z} \left( {\bar \rho} w \right),
\label{eq:pert_rho}
\end{equation}
\begin{equation}
\tilde{p}=\alpha\tilde{\rho} .
\label{eq:pert_p}
\end{equation}
Equations~(\ref{eq:pert_v}) -- (\ref{eq:pert_p}) can be manipulated to give the following second-order ordinary differential equation for the vertical velocity $w$:
\begin{align}
\sigma^2 {\bar \rho} w &= \frac{k^2 \Lambda {\bar B}^2}{\sigma^2 + k^2 F(z)} \left( \frac{1}{H_\rho} - \frac{1}{H_B} \right) w \nonumber \\
&+ \frac{\sigma^2 {\bar \rho}}{\sigma^2 + k^2 F(z)} \left( w \frac{1}{H_\rho} + \frac{\mathrm{d} w}{\mathrm{d} z} \right) + \nonumber \\
&+ \frac{\mathrm{d}}{\mathrm{d} z} \left( \frac{\sigma^2 {\bar \rho} F(z)}{\sigma^2 + k^2 F(z)} \frac{\mathrm{d} w}{\mathrm{d} z} \right. \nonumber \\
&\left. - \frac{\sigma^2}{\sigma^2 + k^2 F(z)} \left( \Lambda {\bar B}^2 \left( \frac{1}{H_\rho} - \frac{1}{H_B} \right) -{\bar \rho} F(z) \frac{1}{H_\rho} \right) w \right) ,
\label{eq:bvp}
\end{align}
where the ($z$-dependent) inverse scale heights for density and magnetic field are given by $H_\rho^{-1} (z) = {\bar \rho} ^{-1} \mathrm{d} {\bar \rho} / \mathrm{d} z$, $H_B^{-1} (z) = {\bar B}^{-1} \mathrm{d} {\bar B} / \mathrm{d} z$, and where $F(z) = \alpha\mathcal{P} + \Lambda {\bar B}^2 / {\bar \rho}$. Thus the MHD stability problem reduces to a two-point boundary problem, with boundary conditions on the vertical velocity at the horizontal boundaries; the simplest choice is to impose impermeability, namely $w=0$ at $z=0, 1$. The growth rate $\sigma$ is determined as an eigenvalue of the problem. For a general $z$-dependent basic state, equation~(\ref{eq:bvp}) requires a numerical solution.
Gilman's (1970) analysis proceeded by considering the limit as $k \to \infty$ in the governing equations (here (\ref{eq:pert_v}) -- (\ref{eq:pert_p})), but with no explicit assumption about the locality in $z$ of the perturbations. This leads, from (\ref{eq:pert_v}), to the vanishing of the total pressure perturbation, thus giving a direct relation between $b_x$ and $\tilde p$. Similarly, from equation~(\ref{eq:pert_w}), the vanishing of the total pressure perturbation leads to a relation between $w$ and $\tilde \rho$. Thus, using also equation~(\ref{eq:pert_p}), $b_x$, $\tilde \rho$ and $\tilde p$ can all be expressed in terms of $w$. On eliminating the finite product $kv$ between equations~(\ref{eq:pert_b}) and (\ref{eq:pert_rho}), the terms in $\mathrm{d} w/\mathrm{d} z$ disappear. Substitution for $b_x$, $\tilde \rho$ and $\tilde p$ in terms of $w$ then leads to the following expression:
\begin{equation}
\left[ \sigma^2 F(z) + \frac{\Lambda {\bar B}^2}{{\bar \rho}} \left( \frac{1}{H_B} - \frac{1}{H_\rho} \right) \right] w = 0 .
\label{eq:Gilman}
\end{equation}
This is a simplified version of expression~(14) in \cite{Gilman_1970}, excluding the effects of rotation and three-dimensional perturbations. Expression~(\ref{eq:Gilman}) may be regarded as a depth-dependent dispersion relation, in that it implies that the instability growth rate $\sigma$ (a constant) takes a different value at each height $z$. With this interpretation, it leads to the following criterion for instability ($\sigma^2 > 0$):
\begin{equation}
- \frac{\mathrm{d}}{\mathrm{d} z} \ln \left( \frac{{\bar B}}{\rho} \right) > 0.
\label{eq:inst_criterion}
\end{equation}
This is consistent with the result of \cite{Acheson_1979}, obtained under the assumption that perturbations are localized in both the horizontal \textit{and} vertical directions; in the derivation of expression~(\ref{eq:Gilman}), however, there is no explicit assumption of locality in $z$. Expression~(\ref{eq:Gilman}) is clearly consistent with formally letting $k \to \infty$ in~(\ref{eq:bvp}); our aim in this paper is to relate the solutions of the full governing ordinary differential equation~(\ref{eq:bvp}) to expressions~(\ref{eq:Gilman}) and (\ref{eq:inst_criterion}). For later work it will be helpful formally to define the function $\sigma(z)$ by
\begin{equation}
\sigma(z)= \left( \frac{\Lambda {\bar B}^2}{{\bar \rho} F(z)} \left( \frac{1}{H_\rho} - \frac{1}{H_B} \right) \right)^{1/2} .
\label{eq:sigma(z)}
\end{equation}
We shall retain the symbol $\sigma$ to denote the true eigenvalues.
\section{An example problem} \label{sec:example}
In this section we introduce a simplified example problem designed to mimic some of the key properties of the perturbation equations for the magnetic buoyancy instability in the short wavelength limit. We consider a linear PDE, first-order in time and second-order in space ($z$), containing a parameter $k$ to represent the wavenumber of the system. With the introduction of a growth rate $\sigma$ this becomes a second-order ODE analogous to (\ref{eq:bvp}). Just as for the full ODE~(\ref{eq:bvp}), our example reduces to a purely algebraic equation determining $\sigma$ as a function of $z$ in the formal limit of $k\to\infty$ (and where the gradient terms are small in comparison with this limit). The example highlights the meaning of the growth rate in such a limit.
We consider the following PDE for the function $f(z,t)$ on the spatial domain $0 \le z \le 1$:
\begin{equation}
\frac{\partial f}{\partial t} = \left[ \sigma_{\mathrm{max}} - (z-z_{\mathrm{max}})^2 \right] f + \frac{1}{k^2} \frac{\partial ^2 f}{\partial z^2} .
\label{eq:exPDE}
\end{equation}
For simplicity, we impose the boundary conditions $f(0,t) = f(1,t) =0$, although this specific choice is not crucial for the arguments that follow. Equation~(\ref{eq:exPDE}) contains the parameters $\sigma_{\mathrm{max}}$, which we shall assume to be positive, and $z_{\mathrm{max}}$, which we assume satisfies $0 < z_{\mathrm{max}} < 1$.
On expressing the time dependence of the function $f$ as $f\propto e^{\sigma t}$, equation~(\ref{eq:exPDE}) becomes the second-order ODE
\begin{equation}
\frac{\mathrm{d}^2 f}{\mathrm{d} z^2} - k^2 \left[ (\sigma - \sigma_{\mathrm{max}}) + (z - z_{\mathrm{max}})^2 \right] f = 0,
\label{eq:exODE}
\end{equation}
with boundary conditions $f(0)=f(1)=0$. As $k$ is increased, the derivative term in equation~(\ref{eq:exODE}) becomes increasingly unimportant, except in the regions where the coefficient of $f$ could be comparable with the second derivative. In the large $k$ limit described earlier, equation~(\ref{eq:exODE}) reduces to
\begin{equation}
\sigma = \sigma_{\mathrm{max}} - (z-z_{\mathrm{max}})^2,
\label{eq:exGR}
\end{equation}
an equation for the growth rate as a function of $z$. This quadratic function is clearly maximized at $z=z_{\mathrm{max}}$, with $\sigma=\sigma_{\mathrm{max}}$. We might therefore expect to see the fastest growing solutions of equation~(\ref{eq:exODE}) become increasingly localized about $z=z_{\mathrm{max}}$ as $k$ is increased. For the particular equation~(\ref{eq:exODE}) this idea can be put on a rigorous footing.
Equation~(\ref{eq:exODE}) can be recast in the standard form of the parabolic cylinder equation \begin{equation}
\frac{\mathrm{d}^2 f}{\mathrm{d} x^2} - \left( \frac{1}{4} x^2 + a \right) f = 0
\label{eq:exPC}
\end{equation}
by writing $x=(z-z_{\mathrm{max}})\sqrt{2k}$ and $a=k(\sigma-\sigma_{\mathrm{max}})/2$. The boundary conditions are now $f=0$ at both $x_1 = -z_{\mathrm{max}}\sqrt{2k}$ and $x_2 = (1-z_{\mathrm{max}})\sqrt{2k}$.
To determine the permissible values of $a$ we note that $|x_1|, |x_2| \gg 1$ in the large $k$ limit, provided that $z_{\mathrm{max}}\gg k^{-1/2}$ and $1-z_{\mathrm{max}}\gg k^{-1/2}$. The problem is then the familiar one of a quantum harmonic oscillator for the wave function $\psi$ with $\psi \to 0$ as $x \to \pm \infty$. The most general solution to equation~(\ref{eq:exPC}) \citep[see, for example,][]{BenderOrszag} may be expressed in terms of parabolic cylinder functions $D_\nu$ as
\begin{equation}
f(x)=c_1 D_\nu (x) + c_2 D_{-\nu -1}(-ix) ,
\label{eq:qho}
\end{equation}
where $\nu = -1/2-a$. For $|\arg z| < 3 \pi/4$, the asymptotic form of $D_\nu(z)$ as $z \to \infty$ is given by
\begin{align}
\nonumber
D_\nu (z) \sim z^\nu e^{-z^2/4}
&\left( 1 - \frac{\nu(\nu-1)}{2z^2} + \right. \\
&\left. \frac{\nu(\nu-1)(\nu-2)(\nu-3)}{2\times 4 z^4} - \cdots \right) .
\label{eq:D_asymp_1}
\end{align}
Thus $D_{-\nu -1}(-ix)$ grows exponentially as $x \to \infty$ for all $\nu$; hence to satisfy $f \to 0$ as $x \to \infty$ it follows that $c_2=0$. Since $D_\nu (x) \to 0$ as $x \to \infty$, the most general solution of equation~(\ref{eq:qho}) satisfying $f \to 0$ as $x \to \infty$ is $f = c_1 D_\nu(x)$. Now for $3\pi/4 < \arg z < 5 \pi/4$, the asymptotic form of $D_\nu(z)$ as $z \to \infty$ becomes
\begin{align}
\nonumber
D_\nu (z) \sim &- \frac{(2 \pi)^{1/2}}{\Gamma(-\nu)} e^{i \pi \nu} z^{-\nu-1} e^{z^2/4}
\left( 1 + \frac{(\nu+1)(\nu+2)}{2z^2} + \right. \\
& \left. \frac{(\nu+1)(\nu+2)(\nu+3)(\nu+4)}{2 \times 4z^4} + \cdots \right) .
\label{eq:D_asymp_2}
\end{align}
Hence $D_\nu(x)$ grows exponentially as $x \to - \infty$ unless $\Gamma(-\nu)$ is infinite; this occurs only if $\nu$ is a non-negative integer. Therefore the only allowable values of $a$ are $a=-1/2$, $-3/2$, $-5/2, \, \ldots$. When $\nu = n$ $= 0, 1, 2, \ldots$,
\begin{equation}
D_n(x)= \exp(-x^2/4) \textrm{He}_n (x) ,
\label{eq:Dn}
\end{equation}
where $\textrm{He}_n (x)$ is the $n$th-degree Hermite polynomial. The first few $\textrm{He}_n (x)$ are given by $\textrm{He}_0 (x) =1$, $\textrm{He}_1 (x) =x$, $\textrm{He}_2 (x) =x^2-1$, $\textrm{He}_3 (x) =x^3-3x$; in general $\textrm{He}_n (x)$ is even (odd) in $x$ for even (odd) $n$. Thus there is a solution for each $n\in\mathbb{N}\cup\{0\}$, with $n+1$ turning points. As $k$ is increased, the spacing between the turning points decreases and the functions become increasingly localized about $z=z_{\mathrm{max}}$. This is illustrated by Figure~\ref{fig:f_eigenmode}, which shows the eigenfunctions for $n=0$ and $n=1$ for increasing values of $k$.
\begin{figure}
\plotone{a12}
\\
\plotone{a32}
\caption{Eigenfunctions of equation~(\ref{eq:exPC}) for (a) $a=-1/2$, (b) $a=-3/2$, for $k= 10$, $10^3$ and $10^5$. The eigenfunctions become increasingly peaked about $z= z_{\mathrm{max}}=0.4$ as $k$ is increased.}
\label{fig:f_eigenmode}
\end{figure}
\begin{figure}
\plotone{sinpizex}
\\
\plotone{2ndex}
\caption{Temporal evolution of the solution of equation~(\ref{eq:exPDE}) towards the maximally growing $a=-1/2$ eigenfunction for a moderately high wavenumber, $k=10^5$. In (a) the initial condition is $f(z)= \sin \pi z$ and the solution is shown at $t=0$, $20$, $40$ and $60$. In (b) the initial condition is $f(z)= z(1-z)^3$ and the solution is shown at $t=0$, $150$, $300$ and $450$. All solutions are normalized so that $\max(f)=1$.}
\label{fig:f_ivp}
\end{figure}
The eigenvalue analysis above can now be related to the original evolution PDE (\ref{eq:exPDE}). Appealing to the completeness of the eigenfunctions of the linear operator, the solution $f(z,t)$ can be expressed as a sum of exponentially growing eigenfunctions (\ref{eq:Dn}). Thus, from an arbitrary initial condition $f(z,0)$, the long term solution will have the spatial form $D_0(x)$, corresponding to the maximum growth rate of $\sigma = \sigma_{\mathrm{max}} - 1/k$. The eigenfunction is peaked at $x=0$, i.e.\ at $z=z_\mathrm{max}$. This is illustrated by Figure~\ref{fig:f_ivp}, which shows the temporal evolution of the spatial dependence of $f(z,t)$ starting from two initial conditions.
\section{The Magnetic Buoyancy Equation} \label{sec:MBE}
Armed with the findings of Section~\ref{sec:example}, we are now in a position to analyze the eigenvalue problem~(\ref{eq:bvp}) determining the magnetic buoyancy instability. Clearly, just as for equation~(\ref{eq:exODE}), the $k \to \infty$ limit is singular, with the coefficient of the highest (second) derivative (i.e.\ $\sigma^2 {\bar \rho}^2 F(z) / \left( \sigma^2 + k^2 F(z) \right)$) tending to zero. This suggests that the eigenmodes become localized when $k$ becomes large and thus we may use singular perturbation techniques (boundary layer techniques) to solve the eigenvalue problem. For a certain eigenmode and eigenvalue $\sigma$ the main flow equation in the region where the derivatives are of order unity (i.e.\ outside any boundary/internal layer) is given, as in \cite{Gilman_1970}, by expression~(\ref{eq:Gilman}), i.e.\
\begin{equation}
\sigma^2 w = \frac{\Lambda {\bar B}^2}{{\bar \rho} F(z)}
\left( \frac{1}{H_\rho} - \frac{1}{H_B} \right) w .
\label{eq:Lklimitgrate}
\end{equation}
Since $\sigma$ is a constant, whereas $\Lambda {\bar B}^2 \left( H_\rho^{-1} - H_B^{-1} \right) / {\bar \rho} F(z)$ is a function of $z$, the only way to satisfy equation (\ref{eq:Lklimitgrate}) is for $w$ to be zero in the main flow, i.e.\ in the region where the $z$-derivatives of $w$ are negligible. Thus for $k\gg1$, we seek a mode associated with a specific eigenvalue $\sigma$ that is localized in the vicinity of $z=z_0$, defined as the value of $z$ where $\sigma^2 = \Lambda {\bar B}^2 \left( H_\rho^{-1} - H_B^{-1} \right) / {\bar \rho} F(z)$. (Another possibility is for the second derivative of $w$ to be large not only locally, but in a significant part of the fluid domain, i.e.\ that the solutions are strongly oscillatory; see Section~\ref{subsec:no_max}.) Hence we define a scaled length variable by
\begin{equation}
\xi=\frac{z-z_0}{\delta},
\label{eq:var_xi}
\end{equation}
where $\delta(k)$ is a measure of the thickness of the boundary/internal layer. It follows from introducing the new boundary layer variable into equation (\ref{eq:bvp}) and expanding all functions of $\xi$ and the growth rate $\sigma = \sigma_0 + \delta \sigma_1 + \delta^2 \sigma_2 + \ldots$ in powers of $\delta$, that the only distinguished limits possible require $\delta\sim k^{-2/(2+n)}$, with a balance between the second derivative and the terms proportional to $w$ of the form
\begin{subequations}\label{eq:possible_balances}
\begin{equation}
\left[\sigma_0^2 - \frac{\Lambda {\bar B}^2\left( H_\rho^{-1} - H_B^{-1} \right)}{{\bar \rho} F(z)}\right]w = \frac{\sigma_0^2} {\delta^2 k^2}
\frac{\mathrm{d} ^2 w}{\mathrm{d} \xi^2} ,
\label{eq:possible_balances_1}
\end{equation}
\begin{equation}
\left[2\sigma_0\sigma_1-\xi\frac{\mathrm{d} }{\mathrm{d} z}
\left( \frac{\Lambda {\bar B}^2 \left( H_\rho^{-1} - H_B^{-1} \right)}
{{\bar \rho} F(z)}\right)\right]w = \frac{\sigma_0^2}{\delta^3 k^2}
\frac{\mathrm{d} ^2 w}{\mathrm{d} \xi^2} ,
\label{eq:possible_balances_2}
\end{equation}
\begin{equation}
\left[\sigma_1^2 + 2 \sigma_0 \sigma_2 -
\frac{1}{2} \xi^2 \frac{\mathrm{d} ^2}{\mathrm{d} z^2}
\left(\frac{\Lambda {\bar B}^2 \left( H_\rho^{-1} - H_B^{-1} \right)}
{{\bar \rho} F(z) }\right)\right]w
\nonumber
\end{equation}
\begin{equation}
\qquad \qquad \qquad \qquad \qquad \qquad
=\frac{\sigma_0^2}{\delta^4 k^2}
\frac{\mathrm{d} ^2 w}{\mathrm{d} \xi^2} , \quad \textrm{etc.},
\label{eq:possible_balances_3}
\end{equation}
\end{subequations}
where all the functions of $z$ are evaluated at $z=z_0$.
It is important to realize that in equations (\ref{eq:possible_balances}) we have neglected higher order terms; in particular, terms of $\mathrm{O}(\xi\delta)$ in (\ref{eq:possible_balances_1}), of $\mathrm{O}(\xi^2\delta)$ in (\ref{eq:possible_balances_2}) and of $\mathrm{O}(\xi^3\delta)$ in (\ref{eq:possible_balances_3}), which are, of course, negligible only if $\xi$ is not too large. The fact, that these terms become large and non-negligible as $\xi$ becomes comparable with $\delta^{-1}$ means that the perturbation problem is singular and that perturbation series such as $\sigma = \sigma_0 + \delta \sigma_1 + \delta^2\sigma_2 + \ldots$ are strictly asymptotic; thus we limit ourselves to calculation of the few first terms of the series.
Starting with $\sigma=\sigma_0+\mathrm{o}(1)$ and $\delta = k^{-1}$ we obtain
\begin{equation}
\frac{\mathrm{d} ^2w}{\mathrm{d} \xi^2} -
\left[1-\frac{\Lambda {\bar B}^2}{\sigma_0^2{\bar \rho} F}
\left(\frac{1}{H_\rho} - \frac{1}{H_B} \right) \right] w = 0,
\label{eq:IntLeq_1}
\end{equation}
where, again, $H_\rho$, $H_B$, ${\bar B}$, ${\bar \rho}$ and $F(z)$ are evaluated at $z=z_0$. It follows that the only continuous solution with continuous derivatives is $w = \textrm{const.} =W$, with $W$ independent of $\xi$ and
\begin{equation}
\sigma_0^2 = \left. \frac{\Lambda {\bar B}^2}{{\bar \rho} F}
\left( \frac{1}{H_\rho} - \frac{1}{H_B} \right) \right|_{z=z_0}.
\label{eq:sigma0}
\end{equation}
Internal layers must then be introduced in order to match this $w = \textrm{const.}$ solution to $w=0$ in the main flow. Importantly, in the analysis that follows, we assume that the basic state is such that $\sigma_0^2 > 0$, i.e.\ the system is unstable.
At this stage it is important to distinguish between various cases requiring different treatments; we consider these in turn below. In Section~\ref{subsec:max_in_interior} we examine the case where $\sigma$, defined by equation~(\ref{eq:Gilman}), has a local maximum at $z_{\mathrm{max}}$, with $0 < z_{\mathrm{max}} < 1$; i.e.\ the growth rate is maximized strictly within the layer. Section~\ref{subsec:max_at_bdry} considers the case where the growth rate is maximized at the boundary ($z=0$ or $z=1$), analyzing separately the generic case, when to leading order $\sigma$ varies linearly with $z$, and the special case when a true maximum of $\sigma$ happens to occur at the boundary. If we consider the evolution of the instability in terms of an initial value problem then, ultimately, the mode of maximum growth rate will prevail. That said, other modes may be significant at earlier times, depending on the initial perturbation. Thus in Section~\ref{subsec:no_max} and the Appendix we explore the large $k$ asymptotic solutions about a general point in the layer where the growth rate is not maximized.
\subsection{The most unstable modes when $z_0=z_{\mathrm{max}}$}
\label{subsec:max_in_interior}
Here we are interested in the case where the function $\sigma(z)$ defined by equation~(\ref{eq:sigma(z)}) has a quadratic maximum at $z=z_{\mathrm{max}}$, with $0 < z_{\mathrm{max}} < 1$. Thus $\sigma_1 = 0$ and the point $z = z_{\mathrm{max}}$ is surrounded by a layer of thickness $k^{-1/2}$; thus $\delta^{\prime\prime} = k^{-1/2}$. On introducing $\varsigma = (z -z_0) / \delta^{\prime\prime}$ and $\sigma = \sigma_0 + \delta^{\prime\prime^2} \sigma_2 + \mathrm{o}(\delta^{\prime\prime^2})$ (with $\sigma_0$ given by (\ref{eq:sigma0})), equation~(\ref{eq:possible_balances_3}) takes the following form:
\begin{equation}
\frac{\mathrm{d}^2 w}{\mathrm{d} \varsigma^2} - \left[ 2 \frac{\sigma_2}{\sigma_0} - \frac{1}{2} \varsigma^2 \Upsilon \right] w = 0,
\label{eq:IntL_max_eq}
\end{equation}
where
\begin{equation}
\Upsilon=\frac{1}{\sigma_0^2} \left. \frac{\mathrm{d}^2 \sigma^2}{\mathrm{d} z^2} \right|_{z=z_0}<0,
\label{eq:UPSILON_Lmax}
\end{equation}
and where the requirement that $\Upsilon<0$ comes from the assumption that we are considering a maximum (rather than a minimum) of $\sigma_0$ at $z=z_{\mathrm{max}}$. Equation~(\ref{eq:IntL_max_eq}) is readily transformed into the parabolic cylinder equation~(\ref{eq:exPC}) by introducing
\begin{equation}
x = \left( -2 \Upsilon \right)^{1/4} \varsigma, \qquad a=\frac{\sigma_2}{\sigma_0} \sqrt{-\frac{2}{\Upsilon}} \, .
\label{eq:chv_z_max}
\end{equation}
Hence the modes localized in the vicinity of $z_{\mathrm{max}}$ take the same form as in the example problem of Section~\ref{sec:example}, with $\sigma_2$ determined by the second relation in (\ref{eq:chv_z_max}) and by the fact that $a=-n-1/2$ with $n\in\mathbb{N}\cup \{0\}$.
\subsection{The most unstable modes when $z_0 =0$ or $1$} \label{subsec:max_at_bdry}
In the case when the maximum of the function $\sigma(z)$ given by (\ref{eq:sigma(z)}) lies strictly outside the layer, or when the function has no maximum, the fastest growing mode has growth rate $\sigma$ given by its value on the upper or lower boundary, depending whether $\sigma(z)$ is an increasing or decreasing function. Furthermore, since now $\sigma (z-z_0)$ varies linearly in $z-z_0$ (not quadratically as in Section~\ref{subsec:max_in_interior}) then this implies a different scaling for the boundary layer.
Here we take $\delta^{\prime} = k^{-2/3}$, $\zeta=(z-z_0)/\delta^{\prime}$, with $z_0=1$ or $z_0=0$ and $\sigma = \sigma_0 + \delta^{\prime} \sigma_1 + \mathrm{o}(\delta^{\prime})$ (with $\sigma_0$ given by expression~(\ref{eq:sigma0})), which, by the use of equation~(\ref{eq:possible_balances_2}), leads to
\begin{equation}
\frac{\mathrm{d}^2 w} {\mathrm{d} \zeta^2} - \left( 2 \frac{\sigma_1}{\sigma_0} -\zeta \Sigma \right) w = 0,
\label{eq:IntLeq_2}
\end{equation}
where
\begin{equation}
\Sigma = \frac{1}{\sigma_0^2} \left. \frac{\mathrm{d} \sigma^2}{\mathrm{d} z} \right|_{z=z_0}.
\label{eq:SIGMA}
\end{equation}
On introducing the new variable $s = \left( 2\sigma_1/\sigma_0 - \zeta \Sigma \right) \Sigma^{-2/3}$, equation~(\ref{eq:IntLeq_2}) becomes Airy's equation,
\begin{equation}
\frac{\mathrm{d}^2 w}{\mathrm{d} s^2} - s w = 0,
\label{eq:Besseleq_1}
\end{equation}
with solution \citep{AS_1972},
\begin{equation}
w=C_{1}\mathrm{Ai}(s) + C_2\mathrm{Bi}(s),
\label{eq:General_SolB_1}
\end{equation}
where $C_{1}$ and $C_{2}$ are complex constants.
To fix ideas, let us assume that the function $\sigma^2(z)$ is an increasing function of $z$ at $z=z_0$, i.e.\ $\Sigma>0$; thus we concentrate on $z_0 = 1$. The solution in the region $\zeta<0$ takes the form
\begin{align}
w =
C_1 \mathrm{Ai} \left[ \frac{1}{\Sigma^{2/3}}
\left( 2 \frac{\sigma_1}{\sigma_0} - \zeta \Sigma \right) \right] &+ \nonumber \\
C_2 \mathrm{Bi} \Bigg[ \frac{1}{\Sigma^{2/3}} &
\left( 2 \frac{\sigma_1}{\sigma_0} - \zeta \Sigma \right) \Bigg] .
\label{eq:General_SolB_2}
\end{align}
Matching with the main flow solution requires that $w(\zeta \to -\infty)=0$. Since $ \mathrm{Ai} (s)$ decays exponentially and $ \mathrm{Bi} (s)$ grows exponentially as $s \to \infty$, it follows that $C_2$ must be zero and hence
\begin{equation}
w = C_1 \mathrm{Ai} (s) =
C_1 \mathrm{Ai} \left[ \frac{1}{\Sigma^{2/3}}
\left( 2 \frac{\sigma_1}{\sigma_0} - \zeta \Sigma \right) \right].
\label{eq:uz_sol_L2_Hankel}
\end{equation}
Furthermore, solution (\ref{eq:uz_sol_L2_Hankel}) must also satisfy the impermeability condition at the upper boundary, i.e.\ $w(z=1)=0$. Thus
\begin{equation}
\mathrm{Ai} \left( \frac{2}{\Sigma^{2/3}}
\frac{\sigma_1}{\sigma_0} \right) = 0 ,
\label{eq:Constants_L2_2}
\end{equation}
which determines $\sigma_1$ as
\begin{equation}
\sigma_1 = \frac{\mathcal{Z} \sigma_0 \Sigma^{2/3}}{2} ,
\label{eq:sig1_final_value}
\end{equation}
where $\mathcal{Z}<0$ denotes a zero of $\mathrm{Ai}(z)$ (all of the zeros of $\mathrm{Ai}(z)$ lie on the negative real axis, the smallest in magnitude being $-2.33811$).
For completeness, let us also consider the special case when the maximum of the function $\sigma(z)$ in (\ref{eq:sigma(z)}) is located at the boundary, say the upper boundary, $z_0=1$. The thickness of the boundary layer in this case is $\mathrm{O} (k^{-1/2})$; we thus take $\delta^{\prime\prime} \sim k^{-1/2}$ and introduce $\varsigma = (1-z)/\delta^{\prime\prime} > 0$ and $\sigma = \sigma_0 + \delta^{\prime\prime^2} \sigma_2 + \mathrm{o} (\delta^{\prime\prime^2})$ (with $\sigma_0$ given by (\ref{eq:sigma0}) and $\sigma_1=0$). By the use of (\ref{eq:possible_balances_3}) we obtain
\begin{equation}
\frac{\mathrm{d}^2 w}{\mathrm{d} \varsigma^2} - \left( 2 \frac{\sigma_2}{\sigma_0} - \frac{1}{2} \varsigma^2 \Upsilon \right) w = 0,
\label{eq:IntLeq_3}
\end{equation}
where
\begin{equation}
\Upsilon=\frac{1}{\sigma_0^2} \left. \frac{\mathrm{d}^2 \sigma^2}{\mathrm{d} z^2} \right|_{z=1}<0,
\label{eq:UPSILON}
\end{equation}
so that $\sigma(z)$ has a maximum at $z=1$. The solution is most easily expressed in terms of the parabolic cylinder function \citep{AS_1972}
\begin{equation}
w = C D_{-a-1/2} \left[\left( -2 \Upsilon \right)^{1/4} \varsigma \right],
\label{eq:uzsol_L3}
\end{equation}
where
\begin{equation}
a = \frac{\sigma_2}{\sigma_0} \sqrt{-\frac{2}{\Upsilon}},
\label{eq:L3_constant_a}
\end{equation}
$C$ is a constant, and where the second linearly independent solution has been omitted since it blows up exponentially in the main flow. The matching condition with the main flow, $w(\varsigma \to +\infty)=0$,
is satisfied since the asymptotic expansion of (\ref{eq:uzsol_L3}) for $\varsigma\gg1$ takes the form
\begin{equation}
w \sim C \left( -\frac{1}{2\Upsilon} \right)^{\frac{a+1/2}{4}} \frac{1}{\varsigma^{(a+1/2)}} \times
\exp \Bigg[ - \left( -\frac{\Upsilon }{8} \right)^{1/2} \varsigma^2 \Bigg] .
\label{eq:asymptform_L3}
\end{equation}
The impermeability condition at the upper boundary gives
\begin{equation}
D_{-a-1/2} \left(0 \right ) = 0 ,
\label{eq:constants_L3}
\end{equation}
which yields a condition for the permitted values of the parameter $a$ and hence for the correction to the growth rate, $\sigma_2$. Expression~(\ref{eq:constants_L3}) can be written as
\begin{equation}
D_{-a-1/2}(0) = \frac{\sqrt{\pi}}{2^{(2a+1)/4}\Gamma\left(\frac{3}{4}+\frac{1}{2}a\right)} = 0 ,
\label{eq:Uprime_expansion}
\end{equation}
where the requirement that the Gamma function is infinite implies $a=-2n-3/2$, where $n \in \mathbb{N} \cup \{0\}$. Taking $a=-3/2$, i.e.\ the smallest absolute value, gives
\begin{equation}
\sigma_{2} = -\frac{3}{2} \sigma_0 \sqrt{-\frac{\Upsilon}{2}} .
\label{eq:sig2_final_expr}
\end{equation}
\subsection{The modes at $z_0\neq z_{\mathrm{max}}$ within the layer} \label{subsec:no_max}
The preceding analysis has demonstrated that in the $k\gg 1$ regime the fastest growing modes are strongly localized. Although in any temporal (and linear) evolution the mode of maximum growth rate, localized at $z_0= z_{\mathrm{max}}$, will ultimately dominate, initial conditions may be such that other modes prevail for some time. Thus it is of some interest also to consider the general case of eigenmodes associated with growth rates not in the vicinity of any extremum of the function $\sigma(z)$ defined by expression~(\ref{eq:sigma(z)}). In this case, however, it is not possible to produce a boundary layer solution varying strongly only in the locality of an isolated $z_0$ determined by $\sigma = \sigma(z_0)$. Instead, the eigenmodes exhibit strong oscillations throughout the bulk of the layer, a possibility that was noted earlier. This behavior can be captured only through a WKB (physical optics) analysis, which is more general, but considerably more complicated than the analysis of Sections~\ref{subsec:max_in_interior} and \ref{subsec:max_at_bdry}. For completeness we have included the detailed analysis for this case in the Appendix.
\section{Time Evolution}\label{sec:TE}
In Section~\ref{sec:MBE} we addressed the problem of magnetic buoyancy instability as an eigenvalue problem for the growth rate $\sigma$. Here we consider a different approach, solving the governing equations~(\ref{eq:pert_v}) -- (\ref{eq:pert_p}) as an initial value problem, in order to illuminate further what the concept of a growth rate that varies with height means for a physical system. For any given basic state we can determine the growth rate function $\sigma(z)$ in the $k \to \infty$ limit via equation~(\ref{eq:Gilman}). This can then be used to predict the long-term behavior of the perturbation variables for large, finite wavenumbers. Suppose we denote the maximum value of $\sigma$ in the range $0\leq z\leq 1$ by $\sigma_{\mathrm{max}}$, attained at $z=z_{\mathrm{max}}$. We study the temporal evolution of the perturbation variables at high wavenumber ($k=1000$), with the aim of determining two things --- first, whether the eigenfunctions (and in particular, $w$) become localized around $z_{\mathrm{max}}$, and, second, whether they ultimately grow at a rate close to $\sigma_{\mathrm{max}}$. We solve four different problems in order to demonstrate all aspects of the system. Adopting $iv$ as one of the variables, the problem can be solved, without loss of generality, purely in terms of real variables \citep[see][]{HC_1987}.
Equations~(\ref{eq:pert_v}) -- (\ref{eq:pert_p}) (in which we assume $\alpha=1$ for simplicity and replace the growth rate $\sigma$ with $\partial/\partial t$) are solved using a second-order Adams-Bashforth timestep, employing second-order finite differences to approximate spatial derivatives. Although such a timestepping scheme does introduce numerical diffusion, the solutions have proven to be of a satisfactory quality. We impose impermeability boundary conditions, $w=0$ at $z=0$ and $z=1$. Growth rates are calculated by analyzing the growth of the energy in $w$, measured by the integral of $|w|^2/2$ across the layer.
To cover the three main cases as described in Section~\ref{sec:MBE}, three different basic state configurations are considered --- specifically, the cases with $z_\mathrm{max}$ located strictly within the layer, $z_\mathrm{max}$ located at the boundary with $\sigma'(z_{\mathrm{max}})$ non-zero, and $z_\mathrm{max}$ located at the boundary with $\sigma'(z_{\mathrm{max}}) = 0$. For the basic state, we adopt a linearly stratified magnetic field, of the form ${\bar B}(z) = 1+\lambda (1-z)$. Case~1 has parameter values $\Lambda=0.2$, $\lambda=1.35$ and $\mathcal{P}=1.9$, and has maximal growth rate $\sigma_\mathrm{max}=0.310$ at $z_\mathrm{max}=0.698$; Case~2 has $\Lambda=0.3$, $\lambda=1.35$, $\mathcal{P}=1.0$, with $\sigma_\mathrm{max}=0.417$ at $z_\mathrm{max}=1.0$; Case~3 has $\Lambda=0.2$, $\lambda=1.35$, $\mathcal{P}=1.48$, with $\sigma_\mathrm{max}=0.319$ at $z_\mathrm{max}=1.0$. The three graphs of $\sigma$ against $z$ (scaled to be displayed on a single set of axes) are shown in Figure~\ref{fig:theorsigma}.
The first calculation is simply intended to show that, in the long term, the solutions do indeed peak at $z_\mathrm{max}$ and that they grow with a rate close to $\sigma_\mathrm{max}$. We adopt the basic state of Case~1, implying that the growth rate is peaked within the layer; we would thus expect the solutions eventually to be localized close to $z=0.698$. The initial conditions are an `educated guess' based on the results of \citet{HC_1987} for an $O(1)$ value of $k$; $iv=-0.02\cos(\pi z)$, $w=-0.07\sin(\pi z)$, $b_x=-\sin(\pi z)$ and $\rho=0.005\sin(\pi z)$.
\begin{figure}
\plotone{sigmas}
\caption{Growth rate function $\sigma(z)$, normalized by its maximum value, versus $z$ for three sets of parameters. Case~1 (solid line), with $\Lambda=0.2$, $\lambda=1.35$, $\mathcal{P}=1.9$; Case~2 (dotted line), with $\Lambda=0.3$, $\lambda=1.35$, $\mathcal{P}=1.0$; Case~3 (dashed line), with $\Lambda=0.2$, $\lambda=1.35$, $\mathcal{P}=1.48$.}
\label{fig:theorsigma}
\end{figure}
The spatial forms of the velocity, magnetic field and density (normalized so that $\max(|w|)=1$) after $1000$ time units are shown in Figure~\ref{fig:allefns}. The vertical velocity $w$ is peaked at $z=0.677$, close to the $k\to\infty$ value of $z_{\mathrm{max}}=0.698$. The growth rate (calculated at the end of the simulation to ensure that the initial conditions have minimal effect) is $0.309$, also extremely close to the predicted value. In addition to $w$ being peaked close to $z=z_\mathrm{max}$, we see that $b_x$ and $\rho$ are also peaked at $z \approx z_\mathrm{max}$, with $v$ (which is out of phase with the other variables) going through zero.
\begin{figure}
\plotone{wefns}
\caption{Variables $iv$, $w$, $b_x$ and $\rho$ after $1000$ time units, plotted as functions of $z$; $k=1000$, $\Lambda=0.2$, $\lambda=1.35$, $\mathcal{P}=1.9$. The functions are normalized so that $\max(|w|)=1$.}
\label{fig:allefns}
\end{figure}
A second calculation was then performed using the same parameter values but different initial conditions. For such a problem, we anticipate that the initial conditions should have no bearing on the final result, since the fastest growing mode should ultimately come to dominate. Initially, $w$ is sharply peaked at $z=0.12$, so that its peak is displaced far from the region in which we predict the solutions will eventually localize. Figure~\ref{fig:winternal} clearly shows the migration of the peak of the solution for $w$ towards $z=z_{\mathrm{max}}$; after $3000$ time units the peak is located at $z=0.677$. The growth rate (calculated at the end of the simulation) is $0.310$, indicating that the initial conditions do not affect the ultimate growth rate, which is indeed determined by the basic state parameters.
\begin{figure}
\plotone{winternal}
\caption{Vertical velocity $w$ (normalized so that $\max(|w|)=1$) plotted as a function of $z$ at $t=0$ (leftmost plot), $t=750$, $t=1500$, $t=2250$ and $t=3000$ (rightmost plot); $k=1000$, $\Lambda=0.2$, $\lambda=1.35$, $\mathcal{P}=1.9$.}
\label{fig:winternal}
\end{figure}
We now turn our attention to Case~2, in which the true maximum of the growth rate function $\sigma(z)$ is located outside the region $0<z<1$; within the layer, the growth rate is thus maximized at the boundary (in this case at $z=1$). In this situation there is a conflict between satisfying the boundary condition and the fact that the eigenfunction wishes to localize at this point. We again start with initial conditions in which $w$ is peaked at $z=0.12$. Figure~\ref{fig:wexternal} shows the peak of the vertical velocity $w$ moving towards the upper boundary and then becoming increasingly localized with time. As predicted by the analysis of Section~\ref{subsec:max_at_bdry}, the solution in this case adopts the form of an Airy function. Owing to the conflicting effects of the boundary condition and the system's desire to localize the eigenfunctions at the boundary, this is a more challenging numerical problem --- for this reason the calculation was continued for only $1000$ time units. The growth rate of the system is $0.408$, which is still close to $\sigma_{\mathrm{max}}$ but is less accurate than for the calculation of Case~1; after $1000$ time units, the peak of $w$ is located at $z=0.931$.
\begin{figure}
\plotone{wexternal}
\caption{Normalized vertical velocity $w$ at $t=0$ (leftmost plot), $t=250$, $t=500$, $t=750$ and $t=1000$ (rightmost plot); $k=1000$, $\Lambda=0.3$, $\lambda=1.35$, $\mathcal{P}=1.0$.}
\label{fig:wexternal}
\end{figure}
Finally we consider Case~3, with $\sigma(z)$ truly maximized on the boundary (i.e.\ $\sigma(z)$ has vanishing derivative at $z=1$). In this case we expect the localized solution to be a parabolic cylinder function of the type displayed in Figure~\ref{fig:f_eigenmode}(b). Again, there is a competition between satisfaction of the homogeneous boundary condition and the system's desire to localize the eigenfunction at the boundary. Starting with the initial condition peaked at $z=0.12$, the system was evolved for $2000$ time units. The evolution of the solution towards the eigenfunction for $w$ can be seen in Figure~\ref{fig:wedge}; the growth rate towards the end of the simulation is $\sigma=0.316$, with the solution peaked at $z=0.912$.
\begin{figure}
\plotone{wedge}
\caption{Normalized vertical velocity $w$ at $t=0$ (leftmost plot), $t=500$, $t=1000$, $t=1500$ and $t=2000$ (rightmost plot); $k=1000$, $\Lambda=0.2$, $\lambda=1.35$, $\mathcal{P}=1.48$.}
\label{fig:wedge}
\end{figure}
\section{The 3D case}\label{sec:3D}
In this section we extend our earlier analysis so as to consider three-dimensional perturbations to the basic state described in Section~\ref{sec:Math_form}. We first show how the ideas of localized solutions developed for two-dimensional interchange modes carry over to the more general case. We then go on to elucidate the conditions under which two- or three-dimensional modes are preferred.
\subsection{Asymptotic analysis of 3D modes}\label{sec:3dasymp}
We now consider fully 3D perturbations of the form
\begin{subequations}\label{eq:pert_form_3D}
\begin{equation}
\mbox{\boldmath$u$}=\left(u\left(z\right),v\left(z\right),w\left(z\right)\right)\textrm{e}^{\sigma t}\textrm{e}^{i (k_{x} x+k_{y} y)},
\end{equation}
\begin{equation}
\mbox{\boldmath$b$}=\left(b_x\left(z\right),b_y\left(z\right),b_z\left(z\right)\right)\textrm{e}^{\sigma t}\textrm{e}^{i (k_{x} x+k_{y} y)},
\end{equation}
\begin{equation}
p=\tilde{p}\left(z\right)\textrm{e}^{\sigma t}\textrm{e}^{i (k_{x} x+k_{y} y)},\qquad\rho=\tilde{\rho}\left(z\right)\textrm{e}^{\sigma t}\textrm{e}^{i (k_{x} x+k_{y} y)}.\end{equation}
\end{subequations}
Introducing such perturbations into equations (\ref{eq:NS})--(\ref{eq:EQSTATE}) and linearizing about the basic state given by expressions~(\ref{eq:BS_field}), (\ref{eq:isotherm_BS_eq_2}) and (\ref{eq:BS_rho}), leads to the following set of equations:
\begin{equation}
\sigma {\bar \rho} u=-i k_x\mathcal{P}\tilde{p}+\Lambda {\bar B}' b_z,\label{eq:Pert3D_u}
\end{equation}
\begin{equation}
\sigma {\bar \rho} v=-i k_y\left(\mathcal{P}\tilde{p}+\Lambda {\bar B} b_x\right)+i k_x \Lambda {\bar B} b_y,\label{eq:Pert3D_v}
\end{equation}
\begin{equation}
\sigma {\bar \rho} w=-\frac{\mathrm{d}}{\mathrm{d} z}\left(\mathcal{P}\tilde{p}+\Lambda {\bar B} b_x\right)+i k_x \Lambda {\bar B} b_z-\tilde{\rho},\label{eq:Pert3D_w}
\end{equation}
\begin{equation}
\sigma b_x=-\left(i k_y v+\frac{\mathrm{d} w}{\mathrm{d} z}\right){\bar B} -{\bar B}' w,\label{eq:Pert3D_a}
\end{equation}
\begin{equation}
\sigma b_y=i k_x {\bar B} v,\qquad \sigma b_z=i k_x {\bar B} w,\label{eq:Pert3D_bc}
\end{equation}
\begin{equation}
\sigma\tilde{\rho}=-{\bar \rho} \left(i k_x u+ i k_y v\right)-\frac{\mathrm{d}}{\mathrm{d} z}\left({\bar \rho} w\right),\label{eq:Pert3D_mass}
\end{equation}
\begin{equation}
0= i k_x b_x + i k_y b_y + \frac{\mathrm{d} b_z}{\mathrm{d} z},\label{eq:Pert3D_solB}
\end{equation}
\begin{equation}
\tilde{p}=\alpha\tilde{\rho}.\label{eq:Pert3D_state}
\end{equation}
Taking equations~(\ref{eq:Pert3D_u}) -- (\ref{eq:Pert3D_state}), manipulating them to produce a single second-order ODE for $w$, and then applying the $k_y\gg1$ limit, yields (cf.\ equation~(\ref{eq:IntLeq_1}))
\begin{equation}
\frac{\mathrm{d}^2w}{\mathrm{d} z^2} - k_y^2(A_1A_2)^{-1} \left( \sigma^4 + A_1A_3\sigma^2 + A_1A_4 \right) w = 0,
\label{eq:3DODE}
\end{equation}
where
\begin{align*}
A_1 &= \frac{\Lambda{\bar B}^2}{\alpha\calP{\bar \rho}+\Lambda{\bar B}^2},\\
A_2 &= \left(\frac{\sigma^2{\bar \rho}}{\Lambda{\bar B}^2}+k_x^2\right)\left(\sigma^2\left[\alpha\calP+\Lambda\frac{{\bar B}^2}{{\bar \rho}}\right]+\alpha\calP\Lambda\frac{{\bar B}^2}{{\bar \rho}}\right),\\
A_3 &= k_x^2\Lambda\frac{{\bar B}^2}{{\bar \rho}}\left(2\frac{\alpha\calP}{\Lambda}\frac{{\bar \rho}}{{\bar B}^2}+1\right)+\frac{1}{H_B}-\frac{1}{H_\rho}, \\
A_4 &= k_x^2\Lambda\frac{{\bar B}^2}{{\bar \rho}}\left(\alpha\calP k_x^2+\frac{1}{H_B}\right).
\end{align*}
Note that the term proportional to the first derivative in (\ref{eq:3DODE}) has been neglected, since, as in the 2D case, it is always much smaller than the second derivative term, which must be of comparable magnitude to the term proportional to $w$, at least in part of the domain.
As for the interchange modes, letting $k_y \to \infty$ yields a purely algebraic equation for the growth rate, namely
\begin{equation}
\sigma^4 + A_1 A_3 \sigma^2 + A_1 A_4 = 0.
\label{eq:3d_disp}
\end{equation}
Alternatively, this can be derived by writing $\xi=(z-z_0)/\delta$, with $\delta\sim k_y^{-1}$, $\sigma=\sigma_0+\mathrm{o} (1)$, and evaluating the functions $A_1$, $A_2$, $A_3$ and $A_4$ at $z=z_0$. Equation~(\ref{eq:3d_disp}) is of biquadratic form and hence gives two solutions, $\sigma_{+}$ and $\sigma_{-}$, say, for the square of the `depth-dependent growth-rate function', where the subscript indicates which sign is taken in the quadratic formula. Written explicitly, the two roots are
\begin{equation}
\sigma^2_{0,\pm} = -\frac{A_1A_3}{2} \pm \frac{1}{2} \left[ A_1 ( A_1A_3^2-4A_4 ) \right]^{1/2}. \label{eq:sigma3D}
\end{equation}
This defines two functions for the square of the growth rate. We will now assume that the maximal value of $\sigma$ is attained within the layer, at, say, $0 < z_{\mathrm{max}} < 1$, which implies
\begin{equation}
\sigma^2 \frac{\mathrm{d}}{\mathrm{d} z} \left. \left(A_1 A_3 \right) \right|_{z=z_\mathrm{max}} +
\frac{\mathrm{d}}{\mathrm{d} z} \left. \left( A_1 A_4 \right) \right|_{z=z_\mathrm{max}}=0;
\label{eq:maximum_grate_3D_condition}
\end{equation}
for now, it does not matter which root ($\sigma_{+}$ or $\sigma_{-}$) attains this value, which we will call $\sigma_{\mathrm{max}}$. Following the analysis of Section~\ref{subsec:max_in_interior}, we seek solutions localized around $z_0=z_{\mathrm{max}}$, where $\sigma_0=\sigma_{\mathrm{max}}.$ As in the 2D analysis, we expect to find localized solutions on applying the scaling $\delta^{\prime\prime}\sim k_y^{-1/2}$. On writing $\varsigma=(z-z_0)/\delta^{\prime\prime}$, with $\sigma=\sigma_0+\delta^{\prime\prime2}\sigma_2+ \mathrm{o} (\delta^{\prime\prime2})$, equation~(\ref{eq:3DODE}) yields
\begin{align}
&\frac{\mathrm{d}^2w}{\mathrm{d}\varsigma^2} - \frac{1}{\left.(A_1A_2)\right|_{z=z_0}}
\bigg[ 2\sigma_0 \sigma_2\left( 2\sigma_0^2 + \left. \left( A_1 A_3 \right) \right|_{z=z_0} \right) + \bigg.\\
&+ \left. \frac{\varsigma^2}{2} \left( \sigma_0^2 \frac{\mathrm{d}^2}{\mathrm{d} z^2} \left. (A_1 A_3)
\right|_{z=z_0} +\frac{\mathrm{d}^2}{\mathrm{d} z^2}\left.(A_1A_4)\right|_{z=z_0}\right)\right]w=0, \nonumber
\end{align}
which can be recast into the parabolic cylinder equation~(\ref{eq:exPC}) under the coordinate transformation
\begin{align*}
x &=\left[\frac{2(\sigma_0^2\frac{\mathrm{d}^2}{\mathrm{d} z^2}\left.(A_1A_3)\right|_{z=z_0}+\frac{\mathrm{d}^2}{\mathrm{d} z^2}\left.(A_1A_4)\right|_{z=z_0})}{\left.(A_1A_2)\right|_{z=z_0}}\right]^{1/4}\varsigma , \\
a &= \frac{\sigma_0\sigma_2(2\sigma_0^2+\left.(A_1A_3) \right|_{z=z_0})}
{\left. \left( A_1 A_2 \right)^{1/2} \right|_{z=z_0} }\times \\
&\ \ \ \ \ \frac{\sqrt{2}}{\left(\sigma_0^2\frac{\mathrm{d}^2}{\mathrm{d} z^2}\left.(A_1A_3)\right|_{z=z_0}+\frac{\mathrm{d}^2}{\mathrm{d} z^2}\left.(A_1A_4)\right|_{z=z_0}\right)^{1/2}} .
\end{align*}
It exhibits localized solutions in the case where
\begin{equation}
\sigma_0^2\frac{\mathrm{d}^2}{\mathrm{d} z^2}\left.(A_1A_3)\right|_{z=z_0}+\frac{\mathrm{d}^2}{\mathrm{d} z^2}\left.(A_1A_4)\right|_{z=z_0}
\end{equation}
is positive.
Thus the general ideas and analysis of the localized solutions explored in Section~\ref{subsec:max_in_interior} for 2D interchange perturbations carry through, albeit in a more involved fashion, to the general 3D case. Similarly, we anticipate that the 2D results of Sections~\ref{subsec:max_at_bdry} and \ref{subsec:no_max} could be extended to cover the 3D instability.
\subsection{Preferred mode of instability}\label{sec:preferred_mode}
Having derived the depth-dependent dispersion relations for 2D (interchange) and 3D perturbations, it is of interest to tie these together in order to determine the preferred mode of instability, and to validate them against previous magnetic buoyancy analyses. Writing the dispersion relation~(\ref{eq:3d_disp}) in full gives
\begin{align}
0 = &\left(\frac{\alpha\mathcal{P}}{\Lambda}\frac{{\bar \rho}}{{\bar B}^2}+1\right)\sigma^4+\left[k_x^2\Lambda \frac{{\bar B}^2}{{\bar \rho}}\left(2\frac{\alpha\mathcal{P}}{\Lambda}\frac{{\bar \rho}}{{\bar B}^2}+1\right)\right.\nonumber \\
& \bigg.+\frac{1}{H_B}-\frac{1}{H_\rho}\bigg]\sigma^2+k_x^2\Lambda \frac{{\bar B}^2}{{\bar \rho}}\left(\alpha \mathcal{P} k_x^2 +\frac{1}{H_B}\right).
\label{eq:3D_dispersion}
\end{align}
Since magnetic buoyancy instabilities are of interest only for bottom-heavy equilibria, we shall only consider the case of $H_\rho<0$. From the dispersion relation~(\ref{eq:3D_dispersion}) it can be shown that, as expected on physical grounds, instability sets in only as a direct mode, with $\sigma$ passing through zero. Instability to 3D modes occurs provided that \begin{equation}
k_x^2<-\frac{1}{\alpha\mathcal{P }H_B}.
\label{eq:3D_instability}
\end{equation}
Thus long wavelength (in the $x$-direction) modes are favored, and these can be destabilized solely by a decrease in height of the magnetic field ($H_B<0$), rather than the decrease in height of $B/\rho$ required to destabilize interchange modes (equation~(\ref{eq:inst_criterion})). This simply confirms the original results of \citet{Newcomb_1961}, derived from the energy principle of ideal MHD. The physical explanation underlying why 3D modes can be more readily destabilized than interchange modes, despite having to do work against magnetic tension, is discussed at length in \citet{HC_1987}.
It is also of interest to determine the mode of maximum growth rate once the instability criterion (\ref{eq:3D_instability}) is satisfied. We find the wavenumbers $k_x$ at which the function $\sigma^2(k_x)$ achieves extremal values by solving $\mathrm{d} \sigma^2/\mathrm{d} k_x=0$. Calculation of the second derivative $\mathrm{d}^2\sigma^2/\mathrm{d} k_x^2$ then allows us to determine the positive extrema. By fairly simple algebraic manipulations, one can demonstrate that the extremal values of $\sigma^2(k_x)$ are achieved for $k_{0x}=0$ and $k_{1x}$, where the latter is defined by the real roots of the biquadratic equation
\begin{align}
0 = k_{1x}^4 &- \frac{2}{\Lambda}\frac{{\bar \rho}}{{\bar B}^2}\left[\frac{1}{H_B}+\frac{1}{H_\rho}\left(2\frac{\alpha\mathcal{P}}{\Lambda}\frac{{\bar \rho} }{{\bar B}^2}+1\right)\right]k_{1x}^2 \nonumber \\
& + \frac{1}{\Lambda^2H_B}\frac{{\bar \rho} ^2}{{\bar B}^4}\left[\frac{1}{H_B}-\frac{1}{H_\rho}\left(2+\frac{\Lambda}{\alpha\mathcal{P}}\frac{{\bar B}^2}{{\bar \rho} }\right)\right],\label{eq:k_at_extremum}
\end{align}
where all the functions are evaluated at a fixed $z=z_0$; thus $k_{1x}$ depends on height. For sufficiently strong field gradients, i.e.\ for
\begin{equation}
-H_B^{-1}>-\left(2+\chi\right)H_\rho^{-1},
\label{strong_Xi}
\end{equation}
where
\begin{equation}
\chi=\frac{\Lambda}{\alpha\mathcal{P}}\frac{{\bar B}^2}{{\bar \rho} },
\label{chi}
\end{equation}
the growth rate is maximized at $k_{0x}=0$ and
\begin{equation}
\sigma_{\mathrm{max}}^2 = \frac{\Lambda {\bar B}^2}{{\bar \rho} \left(\alpha\mathcal{P} + \Lambda {\bar B}^2/{\bar \rho} \right)} \left(\frac{1}{H_\rho} - \frac{1}{H_B} \right),
\label{sigma_at_strong_Xi}
\end{equation}
in agreement with (\ref{eq:sigma0}). In other words, for sufficiently strong field gradients, interchange modes possess the largest growth rate. For $-H_\rho^{-1} < -H_B^{-1} < -(2+\chi)H_\rho^{-1}$, the growth rate $\sigma$ has a local minimum at $k_{0x} = 0$; in this range, $\sigma^2(k_x)$ is maximized at $k_x = k_{1x}$. For $0< -H_B^{-1} < -H_\rho^{-1}$, $\sigma^2$ given by expression~(\ref{sigma_at_strong_Xi}) is no longer positive, but the maximum of $\sigma^2(k_x)$ at $k_{1x}$ remains positive. These findings are in keeping with those of \citet{Hughes_1985}, who examined magnetic buoyancy instabilities under the magneto-Boussinesq approximation.
Finally, since $\chi\sim (\alpha\beta)^{-1}$, we note that this parameter would become negligibly small in the parameter regime of stellar interiors ($\beta\gg1$, $\alpha\sim 1$). The above results then simplify to saying that for $-H_B^{-1}\gtrsim-2H_\rho^{-1}$ the dominant perturbations are two-dimensional interchange modes (the growth rate has a maximum at $k_x=0$), whereas for $-H_B^{-1} \lesssim -2 H_\rho^{-1}$, finite wavelengths in the $x$ direction are preferred (growth rate maximized at $k_{1x}$).
\section{Discussion}\label{sec:Disc}
Through the use of a Rayleigh-Schr\"odinger perturbation analysis to exploit the large wavenumber in the horizontal transverse direction, we have revisited the problem introduced in the pioneering works of \citet{Gilman_1970} and \cite{Acheson_1979} of the magnetic buoyancy instability in a plane layer of fluid permeated by an external horizontal magnetic field decreasing with height. Such an approach is simpler and more physically appealing than a WKB analysis, although is less general in its scope. We have studied in detail the instability of interchange modes (no bending of the field lines) and, consequently, have been able to provide a thorough mathematical explanation of the results of \citet{Gilman_1970}, who obtained instability growth rates as functions of the vertical coordinate $z$. Via an asymptotic analysis of the governing equations, we have shown how the fastest growing eigenmodes may become strongly localized in the vertical direction. The growth rate function $\sigma(z)$ is maximized either strictly within the layer (at $0 < z_\mathrm{max} < 1$, say) or else at the boundary of the domain. In the former case, the mode of maximum growth rate is localized at $z_\mathrm{max}$ and adopts the form of a parabolic cylinder function. In the latter case, the eigenfunction is localized close to the boundary and takes the form of an Airy function. The modes associated with growth rates that are less than maximal exhibit strong oscillations; these can be captured only via a WKB approach, as presented in the Appendix.
We have confirmed the results of the asymptotic analysis by numerical simulations of the time evolution of the perturbations, posed as an initial value problem. These clearly show that, starting from an initial condition composed of a set of eigenmodes, the mode that eventually dominates the evolution, in the large $k$ limit, is that with growth rate corresponding to the maximum of the growth rate function $\sigma(z)$. Furthermore, this mode is localized in the vicinity of $z_{\mathrm{max}}$, the height in the layer at which $\sigma(z)$ is maximized.
The asymptotic analysis carries over, albeit with more algebraic complications, to the case of three-dimensional perturbations that are infinitesimally thin in the horizontal direction perpendicular to the imposed field, but have a finite (though long) wavelength along the field. From an analysis of the dispersion relation~(\ref{eq:3D_dispersion}), we have shown how 3D perturbations are the only unstable modes for weakly unstable magnetic field gradients, but that for sufficiently strong field gradients, interchange modes have the largest growth rate.
It is of interest to consider what the analysis tells us about the specific physics of the magnetic buoyancy instability and, more generally, to enquire into the wider class of instabilities that can be so analyzed. Magnetic buoyancy instabilities are driven by a destabilizing magnetic field gradient and inhibited by a stabilizing entropy gradient. The latter is eroded by a combination of high thermal diffusivity and small wavelengths transverse to the imposed field; in the limiting case we have considered here, the thermal diffusion tends to infinity and the wavelength to zero. The crucial feature revealed by the analysis of Section~\ref{sec:MBE} is that the assumption of locality in the horizontal direction implies locality in the vertical. For a case when the growth rate function $\sigma(z)$ is positive in only a small section of the layer, it is perhaps not surprising that the eigenfunction of the fastest growing mode is peaked in that region; the important feature revealed here is that the fastest growing eigenfunction is always peaked where $\sigma(z)$ is maximized, even when $\sigma(z)$ is positive throughout the layer.
In a wider context, is it possible to identify, simply from general considerations, other instabilities that can be analyzed within the same framework? Certainly two necessary conditions can be identified: one is a physical reason for an instability to have a small wavelength in one specific direction; the other is an inhomogeneity of the basic state in an orthogonal direction. For magnetic buoyancy instability as discussed here, the small wavelength in the $y$-direction arises from the erosion of the stabilizing gradient through thermal diffusion, whereas the inhomogeneity in the vertical direction is a consequence of the $z$-dependence of the basic state magnetic field and thermodynamic variables. In a Boussinesq atmosphere, in which stratification is uniform with depth, we would therefore not expect any localization in the vertical --- this is consistent with the magneto-Boussinesq analysis of \cite{Hughes_1985}. For the very different problem of the instability of a rotating, stratified flow with arbitrary horizontal cross-stream shear, considered by \cite{Griffiths_2008}, the assumed small scale is in the vertical, resulting from the stable stratification, with the inhomogeneity resulting from the shear flow.
In the present analysis we have neglected viscosity and magnetic diffusivity and assumed infinitely fast thermal diffusion. The effects of finite diffusion will of course influence the dynamics at sufficiently large $k$, and will act to establish a finite magnitude of the wave vector for unstable modes. Thermal diffusion will, however, still greatly dominate and thus, although consideration of the full dynamics will lead to modification of the quantitative results, we expect that the short-wavelength nature in the horizontal direction perpendicular to the field and the strong localization in the vertical direction of the most unstable perturbations to persist.
\acknowledgments
We are grateful to Douglas Gough, Stephen Griffiths, Chris Jones, Michael Proctor and Steve Tobias for valuable discussions. This work was funded by the STFC rolling grant held at the University of Leeds.
|
1,116,691,497,795 | arxiv | \section{Introduction}
Faster-than-Nyquist (FTN) signaling \cite{FTN_Mazo,Anderson_FTN,Detection-li2018superposition} has long been one of the promising communication paradigms for future high data rate wireless networks.
Different from the conventional methods of enhancing the data rate, which normally requires more time/bandwidth/spatial resources, FTN signaling
enhances the spectral efficiency by intentionally transmitting the symbols faster than the Nyquist rate.
More importantly, FTN signaling is able to achieve the ultimate capacity for the signal power spectral density (PSD) \cite{Capacity_FTN_1}.
Therefore, FTN signaling has been widely proposed for various communication applications, such as satellite communications \cite{TFP}, and beyond 5G communications \cite{towards2019lee, yuan2020iterative}.
A major drawback of FTN signaling is that the higher symbol rate induces inevitable and severe intersymbol interference (ISI) at the transmitter side. Consequently, a high complexity detector is usually required at the receiver side \cite{Anderson_FTN,TE_First,TE_Journal,yuan2019iterative}. For example, the number of trellis states in a Bahl-Cocke-Jelinek-Raviv (BCJR) detector increases exponentially with the constellation size and number of ISI taps.
Moreover, for a coded FTN system, Turbo equalization is usually applied at the receiver, where iterations are performed between the BCJR detector and channel decoder. The overall detection/decoding complexity further increases with respect to the number of iterations.
Therefore, designing practical detectors with reduced-complexity is a major research topic for FTN signaling \cite{Shuangyang,ReducedMBCJR,MBCJR_Covalope}.
For example, two M-algorithm BCJR (M-BCJR) detectors were proposed for FTN signaling in \cite{Shuangyang} based on the Ungerboeck observation model \cite{Ungerboeck}, and they show promising error performance for coded FTN systems by applying Turbo equalization.
Other than the BCJR algorithm, sum-product algorithms (SPA) have also been recognized as an efficient method to compute the marginal probabilities with a low complexity \cite{Kschischang2019factor} and it has been
widely used in the decoding of channel codes \cite{DE_LDPC} and signal processing \cite{AMP}. A soft-input soft-output (SISO) detection algorithm for linear ISI channels was proposed in \cite{SISO}, where the sum-product algorithm is applied to a factor graph (FG) of the coded FTN system.
The complexity of the algorithm is linear to the number of interferers during each iteration.
However, reduced-complexity detection algorithms usually undermine the error performance.
For instance, there are mainly two different aspects that may contribute to the performance loss if we apply the SISO algorithm in \cite{SISO} for FTN detection.
Firstly, due to the complexity limitation, the SISO algorithm can only consider a limited number of ISI taps.
Note that the number of ISI taps of FTN signaling can be infinite in theory \cite{Anderson_FTN}. Therefore,
the unconsidered residual ISI will degrade the error performance of the system.
Secondly, the FG generated from the Ungerboeck observation model in \cite{SISO} has shortest cycles of length-6 and the cycles may accumulate the correlation between the messages during the iterations of detection. This accumulated correlation is difficult to be predicted by mathematical models and can undermine the error performance.
In light of these two aspects, we consider to utilize a neural network (NN) to compensate for the performance loss of the SISO algorithm for FTN detection.
Recently, deep learning supplemented communication systems have shown the potential to further enhance the system's performance \cite{DL_Comm_1,DL_Comm_2,DL_Comm_4,DL_Comm_3, Bryan_ITW2019}. In particular, for the detection and decoding algorithms, the research on autoencoders \cite{Autoencoder_1} and the NN optimization schemes which transform the FGs into NN systems \cite{DLLC, BP_adapt}, has drawn significant interests.
For the autoencoders, an NN system with multiple layers is employed and trained to overcome the issues such as multipath interferences and signal distortions \cite{Autoencoder_1}.
However, since the connection among the multiple layers of the NN model does not rely on the mathematical models of the channel, the NN usually needs a large number of training samples. Generally, more than $2^K$ samples are needed to converge to a good performance \cite{DLSTB}, where $K$ is the information sequence length.
On the other hand, the "unfolded" NN detection or decoding algorithms take advantage of the well-developed channel models \cite{DLLC}, which lead to specific NN connections. However, the flexibility of the NN designs is neglected. Such an NN can only optimize the performance based on a specific graph, which may not lead to the globally optimized performance.
Other than the autoencoders, NN aided algorithms for Turbo equalization systems have also been reported in the literature.
In \cite{TENN}, a deep NN (DNN) was proposed for Turbo equalization by assuming that the log-likelihood ratios (LLRs) of the extrinsic information from the decoder follow a consistent Gaussian distribution with a variance computed by the inverse $J$-function as introduced in \cite{EXIT}.
However, the DNN in \cite{TENN} is constructed by a fully-connected network that has inputs of both the decoder's extrinsic information and the channel information, which indicates that the extrinsic information from the decoder is fed into the NN separately from the channel information. Consequently, this may largely increase the number of neurons in the NN, leading to a high training complexity.
In this paper, we propose an NN assisted approach to compensate the performance loss for the SISO algorithm (referred to as the sum-product detection algorithm (SPDA) thereafter) proposed in \cite{SISO} for detecting coded FTN signals.
We modify the FG of the SPDA by connecting an arbitrary NN to the variable nodes (VNs) via additional function nodes (FNs), where tunable parameters over the edges of the FG are optimized through training.
The proposed algorithm is constructed from a neural network system and it can be trained off-line. Therefore, we name it as deep learning assisted sum-product detection algorithm (DL-SPDA). The inaccuracy of the messages over the edges of the conventional FG due to the limited number of ISI taps and short cycles is expected to be compensated by the NN.
The main contributions of this paper are summarized as follows.
\begin{itemize}
\item We propose a novel DL-SPDA for FTN detection in a modified FG, where an arbitrary NN is concatenated to the original FG of the FTN signaling. We also introduce a new message updating rule to facilitate the application of the proposed DL-SPDA in Turbo equalization receivers. Consequently, the proposed DL-SPDA can be directly applied to a Turbo equalization without considering any particular channel decoding algorithms.
\item We propose a new compatible training technique for NN that is specifically designed for Turbo equalization receivers. The compatible training technique combines the channel information and the extrinsic information from the decoder before training. The extrinsic information is sampled from a consistent Gaussian distribution, where the variance is computed by the inverse $J$-function \cite{EXIT}.
As a result, the NN is more robust to the variations of extrinsic information from the actual decoder while the training complexity is acceptable.
\item Besides demonstrating that the proposed DL-SPDA can obtain a BER performance gain compared to the conventional FG based FTN detection algorithm, we show the computational complexity analysis of the proposed NN structure.
\item We also derive the theoretical bit error rate (BER) performances for convolutionally encoded FTN systems in order to demonstrate the effectiveness of the proposed DL-SPDA. Specifically, we focus on the case where the detector is imperfect in the sense that only a limited number of ISI taps is considered. By applying approximations, we show that the derived analytical BERs are accurate even for a short block with 250 bits. Meanwhile, the derived analytical BERs also verify that the proposed DL-SPDA approaches the maximum-likelihood (ML) performance.
\end{itemize}
The rest of this paper is organized as follows. In the next section, we will briefly introduce the system model and review the sum-product detection algorithm proposed in \cite{SISO}. Then, in Section \Romannum{3}, we will detail the proposed DL-SPDA, including the new FG model, the modified message updating rules and the compatible training method. The performance analysis for the convolutionally encoded FTN system is discussed in Section \Romannum{4}. The numerical results for both the proposed detection algorithm and the analytical estimates are depicted in Section \Romannum{5} and lastly, the concluding remarks are given in Section \Romannum{6}.
\color{black}
\begin{figure}[t!]
\centering
\includegraphics[width=130mm]{Turbo_Equalization.eps}
\caption{System model.}
\label{TE}
\end{figure}
\section{Preliminaries}
\subsection{System model}
Without loss of generality, the considered coded FTN system model is shown in Fig. \ref{TE}. Let $\bm{b}$ denote the binary source data with length $K$.
At the transmitter, $\bm{b}$ is convolutionally encoded, resulting in a binary codeword $\bm{c}$ of length $N$.
A sequence of $N$ binary phase-shift keying (BPSK) symbols $\mathbf{x} = [x_1, x_2, ..., x_N]^\text{T}$ is generated after interleaving the bits in $\bm{c}$ with $x_i = (-1)^{c_i}$ for $i \in \{1,..., N\}$.
The transmitted FTN signals are linear modulation signals of the form $s(t) = \sum_n{x_n}{h(t-n\tau T)}$, where $\tau$ is the time acceleration factor of the FTN signaling \cite{FTN_Mazo} and $h(t)$ is a $T$-orthogonal root raised cosine pulse with a roll-off factor $\alpha$.
Assume that the channel is modelled by additive white Gaussian noises (AWGN) with a zero mean and a variance of $\sigma^2$.
The received sequence $\bm{y}$ after a matched filtering and FTN rate sampling is given by $\bm{y} = \mathbf{G}\mathbf{x} + \bm{\eta}$, where $\mathbf{G}$ is a Toeplitz generator matrix, as shown below,
\begin{equation}
\bf{G}=
\begin{pmatrix}
{{g_0}} & {{g_{1}}} & \cdots & {{g_{N-1}}}\\
{{g_{-1}}} & {{g_0}} & \cdots & {{g_{N-2}}}\\
\vdots & \vdots & \ddots & \vdots \\
{{g_{1-N}}} & {{g_{2-N}}} & \cdots & {{g_0}}
\end{pmatrix}.
\end{equation}
The generator matrix consists ISI taps ${g_i} = \int_{ - \infty }^\infty {h\left( t \right){h^*}\left( {t - i\tau T} \right){\rm{d}}t} $. Specifically, we define $L$ as the number of channel responses with significant energy. The rest ISI taps with insignificant energy are therefore negligible and then set to zeros for simplicity, i.e., ${g_i} = 0$, for $|i| >L$.
Meanwhile, the term $\bm{\eta}$ represents the colored noise samples whose autocorrelation matrix is $\mathbb{E}[\bm{\eta} {\bm{\eta}}^{\rm{H}}] = \sigma^2\mathbf{G}$.
Once the sequence $\bm{y}$ is observed, the receiver performs the Turbo equalization,
where the extrinsic information from the detector and decoder is exchanged iteratively via the interleaver $\mathbf{\Pi}$ or the deinterleaver ${\mathbf{\Pi} ^{ - 1}}$
until the maximum iteration number is reached. The sequence \(\hat {\bm{b}}\) as the estimate of $\bm{b}$ is generated after the Turbo equalization iterations, which is regarded as the output for the receiver.
\subsection{Sum-product detection algorithm}
In this subsection, we briefly review the SPDA proposed in \cite{SISO}. Given the received sequence $\bm{y}$, the SPDA factorizes the $a \ posterior$ probabilities (APPs) $P(\bm{x}|\bm{y})$ of the transmitted sequence $\bm{x}$ mainly based on three types of FNs:
$\bullet$ $O_i(x_i)$ for $i \in \{1,...N\}$: The $a \ priori$ probability for symbol $x_i$ being transmitted.
$\bullet$ $T_i(x_i)$ for $i \in \{1,...N\}$: The symbol likelihood function for symbol $x_i$ being transmitted based on the received symbol $y_i$.
$\bullet$ $I_{i,j}(x_i, x_j)$ for $j \in \{1,...N\}$ and $i \in \{j,...N\}$: The FN that conveys the messages from node $i$ to the interfering node $j$.
The functions of $T_i(x_i)$ and $I_{i,j}(x_i, x_j)$ are defined as \cite{SISO}:
\begin{gather}\label{T}
T_i(x_i) = \text{exp}\bigg{[}\frac{1}{\sigma^2}\text{Re}\bigg{\{}y_ix_i^*-\frac{{\bf{G}}_{i,i}}{2}|x_i|^2\bigg{\}}\bigg{]},\vspace{-5mm}
\vspace{-3mm}
\end{gather}
\begin{gather}\label{I}
I_{i,j}(x_i, x_j) = \text{exp}\bigg{[}-\frac{1}{\sigma^2}\text{Re}\big{\{}{\bf{G}}_{i,j}x_ix_j^*\big{\}}\bigg{]},
\end{gather}
where $x_i^*$ refers to the conjugate of the symbol $x_i$, $\text{Re}\{ \cdot \}$ represents the function that returns the real part of a value, and ${\bf{G}}_{i,j}=g_{i-j}$ is the ${(i-j)}$-th ISI tap. It is derived in \cite{SISO} that $P(\bm{x}|\bm{y}) \propto \prod_{i=1}^{N}\bigg{[}O_i(x_i)T_i(x_i)\prod_{j<i}I_{i,j}(x_i,x_j) \bigg{]}$.
\begin{figure}[t!]
\centering
\includegraphics[width=90mm]{SPDA.eps}
\vspace{-7mm}
\caption{Message updating in the factor graph of SPDA.}
\label{SPDA_Graph}
\vspace{-4mm}
\end{figure}
Define $q_{i,j}(x_i)$ as the message from the FN $I_{i,j}$ to the VN $x_i$, $p_{i,j}(x_i)$ as the message from the VN $x_i$ to the FN $I_{i,j}(x_i,x_j)$, $o(x_i)$ as the message from the VN $x_i$ to the FN $O_i$, and $Q_i(x_i)$ as the product of all messages incoming to the VN $x_i$, respectively. Here, $Q_i(x_i)$ indicates the proportional probability of the (approximated) APP $P(x_i|\bm{y})$ \cite{SISO}.
An example of part of the FG is given in Fig. \ref{SPDA_Graph}, where the messages are updated according to \cite{SISO}:
\begin{gather}\label{Q_update}
Q_i(x_i) = O_i(x_i)T_i(x_i)\prod_{j \neq i}{q_{i,j}(x_i)},
\end{gather}
\begin{gather}\label{o_update}
o_i(x_i) = \frac{Q_i(x_i)}{O_i(x_i)},
\end{gather}
\begin{gather}\label{p_update}
p_{i,j}(x_i) = \frac{Q_i(x_i)}{q_{i,j}(x_i)},
\end{gather}
\begin{gather}\label{q_update}
q_{i,j}(x_i) = \sum_{x_j}I_{i,j}(x_i, x_j)p_{j,i}(x_j)
\end{gather}
The messages $\{p_{i,j}\}$ and $\{q_{i,j}\}$ are initialized to the same positive values and the messages are updated iteratively until the maximum number of iterations is reached.
Note that the detection complexity of the SPDA is linear to the degree of VNs, i.e., the number of the FNs $I_{i,j}(x_i)$ linked to each VN $x_i$.
Therefore, the SPDA provides a trade-off between the detection performance and detection complexity by intentionally choosing the degree $L_E$ of VNs, where the VN degree $L_E \le L$ corresponds to the number of ISI taps considered by the SPDA. On the other hand, as we will discuss in Section \Romannum{4}, the coded FTN system's performance is restrained by the number of ISI responses considered by the detector. If only a small number of ISI responses is considered by the detector, an error floor may occur in the high signal-to-noise ratio (SNR) region due to the existence of residual ISI \cite{SY_WJ_FTN}.
It should also be noted that the short cycles contained in the FG will accumulate the correlation between the messages during the iterative update, which may undermine the overall error performance \cite{MCT}. Therefore, to compensate for the potential performance degradation, we propose a deep learning assisted SPDA, the details of which will be discussed in the following section.
\section{Deep Learning Assisted Sum-Product Detection Algorithm}
In this section, the proposed DL-SPDA is introduced.
The basic idea of the proposed DL-SPDA is to transform the SPDA into an NN with additional tuneable multiplicative weights and neuron FNs nested to the VNs.
More specifically, we modify the FG by connecting an NN FN on top of the original FG and then unfold the modified FG into an NN system for training.
Note that the NN FN is directly connected to all VNs.
Therefore, the influence of the residual ISI and short cycles can be compensated by the NN.
Furthermore, we also modify the message-passing rules to make the DL-SPDA be suitable for Turbo equalization receivers. This allows us to train the DL-SPDA without the prior knowledge of the decoder.
Moreover, we propose a compatible training method to improve the performance of the DL-SPDA for Turbo equalization receivers. With the compatible training method, the NN is optimized in terms of the mutual information between the extrinsic information and the transmitted sequence with an acceptable training complexity.
\color{black}
\subsection{New FG model and modified message updating rule}
Conventional NN assisted detection or decoding algorithms introduce tuneable multiplicative weights to the FG and then unfold the message-passing algorithm to an NN system for training and optimization \cite{DLLC}. In terms of the SPDA, trainable weights can be attached to the messages $p_{i,j}(x_i)$ in the corresponding FG, so the updating rule in (\ref{q_update}) is modified as:
\begin{gather}\label{q_update_new}
q_{i,j}(x_i) = \sum_{x_j}I_{i,j}(x_i, x_j)\varsigma_{j,i}p_{j,i}(x_j),
\end{gather}
where $\varsigma_{j,i}$ is the weight attached to the message $p_{j,i}(x_j)$ in the corresponding FG.
During the off-line training, since $I_{i,j}(x_i, x_j)$ is the interfering node that is irrelevant to the message $p_{j,i}(x_j)$, $I_{i,j}(x_i, x_j)\varsigma_{j,i}$ is treated as the trainable parameters in the neural network.
Training with the additional weights has shown to improve the sum-product algorithm's performance in a high SNR region \cite{DLLC}.
However, merely tuning the additional weight $\varsigma_{j,i}$ attached to the message $p_{j,i}(x_j)$ will not change the connections in the FG. Therefore, the performance improvement by attaching tuneable weights to the NN is limited.
To this end, we propose to connect an NN FN $\Phi(x_1,...,x_N)$ to the VNs in the FG to compensate for the effects of the residual ISI responses and the correlation induced along the short cycles.
As shown in Fig. \ref{DL-SPDA}, different from the traditional FG, we nest an NN FN to the VNs $x_i$ of the FG, for $i \in \{1,...,N\}$.
The aim of nesting an NN to the FG is two-fold:
$\bullet$ The NN connects to all the VNs. It is expected that all the ISI components among the VNs are considered by the NN.
$\bullet$ The correlation induced during the iteration is expected to be compensated by the NN. The APPs computation for all the VNs in each iteration can be optimized after tuning the parameters in the NN.
Define $u_i(x_i)$ as the message from the variable node $x_i$ to the FN $\Phi(x_1,...,x_N)$ and $v_i(x_i)$ as the message from the FN $\Phi(x_1,...,x_N)$ to the variable node $x_i$ as shown in Fig. \ref{DL-SPDA}.
The conventional sum-product algorithm sums all the intrinsic information for each variable node before passing the extrinsic information to the FN for further processing. This indicates that in a conventional sum-product algorithm, $u_i(x_i)=O_i(x_i)T_i(x_i)\prod_{j \neq i}{q_{i,j}(x_i)}$.
However, in terms of Turbo equalization, the extrinsic information from the decoder will be passed to the detector.
Therefore, the NN needs to be trained with respect to the specific decoder, i.e., the decoding algorithm becomes part of the NN, so that the global optimality can be obtained.
Nevertheless, optimizing the NN which consists of both the detector and decoder has two major problems.
Firstly, the training complexity will be largely increased if the decoder is also included by the NN layers.
Secondly, the optimization of the NN needs to consider the specific channel decoding algorithms, which is inflexible from the design perspective, and undermines the generality of the ISI detector.
Therefore, in order to deal with the extrinsic information from the decoder for Turbo equalization, without introducing any extra off-line training complexity, we propose a new message updating rule for the messages to be passed to the neural network as follows:
\begin{gather}\label{u_update}
u_{i}(x_i) = \prod_{j \neq i}q_{i,j}(x_i).
\end{gather}
Correspondingly, the accumulated APPs for each symbol becomes:
\begin{gather}\label{Q_update_new}
Q_i(x_i) = O_i(x_i)T_i(x_i)v_i(x_i)\prod_{j \neq i}{q_{i,j}(x_i)}.
\end{gather}
The message passed from each VN $x_i$ to the NN FN does not include the messages from $O_i(x_i)$ or $T_i(x_i)$, only the message from the FN $I_{i,j}(x_i,x_j)$ is passed to the NN FN.
\begin{figure}[t!]
\centering
\includegraphics[width=150mm]{DLSP_Tanner.eps}
\vspace{-6mm}
\caption{Deep learning assisted sum-product algorithm with $L_E$ = 2.}
\vspace{-3mm}
\label{DL-SPDA}
\end{figure}
According to the proposition, the APP $Q_i(x_i)$ is given by:
\begin{gather}\label{Q_update_new}
Q_i(x_i) = O_i(x_i)T_i(x_i)v_i(x_i)\prod_{j \neq i}{q_{i,j}(x_i)}.
\end{gather}
Compared with the APP update rule in (\ref{Q_update}), (\ref{Q_update_new}) contains the message $v_i(x_i)$ from the NN to the VN $x_i$. Since in each iteration of the Turbo equalization, the $priori$ information $T_i(x_i)$ is constant, the messages from $T_i(x_i)$ will not be passed to the NN for further processing.
We expect the new message updating rule tunes the APPs during each iteration according to the output messages $v_i(x_i)$ of the NN FN.
We summarize the DL-SPDA as follows:
\begin{enumerate}[Step 1:]
\item Update all the $a \ posterior$ probabilities $\{Q_i\}$ as in (\ref{Q_update_new});
\item Update the messages $\{p_{i,j}\}$ from the VN $x_i$ to the FN $I_{j,i}$, as in (\ref{p_update});
\item Update the messages $\{q_{i,j}\}$ from the FN $I_{j,i}$ to the VN $x_i$, as in (\ref{q_update});
\item Update the messages $\{u_i\}$ from the VN $x_i$ to the NN FN $\Phi(x_1,...,x_N)$, as in (\ref{u_update});
\item Compute the messages $\{v_i\}$ based on the trained $\Phi(x_1,...,x_N)$;
\item If the maximum number of iterations is not reached, then go back to Step 1;
\end{enumerate}
\subsection{DL-SPDA with a convolutional NN and its training procedure}
As introduced in Section. \Romannum{2}, the generator matrix of the FTN system is a Toeplitz matrix, which follows a convolutional structure. To explore the convolutional structure of the FTN signaling, we propose to employ a simplified convolutional NN (CNN) to be performed as the NN FN to assist the SPDA \cite{CNN,CNNDetection, CNNRNN_Detection}.
CNNs are widely used in image recognition systems \cite{CNN}.
Traditional CNNs usually involve several convolutional layers (Conv) and max-pooling layers.
The convolutional layer performs the convolution operation of the filters (kernels), where the filters convolve and stride over the input. The max-pooling layer performs downsampling to reduce the spatial size of the convolved features.
A dense layer is appended after the max-pooling layer to provide possibly nonlinear functions \cite{CNN}.
In \cite{CNNDetection}, a pure CNN based detection algorithm was proposed, where both the max-pooling layers and the dense layer are removed to reduce the training complexity, but a large number of convolutional layers and filters are kept.
In this paper, for the sake of reducing the training complexity of the additional NN FN, we remove the max-pooling layer and simplify the CNN to have only two convolutional layers and one dense layer. The convolutional layer explores the features of the NN FN input message $u_i(x_i)$.
The dense layer is appended to process the features and then delivers the output to the VNs.
As shown in Fig. \ref{CNN}, the first convolutional layer has $f_{n_1}$ filters and each filter has a size of $ f_{l_1}$, where $f_{l_1}$ indicates that $f_{l_1}$ messages are considered by the filter during each convolutional step.
Let $f_{s_1}$ denote the stride of the sliding window in the first convolutional layer.
With the attachment of zero padding, the output dimension of the first convolutional layer reduces to $ \lceil N/f_{s_1}\rceil$.
The filter convolves with the messages from $u_1(x_1)$ to $u_N(x_N)$ to compensate any detrimental effect in the message passing that may degrade the detection performance.
A similar operation of filtering is performed in the second convolutional layer which has a corresponding hyper-parameter set of $(f_{n_2}, f_{l_2}, f_{s_2})$.
The output of the convolutional layer is sent to the dense layer to combine and process the filters' results before delivering the messages to the VNs.
The dense layer processes all the filters' results and outputs the message $v_i(x_i)$ for $i \in \{1,...,N\}$.
The negative of a rectified linear activation function (ReLU) is used as the activation function in the dense layer \cite{ReLU}. During each iteration, the CNN has initial weights and biases randomly generated from a truncated normal distribution with a standard deviation of $\sigma_{\text{CNN}_1}$ and $\sigma_{\text{CNN}_2}$ for the first and second CNN layers, respectively.
For each iteration, messages $u_i(x_i)$ will be updated by using (\ref{u_update}) then passed to the CNN.
Messages are sent back from the CNN to join the APPs accumulation according to (\ref{Q_update_new}).
By unfolding the FG including the NN FN to an NN system with multiple layers, the overall detector can be trained and optimized.
\begin{figure}[t!]
\centering
\includegraphics[width=110mm]{CNN.eps}
\vspace{-5mm}
\caption{Simplified CNN function node $\Phi(x_1,...,x_N)$.}
\label{CNN}
\vspace{-6mm}
\end{figure}
A loss function in an NN training is defined as the errors between the ground truth label and the estimated output.
The cross-entropy loss function has been widely used for the classification problems \cite{DL_Textbook} to minimize the errors between two probability distributions.
By considering the BPSK constellation, we define $\hat{\mathcal{D}}_i^m(x_i)$ as the LLR of the VN $x_i$ at the $m$-th iteration and $m_{max}$ as the maximum number of iterations of DL-SPDA.
The cross-entropy function can be defined as $\mathcal{F}_{ce}(\bm{\mathcal{D}}, \bm{\hat{\mathcal{D}}}^m) = -\frac{1}{N}\sum_{i=1}^{N}\big{(}\mathcal{D}_{x_i}\text{log}(\frac{1}{1+e^{-\hat{\mathcal{D}}^m_{x_i}}}) + (1-\mathcal{D}_{x_i})\text{log}(1-\frac{1}{1+e^{-\hat{\mathcal{D}}^m_{x_i}}})\big{)}$, where $\bm{\mathcal{D}}$ is the ground-truth label of the transmitted bits.
The multi-loss function introduced in \cite{DLLC} and \cite{Syndrome_based} can also be setup to train the tuneable parameters in the FG, which includes the tuneable weights in (\ref{q_update_new}) and the weights and biases introduced in the CNN.
The final loss function for training the NN is then given by:
\begin{gather}\label{loss_func}
\Lambda = \sum_{m=1}^{m_{max}}\gamma^{m_{max}-m}\mathcal{F}_{ce}(\bm{\mathcal{D}}, \bm{\hat{\mathcal{D}}}^m),
\end{gather}
where $\gamma < 1$ is a discount factor to adjust the loss at each iteration. During the training phase, a batch of random transmitted symbols over a range of SNRs is generated as the samples to train the NN.
Once the NN is fully trained, the DL-SPDA can be employed to the Turbo equalization receiver to perform as an ISI detector by exchanging the extrinsic information with the decoder. The extrinsic information from the decoder is passed to the ISI-detector without further tuning the parameters in the NN.
\subsection{Compatible DL-SPDA Training with Mutual Information Compensation}
We have introduced the proposed DL-SPDA in Section \Romannum{3}.A, where the FG is modified by connecting an NN FN and the message updating rule is changed accordingly.
However, in our training phase, we have not considered the extrinsic information passed from the specific decoder. In other words, the extrinsic information to be conveyed to the DL-SPDA at the $\rho$-th iteration of the Turbo equalization is assumed to be an all-zero sequence, i.e. $\mathbf{\Theta}^\rho$ = $\mathbf{0}$.
It can be observed that the DL-SPDA is optimized in the sense of ML detection. Nevertheless, it is well-known that the maximum \emph{a posterior} (MAP) detection outperforms the ML detection for coded systems. Therefore, we propose a compatible training technique which considers the extrinsic information from the decoder.
Inspired by the training technique introduced in \cite{TENN}, we are motivated to deliver the expected distribution of the extrinsic information from the decoder to the NN by assuming that the extrinsic information follows a consistent Gaussian distribution \cite{Anderson_BestCC}, where the mean and variance can be computed from the inverse $J$-function as introduced in \cite{EXIT}.
Note that in \cite{TENN}, the DNN treats the extrinsic information from the decoder as separate input neurons to the NN, i.e., the channel information and the extrinsic information from the decoder are fed into the NN separately. However, this largely increases the number of neurons in the NN and overloads the training process of the whole NN. Therefore, we propose to combine the LLRs of the channel information and the extrinsic information before sending the LLRs to the DL-SPDA, such that the DL-SPDA uses the combined information for further training.
Let $\mathbf{\Psi}^{(\cdot)}$ and $\mathbf{\Upsilon}^{(\cdot)}$ be the LLR realizations for the channel information and the extrinsic information to be passed to the detector, respectively.
Note that the DL-SPDA accepts both $\mathbf{\Psi}^{(\cdot)}$ and $\mathbf{\Upsilon}^{(\cdot)}$ as the inputs.
Each realization of the channel information $\mathbf{\Psi}^{(\cdot)}$ can be generated by transmitting a known data sequence $\mathbf{x}$.
On the other hand, to generate the extrinsic information $\mathbf{\Upsilon}^{(\cdot)}$, we adopt the idea from the previous work in \cite{TENN}.
We assume that the extrinsic information $\mathbf{\Theta}^\rho$ follows a consistent Gaussian probability density function, i.e.,
$\mathbf{\Theta}_i^\rho \sim \mathcal{N}\big{(}(-1)^{x_i} \sigma_E^2/2,\sigma_E^2\big{)}$ for $i \in \{1,...N\}$, where the variance $\sigma_E^2$ can be computed by using the inverse $J$-function \cite{EXIT}, given the mutual information $\mathcal{M}(\mathbf{\Theta}^\rho, \mathbf{x})$ between the extrinsic information $\mathbf{\Theta}^\rho$ and the transmitted sequence $\mathbf{x}$.
To maintain the generality of the NN, we assume that the mutual information $\mathcal{M}(\mathbf{\Theta}^\rho, \mathbf{x}))$ is uniformly ranged between $ [0, 1]$.
In practice, it is sufficient to consider only a set of mutual information $\Omega = \{0, 1/(|\Omega|-1), ..., 1\}$, where $|\Omega|$ is the cardinality of $\Omega$.
Thus, a set of variances can be then determined as $\Xi = \{J^{-1}(\partial), \ \text{for} \ \partial \in \Omega\}$.
Based on the variance set $\Xi$ and the known transmitted sequence $\mathbf{x}$ in the training phase, we are able to randomly generate a realization of the LLR sequence $\mathbf{\Upsilon}^{(\cdot)}_i \sim \mathcal{N}\big{(}(-1)^{x_i} \sigma_E^2/2,\sigma_E^2\big{)}$, for $i \in \{1,...,N\}$.
\begin{figure}[t!]
\centering
\includegraphics[width=105mm]{Compatible_Training.eps}
\vspace{-4mm}
\caption{Graphical illustration for the compatible training technique.}
\label{Compatible_Training}
\vspace{-4mm}
\end{figure}
Fig. \ref{Compatible_Training} illustrates the concept of the compatible training technique to improve the compatibility of the DL-SPDA in the sense of Turbo equalization, where the process is divided into two stages, namely the generation stage and the preprocessing stage.
In Fig. \ref{Compatible_Training}, as an example, $\Omega$ is set to be $\Omega=\{0, 0.25, 0.5, 0.75, 1\}$ and one LLR realization of the extrinsic information is randomly sampled by a specific variance $\sigma_E^2 \in \Xi$ for each realization of the channel information.
Note that during the training phase, based on each variance $\sigma^2_E \in \Xi$, a factor $\mathcal{V}$ can be preset so that $\mathcal{V}$ realizations of the extrinsic information $\mathbf{\Upsilon}^{(\cdot)}$ are randomly sampled based on one realization of the channel information.
This results in that $\mathcal{V} \times |\Omega|$ realizations of the extrinsic information will be randomly sampled based on one realization of the channel information.
We call the stage above as the generation stage.
As for the preprocessing stage, we combine the extrinsic information and the channel information. Specifically, let $\mathbf{\Upsilon}^{(i,j)}$ be the $j$-th realization of the extrinsic information for the
$i$-th realization of the channel information $\mathbf{\Psi}^{(i)}$, for $j \in \{1,..., \mathcal{V} \times |\Omega|\}$.
The message that is passed to the VNs is the combined information sequence, where the elements in $\mathbf{\Psi}^{(i)}$ and $\mathbf{\Upsilon}^{(i,j)}$ are simply added together.
As Fig. \ref{Compatible_Training} demonstrated, each $i$-th realization of the channel information is separately added by the sampled extrinsic information $\mathbf{\Upsilon}^{(i,j)}$ for $j \in \{1,..., 1 \times |\Omega|\}$.
After all the realizations have been preprocessed, the batch of sequences are fed into the NN for further training.
It can be seen that compared to the training method in Section \Romannum{3}.B, the compatible training technique includes two more stages for processing.
This indicates that the input training sequences $\{\mathbf{\Psi}^{(1)} + \mathbf{0},\mathbf{\Psi}^{(2)}+ \mathbf{0},...\}$ in the previous sections are now changed to $\{\mathbf{\Psi}^{(1)}+\mathbf{\Upsilon}^{(1,1)},...,\mathbf{\Psi}^{(1)}+\mathbf{\Upsilon}^{(1,|\Omega|)},\mathbf{\Psi}^{(2)}+\mathbf{\Upsilon}^{(2,1)},...\}$.
These two stages are the preprocessing of the NN training.
It should be noted that the preprocessing does not add any extra input neurons to the NN while it provides the \emph{priori} information for the training.
Therefore, a better error performance can be expected for the DL-SPDA with the compatible training than that without the compatible training.
\section{Performance Analysis for Convolutionally Encoded FTN system}
As we previously discussed, the DL-SPDA is expected to reduce the detrimental effect of the residual ISI responses and the accumulated correlations of the detecting messages in the FG.
Therefore, in this section, we aim to theoretically derive the error performance of coded FTN systems in order to evaluate the error performance of the proposed DL-SPDA.
In specific, we focus on the BER performance of the convolutionally encoded FTN systems with a finite block length, where the detection algorithm considers a limited number of ISI taps.
As the generation of the FTN signaling follows the convolutional structure, which can be interpreted by a trellis \cite{Anderson_FTN}.
For the detection of FTN signaling, we define an \emph{error event} as a segment of a path on the FTN trellis that diverges/remerges from/to the correct path \cite{ShuLinBook}.
For an FTN system with a block length of $N$, multiple error events can happen.
Specifically, suppose that the output of a correct path on the FTN trellis for a legitimate transmitted sequence is denoted by $\mathbf{x}$ and the output of the erroneous path is another legitimate transmitted sequence denoted by $\mathbf{x}'$.
Then, for BPSK modulation, we define an error sequence as $\mathbf{e} = \mathbf{x} - \mathbf{x}'$, where $\mathbf{e} \in \mathcal{S}^{(N \times 1)}$ and $\mathcal{S}=\{0,+2, -2\}$.
According to the derivations in \cite{Rusek_ML, SY_WJ_FTN}, we find the following definitions and lemma are useful.
\begin{defi}
\emph{(Normalized Euclidean distance)}:
Let $E_b$ be the average energy per information bit. The normalized Euclidean distance $d^2(\mathbf{e})$ of the error sequence $\mathbf{e}$ is defined as
$d^2(\mathbf{e}) \triangleq \frac{1}{2E_b}\mathbf{e}^\text{H}\mathbf{G}\mathbf{e}$.
\end{defi}
\begin{defi}
\emph{(Operative Euclidean distance \cite{SY_WJ_FTN})}:
Denote by $\mathbf{F}$ the truncated version of $\mathbf{G}$, where $\mathbf{F}$ is a Toeplitz matrix constructed by circular shift of the vector $\{0,...,0,g_{-L_E},...,g_0,...,g_{L_E},0,...,0\}$.
The operative Euclidean distance $d_{\text{ope}}^2(\mathbf{e}) $ of the error sequence $\mathbf{e}$ is defined as
$d_{\text{ope}}^2(\mathbf{e}) \triangleq \frac{1}{2E_b}\mathbf{e}^{\rm{H}}\mathbf{F}\mathbf{e}$.
\end{defi}
\begin{defi}
\emph{(Hamming error weight)}:
The Hamming error weight $w$ is defined as the number of non-zero elements of an error sequence $\mathbf{e}$.
\end{defi}
\begin{lemma}
\emph{(Error probability for $\mathbf{e}$ \cite{Ungerboeck})}:
Given the transmitted sequence $\mathbf{x}$, the probability of the ML detector declares the sequence $\mathbf{x}' = \mathbf{x} + \mathbf{e}$ as the detection output is given by
$P_{\mathbf{e}} = Q(\sqrt{\frac{d^2(\mathbf{e})E_b}{N_0}})$.
\end{lemma}
In the following, we will evaluate the ML performance of coded FTN system with a short block length (finite-length). In particular, we consider truncated ISI responses (finite-taps) and approximately full ISI responses (full-taps).
The approximation of full ISI responses means that we consider $L$ most recent ISI taps as described in Section \Romannum{2}.A.
\subsection{Finite-Length Full-Taps Coded FTN system}
To begin with, we start from the ML estimation of the \textbf{uncoded} FTN system.
Let us consider that the error sequence $\mathbf{e}$ has a Hamming error weight $w$.
Therefore, it is obvious that there are in total $\mathcal{A}=\binom{N}{w}2^w$ combinations for $\mathbf{e}$.
For the convenience of notation, we use $\mathbf{e}^{(w,j)} \in \mathcal{S}^{(N \times 1)}$ to denote a specific error sequence $\mathbf{e}$ with a Hamming error weight $w$, where $j \in \{1,...,\mathcal{A} \}$ indicates the $j$-th instance of all possible combinations of $\mathbf{e}$.
Then, based on Lemma 1, the BER $P_b$ can be upper-bounded by applying the union bound \cite{ShuLinBook}, such as:
\begin{align}
P_b < \sum\limits_{w = 1}^N {{}}{\sum\limits_j^{\rm \mathcal{A}} \frac{w}{{2^wN}}{Q\Bigg{(}\sqrt {\frac{{{d^2(\mathbf{e}^{(w,j)})}{E_b}}}{{{N_0}}}} \Bigg{)}} } \label{Pb_FLFull_UB},
\end{align}
where
\begin{align}\label{Q_Func}
Q(\alpha)=\int\limits_\alpha ^\infty {\frac{1}{{\sqrt {2\pi } }}\exp ( - {\rho ^2}/2)d} \rho,
\end{align}
is the well-known Q-function \cite{ShuLinBook}.
We now consider the \textbf{convolutionally encoded} FTN system, where a uniform interleaver is assumed to be applied between the convolutional encoder and the BPSK modulator.
Suppose a rate-$R$ binary terminated convolutional code (CC) $\mathcal{C}'$ is utilized. We define $K_b$ bit streams as the encoder's inputs and $N_b$ bit streams as outputs, i.e., $R=K_b/N_b$.
According to (\ref{Pb_FLFull_UB}), the BER performance can be estimated by considering all possible error sequences, i.e., the distance spectrum.
However, conventional methods, which assume that the all-zero sequence is transmitted, to estimate the BER performance of CC over AWGN channels are not applicable \cite{ShuLinBook}. This is because the Euclidean distance between coded FTN signals can be different even for error sequences with the same Hamming error weight \cite{Li2019code}.
For the ease of presentation, we refer to the error sequence $\mathbf{e} = \mathbf{x} - \mathbf{x}'$ after the FTN modulation as the \textbf{FTN error sequence}, and refer to the binary error vector of the difference between two legitimate CC codewords as the \textbf{CC error sequence}.
It is obvious that given the pair of CC codewords, the FTN error sequence and CC error sequence share the same Hammming error weight.
Therefore, we aim to list all the possible CC error sequences and estimate the BER according to the corresponding FTN error sequences.
In the following, we consider that only a \textbf{single} error event is occurred over the CC trellis. Note that this simplification is widely considered in the literature \cite{ShuLinBook}.
Therefore, we intend to find out all the possible legitimate FTN error sequences that are induced by a single error event in the CC trellis.
Without loss of generality, we consider CC codewords with information bit length $K'$, which is long enough to contain all the possible codewords
covering the CC error events with a relatively small Hamming error weight $w$.
Define $\mathbf{c},\mathbf{c}' \in \mathcal{C}'$ as the instances of the $2^{K'}$ legitimate codeword.
By listing all the possible codewords, we are able to find the CC error sequences corresponding to each single error event $\varepsilon$ and $\mathbf{c}$.
We use $U(\mathbf{c}, \varepsilon) = \{\varepsilon | \mathcal{I}_{\varepsilon} = i, \mathcal{L}_{\varepsilon} = l, \mathcal{W}_{\varepsilon} = w \}$ to denote the set of possible single error events for a legitimate codeword $\mathbf{c}$, where each error event $\varepsilon$ in $U(\mathbf{c}, \varepsilon)$ has an event error length $\mathcal{L}_{\varepsilon} = l$ and the corresponding CC error sequence has Hamming error weight $\mathcal{W}_{\varepsilon} = w$ between $\mathbf{c}$ and $\mathbf{c}'$, corresponding to $\mathcal{I}_{\varepsilon} = i$ different information bits.
As a uniform interleaver is assumed to be employed between the convolutional encoder and the FTN modulator, each possible FTN error sequence corresponding to the CC error sequence is therefore, equally likely to be considered. Define $\mathbf{e}^{(w,o)}|\varepsilon$ as the $o$-th legitimate FTN error sequence with a Hamming error weight $w$ for a given CC event error $\varepsilon$ and $\mathcal{A}_{\varepsilon}$ as the total number of legitimate FTN error sequences for a given CC error event $\varepsilon$.
The BER for the convolutionally encoded FTN system can now be upper bounded by:
\begin{align}\label{Pb_FLFullCoded_FTN}
{P_b} \lesssim \sum\limits_{w = d_{\text{min}}}^N\sum\limits_{\mathbf{c}}^{\mathcal{C}'}\sum\limits_{\varepsilon}^{U(\mathbf{c}, \varepsilon)}\sum\limits_o^{\mathcal{A}_{\varepsilon}} \frac{{{ {\frac{\mathcal{I}_{\varepsilon}}{{NR}}{NR/K_b-\mathcal{L}_\varepsilon+1\choose 1}Q\Bigg{(}\sqrt {\frac{{{d^2\big{(}\mathbf{e}^{(w,o)}|\varepsilon\big{)}}{E_b}}}{{{N_0}}}} \Bigg{)}} } }}{\mathcal{A}_{\varepsilon}2^{K'}},
\end{align}
It can be seen that the denominator part in (\ref{Pb_FLFullCoded_FTN}) is due to the assumptions of binary symmetric inputs of the convolutional encoder and the employment of the uniform interleaver between the convolutional encoder and the FTN signaling.
\subsection{Finite-Length Finite-Taps Coded FTN system}
With the analysis of the finite length coded FTN system which considers full ISI taps, we can now investigate the error performance for the ML estimate when a finite number of ISI taps $L_E < L$ is considered by the detector.
As in the previous section, we firstly consider the case of \textbf{uncoded} finite-taps FTN signaling. Following \cite{SY_WJ_FTN} and Definition 2,
we intend to find the error probability $P_{\mathbf{e}}^{'}$ under the condition that a finite number of ISI responses is considered at the receiver, which is equivalent of finding the probability of $P(\text{Re}\big{\{} \mathbf{e}^\text{H}\bm{\eta} + \mathbf{e}^\text{H}(\mathbf{G}-\mathbf{F})\mathbf{x}\big{\} > E_bd^2_{\text{ope}}(\mathbf{e})})$.
It can be seen that $\mathbf{e}^\text{H}(\mathbf{G}-\mathbf{F})\mathbf{x}$ represents the influence of the residual ISI that is not considered by the matrix $\mathbf{F}$.
Define the term $\wp \triangleq \text{Re}\big{\{} \mathbf{e}^\text{H}\bm{\eta} + \mathbf{e}^\text{H}(\mathbf{G}-\mathbf{F})\mathbf{x}\big{\}}$.
In particular, the term $\wp$ can be modeled as a Gaussian variable according to the central limit theorem and it can be shown that $\mathbb{E}[\wp] = 0$ and the variance $\mathbb{E}[\wp\wp^{\rm{H}}] = N_0 E_b d^2(\mathbf{e})+2E_b\sigma^2_{\text{R},\mathbf{e}}$, where $\sigma^2_{\text{R},\mathbf{e}}$ refers to the variance of the term $\mathbf{e}^\text{H}(\mathbf{G}-\mathbf{F})\mathbf{x}$ induced by the residual ISI taps for a given error sequence $\mathbf{e}$. In specific, we have $\sigma^2_{\text{R},\mathbf{e}} = \frac{1}{2E_b} \mathbb{E}[\mathbf{x}^\text{H}(\mathbf{G}-\mathbf{F})\mathbf{e}\mathbf{e}^\text{H}(\mathbf{G}-\mathbf{F})^\text{H}\mathbf{x}]$ \cite{SY_WJ_FTN}.
Based on the derivations in \cite{SY_WJ_FTN}, the error probability for an ML detection if finite taps of ISI responses are considered by the detector is approximated by the following lemma:
\begin{lemma}
\emph{(Error probability for $\mathbf{e}$ with finite ISI responses) }:
Given the transmitted sequence $\mathbf{x}$ and assume that the ML detector considers finite ISI responses with a length of $L_E$, the probability of the ML detector declares the sequence $\mathbf{x}' = \mathbf{x} + \mathbf{e}$ as the detection output is given by
${P_{\mathbf{e}}^{'}} \approx Q\Bigg{(}\sqrt {\frac{{{E_b}d_{{\rm{ope}}}^2(\mathbf{e})}}{{{N_0}}}\frac{{d_{{\rm{ope}}}^2(\mathbf{e})}}{{{d^2}(\mathbf{e}) + \frac{{2\sigma _{{\text{R},\mathbf{e}}}^2}}{{{N_0}}}}}} \Bigg{)}$.
\end{lemma}
Finding the expectation over all the possible transmitted sequence $\mathbf{x}$ for a given $\mathbf{e}$ can be computationally prohibitive.
In the following, we introduce a theorem that obtains the lower bound $\sigma^2_{\text{R}_L,\mathbf{e}}$ for $\sigma^2_{\text{R},\mathbf{e}}$.
\begin{theo}
For BPSK modulation, given an error sequence $\mathbf{e} \in \mathcal{S}^{(N \times 1)}$,
define $\mathcal{P}$ as the set of non-zero elements' positions in $\mathbf{e}$,
the normalized variance of the term $\mathbf{x}^{\rm{H}}(\mathbf{G}-\mathbf{F})\mathbf{e}$ is \textbf{lower bounded} by $\sigma^2_{\text{R},\mathbf{e}} \geq \sigma^2_{\text{R}_L,\mathbf{e}} = \frac{1}{2E_b}\big{[}\sum_{j \in \mathcal{P}}x_j \cdot [(\mathbf{G}-\mathbf{F})\mathbf{e}]_j\big{]}^2$.
\end{theo}
\begin{prf}
Please refer to Appendix \ref{ProofTheorem_1} for the details.
\end{prf}
Based on Theorem 1, we further obtain the following corollary.
\begin{corollary}
Any two error sequences $\mathbf{e}$ and $\mathbf{e}'$ with the same normalized Euclidean distance, $d^2(\mathbf{e}) = d^2(\mathbf{e}')$, and operative Euclidean distance, $d^2_{\text{ope}}(\mathbf{e}) = d^2_{\text{ope}}(\mathbf{e}')$, have identical values of lower bounds $\sigma^2_{\text{R}_L,\mathbf{e}}$ and $\sigma^2_{\text{R}_L,\mathbf{e}'}$, i.e., $\sigma^2_{\text{R}_L,\mathbf{e}}=\sigma^2_{\text{R}_L,\mathbf{e}'}$.
\end{corollary}
\begin{prf}
Please refer to Appendix \ref{ProofCorollary} for the details.
\end{prf}
Based on Lemma 2, we are able to derive the analytical BERs for the coded FTN signaling, where the ML detector considers finite ISI responses.
As previously described, $\sigma _{{\text{R},\mathbf{e}}}^2$ is derived by computing the expectation over all the possible transmitted sequences $\mathbf{x}$, while $\sigma _{{\text{R}_L,\mathbf{e}}}^2$ is determined based on the error sequence $\mathbf{e}$.
It is reasonable to employ $\sigma _{{\text{R}_L,\mathbf{e}}}^2$ to estimate the error probability for $\mathbf{e}$.
Moreover, to simplify the computation complexity, the distance spectrum of the FTN signaling can be searched based on the FTN error events \cite{Shuangyang,ISI_Distance_Spec,Modulated_Coding}.
For instance, given a CC error event $\varepsilon$ and a Hamming error weight $w$, let $\Big{\{}\big{(}D_{1_\varepsilon}, d^2(\mathbf{\tilde{e}}^{(w,1)}|\varepsilon),d^2_{\text{ope}}(\mathbf{\tilde{e}}^{(w,1)}|\varepsilon)\big{)},...,\big{(}D_{M_\varepsilon},d^2(\mathbf{\tilde{e}}^{(w,M)}|\varepsilon),,d^2_{\text{ope}}(\mathbf{\tilde{e}}^{(w,M)}|\varepsilon)\big{)}\Big{\}}$ be the distance spectrum for the CC error event $\varepsilon$, where $D_{o_\varepsilon}$ denotes the corresponding number of error sequences which have the same Euclidean and operative distance as $\mathbf{\tilde{e}}^{(w,o)}$ and $M_\varepsilon$ denotes the total number of considered error sequences.
According to Corollary 1, any two error sequences with the same Euclidean distance and operative Euclidean distance have the same values of $\sigma_{\text{R}_L,\mathbf{e}}^2$.
Therefore, the BER for a convolutionally encoded FTN system with a finite number of ISI responses can be approximated by:
\begin{align}\label{FiniteTaps_coded_BER}
{P_b} &\approx \sum\limits_{w = d_{\text{min}}}^N\sum\limits_{\mathbf{c}}^{\mathcal{C}'}\sum\limits_{\varepsilon}^{U(\mathbf{c}, \varepsilon)}\sum\limits_o^{M_{\varepsilon}} \frac{1}{\mathcal{A}_{\varepsilon}2^{K'}} \frac{\mathcal{I}_{\varepsilon}}{{NR}}{NR/K_b-\mathcal{L}_\varepsilon+1\choose 1}D_{o_\varepsilon} \cdot
\notag
\\
& Q\Bigg{(}\sqrt {\frac{{{E_b}d_{{\rm{ope}}}^2(\tilde{\mathbf{e}}^{(w,o)}|\varepsilon)}}{{{N_0}}}\frac{{d_{{\rm{ope}}}^2(\tilde{\mathbf{e}}^{(w,o)}|\varepsilon)}}{{{d^2}(\tilde{\mathbf{e}}^{(w,o)}|\varepsilon) + \frac{{2\sigma _{{\text{R}_L,(\tilde{\mathbf{e}}^{(w,o)}|\varepsilon)}}^2}}{{{N_0}}}}}}\Bigg{)}
\end{align}
Note that the analytical BER in (\ref{FiniteTaps_coded_BER}) is an approximation of the real BER due to the use of the lower bound of $\sigma^2_{\text{R},\mathbf{e}}$. As will be shown in the numerical results, in a high SNR region, using $\sigma _{{\text{R}_L,\mathbf{e}}}^2$ has achieved a close estimate to the error performance of the ML detection and decoding.
More importantly, the lower bound $\sigma _{{\text{R}_L,\mathbf{e}}}^2$ becomes closer to $\sigma _{{\text{R},\mathbf{e}}}^2$, if the influence of the residual ISI tends to be insignificant.
Furthermore, (\ref{FiniteTaps_coded_BER}) demonstrates that the ML performance is restricted by the number of ISI taps considered by the detection algorithm. If full-taps are considered, then $d^2_{\text{ope}(\mathbf{e})}$ is identical to $d^2(\mathbf{e})$ and the residual ISI term will vanish.
In the following numerical results, we will show that when finite taps are considered by the conventional SPDA, the performance is limited by the residual ISI taps and an error floor occurs in a high SNR region when the noise has less impact to the performance than the residual ISI. On the other hand, simulation results show that the performance of the DL-SPDA approaches to that of the ML detection and decoding which is consistent with our analysis.
\section{Numerical Results}
\subsection{Bit-error-rate performance of the proposed DL-SPDA}
In this section, we evaluate the performance of the proposed DL-SPDA scheme for convolutionally encoded FTN systems.
Without loss of generality, we consider the coded FTN systems
with $\tau=0.5$ and $\tau=0.6$, respectively, where the (7, 5) 4-state rate-1/2 non-recursive CC is used.
At the receiver, Turbo equalization is performed, where the extrinsic information is exchanged between the DL-SPDA detector and the BCJR decoder for CC. Compatible training method is applied to the DL-SPDA.
To have a fair comparison, we also perform the SPDA \cite{SISO} and the channel shortening (CS) method as proposed in \cite{CS}.
Furthermore, truncated-BCJR detection algorithm \cite{MAP_Covalope} for FTN detection is also evaluated to demonstrate the accuracy of the analytical bound, i.e., (\ref{Pb_FLFullCoded_FTN}) and (\ref{FiniteTaps_coded_BER}), when a limited number of ISI taps is considered by the detection algorithm.
Note that, in order to improve the error perforamnce, the BCJR algorithm usually requires a terminated trellis \cite{ShuLinBook}.
Therefore, in our simulation, we intentionally transmit additional symbols to terminate the ISI trellis.
Specifically, four additional code symbols are transmitted for the BCJR decoder for CC, while $2 \times L_E$ additional symbols are transmitted for the truncated-BCJR algorithm.
The analytical BER of a specific code rate $R$ is summed over the CC minimum Hamming distance $w = d_{\text{min}} = 5$ to a Hamming error weight 8 and the approximate full-taps BERs consider $L = 11$ ISI responses.
The corresponding code rate $R$ with the implementation of the truncated-BCJR detection algorithm and BCJR decoding algorithm is provided in the figures for reference, which is also used for the calculation of analytical BERs.
In this paper, the hyper-parameters to train the NN system are shown in Table \ref{table:1}.
\vspace{-0mm}
\begin{table}[t!]
\centering
\begin{center}
\begin{tabular}{c| c}
\hline
$\sigma_{\text{CNN}_1}$ & 0.03 \\
\hline
$\sigma_{\text{CNN}_2}$ & 0.03 \\
\hline
$(f_{n_1}, f_{l_1}, f_{s_1})$ & (3, 8, 5) \\
\hline
$(f_{n_2}, f_{l_2}, f_{s_2})$ & (1, 3, 1)\\
\hline
Optimizer & Root Mean Square Propagation\\
\hline
SNR Range (dB) & [6, 8] \\
\hline
Learning Rate & 0.001 \\
\hline
Batch per SNR & 60 \\
\hline
Compatible Training Factor $\mathcal{V}$ & 12 \\
\hline
$\Omega$ & \{0.2, 0.4, 0.6, 0.8, 1\} \\
\hline
$\gamma$ & 0.9 \\
\hline
$m_{max}$ & 6 \\
\hline
\end{tabular}
\end{center}
\caption{Hyper-parameters for training the DL-SPDA.}
\label{table:1}
\vspace{-10mm}
\end{table}
Note that for a BPSK modulation, the LLR values can be computed by $u_i(x_i)$ for $i \in \{1,..., N\}$, where the LLR values serve as the input of the CNN FN.
The error performance of various detection algorithms for the FTN systems is shown in Figs. \ref{N250_Bound_Performane}, \ref{N250_Tau06}, \ref{N250_Tau06_Trunc2Taps} and \ref{N250_Tau05}.
Let $\rho_{max}$ indicate the number of iterations of the Turbo equalization and $L_E$ denotes the number of truncated ISI taps.
DL-SPDA($\rho_{max}$, $L_E$) indicates the proposed DL-SPDA detection method, and BCJR($\rho_{max}$, $L_E$) refers to the corresponding truncated-BCJR detection algorithm \cite{MAP_Covalope}.
Both DL-SPDA and SPDA utilize 6 iterations for updating the messages. Unless specially notified, we compare the error performance at BER $\approx 1 \times 10^{-5}$.
In Fig. \ref{N250_Bound_Performane}, the comparison between analytical bound and the proposed algorithm of an FTN system with $\tau=0.6$ and $N = 250$ is shown, where we can observe that DL-SPDA(15, 3) has achieved a closed performance to the analytical bound with a code rate $R = 0.492$.
Furthermore, we demonstrate the correctness of the derived analytical BERs by performing BCJR(15, 7), where the error performance has a close match to the analytical BERs with a code rate $R=0.466$.
In Fig. \ref{N250_Tau06}, the simulation results of an FTN system with $\tau=0.6$ and $N = 250$ are provided.
It can be seen that DL-SPDA(15, 3) provides 0.35 dB gain compared to the SPDA(15, 3) at a BER = $3.2 \times 10^{-6}$, where both algorithms consider the same number of ISI taps in the original Tanner graph.
This is due to the fact that after the off-line training, the DL-SPDA algorithm has ``learned'' the residual ISIs which are not considered by the original FG.
Meanwhile, the error performance of CS(15, 6) is also provided in Fig. \ref{N250_Tau06}. As observed from the figure, both DL-SPDA(15, 3) and CS(15, 6) achieve almost the same error performance that is close to the analytical bound.
However, the proposed DL-SPDA requires less complexity, which will be discussed in Section \Romannum{5}.B.
In Fig. \ref{N250_Tau06_Trunc2Taps}, we provide the comparison between the analytical BERs for finite-length finite-tap ML estimate and the simulation results for coded FTN signaling. The truncated-BCJR detection algorithm and BCJR decoding algorithm are employed. The codeword length is $N = 250$ and $\tau=0.6$.
In order to demonstrate that an NN FN is capable to capture the interference characteristics throughout the off-line training, we set the DL-SPDA to consider 2 ISI taps based on the Tanner graph and we expect the NN FN learns all the ISI taps leading to an error performance that approachs the analytical bound.
The simulation results in the low SNR region show close performances to the analytical BERs.
Since a lower bound for $\sigma^2_{\text{R},\mathbf{e}}$ is implemented as in (\ref{FiniteTaps_coded_BER}), the BERs are slightly lower than the simulation results of the BCJR algorithm in the high SNR region, when 2 taps are considered by the truncated-BCJR algorithm.
It can be seen that the SPDA(15, 2) has an error floor at a BER = $2 \times 10^{-7}$ due to the insufficient considered ISI taps, while with the help of the concatenated neural network, DL-SPDA(15, 2) shows a promising performance with no noticeable error floor at a BER $< 10^{-7}$.
Fig. \ref{N250_Tau05} depicts the error performance of an FTN system with a more difficult ISI, e.g. $N = 250$ and $\tau=0.5$.
The DL-SPDA(15, 3) outperforms SPDA(15, 3) by 0.75 dB at a BER = $2.5 \times 10^{-5}$.
We observe that DL-SPDA(15, 3) has a close performance to the CS(15, 6).
From Fig. \ref{N250_Tau05}, we observe that for conventional SPDA, the performance is poor due to the strong ISI and the short codeword length.
On the other hand, DL-SPDA can compensate the strong correlation that exists in the detecting messages and help the detector converges to a better performance.
\begin{figure}[t!]
\centering
\includegraphics[width=90mm]{N250_Bound_Performane.eps}
\vspace{-6mm}
\caption{The BER of FTN signaling with $\tau = 0.6$, $\alpha = 0.3$, CC(7, 5) and $N$ = 250.}
\label{N250_Bound_Performane}
\vspace{-7mm}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=90mm]{N250_Tau06.eps}
\vspace{-6mm}
\caption{The BER of FTN signaling with $\tau = 0.6$, $\alpha = 0.3$, CC(7, 5) and $N$ = 250.}
\label{N250_Tau06}
\vspace{-7mm}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=90mm]{N250_Tau06_Trunc2Taps.eps}
\vspace{-6mm}
\caption{The BER of FTN signaling with $\tau = 0.6$, $\alpha = 0.3$, CC(7, 5) and $N$ = 250.}
\label{N250_Tau06_Trunc2Taps}
\vspace{-7mm}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=90mm]{N250_Tau05.eps}
\vspace{-6mm}
\caption{The BER of FTN signaling with $\tau = 0.5$, $\alpha = 0.3$, CC(7, 5) and $N$ = 250.}
\label{N250_Tau05}
\vspace{-7mm}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=90mm]{Loss.eps}
\vspace{-6mm}
\caption{Normalized average loss for every $10^3$ batches of training samples and the absolute change of loss for every $5 \times 10^3$ batches. FTN signaling with $\tau = 0.5$, $\alpha = 0.3$, $N = 250$.}
\label{Training_Loss}
\vspace{-5mm}
\end{figure}
In Fig. \ref{Training_Loss}, the normalized average training loss for every $10^3$ training batches for an FTN signaling of $\tau = 0.5$ and $N = 250$ with the DL-SPDA(15, 3) is shown.
Define $\xi^a_{\text{avg}}$ as the average loss from $(a-1) \times 5 \times 10^3$ to $a \times 5 \times10^3$ batches ($\xi^0_{\text{avg}} = 1$) and define $\xi^a_{\text{cg}} = |(\xi^a_{\text{avg}}-\xi^{a-1}_{\text{avg}})/\xi^{a-1}_{\text{avg}}|$ as the percentage of the absolute change on the average loss $(\xi^0_{\text{cg}} = 0)$, where $a \in \mathbb{Z}$.
Define that a stable performance of the training is reached after $a \times 5 \times 10^3$ batches, if $\xi^{a'}_{\text{cg}} < 0.1$ for any integer $a' > a$.
From Fig. \ref{Training_Loss}, we can see that the training phase of the proposed algorithm takes roughly $180 \times 70000$ = $1.26 \times 10^7$ training samples to converge to a relative stable performance.
Benefited from the derived FG and simplified NN model, this number of training samples is much smaller than the conventional NN decoders which generally need more than $2^K$ training samples to converge to a good performance.
\subsection{Computational complexity of the proposed DL-SPDA}
Previous simulation results demonstrate the potential BER performance gain that can be acquired by the proposed DL-SPDA based on the structure of conventional SPDA.
In this section, we discuss the computational complexity of the proposed DL-SPDA.
We mainly compare the computational complexity with the SPDA and the maximum \emph{a posteriror} (MAP) detection algorithm, i.e., log-BCJR algorithm \cite{BCJR_Complexity}.
Compared to the conventional SPDA, DL-SPDA has extra computational complexity introduced by the additional neural network function node. The multiplicative weights attached to the interference node $I_{i,j}\{x_i, x_j\}$ and the interference node itself can be considered as one parameter during the message computation. Therefore, no additional complexity is added by the multiplicative weights of the messages $p_{j,i}(x_j)$ in Section \Romannum{3}.
In Table \ref{Complexity_table}, we show the approximated computational complexity comparison by the number of addition and look-up table accesses, where the operations of comparison, multiplication and non-linear process are assumed to be executed through the look-up table access.
According to \cite{BCJR_Complexity}, the log-MAP algorithm has a computational complexity that grows exponentionally with the considered number of ISI taps.
Comparatively, the SPDA has a linear complexity with the number of ISI per message-passing iteration.
In Table \ref{Complexity_table}, the extra computation complexity that the DL-SPDA requires on top of the SPDA is shown.
For the additions, there are mainly two processes in the DL-SPDA, which are the extra message $v_i(x_i)$ at VN $x_i$ and the process of the neural function node, respectively. Similar to the additions, the processes of the convolutional layer, the ReLu non-linear activation function and the fully connected layer are considered by the look-up table accesses.
In our numerical results, with 6 iterations, DL-SPDA(15, 3) has approximately $2.4 \times 10^5$ additions and $1.0 \times 10^5$ look-up table accesses.
Comparatively, CS(15, 6) performs linear filtering and the BCJR algorithm. In specific, the BCJR algorithm requires $2.42 \times 10^5$ additions and $1.59 \times 10^5$ look-up table accesses, which is 60$\%$ more than the DL-SPDA algorithm.
Since FTN signaling theoretically introduces infinite number of ISI responses, the difference of computational complexity becomes more significant as more ISI responses are considered by the log-MAP detection algorithm.
More importantly, the proposed algorithm offers the flexility of designing the NN, where the complexity of the proposed algorithm might be further mitigated with advanced NN designs.
\vspace{-0mm}
\begin{table}[h!]
\centering
\begin{center}
\begin{tabular}{c| c|c}
\hline
Operations & Additions & Look-up table accesses \\
\hline
Log-MAP & $N (15 \times 2^{L_E} + 9)$ & $N(10 \times 2^{L_E} - 4$)\\
\hline
SPDA & $N(32L_E+6)$ & $4N \times L_E$ \\
\hline
\multirow{3}{*}{DL-SPDA (extra complexity)}& \multirow{3}{*}{\begin{tabular}{@{}c@{}}$2N + \lceil N/f_{s_1}\rceil (f_{l_1}f_{n_1}+1)+ $ \\ $ \big{\lceil} \lceil N/f_{s_1} \rceil / f_{s_2} \big{\rceil} (f_{l_2}f_{n_1}f_{n_2}+1) + $\\ $\big{\lceil}\lceil N/f_{s_1} \rceil / f_{s_2}\big{\rceil} f_{n_2}N$\end{tabular}}
& \multirow{3}{*}{\begin{tabular}{@{}c@{}}$\lceil N/f_{s_1}\rceil f_{l_1}f_{n_1} + $\\ $\big{\lceil} \lceil N/f_{s_1} \rceil / f_{s_2} \big{\rceil} (f_{l_2}f_{n_1}f_{n_2}+1) + $ \\ $\big{\lceil}\lceil N/f_{s_1} \rceil / f_{s_2}\big{\rceil} f_{n_2}N$\end{tabular}} \\
& & \\
& & \\
\hline
\end{tabular}
\end{center}
\vspace{-0mm}
\caption{Computational Complexity Comparison Per Iteration.}
\label{Complexity_table}
\vspace{-10mm}
\end{table}
\section{Conclusion}
In this paper, we proposed a DL-SPDA for FTN signaling.
An NN FN is concatenated to the FG of conventional FTN systems to compensate for the detrimental effect of the short cycles in the FG and the residual ISI that are not considered by the FG. The proposed DL-SPDA computes the \emph{a posterior} probability with the assistance of the NN.
A new message updating rule is proposed so that the proposed detection algorithm does not need to be optimized with respect to any particular channel decoder.
Meanwhile, we propose a compatible training technique to improve the compatibility of the DL-SPDA in Turbo equalization.
Moreover, we analysis the finite length coded FTN system's ML BER performance.
The proposed DL-SPDA for coded FTN systems has approached the analytical BERs and the MAP detection and decoding performances.
\appendices
\section{Proof of Theorem 1}
\label{ProofTheorem_1}
We rewrite the residual ISI taps' variance term $\mathbf{x}^\text{H}(\mathbf{G}-\mathbf{F})\mathbf{e}\mathbf{e}^\text{H}(\mathbf{G}-\mathbf{F})^\text{H}\mathbf{x}$ into a format of:
\begin{align}\label{Mtx_e} & \mathbf{x}^\text{H}(\mathbf{G}-\mathbf{F})\mathbf{e}\mathbf{e}^\text{H}(\mathbf{G}-\mathbf{F})^\text{H}\mathbf{x} =[\mathbf{x}^\text{H}(\mathbf{G}-\mathbf{F})\mathbf{e}]^2\notag \\
&= \bigg{[}\sum_{j \in \mathcal{P}}x_j \cdot [(\mathbf{G}-\mathbf{F})\mathbf{e}]_j + \sum_{j \not\in \mathcal{P}}x_j \cdot [(\mathbf{G}-\mathbf{F})\mathbf{e}]_j\bigg{]}^2,
\end{align}
where $\mathcal{P}$ indicates the set of positions of the non-zero elements in $\mathbf{e}$, $[(\mathbf{G}-\mathbf{F})\mathbf{e}]_j$ indicates the $j$-th element of the sequence and $\cdot$ denotes the element-wise multiplication.
The variance of the term $\mathbf{e}^{\rm{H}}\mathbf{G}\mathbf{e}$ induced by the residual ISI taps for a given error sequence $\mathbf{e}$ can now be derived as $\sigma^2_{\text{R},\mathbf{e}} = \frac{1}{2E_b}\mathbb{E}\bigg{\{}\Big{[}\sum_{j \in \mathcal{P}}x_j \cdot [(\mathbf{G}-\mathbf{F})\mathbf{e}]_j + \sum_{j \not\in \mathcal{P}}x_j \cdot [(\mathbf{G}-\mathbf{F})\mathbf{e}]_j\Big{]}^2\bigg{\}}$.
Applying the Jensen's inequality, we have
\begin{align}
\mathbb{E}_{\mathbf{x}|\mathbf{e}}\Big{\{}[\mathbf{x}^\text{H}(\mathbf{G}-\mathbf{F})\mathbf{e}]^2\Big{\}} \geq \big{[}[\mathbb{E}_{\mathbf{x}|\mathbf{e}}(\mathbf{x})]^\text{H}(\mathbf{G}-\mathbf{F})\mathbf{e}\big{]}^2
\end{align}
The second term in the second line of (\ref{Mtx_e}) vanishes as the expectation over the $\{x_j, \ \text{for } \ j \not\in \mathcal{P}\}$ is 0 and the first term is determined on $\mathbf{e}$. Therefore, $\sigma^2_{\text{R},\mathbf{e}} \geq \sigma^2_{\text{R}_L,\mathbf{e}} = \frac{1}{2E_b}\big{[}\sum_{j \in \mathcal{P}}x_j \cdot [(\mathbf{G}-\mathbf{F})\mathbf{e}]_j\big{]}^2$.
\section{Proof of Corollary 1}
\label{ProofCorollary}
Given two error sequence $\mathbf{e}$ and $\mathbf{e}'$ have the same Euclidean distance and operative Euclidean distance, then we have $\frac{1}{2E_b}(\mathbf{e})^\text{H}\mathbf{G}\mathbf{e} = \frac{1}{2E_b}(\mathbf{e}')^\text{H}\mathbf{G}\mathbf{e}'$. Similar to Theorem 1, the equation can be transformed into the element-wise summation by:
\begin{equation}
\frac{1}{2E_b}\sum_{j \in \mathcal{P}_1}e_{j} \cdot [\mathbf{G}\mathbf{e}]_j = \frac{1}{2E_b}\sum_{j \in \mathcal{P}_2}e'_{j} \cdot [\mathbf{G}\mathbf{e}']_j,
\end{equation}
where $\mathcal{P}_1$ and $\mathcal{P}_2$ are the sets of positions of the non-zero elements of $\mathbf{e}$ and $\mathbf{e}'$, respectively. Then, given $\mathbf{F}$ is a truncated matrix of $\mathbf{G}$ that the elements out of the truncation are filled with 0s, we have $\frac{1}{2E_b}\sum_{j \in \mathcal{P}_1}e_{j} \cdot [\mathbf{F}\mathbf{e}]_j = \frac{1}{2E_b}\sum_{j \in \mathcal{P}_2}e'_{j} \cdot [\mathbf{F}\mathbf{e}']_j$, which results in $\frac{1}{2E_b}\sum_{j \in \mathcal{P}_1}\big{(}e^{(1)}_{j} \cdot [\mathbf{G}\mathbf{e}]_j - e^{(1)}_{j}[\mathbf{F}\mathbf{e}]_j\big{)} = \frac{1}{2E_b}\sum_{j \in \mathcal{P}_2}\big{(}e'_{j} \cdot [\mathbf{G}\mathbf{e}']_j - e'_{j} \cdot [\mathbf{F}\mathbf{e}']_j\big{)}$. It can be modified to
\begin{equation}
\frac{1}{2E_b}\sum_{j \in \mathcal{P}_1}\big{(}e_{j} \cdot [\mathbf{(G-F)}\mathbf{e}]_j\big{)} = \frac{1}{2E_b}\sum_{j \in \mathcal{P}_2}\big{(}e'_{j} \cdot [\mathbf{(G-F)}\mathbf{e}']_j\big{)}.
\end{equation}
The error sequence $\mathbf{e}$ and the transmitted sequence $\mathbf{x}$ follow a one-to-one mapping at the non-zero elements' positions $\mathcal{P}_1$ and $\mathcal{P}_2$. Let $\mathbf{x}$ and $\mathbf{x}'$ be the corresponding transmitted sequences for $\mathbf{e}$ and $\mathbf{e}'$, respectively. Then, the corollary is proved that:
\begin{equation}
\frac{1}{2E_b}\big{[}\sum_{j \in \mathcal{P}_1}x_{j} \cdot [(\mathbf{G}-\mathbf{F})\mathbf{e}]_j\big{]}^2 = \frac{1}{2E_b}\big{[}\sum_{j \in \mathcal{P}_2}x'_{j} \cdot [(\mathbf{G}-\mathbf{F})\mathbf{e}']_{j}\big{]}^2
\end{equation}
\label{SecondAppendix}
|
1,116,691,497,796 | arxiv | \section{Introduction}
\label{sec:intro}
Distances are very important in statistics: a class of hypotheses testing methods are based on distances, such as the energy statistics \citep{szekely2004testing}, the distance covariance \citep{szekely2007measuring, szekely2009brownian,lyons2013distance}, and many others.
This type of testing statistics usually belong to the class of U-statistics or the V-statistics \citep{mises1947asymptotic, hoeffding1992class, korolyuk2013theory}, which require the calculation of all pairwise distances within the sample.
When variables are univariate, assuming the sample size is $m$, both \citet{huo2016fast} and \citet{chaudhuri2018fast} proposed fast algorithms with computational complexity $O(m \mbox{log}(m))$ where $m$ is the sample size.
Recall that the computational complexity is $O(m^2)$ when the statistics are computed directly based on their definitions.
When variables are multivariate, especially when they are high-dimensional, the calculation of the pairwise distances among these multivariate variables can not be implemented directly by the algorithm in \cite{huo2016fast}, and therefore becomes a potential bottleneck.
Our paper is aimed at reducing the computation complexity in the multivariate case by projecting the variables along a set of pre-specified optimal directions.
When the number of pre-specified optimal directions $n \ll m/\log(m)$, computational savings can be achieved, since the computational complexity is $O(nm\cdot\mbox{log}(m))$, which would be less than $O(m^2)$.
We use the energy distances \citep{szekely2004testing} as an example to solidify our motivation.
The energy statistic is used to test the equality between two distributions.
More precisely, suppose $X_1, . . . , X_{n_1} \in\mathbb{R}^p, p \geq 1$ are independent and identically distributed (i.i.d.), sampled from the distribution $F_X$, and $Y_1, . . . , Y_{n_2} \in\mathbb{R}^p$ are i.i.d., sampled from the distribution $F_Y$.
The two-sample test statistic (also called the energy statistic) for testing the two-sample hypothesis
$$
H_0: F_X=F_Y
$$
is defined as \citep{szekely2004testing}:
\begin{equation}
\label{energy}
\mathcal{E}_{n_1,n_2} \triangleq
\frac{2}{n_1 n_2}\sum\limits_{i=1}^{n_1}\sum\limits_{j=1}^{n_2}\left\|X_i-Y_j\right\|
-\frac{1}{n_1^2}\sum\limits_{i=1}^{n_1}\sum\limits_{k=1}^{n_1}\left\|X_i-X_k\right\|
-\frac{1}{n_2^2}\sum\limits_{j=1}^{n_2}\sum\limits_{k=1}^{n_2}\left\|Y_j-Y_k\right\|,
\end{equation}
where $\left\|X_i-Y_j\right\|, \left\|X_i-X_k\right\|, \left\|Y_j-Y_k\right\|$ are the distances from the two samples.
Note that the statistic $\mathcal{E}_{n_1,n_2}$ solely depends on three types of inter-point distances:
$\left\|X_i-Y_j\right\|,
\left\|X_i-X_k\right\|,
\left\|Y_j-Y_\ell\right\|,
i,k = 1,\ldots,n_1,
j,\ell=1,\ldots,n_2.$
Denote $m = n_1+n_2$.
\cite{huang2017efficient} have showed that it can be efficiently computed with computational complexity $O(m\mbox{log}(m))$ in the univariate case (i.e., $p=1$).
When $X_i$\rq s and $Y_j$\rq s are multivariate (i.e., we have $p>1$), random projections have been proposed to find a fast approximation to the statistic $\mathcal{E}_{n_1,n_2}$.
For example, \cite{huang2017efficient} gave a fast algorithm that is based on random projections, which can achieve $O(n m \cdot \mbox{log}(m))$ computational complexity, where $n$ is the number of random projections.
Note that the approach in \cite{huang2017efficient} is a pure Monte Carlo approach.
The recent advances in the quasi-Monte Carlo methods \citep{niederreiter1992random, morokoff1995quasi} have demonstrated that in some settings, utilizing pre-determined projections can lead to better performance than the completely random ones in the pure Monte Carlo approach.
Quasi-Monte Carlo methods sometimes enjoy faster rate of convergence, e.g., \cite{asmussen2007stochastic}.
Our approach turns a distance calculation in a multivariate situation to the one in a univariate situation.
The proposed approach
\begin{enumerate}\setlength\itemindent{1em}
\item[{\bf P1.}] first projects each multivariate variable along some pre-specified optimal directions to corresponding one-dimensional subspaces (the projected values are univariate),
\item[{\bf P2.}] then the sum of the $\ell_1$ norm of the projected values is used to approximate the associated distance in the multivariate setting.
\end{enumerate}
More specifically, let's suppose the multivariate variable is $v = (v_1,...,v_p) \in \mathbb{R}^p$.
Recall that the norm of $v$ is
$$
||v||=\sqrt{\sum\limits_{i=1}^p v_i^2}.
$$
For $n \geq 1$, our objective is to identify the projection directions, which can be represented by vectors $u_1, u_2, ..., u_n \in \mathbb{R}^p$, and a predetermined constant $C_n \in \mathbb{R}$, such that for any $v\in \mathbb{R}^p$, we have
\begin{equation
||v||\approx C_n\sum\limits_{i=1}^n\left|u_i^Tv\right|.
\end{equation}
Consequently in step {\bf P2.}, when one needs to compute a distance $\|X_i - Y_j\|$, one can alternatively compute $C_n \sum\limits_{i=1}^n \left|u_i^T X_i - u_i^T Y_j\right|$.
Note that $u_i^T X_i$ and $u_i^T Y_j$ are univariate.
Therefore, the fast algorithm in the one-dimensional case can be utilized.
We continue with the example of the energy distances.
Recall that the pre-specified directions are supposed to be $u_1, . . . , u_n$.
The projected values of the corresponding multivariate variables then become
\begin{eqnarray*}
X_{wi} &=& u_w^TX_i\in\mathbb{R}, w=1, . . ., n; i = 1, . . ., n_1; \mbox{ and } \\
Y_{wj} &=& u_w^TY_j\in\mathbb{R}, w=1, . . ., n; j = 1, . . ., n_2.
\end{eqnarray*}
The distance between any two multivariate variables can be approximated by the sum of these projections multiplying by a constant:
$$
\|X_i - Y_j\| \approx C_n\sum\limits_{w=1}^n|X_{wi}-Y_{wj}|.
$$
Therefore, the statistic $\mathcal{E}_{n_1,n_2}$ in \eqref{energy} can be approximated by
\begin{eqnarray}
\mathcal{E}_{n_1,n_2} &\approx&
C_n\Big(\frac{2}{n_1n_2}\sum\limits_{i=1}^{n_1}\sum\limits_{j=1}^{n_2}\sum\limits_{w=1}^{n}\left\|X_{wi}-Y_{wj}\right\|
-\frac{1}{n_1^2}\sum\limits_{i=1}^{n_1}\sum\limits_{k=1}^{n_1}\sum\limits_{w=1}^{n}\left\|X_{wi}-X_{wk}\right\| \nonumber \\
&& -\frac{1}{n_2^2}\sum\limits_{j=1}^{n_2}\sum\limits_{k=1}^{n_2}\sum\limits_{w=1}^{n}\left\|Y_{wj}-Y_{wk}\right\|\Big) \nonumber \\
&=& C_n\Big(\frac{2}{n_1n_2}\sum\limits_{i=1}^{n_1}\sum\limits_{j=1}^{n_2}\sum\limits_{w=1}^{n}\left|X_{wi}-Y_{wj}\right|
-\frac{1}{n_1^2}\sum\limits_{i=1}^{n_1}\sum\limits_{k=1}^{n_1}\sum\limits_{w=1}^{n}\left|X_{wi}-X_{wk}\right| \label{eq:energy_p} \\
&&-\frac{1}{n_2^2}\sum\limits_{j=1}^{n_2}\sum\limits_{k=1}^{n_2}\sum\limits_{w=1}^{n}\left|Y_{wj}-Y_{wk}\right|\Big). \nonumber
\end{eqnarray}
The second equation is true because in the one-dimensional case, the $\ell_2$ norm becomes the absolute value.
Then one can apply the fast algorithms for univariate variables to calculate the energy statistic in \eqref{eq:energy_p}.
\textbf{Remark:} \textit{Our method is not restricted to the calculation of the energy statistic, or other distance-based statistics.
It can also be applied to the calculation of the distance-based smooth kernel functions.}
In this paper, we first give a detailed description of our strategy to find the optimal pre-specified projection directions.
We formulate the searching for optimal projection directions problem as a minimax optimization problem.
Let $\left\{u_1, u_2, \cdots, u_n\right\}$ denote the optimal set of projection directions, they should minimize the worst-case difference between the true distance and the approximate distance.
Equation \eqref{eq:intuit} below shows this idea in the mathematical form:
\begin{equation}
\label{eq:intuit}
\min\limits_{\begin{subarray}{c}
C_n, u_i: \\
\|u_i\|=1, i=1,\cdots, n\end{subarray}}
\max\limits_{v: \|v\|_2\le 1}
\left|C_n\sum_{w=1}^n\left|u_w^T v \right|-\|v\|\right|.
\end{equation}
Discussion on how to solve the above problem is presented in Section \ref{sec:formulation}.
In general, the problem in \eqref{eq:intuit} is a nonconvex optimization problem, which is potentially NP-hard.
We found that in two special cases, the optimal directions can be derived analytically:
(a) the $2$-dimensional case and
(b) when the dimension is equal to the number of projections.
More details on these two special cases are presented in Section \ref{sec:anal}.
In general cases, we propose a greedy algorithm to find the projection directions.
Note that the greedy algorithm terminates at a local optimal solution to \eqref{eq:intuit}.
In this case, we cannot theoretically guaranteed that the found directions correspond to the global solution to the problem in \eqref{eq:intuit}, which is the case in most nonconvex optimization problems.
At the same time, the simulations show that our approach can still outperform the pure Monte Carlo approach in many occasions.
The rest of this paper is organized as follows.
Section \ref{sec:formulation} shows the formulation of our problem.
Section \ref{sec:anal} provides the analytical solutions to the problem in \eqref{eq:intuit} in two special cases.
Section \ref{sec:general} presents the numerical algorithm for the general cases.
In Section \ref{sec:simulation}, the simulation results of our method are furnished.
Section \ref{sec:conc} contains the conclusion and a summary of our work.
All the technical proofs are relegated to the appendix (Section \ref{sec:appendix}).
We adopt the following notations.
Throughout this paper, we use $p$ to denote the dimension of the data.
The sample size is denoted by $m$.
The number of projections is denoted by $n$.
\section{Problem formulation}
\label{sec:formulation}
As mentioned above, in order to estimate the distance between two multivariate variables, we project them onto some pre-specified one-dimensional linear subspaces.
We present details in the following.
Suppose the multivariate variable is $v = (v_1,...,v_p) \in \mathbb{R}^p$.
Recall that the norm of vector $v$ is
$$
||v||=\sqrt{\sum\limits_{i=1}^pv_i^2}.
$$
Our objective is to design $u_1, u_2, ..., u_n \in \mathbb{R}^p$, for $n \geq 1,$ and $C_n \in \mathbb{R}$, such that for any $v\in \mathbb{R}^p$, we have
\begin{equation}\label{appr}
||v||\approx C_n\sum\limits_{i=1}^n\left|u_i^Tv\right|.
\end{equation}
We would like to turn a distance (i.e., norm) of a multivariate variable $v$ into a weighted sum of the absolute values of some of its one dimensional projections (i.e., $u_i^Tv$\rq s), knowing that the one dimensional projections may facilitate efficient numerical algorithms.
Without loss of generality, we may assume $||v|| = 1$.
The approximation problem in \eqref{appr} can be formulated into the following problem:
\begin{equation}\label{obj}
\min\limits_{C_n, u_1, ..., u_n}\max\limits_{v: ||v||_2=1}\left|C_n\sum_{i=1}^n\left|u_i^Tv\right|-1\right|.
\end{equation}
In words, we would like to select $u_1,..., u_n$ and $C_n$ such that the approximation in \eqref{appr} has the minimal discrepancy in the worst case.
One can verify that the problem in \eqref{obj} and the problem in \eqref{eq:intuit} share the same solution.
To solve the problem in \eqref{obj}, the following two quantities are needed.
For fixed $u_1, u_2, ..., u_n$, we define
\begin{eqnarray}
\label{eq:vmax}
V_{\max} &=& \max\limits_{v:||v||_2=1}\sum\limits_{i=1}^n\left|u_i^Tv\right|, \\
V_{\min} &=& \min\limits_{v:||v||_2=1}\sum\limits_{i=1}^n\left|u_i^Tv\right|, \label{eq:vmin}
\end{eqnarray}
where $V_{\max}$ and $V_{\min}$ are the maximum and minimum of
$\sum\limits_{i=1}^n\left|u_i^Tv\right|$
among all possible $v$ under the constraint $||v||_2=1$, respectively.
With these two quantities (i.e., $V_{\max}\mbox{ and }V_{\min}$), we have the following result.
\begin{theorem}
\label{Cn}
For given $u_1, u_2,...,u_n\in\mathbb{R}^p$, the optimal value for $C_n$ in the problem \eqref{obj} is
$$
C_n = \frac{2}{V_{\min}+V_{\max}}.
$$
Furthermore, the solutions of $u_1, u_2,...,u_n$ in problem \eqref{obj} are identical to the solutions to the following problem:
\begin{equation}
\label{obj_f}
\max\limits_{\begin{subarray}{c}u_1,...,u_n:\\ ||u_i||=1,\forall i, 1\leq i\leq n\end{subarray}}\frac{V_{\min}}{V_{\max}}.
\end{equation}
\end{theorem}
The above theorem indicates that the minimax problem in \eqref{obj} is equivalent to the maximization problem in \eqref{obj_f}.
Note that in general, both problems are nonconvex, therefore potentially NP-hard.
In our analysis, we found that both formulations (in \eqref{obj} and \eqref{obj_f}) are convenient in various steps of derivation.
Both of them are used in later analysis.
\section{Derivable analytical results}
\label{sec:anal}
We present the two special cases where analytical solutions are derivable.
When the dimension is $2$ (i.e., $p=2$), we show in Section \ref{sec:p=2} that an analytical solution to the problem in \eqref{obj_f} is available.
In Section \ref{sec:p=n}, we present another case (when the dimension of the data is equal to the number of projections, that is we have $n=p$) where an analytic solution to the problem in \eqref{obj_f} is derivable.
\subsection{Special case when the dimension is $2$}
\label{sec:p=2}
When the multivariate variables are two-dimensional, we can get the exact optimal projections that minimize the worse-case discrepancy.
The following theorem describes such a result.
\begin{theorem}
\label{dim2}
When $p = 2$, the $2$-dimensional vectors $u_1, u_2, ..., u_n$ can be represented by
$$
u_i = e^{\sqrt{-1}\theta_i}, i = 1,...,n.
$$
The optimal solution in \eqref{obj_f} has the form
\begin{equation}
\label{optdim2}
\theta_i=\frac{(i-1)\pi}{n} + k_i\pi, i = 1,...,n
\end{equation}
where each $k_i\in\mathbb{N}$.
\end{theorem}
Specially, when $n$ is odd, the optimal solutions can be represented by the equally spaced points on the circle.
Furthermore, we can get the error rate in the $2$-dimensional case, as in the following theorem.
\begin{theorem}
\label{ese_O}
If $u_1, \cdots, u_n$ are chosen according to Theorem \ref{dim2}, we have
$$
\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left\{\left|C_n\sum\limits_{i=1}^n\left|u_i^Tv\right|-1\right|^2\right\} = O\left(\frac{1}{n^2}\right).
$$
\end{theorem}
\textbf{Remark:}\textit{Theorem \ref{ese_O} can be used as a guidance of choosing the number of directions.
Assume we would like to control the squared error to be $\epsilon$.
Then, we can get $\frac{1}{n^2} = \epsilon$, and therefore the number of directions should be larger than $\frac{1}{\sqrt{\epsilon}}$.}
In the above theorem, the random vector $v$ is sampled independently from the Uniform distribution on the unit circle $S^1$.
Note that the squared error rate is $O(1/n^2)$.
The following theorem presents the corresponding rate for the pure random projections.
\begin{theorem}
\label{ese_M}
If $u_1, \cdots, u_n$ are selected base on Monte Carlo, we have
$$
\mathop{\mathbb{E}}\limits_{u_i, v\sim \mbox{Unif}(S^1)}\left\{\left|C_n\sum\limits_{i=1}^n\left|u_i^Tv\right|-1\right|^2\right\} = O\left(\frac{1}{n}\right).
$$
\end{theorem}
In the above theorem, both random vector $v$ and vectors $u_i$'s are independently sampled from the Uniform distribution on the unit circle ($S^1$).
The squared error rate in the pure Monte Carlo case is $O(1/n)$.
These two theorems illustrate the theoretical advantage of adopting the pre-calculated projection directions (in relative to the random projections).
Such a phenomenon has been discovered in the literature regarding the quasi-Monte Carlo methodology.
\subsection{Second special case with provable result}
\label{sec:p=n}
When the dimension is larger than $2$, the problem in \eqref{obj} is challenging.
There is some potentially relevant literature in mathematics, such as the searching for algorithms to locate the equally-distributed points on the surfaces of some high-dimensional spheres \citep{sloan2004extremal, hesse2010numerical, brauchart2014qmc}.
We fail to locate the exact solutions to our problem.
Our analysis indicates that when the number of projections is equal to the dimension, an analytical solution to the problem in \eqref{obj} is derivable.
We present details in the following.
To derive our analytical solution in a special case, we need to revisit two quantities, $V_{\min}$ and $V_{\max}$, which have been introduced in \eqref{eq:vmax} and \eqref{eq:vmin}.
The following lemma is about $V_{\max}$.
\begin{lemma}\label{V_max}
For fixed $u_1, u_2,...,u_n\in\mathbb{R}^p$, we have
\begin{equation}\label{Vmax}
V_{\max} = \max\limits_{s_i\in\{1, -1\}}\left\|\sum\limits_{i=1}^ns_iu_i\right\|.
\end{equation}
\end{lemma}
Lemma \ref{V_max} points out a way to calculate $V_{\max}$, that is, given binary $s_i$'s, finding out the linear combination $\sum\limits_{i=1}^ns_iu_i$ with the maximal norm out of the all possible $2^n$ linear combinations.
Let $\{s_i^{\max}\in\{1,-1\}: i = 1, ..., n\}$ denote the solution for \eqref{Vmax} when $u_1,\cdots, u_n$ are given.
The Algorithm \ref{alg:s} formally presents the aforementioned approach.
Assume we are in the $k$-th loop, where the $u_j$\rq s are known, which are denoted by $u_1^{(k)}, u_2^{(k)}, ..., u_n^{(k)}$.
Let $s_i^{(k)}$\rq s denote the $s_i$'s that can achieve $V_{\max}$ in the $k$-th loop.
We have the Algorithm \ref{alg:s}.
\begin{algorithm}[htbp]
\caption{Find $s_i^{\max}$\rq s in the $k$-loop}
\label{alg:s}
\begin{algorithmic}[1]
\REQUIRE{Unit vectors $u_1^{(k)}, u_2^{(k)}, \dots, u_n^{(k)}\in S^{p-1}$ are given.}
\ENSURE {$s_i^{(k)}$\rq s.}
\FORALL {binary combination of $s_i^{(k)}$\rq s}
\STATE Calculate the value $\left\|\sum\limits_{i=1}^ns_iu_i\right\|$.
\ENDFOR
\STATE The binary combination that can make the value of $\left\|\sum\limits_{i=1}^ns_iu_i\right\|$ be the maximum among all the possible values,
is the $s_i^{\max}$\rq s, which is denoted as $s_i^{(k)}$\rq s.
\end{algorithmic}
\end{algorithm}
As for $V_{\min}$, suppose $v_{\min}$ is a minimizer of $V_{\min}$.
We have the following property for $v_{\min}$.
\begin{lemma}
\label{prop}
For fixed $u_1, u_2,...,u_n\in\mathbb{R}^p$, if $\Omega$ is an intersection of $S^{p-1}$ and a linear subspace with at least 2 dimensions, then the solution to the minimization problem
$$
\min_{v\in \Omega} f(v) = \sum\limits_{i=1}^n\left|u_i^Tv\right|
$$
must have $u_j^Tv_{\min}=0$ for at least one $j$ $(1\leq j\leq n)$.
\end{lemma}
Geometrically, the above lemma indicates that vector $v_{\min}$ should be orthogonal to at least one of the projection vector $u_j$.
For vector $v_{\min}$, we will need the following definition to further our derivation.
\begin{definition}[maximal subset]
\label{def:max-subset}
We call $\Omega(v_{\min})$ a maximal subset of the set $\{u_1,...,u_n\}$ if it satisfies
$$
\Omega(v_{\min}) = \left\{u_j: u_j^Tv_{\min}=0\right\} \subset \{u_1,...,u_n\},
$$
and it cannot be a strict subset for another $\Omega(v^\prime_{\min})$ where $v^\prime_{\min}$ is a minimizer that is different from $v_{\min}$.
\end{definition}
Lemma \ref{prop} ensures that the set $\Omega(v_{\min})$ cannot be empty.
The following lemma shows that the linear subspace that is spanned by the elements of $\Omega(v_{\min})$ must have certain dimensions.
\begin{lemma}
\label{Omega_0}
If $\Omega(v_{\min})$ is a maximal subset of $u_1,...,u_n$,
we must have
$$
\mbox{rank}\left(\Omega(v_{\min})\right)=p-1,
$$
for any minimizer $v_{\min}$.
\end{lemma}
Recall $p$ is the dimension of the data.
The above lemma essentially states that the space that is spanned by the elements of $\Omega(v_{\min})$ is the orthogonal complement subspace of the one-dimensional space that is spanned by the vector $v_{\min}$.
One direct corollary of Lemma \ref{Omega_0} is that the cardinality of the set $\Omega(v_{\min})$ is at least $p-1$.
Consequently, the total number of possible sets (of $\Omega(v_{\min})$) is no more than $n \choose p-1$.
This inspires us to use Algorithm \ref{alg:vmin} to find $v_{\min}$ as well as $\Omega(v_{\min})$ if all the $u_j$\rq s are given.
Here suppose we are in the $k$-th loop where the $u_j$\rq s are known, which are $u_1^{(k)}, u_2^{(k)}, ..., u_n^{(k)}$.
\begin{algorithm}[htbp]
\caption{Find $v_{\min}$ and $\Omega(v_{\min})$ in the $k$-loop}
\label{alg:vmin}
\begin{algorithmic}[1]
\REQUIRE{Unit vectors $u_1^{(k)}, u_2^{(k)}, \dots, u_n^{(k)}\in S^{p-1}$ are given.}
\ENSURE {$v^{(k)}$ and $\Omega(v^{(k)})$.}
\FORALL {$(p-1)$ combination of $u_i^{(k)}$\rq s, denoted as $S^u_t$}
\WHILE {$\mbox{rank}(S^u_t)<p-1$}
\STATE Add another $u_j$ that is not in the set $S_t^u$;
\ENDWHILE
\STATE Find the orthogonal direction of the set $S_t^u$, which is one of the candidates of $v^{(k)}$, denoted as $v^{(k)}_t$,
and calculate the value of $f(v^{(k)}_t) = \sum\limits_{i=1}^{n} \left|\left(u_i^{(k)}\right)^Tv^{(k)}_t \right|$.
\ENDFOR
\STATE The $v^{(k)}_t$, that can make the value of $f(v^{(k)}_t)$ be the minimum among all the possible $f(v^{(k)}_t)$ values,
is the $v_{\min}$, which is denoted as $v^{(k)}$,
and the corresponding $S_t^u$ set is the set $\Omega(v_{\min})$, which is denoted as $\Omega(v^{(k)})$.
\end{algorithmic}
\end{algorithm}
From Lemma \ref{Omega_0} we can get the exact solution for the special case when the number of projection directions is equal to the dimension of the multivariate variables, which is described in the following theorem.
\begin{theorem}
\label{dim=n}
When the number of projections is equal to the dimension of the data, i.e., we have $n = p$, the optimal solution in \eqref{obj_f} satisfies the following condition:
\begin{equation}\label{optdim=n}
u_i^Tu_j=0, \forall i \not= j.
\end{equation}
The above is equivalent to stating that the set $\left\{u_1, u_2, \cdots, u_n\right\}$ forms an orthonormal basis in $\mathbb{R}^p$.
\end{theorem}
\section{Numerical approach in general cases}
\label{sec:general}
When $p>2$ and $n\neq p$, we propose an algorithm to identify the optimal projections $u_1, u_2, ..., u_n$, such that they solve \eqref{obj_f}.
Per Lemma \ref{V_max} and the definition of $s_i^{\max}$\rq s,
the $V_{\max}$ can be written as:
$$
V_{\max} = \left\|\sum\limits_{i=1}^ns_i^{\max}u_i\right\|.
$$
According to Lemma \ref{Omega_0}, we have
\begin{eqnarray*}
V_{\min} = \sum\limits_{i=1}^n\left|u_i^Tv_{\min}\right|
&=& \sum\limits_{u_i\in\Omega(v_{\min})}\left|u_i^Tv_{\min}\right|
+ \sum\limits_{u_i\not\in\Omega(v_{\min})}\left|u_i^Tv_{\min}\right| \\
&=& \sum\limits_{u_i\not\in\Omega(v_{\min})}\left|u_i^Tv_{\min}\right|.
\end{eqnarray*}
So when $u_1,\cdots, u_n$ are given, $\frac{V_{\min}}{V_{\max}}$ can be written as
\begin{equation}\label{VminVmax}
\frac{V_{\min}}{V_{\max}}
= \frac{\sum\limits_{u_i\not\in\Omega(v_{\min})}\left|u_i^Tv_{\min}\right|}{\left\|\sum\limits_{i=1}^ns_i^{\max}u_i\right\|},
\end{equation}
where $v_{\min}$ and $\Omega(v_{\min})$ are defined in Section \ref{sec:p=n}.
We assume that the set $\Omega(v_{\min})$ corresponds to the minimum over all $n \choose p-1$ possible sets, and ($s_i^{\max}$)\rq s maximize the norm of $\sum\limits_{i=1}^n s_i^{\max}u_i$.
We use a method that is similar to the coordinate descent algorithm \citep{nesterov2012efficiency,wright2015coordinate} to search for the optimal solutions of \eqref{obj_f}.
Details of our algorithm can be found in Algorithm \ref{alg:general}.
The optimal solution can be achieved in circular iterations: maximizing \eqref{VminVmax} with respect to one $u_i$, while the others are fixed.
We then iteratively maximize the objective function in \eqref{VminVmax} until the value of the objective function \eqref{VminVmax} cannot be increased.
We derive the iteration strategy in the following.
Let $v^{(k)}$ be the minimizer of $\sum\limits_{i=1}^n\left|u_i^Tv\right|$ at the $k$th iteration.
Let $\Omega^{(k)}$ denote the minimum over all $n \choose p-1$ possible sets at the $k$th iteration.
For any $u_j^{(k)}\not\in\Omega^{(k)}$, without loss of generality, we assume that $u_1\not\in\Omega^{(k)}$.
The objective function in \eqref{VminVmax} can be written as
\begin{equation}\label{VminVmax_iter}
\frac{V_{\min}}{V_{\max}}
= \frac{\left|u_1^Tv^{(k)}\right|
+\sum\limits_{i>1, u_i\not\in\Omega^{(k)}}\left|u_i^Tv^{(k)}\right|}{\left\|s_1^{\max}u_1
+ \sum\limits_{i=2}^ns_i^{\max}u_i\right\|}.
\end{equation}
Without loss of generality, we can assume $s_1^{\max} = 1$.
This is because, recalling that ($s_i^{\max}$)'s are binary, we have
$$
\left\|s_1^{\max}u_1 + \sum\limits_{i=2}^ns_i^{\max}u_i\right\|
= \left\|u_1 + \sum\limits_{i=2}^ns_1^{\max}s_i^{\max}u_i\right\|.
$$
The expression in \eqref{VminVmax_iter} can then be rewritten as
\begin{equation}
\label{eq:u1}
\frac{\left|u_1^Tv^{(k)}\right|+A}{\left\|u_1 +B\right\|},
\end{equation}
where
$$
A = \sum\limits_{i>1, u_i\not\in\Omega^{(k)}}\left|u_i^Tv^{(k)}\right|, \mbox{ and }
B = \sum\limits_{i=2}^ns_i^{\max}u_i.
$$
Note that quantities $A$ and $B$ do not depend on $u_1$.
Our objective is to derive a strategy to maximize the quantity in \eqref{eq:u1} as a function of the vector variable $u_1$.
We first solve a constrained version of the above maximization problem.
We define
$\Sigma(v, \theta) = \left\{x:\|x\| = 1, \left<x,v\right>=\theta\right\}$, for any fixed $\theta\in[0,\pi)$,
where $\left<\cdot,\cdot \right>$ denote the angle between two vectors.
Conditioning on $u_1 \in \Sigma(v, \theta)$, and $v=v^{(k)}$, maximizing the function in \eqref{eq:u1} is equivalent to maximizing the following function:
\begin{equation}\label{opt_u1}
\frac{\left|\cos\theta\right|+A}{\left\|u_1 +B\right\|}.
\end{equation}
Note that the numerator is not a function of $u_1$.
Consequently, it is equivalent to minimizing
$$
\left\|x+B\right\|, \mbox{ where } x\in\Sigma(v, \theta).
$$
The following lemma presents an analytical solution to the above minimization problem.
\begin{lemma}
\label{u_1}
Given a vector $B$, a constant $\theta \in [0, \pi)$, and a unit-norm vector $v$, the solution to the following problem
\begin{equation}\label{u1}
\min_{x:\|x\|=1,\left<x,v\right>=\theta}\left\|x+B\right\|^2
\end{equation}
is
\begin{equation}\label{x}
x = v\cos\theta + \frac{|\sin\theta|}{\sqrt{B^TB-(v^TB)^2}}\left[(v^TB)v-B\right].
\end{equation}
\end{lemma}
Using the solution in \eqref{x} to substitute the $u_1$ in \eqref{opt_u1}, we have
\begin{equation}\label{opt_u1_2}
\frac{\left|\cos\theta\right|+A}{\left\|u_1 +B\right\|}
= \frac{\left|\cos\theta\right|+A}{\left\|v\cos\theta +B
+ \frac{|\sin\theta|}{\sqrt{B^TB-(v^TB)^2}}\left[(v^TB)v-B\right]\right\|}.
\end{equation}
Maximizing \eqref{opt_u1} with respect to $\theta$ is equivalent to maximizing \eqref{opt_u1_2}.
For fixed $A$, $B$, and $v$, the right hand side of \eqref{opt_u1_2} is a function of $\theta$.
The following Theorem \ref{gtheta} gives the solution to the above problem.
\begin{theorem}
\label{gtheta}
The solutions of maximizing \eqref{opt_u1} with respect to $\theta$ are the zeros of the following function:
\begin{equation}\label{eq:g}
g(\theta) =
\left\{ \begin{array}{ll}
\sqrt{B^TB}\left[\cos\alpha
+ A\cos(\alpha-\theta) - \sin\theta\sin(\alpha - \theta)\right] & \\
-(1+B^TB)\sin\theta, & \mbox{ if } \theta\in[0,\frac{\pi}{2}),\\
\sqrt{B^TB} \left[-\cos\alpha + A\cos(\alpha-\theta) + \sin\theta\sin(\alpha - \theta)\right] & \\
+ (1+B^TB)\sin\theta &\mbox{ if } \theta\in[\frac{\pi}{2},\pi),
\end{array}
\right.
\end{equation}
where $\alpha$ satisfies
$\sin\alpha = \frac{v^TB}{\sqrt{B^TB}},$ and
$\cos\alpha = \frac{\sqrt{B^TB-(v^TB)^2}}{\sqrt{B^TB}}.$
\end{theorem}
The above theorem indicates that one can adopt a line search algorithm to compute for $\theta$.
Based on all the above, the Algorithm \ref{alg:general} (below) furnishes a coordinate ascent scheme to maximize the objective in \eqref{obj_f}.
\begin{algorithm}[htbp]
\caption{Optimal projection algorithm}
\label{alg:general}
\begin{algorithmic}[1]
\REQUIRE {Set a threshold $\Delta>0$, initial unit vectors $u_1^{(0)}, u_2^{(0)}, ..., u_n^{(0)}\in S^{p-1}$.
Thus, by Algorithm \ref{alg:s} and \ref{alg:vmin}, we can get the corresponding values $v^{(0)}$,$\Omega^{(0)}(v^{(0)})$, and $s_i^{(0)}$\rq s.}
\REPEAT
\STATE In the $k$-th loop, suppose the previous $u_1^{(k-1)}, u_2^{(k-1)}, \ldots, u_n^{(k-1)}$ are known.
\FORALL{$u_j^{(k-1)}\not\in\Omega^{(k-1)}(v^{(k-1)})$}
\STATE Find the zeros of the function $g(\theta)$ in \eqref{eq:g} in Theorem \ref{gtheta}, where
$v = v^{(k-1)},$
$B = \sum\limits_{i\not=j}s_j^{(k-1)} s_i^{(k-1)} u_i^{(k-1)}$,
and denote the zeros as $\theta^*$.
\STATE According to Lemma \ref{u_1}, the new $u_j^{(k)}$ would be
$v\cos\theta^* + \frac{|\sin\theta^*|}{\sqrt{B^TB-(v^TB)^2}}\left[(v^TB)v-B\right].$
\STATE By Algorithm \ref{alg:s} and \ref{alg:vmin}, we can get the corresponding values $v^{(k)}$,$\Omega^{(k)}(v^{(k)})$, and $s_i^{(k)}$\rq s,
based on the newly updated $u_j$\rq s, which also give us the value of $V_{\min}$ and $V_{\max}$.
\STATE Compute $V_{\min}/V_{\max}$.
\ENDFOR
\STATE Pick the $u_j^{(k)}\not\in\Omega^{(k-1)}(v^{(k-1)})$ that gives the maximal value of $V_{\min}/V_{\max}$ in the above loop.
\IF {The value of $V_{\min}/V_{\max}$ decreases}
\STATE Go back to $u_j^{(k-1)}$.
\ENDIF
\UNTIL{The increment of $V_{\min}/V_{\max}$ is less than $\Delta$}.
\end{algorithmic}
\end{algorithm}
\section{Simulations}
\label{sec:simulation}
In the previous section, the optimal projections for both the special cases and the general case are provided.
The simulations will follow the same order.
The simulations are about the comparison of the Monte Carlo method and our method for the special cases and then for a general case.
According to \cite{huang2017efficient}, Monte Carlo method is to select some random directions, denoted as $w_i$, $i=1, \dots, n$, on the unit sphere $S^{p-1}$ and project the vector we would like to estimate, that is $v$, along these directions, so the norm of the vector $v$ could be estimated as
$$
\|v\| \approx C^\prime_p\frac{1}{n}\sum\limits_{i=1}^n |w_i^Tv|,$$
where $C^\prime_p = \frac{\sqrt{\pi}\Gamma(\frac{p+1}{2})}{\Gamma(\frac{p}{2})}$.
In all the experiments, we randomly select $100$ unit vectors on the sphere as the vectors that we would like to estimate, in order to get the mean squared error for comparison between the Monte Carlo method and the method we propose.
\subsection{When the dimension is $2$}
When the dimension is equal to $2$, the exact solution can be found as well as the mean squared error rate.
So we randomly select $100$ unit vectors on the sphere as the vectors that we would like to estimate.
For both the Monte Carlo method and our optimal projection method, we calculate the mean squared error over these $100$ vectors.
More specifically, the squared error between the true norm of the vector, which is $1$, and the estimated norm is calculated for each of the $100$ unit vectors when the number of directions is fixed.
By taking the mean of the $100$ squared errors from the previous step, we get the mean squared error for given number of directions.
The number of directions used in our simulation is from $2$ to $10000$.
Figure \ref{fig:dim2} shows the comparison between our method and Monte Carlo method regarding the logarithm of the mean squared error and the number of projection directions.
From the figure, we can see that our method performs better than the Monte Carlo, and the advantage becomes more obvious when the number of projection directions increases.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{fig/dim2_new.png}
\caption{Optimal projection vs. Monte Carlo in the 2 dimensional case}
\label{fig:dim2}
\end{figure}
\subsection{When we have $n=p$}
When the dimension $p$ is equal to the number of projection directions $n$, recall that in Theorem \ref{dim=n},
we give the exact solution of the pre-specified directions.
Similar to what we have done in the $2$-dimensional case, we randomly select 100 unit vectors on the sphere $S^{p-1}$,
with dimension $p$ varying from $8$ to $11$.
So the number of projection directions is varying from $8$ to $11$ correspondingly.
We calculate the mean squared error of both the Monte Carlo method and our optimal projection method for each $p$ using the same strategy as before.
The details are in the Figure \ref{fig:np}, where the $x$-axis represents the dimension, and $y$-axis represents the mean squared error.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{fig/np.png}
\caption{Optimal projection vs. Monte Carlo in the $n=p$ case}
\label{fig:np}
\end{figure}
\subsection{General setting: $n > p$}
When the dimension $p$ is larger than 2 and $n\not=p$, the exact solution of \eqref{obj_f} can not be obtained.
Therefore, we adopt the Algorithm \ref{alg:general}.
Like in previous simulations, we randomly select 100 unit vectors on the sphere $S^{p-1}$, with dimension $p$ varying from $3$ to the number of directions minus $1$, and the fixed number of directions to be 8, 9, 10, 11, respectively, and calculate the mean squared error of both the Monte Carlo method and our optimal projection method for each $p$ using the same strategy as before.
Figure \ref{fig:n8}, \ref{fig:n9}, \ref{fig:n10} and \ref{fig:n11} show the comparison, where the $x$-axis represents the dimension, and $y$-axis represents the mean squared error.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{fig/n8.png}
\caption{Optimal projection vs. Monte Carlo for dimension varying from $3$ to $7$ in the case $n = 8$}
\label{fig:n8}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{fig/n9.png}
\caption{Optimal projection vs. Monte Carlo for dimension varying from $3$ to $8$ in the case $n = 9$}
\label{fig:n9}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{fig/n10.png}
\caption{Optimal projection vs. Monte Carlo for dimension varying from $3$ to $9$ in the case $n = 10$}
\label{fig:n10}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{fig/n11.png}
\caption{Optimal projection vs. Monte Carlo for dimension varying from $3$ to $10$ in the case $n = 11$}
\label{fig:n11}
\end{figure}
Overall, we can see that our method performs better than the Monte Carlo method.
\section{Conclusion}
\label{sec:conc}
We propose a new method to calculate the distance, which is critical in computing the distance-based statistics, and can also be utilized in the calculation of the kernel functions that are distance-based and smooth.
The main idea is to use the sum of the norms of the projections along a set of pre-calculated directions to approximate the original norm.
By doing so, one can utilize the fast algorithm for univariate variables that has been proposed by \citet{huo2016fast}.
The advantage is that the computational complexity is reduced from $O(m^2)$ to $O(m \mbox{log}(m))$ where $m$ is the sample size.
These pre-specified directions can be found by minimizing the difference between the estimated distance and the true value in the worst case.
The associated problem is eventually a nonconvex optimization problem.
We derive the exact solutions when dimension is equal to either $2$ or the number of projection directions.
In general cases, we propose an algorithm to find the projection directions.
The simulations show the advantage of the proposed method versus the pure Monte Carlo approach, via comparing the mean squared errors.
\begin{acknowledgement}
This project is partially supported by the Transdisciplinary Research Institute for Advancing Data Science (TRIAD), http://
triad.gatech.edu, which is a part of the TRIPODS program at NSF and locates at Georgia Tech, enabled by the NSF grant
CCF-1740776.
Both authors are also partially supported by the NSF grant DMS-1613152.
\end{acknowledgement}
\section{Appendix}
\addcontentsline{toc}{section}{Appendix}
\label{sec:appendix}
All the proofs are included in this section, including
a proof of Theorem \ref{Cn} (Section \ref{sec:app:proof-1})),
a proof of Theorem \ref{dim2} (Section \ref{sec:app:proof-2}),
a proof of Theorem \ref{ese_O} (Section \ref{sec:app:proof-3}),
a proof of Theorem \ref{ese_M} (Section \ref{sec:app:proof-4}),
a proof of Lemma \ref{V_max} (Section \ref{sec:app:proof-5}),
a proof of Lemma \ref{prop} (Section \ref{sec:app:proof-6}),
a proof of Lemma \ref{Omega_0} (Section \ref{sec:app:proof-7}),
a proof of Theorem \ref{dim=n} (Section \ref{sec:app:proof-8}),
a proof of Lemma \ref{u_1} (Section \ref{sec:app:proof-9}), and
a proof of Theorem \ref{gtheta} (Section \ref{sec:app:proof-10}).
Some of these proofs involves detailed and potentially tedious derivations.
We try to furnish as much details as deemed reasonable.
\subsection{Proof of Theorem \ref{Cn}} \label{sec:app:proof-1}
\begin{proof}
By definition of $V_{\min}$ and $V_{\max}$, we have
$$C_nV_{\min}-1\leq C_n\sum_{i=1}^n\left|u_i^Tv\right|-1\leq C_nV_{\max}-1.$$
The above leads to the following
\begin{equation}\label{max}
\max\limits_{v: ||v||_2=1}\left|C_n\sum_{i=1}^n|u_i^Tv|-1\right|
= \max\left\{|C_nV_{\min}-1|, |C_nV_{\max}-1|\right\}.
\end{equation}
Consider the right hand side of the above as a function of $C_n$, it is verifiable that the minimum is achieved when
$$
1-C_nV_{\min}=C_nV_{\max}-1\mbox{, which leads to, }C_n = \frac{2}{V_{\min}+V_{\max}}.
$$
Bringing the above to \eqref{max}, we have
\begin{equation}\label{max_2}
\left|\frac{2}{V_{\min}+V_{\max}}V_{\min}-1\right|
= \frac{V_{\max}-V_{\min}}{V_{\max}+V_{\min}}
= \frac{2}{1+\frac{V_{\min}}{V_{\max}}}-1.
\end{equation}
From the above, it is evident that minimizing the right hand of \eqref{max_2} is equivalent to the following
$$
\max\limits_{u_1,...,u_n: ||u_i||_2=1}\frac{V_{\min}}{V_{\max}}.
$$
From all the above, the lemma is proved. \qed
\end{proof}
\subsection{Proof of Theorem \ref{dim2}}
\label{sec:app:proof-2}
\begin{proof}
Without loss of generality, we assume $\theta_i = \alpha_i + k_i\pi, \mbox{ where }\alpha_1\leq\alpha_2\leq ...\leq\alpha_n\in[0,\pi).$ Then the problem in \eqref{obj_f} can be written as
$$
\max\limits_{\alpha_i: i=1,...,n}\frac{\min\limits_\theta f(\theta)}{\max\limits_\theta f(\theta)},
$$
where $f(\theta) = \sum\limits_{i =1}^n|cos(\alpha_i-\theta)|.$
Let $\delta_i = \alpha_{i+1} - \alpha_i, i = 1, ..., n-1, \mbox{ and } \delta_n = \alpha_1 - \alpha_n + \pi.$
We have
$$
\sum\limits_{i=1}^n\delta_i = \pi.
$$
For given $\alpha_i$, the minimum and the maximum of $f(\theta)$ satisfy
\begin{eqnarray}
\frac{1}{n}\min\limits_\theta f(\theta) &\leq& \frac{1}{n}f(\alpha_i-\frac{\pi}{2}), \mbox{ for } i = 1, \dots, n, \label{eq:f1}\\
\frac{1}{n}\max\limits_\theta f(\theta) &\geq& \frac{1}{n}f\left(\frac{\alpha_i+\alpha_{i+1}}{2}-\frac{\pi}{2}\right), \mbox{ for } i = 1, \dots, n-1, \label{eq:f2-1}\\
\frac{1}{n}\max\limits_\theta f(\theta) &\geq& \frac{1}{n}f\left(\frac{\alpha_n+\alpha_1}{2}\right). \label{eq:f2-2}
\end{eqnarray}
By summing up each side of \eqref{eq:f1} with $i$ from $1$ through $n$, we get
\begin{equation}\label{eq:minf}
\min\limits_\theta f(\theta) \leq \frac{1}{n}\sum\limits_{i =1}^nf(\alpha_i-\frac{\pi}{2}).
\end{equation}
By summing up each side of \eqref{eq:f2-1} with $i$ from $1$ through $n-1$ and adding it to \eqref{eq:f2-2}, we have
\begin{equation}\label{eq:maxf}
\max\limits_\theta f(\theta) \geq \frac{1}{n}\left[\sum\limits_{i=1}^{n-1}f\left(\frac{\alpha_i+\alpha_{i+1}}{2}-\frac{\pi}{2}\right) + f\left(\frac{\alpha_n+\alpha_1}{2}\right)\right].
\end{equation}
Based on \eqref{eq:minf} and \eqref{eq:maxf}, for given $\alpha_i$, we have
$$
\frac{\min\limits_\theta f(\theta)}{\max\limits_\theta f(\theta)} \leq \frac{\frac{1}{n}\sum\limits_{i =1}^nf(\alpha_i-\frac{\pi}{2})}{\frac{1}{n}\left[\sum\limits_{i=1}^{n-1}f\left(\frac{\alpha_i+\alpha_{i+1}}{2}-\frac{\pi}{2}\right) + f\left(\frac{\alpha_n+\alpha_1}{2}\right)\right]}.
$$
Therefore, one can verify the following:
\begin{eqnarray}
\max\limits_{\alpha_i: i=1,...,n}\frac{\min\limits_\theta f(\theta)}{\max\limits_\theta f(\theta)}&\leq&
\max\limits_{\alpha_i: i=1,...,n}\frac{\frac{1}{n}\sum\limits_{i =1}^nf(\alpha_i-\frac{\pi}{2})}{\frac{1}{n}\left[\sum\limits_{i=1}^{n-1}f\left(\frac{\alpha_i+\alpha_{i+1}}{2}-\frac{\pi}{2}\right) + f\left(\frac{\alpha_n+\alpha_1}{2}\right)\right]} \nonumber \\
& = &\max\limits_{\alpha_i: i=1,...,n}\frac{\sum\limits_{i =1}^nf(\alpha_i-\frac{\pi}{2})}{\left[\sum\limits_{i=1}^{n-1}f\left(\frac{\alpha_i+\alpha_{i+1}}{2}-\frac{\pi}{2}\right) + f\left(\frac{\alpha_n+\alpha_1}{2}\right)\right]}. \label{less}
\end{eqnarray}
Denote the numerator of the right hand side of \eqref{less} as $N_n$, and the denominator as $D_n$.
Thus, we have
\begin{equation*}
N_n =
\left\{ \begin{array}{ll}
2\sum\limits_{i=1}^n\left|\sin\delta_i\right| + 2\sum\limits_{i=1}^{n-2}\left|\sin(\delta_i+\delta_{i+1})\right| + 2\sum\limits_{i=1}^{n-3}\left|\sin(\delta_i+\delta_{i+1} +\delta_{i+2})\right| + ... & \\
+ 2\sum\limits_{i=1}^{2}\left|\sin(\delta_i+\delta_{i+1} +... +\delta_{i+n-3})\right|, & \mbox{ if }n\geq 4,\\
2\sum\limits_{i=1}^n\left|\sin\delta_i\right| & \mbox{ if }n=3;
\end{array}
\right.
\end{equation*}
and
\begin{eqnarray*}
D_n &= &\sum\limits_{i=1}^n2\left|\sin\frac{\delta_i}{2}\right| + \sum\limits_{i=2}^{n-1}\sum\limits_{j=1}^{i-1}\left|\sin\left(\frac{\delta_i}{2} + \delta_{i-1} +\delta_{i-2} +...+\delta_j\right)\right| \\
&&+ \sum\limits_{i=1}^{n-2}\sum\limits_{j=i+1}^{n-1}\left|\sin\left(\frac{\delta_i}{2}+\delta_{i+1} + ... +\delta_{j}\right)\right|
+ \sum\limits_{j=2}^{n-1}\left|\sin\left(\frac{\delta_n}{2}+\delta_1 + ... +\delta_{j-1}\right)\right|.
\end{eqnarray*}
We would like to show that when all the $\theta_i$\rq s satisfies \eqref{optdim2}, $\frac{\min\limits_\theta f(\theta)}{\max\limits_\theta f(\theta)}$ is equal to the right hand side of \eqref{less}, which means \eqref{optdim2} is the optimal solution.
In order to do that, we first need to figure out what value the right hand side of \eqref{less} is.
In the following we use perturbation analysis to show that when $\delta_i = \frac{\pi}{n},$ which is equivalent to \eqref{optdim2}, the right hand side achieves the maximum value.
And then we show that the left side is equal to the right side under the condition of \eqref{optdim2}.
Therefore our proof can be completed.
For $n\geq4,$ $N_n$ and $D_n$ are treated as functions of $\Delta$.
Then we have
\begin{eqnarray*}
N_n(\delta_1+\Delta,\delta_2-\Delta,\delta_3,...,\delta_n) &=&
2\left|\sin(\delta_1+\Delta)\right| + 2\left|\sin(\delta_2-\Delta)\right| \\
&&+2\sum\limits_{j=3}^{n-1}\left|\sin(-\Delta+\sum\limits_{i=2}^j\delta_i)\right| + Const,
\end{eqnarray*}
and
\begin{eqnarray*}
\frac{\partial N_n(\delta_1+\Delta,\delta_2-\Delta,\delta_3,...,\delta_n)}{\partial \Delta}\bigg|_{\Delta=0}&=&
2\cos\delta_1\mbox{sign}(\sin\delta_1)-2\cos\delta_2\mbox{sign}(\sin\delta_2) \\
&&-2\sum\limits_{j=3}^{n-1}\cos\left(\sum\limits_{i=2}^j\delta_i\right)
\mbox{sign}\left(\sin\left(\sum\limits_{i=2}^j\delta_i\right)\right).
\end{eqnarray*}
When $\delta_i = \frac{\pi}{n}, i = 1,...,n,$ we have
\begin{eqnarray}
\frac{\partial N_n(\delta_1+\Delta,\delta_2-\Delta,\delta_3,...,\delta_n)}{\partial \Delta}\bigg|_{\Delta=0}&=&
0-2\sum\limits_{j=3}^{n-1}\cos\left(\frac{(j-1)\pi}{n}\right)\mbox{sign}\left(\sin\left(\frac{(j-1)\pi}{n}\right)\right) \nonumber \\
&=&0. \label{eq:Nn}
\end{eqnarray}
Similarly, for $D_n,$ we have
\begin{eqnarray*}
&&D_n(\delta_1+\Delta,\delta_2-\Delta,\delta_3,...,\delta_n) \\
&=& 2\left|\sin\left(\frac{\delta_1+\Delta}{2}\right)\right| + 2\left|\sin\left(\frac{\delta_2-\Delta}{2}\right)\right|
+ \left|\sin\left(\frac{\Delta}{2} + \delta_1 + \frac{\delta_2}{2}\right)\right|\\
&&+\sum\limits_{j=3}^{n-1}\left|\sin\left(-\Delta+\sum\limits_{i=2}^{j-1}\delta_i + \frac{\delta_j}{2}\right)\right|
+\sum\limits_{j=3}^{n}\left|\sin\left(-\frac{1}{2}\Delta+ \frac{\delta_1}{2}+\sum\limits_{i=2}^{j-1}\delta_i\right)\right|\\
&&+\sum\limits_{j=3}^{n-1}\left|\sin\left(-\frac{1}{2}\Delta+ \frac{\delta_2}{2}+\sum\limits_{i=3}^{j}\delta_i\right)\right|+\left|\sin\left(\frac{\delta_n}{2}+\delta_1+\Delta\right)\right| +Const,
\end{eqnarray*}
and
\begin{eqnarray*}
&&\frac{\partial D_n(\delta_1+\Delta,\delta_2-\Delta,\delta_3,...,\delta_n)}{\partial \Delta}\bigg|_{\Delta=0}\\
&=&\cos\frac{\delta_1}{2}\mbox{sign}(\sin\frac{\delta_1}{2})
-\cos\frac{\delta_2}{2}\mbox{sign}(\sin\frac{\delta_2}{2})
+\frac{1}{2}\cos\left(\frac{\delta_2}{2}+\delta_1\right)\mbox{sign}\left(\sin\left(\frac{\delta_2}{2}
+\delta_1\right)\right)\\
&&-\sum\limits_{j=3}^{n-1}\cos\left(\sum\limits_{i=2}^{j-1}\delta_i+ \frac{\delta_j}{2}\right)\mbox{sign}\left(\sin\left(\sum\limits_{i=2}^{j-1}\delta_i+ \frac{\delta_j}{2}\right)\right)\\
&&-\frac{1}{2}\sum\limits_{j=2}^{n-1}\cos\left(\frac{\delta_1}{2} + \sum\limits_{i =2}^{j}\delta_i\right)\mbox{sign}\left(\sin\left(\frac{\delta_1}{2}
+ \sum\limits_{i =2}^{j}\delta_i\right)\right)\\
&&-\frac{1}{2}\sum\limits_{j=3}^{n-1}\cos\left(\frac{\delta_2}{2} + \sum\limits_{i =3}^{j}\delta_i\right)\mbox{sign}\left(\sin\left(\frac{\delta_2}{2}
+ \sum\limits_{i =3}^{j}\delta_i\right)\right)\\
&&+\cos\left(\frac{\delta_n}{2}+\delta_1\right)\mbox{sign}\left(\sin\left(\frac{\delta_n}{2}+\delta_1\right)\right).
\end{eqnarray*}
When $\delta_i = \frac{\pi}{n}, i = 1,...,n,$ we have
\begin{eqnarray}
&&\frac{\partial D_n(\delta_1+\Delta,\delta_2-\Delta,\delta_3,...,\delta_n)}{\partial \Delta}\bigg|_{\Delta=0} \nonumber \\
&=&0+\frac{1}{2}\cos\left(\frac{3\pi}{2n}\right)\mbox{sign}\left(\sin\left(\frac{3\pi}{2n}\right)\right)
-\sum\limits_{j=3}^{n-1}\cos\left(\frac{(2j-3)\pi}{2n}\right)\mbox{sign}\left(\sin\left(\frac{(2j-3)\pi}{2n}\right)\right) \nonumber \\
&&-\frac{1}{2}\sum\limits_{j=2}^{n-1}\cos\left(\frac{(2j-1)\pi}{2n}\right)
\mbox{sign}\left(\sin\left(\frac{(2j-1)\pi}{2n}\right)\right) \nonumber \\
&&-\frac{1}{2}\sum\limits_{j=3}^{n-1}\cos\left(\frac{(2j-3)\pi}{2n}\right)
\mbox{sign}\left(\sin\left(\frac{(2j-3)\pi}{2n}\right)\right) \nonumber \\
&&+\cos\left(\frac{3\pi}{2n}\right)\mbox{sign}\left(\sin\left(\frac{3\pi}{2n}\right)\right) \nonumber \\
&=&0. \label{eq:Dn}
\end{eqnarray}
Define $g(\Delta)$ as the following
$$
g(\Delta) = \frac{N_n(\delta_1+\Delta,\delta_2-\Delta,\delta_3,...,\delta_n)}{D_n(\delta_1+\Delta,\delta_2-\Delta,\delta_3,...,\delta_n)}.
$$
Then we have
$$
\frac{\partial g(\Delta)}{\partial \Delta}\bigg |_{\Delta = 0} = \frac{N_n\rq\bigg|_{\Delta = 0}}{D_n(0)}-\frac{N_n(0)D_n\rq\bigg|_{\Delta = 0}}{D_n(0)^2},
$$
where
\begin{eqnarray*}
N_n\rq\bigg|_{\Delta = 0} &=& \frac{\partial N_n(\delta_1+\Delta,\delta_2-\Delta,\delta_3,...,\delta_n)}{\partial \Delta}\bigg|_{\Delta=0}, \\
D_n\rq\bigg|_{\Delta = 0} &=& \frac{\partial D_n(\delta_1+\Delta,\delta_2-\Delta,\delta_3,...,\delta_n)}{\partial \Delta}\bigg|_{\Delta=0};\\
N_n(0) &=& N_n(\delta_1,\delta_2,\delta_3,...,\delta_n), \\
D_n(0) &=& D_n(\delta_1,\delta_2,\delta_3,...,\delta_n).
\end{eqnarray*}
According to \eqref{eq:Nn} and \eqref{eq:Dn}, we have
$$
N_n\rq\bigg|_{\Delta = 0} = D_n\rq\bigg|_{\Delta = 0} = 0.
$$
So we can get $\frac{\partial g(\Delta)}{\partial \Delta}\bigg |_{\Delta = 0} = 0-0=0.$
Similarly, for any two $\delta_i, \delta_j,$ simply give some perturbation to them, we can get the same result as above.
Therefore we can conclude that, for $n\geq 4$, $\left\{\delta_i = \frac{\pi}{n}, i = 1,...,n\right\}$ can maximize the function $\frac{N_n}{D_n}.$
Furthermore, we can get the maximum of $\frac{N_n}{D_n}$ by letting each $\delta_i$ be $\frac{\pi}{n}$:
\begin{equation}\label{eq:NnDn}
\left(\frac{N_n}{D_n}\right)_{\max} =
\frac{2n\sin\frac{\pi}{n} + 2\sum\limits_{r=2}^{n-2}(n-r)\sin\frac{r\pi}{n}}{2n\sin\frac{\pi}{n}
+\sum\limits_{r=1}^{n-2}\left[2(n-r)-1\right]\sin\frac{(2r+1)\pi}{2n}}.
\end{equation}
Next, we would like to show that when $\delta_i = \frac{\pi}{n}, i = 1,...,n$, we have
$$
\frac{\min\limits_\theta f(\theta)}{\max\limits_\theta f(\theta)} = \left(\frac{N_n}{D_n}\right)_{\max}.
$$
As $f(\theta) = \sum\limits_{i =1}^n \left|\cos\left(\theta - \frac{(i-1)\pi}{n}\right)\right|$,
we know $f(\theta)=f\left(\theta - \frac{\pi}{n}\right)$.
So we only need to consider $\theta\in[0,\frac{\pi}{n}]$ to get the maximum.
Recall $f(\theta)$ is linear, so the minimum and maximum must be either $\theta=0$ or $\theta=\frac{\pi}{n}$.
By observing the periodicity of the function $f(\theta)$, we can get
\begin{eqnarray}
\min\limits_{\theta}f(\theta) &=&
\begin{cases}
f(0) = 2\sum\limits_{r=1}^{a-1}\sin\frac{r\pi}{2a}+1&\mbox{ if $n=2a$,}\\\nonumber
f\left(\frac{\pi}{2(2a+1)}\right) = 2\sum\limits_{r=1}^{a}\sin\frac{r\pi}{2a+1}&\mbox{ if $n=2a+1$.} \\
\end{cases}\\
\max\limits_{\theta}f(\theta) &=&
\begin{cases}
f\left(\frac{\pi}{4a}\right) = 2\sum\limits_{r=1}^{a}\sin\frac{(2r-1)\pi}{4a}&\mbox{ if $n=2a$.} \\
f(0) = 2\sum\limits_{r=1}^{a}\sin\frac{(2r-1)\pi}{2(2a+1)}+1&\mbox{ if $n=2a+1$,}\nonumber
\end{cases}
\end{eqnarray}
From \eqref{eq:NnDn} we can get
\begin{eqnarray}
\label{eq:fmin/fmax}
\left(\frac{N_n}{D_n}\right)_{\max} =
\begin{cases}
\frac{2\sum\limits_{r=1}^{a-1}\sin\frac{r\pi}{2a}+1}{2\sum\limits_{r=1}^{a}\sin\frac{(2r-1)\pi}{4a}}&\mbox{ if $n=2a$.} \\
\frac{2\sum\limits_{r=1}^{a}\sin\frac{r\pi}{2a+1}}{2\sum\limits_{r=1}^{a}\sin\frac{(2r-1)\pi}{2(2a+1)}+1}&\mbox{ if $n=2a+1$,} \\
\end{cases}
\end{eqnarray}
Therefore, we can conclude that when $\delta_i = \frac{\pi}{n}, i = 1,...,n$,
$$
\frac{\min\limits_\theta f(\theta)}{\max\limits_\theta f(\theta)} = \left(\frac{N_n}{D_n}\right)_{\max}.
$$
Recall the definition of $\delta_i$\rq s, we know that \eqref{optdim2} is the optimal solution for $n\geq4.$
For $n=3\mbox{ and }2,$ by applying the similar strategy, we can get the same result as above.
\qed
\end{proof}
\subsection{Propositions we need in order to prove Theorem \ref{ese_O}}
Before proceeding to the proof of Theorem \ref{ese_O}, we need the following Proposition \ref{prop_2} and \ref{prop_1}:
\begin{proposition}
\label{prop_2}
\begin{eqnarray*}
&&\sum\limits_{s=1}^{n-1}\sin\frac{s}{n}\pi=\mbox{cot}\frac{\pi}{2n},\\
&&\sum\limits_{s=1}^{n-1}\cos\frac{s}{n}\pi=0,\\
&&\sum\limits_{s=1}^{n-1}s\sin\frac{s}{n}\pi=\frac{n}{2}\mbox{cot}\frac{\pi}{2n},\\
&&\sum\limits_{s=1}^{n-1}s\cos\frac{s}{n}\pi=-\frac{1}{2}\mbox{cot}^2\frac{\pi}{2n}+\frac{n-1}{2},\\
&&\sum\limits_{s=1}^{n-1}s^2\cos\frac{s}{n}\pi=-\frac{n}{2}\mbox{cot}^2\frac{\pi}{2n}+\frac{n(n-1)}{2}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
As the following holds true
$$
\sin\frac{s\pi}{N}\sin\frac{\pi}{2n}
=\frac{1}{2}\left(\cos\frac{(2s-1)\pi}{2n}-\cos\frac{(2s+1)\pi}{2n}\right),
$$
we have
\begin{eqnarray*}
&&\left(\sum\limits_{s=1}^{n-1}\sin\frac{s}{n}\pi\right)\cdot\sin\frac{\pi}{2n} \\
&=&\frac{1}{2}\sum\limits_{s=1}^{n-1}\left(\cos\frac{(2s-1)\pi}{2n}-\cos\frac{(2s+1)\pi}{2n}\right)
=\frac{1}{2}\left(\cos\frac{\pi}{2n}-\cos\frac{(2n-1)\pi}{2n}\right)=\cos\frac{\pi}{2n}.
\end{eqnarray*}
So by dividing $\sin\frac{\pi}{2n}$ for both sides, we can get
\begin{equation}
\label{sum_sin}
\sum\limits_{s=1}^{n-1}\sin\frac{s}{n}\pi=\mbox{cot}\frac{\pi}{2n}.
\end{equation}
As we also have
$$
\cos\frac{s\pi}{N}\sin\frac{\pi}{2n}
=\frac{1}{2}\left(\sin\frac{(2s+1)\pi}{2n}-\sin\frac{(2s-1)\pi}{2n}\right).
$$
Therefore, we can get
\begin{eqnarray*}
&&\left(\sum\limits_{s=1}^{n-1}\cos\frac{s}{n}\pi\right)\cdot\sin\frac{\pi}{2n} \\
&=&\frac{1}{2}\sum\limits_{s=1}^{n-1}\left(\sin\frac{(2s+1)\pi}{2n}-\sin\frac{(2s-1)\pi}{2n}\right)
=\frac{1}{2}\left(\sin\frac{(2n-1)\pi}{2n}-\sin\frac{\pi}{2n}\right)=0,
\end{eqnarray*}
which implies
\begin{equation}
\label{sum_cos}
\sum\limits_{s=1}^{n-1}\cos\frac{s}{n}\pi=0.
\end{equation}
As we also have
$$
\sin\frac{s}{n}\pi\cdot\sin\frac{\pi}{2n} = \cos\frac{(2s-1)\pi}{2n}-\cos\frac{(2s+1)\pi}{2n},
$$
the following can be derived:
\begin{equation}\label{eq:sumsinsin}
\left(\sum\limits_{s=1}^{n-1}s\sin\frac{s}{n}\pi\right)\cdot\sin\frac{\pi}{2n}
= \frac{1}{2}\sum\limits_{s=1}^{n-1}s\cdot\left(\cos\frac{(2s-1)\pi}{2n}-\cos\frac{(2s+1)\pi}{2n}\right).
\end{equation}
Since we have
\begin{eqnarray*}
&&\sum\limits_{s=1}^{n-1}s\cdot\left(\cos\frac{(2s-1)\pi}{2n}-\cos\frac{(2s+1)\pi}{2n}\right)\\
&=& \sum\limits_{s=1}^{n-1}\cos\frac{(2s-1)\pi}{2n}-(n-1)\cos\frac{(2n-1)\pi}{2n} \\
&=& \sum\limits_{s=1}^{n-1}\left(\cos\frac{s}{n}\pi\cos\frac{\pi}{2n}
+\sin\frac{s}{n}\pi\sin\frac{\pi}{2n}\right)+(n-1)\cos\frac{\pi}{2n},
\end{eqnarray*}
by plugging the above as well as \eqref{sum_sin} and \eqref{sum_cos} into \eqref{eq:sumsinsin}, we can get
\begin{eqnarray}
&&\left(\sum\limits_{s=1}^{n-1}s\sin\frac{s}{n}\pi\right)\cdot\sin\frac{\pi}{2n} \nonumber \\
&=&\frac{1}{2}\left(\sum\limits_{s=1}^{n-1}\left(\cos\frac{s}{n}\pi\cos\frac{\pi}{2n}
+\sin\frac{s}{n}\pi\sin\frac{\pi}{2n}\right)+(n-1)\cos\frac{\pi}{2n}\right) \nonumber \\
&=&\frac{1}{2}\left(0+\cos\frac{\pi}{2n}\right)+\frac{n-1}{2}\cos\frac{\pi}{2n}
=\frac{n}{2}\cos\frac{\pi}{2n}. \label{sum_times_sin}
\end{eqnarray}
Similarly, since we have
$$
\cos\frac{s}{n}\pi\cdot\sin\frac{\pi}{2n} = \sin\frac{(2s+1)\pi}{2n}-\sin\frac{(2s-1)\pi}{2n},
$$
by using the similar strategy, we can get
\begin{equation}
\left(\sum\limits_{s=1}^{n-1}s\cos\frac{s}{n}\pi\right)\cdot\sin\frac{\pi}{2n}
=-\frac{1}{2}\cos\frac{\pi}{2n}\mbox{cot}\frac{\pi}{2n}+\frac{n-1}{2}\sin\frac{\pi}{2n}. \label{sum_times_sin_2}
\end{equation}
Therefore, dividing both the equations \eqref{sum_times_sin} and \eqref{sum_times_sin_2}, we can get
\begin{eqnarray}
&&\sum\limits_{s=1}^{n-1}s\sin\frac{s}{n}\pi=\frac{n}{2}\mbox{cot}\frac{\pi}{2n}, \label{eq:sum_ssin}\\
&&\sum\limits_{s=1}^{n-1}s\cos\frac{s}{n}\pi=-\frac{1}{2}\mbox{cot}^2\frac{\pi}{2n}+\frac{n-1}{2}. \label{eq:sum_scos}
\end{eqnarray}
Since the following holds true,
$$
\left(\sum\limits_{s=1}^{n-1}s^2\cos\frac{s}{n}\pi\right)\cdot\sin\frac{\pi}{2n}
=\frac{1}{2}\left\{-\sum\limits_{s=1}^{n-1}(2s-1)\sin\frac{(2s-1)\pi}{2n}+(n-1)^2\sin\frac{2n-1}{2n}\pi\right\},
$$
by simplifying the above equation we can get
\begin{eqnarray*}
&&\left(\sum\limits_{s=1}^{n-1}s^2\cos\frac{s}{n}\pi\right)\cdot\sin\frac{\pi}{2n} \\
&=&-\sum\limits_{s=1}^{n-1}s\left(\sin\frac{s}{n}\pi\cos\frac{\pi}{2n}
-\cos\frac{s}{n}\pi\sin\frac{\pi}{2n}\right)
+\frac{1}{2}\sum\limits_{s=1}^{n-1}\left(\sin\frac{s}{n}\pi\cos\frac{\pi}{2n}
-\cos\frac{s}{n}\pi\sin\frac{\pi}{2n}\right) \\
&&+\frac{(n-1)^2}{2}\sin\frac{\pi}{2n} \\
&=&-\left(\sum\limits_{s=1}^{n-1}s\sin\frac{s}{n}\pi\right)\cos\frac{\pi}{2n}
+ \left(\sum\limits_{s=1}^{n-1}s\cos\frac{s}{n}\pi\right)\sin\frac{\pi}{2n}
+\frac{1}{2}\left(\sum\limits_{s=1}^{n-1}\sin\frac{s}{n}\pi\right)\cos\frac{\pi}{2n} \\
&&-\frac{1}{2}\left(\sum\limits_{s=1}^{n-1}\cos\frac{s}{n}\pi\right)\sin\frac{\pi}{2n}
+\frac{(n-1)^2}{2}\sin\frac{\pi}{2n}. \\
\end{eqnarray*}
Plugging \eqref{sum_sin}, \eqref{sum_cos}, \eqref{eq:sum_ssin}, and \eqref{eq:sum_scos} into the above, we can get
$$
\left(\sum\limits_{s=1}^{n-1}s^2\cos\frac{s}{n}\pi\right)\cdot\sin\frac{\pi}{2n}
=-\frac{1}{2}\mbox{cot}^2\frac{\pi}{2n}\sin\frac{\pi}{2n}
+\frac{n(n-1)}{2}\sin\frac{\pi}{2n}.
$$
Therefore, dividing $\sin\frac{\pi}{2n}$ on each side, we get
$$
\sum\limits_{s=1}^{n-1}s^2\cos\frac{s}{n}\pi
=-\frac{n}{2}\mbox{cot}^2\frac{\pi}{2n}+\frac{n(n-1)}{2}.
$$ \qed
\end{proof}
\begin{proposition}
\label{prop_1}
$$
2\sum\limits_{s=1}^{n-1}(n-s)f(s)
=\frac{n}{\pi}\mbox{cot}\frac{\pi}{2n}+\frac{1}{2}\mbox{cot}^2\frac{\pi}{2n}-\frac{n}{2}+\frac{1}{2}.
$$
\end{proposition}
\begin{proof}
According to the definition of function $f(s)$ in \eqref{eq:fs}, we have
\begin{eqnarray*}
&&2\sum\limits_{s=1}^{n-1}(n-s)f(s) \\
&=&2\sum\limits_{s=1}^{n-1}(n-s)\left(\frac{1}{\pi}\sin\frac{s}{n}\pi+\left(\frac{1}{2}-\frac{s}{n}\right)\cos\frac{s}{n}\pi\right) \\
&=&\frac{2n}{\pi}\sum\limits_{s=1}^{n-1}\sin\frac{s}{n}\pi-\frac{2}{\pi}\sum\limits_{s=1}^{n-1}s\sin\frac{s}{n}\pi
+n\sum\limits_{s=1}^{n-1}\cos\frac{s}{n}\pi-3\sum\limits_{s=1}^{n-1}s\cos\frac{s}{n}\pi
+\frac{2}{n}\sum\limits_{s=1}^{n-1}s^2\cos\frac{s}{n}\pi.
\end{eqnarray*}
Applying Proposition \ref{prop_2},we have
$$
2\sum\limits_{s=1}^{n-1}(n-s)f(s)
=\frac{n}{\pi}\mbox{cot}\frac{\pi}{2n}+\frac{1}{2}\mbox{cot}^2\frac{\pi}{2n}-\frac{n}{2}+\frac{1}{2}. \qquad \qquad \qed
$$
\end{proof}
\subsection{Proof of Theorem \ref{ese_O}}
\label{sec:app:proof-3}
\begin{proof}
Recall that $u_i$ can be rewritten as $$u_i = e^{\sqrt{-1} \frac{i\pi}{n}}, i = 0,1,\dots,n-1.$$
And we have
\begin{eqnarray}
&&\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left\{\left|C_n\sum\limits_{i=1}^n\left|u_i^Tv\right|-1\right|^2\right\} \nonumber \\
&=&C_n^2\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left\{\left(\sum\limits_{i=1}^n\left|u_i^Tv\right|\right)^2\right\}-2C_n\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left\{\sum\limits_{i=1}^n\left|u_i^Tv\right|\right\}+1 \nonumber \\
&=&C_n^2\sum\limits_{i=1}^n\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|^2\right)+2C_n^2\sum\limits_{1\leq i<j\leq N}\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|\left|u_j^Tv\right|\right) \nonumber \\
&&-2C_n\sum\limits_{i=1}^n\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left\{\left|u_i^Tv\right|\right\}+1 \label{eq_0}.
\end{eqnarray}
So we will find out the expected squared error, if for all $i,j = 1, \dots, n$, we can get the values of
$$
\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|^2\right), ~~~
\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|\left|u_j^Tv\right|\right), ~~~
\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left\{\left|u_i^Tv\right|\right\}.
$$
In order to calculate $\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|^2\right)$, we let $u_i = (1,0)^\prime$ and $v = (\cos\theta, \sin\theta)^\prime$ without loss of generality.
Then,
$$
\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|^2\right)
= \mathop{\mathbb{E}}\limits_{\theta\sim \mbox{Unif}(0,2\pi)}\cos^2\theta
= \frac{1}{2}+\frac{1}{2}\mathop{\mathbb{E}}\limits_{\theta\sim \mbox{Unif}(0,2\pi)}\cos2\theta
= \frac{1}{2}.
$$
Without loss of generality, assume $\langle u_i, u_j\rangle= \frac{s}{n}\pi, \mbox{ for all }1\leq i,j\leq n, i\not=j, $ which means we can assume
$$
u_i = (1,0)^\prime, u_j = (\cos\frac{s}{n}\pi, \sin\frac{s}{n}\pi)^\prime, s = 1,2,\dots, n-1.
$$
Therefore, we have
\begin{eqnarray*}
\left|u_i^Tv\right|\cdot\left|u_j^Tv\right| &=& \left|\cos\theta\right|\left|\cos\theta\cos\frac{s}{n}\pi
+\sin\theta\sin\frac{s}{n}\pi\right| \\
&=& \left|\cos^2\theta\cos\frac{s}{n}\pi
+ \cos\theta\sin\theta\sin\frac{s}{n}\pi\right|.
\end{eqnarray*}
As the following equations hold,
$$
\cos^2\theta=\frac{1+\cos2\theta}{2} \mbox{ and } \cos\theta\sin\theta = \frac{\sin2\theta}{2},
$$
quantity $\left|u_i^Tv\right|\cdot\left|u_j^Tv\right|$ can be further written as
\begin{eqnarray*}
\left|u_i^Tv\right|\cdot\left|u_j^Tv\right|&=&\frac{1}{2}\left|\cos2\theta\cos\frac{s}{n}\pi
+\sin2\theta\sin\frac{s}{n}\pi+\cos\frac{s}{n}\pi\right| \\
&=& \frac{1}{2}\left|\cos\left(2\theta-\frac{s}{n}\pi\right)
+\cos\frac{s}{n}\pi\right|.
\end{eqnarray*}
So $\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|\cdot\left|u_j^Tv\right|\right)$ can be rewritten as follows:
\begin{eqnarray*}
&&\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|\cdot\left|u_j^Tv\right|\right) \\
&=&\frac{1}{2}\mathop{\mathbb{E}}\limits_{\theta\sim \mbox{Unif}(0,2\pi)}\left\{\left|\cos\left(2\theta-\frac{s}{n}\pi\right)
+\cos\frac{s}{n}\pi\right|\right\}\\
&=&\frac{1}{2}\times\frac{1}{2\pi}\left(\int\limits_0^\pi
+\int\limits_\pi^{2\pi}\right)\left|\cos\left(2\theta-\frac{s}{n}\pi\right)+\cos\frac{s}{n}\pi\right|\mbox{d}\theta.\\
\end{eqnarray*}
As we have
\begin{eqnarray*}
&&\int\limits_\pi^{2\pi}\left|\cos\left(2\theta-\frac{s}{n}\pi\right)+\cos\frac{s}{n}\pi\right|\mbox{d}\theta \\
&=& \int\limits_0^\pi\left|\cos\left(2\theta-\frac{s}{n}\pi\right)+\cos\frac{s}{n}\pi\right|\mbox{d}\theta
= \int\limits_0^\pi\left|\cos\left(2\theta\right)+\cos\frac{s}{n}\pi\right|d\theta \\
&=&\int_{-\frac{\pi}{2}+\frac{s}{2n}\pi}^{\frac{\pi}{2}+\frac{s}{2n}\pi}\left|\cos\left(2\theta\right)+\cos\frac{s}{n}\pi\right|d\theta,
\end{eqnarray*}
we can get
\begin{equation}\label{eq:Euv}
\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|\cdot\left|u_j^Tv\right|\right)
=\frac{1}{2\pi}\int_{-\frac{\pi}{2}+\frac{s}{2n}\pi}^{\frac{\pi}{2}-\frac{s}{2n}\pi}\left|\cos\left(2\theta\right)+\cos\frac{s}{n}\pi\right|d\theta.
\end{equation}
By breaking the integral interval $(-\frac{\pi}{2}+\frac{s}{2n}\pi, \frac{\pi}{2}+\frac{s}{2n}\pi)$ into two subintervals,
$(-\frac{\pi}{2}+\frac{s}{2n}\pi, \frac{\pi}{2}-\frac{s}{2n}\pi)$ and
$(\frac{\pi}{2}-\frac{s}{2n}\pi, \frac{\pi}{2}+\frac{s}{2n}\pi)$, we have
\begin{eqnarray*}
\left|\cos2\theta+\cos\frac{s}{n}\pi\right| =
\begin{cases}
\cos2\theta+\cos\frac{s}{n}\pi, & \theta\in(-\frac{\pi}{2}+\frac{s}{2n}\pi, \frac{\pi}{2}-\frac{s}{2n}\pi),\\
-\left(\cos2\theta+\cos\frac{s}{n}\pi\right), &\theta\in(\frac{\pi}{2}-\frac{s}{2n}\pi, \frac{\pi}{2}+\frac{s}{2n}\pi).
\end{cases}
\end{eqnarray*}
Combining \eqref{eq:Euv}, we get
\begin{eqnarray*}
&&\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|\cdot\left|u_j^Tv\right|\right)\\
&=&\frac{1}{2\pi}\left(\int_{-\frac{\pi}{2}+\frac{s}{2n}\pi}^{\frac{\pi}{2}-\frac{s}{2n}\pi}
+\int_{\frac{\pi}{2}-\frac{s}{2n}\pi}^{\frac{\pi}{2}+\frac{s}{2n}\pi}\right)\left|\cos2\theta+\cos\frac{s}{n}\pi\right|d\theta\\
&=&\frac{1}{2\pi}\left\{\int_{-\frac{\pi}{2}+\frac{s}{2n}\pi}^{\frac{\pi}{2}-\frac{s}{2n}\pi}\left(\cos2\theta
+\cos\frac{s}{n}\pi\right)d\theta-\int_{\frac{\pi}{2}-\frac{s}{2n}\pi}^{\frac{\pi}{2}+\frac{s}{2n}\pi}\left(\cos2\theta
+\cos\frac{s}{n}\pi\right)d\theta\right\} \\
&=&\frac{1}{2\pi}\left\{2\sin\frac{s}{n}\pi+\left(\pi-\frac{2s}{N}\pi\right)\cos\frac{s}{n}\pi\right\} \\
&=&\frac{1}{\pi}\sin\frac{s}{n}\pi+\left(\frac{1}{2}-\frac{s}{n}\right)\cos\frac{s}{n}\pi.
\end{eqnarray*}
If we define
\begin{equation}\label{eq:fs}
f(s)=\frac{1}{\pi}\sin\frac{s}{n}\pi+\left(\frac{1}{2}-\frac{s}{n}\right)\cos\frac{s}{n}\pi, s = 0,1,2,\cdots,n-1.
\end{equation}
Then we will get
\begin{eqnarray}
\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|^2\right) &=& f(0), \nonumber \\
\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|\cdot\left|u_j^Tv\right|\right) &=&f(s),
\mbox{ where }\langle u_i, u_j\rangle=\frac{s}{n}\pi, s = 1,2,\cdots,n-1. \label{eq_1}
\end{eqnarray}
Similarly, without loss of generality, if we assume
$u_i = (1,0)^\prime, v = (\cos\theta,\sin\theta)^\prime,$ the following holds,
\begin{equation}
\label{eq_2}
\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left\{\left|u_i^Tv\right|\right\}
= \mathop{\mathbb{E}}\limits_{\theta\sim \mbox{Unif}(-\pi,\pi)}\left|\cos\theta\right|
= 2\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{1}{2\pi}\cos\theta d\theta = \frac{2}{\pi}.
\end{equation}
Recall that we have
$$
C_n = \frac{2}{\mbox{V}_{\min}+\mbox{V}_{\max}},
\mbox{ where } \mbox{V}_{\min}=\min\limits_{v:\|v\|=1}\sum\limits_{i=1}^n\left|u_i^Tv\right|,
\mbox{V}_{\max}=\max\limits_{v:\|v\|=1}\sum\limits_{i=1}^n\left|u_i^Tv\right|.
$$
From \eqref{eq:fmin/fmax} we can easily verify that
$$
\mbox{V}_{\min}+\mbox{V}_{\max}=2\sum\limits_{k=1}^{n-1}\sin\frac{k\pi}{2n} +1.
$$
Therefore, $C_n$ can be derived:
\begin{equation}\label{eq:Cn1}
C_n = \frac{2}{2\sum\limits_{k=1}^{n-1}\sin\frac{k\pi}{2n} +1}.
\end{equation}
As we have
$$
\sin\frac{k\pi}{2n}\cdot\sin\frac{\pi}{4n}
=\frac{1}{2}\left(\cos\frac{(2k-1)\pi}{4n}-\cos\frac{(2k+1)\pi}{4n}\right),
$$
we can get
$$
\sin\frac{\pi}{4n}\left(\sum\limits_{k=1}^{n-1}\sin\frac{k\pi}{2n}\right)
=\frac{1}{2}\sum\limits_{k=1}^{n-1}\left(\cos\frac{(2k-1)\pi}{4n}-\cos\frac{(2k+1)\pi}{4n}\right)
=\frac{1}{2}\left(\cos\frac{\pi}{4n}-\sin\frac{\pi}{4n}\right),
$$
which leads to
\begin{equation}
\label{eq:triangle1}
\sum\limits_{k=1}^{n-1}\sin\frac{k\pi}{2n}
= \frac{\frac{1}{2}\left(\cos\frac{\pi}{4n}-\sin\frac{\pi}{4n}\right)}{\sin\frac{\pi}{4n}}
=\frac{1}{2}\mbox{cot}\frac{\pi}{4n}-\frac{1}{2}.
\end{equation}
Therefore, by plugging \eqref{eq:triangle1} into \eqref{eq:Cn1}, we have
$$
C_n=\frac{2}{\mbox{cot}\frac{\pi}{4n}} = 2\mbox{tan}\frac{\pi}{4n}.
$$
If we plug in \eqref{eq_0} with \eqref{eq_1} and \eqref{eq_2}, we can get
\begin{eqnarray}
&&\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left\{\left|C_n\sum\limits_{i=1}^n\left|u_i^Tv\right|-1\right|^2\right\} \nonumber \\
&=&C_n^2\sum\limits_{i=1}^n\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|^2\right)+2C_n^2\sum\limits_{1\leq i<j\leq N}\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|\left|u_j^Tv\right|\right) \nonumber \\
&&-2C_n\sum\limits_{i=1}^n\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left\{\left|u_i^Tv\right|\right\}+1 \nonumber \\
&=&4\mbox{tan}^2\frac{\pi}{4n}\left(\frac{n}{2}+2\sum\limits_{s=1}^{n-1}(n-s)f(s)\right)-\frac{8N}{\pi}\mbox{tan}\frac{\pi}{4n} +1. \label{eq_3}
\end{eqnarray}
In order to calculate the part $\sum\limits_{s=1}^{n-1}(n-s)f(s)$ in \eqref{eq_3}, we need the Proposition \ref{prop_1}.
Applying Proposition \ref{prop_1} on \eqref{eq_3}, we get
\begin{eqnarray}
&&\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left\{\left|C_n\sum\limits_{i=1}^n\left|u_i^Tv\right|-1\right|^2\right\} \nonumber \\
&=&4\mbox{tan}^2\frac{\pi}{4n}\left(\frac{n}{2}+\frac{n}{\pi}\mbox{cot}\frac{\pi}{2n}+\frac{1}{2}\mbox{cot}^2\frac{\pi}{2n}-\frac{n}{2}
+\frac{1}{2}\right)-\frac{8n}{\pi}\mbox{tan}\frac{\pi}{4n} +1 \nonumber \\
&=&2\mbox{tan}^2\frac{\pi}{4n}\mbox{cot}^2\frac{\pi}{2n}+\frac{4n}{\pi}\mbox{tan}^2\frac{\pi}{4n}\mbox{cot}\frac{\pi}{2n}
+2\mbox{tan}^2\frac{\pi}{4n}-\frac{8n}{\pi}\mbox{tan}\frac{\pi}{4n} +1. \label{eq_4}
\end{eqnarray}
As $\mbox{tan}x\rightarrow x,\mbox{ as } x\rightarrow 0$, we can get
\begin{eqnarray*}
\mathop{\mathbb{E}}\limits_{v\sim \mbox{Unif}(S^1)}\left\{\left|C_n\sum\limits_{i=1}^n\left|u_i^Tv\right|-1\right|^2\right\}
&\longrightarrow & 2\frac{\pi^2}{16n^2}\frac{4n^2}{\pi^2}+\frac{4n}{\pi}\frac{\pi^2}{16n^2}\frac{2n}{\pi}+2\frac{\pi^2}{16n^2}
-\frac{8n}{\pi}\frac{\pi}{4n}+1 \\
&=&\frac{\pi^2}{8n^2}. \qquad \qquad \qed
\end{eqnarray*}
\end{proof}
\subsection{Proof of Theorem \ref{ese_M}}
\label{sec:app:proof-4}
\begin{proof}
Monte Carlo method uses random directions to approximate the norm, which means
$$
u_i\sim \mbox{Unif}(S^1), i.i.d.
$$
We also know that
\begin{eqnarray}
&&\mathop{\mathbb{E}}\limits_{u_i, v\sim \mbox{Unif}(S^1)}\left\{\left|C_n\sum\limits_{i=1}^n\left|u_i^Tv\right|-1\right|^2\right\} \nonumber \\
&=&C_n^2\mathop{\mathbb{E}}\limits_{u_i, v\sim \mbox{Unif}(S^1)}\left\{\left(\sum\limits_{i=1}^n\left|u_i^Tv\right|\right)^2\right\}-2C_n\mathop{\mathbb{E}}\limits_{u_i,v\sim \mbox{Unif}(S^1)}\left\{\sum\limits_{i=1}^n\left|u_i^Tv\right|\right\}+1 \nonumber \\
&=&C_n^2\sum\limits_{i=1}^n\mathop{\mathbb{E}}\limits_{u_i, v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|^2\right)+2C_n^2\sum\limits_{1\leq i<j\leq N}\mathop{\mathbb{E}}\limits_{u_i, u_j, v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|\left|u_j^Tv\right|\right) \nonumber \\
&&-2C_n\sum\limits_{i=1}^n\mathop{\mathbb{E}}\limits_{u_i,v\sim \mbox{Unif}(S^1)}\left\{\left|u_i^Tv\right|\right\}+1, \label{eq_M}
\end{eqnarray}
where
$C_n$ satisfies
$$
C_n\cdot\int\limits_{u_i\in S^1}\sum\limits_{i=1}^n\left|u_i^Tv\right|du_i=1,
$$
which implies
$$
C_n = \frac{\pi}{2n}.
$$
We can find out the expected squared error if we can get the values of
\begin{eqnarray*}
&&\mathop{\mathbb{E}}\limits_{u_i, v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|^2\right),\\
&&\mathop{\mathbb{E}}\limits_{u_i, u_j, v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|\left|u_j^Tv\right|\right), \\
&&\mathop{\mathbb{E}}\limits_{u_i, v\sim \mbox{Unif}(S^1)}\left\{\left|u_i^Tv\right|\right\}, ~~~
\mbox{for all }i,j = 1, \cdots, n.
\end{eqnarray*}
Let $u_i=(\cos\phi, \sin\phi)^\prime, v=(\cos\theta,\sin\theta)^\prime,$ where $\phi\sim \mbox{Unif}(0,2\pi), \theta\sim \mbox{Unif}(0,2\pi).$
Then the above three can be computed as follows:
\begin{eqnarray*}
&&\mathop{\mathbb{E}}\limits_{u_i, v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|^2\right) \\
&=&\mathop{\mathbb{E}}\limits_{\phi,\theta\sim \mbox{Unif}(0,2\pi)}\cos^2(\phi-\theta)
=\mathop{\mathbb{E}}\limits_{\phi\sim \mbox{Unif}(0,2\pi)}\left[\mathop{\mathbb{E}}\limits_{\theta\sim \mbox{Unif}(0,2\pi)}\left[\cos^2(\phi-\theta)\vert\phi\right]\right] \\
&=&\frac{1}{2},
\end{eqnarray*}
and
\begin{eqnarray*}
&&\mathop{\mathbb{E}}\limits_{u_i, u_j, v\sim \mbox{Unif}(S^1)}\left(\left|u_i^Tv\right|\left|u_j^Tv\right|\right) \\
&=&\mathop{\mathbb{E}}\limits_{\phi_i,\phi_j,\theta\sim \mbox{Unif}(0,2\pi)}\left\{\left|\cos(\theta-\phi_i)\right|\left|\cos(\theta-\phi_j)\right|\right\} \\
&=&\mathop{\mathbb{E}}\limits_{\phi_j,\theta\sim \mbox{Unif}(0,2\pi)}\left\{\mathop{\mathbb{E}}\limits_{\phi_i\sim \mbox{Unif}(0,2\pi)}\left[\left|\cos(\theta-\phi_i)\right|[\left|\cos(\theta-\phi_j)\right|\vert\phi_j,\theta\right]\right\} \\
&=&\mathop{\mathbb{E}}\limits_{\phi_j,\theta\sim \mbox{Unif}(0,2\pi)}\left\{\left|\cos(\theta-\phi_j)\right|\cdot\frac{2}{\pi}\right\} \\
&=&\mathop{\mathbb{E}}\limits_{\theta\sim \mbox{Unif}(0,2\pi)}\left\{\mathop{\mathbb{E}}\limits_{\phi_j\sim \mbox{Unif}(0,2\pi)}\left[|\cos(\theta-\phi_j)|\cdot\frac{2}{\pi}\vert\theta\right]\right\}
=\mathop{\mathbb{E}}\limits_{\theta\sim \mbox{Unif}(0,2\pi)}\left[\frac{2}{\pi}\cdot\frac{2}{\pi}\right] \\
&=&\frac{4}{\pi^2},
\end{eqnarray*}
and
\begin{eqnarray*}
&&\mathop{\mathbb{E}}\limits_{u_i, v\sim \mbox{Unif}(S^1)}\left\{\left|u_i^Tv\right|\right\} \\
&=&\mathop{\mathbb{E}}\limits_{\phi,\theta\sim \mbox{Unif}(0,2\pi)}\left|\cos(\phi-\theta)\right|=\mathop{\mathbb{E}}\limits_{\phi\sim \mbox{Unif}(0,2\pi)}\left[\mathop{\mathbb{E}}\limits_{\theta\sim \mbox{Unif}(0,2\pi)}\left[\left|\cos(\phi-\theta)\right|\vert\phi\right]\right] \\
&=& \mathop{\mathbb{E}}\limits_{\phi\sim \mbox{Unif}(0,2\pi)}\frac{2}{\pi}
=\frac{2}{\pi}.
\end{eqnarray*}
Therefore by plugging the above results into \eqref{eq_M}, we eventually get
\begin{eqnarray*}
&&\mathop{\mathbb{E}}\limits_{u_i, v\sim \mbox{Unif}(S^1)}\left\{\left|C_n\sum\limits_{i=1}^n\left|u_i^Tv\right|-1\right|^2\right\} \\
&=&\frac{\pi^2}{4N^2}\left(N\cdot\frac{1}{2}+2\frac{N(N-1)}{2}\frac{4}{\pi^2}\right)-2\frac{\pi}{2n}\cdot N\cdot\frac{2}{\pi}+1
=\frac{\pi^2-8}{8N}. \qquad \qquad \qed
\end{eqnarray*}
\end{proof}
\subsection{Proof of Lemma \ref{V_max}}
\label{sec:app:proof-5}
\begin{proof}
Recall that we have
\begin{equation}\label{maxmax}
V_{\max} = \max\limits_{v:||v||_2=1}\sum\limits_{i=1}^n\left|u_i^Tv\right|
= \max\limits_{v:||v||_2=1}\max\limits_{s_i\in\{1, -1\}}\left(\sum\limits_{i=1}^ns_iu_i^T\right)v,
\end{equation}
where the second equality is based on a standard trick in optimization \cite[Chapter 9.2(ii)]{bradley1977applied}.
The following is an application of the Cauchy-Schwartz inequality:
$$
\left(\sum\limits_{i=1}^ns_iu_i^T\right)v\leq \sqrt{\left\|\sum\limits_{i=1}^n s_iu_i\right\|_2^2||v||_2^2}
= \left\|\sum\limits_{i=1}^n s_iu_i\right\|,
$$
where the equality is due to the condition $\left\|v\right\|=1$.
In the first part, the equality holds if and only if $|v_j| = c\left|\left(\sum\limits_{i=1}^ns_iu_i\right)_j\right|, j = 1,...,p$.
Apparently, we must have $c = \left\|\sum\limits_{i=1}^ns_iu_i\right\|^{-1}$ (because of $\left\|v\right\|=1$).
So we can have
\begin{equation}\label{v}
v= \frac{\sum\limits_{i=1}^ns_iu_i}{\left\|\sum\limits_{i=1}^ns_iu_i\right\|}.
\end{equation}
Combining \eqref{v} and \eqref{maxmax}, we have \eqref{Vmax}.
\qed
\end{proof}
\subsection{Proof of Lemma \ref{prop}}
\label{sec:app:proof-6}
\begin{proof}
We start with a special case: the linear subspace is $\mathbb{R}^p$(the entire space). Obviously the $n$ hyperplanes
$$\left\{y: u_i^Ty=0\right\}\mbox{, for } i = 1,2,...,n$$
divide the sphere $S^{p-1}$ into at most $2^n$ sectors. Within each sector, function$f(v)$ is strictly linear, therefore the minima cannot be an interior point.
Recall a boundary point $v$ must have $u_j^Tv=0$ for at least one $j, 1\leq j\leq n$.
Now we consider a linear subspace with dimension less than $p$, say, $k$.
Let $b_1,...,b_k$ be the orthonormal basis of such a linear subspace, we have $\forall x\in \Omega$,
$$
x = \sum\limits_{j=1}^kc_jb_j,
$$
and
$$
\sum\limits_{j=1}^kc_j^2=1, (\mbox{Because we have }\|x\|=1).
$$
Therefore, we have
$$
f(v) = \sum\limits_{i=1}^n\left|u_i^Tv\right|
= \sum\limits_{i=1}^n\left|u_i^T\sum\limits_{j=1}^kc_jb_j\right|
= \sum\limits_{i=1}^n\left|\sum\limits_{j=1}^kc_j(u_i^Tb_j)\right|
= \sum\limits_{i=1}^n\left|h_i^Tc\right|,
$$
where $c = (c_1,...,c_k)^T$ and $h_i^T = \left(u_i^Tb_1,...,u_i^Tb_k\right), i=1, ..., n.$
Note that in the early part of this proof, the $u_i$ can be arbitrary.
The above derivation indicates that the latter case can be converted into the former case, as $c\in\mathbb{R}^k$ is from the entire space.
So we can get
$$
h_i^Tc = 0\mbox{ for at least one }i, 1 \leq i \leq n.
$$
As $h_i^Tc = u_i^T\left(\sum\limits_{j=1}^kb_jc_j\right)$, the above is equivalent to
$$
u_i^T\left(\sum\limits_{j=1}^kb_jc_j\right) =0\mbox{ for at least one }i, 1 \leq i \leq n.
$$
Quantity $\sum\limits_{j=1}^kb_jc_j$ can also be denoted as $v$, because any vector on the space is a linear combination of the orthonormal basis $b_1, ..., b_k.$
From all the above, we proved the lemma. \qed
\end{proof}
\subsection{Proof of Lemma \ref{Omega_0}}
\label{sec:app:proof-7}
\begin{proof}
For notational simplicity, let us donate $\Omega = \Omega(v_{\min})$. We can easily verify the following
$$
\mbox{rank}(\Omega)\leq p-1.
$$
Otherwise (i.e., $\mbox{rank}(\Omega) = p$), by the definition of $\Omega$, we will have $v_{\min} = 0.$
Now we show that
$$
\mbox{rank}(\Omega)\geq p-1.
$$
We use contradiction.
Let us assume that $\mbox{rank}(\Omega)<p-1.$
Define the following complementary set
$$
\Omega^{\perp} = \left\{x:\|x\| = 1, x\perp \Omega\right\},
$$
where $x\perp\Omega$ stands for that $x$ is perpendicular to the linear space that is spanned by all the $u_j$\rq s in $\Omega$.
Because $v_{\min}$ is a minimizer, we have that
$$
f(v_{\min}) = \min\limits_{v\in\Omega^\perp}f(v)
= \min\limits_{v\in\Omega^\perp}\sum\limits_{i=1}^n\left|u_i^Tv\right|
= \min\limits_{v\in\Omega^\perp}\sum\limits_{u_i\not\in\Omega}\left|u_i^Tv\right|
$$
Note that if $\mbox{rank}(\Omega)<p-1$, we have $\mbox{dim}(\Omega^\perp)\geq 2$.
By Lemma \ref{prop}, we can declare that there exists $u_j\not\in\Omega$, $u_j^Tv_{\min}=0$.
However, this contradicts to the definition of $\Omega$, which is supposed to be the maximal subset. \qed
\end{proof}
\subsection{Proof of Theorem \ref{dim=n}}
\label{sec:app:proof-8}
\begin{proof}
When $n=p$, we have
$$
f(v) = |u_1^Tv| + |u_2^Tv| + ... + |u_p^Tv|, \mbox{ for } u_1,...u_p, v\in \mathbb{S}^{p-1}.
$$
According to the Lemma \ref{Omega_0}, we have
$$
\mbox{rank}\left(\Omega(v_{\min})\right) = p-1,
$$
where $\Omega(v_{\min}) = \left\{u_j:u_j^Tv_{\min}=0\right\}$, and $v_{\min}$ is the minimizer of $f(v).$
So the minimizer of $f(v)$ must satisfy that it is orthogonal to $p-1$ linearly independent $u_j$\rq s.
Assume every $p-1$ $u_j$\rq s are linearly independent.
Then the minimizer is among the vectors that are orthogonal to any $p-1$ $u_j$\rq s.
We know there are ${p \choose p-1}=p$ different combinations of $u_j$\rq s, and each combination is correspond to $2$ unit vectors orthogonal to one of the $p-1$ $u_j$\rq s.
(These $2$ unit vectors are the two directions that are orthogonal to a $p-1$ spaces in $\mathbb{R}^p.$)
Thus there are totally $2p$ unit vectors that might be the minimizer of $f(v).$
Suppose $p$ of the $2p$ unit vectors are those whose first nonzero entry is positive.
Denote them as $v^{-(1)}, v^{-(2)}, ..., v^{-(p)}$.
Then the other $p$ unit vectors would be $-v^{-(1)}, -v^{-(2)}, ..., -v^{-(p)}.$
Suppose that for any $i\in\{1,2,...,p\},$ $v^{-(1)}, v^{-(2)}, ..., v^{-(p)}$ satisfy
$$
\left(v^{-(i)}\right)^Tu_j =0, \forall j\not=i, j\in\{1,2,...,p\}.
$$
Thus the minimum value of $f(v)$ can be upper bounded by the average of the function values of the $p$ unit vectors:
\begin{equation}\label{fvmin}
\min_vf(v) \leq \frac{1}{p}\sum\limits_{i = 1}^pf(v^{-(i)}).
\end{equation}
We can also bound the maximum value of $f(v)$ by some value:
\begin{equation}\label{fvmax1}
\max_vf(v) \geq \max_{s_i = \pm1}f\left(\frac{\sum\limits_{i = 1}^ps_iv^{-(i)}}{\|\sum\limits_{i = 1}^ps_iv^{-(i)}\|}\right).
\end{equation}
Because we have
$$
f\left(\frac{\sum\limits_{i = 1}^ps_iv^{-(i)}}{\|\sum\limits_{i = 1}^ps_iv^{-(i)}\|}\right)
= \frac{f\left(\sum\limits_{i = 1}^ps_iv^{-(i)}\right)}{\|\sum\limits_{i = 1}^ps_iv^{-(i)}\|},
$$
and
\begin{eqnarray*}
f\left(\sum\limits_{i = 1}^ps_iv^{-(i)}\right)
&=& \sum\limits_{j=1}^p\left|u_j^T\left(\sum\limits_{i = 1}^ps_iv^{-(i)}\right)\right|
= \sum\limits_{j=1}^p\left|\sum\limits_{i = 1}^ps_iu_j^Tv^{-(i)}\right|
= \sum\limits_{j=1}^p\left|s_ju_j^Tv^{-(j)}\right| \\
&=& \sum\limits_{j=1}^p\left|u_j^Tv^{-(j)}\right| = \sum\limits_{j=1}^pf\left(v^{-(j)}\right),
\end{eqnarray*}
we can get
$$
f\left(\frac{\sum\limits_{i = 1}^ps_iv^{-(i)}}{\|\sum\limits_{i = 1}^ps_iv^{-(i)}\|}\right)
= \frac{\sum\limits_{j=1}^pf\left(v^{-(j)}\right)}{\|\sum\limits_{i = 1}^ps_iv^{-(i)}\|}.
$$
So \eqref{fvmax1} becomes
\begin{equation}\label{fvmax2}
\max_vf(v) \geq \max_{s_i = \pm1}\frac{\sum\limits_{i = 1}^pf(v^{-(i)})}{\left\|\sum\limits_{i = 1}^ps_iv^{-(i)}\right\|}
= \frac{\sum\limits_{i = 1}^pf(v^{-(i)})}{\min\limits_{s_i = \pm 1}\left\|\sum\limits_{i = 1}^ps_iv^{-(i)}\right\|}.
\end{equation}
Based on \eqref{fvmin} and \eqref{fvmax2}, we can get
$$
\frac{\min\limits_vf(v)}{\max\limits_vf(v)} \leq
\frac{\frac{1}{p}\sum\limits_{i = 1}^pf(v^{-(i)})}{\frac{\sum\limits_{i = 1}^pf(v^{-(i)})}{\min\limits_{s_i = \pm 1}\left\|\sum\limits_{i = 1}^ps_iv^{-(i)}\right\|}}
= \frac{1}{p}\min\limits_{s_i = \pm 1}\left\|\sum\limits_{i = 1}^ps_iv^{-(i)}\right\|.
$$
So we have
\begin{equation}\label{bound}
\max\limits_{u_1,...u_p}\frac{\min\limits_vf(v)}{\max\limits_vf(v)} \leq
\frac{1}{p}\max_{u_1,...u_p}\min\limits_{s_i = \pm 1}\left\|\sum\limits_{i = 1}^ps_iv^{-(i)}\right\|.
\end{equation}
Since solving the problem
$$
\max_{u_1,...u_p}\min\limits_{s_i = \pm 1}\left\|\sum\limits_{i = 1}^ps_iv^{-(i)}\right\|
$$
is equivalent to solving $$\max_{u_1,...u_p}\min\limits_{s_i = \pm 1}\left\|\sum\limits_{i = 1}^ps_iv^{-(i)}\right\|^2,
$$
we will try to solve the latter one in the following.
We have
$$
\max_{u_1,...u_p}\min\limits_{s_i = \pm 1}\left\|\sum\limits_{i = 1}^ps_iv^{-(i)}\right\|^2 = \max_{u_1,...u_p}\min\limits_{s_i = \pm 1}s^T\Sigma s,
$$
where we have $\Sigma \in\mathbb{R}^{p\times p}$ and
\begin{eqnarray*}
\Sigma &=& \left(\begin{array}{ccccc}
1& \left(v^{-(1)}\right)^Tv^{-(2)} & \left(v^{-(1)}\right)^Tv^{-(3)} & \cdots & \left(v^{-(1)}\right)^T v^{-(p)} \\
\left(v^{-(2)}\right)^T v^{-(1)} & 1 & \left(v^{-(2)}\right)^Tv^{-(3)} & \cdots &\left(v^{-(2)}\right)^T v^{-(p)} \\
\cdots&\cdots&\cdots&\cdots&\cdots \\
\left(v^{-(p)}\right)^T v^{-(1)} & \left(v^{-(p)}\right)^Tv^{-(2)} & \left(v^{-(p)}\right)^Tv^{-(3)} & \cdots &1 \\
\end{array}
\right).
\end{eqnarray*}
We claim that $\min\limits_{s_i = \pm 1}s^T\Sigma s$ is upper bounded by $p$, and $\min\limits_{s_i = \pm 1}s^T\Sigma s=p$ when
$$
\left(v^{-(i)}\right)^Tv^{-(j)}=0, \forall i\not=j.
$$
We can see that if there are some $i,j$ $(i\not=j)$, such that $\left(v^{-(i)}\right)^Tv^{-(j)}\not=0,$ then there exists some $s$, such that $s^T\Sigma s\leq p$.
Suppose there does not exist such $s$, which means for any $s$, the following holds,
\begin{equation}\label{eq:sSs}
s^T\Sigma s>p.
\end{equation}
Since we have
\begin{eqnarray*}
\sum\limits_{s_i = \pm 1} s^T\Sigma s
&=&\sum\limits_{s\in\{s:s_k=\pm1\}}\sum\limits_{i,j}s_is_j\Sigma_{ij}
= \sum\limits_{s\in\{s:s_k=\pm1\}}\left(p + \sum\limits_{i\not=j}s_is_j\Sigma_{ij}\right) \\
&=& 2^pp + \sum\limits_{s\in\{s:s_k=\pm1\}}\sum\limits_{i\not=j}s_is_j\Sigma_{ij} = 2^pp,
\end{eqnarray*}
this will lead to $\sum\limits_{s_i = \pm 1} s^T\Sigma s>2^pp,$ which is a contradiction of \eqref{eq:sSs}.
So we proved that our claim is true, which says
$$
\min\limits_{s_i = \pm 1}s^T\Sigma s\leq p,
$$
and when $\left(v^{-(i)}\right)^Tv^{-(j)}=0, \forall i\not=j,$ which means $\Sigma = I_p$, we have
$\min\limits_{s_i = \pm 1}s^T\Sigma s =p.$
We know that $v^{-(i)}$\rq s only depends on $u_i$\rq s, and when
$u_i^Tu_j = 0, \forall i\not=j,$ we have
$\left(v^{-(i)}\right)^Tv^{-(j)}=0, \forall i\not=j.$
So when the following holds,
$$
u_i^Tu_j = 0, \forall i\not=j,
$$
$\min\limits_{s_i = \pm 1}s^T\Sigma s$ achieves the maximum value, which is $p$.
Therefore we get
$$
\max_{u_1,...u_p}\min\limits_{s_i = \pm 1}\left\|\sum\limits_{i = 1}^ps_iv^{-(i)}\right\|^2
= \max_{u_1,...u_p}\min\limits_{s_i = \pm 1}s^T\Sigma s =p,
$$
which leads to
\begin{equation}\label{eq:maxminnorms}
\max_{u_1,...u_p}\min\limits_{s_i = \pm 1}\left\|\sum\limits_{i = 1}^ps_iv^{-(i)}\right\| = \sqrt{p}.
\end{equation}
Based on \eqref{bound} and \eqref{eq:maxminnorms}, we have
\begin{equation}\label{upper}
\max\limits_{u_1,...u_p}\frac{\min\limits_vf(v)}{\max\limits_vf(v)} \leq \frac{\sqrt{p}}{p}.
\end{equation}
Next if we can prove that when $u_i^Tu_j = 0, \forall i\not=j, $
the following holds,
$\frac{\min\limits_vf(v)}{\max\limits_vf(v)} = \frac{\sqrt{p}}{p}$;
combined with \eqref{upper}, we can arrive at the conclusion and finish the proof of the Lemma.
Let us assume
$$
u_i^Tu_j = 0, \forall i\not=j.
$$
Without loss of generality, we can assume $u_i = e_i, \forall i\not=j,$ where $e_i$\rq s are the basic vectors of $\mathbb{R}^p.$
Then the following holds,
$$
f(v) = \sum\limits_{i=1}^p|v_i|, v\in \mathbb{S}^{p-1}.
$$
We can easily verify the following,
$\min\limits_vf(v) = 1,$ and $\max\limits_vf(v) = \sqrt{p}.$
So when $u_i^Tu_j = 0, \forall i\not=j$, we have
$$
\frac{\min\limits_vf(v)}{\max\limits_vf(v)} = \frac{\sqrt{p}}{p}.
$$
Combined what we get from \eqref{upper}, that is, $\frac{\sqrt{p}}{p}$ is the upper bound of $\max\limits_{u_1,...u_p}\frac{\min\limits_vf(v)}{\max\limits_vf(v)},$ we finished the proof. \qed
\end{proof}
\subsection{Proof of Lemma \ref{u_1}}
\label{sec:app:proof-9}
\begin{proof}
As we have
$$
\min_{x:\|x\|=1,\left<x,v\right>=\theta}\left\|x+B\right\|^2
= \min_{x:\|x\|=1,\left<x,v\right>=\theta}1 + \|B\|^2 + 2\left<x,B\right>,
$$
the problem \eqref{u1} is equivalent to
\begin{equation}\label{u1_2}
\min_{x:\|x\|=1,\left<x,v\right>=\theta}\left<x,B\right>.
\end{equation}
Suppose $x^*$ is the solution to the above problem \eqref{u1}. Then $x^*$ is the farthest point to $B$ on the circle that satisfies the constraints $\|x\|=1,\left<x,v\right>=\theta$. The three points $x^*, v, \mbox{ and }B$ must be on a same plane. Therefore, we can assume
\begin{equation}\label{eq:xstar}
x^* = av+bB.
\end{equation}
Bringing \eqref{eq:xstar} into \eqref{u1_2}, we have
\begin{equation*}\label{u1_3}
\min_{x:\|x\|=1,\left<x,v\right>=\theta}\left<x,B\right>=\min_{a,b:\|av+bB\|=1,\left<av+bB,v\right>=\theta}\left<av+bB,B\right>,
\end{equation*}
which is equivalent to
\begin{eqnarray}
\min_{a,b} & av^TB+bB^TB \label{opt}\\
\text{s.t.} &
\begin{cases}\label{cond}
a^2 + b^2\|B\|^2 + 2abv^TB&=1\\
a+bv^TB& =\cos\theta.\\
\end{cases}
\end{eqnarray}
Bringing the second equation in the constraints \eqref{cond}, that is,
\begin{equation}\label{a}
a = \cos\theta - bv^TB
\end{equation}
into \eqref{opt}, we have
\begin{eqnarray}
\label{opt2}
\min_{b} && v^TB\cos\theta+b\left(B^TB-(v^TB)^2\right) \nonumber \\
\text{s.t.}&&
b^2\left(B^TB-(v^TB)^2\right) + \cos\theta^2=1.
\end{eqnarray}
Then the solution to \eqref{opt2} is
\begin{equation*}\label{b}
b = \pm\frac{\sin\theta}{\sqrt{B^TB-(v^TB)^2}}.
\end{equation*}
Since $B^TB-(v^TB)^2\geq 0$, the minimum is achieved when
\begin{equation}\label{b_final}
b = -\frac{|\sin\theta|}{\sqrt{B^TB-(v^TB)^2}}.
\end{equation}
Combining \eqref{b_final} with \eqref{a}, we can get the solution. \qed
\end{proof}
\subsection{Proof of Theorem \ref{gtheta}}
\label{sec:app:proof-10}
\begin{proof}
If $\theta \in [0,\pi)$, the square of the denominator of \eqref{x} becomes
$$
1 + 2v^TB\cos\theta + B^TB - 2\sin\theta\sqrt{B^TB-(v^TB)^2}
= 1 + B^TB + 2\sqrt{B^TB}\sin(\alpha - \theta),
$$
where
\begin{eqnarray*}
\sin\alpha &=& \frac{v^TB}{\sqrt{B^TB}}, \\
\cos\alpha &=& \frac{\sqrt{B^TB-(v^TB)^2}}{\sqrt{B^TB}}.
\end{eqnarray*}
Similarly, if $\theta \in [\pi, 2\pi)$, then the square of the denominator of \eqref{x} becomes
$$
1 + 2v^TB\cos\theta + B^TB + 2\sin\theta\sqrt{B^TB-(v^TB)^2}
= 1 + B^TB + 2\sqrt{B^TB}\sin(\alpha + \theta),
$$
where $\alpha$ is the same defined as above.
Hence, for $\theta \in [0,\pi)$, we have
$$
f(\theta) = \frac{|\cos\theta| + A}{\sqrt{1 + B^TB + 2\sqrt{B^TB}\sin(\alpha - \theta)}};
$$
for $\theta \in [\pi,2\pi)$, we have
$$
f(\theta) = \frac{|\cos\theta| + A}{\sqrt{1 + B^TB + 2\sqrt{B^TB}\sin(\alpha + \theta)}},
$$
which is equivalent to
$$
f(\theta) = \frac{|\cos\theta| + A}{\sqrt{1 + B^TB + 2\sqrt{B^TB}\sin(\alpha + \theta)}},
$$
where $\theta \in [-\pi,0),$
which is also equivalent to
$$
f(\theta) = \frac{|\cos\theta| + A}{\sqrt{1 + B^TB + 2\sqrt{B^TB}\sin(\alpha - \theta)}},
$$
where $\theta \in [0,\pi).$
So the problem we want to solve is actually to maximize
$$
f(\theta) = \frac{|\cos\theta| + A}{\sqrt{1 + B^TB + 2\sqrt{B^TB}\sin(\alpha - \theta)}}
$$
on $\theta \in [0,\pi)$.
Under the first order condition, we have that if $\theta^*$ maximizes $f(\theta)$, then
$0 = f\rq(\theta^*).$
When $\theta \in [0,\frac{\pi}{2})$, the first order differentiable function of $f(\theta)$ can be written as
$$
f\rq(\theta) = \frac{-(1+B^TB)\sin\theta + \sqrt{B^TB}\left[\cos\alpha
+ A\cos(\alpha-\theta)
- \sin\theta\sin(\alpha - \theta)\right]}{\left(1 + B^TB + 2\sqrt{B^TB}\sin(\alpha - \theta)\right)^{3/2}};
$$
When $\theta \in [\frac{\pi}{2},\pi)$, the first order differentiable function of $f(\theta)$ can be written as
$$
f\rq(\theta) = \frac{(1+B^TB)\sin\theta
+ \sqrt{B^TB}\left[-\cos\alpha + A\cos(\alpha-\theta)
+ \sin\theta\sin(\alpha - \theta)\right]}{\left(1 + B^TB + 2\sqrt{B^TB}\sin(\alpha - \theta)\right)^{3/2}}.
$$
If we define function $g(\theta)$ as the following
\begin{equation*}
g(\theta) =
\left\{ \begin{array}{ll}
\sqrt{B^TB}\left[\cos\alpha
+ A\cos(\alpha-\theta) - \sin\theta\sin(\alpha - \theta)\right] & \\
-(1+B^TB)\sin\theta, & \mbox{ if } \theta\in[0,\frac{\pi}{2}),\\
\sqrt{B^TB} \left[-\cos\alpha + A\cos(\alpha-\theta) + \sin\theta\sin(\alpha - \theta)\right] & \\
+ (1+B^TB)\sin\theta &\mbox{ if } \theta\in[\frac{\pi}{2},\pi),
\end{array}
\right.
\end{equation*}
Then our goal becomes to find the zeros of the function $g(\theta).$ \qed
\end{proof}
\bibliographystyle{plainnat}
|
1,116,691,497,797 | arxiv | \section{\label{sec:The-diagonaization}The diagonalization of the LRK Hamiltonian
and the generator for parameter estimation}
We note that the Hamiltonian in the momentum space is~\citep{pezz`e2017multipartite}
\begin{align}
H_{\theta} & =-\frac{J}{2}\sum_{k}\left[a^{\dagger}(k)a(k)+a^{\dagger}(-k)a(-k)\right]\cos k-\frac{\mu}{2}\left[\sum_{k}a^{\dagger}(k)a(k)+\sum_{-k}a^{\dagger}(-k)a(-k)\right]\nonumber \\
& +\frac{\text{i}\Delta}{4}\sum_{k}\left[a(-k)a(-k)-a^{\dagger}(-k)a^{\dagger}(-k)\right]f_{\alpha}(k).\label{eq:LRK-p-space}
\end{align}
We note that the unitary Fourier transformation that related the original
Hamiltonian~(\ref{eq:LRK}) to above Hamiltonian does not depend
on any estimation parameters. Thus, the Fisher information is preserved
by the transformation. Eq.~(\ref{eq:LRK-p-space}) can be diagonalized
as follows
\begin{equation}
H_{\theta}=\frac{1}{2}\sum_{k}\epsilon_{\theta}(k)\begin{bmatrix}a^{\dagger}(k),\, & a(-k)\end{bmatrix}U_{\theta}^{\dagger}(k)\sigma_{z}U_{\theta}(k)\begin{bmatrix}a(k)\\
a^{\dagger}(-k)
\end{bmatrix}
\end{equation}
through the Bogoliubov transformation $U_{\theta}(k)$ , where
\begin{align}
U_{\theta}(k) & \equiv\begin{pmatrix}\cos\left[\phi_{\theta}(k)/2\right] & \text{i}\sin\left[\phi_{\theta}(k)/2\right]\\
\text{i}\sin\left[\phi_{\theta}(k)/2\right] & \cos\left[\phi_{\theta}(k)/2\right]
\end{pmatrix},\\
\sin\phi_{\theta}(k) & =-\frac{\Delta}{2}\frac{f_{\alpha}(k)}{\epsilon_{\theta}(k)},\\
\cos\phi_{\theta}(k) & =-\frac{(J\cos k+\mu)}{\epsilon_{\theta}(k)}.
\end{align}
We denote
\begin{equation}
\begin{bmatrix}\eta_{\theta}(k)\\
\eta_{\theta}^{\dagger}(-k)
\end{bmatrix}\equiv U_{\theta}(k)\begin{bmatrix}a(k)\\
a^{\dagger}(-k)
\end{bmatrix}.
\end{equation}
The Hamiltonian can be rewritten as
\begin{equation}
H_{\theta}=\sum_{k}\epsilon_{\theta}(k)\left[\eta_{\theta}^{\dagger}(k)\eta_{\theta}(k)-\frac{1}{2}\right].\label{eq:H-diagonal}
\end{equation}
Once again, the parameter-dependent constant $\sum_{k}\epsilon_{\theta}(k)/2$
does not contribute the Fisher information and will be suppressed.
The generator for parameter estimation is~\citep{pang2017optimal,pang2014quantum}
\begin{equation}
G=\int_{0}^{T}\mathcal{U}_{\theta}^{\dagger}(\tau)\partial_{\theta}H_{\theta}\mathcal{U}_{\theta}(\tau)d\tau,\label{eq:G-def}
\end{equation}
where the evolution operator is $\mathcal{U}_{\theta}(\tau)=e^{-\text{i}H_{\theta}\tau}$.
Apparently, according to Eq.~(\ref{eq:H-diagonal}), the generator
contains two parts: The first part is due to the $\partial_{\theta}\epsilon_{\theta}(k)$
and the other is due to $\partial_{\theta}\eta_{\theta}^{\dagger}$
and $\partial_{\theta}\eta_{\theta}(k)$. It is readily checked that
\begin{gather}
\begin{bmatrix}\partial_{\theta}\eta_{\theta}(k)\\
\partial_{\theta}\eta_{\theta}^{\dagger}(-k)
\end{bmatrix}=\frac{d\phi_{\theta}}{d\theta}[\partial_{\phi}U_{\theta}(k)U_{\theta}^{-1}(k)]\begin{bmatrix}\eta_{\theta}(k)\\
\eta_{\theta}^{\dagger}(-k)
\end{bmatrix}\nonumber \\
=\frac{\text{i}}{2}\frac{d\phi_{\theta}(k)}{d\theta}\sigma_{x}\begin{bmatrix}\eta_{\theta}(k)\\
\eta_{\theta}^{\dagger}(-k)
\end{bmatrix}=-\frac{\text{i}\partial_{\theta}\cos\phi_{\theta}(k)}{2\sin\phi_{\theta}(k)}\sigma_{x}\begin{bmatrix}\eta_{\theta}(k)\\
\eta_{\theta}^{\dagger}(-k)
\end{bmatrix}.
\end{gather}
Therefore, we find
\begin{align}
\partial_{\theta}H_{\theta} & =\sum_{k}\partial_{\theta}\epsilon_{\theta}(k)\eta_{\theta}^{\dagger}(k)\eta_{\theta}(k)\nonumber \\
& +\frac{\text{i}}{2}\sum_{k}\xi_{\theta}(k)\epsilon_{\theta}(k)\left(\eta_{\theta}(-k)\eta_{\theta}(k)-\eta_{\theta}^{\dagger}(k)\eta_{\theta}^{\dagger}(-k)\right).\label{eq:dHdtheta}
\end{align}
Substituting Eq.~(\ref{eq:dHdtheta}) into Eq.~(\ref{eq:G-def}),
we find
\begin{equation}
G_{\theta}=T\sum_{k}\partial_{\theta}\epsilon_{\theta}(k)\eta_{\theta}^{\dagger}(k)\eta_{\theta}(k)+\frac{\text{i}}{2}\sum_{k}\xi_{\theta}(k)\epsilon_{\theta}(k)\int_{0}^{T}d\tau e^{\text{i}\sum_{p}\epsilon_{\theta}(p)\eta_{\theta}^{\dagger}(p)\eta_{\theta}(p)\tau}\left(\eta_{\theta}(-k)\eta_{\theta}(k)-\eta_{\theta}^{\dagger}(k)\eta_{\theta}^{\dagger}(-k)\right)e^{-\text{i}\sum_{p^{\prime}}\epsilon_{\theta}(p^{\prime})\eta_{\theta}^{\dagger}(p^{\prime})\eta_{\theta}(p^{\prime})\tau}.\label{eq:G}
\end{equation}
Note that the product of two Fermionic creation and annihilation operators
behaves like a $c$-number when commuting with Fermionic operators
in other modes. We further note that the negative modes are equivalent
to the positive modes via $-k\sim2\pi-k$. With these two observations,
it is readily checked that
\begin{align}
& e^{\text{i}\sum_{p}\epsilon_{\theta}(p)\eta_{\theta}^{\dagger}(p)\eta_{\theta}(p)\tau}\left(\eta_{\theta}(-k)\eta_{\theta}(k)-\eta_{\theta}^{\dagger}(k)\eta_{\theta}^{\dagger}(-k)\right)e^{-\text{i}\sum_{p^{\prime}}\epsilon_{\theta}(p^{\prime})\eta_{\theta}^{\dagger}(p^{\prime})\eta_{\theta}(p^{\prime})\tau}\nonumber \\
= & e^{\text{i}[\epsilon_{\theta}(k)\eta_{\theta}^{\dagger}(k)\eta_{\theta}(k)+\epsilon_{\theta}(2\pi-k)\eta_{\theta}^{\dagger}(2\pi-k)\eta_{\theta}(2\pi-k)]\tau}\left(\eta_{\theta}(-k)\eta_{\theta}(k)-\eta_{\theta}^{\dagger}(k)\eta_{\theta}^{\dagger}(-k)\right)e^{-\text{i}[\epsilon_{\theta}(k)\eta_{\theta}^{\dagger}(k)\eta_{\theta}(k)+\epsilon_{\theta}(2\pi-k)\eta_{\theta}^{\dagger}(2\pi-k)\eta_{\theta}(2\pi-k)]\tau}\nonumber \\
= & e^{\text{i}\epsilon_{\theta}(k)\tau[\eta_{\theta}^{\dagger}(k)\eta_{\theta}(k)+\eta_{\theta}^{\dagger}(-k)\eta_{\theta}(-k)]}\left(\eta_{\theta}(-k)\eta_{\theta}(k)-\eta_{\theta}^{\dagger}(k)\eta_{\theta}^{\dagger}(-k)\right)e^{-\text{i}\epsilon_{\theta}(k)\tau[\eta_{\theta}^{\dagger}(k)\eta_{\theta}(k)+\eta_{\theta}^{\dagger}(-k)\eta_{\theta}(-k)]}.\label{eq:exp-fermionic}
\end{align}
Using the relation
\begin{equation}
e^{i\theta\eta^{\dagger}\eta}=1+\eta^{\dagger}\eta(e^{\text{i}\theta}-1)=e^{\text{i\ensuremath{\theta}}}-\eta\eta^{\dagger}(e^{\text{i}\theta}-1)
\end{equation}
for Fermionic operators, Eq.~(\ref{eq:exp-fermionic}) becomes
\begin{align}
& e^{\text{i}\epsilon_{\theta}(k)\tau[\eta_{\theta}^{\dagger}(k)\eta_{\theta}(k)+\eta_{\theta}^{\dagger}(-k)\eta_{\theta}(-k)]}\eta_{\theta}(-k)\eta_{\theta}(k)e^{-\text{i}\epsilon_{\theta}(k)\tau[\eta_{\theta}^{\dagger}(k)\eta_{\theta}(k)+\eta_{\theta}^{\dagger}(-k)\eta_{\theta}(-k)]}\nonumber \\
= & [1+\eta_{\theta}^{\dagger}(k)\eta_{\theta}(k)(e^{\text{i}\epsilon_{\theta}(k)\tau}-1)][1+\eta_{\theta}^{\dagger}(-k)\eta_{\theta}(-k)(e^{\text{i}\epsilon_{\theta}(k)\tau}-1)]\eta_{\theta}(-k)\eta_{\theta}(k)\nonumber \\
\times & [e^{-\text{i}\epsilon_{\theta}(k)\tau}-\eta_{\theta}(k)\eta_{\theta}^{\dagger}(k)(e^{-\text{i}\epsilon_{\theta}(k)\tau}-1)][e^{-\text{i}\epsilon_{\theta}(k)\tau}-\eta_{\theta}(-k)\eta_{\theta}^{\dagger}(-k)(e^{-\text{i}\epsilon_{\theta}(k)\tau}-1)]\nonumber \\
= & e^{-\text{i}\epsilon_{\theta}(k)\tau}[1+\eta_{\theta}^{\dagger}(k)\eta_{\theta}(k)(e^{\text{i}\epsilon_{\theta}(k)\tau}-1)]\eta_{\theta}(-k)\eta_{\theta}(k)[e^{-\text{i}\epsilon_{\theta}(k)\tau}-\eta_{\theta}(-k)\eta_{\theta}^{\dagger}(-k)(e^{-\text{i}\epsilon_{\theta}(k)\tau}-1)]\nonumber \\
= & -e^{-\text{i}\epsilon_{\theta}(k)\tau}[1+\eta_{\theta}^{\dagger}(k)\eta_{\theta}(k)(e^{\text{i}\epsilon_{\theta}(k)\tau}-1)]\eta_{\theta}(k)\eta_{\theta}(-k)[e^{-\text{i}\epsilon_{\theta}(k)\tau}-\eta_{\theta}(-k)\eta_{\theta}^{\dagger}(-k)(e^{-\text{i}\epsilon_{\theta}(k)\tau}-1)]\nonumber \\
= & -e^{-2\text{i}\epsilon_{\theta}(k)\tau}\eta_{\theta}(k)\eta_{\theta}(-k)=e^{-2\text{i}\epsilon_{\theta}(k)\tau}\eta_{\theta}(-k)\eta_{\theta}(k),
\end{align}
where we have used $\eta_{\theta}^{2}(-k)=\eta_{\theta}^{2}(k)=0$.
The generator now becomes
\begin{align}
G_{\theta} & =\sum_{k}\left\{ T\partial_{\theta}\epsilon_{\theta}(k)\eta_{\theta}^{\dagger}(k)\eta_{\theta}(k)\right.\nonumber \\
& \left.+\frac{\xi_{\theta}(k)}{4}\left[(1-e^{-2\text{i}\epsilon_{\theta}(k)T})\eta_{\theta}(-k)\eta_{\theta}(k)+(1-e^{2\text{i}\epsilon_{\theta}(k)T}))\eta_{\theta}^{\dagger}(k)\eta_{\theta}^{\dagger}(-k)\right]\right\} .\label{eq:G-general}
\end{align}
We rewrite Eq.~(\ref{eq:G-general}) in a more compact form
\begin{equation}
G_{\theta}=\frac{1}{2}\sum_{k}\begin{bmatrix}\eta_{\theta}^{\dagger}(k),\, & \eta_{\theta}(-k)\end{bmatrix}\mathscr{G}_{\theta}(k)\begin{bmatrix}\eta_{\theta}(k)\\
\eta_{\theta}^{\dagger}(-k)
\end{bmatrix},
\end{equation}
where the matrix $\mathscr{G}_{\theta}(k)$ is defined as
\begin{equation}
\mathscr{G}_{\theta}(k)\equiv T\partial_{\theta}\epsilon_{\theta}(k)\sigma_{z}+\frac{\xi_{\theta}(k)}{2}\left(1-\cos[2\epsilon_{\theta}(k)T]\right)\sigma_{x}+\frac{\xi_{\theta}(k)}{2}\sin[2\epsilon_{\theta}(k)T]\sigma_{y}.
\end{equation}
Upon noting that $\mathscr{G}_{\theta}(k)=\mathscr{E}_{\theta}(k)\bm{n}_{\theta}(k)\cdot\bm{\sigma}$,
we obtain the spectrum~(\ref{eq:scrE-def}) in the main text.
\section{\label{sec:Euler-Maclaurin-formula}The Euler-Maclaurin formula}
\begin{lem}\label{lem:series-integral-rep}(Euler-Maclaurin formula)
For arbitrary function $g(x)$ with continuous derivatives, the infinite
series $\sum_{n=a}^{b}g(m)$ can be converted the corresponding integral
plus remainder terms via the Euler-Maclaurin formula~\citep{knopp1990theoryand}.
\begin{equation}
\sum_{n=a}^{n=b}g(n)=\int_{a}^{b}g(x)dx+R,\label{eq:E-M}
\end{equation}
where the remainder is
\begin{align}
R & =\frac{1}{2}[g(b)-g(a)]+\sum_{m=1}^{M}\frac{b_{2m}}{(2m)!}\left[g^{(2m-1)}(b)-g^{(2m-1)}(a)\right]\nonumber \\
& +\int_{a}^{b}\frac{1}{(2M+1)!}P_{2M+1}(x)g^{(2M+1)}(x)dx.
\end{align}
Here $M$ can be arbitrarily chosen from the natural numbers $0,1,\,2,\cdots$
, $b_{2m}$ is the Bernoulli number and $P_{0}(x)=1$ and for $M>0$
\begin{equation}
P_{M}(x)=\frac{1}{M!}B_{M}(\{x\}),\label{eq:P-def}
\end{equation}
where$\{x\}\equiv x-[x]$ and $B_{M}$ is the Bernoulli polynomial.\end{lem}
We can use the Euler-Maclaurin formula to approximate a series
\begin{equation}
\sum_{n=a}^{b}f\left(\frac{(2n+1)\pi}{N}\right)=\sum_{k=k_{a}}^{k_{b}}f(k),
\end{equation}
where we $k=(2n+1)\pi/N$. We assume $f(k)$ is piecewise smooth on
$[k_{a},\,k_{b}]$ and does not blow up on $[k_{a},\,k_{b}]$. \textit{We
allow some discontinuities in the first derivatives if $M=1$ so that
$f(k)$ may contain absolute or square root}. Without loss of generality,
we can assume $f(k)$ is smooth on interval $F_{j}$'s where $\cup_{j}F_{j}=[k_{a},\,k_{b}]$.
When denoting the function in terms of the variable , these intervals
are denotes as $E_{j}$'s. applying the Euler-Maclaurin formula for
these intervals respectively with $M=0$, we find
\begin{equation}
\sum_{n=0}^{N-1}f\left[\frac{(2n+1)\pi}{N}\right]=\sum_{j}\left\{ \int_{E_{j}}f\left[\frac{(2x+1)\pi}{N}\right]dx+R_{j}\right\} ,
\end{equation}
where
\begin{equation}
R_{j}=\frac{2\pi}{N}\int_{E_{j}}P_{1}(x)\sin\left[\frac{(2x+1)\pi}{N}\right]dx+\text{boundary terms},
\end{equation}
and $P_{1}(x)$ is defined in Eq.~(\ref{eq:P-def}). Since the boundary
terms remains finite and does not scale with $N$. They will be omitted
subsequently. Now change $x$ back to $k$, we find
\begin{align}
\sum_{n=0}^{N-1}f\left(\frac{[2n+1]\pi}{N}\right) & =\frac{N}{2\pi}\int_{\pi/N}^{2\pi-\pi/N}f(k)dk+\sum_{j}R_{j},
\end{align}
where
\begin{equation}
R_{j}=\int_{F_{j}}P_{1}\left(\frac{Nk}{2\pi}-\frac{1}{2}\right)f^{\prime}(k)dk.
\end{equation}
Since $f(k)$ is differentiable on $F_{j}$, $f^{\prime}(k)$ is regular
on $F_{j}$. On the other hand, $P_{1}(x)\in[-1/2,\,1/2]$ is bounded.
We find that $R_{j}$ remains finite as long as the number of the
$F_{j}$'s does not scale with $N$. \textit{So we conclude that when
$f(k)$ is regular, $\sum_{k=k_{a}}^{k=k_{b}}f(k)\sim N$.} For example,
in Eq.~(\ref{eq:I0-J}), we take $f(k)=|\cos k|$. Apparently $|\cos k|$
is differentiable on $F_{1}=[\pi/N,\,\pi/2-\pi/N]$, $F_{2}=[\pi/2+\pi/N,\,3\pi/2-\pi/N]$
and $F_{3}=[3\pi/2+\pi/N,\,2\pi-\pi/N]$ respectively.
However, we note that if $f(k)$ has a singularity in $[0,\,2\pi]$,
the remainder may not be necessarily stay as a constant as $N\to\infty$.
For example, if we take
\begin{equation}
f(k)=\cot\left[\frac{k}{2}\right],\label{eq:cot}
\end{equation}
where $k\in[\pi/N,\,\pi-\pi/N]$. Then we obtain
\begin{equation}
\sum_{n=0}^{N/2-1}\cot\left[\frac{(2n+1)\pi}{2N}\right]=\frac{N}{\pi}\int_{\pi/N}^{\pi-\pi/N}\cot\left(\frac{k}{2}\right)dk+R,\label{eq:gamma0-EM}
\end{equation}
where
\begin{align}
R & =\frac{1}{2}\left[\cot\left(\frac{\pi}{2}-\frac{\pi}{2N}\right)-\cot\left(\frac{\pi}{2N}\right)\right]\nonumber \\
& -\int_{\pi/N}^{\pi-\pi/N}P_{1}\left(\frac{Nk}{2\pi}-\frac{1}{2}\right)\frac{1}{\sin^{2}(k/2)}dk,
\end{align}
where $P_{1}(t)$ is defined in Eq.~(\ref{eq:P-def}). One can see
the integrand in the remainder has a singularity around $k=0$ and
there main integral is no longer a good approximation of the sum.
Nevertheless we can find upper bound of the scaling of integral in
the remainder, i.e.,
\begin{align}
& \bigg|\int_{\pi/N}^{\pi-\pi/N}P_{1}\left(\frac{Nk}{2\pi}-\frac{1}{2}\right)\frac{1}{\sin^{2}(k/2)}dk\bigg|\nonumber \\
\le & \int_{\pi/N}^{\pi-\pi/N}\frac{1}{\sin^{2}(k/2)}\sim\cot\left(\frac{\pi}{2N}\right).\label{eq:R0-bound}
\end{align}
So we conclude the remainder will scale at most as $N$. Since $\int_{\pi/N}^{\pi-\pi/N}\cot(k/2)dk\sim N\ln N$,
we obtain the scaling of $\gamma_{0}(N)$ in the main text. We see
that in the current case\textit{ }the remainder depends on $N$ instead
of a constant as indicated in Eq.~(\ref{eq:R0-bound}). We would
like to emphasize that \textit{when the summand of a sum has a singularity
in the limit $N\to\infty$, it is not rigorous to analyze the scaling
of the sum only with the main integral because the remainder may also
scale. }
\section{\label{sec:Proof-Scaling-Gamma} The scaling of $\gamma_{\alpha}(N)$
for $f_{\alpha}(k)\le\mathcal{O}(1/k)$ near $k=0$.}
\begin{thm}We shall assume the only possible singularity of $f_{\alpha}(k)$
is near $k=0$, a fact which we will prove in Corollary~\ref{cor:sing-at-zero}.
Then the scaling of $\gamma_{\alpha}(N)$ is controlled by the main
integral if $f_{\alpha}(k)\le\mathcal{O}(1/k)$ near $k=0$.
\end{thm}
\begin{proof}
Let us first focus on the case $f_{\alpha}(k)$ is strictly slower
than $1/k$ near $k=0$. We denote $E_{j}$ as the intervals where
$f_{\alpha}([2x+1]\pi/N)$ is smooth as function $x$. Similar as
Sec.~\ref{sec:Euler-Maclaurin-formula}, this denomination allows
$f_{\alpha}(k)$ to be piecewise functions joined by smooth functions,
as long as there are no singularities at the joints. The intervals
$E_{j}$becomes $F_{j}$ when the function is written in terms of
the variable $k$. In particular, one can easily show that $f_{\alpha}(\pi/N)$
is positive. Applying Euler-Maclaurin formula~(\ref{eq:E-M}) to
each of these intervals, we find
\begin{equation}
\gamma_{\alpha}(N)=2\sum_{j}\left[\int_{E_{j}}(-1)^{j-1}f_{\alpha}\left(\frac{[2x+1]\pi}{N}\right)dx+R_{\alpha j}\right],
\end{equation}
where the remainder is
\begin{equation}
R_{\alpha j}=\int_{E_{j}}P_{1}\left(x\right)(-1)^{j-1}f^{\prime}\left(\frac{[2n+1]x\pi}{N}\right)dx+\text{boundary terms}.
\end{equation}
Since the boundary terms does not scale with $N$, we shall suppress
them in subsequent analysis. Now we change $x$ back to $k$, we find
\begin{equation}
\gamma_{\alpha}(N)=2\left[\frac{N}{2\pi}\int_{\pi/N}^{\pi-\pi/N}\big|f_{\alpha}(k)\big|dk+\sum_{j}R_{\alpha j}\right],\label{eq:gamma-EM}
\end{equation}
where
\begin{equation}
R_{\alpha j}=\int_{F_{j}}P_{1}\left(\frac{Nk}{2\pi}-\frac{1}{2}\right)(-1)^{j-1}f_{\alpha}^{\prime}(k)dk.
\end{equation}
For the remainders, if $F_{j}$ does not contain the origin, then
the integral in $R_{j}$ is regular and does not scale with constant.
For $F_{j}$ contains the origin, we use a common trick in asymptotic
analysis~\citep{bender2013advanced}: The leading order of a singular
integral can be found by replacing the integrand with its leading
order Laurent expansion near the singular point. In our current case,
since
\begin{equation}
f_{\alpha}(k)<\mathcal{O}\left(\frac{1}{k}\right),\label{eq:f-singu}
\end{equation}
we find
\begin{align}
& \bigg|\int_{F_{j}}P_{1}\left(\frac{Nk}{2\pi}-\frac{1}{2}\right)(-1)^{j-1}f_{\alpha}^{\prime}(k)dk\bigg|\nonumber \\
< & \bigg|\int_{F_{j}}P_{1}\left(\frac{Nk}{2\pi}-\frac{1}{2}\right)(-1)^{j-1}\left(\frac{1}{k}\right)^{\prime}dk\bigg|\nonumber \\
\leq & \bigg|\int_{\pi/N}\left(\frac{1}{k}\right)^{\prime}dk\bigg|\sim N,\label{eq:R-bound}
\end{align}
where we have used that $P_{1}(x)\in[-1/2,\,1/2]$ is bounded. That
is, the remainder scale scale strictly slower than $N$, which is
subleading order compared to the first term on the r.h.s. of Eq.~(\ref{eq:gamma-EM}).
When $f_{\alpha}(k)\sim\mathcal{O}(1/k)$, one can go through the
same argument and will find that the main integral will scale as $N\ln N$
while the upper bound of the scaling of the remainder is $N$. Therefore,
we conclude that the leading order scaling of $\gamma_{\alpha}(N)$
is only given by the main integral if $f_{\alpha}(k)\le\mathcal{O}(1/k)$
near $k=0$.
\end{proof}
We conclude this section by note that the condition $f_{\alpha}(k)\le\mathcal{O}(1/k)$
is non-trivial and essential: \textit{Had $f_{\alpha}(k)$ scaled
as $1/k^{1+\varepsilon}$ near $k=0$, where $\varepsilon$ is arbitrary
positive number, the above proof would yield that both the main integral
and the upper bound of the remainder $R_{\alpha j}$ scales $N^{1+\varepsilon}$}.
The analysis of the scaling of $\gamma_{\alpha}(N)$ would be subtle
because the leading order scaling of the main integral and the remainder
$R_{\alpha j}$ might cancel each other. Fortunately, we see such
a situation does not occur because we have shown in the main text
that $f_{\alpha}(k)\le\mathcal{O}(1/k)$.
\section{\label{sec:Singularity-f-alpha-power}The singularity of $f_{\alpha}(k)$
for $\kappa_{l,\,\alpha}=l^{-\alpha}$ with $\alpha\in(0,\,1]$}
In this section, we prove an analytic property of $f_{\alpha}(k)$
for the particular case where $\kappa_{l,\,\alpha}=l^{-\alpha}$:
\begin{equation}
f_{\alpha}(k)\sim\frac{1}{k^{1-\alpha}},\,\alpha\in(0,\,1].\label{eq:f-alpha-ana-power}
\end{equation}
This result can be shown using the singularity of the the polylogarithm
functions~\citep{olver2010nisthandbook,vodola2014kitaevchains}.
However, this approach does not allows to obtain general property
of $f_{\alpha}(k)$ when $\kappa_{l,\,\alpha}$ takes a more general
class of functions. Now we shall we shall explicitly show the singularity
of $f_{\alpha}(k)\sim1/k^{1-\alpha}$ around $k=0$ for $\alpha\in(0,\,1]$
without resorting to the polylogarithm functions. Recall
\begin{equation}
f_{\alpha}(k)\equiv2\sum_{l=1}^{N/2-1}\kappa_{l,\,\alpha}\sin(kl)+\kappa_{N/2,\,\alpha}\label{eq:f-alpha-def}
\end{equation}
Note that due to the regularity condition~(\ref{eq:kappa-reg-ends}),
we know that $\kappa_{N/2,\,\alpha}$ is finite as $N\to\infty$.
Therefore in what follows we shall omit $\kappa_{N/2,\,\alpha}$ in
the definition of $f_{\alpha}(k)$ because it does not affect the
analytic property of $f_{\alpha}(k)$. Now we are in a position to
prove Eq.~(\ref{eq:f-alpha-ana-power}):
\begin{proof}
Apparently $f_{0}(k)$ can be exactly calculated to be $\cot(k/2)$
which scales as $1/k$ near $k=0$. For the case $\alpha\in(0,\,1]$,
after applying the Euler-Maclaurin formula~(\ref{eq:E-M}), $f_{\alpha}(k)$
becomes
\begin{equation}
f_{\alpha}(k)=2\mathscr{F}_{\alpha}(k)+\mathscr{R}_{\alpha}(k),\label{eq:falpha-EM}
\end{equation}
where
\begin{equation}
\mathscr{F}_{\alpha}(k)=\int_{1}^{N/2-1}\frac{\sin(kx)}{x^{\alpha}}dx,
\end{equation}
and the remainder is
\begin{equation}
\mathscr{R}_{\alpha}(k)=2k\int_{1}^{N}P_{1}(\{x\})\frac{\cos(kx)}{x^{\alpha}}-2\int_{1}^{N}P_{1}(\{x\})\frac{\sin(kx)}{x^{\alpha+1}},\label{eq:R-f}
\end{equation}
where we have again ignored the finite boundary terms. Apparently
the second term in Eq.~(\ref{eq:R-f}) is finite and therefore will
not contribute to the singularity of $f_{\alpha}(k)$, since
\begin{equation}
2\bigg|\int_{1}^{N}P_{1}(\{x\})\frac{\sin(kx)}{x^{\alpha+1}}\bigg|<\bigg|\int_{1}^{N}\frac{1}{x^{\alpha+1}}\bigg|<\infty.
\end{equation}
Our goal now is to determine the asymptotic behavior of the first
term of Eq.~(\ref{eq:R-f}). Applying Fourier transform of $P_{1}(\{x\})$~\citep{knopp1990theoryand}
\begin{equation}
P_{1}(\{x\})=-\sum_{m=1}^{\infty}\frac{\sin(2m\pi x)}{m\pi},
\end{equation}
we obtain
\begin{align}
\int_{1}^{N}P_{1}(\{x\})\frac{\cos(kx)}{x^{\alpha}}dx & =-\frac{1}{2}\sum_{m=1}^{\infty}\frac{1}{m\pi}\int_{1}^{N}\left\{ \frac{\sin[(2m\pi+k)x]}{x^{\alpha}}\right.\nonumber \\
& +\left.\frac{\sin[(2m\pi-k)x]}{x^{\alpha}}\right\} dx.\label{eq:R-f-first}
\end{align}
Integrating by parts, we find that
\begin{align}
\int_{1}^{N}\frac{\sin[(2m\pi+k)x]}{x^{\alpha}} & =\frac{1}{2m\pi+k}\left\{ \frac{\cos[(2m\pi+k)x]}{x^{\alpha}}\big|_{x=1}^{N}\right.\nonumber \\
& +\left.\alpha\int_{1}^{N}\frac{\cos[(2m\pi+k)x]}{x^{\alpha+1}}dx\right\} .\label{eq:sin-power-scaling1}
\end{align}
Apparently, the integral on the r.h.s is bounded in the limit $N\to\infty$
as long as $\alpha>0$. So in the limit $N\to\infty$, we find
\begin{equation}
\int_{1}^{N}\frac{\sin[(2m\pi+k)x]}{x^{\alpha}}\lesssim\frac{1}{2m\pi+k}.
\end{equation}
By similar argument, one can show
\begin{equation}
\int_{1}^{N}\frac{\sin[(2m\pi-k)x]}{x^{\alpha}}\lesssim\frac{1}{2m\pi-k}.
\end{equation}
Note if $k$ is resonant with $2m\pi$ then the original integral
vanishes and there is no need to do the scaling analysis for the second
integral on the r.h.s. of Eq.~(\ref{eq:R-f-first}). Substituting
above results into Eq.~(\ref{eq:R-f-first}), we find
\begin{equation}
\int_{1}^{N}P_{1}(\{x\})\frac{\cos(kx)}{x^{\alpha}}\lesssim\sum_{m=1}^{\infty}\frac{1}{m^{2}}<\infty.\label{eq:sin-power-scaling2}
\end{equation}
Up to now, we have shown that there is no singularity in the remainder
as long as $\alpha>0$. To see the singularity in the first term of
Eq.~(\ref{eq:falpha-EM}), we make change of variables $kl=s$ and
obtain
\begin{equation}
\mathscr{F}_{\alpha}(k)=\frac{1}{k^{1-\alpha}}\int_{k}^{n\pi+\pi/2}\frac{\sin s}{s^{\alpha}}ds,\label{eq:integral-s}
\end{equation}
where we note $Nk=(2n+1)\pi$, where $n=0,\,1\cdots,\,N-1$. In the
limit $N\to\infty$, if finite $n$is finite, apparently the singularity
of $f_{\alpha}(k)\sim1/k^{1-\alpha}$. On the other hand, if $n\to\infty$,
the integral in Eq.~(\ref{eq:integral-s}) is still finite since
\begin{equation}
\int_{0}^{\infty}ds\frac{\sin s}{s^{\alpha}}=\Gamma(1-\alpha)\cos\left(\frac{\pi\alpha}{2}\right),\text{for},\,\alpha\in(0,\,1]\label{eq:SIntegral-convergence}
\end{equation}
as we will now show. We note that
\begin{equation}
\lim_{n\to\infty}\int_{0}^{n\pi+\pi/2}\frac{\sin s}{s^{\alpha}}ds=\int_{0}^{\infty}ds\frac{\sin s}{s^{\alpha}}=\text{Im}\int_{0}^{\infty}dss^{-\alpha}e^{\text{i}s}.\label{eq:S-infty}
\end{equation}
One can evaluate Eq.~(\ref{eq:S-infty}) by first replacing $s\to\text{i}t$
and obtain
\begin{align}
\int_{0}^{\infty}dss^{-\alpha}e^{\text{i}s} & =\text{i}^{1-\alpha}\int_{0}^{\text{i}\infty}dtt^{-\alpha}e^{-t}\nonumber \\
& =\text{i}^{1-\alpha}\lim_{\varepsilon\to0}\left[\int_{0}^{\text{i}\varepsilon}dtt^{-\alpha}e^{-t}+\int_{\text{i}\varepsilon}^{\text{i}\infty}dtt^{-\alpha}e^{-t}\right].\label{eq:s-to-t}
\end{align}
The convergence of the first integral on the r.h.s. of Eq.~(\ref{eq:s-to-t})
requires that $\alpha<1$. The convergence of the second integral
on the r.h.s. of Eq.~(\ref{eq:s-to-t}) requires the integrand vanishes
at $t=\text{i}\infty$, which leads to $\alpha>0$. Now we take advantage
of the analyticity of the integrand for $\alpha\in(0,\,1]$ and rotate
the integral from positive imaginary $t-$axis to positive real $t-$axis,
which yields,
\begin{equation}
\int_{0}^{\text{i}\infty}dtt^{-\alpha}e^{-t}=\int_{0}^{\infty}dtt^{-\alpha}e^{-t}=\Gamma(1-\alpha)
\end{equation}
which concludes the proof of Eq.~(\ref{eq:SIntegral-convergence})
for $\alpha\in(0,\,1)$. In fact Eq.~(\ref{eq:SIntegral-convergence})
also holds for $\alpha=1$ since $\int_{0}^{\infty}ds\sin s/s=\frac{\pi}{2}$
which can be evaluated by the residue theorem is actually $\lim_{\alpha\to1}\Gamma(1-\alpha)\cos\left(\frac{\pi\alpha}{2}\right)$.
Therefore, we have successfully shown that the singularity of $f_{\alpha}(k)$
only lies in the main term of Euler-Maclaurin formula, which is Eq.~(\ref{eq:falpha-EM}).
\end{proof}
\section{\label{sec:Proof-General-D}An integral approximation to $f_{\alpha}(k)$}
We show in Sec.~\ref{sec:Singularity-f-alpha-power} that the singularity
of $f_{\alpha}(k)$ when $\kappa_{l,\,\alpha}=l^{-\alpha}$ can be
explicitly found with only elementary techniques, without resorting
to the polylogarithmic function as in the original proposal of the
LRK~\citep{vodola2014kitaevchains}. The advantage of this approach
is that it will allows us to prove the following theorem for general
functions $\kappa_{l,\,\alpha}$ that that satisfy the regularity
conditions~(\ref{eq:kappa-reg-ends},~\ref{eq:kappa-regularity}):
\begin{thm}\label{thm:General-d}We consider a general piecewise
smooth function $\kappa_{x,\,\alpha}$ that satisfies the regularity
conditions in the main text, i.e., $\kappa_{x,\,\alpha}$ satisfies
(i)
\begin{align}
\big|\kappa_{x,\,\alpha}^{(q)} & \big|<\infty,\,q=0,\,1,\,\cdots2Q,\label{eq:kappa-reg-ends}
\end{align}
which holds piecewisely on $[1,\,\infty]$, and
(ii)
\begin{equation}
\big|\int_{1}^{\infty}\kappa_{x,\,\alpha}^{(2Q+1)}dx\big|<\infty,\label{eq:kappa-regularity}
\end{equation}
where $Q$ is a non-negative integer and the superscript $(q)$ denotes
the $q$-th derivative with respect to $x$.
Then the singularity of $f_{\alpha}(k)$ near $k=0$ is controlled
by the main integral in the Euler-Maclaurin formula, i.e., the first
term in
\begin{equation}
f_{\alpha}(k)=2\mathscr{F}_{\alpha}(k)+\mathscr{R}_{\alpha}(k).\label{eq:f-rep}
\end{equation}
\end{thm}
Before we start the proof, let us first note that for the long-range
decay function $d_{x,\,\alpha}$, we allow not only smooth functions
of $x$, but also piecewise functions consisting of several smooth
functions. This is because, as we have seen in Sec.~\ref{sec:Euler-Maclaurin-formula}
and \ref{sec:Proof-Scaling-Gamma}, one can apply the Euler-Maclaurin
in a piecewise way. The condition~(\ref{eq:kappa-reg-ends}) indicates
there can be only discontinuities at the joints, but no singularities.
Nevertheless, in what follows, we shall prove for the case when $d_{x,\,\alpha}$
is smooth $[1,\,\infty]$, which can be easily generalized to the
case of piecewise smoothness without any difficulty.
\begin{proof}
We take $M=Q$ in the Euler-Maclaurin formula~(\ref{eq:E-M}), and
obtain
\begin{equation}
f_{\alpha}(k)=2\mathscr{F}_{\alpha}(k)+\mathscr{R}_{\alpha}(k),
\end{equation}
where
\begin{equation}
\mathscr{F}_{\alpha}(k)=\int_{1}^{N/2-1}\sin(kx)\kappa_{x,\,\alpha}dx,
\end{equation}
\begin{equation}
\mathscr{R}_{\alpha}(k)=\sum_{q=0}^{2Q+1}\mathscr{R}_{\alpha,\,q}(k)+\sum_{m=1}^{Q}\frac{b_{2m}}{(2m)!}\left[\sin(kx)\kappa_{x,\,\alpha}\right]^{(2m-1)}\bigg|_{x=1}^{x=N/2-1},
\end{equation}
\begin{equation}
\mathscr{R}_{\alpha,\,q}(k)\equiv\mathcal{C}_{Q,\,q}\int_{1}^{N/2-1}P_{2Q+1}(x)[\sin(kx)]^{(q)}\left[\kappa_{x,\,\alpha}\right]^{(2Q+1-q)}dx,\label{eq:scrR-q}
\end{equation}
\begin{equation}
\mathcal{C}_{Q,\,q}\equiv\begin{pmatrix}2Q-1\\
q
\end{pmatrix}\frac{1}{(2Q+1)!}.
\end{equation}
Apparently, the boundary term is finite due to the regularity condition~(\ref{eq:kappa-reg-ends}).
Thus we shall focus on the integral in the remainder $\mathscr{R}_{\alpha}(k)$
subsequently. For $q=0$, i we find
\begin{align}
\mathscr{R}_{\alpha,\,q}(k)\leq & \mathcal{C}_{Q,\,q}\mathcal{C}_{2Q+1}\bigg|\int_{1}^{N/2-1}\kappa_{x,\,\alpha}^{(2Q+1)}dx\bigg|<\infty,
\end{align}
where we have used the fact that $P_{2Q+1}(x)$ is bounded, $\big|P_{2Q+1}(x)\big|\le\mathcal{C}_{2Q+1}$
and Eq.~(\ref{eq:kappa-regularity}). When $q\ge1$, we apply the
Fourier transform of $P_{2Q+1}(x)$~\citep{knopp1990theoryand}
\begin{equation}
P_{2Q+1}(x)=\sum_{m=1}^{\infty}(-1)^{Q-1}\frac{2\sin(2m\pi x)}{(2m\pi)^{2Q+1}}.
\end{equation}
to Eq.~(\ref{eq:scrR-q}). For for even $q>0$, we obtain
\begin{gather}
\mathscr{R}_{\alpha,\,q}(k)=\mathcal{C}_{Q,\,q}k^{q}(-1)^{Q-1+q/2}\sum_{m=1}^{\infty}\frac{1}{(2m\pi)^{2Q+1}}\times\nonumber \\
\int_{1}^{N/2-1}\left(\cos[(2m\pi-k)x]-\cos[(2m\pi+k)x]\right)\kappa_{x,\,\alpha}^{(2Q+1-q)}dx,\label{eq:scrR-even}
\end{gather}
and for odd $q\ge1$, we obtain
\begin{gather}
\mathscr{R}_{\alpha,\,q}(k)=\mathcal{C}_{Q,\,q}k^{q}(-1)^{Q-1+(q-1)/2}\sum_{m=1}^{\infty}\frac{1}{(2m\pi)^{2Q+1}}\times\nonumber \\
\int_{1}^{N/2-1}\left\{ \sin[(2m\pi+k)x]+\sin[(2m\pi-k)x]\right\} \kappa_{x,\,\alpha}^{(2Q+1-q)}dx.\label{eq:scrR-odd}
\end{gather}
Since the convergence of the series is determined by the behavior
of the general term at large values of the index, we shall focus on
the case of large $m$ in the series in Eqs.~(\ref{eq:scrR-even},
\ref{eq:scrR-odd}) subsequently. Similarly with Eqs.~(\ref{eq:sin-power-scaling1}-\ref{eq:sin-power-scaling2}),
one can perform integration by parts until one gets an integrand that
contains $(\kappa_{x,\,\alpha})^{(2Q+1)}$, which yields
\begin{align}
& \int_{1}^{N/2-1}\cos[(2m\pi\pm k)x]\left(\kappa_{x,\,\alpha}\right)^{(2Q+1-q)}dx\nonumber \\
= & \frac{1}{(2m\pi\pm k)}\left\{ \sin[(2m\pi\pm k)x]\left(\kappa_{x,\,\alpha}\right)^{(2Q+1-q)}\big|_{x=1}^{x=\infty}\right.\nonumber \\
+ & \left.\frac{1}{(2m\pi\pm k)}\left[\cos[(2m\pi\pm k)x]\left(\kappa_{x,\,\alpha}\right)^{(2Q+2-q)}\big|_{x=1}^{x=\infty}\right]+\cdots\right\} \nonumber \\
+ & \frac{(-1)^{(q-1)/2}}{(2m\pi\pm k)^{q}}\int_{1}^{N/2-1}\cos[(2m\pi\pm k)x]\kappa_{x,\,\alpha}^{(2Q+1)}dx,
\end{align}
Apparently the last integral is bounded, according to Eq.~(\ref{eq:kappa-regularity}).
Therefore, for large $m$ we know
\begin{align}
\bigg|\int_{1}^{N/2-1}\cos[(2m\pi\pm k)x]\left(\kappa_{x,\,\alpha}\right)^{(2Q+1-q)}dx\bigg| & \lesssim\frac{\mathcal{C}_{2Q+1-q}(\alpha)}{(2m\pi\pm k)},
\end{align}
where
\begin{equation}
\mathcal{C}_{q}(\alpha)\equiv\big|\left(\kappa_{x,\,\alpha}^{(q)}\right)_{x=1}\big|+\big|\left(\kappa_{x,\,\alpha}^{(q)}\right)_{x=\infty}\big|.
\end{equation}
which is finite according to the regularity condition Eq.~(\ref{eq:kappa-reg-ends}).
With similar argument, we obtain
\begin{equation}
\bigg|\int_{1}^{N/2-1}\sin[(2m\pi\pm k)x]\kappa_{x,\,\alpha}^{(2Q+1-q)}dx\bigg|\lesssim\frac{\mathcal{C}_{2Q+1-q}(\alpha)}{(2m\pi\pm k)}
\end{equation}
for large $m$. So we conclude
\begin{equation}
\big|\mathscr{R}_{\alpha,\,q}(k)\big|\lesssim\mathcal{C}_{Q,\,q}\mathcal{C}_{2Q+1-q}(\alpha)k^{q}\sum_{m=1}^{\infty}\frac{1}{(2m\pi)^{2Q+2}},
\end{equation}
which is bounded for all finite values of $k$. Now we have shown
that the remainder $R_{\alpha,\,q}(k)$ is regular with no singularity
in $k$, which concludes the proof.
\end{proof}
\section{\label{sec:Singularity-f}The possible singularity of $\mathscr{F}_{\alpha}(k)$
or $f_{\alpha}(k)$ for $d_{l,\,\alpha}$ satisfying the regularity
conditions}
\begin{cor}\label{cor:sing-at-zero}
If $\kappa_{l,\,\alpha}$ satisfies the regularity conditions the
regularity condition ~(\ref{eq:kappa-reg-ends}-\ref{eq:kappa-regularity})
in the main text then $\mathscr{F}_{\alpha}(k)$ is regular as long
as $k\neq0$.
\end{cor}
\begin{proof}The proof is straightforward: performing integration
by part for the main integral in Eq.~(\ref{eq:f-rep}), we find
\begin{align}
\int_{1}^{N/2-1}\sin(kx)\kappa_{x,\,\alpha}dx & =-\frac{\cos(kx)}{k}\kappa_{x,\,\alpha}|_{x=1}^{x=N/2-1}+\frac{\sin(kx)}{k^{2}}\kappa_{x,\,\alpha}^{(1)}\big|_{x=1}^{x=N/2-1}+\cdots\nonumber \\
& +\frac{(-1)^{Q}}{k^{2Q}}\int_{1}^{N/2-1}\cos(kx)\kappa_{x,\,\alpha}^{(2Q+1)}.
\end{align}
Since the integral is bounded according to Eq.~\ref{eq:kappa-regularity},
we find
\begin{equation}
\big|\mathscr{F}_{\alpha}(k)\big|\leq\sum_{m=1}^{2Q}\frac{\mathcal{C}_{m-1}(\alpha)}{k^{m}}.
\end{equation}
So $f_{\alpha}(k)$ or $\mathscr{F}_{\alpha}(k)$ is bounded as long
as $k\neq0$.
\end{proof}
From this proof, we immediately see that the only possible singularity
of $\mathscr{F}_{\alpha}(k)$ is at $k=0$. As we have mentioned in
the main text, we can introduce a trick to get a rough estimate about
the possible singularity of the main integral $\mathscr{F}_{\alpha}(k)$
near $k=0$, : We integrate over $k$ from $1/N$ to $\Lambda$, where
$\Lambda$ is finite. This yields
\begin{equation}
\int_{1/N}^{\Lambda}dk\mathscr{F}_{\alpha}(k)=\int_{1}^{N/2-1}\frac{\kappa_{x,\,\alpha}}{x}dx-\int_{1}^{N/2-1}\frac{\kappa_{x,\,\alpha}\cos(\Lambda x)}{x}dx,\label{eq:int-scrF}
\end{equation}
where we have interchanged order of integration. According to Sec.~\ref{sec:The-convergence-integration-by-parts},
the second integral on the r.h.s. of Eq.~(\ref{eq:int-scrF}) is
bounded with the exact scaling of with respect to $N$ can be easily
found by integrating by parts. Therefore the scaling of $\int_{1/N}^{\Lambda}\mathscr{F}_{\alpha}(k)dk$
is totally controlled by the first integral on the r.h.s of Eq.~(\ref{eq:int-scrF}).
If the scaling of $\int_{1/N}^{\Lambda}\mathscr{F}_{\alpha}(k)dk$
can be computed, the it can reveal \textit{some partial} information
about the singularity of $\mathscr{F}_{\alpha}(k)$ around $k=0$.
For example, if $\int_{1/N}^{\Lambda}\mathscr{F}_{\alpha}(k)dk\sim\text{constant}$
or $\int_{1/N}^{\Lambda}\mathscr{F}_{\alpha}(k)dk\sim\ln N$, then
we know the singularity of $\mathscr{F}_{\alpha}(k)$ at $k=0$ is
at most $1/k^{1-\varepsilon}$ or $1/k$ respectively, where $\varepsilon$
is arbitrary small positive number.
\section{\label{sec:The-convergence-integration-by-parts}The convergence
of the integral $\int_{1}^{\infty}\kappa_{x,\,\alpha}\cos(\Lambda x)/x]dx$}
One can prove the integral $\int\kappa_{x,\,\alpha}\cos(\Lambda x)/x]dx$
is bounded via integration by parts. First, it is readily checked
that
\begin{equation}
\int_{1}^{\infty}\kappa_{x,\,\alpha}\cos(\Lambda x)/x]dx=\frac{1}{\Lambda}\sin(\Lambda x)\frac{\kappa_{x,\,\alpha}}{x}\big|_{x=1}^{x=\infty}+\frac{1}{\Lambda}\int_{1}^{\infty}\sin(\Lambda x)\frac{\kappa_{x,\,\alpha}}{x^{2}}-\frac{1}{\Lambda}\int_{1}^{\infty}\sin(\Lambda x)\frac{\kappa_{x,\,\alpha}^{(1)}}{x}.
\end{equation}
Apparently, according to regularity condition~(\ref{eq:kappa-reg-ends})
of $\kappa_{x,\,\alpha}$, we know
\begin{equation}
\bigg|\int_{1}^{\infty}\sin(\Lambda x)\frac{\kappa_{x,\,\alpha}}{x^{2}}\bigg|\le\mathcal{C}_{\max,\,\alpha}\bigg|\int_{1}^{\infty}\frac{1}{x^{2}}\bigg|<\infty,\label{eq:q-zero}
\end{equation}
where
\begin{equation}
\mathcal{C}_{\max,\,\alpha}\equiv\max_{x\in[1,\,\infty]}\kappa_{x,\,\alpha}.
\end{equation}
So the convergence of $\int_{1}^{\infty}\kappa_{x,\,\alpha}\cos(\Lambda x)/x]dx$
will depends on the convergence $\int_{1}^{\infty}\sin(\Lambda x)\kappa_{x,\,\alpha}^{(1)}/x$.
Further integrating by parts and apply the same argument, one can
show that the convergence of $\int_{1}^{\infty}\sin(\Lambda x)\kappa_{x,\,\alpha}^{(1)}/x$,
will depend on $\int_{1}^{\infty}\cos(\Lambda x)\kappa_{x,\,\alpha}^{(2)}/x$.
We continue to apply integration by parts until we obtain $\int_{1}^{\infty}\sin(\Lambda x)\kappa_{x,\,\alpha}^{(2Q+1)}/x$,
which yields,
\begin{align}
\int_{1}^{\infty}\kappa_{x,\,\alpha}\cos(\Lambda x)/x]dx & =\frac{1}{\Lambda}\sin(\Lambda x)\frac{\kappa_{x,\,\alpha}}{x}\big|_{x=1}^{x=\infty}+\frac{1}{\Lambda}\int_{1}^{\infty}\sin(\Lambda x)\frac{\kappa_{x,\,\alpha}}{x^{2}}\nonumber \\
& +\frac{1}{\Lambda^{2}}\cos(\Lambda x)\frac{\kappa_{x,\,\alpha}^{(1)}}{x}\big|_{x=1}^{x=\infty}+\frac{1}{\Lambda^{2}}\int_{1}^{\infty}\cos(\Lambda x)\frac{\kappa_{x,\,\alpha}^{(1)}}{x^{2}}\nonumber \\
& +\cdots+\frac{1}{\Lambda^{2Q+2}}\int_{1}^{\infty}\cos(\Lambda x)\frac{\kappa_{x,\,\alpha}^{(2Q+1)}}{x^{2}}.
\end{align}
Once again all the boundary terms in the above equation is bounded
thanks the regularity condition~(\ref{eq:kappa-reg-ends}). Furthermore,
Similar with Eq.~(\ref{eq:q-zero}), we note,
\begin{equation}
\bigg|\int_{1}^{\infty}\sin(\Lambda x)\frac{\kappa_{x,\,\alpha}^{(2q)}}{x^{2}}\bigg|\le\mathcal{C}_{\max,\,\alpha}^{(2q)}\bigg|\int_{1}^{\infty}\frac{1}{x^{2}}\bigg|<\infty,
\end{equation}
\begin{equation}
\bigg|\int_{1}^{\infty}\cos(\Lambda x)\frac{\kappa_{x,\,\alpha}^{(2q+1)}}{x^{2}}\bigg|\le\mathcal{C}_{\max,\,\alpha}^{(2q+1)}\bigg|\int_{1}^{\infty}\frac{1}{x^{2}}\bigg|<\infty,
\end{equation}
where $q=1,2,\,\cdots,\,Q$ and
\begin{equation}
\mathcal{C}_{\max,\,\alpha}^{(q)}\equiv\max_{x\in[1,\,\infty]}\big|\kappa_{x,\,\alpha}^{(q)}\big|.
\end{equation}
Thus the convergence of $\int_{1}^{\infty}\kappa_{x,\,\alpha}\cos(\Lambda x)/x]dx$
depends on $\int_{1}^{\infty}\sin(\Lambda x)\kappa_{x,\,\alpha}^{(2Q+1)}/x$.
It is readily checked that
\begin{equation}
\bigg|\int_{1}^{\infty}\cos(\Lambda x)\frac{\kappa_{x,\,\alpha}^{(2Q+1)}}{x^{2}}\bigg|\le\bigg|\int_{1}^{\infty}\kappa_{x,\,\alpha}^{(2Q+1)}\bigg|<\infty
\end{equation}
according to the regularity condition~(\ref{eq:kappa-regularity}).
We conclude that the integral $\int_{1}^{\infty}\kappa_{x,\,\alpha}\cos(\Lambda x)/x]dx$
is convergent.
\section{\label{sec:spin-rep}The LRK Hamiltonian in the spin representation}
With the Jordan-Wigner transformation~\citep{coleman2015introduction},
\begin{equation}
a_{j}^{\dagger}=(-1)^{j-1}\prod_{k=1}^{j-1}\sigma_{k}^{z}\sigma_{j}^{+},\label{eq:JW1}
\end{equation}
\begin{equation}
a_{j}=(-1)^{j-1}\prod_{k=1}^{j-1}\sigma_{k}^{z}\sigma_{j}^{-},\label{eq:JW2}
\end{equation}
where $\sigma_{j}^{z}$ is the standard Pauli $z$-matrix
\begin{align}
\sigma_{j}^{+} & \equiv\begin{bmatrix}0 & 1\\
0 & 0
\end{bmatrix},\\
\sigma_{j}^{-} & \equiv\begin{bmatrix}0 & 0\\
1 & 0
\end{bmatrix},
\end{align}
it is readily checked that
\begin{align}
a_{j}^{\dagger}a_{j} & =\sigma_{j}^{+}\sigma_{j}^{-}=\frac{1}{2}(\sigma_{j}^{z}+1),
\end{align}
\begin{align}
a_{j+1}^{\dagger}a_{j} & =-\sigma_{j+1}^{+}\sigma_{j}^{z}\sigma_{j}^{-}=\sigma_{j+1}^{+}\sigma_{j}^{-},
\end{align}
where we have used the fact $\sigma_{j}^{z}\sigma_{j}^{\pm}=\pm\sigma_{j}^{\pm}$
in the second equation. Furthermore,
\begin{align}
a_{j}a_{j+l} & =(-1)^{j-1}\prod_{k=1}^{j-1}\sigma_{k}^{z}\sigma_{j}^{-}\times(-1)^{j+l-1}\prod_{m=1}^{j-1+l}\sigma_{m}^{z}\sigma_{j+l}^{-}\nonumber \\
& =(-1)^{l}\sigma_{j}^{-}\sigma_{j}^{z}\prod_{k=j+1}^{j-1+l}\sigma_{k}^{z}\sigma_{j+l}^{-}\nonumber \\
& =(-1)^{l}\sigma_{j}^{-}\prod_{k=j+1}^{j-1+l}\sigma_{k}^{z}\sigma_{j+l}^{-},
\end{align}
where we have used the fact that $\sigma_{j}^{\pm}\sigma_{j}^{z}=\mp\sigma_{j}^{\pm}$.
Now using the relation
\begin{align}
\sigma_{j}^{+} & \equiv\frac{1}{2}(\sigma_{j}^{x}+\text{i}\sigma_{j}^{y}),\\
\sigma_{j}^{-} & \equiv\frac{1}{2}(\sigma_{j}^{x}-\text{i}\sigma_{j}^{y}),
\end{align}
where $\sigma_{j}^{x}$ and $\sigma_{j}^{y}$ are standard Pauli $x-$
and $y-$ matrices respectively, we find
\begin{align}
\sigma_{j}^{+}\sigma_{j+l}^{-}+\sigma_{j}^{-}\sigma_{j+l}^{+} & =\frac{1}{4}(\sigma_{j}^{x}+\text{i}\sigma_{j}^{y})(\sigma_{j+l}^{x}-\text{i}\sigma_{j+l}^{y})+\frac{1}{4}(\sigma_{j}^{x}-\text{i}\sigma_{j}^{y})(\sigma_{j+l}^{x}+\text{i}\sigma_{j+l}^{y})\nonumber \\
& =\frac{1}{2}(\sigma_{j}^{x}\sigma_{j+l}^{x}+\sigma_{j}^{y}\sigma_{j+l}^{y}),\label{eq:Pauli-Id1}
\end{align}
\begin{align}
\sigma_{j}^{+}\sigma_{j+l}^{+}+\sigma_{j}^{-}\sigma_{j+l}^{-} & =\frac{1}{4}(\sigma_{j}^{x}+\text{i}\sigma_{j}^{y})(\sigma_{j+l}^{x}+\text{i}\sigma_{j+l}^{y})+\frac{1}{4}(\sigma_{j}^{x}-\text{i}\sigma_{j}^{y})(\sigma_{j+l}^{x}-\text{i}\sigma_{j+l}^{y})\nonumber \\
& =\frac{\text{1}}{2}(\sigma_{j}^{x}\sigma_{j+l}^{x}-\sigma_{j}^{y}\sigma_{j+l}^{y}).\label{eq:Pauli-Id2}
\end{align}
So with Eqs.~(\ref{eq:JW1},~\ref{eq:JW2},~\ref{eq:Pauli-Id1}),
the tunneling and kinetic terms become
\begin{equation}
\sum_{j=1}^{N}(a_{j}^{\dagger}a_{j+1}+a_{j+1}^{\dagger}a_{j})=\sum_{j=1}^{N}(\sigma_{j}^{-}\sigma_{j+1}^{+}+\sigma_{j}^{+}\sigma_{j+1}^{-})=\frac{1}{2}\sum_{j=1}^{N}(\sigma_{j}^{x}\sigma_{j+l}^{x}+\sigma_{j}^{y}\sigma_{j+l}^{y}),\label{eq:tunnel-spin}
\end{equation}
\begin{equation}
\sum_{j=1}^{L}(a_{j}^{\dagger}a_{j}-\frac{1}{2})=\frac{1}{2}\sum_{j=1}^{L}\sigma_{j}^{z},\label{eq:kinetic-spin}
\end{equation}
For the long-range superconducting terms, with the anti-periodic boundary
condition, one can easily obtain the following alternative form
\begin{equation}
\sum_{j=1}^{N-1}\sum_{l=1}^{N-j}\kappa_{l,\,\alpha}a_{j}a_{j+l}=\frac{1}{2}\sum_{j=1}^{N}\sum_{l=1}^{N-1}\kappa_{l,\,\alpha}a_{j}a_{j+l},
\end{equation}
and similarly equation for the term $\sum_{j=1}^{N-1}\sum_{l=1}^{N-j}\kappa_{l,\,\alpha}a_{j+l}^{\dagger}a_{j}^{\dagger}$.
On the other hand, with Eqs.~(\ref{eq:JW1},~\ref{eq:JW2},~\ref{eq:Pauli-Id2}),
we find
\begin{equation}
\sum_{j=1}^{N}\sum_{l=1}^{N-1}\kappa_{l,\,\alpha}(a_{j}a_{j+l}+a_{j+l}^{\dagger}a_{j}^{\dagger})=\frac{1}{2}\sum_{j=1}^{N}\sum_{l=1}^{N-1}(-1)^{l}\kappa_{l,\,\alpha}(\sigma_{j}^{x}\sigma_{j+l}^{x}-\sigma_{j}^{y}\sigma_{j+l}^{y})\sigma_{j+1}^{z}\cdots\sigma_{j+l-1}^{z}.\label{eq:superconducting-spin}
\end{equation}
Substituting Eqs.~(\ref{eq:tunnel-spin}-\ref{eq:superconducting-spin})
into Eq.~(\ref{eq:LRK}) in the main text yields the LRK Hamiltonian
in the spin-representation, i.e.,
\begin{align}
H_{\text{spin}} & =-\frac{J}{4}\sum_{j=1}^{N}(\sigma_{j}^{x}\sigma_{j+l}^{x}+\sigma_{j}^{y}\sigma_{j+l}^{y})-\frac{\mu}{2}\sum_{j=1}^{N}\sigma_{j}^{z}\nonumber \\
& +\frac{\Delta}{8}\sum_{j=1}^{N}\sum_{l=1}^{N-1}(-1)^{l}\kappa_{l,\,\alpha}(\sigma_{j}^{x}\sigma_{j+l}^{x}-\sigma_{j}^{y}\sigma_{j+l}^{y})\otimes_{k=1}^{l-1}\sigma_{j+k}^{z}.
\end{align}
\end{document}
|
1,116,691,497,798 | arxiv | \section{INTRODUCTION}
Lung cancer represents the leading cause of cancer deaths worldwide claiming an estimated 1.76 million lives in 2018 \cite{c1}. Chest radiography (CXR) remains widely used for initial radiological examination in patients with suspected lung cancer. However, the National Lung Screening Trial showed that CXR has low sensitivity (73.5\%) and high specificity (91.3\%) in lung cancer detection \cite{c2}. Many factors contribute to this low sensitivity including observer errors in scanning CXR images, recognition of nodules, and decisions regarding the clinical relevance of subtle opacities \cite{c3}. Highly variable nodule characteristics such as size and location add complexity to this task. Computer aided diagnosis (CAD) methods have been shown to improve lung nodule detection performance of radiologists in CXR \cite{c4}.
Recently, deep learning CAD tools using Convolutional Neural Networks (CNN) for lung cancer detection have been proposed and validated \cite{c5}. However, these methods have all applied supervised learning. Supervised deep learning methods face several challenges in medical imaging including the high costs of obtaining large volumes of labelled training data, inaccuracies in data labelling due to intra-observer variability, lack of representativeness due to labelling by one or just a few experts, and difficulty learning the nodule class due to clinical imbalance in healthy and unhealthy data. Unsupervised anomaly detection methods can circumvent many of these challenges by learning instances of a healthy (normal) class and then distinguishing instances of unhealthy data deviating from the learned normal class.
Several previous studies have proposed unsupervised anomaly detection models based on generative adversarial network (GAN) models. Schlegl et al. introduced the well-known f-AnoGAN model by first training a Wasserstein GAN (WGAN) to learn the distribution of normal data and then using a deep convolutional encoder to learn the optimal mapping between images and the GAN latent space to detect abnormalities in optical coherence tomography (OCT) \cite{c6}. Furthermore, Sun et al. reported an abnormal-to-normal translation GAN for unsupervised detection and segmentation of lesions in brain MRI \cite{c7}. Very recently, Nakao et al. presented an auto-encoding GAN model to classify lung opacities in CXR \cite{c8}. However, quantitative results of model performance for lung nodules specifically were not reported. To our understanding, the use of deep anomaly detection techniques for lung nodules in chest radiographs has not systematically been studied. Moreover, previous methods have not validated or tested unsupervised anomaly detection methods using ``external datasets'' and thus cannot shed light on the robustness and generalizability of such methods in medical imaging.
In this paper, we investigate the feasibility of unsupervised anomaly detection for lung nodules in chest radiography. Our contributions can be summarized as follows: (1) performing automated detection of lung nodules in chest radiographs without the pathology labels; (2) extending the popular f-AnoGAN method by using a progressive growing GAN, proposing new robust loss for encoder training, and comparing ResNet and DenseNet encoders; and (3) analyzing our network using external validation and testing datasets.
\section{METHODS}
\subsection{Datasets}
We utilized three datasets for training, validation, and testing obtained from independent sources. The training dataset for our method contained only healthy data instances. The Indiana University CXR collection consists of 8,121 images obtained from 3,996 patients \cite{c9}. De-identified radiology reports with findings by a board certified radiologist were also available for each patient. We manually reviewed radiological reports and included 320 consecutive patients with no acute cardiopulmonary findings in our training dataset.
Our validation dataset consisted of patients with lung nodules (i.e., \emph{abnormality}) and healthy patients obtained from the National Institutes of Health (NIH) ChestX-ray14 dataset \cite{c10}. The NIH dataset contains 108,948 images obtained from 32,717 patients, however the majority of these images were automatically annotated using a natural language processing labeller which is error prone and thus could not be used for evaluation. The dataset contained a small subset of unhealthy patients with specific pathologies annotated using a bounding box by a board certified radiologist. From this subset, we included all 79 patients with lung nodules in our validation dataset. Since no manually labelled healthy patients were available, we randomly sampled 25 automatically annotated patients with “no findings” label. A board certified physician was consulted, and 10 patients confirmed to have no cardiopulmonary findings or medical support devices were included in our validation dataset.
The testing dataset consisted of healthy patients and patients with lung nodules from the Japanese Society of Radiological Technology (JSRT) database \cite{c11}. The JSRT database contains 93 healthy patients and 154 patients with lung nodules. All images were manually labelled by using consensus of a panel of board-certified radiologists and nodule location was provided using image coordinates. Therefore, images in this dataset were suitable for model testing. In all datasets, only frontal (posteroanterior) views were used, and all images were resized to $1024\times 1024$ pixels.
\subsection{Data Preprocessing}
Contrast Limited Adaptive Histogram Equalization (CLAHE) was used to enhance visual local image contrast (kernel size$=128$; clip limit$=0.01$) \cite{c12}. An open-source, robust pre-trained deep network based on criss-cross attention (CCNet) was used to perform image segmentation and extract lung regions of interest (ROI) \cite{c13}. Patients with nodules outside of the lung ROI were excluded (6 validation patients and 13 test patients). As a result, our validation and test datasets contained 73 and 141 patients with nodules, respectively. Uniform patches of size 64$\times$64 were extracted from the lung ROIs. All patches extracted from healthy patients were labelled as normal. All patches from unhealthy patients which intersected the nodule centroid were labelled as abnormal. Thus, we obtained normal training patches ($n=70798$); normal ($n=9010$) and abnormal ($n=1062$) validation patches; and normal ($n=96765$) and abnormal ($n=2105$) test patches. Model training and evaluation were performed using these extracted patches.
\subsection{Model Architecture}
Our model (P-AnoGAN) consisted of a GAN and encoder network (Fig. \ref{model-architecture}) inspired by the f-AnoGAN method \cite{c6}. We utilized the Progressive GAN (PGAN) which consisted of two mirrored CNNs: generator G and discriminator D. The generator maps a random 512-point latent vector to a 64$\times$64 image patch using 3$\times$3 convolution layers, upsampling, leaky ReLU activation, and per-pixel normalization of feature vectors. Simultaneously, the discriminator learns classification of real and fake images using 3$\times$3 convolution layer, downsampling, and leaky ReLU \cite{c14}. Our encoder was based on the ResNet-50 and DenseNet-169 architectures which embedded input image patch $\bold{x}$ to a 512-point latent vector $\bold{z}$. The trained GAN generator and encoder were combined in an encoder-decoder-encoder architecture where the encoder performed the mapping $\bold{z}=E(\bold{x})$ and generator acted as a decoder to reconstruct $\bold{\tilde{x}}=G(\bold{z})$. The encoder then mapped $\bold{\tilde{x}}$ to the latent space $\bold{\tilde{z}}=E(\bold{\tilde{x}})$.
\begin{figure}[t]
\includegraphics[scale=1.0,width=\linewidth]{./figures/Architecture_overview.png}
\caption{Model with Encoder (E), Generator (G), and Discriminator (D).}
\label{model-architecture}
\vspace{-2.5mm}
\vspace{-2.5mm}
\end{figure}
\subsection{Model Training}
P-AnoGAN was implemented using Python (version 3.7) and PyTorch (version 1.8.1) and training was completed on a single Tesla V100 GPU (NVIDIA Corporation, Santa Clara, CA) with CUDA version 11.2. For model training, we used unlabeled patches $\bold{x} \in X$, where $X$ is the manifold of patches containing healthy anatomy from our training dataset.
The PGAN was trained to model the variability in $X$. The generator learned the mapping between a latent space $Z \in \mathbb{R}^{512}$ to $X$ by generating “fake” images $\bold{\tilde{x}}$ from randomly sampled latent vector $\bold{z}$. Adam optimizer with $\beta_1=0.0$, $\beta_2=0.99$, base learning rate of 0.001 was used as in \cite{c14}. Mini-batch size of 16 was set. Training was completed for 48,000 iterations for the $4\times4$ scale; 96,000 iterations for $8\times8$, $16\times16$, and $32\times32$; and 200,000 iterations for the $64\times64$ scale. The generator and discriminator networks were trained progressively at growing resolution scale by upsampling and adding network blocks. Training started at the $4\times4$ scale. Initially, the generator consisted of a single network block comprised of convolutional layers and ReLU activation which outputted $4\times4$ “fake” patches and the discriminator also comprised of a single block comparing real and generated patches. After iterations were completed, we moved to the next $8\times8$ scale by upsampling and adding network blocks. Linear fading was performed whereby output features from $4\times4$ blocks were scaled by a parameter $\alpha$ and concatenated to the output of the $8\times8$ blocks; the $\alpha$ decremented linearly by 1/600 every 32 iterations. This process was repeated until the final scale.
After the completion of GAN training, generator and discriminator networks weights were fixed. Next, the ResNet-50 and DenseNet-169 encoders were trained. Encoders learned the mapping from manifold $X$ to latent space $Z$ such that provided a healthy image $\bold{x}$, the encoded latent vector could be used by the generator to accurately reconstruct the input image $\bold{\tilde{x}}=G(E(\bold{x}))$. Several loss functions were tested for encoder training. Three loss functions $izi$, $ziz$, and $izi_f$ were introduced in Schlegl et al. \cite{c6}. The image-latent-image ($izi$) loss was calculated as the pixel-level mean square error (MSE) between original and reconstructed patches as
\begin{equation}
L_{\bold{izi}} = \frac{1}{n} \lVert \bold{x} - \bold{\tilde{x}} \rVert^2 ,
\end{equation}
where $n$ is the number of pixels in the image patch. To use latent-image-latent ($ziz$) loss function in training, a random latent vector $\bold{z} \in Z$ was sampled and used to generate an image $\bold{\tilde{x}} =G(\bold{z})$ which was subsequently re-encoded to a latent $\bold{\tilde{z}} = E(\bold{\tilde{x}})$. The $ziz$ loss was then the MSE between original and re-encoded latent vectors as
\begin{equation}
L_{\bold{ziz}} = \frac{1}{d} \lVert \bold{z} - \bold{\tilde{z}} \rVert^2 ,
\end{equation}
where $d$ is the number of points in the latent vector. The image-latent-image loss with discriminator guidance ($izi_f$) was similar to $izi$ but included a loss term quantifying MSE between features from an intermediate layer of the discriminator for the original $\bold{D_f}$ and reconstructed $\bold{\tilde{D}_f}$ images, as
\begin{equation}
L_{\bold{izi_f}} = \frac{1}{n} \lVert \bold{x} -\bold{\tilde{x}} \rVert^2 + \frac{\gamma}{n_d} \lVert \bold{D_f} - \bold{\tilde{D}_f} \rVert^2 ,
\end{equation}
where $n_d$ is the number of discriminator features and $\gamma$ is a weighting factor. Finally, we proposed the combination of $ziz$ and $izi_f$ losses to enforce similarity between images and latent vectors. The encoder network was used to embed the reconstructed image to a latent vector $\bold{\tilde{z}} = E(\bold{\tilde{x}})$. Then, we compute the MSE residual in latent space between latent vector $\bold{z}$ from the original patch and the latent vector $\bold{\tilde{z}}$ from the reconstructed patch. This "latent loss" term was added to $izi_f$ to formulate the $iziz_f$ loss as
\begin{equation}
L_{\bold{iziz_f}} = \frac{1}{n} \lVert \bold{x} -\bold{\tilde{x}} \rVert^2 + \frac{\gamma}{n_d} \lVert \bold{D_f} - \bold{\tilde{D}_f} \rVert^2 + \frac{1}{d} \lVert \bold{z} - \bold{\tilde{z}} \rVert^2 .
\end{equation}
Training was performed with ResNet-50 and DenseNet-169 encoders for each loss function over 30 epochs. Adam optimizer with $\beta_1=0.9$, $\beta_2=0.999$ and base learning rate of $0.0001$ was used. Batch size of 64 was set. We set $\gamma=0.1$ based on empirical trials.
\subsection{Model Validation and Testing}
For any query patch $\bold{x_q}$ and its latent vector $\bold{z_q}$, anomaly scores $A_s$ were computed according to the loss function used for model training. For $iziz_f$ loss, we computed mean absolute deviation (MAD) between original and reconstructed patches in image space and latent space as
\begin{equation}
A_s = \frac{1}{n} \lVert \bold{x_q} -\bold{\tilde{x}_q} \rVert + \frac{1}{d} \lVert \bold{z_q} - \bold{\tilde{z}_q} \rVert .
\end{equation}
For $izi$, $ziz$, and $izi_f$, since model training did not include a latent loss term, the anomaly score simplified as
\begin{equation}
A_s = \frac{1}{n} \lVert \bold{x_q} -\bold{\tilde{x}_q} \rVert .
\end{equation}
Model validation was performed after every epoch of encoder training using normal and abnormal patches from the NIH data The receiver operating characteristic (ROC) area under curve (AUC) was computed and the epoch producing the highest AUC value was saved. Model testing was performed using normal and abnormal patches from the JSRT data and test AUC were recorded. In both validation and testing, optimal cut-off point was determined by maximizing the Youden Index J = sensitivity + specificity – 1 and associated sensitivity and specificity were recorded \cite{c15}.
\section{RESULTS}
The validation AUC, sensitivity, and specificity are recorded in Table 1. Training and evaluating with $iziz_f$ loss and anomaly scoring function yielded best validation results for Resnet-50 and DenseNet-169 encoders. Models with ResNet-50 and DenseNet-169 encoders were trained using $iziz_f$ loss. Higher testing AUC was noted for the model with ResNet-50 (AUC: 87.89\%; sensitivity: 88.65\%; specificity: 73.69\%) in comparison to DenseNet-169 (AUC: 85.24\%; sensitivity: 87.94\%; specificity: 69.94\%). ROC curves for testing data are shown in Fig. 2a. Sample visual inspection of normal and abnormal testing patches are shown in Fig. 2b and 2c, respectively. Typical reconstructions for abnormal query images were poorer than for normal. Average processing time per patch was 1.77 ms.
\begin{table}[b]
\vspace{-2.5mm}
\caption{Model performance for validation data}
\vspace{-2.5mm}
\vspace{-2.5mm}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{2}{*}{\textbf{Encoder}} & \multicolumn{4}{c|}{\textbf{Experiments}} \\ \cline{2-5}
& \textbf{Loss} & \textbf{AUC} & \textbf{Sensitivity} & \textbf{Specificity} \\ \hline
\multirow{4}{*}{ResNet-50} & $ziz$ & 88.58\% & 93.15\% & 74.13\% \\ \cline{2-5}
& $izi$ & 86.60\% & 83.56\% & 74.79\% \\ \cline{2-5}
& $izif$ & 88.14\% & 87.67\% & 77.50\% \\ \cline{2-5}
& $iziz_f$ & 91.17\% & 89.04\% & 80.92\% \\ \hline
\multirow{4}{*}{DenseNet-169} & $ziz$ & 88.61\% & 90.41\% & 77.29\% \\ \cline{2-5}
& $izi$ & 86.02\% & 75.34\% & 80.24\% \\ \cline{2-5}
& $izi_f$ & 86.48\% & 84.93\% & 74.47\% \\ \cline{2-5}
& $iziz_f$ & 91.10\% & 91.78\% & 80.60\% \\ \hline
\end{tabular}
\end{center}
\vspace{-2.5mm}
\end{table}
\section{DISCUSSIONS}
We developed an unsupervised anomaly detection model for lung nodules in CXR. This model was trained with only unlabeled healthy data instances. Using external validation and testing data, we demonstrate that our method generalizes well to independent datasets and can robustly discriminate healthy lung patches from those containing lung nodules.
\begin{figure}[t]
\begin{center}
\includegraphics[keepaspectratio,height=8.8cm,width=\linewidth]{./figures/combined-roc-recon3.png}
\vspace{-2.5mm}
\caption{Summary of test results. (a) ROC curves for models trained using $iziz_f$ loss. Original and reconstructed patches using model trained with ResNet-50 and $iziz_f$ loss for (b) normal and (c) abnormal test patches.}
\label{results1}
\vspace{-2.5mm}
\vspace{-2.5mm}
\vspace{-2.5mm}
\end{center}
\end{figure}
P-AnoGAN introduced substantial modifications to the f-AnoGAN framework. CXR patches contain diverse anatomy present at multiple scales (e.g., parenchyma, ribs, chest wall, and vasculature). Compared with previous GAN topologies (DCGAN, WGAN), PGAN enabled progressive learning of large scale features followed by finer details making it suitable for our purpose as suggested in \cite{c14}. Furthermore, we proposed a new loss function and anomaly scoring function ($iziz_f$) to jointly quantify deviations between original and reconstructed images in image space, latent space, and discriminator features which improved model performance. Very recently, Nakao et al. used a progressive growing auto-encoding GAN, a CNN encoder, and a ``code discriminator'' which differentiated randomly sampled latent vectors and latent obtained from real images to detect a variety of abnormalities (including lung masses) in CXR. However, the authors do not provide quantitative results specifically for lung masses. Moreover, their model performance was significantly poorer for subtle pathology (AUC$=70.4\%$) vs. large pneumonia opacities (AUC$=75.2\%$). Furthermore, neither of the above-mentioned studies have performed external validation/testing. In contrast, we demonstrated strong performance on the challenging JSRT dataset (avg. manual expert AUC$= 84.9\%$) \cite{c11} which indicates robust and generalizable detection of subtle lung nodules in external datasets.
We noted that ResNet-50 yielded higher AUC in both external validation and test data when compared to DenseNet-169. Furthermore, the addition of latent loss in $iziz_f$ loss function and anomaly scoring improved performance. During encoder training, enforcing similarity between latent vectors derived from original and reconstructed patches allowed a robust mapping between image and latent space. Moreover, since encoder training was performed using only healthy data, the image-to-latent mapping is accurate only for healthy queries during evaluation; thus, allowing us to better quantify deviations between original and reconstructed unhealthy data instances in both image and latent spaces. Our paper has several limitations. Firstly, we only used frontal view CXR which may cause difficulties in identifying nodules, particularly those found in the hilar region and near the clavicles \cite{c3}. Furthermore, by performing lung segmentation in CXR, we excluded some nodules which may appear outside of the lung region when considering frontal view only; this occurs most commonly when nodules are obscured by surrounding anatomy such as the heart. These challenges could be simultaneously overcome by incorporating lateral view CXR into the algorithm design, training, and testing. Our method detects \emph{any anomaly} appearing in radiographs. Therefore, presence of pathologies appearing alongside lung cancer (e.g., edema, atelectasis, pleural effusion) and medical support devices may also be detected. While we evaluate our method for the task of lung nodule detection, P-AnoGAN could conceivably be extended for detection of multiple pathologies; though this requires further investigation.
Future works should aim to improve detection performance by optimizing hyper-parameters as well as experiment with additional GANs (StyleGAN) and encoders (InceptionNet, SqueezeNet). We anticipate that such unsupervised deep models can potentially be appealing for triaging and diagnostic tasks in high-volume clinical settings.
\addtolength{\textheight}{-12cm}
\vspace{0.1in}
\textbf{ACKNOWLEDGMENTS --} The authors thank Dr. Ana M. Ramón Prados for confirming healthy patient cases included in validation data and Dr. Thomas Willett for feedback in the preparation of this work.
|
1,116,691,497,799 | arxiv | \section{Related Work}
Self-supervised learning can be broken down into two tasks:
\textbf{A pretext task} where no labels are involved and where the model learns image features and finds information in the images that it can use as labels to carry out its mission (e.g. classification).
\textbf{A downstream task} where small-labeled data is used (i.e. only a small percentage of the dataset has labels) to fine-tune the model and perform its initially intended purpose; image captioning in our case.
Generating image captions in a self-supervised way doesn't seem to be a prevalent topic in the existing literature, and we're suspecting that it comes from its difficulty rather than a lack of interest. The work that we have come across seems to be using either an unsupervised or a semi-supervised learning approach. While unsupervised learning deals with no supervision, semi-supervised learning merges between labeled and unlabeled data. We will be presenting some of the work in these two categories for a lack of self-supervised image captioning papers.
\cite{gu} use semi-supervised learning by assuming that they have an image-caption corpus in a different language that they can use to generate captions with before translating them to a target language. In their paper, the source target is Chinese and English is the target one. Chinese here is called a pivot-language, and the task of performing a task through a proxy language, language pivoting. This strategy is used already in machine translation when a language is rich in resources and is used to bridge the cap between two languages that don't have a lot of data connecting them.
\cite{gu} come up with a framework that is composed of an image captioner and a translation model, trained on distinct datasets. The image captioner, called \textit{image-to-pivot} generates the captions and passes them to the translation model, called \textit{pivot-to-target}, which translates the captions to the intended language.
Although language pivoting in the context of image captioning is a creative idea that can definitely be of very good use to languages that don't have image-caption datasets yet, it might not be very useful to English for instance since it relies on the assumption of an image captioning dataset being available in some other language. It is known however, that the biggest image-captioning datasets in English still fall a bit short due to captioning being a tedious annotation task and them introducing only a small subset of objects that are encountered in real-life.
Additionally, machine-translation is known to have its own share of errors, and using it in conjunction with captioning could multiply these errors and accumulate them. This solution is therefore great for reusing data across languages but it does not seem to improve much on the current technology in resource-rich languages.
\cite{chen} bring the attention to how real world applications might not benefit much from the existing image-captioning models due to them being trained on "general" datasets. Indeed, in domain-specific settings, a well-performing model might not achieve satisfying results. For that reason, \cite{chen} use adversarial training to remedy to the issue using two different discriminators. One of them focuses on the domain and decided whether the captions generated are discernible from target domain sentences. The second one is responsible for assessing whether an image and caption are a good match. The two take the usual alternate training with the caption generator, each one of them trying to either distinguish more or generate better captions.
We believe that this approach is very useful and any advances towards more domain-specific and real-world scenarios hint to promising results in the future.
The aim in \cite{feng} is not to use any image-captioning dataset, and to instead have separate sentence and image corpora. Adversarial learning is first used to generate sentences that can't be told apart from the sentences in the corpus. After that, information from a visual concept detector is used to reward based on words in the caption that are related to that visual concept, thus attaching the image concept to its caption.
\cite{feng} use a common latent space between images and captions which motivates the semantic consistency of the sentences. An image feature can help in decoding a caption, and this latter can be used for the reconstruction of the former.
\section{Methods}
\subsection{Self-supervised Learning Methods}
\cite{jigsaw}
The majority of conventional approaches can be classified as either generative or discriminative. Generic methods learn to produce or model pixels in the input space in a variety of ways. Pixel-level generation, on the other hand, is computationally expensive and may not be needed for representation learning. Discriminative methods train networks to execute pretext tasks using objective functions close to those found in supervised learning, but the inputs and marks are obtained from an unlabeled dataset.
Unsupervised learning approaches can be divided into three categories: probabilistic, direct mapping (autoencoders), and manifold learning. Variables in a network are divided into measured and latent variables using probabilistic techniques. After that, learning is linked to evaluating model parameters that optimize the probability of latent variables given the observations. Unfortunately, as several layers are present, these models become intractable and are not optimized to deliver features efficiently.
The direct mapping method, which uses autoencoders, focuses on the latter part. Autoencoders define the feature extraction function (encoder) and the mapping from the feature to the input in a parametric manner (decoder). The direct mappings are learned by minimizing the autoencoder's reconstruction error between the input and the output.
Manifold learning methods may be used if the data structure indicates the data points are likely to cluster along a manifold. This representation enables smooth factor variations to be directly mapped to smooth observation variations. Manifold learning strategies have a range of drawbacks, including the need to compute nearest neighbors, which scales quadratically with the number of samples. Besides also the need for a sufficiently large density of samples across the manifold, that becomes more challenging to obtain for high-dimensional manifolds.
\subsubsection{Rotation}
\cite{rot} uses CNNs to learn image features by recognizing the rotations applied to the images. The intuition behind it is that similar to a human, the CNN will need to understand the image concept and what objects appear in order to recognize the rotation. In particular, the encoder needs to localize objects that are salient, identify which orientation they have and what relations they have with their surroundings.
\begin{figure}
\includegraphics[width=\textwidth]{maps.png}
\caption{Attention maps from AlexNet supervised versus self-supervised using rotations.}
\label{maps}
\end{figure}
This results in similar attention maps to those of a supervised learning model (Fig. \ref{maps}). This proves that although the idea is simple, it does achieve great results
4 rotations are used: 0º, 90º, 180º and 270º. They're preferred as they're very easily implemented using transpose and flip operations which prevents commonly faced transformation issues where low-level artifacts are left and can distract the CNN from the important features. We also find them to be optimal since any other rotation will result in some of the data being lost as depicted in (Fig. \ref{cat}).
They also add that since images are usually taken in upwards positions, the rotation task is well defined except for round objects which don't have a recognizable orientation. We argue however, that it doesn't only concern round object but any radially symmetric ones (e.g. flowers, some sea creatures), or ones that do not have a consistent shape (e.g. a rock)
Rotation has the same computational cost as supervised learning, making it a good candidate for our project. \cite{rot} train their AlexNet CNN on a single GPU in 2 days.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{cat.png}
\caption{Data is lost when performing a rotation other than the four main ones (e.g. 45º)}
\label{cat}
\end{figure}
\subsubsection{Jigsaw}
According to \cite{jigsaw}, one measure of a person's organizational capacity when it comes to visual stimuli, uses Jigsaw puzzles. It thus sounded natural to attempt the same method on CNNs to get them to learn visual representations.
While the CNN trains on solving a Jigsaw puzzle (guessing the correct locations of the shuffled tiles), it learns parts of the objects along with their spatial positioning at the same time.
Since similarities between the puzzle tiles don't help in their placement's decision, the focus is put on the differences instead. Additionally, when the location seems to be unclear, having a view on all the tiles helps to determine the appropriate location.
To solve the puzzle, the CNN starts by extracting features using solely the pixels inside a tile. It then uses only those features to decide on the positioning of the tiles. Doing that ensures that the features learnt by the network are discriminative enough.
At the same time, a number of Jigsaw puzzles (69 on average) are fed to the network at the same time to avoid mapping a certain feature to a position. The tiles also go through configurations for optimal shuffling to make sure that a certain feature can be equally assigned to all of the tiles.
\subsubsection{Colorization}
The primary object of \cite{color} was to develop a colorization method that is realistic and has vibrant colors. It managed however, to also set itself apart as a pretext task.
They achieve nearly photo-realistic results and call their method a "cross-channel encoder" since it's similar to an autoencoder but is inputs and outputs are image channels.
In order to guess the colors correctly, the cross-channel encoder hints such as textures and scene semantics. They also define their objective as successfully fooling a human rather than guessing the exact color, which might not be possible for some objects (e,g. t-shirt). For that reason, using the ground truth images to validate will not reflect the actual performance of the model. They thus use a "colorization Turing test" where individuals try to spot the colorized image, which they failed at at a 32\% rate.
Another issue relating to the wide plausible colors that an object might take is that the traditional colorizing technique is by using regression. As a result, those objects end up with grey-like hues since all the different colors that they could take get averaged. What \cite{color} do different is they use colorization as a classification task and each pixel gets a distribution of potential colors.
\subsubsection{SimCLR}
SimCLR \cite{simclr} is a self-supervised learning framework relying on contrastive loss. The idea behind this loss is to maximize the "agreement" between matching image-label pairs (called positive) and minimize it with all others (negative pairs).
\cite{simclr} experimented with different image transformations and found the most optimal ones to be random cropping and resizing (necessary resizing since the model is trained on ImageNet\cite{simclr} images which have different sizes) and color distortion, offering a combination of spatial transformation (crop) and appearance transformation (color).
The color distortion turned out to be crucial since the CNN learns to use color distribution from the pixel intensities histograms to recognize positive pairs, which result in overfitting.
The transformations are applied in an asymmetric way (applied to only one branch of the network) and although that complicates the predication task, a significant improvement is noticed in the representations quality.
SimCLR managed to achieve and accuracy that is comparable to a pre-trained Resnet-50. It benefits from large batch sizes and longer training periods since both contribute in it seeing more negative pairs and thus making convergence easier.
It uses Resnet \cite{simclr} as its base encoder for feature extraction and uses heavy computational resources: 100 epochs on cloud TPUs, with 32-128 cores and huge batch sizes (4000+).
Although SimCLR is the most performant, it requires computational resources that we do not have access to. We're therefore left with the 3 other and rotnet is the one who's most performant (under XY conditions). We have therefore opted for using the Rotation pretext task, which comes second to SimCLR among the presented tasks and relies on hardware similar to the one on our disposition.
We however have decided to use models pre-trained on each one of these pretext tasks (from \url{https://github.com/facebookresearch/vissl/blob/master/MODEL_ZOO.md}) and fine-tune them on our image captioning downstream task. Doing this allows us to assess our model to make sure whether getting a certain result is due to using self supervised learning in image captioning or if it's due to a potential error in our implementation.
\subsection{Architecture}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{Rot_SSL_Architecture.png}
\caption{Architecture of our image captioning model}
\label{arch}
\end{figure}
\subsubsection{Pretext Task}
In the pretext task, we start by performing 4 rotations on the images: 0º, 90º, 180º and 270º. Each of these rotations is given a "pseudo-label" depending on their angle (label 0 to 0º, label 1 to 90º and so on).
We choose to use these 4 rotations because they're the only ones that don't lose data when rotated. Rotating an image to 45º for instance, will result in some of its edges being cut off (\ref{cat}).
Generally, Convolutional Neural Network (CNN) is a type of deep neural network that is most powerful in image processing tasks, such as sorting images into groups. CNN's consist of layers that process visual information. A CNN first takes in an input image and then passes it through these layers. There are a few different types of layers: convolutional, pooling, and fully-connected layers. A classification CNN takes in an input image and outputs a distribution of class scores, from which we can find the most likely class for a given image.
The encoder we provide extracts features from a batch of pre-processed images using the pre-trained ResNet-50 architecture, without the final fully-connected layer. The output is then transformed to a vector and passed through a Linear layer to make the feature vector of the word embedding's size. The output of a given convolutional layer is a set of feature maps (also called activation maps), which are filtered versions of an original input image.
Convolutional Layer is made of a set of convolutional filters. Each filter extracts a specific kind of feature, ex. a high-pass filter is often used to detect the edge of an object. Pooling Layer takes in an image (usually a filtered image) and output a reduced version of that image. Maxpooling layers look at areas in an input image (like the 4x4 pixel area pictured below) and choose to keep the maximum pixel value in that area, in a new, reduced-size area. Maxpooling is the most common type of pooling layer in CNN's, but there are also other types such as average pooling. A fully-connected layer's job is to connect the input it sees to a desired form of output. Typically, this means converting a matrix of image features into a feature vector whose dimensions are 1xC, where C is the number of classes.
It is known in the field of deep learning that better performance can be achieved with deeper networks but that entails facing gradient vanishing and gradient explosion \cite{SSL}.\\ ResNet was suggested to provide a remedy for this problem through using a “skip connection” in convolution blocks, that sends the previous feature map to the following blocks. Since ResNet is smaller and performs better, it’s very common for it to be used as the base network in other image-related tasks \cite{SSL}. ResNet was used in 33 of 78 reviewed articles in \cite{systematic} for its high efficiency in terms of computation compared to the other convolutional networks. It has also shown the best performance in comparison to AlexNet, VGG and Inception-V1 in both Top-1 and Top-5 accuracy according to \cite{systematic} and has less parameters than VGG, saving in computational resources (Fig. \ref{resnet}).
\begin{figure}[h!]
\centering
\label{resnet} \includegraphics[width=\textwidth]{Resnet-Table.png}
\caption{Table of comparison of CNN architectures from \cite{systematic}}
\end{figure}
Using self-supervised learning, our ResNet-50 encoder learnt the image feature representation by making the model predict image rotations. In the quest of predicting the rotation, the model is able to learn good representation of features from the image. In order to predict how much degree the image is rotated, the model somehow has to look into the image and understand (learns good features to be able to detect their rotation), and from there it detects the whole image rotation. From the proxy task training, the pre-trained model can be transferred (transfer learning) and used on the actual downstream task. The pre-trained model at this stage is with better initialization, after being trained on the proxy task. In our project, we trained on unlabeled subset of OpenImages dataset of 1K images/class. We trained it for 100 epochs with a batch size of 32. We fine-tuned the model afterwards using 10\% of the used subset of OpenImages dataset. For evaluating the pretext task by itself, logistic regression was used. So whatever representations have been learned on the pretext task, we pass that a logistic regression and we perform the classification task to see whether it is doing good or not. Hence, if the representations are good enough, the classifier will be performing good.
We decided to use a subset of OpenImages by only using a maximum of 1000 images per class and removing any class that has less than 5 images, resulting in a total of 563 classes. There are, however, only 135 classes that have 1000 or more images. In retrospect, we believe that as a future work it will be better to balance out the dataset more by applying some data augmentation methods or removing the classes below a higher threshold of number of images.
\subsection{Downstream Task}
In the downstream task, we fine-tune the encoder using supervised learning with small labeled data (only 10\% of the labels are included). We then remove the the final fully connected layer of the CNN that normally gives classification scores in order to get spatial information represented as image feature vectors. These vectors are then passed to the decoder. Our decoder is a special type of recurrent neural networks (RNNs), which is the long short-term memory (LSTM).
The recurrent layer of all RNNs has feedback loops. This allows them to keep details in 'memory' for a long time. However, training regular RNNs to solve problems that involve understanding long-term temporal dependencies can be challenging. This is due to the fact that the loss function's gradient decays exponentially over time (called the vanishing gradient problem). \cite{RNN_LSTM} In addition to regular units, LSTM networks use special units. A 'memory cell' in an LSTM unit can store data for a long time. Series of gates are used to control how the information enters and exists the memory and when it is forgotten. They make it learn long-term dependencies using this architecture.
All inputs are passed as a sequence to an LSTM. A sequence looks like this: first a feature vector that is extracted from an input image, then a start word, then the next word, the next word, and so on. The LSTM is defined such that, as it sequentially looks at inputs, it expects that each individual input in a sequence is of a consistent size and so the feature vector is embedded and each word so that they are of a defined embed size. Linear layer is used at the end to map the hidden state's output dimension. The LSTM cell allows a recurrent system to learn over many time steps without the fear of losing information due to the vanishing gradient problem. It is fully differentiable, therefore gives us the option of easily using backpropagation when updating the weights. So, an LSTM looks at inputs sequentially. In PyTorch, there are two ways to do this. The first is intuitive: for all the inputs in a sequence, which in this case would be a feature from an image, a start word, the next word, the next word, and so on (until the end of a sequence/batch), each input is looped through. The second is when the LSTM is fed with the entire sequence having it producing a set of outputs and the last hidden state. According to the paper reference \cite{show_tell_lessons}, we used the second approach, as it was proven to excel over the first one.
The Microsoft Common Objects in COntext (MS COCO) dataset is a large-scale dataset for scene understanding. The dataset is commonly used to train and benchmark object detection, segmentation, and captioning algorithms. We used the image-caption dataset to train a CNN-LSTM model to generate automatically captions from images. The project is implemented using PyTorch framework with other python modules including NLTK, Numpy, Matplotlib. The COCO API for MS COCO dataset was used to retrieve the data. The API is first initialized, then we load the dataset in batches using a data loader. Doing this, we can access the dataset using the dot operator of the data loader variable "data\_loader.dataset", where "dataset" is an instance of the "CoCoDataset" class.
Using the data loader, we firstly specify some parameters for the tmages pre-processing: using "transform" argument, we converted the COCO training images into PyTorch tensors before being fed to the CNN encoder. Then we resized the images, cropped them to 224x224 from a random location, randomly flipped the images horizontally, and finally normalized them after converting them to tensors. For the CoCoDataset Class, the \_\_getitem\_\_ method is used to determine how an image-caption pair is pre-processed before joining a batch. The image is pre-processed using the same transform that was provided with when the data loader was instantiated, after it has been loaded into the training folder with the name path. In addition, the captions are pre-processed for training. The model predicts the next token of a sentence from previous tokens, so the caption is converted associated with any image into a list of tokenized words before casting it to a PyTorch tensor that we can use to train the network. The \_\_init\_\_ method is where we intialized an instance of the Vocabulary class with defining different parameters: the vocab threshold, the vocab file path, the start word , the end word, the unknown word and the annotations file paths. Using NLTK Library, tokenization is where all string-valued captions are converted into integers list, and this is before casting these tokenized lists to PyTorch tensors. The tokenization process starts by initializing an empty list and appending an integer as a caption start mark "<start>" (its value is integer 0), and the same happens at the end of a caption with a caption end mark "<end>" (its value is integer 1). After the caption's start mark, we append to the list the integers corresponding to the tokens till the end mark . Finally, the integers list is converted to a PyTorch tensor.
Example:
- Sentence (List of Tokens):
[<start>, 'a', 'man', 'riding', 'a', 'bike', 'while', 'listening', 'to', 'music', '.', <end>]
- Corresponding list of integer values:
[0, 2, 74, 12, 2, 18, 56, 865, 7, 66, 369, 1]
The length of the captions in the MS COCO dataset varies. To produce batches of training data, we started by sampling a caption length, having the probability that any length is drawn is proportional to the number of captions with that length in the dataset. Then we got a batch of image-caption pairs with a given batch size, where all captions are the same length as the sampled captions. The get\_train\_indices method samples a caption length, and then samples the indices of the defined bacth size corresponding to training data points with captions of that length. The data loader receives these indices and uses them to retrieve the corresponding data points. The batch's pre-processed images and captions are saved to images and captions.
When we deal with the image captioning task, especially the RNN component, we define some essential hyperparameters. For instance, word embedding size specifies the dimentiontioality of the image and word embedding. Besides, hidden size states the number of features in the hidden state of the RNN decoder. In addition, the vocab threshold defines the minimum word count threshold that each caption should have.
\section{Experiments \& Results}
The training of the CNN-LSTM model is by specifying hyperparameters and setting other options that are important to the training procedure. For instance, we tuned the batch size controlling the image-caption pairs used to update the weights in each step. Besides, we tuned the vocab threshold accepting only words to the vocabulary that exceeds it. In addition, we tuned the hidden size and embed size defining the features number in the LSTM hidden states and dimensionality of the word embeddings and image respectively.
The first part of the architecture is the pre-trained ResNet CNN. As in the papers referenced \cite{show_tell} \cite{show_attend_tell}, the LSTM decoder follows the described architecture. The paper specifies values for a variety of hyperparameters: a vocab threshold of 4, a embed size of 512 and a hidden size of 512. These hyperparameters have been used here. The paper also mentioned the dropout usage, although the value was not specified. However, in PyTorch this requires at least two hidden layers. With a single layer and no dropout, reasonable results have been got here. The paper didn't specify a batch size, so different batch values have been tried. Moreover, the number of epochs was found to be suitable to be 10 concerning the dataset size and the batches taken out of it for every training epoch. For selecting the optimizer, firstly, SGD is started with as specified on the paper. In fact, it appeared to be slow even with 0.1 learning rate. Afterward, Adam was used and it performed better and faster with learning rate of 0.001 and 0.9 momentum.
Using the same training hyperparameters, we introduced self-supervised learning (SSL) to image captioning using rotation prediction for learning image feature representation. The rotation is used as a pretext task to train the encoder for the image captioning task. It was chosen since it’s initially trained in the original paper on similar conditions (single GPU, 100 epochs, 2 days). Moreover, 4 rotations were performed (0º, 90º, 180º, 270º), which yielded the best results in the original paper.
In addition, the following pre-trained VISSL models are used to assess our work: Rotnet, SimCLR, Jigsaw and Colorization. They were selected as they were trained by Facebook research on similar number of epochs and hyperparameters\cite{VISSL}.
PyTorch is definitely a newer framework, but it's fast and intuitive when compared to Tensorflow variables and sessions. PyTorch is designed to look and act like normal Python code: PyTorch neural nets have their layers and feedforward behavior defined in a class. Defining a network in a class means that it is possible to instantiate multiple networks, dynamically change the structure of a model, and these class functions are called during training and testing. PyTorch is also great for testing different model architectures, and the networks are modular, which makes it easy to change a single layer in a network or modify the loss function and see the effect on training \cite{pytorch}.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{resultstable.png}
\caption{Comparison of accuracy, perplexity and loss between the different models. PR: Pre-trained}
\label{results}
\end{figure}
As our optimizer, we use Adam, \cite{adam} since it has the best validation accuracy overtime, with a momentum of 0.9 and a learning rate of 0.001. We chose those values as their combination with the optimizer minimizes the training time. We train our encoder and decoder for 100 and 10 epochs and with a batch size of 32 and 64 respectively.
The results were surprising to us as we expected SimCLR to be the most performant model after the baseline on the downstream task. (Fig. \ref{results}) shows, however, that although SimCLR has a considerably higher accuracy when it comes to the pretext task, it ends up being comparable to the other pre-trained models in both perplexity and loss.
The BLEU score however, is interesting. All of the models score close to each other and around the baseline. We explain a high loss and high perplexity (negative) and a low BLEU score (positive) by looking at the captions in (Fig. \ref{caption_good}) and (Fig. \ref{caption_bad}). One of them describes the image correctly, resulting in a low loss, perplexity and BLEU score. (Fig. \ref{caption_bad}) however, has a caption that doesn't match the image, making its loss and perplexity high but the BLEU score being low has to do with it being a correct sentence that could have been generated by a human.
If we were to compare the captions without their paired images, we wouldn't see a difference between their quality.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{Bad_caption.png}
\caption{Caption from our SSL model}
\label{caption_good}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{Good_caption.png}
\caption{Caption from the baseline}
\label{caption_bad}
\end{figure}
We therefore conclude that having high classification accuracy doesn't always guarantee accurate captions, as demonstrated by the SimCLR performance on the downstream task. We think that SimCLR could potentially perform better if we provide it with the hardware that it needs.
We can also see how the BLEU score is only assessing 1 aspect of the captions and that having multiple metrics to evaluate your model give you a fuller picture of its strengths and limitations.
Our next work is to explore other approaches with image captioning, as the rotation prediction method did not yield satisfactory results, even though it sounded like it could potentially work in theory.
\bibliographystyle{plain}
|
1,116,691,497,800 | arxiv | \section{Introduction}
Optical intensity interferometry (I.I.) offers some considerable advantages over
Michelson interferometry in terms of complexity, cost and wavelength coverage.
Therefore, whilst so far successfully realised for only
one astrophysical instrument (the Narrabri interferometry of Hanbury Brown et al.
\cite{1956Natur.178.1046H}), it has enormous potential for future instruments. Several years ago it was
recognised that the optical requirements for an intensity interferometer match
rather closely those for the arrays of Cherenkov telescopes used in ground-based
$\gamma$-ray astronomy \cite{2006ApJ...649..399L}, namely very large mirrors and modest
optical quality. Furthermore, it seems relatively straightforward
to 'share' the optical systems of such a telescope array between these two
activities. As large arrays of such $\gamma$-ray telescopes are now being planned,
it seems timely to revisit this possibility. The technical details of this
approach are described in a companion paper in these proceedings (Le Bohec et al).
Here we focus on the planned new instrumentation and on the astrophysical
potential of an intensity interferometer utilising such a telescope array.
\section{Cherenkov Telescope Arrays}
The current operational Imaging Atmospheric Cherenkov Telescope (IACT) arrays such as
VERITAS \cite{2006APh....25..391H} and H.E.S.S. \cite{2004NewAR..48..331H}
consist of four 12~m ($\sim100~{\rm m}^{2}$) telescopes compound
mirrors producing on-axis point-spread-functions of $\rm FWHM\approx
0.05^{\circ}$. These telescopes are designed to image the Cherenkov light
produced in the electromagnetic cascades initiated by very-high-energy $\gamma$-rays
(typically $10^{11} - 10^{13}$ electronVolt), and have been used
in the past to do optical measurements. For example the telescopes of
H.E.S.S. have been used to measure the optical light-curve of the Crab pulsar
\cite{2006APh....26...22H} and to search for ultra-fast optical transients from
binary systems (Deil et al. these proceedings).
Two initiatives currently exist aiming for an order of magnitude improvement in the
sensitivity of IACT arrays. To achieve this, both projects aim to construct a
large array with more and/or larger telescopes, covering an area of approximately 1\,$\rm
km^2$. Larger light collecting dishes, important for I.I. sensitivity, are technically feasible and are currently used by
the single dish MAGIC telescope (17\,m) \cite{2005Ap&SS.297..245C}. A 30\,m
dish is under construction as part of the second phase of the H.E.S.S. project. AGIS
(Advanced $\gamma$-ray Imaging System) \cite{AGIS}
is the American-proposed next generation
$\rm 1km^2$ instrument. The Cherenkov Telescope Array (CTA) \cite{CTA}
is the European equivalent.
The science goals of these projects are extremely diverse, covering
many aspects of both galactic and extragalactic high energy astrophysics as well
as topics from particle physics and cosmology. The left panel of Fig.\,1
illustrates a potential layout of CTA with simulated performance that meets
the goal sensitivity for $\gamma$-ray astronomy \cite{bernloehr}. Some 85 identical
telescopes provide 3570 potential pairings, providing 47 closely-spaced, unique
baselines ranging between 50 and 1000\,m. The idea of using CTA as an intensity
interferometer is currently under discussion within the consortium.
\begin{figure}
\includegraphics[height=.275\textheight,width=.275\textheight]{ii_layout.eps}
\includegraphics[height=.3\textheight,width=.4\textheight]{ii_sensitivity.eps}
\caption{{\it Left:} Proposed lay-out for the future CTA. The small dots are
the 85 $\rm 100\,m^2$ dishes, the large dots are the 4 $\rm 600\,m^2$
dishes. {\it Right:} I.I. with
CTA sensitivity estimate as function of the 47 non-redundant baselines for a 5$\sigma$ detection, in a 5 hours integration on a
star with 50\% visibility. Final values depend on signal bandwith and CTA
design details (see Le Bohec et al., these proceedings).}
\end{figure}
\section{Science objectives}
Intensity Interferometry at a future IACT is likely to operate in the visual
wavelength region. Science objectives are therefore similar but not the same as
future 1\,km baseline Michelson interferometers such as \emph{Ohana} \cite{2004SPIE.5491..391P} which operate in
the $K$ band. Limiting $\rm m_{v}$ of a CTA concept is illustrated in the right panel
of Fig.\,1. Targets are limited to a $\rm
m_{v}\approx8.5^{m}$ for a S/N = 5, and a 5 hours integration in case of
50\% visibility (see Le Bohec et al., these proceedings). These specifications
allow important interferometric studies regarding binary stars, stellar radii and
pulsating stars with unprecedented resolution on scales of 10s of $\mu$-arcseconds.
Below we highlight three of the potential science topics.
\paragraph{\bf{Star formation}}
Key questions relating to the physics of mass accretion and pre-main sequence (PMS)
evolution can be addressed by means of very high resolution imaging as provided
by the next generation IACTs and I.I. They involve the
absolute calibration of PMS tracks, the mass accretion process,
continuum emission variability, and stellar magnetic activity. The technique will be
able to resolve features on the stellar surface.
Hot spots deliver direct information regarding the accretion
of material onto the stellar surface. Cool spots on the other hand may cover 50\%
of the stellar surface, and are the product of the slowly decaying rapid
rotation of young stars. Imaging these cool spots will constrain ideas regarding the interplay of
rotation, convection and chromospheric activity and may also
provide a direct practical application in the explanation for the anomalous
photometry observed in young stars \cite{2003AJ....126..833S}.
In practise about 50 young stars with $\rm m_{v}<8^{\rm m}$ are within reach of future
IACTs. In the last decade several young coeval stellar groups have been
discovered within $\sim$50\,pc of the sun. Famous
examples are the TW\,Hydra and $\beta$ Pic comoving groups. The majority of the spectral types
within reach lie between A and G-type. Their ages lie within the range 8
and 50 Myr (for an overview see \cite{2004ARA&A..42..685Z}),
ensuring that a substantial fraction of the low-mass members are still
in their PMS contraction phase. Measurements of their angular size can be used in
the calibration of evolutionary tracks, fundamental in deriving the properties
of star forming regions and young stellar clusters. The proximity
of the comoving groups ensures that their members are bright and also makes
these groups relatively sparse, making them unconfused
targets, even given the large PSF of $\sim3'$. This sparseness is also the
reason for incomplete group memberships, making it likely that the number of
young stars known to lie close to the sun will increase in the years to come.
\paragraph{\bf{Distance scale}}
Measuring the diameters of Cepheids is a basic method with far reaching
implications. A radius estimate of a Cepheid can be obtained using the
Baade-Wesselink method. The Baade-Wesselink method relies on measurement of the
ratio of the star size at times $t_{1}$ and $t_{2}$, based on the luminosity and
colour. Combining this with the measurement of the radial velocity, delivering the
difference in the radius between $t_{1}$ and $t_{2}$, one can derive the radius of the
Cepheid. Combining I.I. angular size measurement with the radius estimate using
the Baade-Wesselink method one obtains the distance to the Cepheid (see
\cite{1994ApJ...432..367S}). This allows the calibration of the all important
Cepheid period-luminosity relation using local Cepheids. A count of Cepheids
observed with Hipparcos \cite{1999A&AS..139..245G} shows that at least 60
Cepheids with $\rm m_{v} <8^{m}$ are within reach of future I.I.-IACTs.
\paragraph{\bf {Rapidly rotating stars}}
As a group classical Be stars are particularly well-known for their proximity to
break-up rotational velocities as deduced from photospheric absorption lines.
In addition these stars show Balmer line emission firmly established as due to
gaseous circumstellar disks, that appear and disappear on timescales of months
to years. These two properties are somehow related, but many open questions
regarding the detailed physical processes at play still exist.
The Be-phenomenon is an important simply due to the number of stars and
the range of stellar physics involved (the fraction of Be stars to normal B-type peaks
at nearly 50\% for B0 stars, \cite{1997A&A...318..443Z}). Absorption lines will
however never provide the final answer to their actual rotational velocity due
to strong gravity darkening at the equator and brightening at the poles.
Direct measurement of the shape of the rotating star is not hampered by
gravity darkening, and provides a direct indication of the rotational speed (see
e.g. $\alpha$ Eri with the VLTI, \cite{2003A&A...407L..47D}).
The Be star disk formation and dissolution activity is little
understood. Photometric observations of Be star disks seem to indicate that they
may actually evolve into ring structures before disappearing into the
interstellar medium (e.g. \cite{2006A&A...456.1027D}). Bremsstrahlung in the disk
can constitute $\sim30\%$ of the total light in $V$-band
\cite{1997A&A...318..443Z}.
There are about 300 Be stars brighter than $m_{v}=8^m$, roughly
corresponding to a distance limit of 700\,pc \cite{Cat},
signifying that Be star phenomena can be probed in depth with IACT based I.I.
\section{Conclusion}
Intensity Interferometry using the telescopes of the future IACT arrays CTA
and/or AGIS has the potential to make a major impact in several areas of stellar astrophysics
and beyond. Whilst much research and development in this area is still required,
such a major instrument could be realised at rather modest cost and as
early as 2015.
\bibliographystyle{aipproc}
|
1,116,691,497,801 | arxiv | \section{Introduction}
Subluminous B stars (subdwarf B stars or sdBs) are stars with thin hydrogen envelopes, currently undergoing helium-core burning, which are found on the extreme horizontal branch (EHB). Their masses were determined to be around $0.47$\,M$_{\rm \odot}$ \citep{heber09, heber16}. About half of the known single-lined sdB stars are found to be members of short-period binaries \citep[P $\lesssim$ 30 d, most even with P $\lesssim$ 10 d,][]{maxted01,napiwotzki04a,kupfer15}. A large mass loss on the red giant branch (RGB) is required to form these stars, which can be caused by mass transfer to the companion, either via stable Roche lobe overflow or the formation and eventual ejection of a common envelope \citep{han02,han03}.
For the existence of apparently single sdB stars binary evolution might play an important role as well, as such stars could be remnants of helium white dwarf mergers \citep{webbink84,ibentutukov84} or from engulfing a substellar object, which might get destroyed in the process \citep{soker98,nelemans98}.
Eclipsing sdB+dM binaries (HW\,Vir systems) having short orbital periods ($0.05-1\,{\rm d}$) and low companion masses between $0.06$\,M$_{\rm \odot}$ and $0.2$\,M$_{\rm \odot}$ \citep[see][for a summary of all known HW Vir systems]{schaffenroth18,schaffenroth19} have been known for decades \citep{menzies} and illustrate that objects close to the nuclear burning limit of $\sim 0.070-0.076$\,M$_{\rm \odot}$ for an object of solar metallicity and up to $0.09\,\rm M_{\rm \odot}$ for metal-poor objects \citep[see][for a review]{2014AJ....147...94D} can eject a common envelope and lead to the formation of an sdB. The light travel-time technique was used to detect substellar companion candidates to sdB stars \citep[e.g.][and references therein]{beuermann12, kilkenny12}. However, in these systems the substellar companions
have wide orbits and therefore cannot have influenced the evolution of the host star.
The short-period eclipsing HW~Vir type binary SDSS J082053.53+000843.4, hereafter J08205+0008, was discovered as part of the MUCHFUSS project \citep{geier11a, geier11b}. \citet{geier11c} derived an orbital solution based on time resolved medium resolution spectra from SDSS \citep{abazajian09} and ESO-NTT/EFOSC2. The best fit orbital period was $P_{\rm orb}=P=0.096\pm0.001\,{\rm d}$ and the radial velocity (RV) semi-amplitude $K=47.4\pm1.9\,{\rm km\,s^{-1}}$ of the sdB. An analysis of a light curve taken with Merope on the Mercator telescope allowed them to constrain the inclination of the system to $85.8^{\rm \circ}\pm0.16$.
The analysis resulted in two different possible solutions for the fundamental parameters of the sdB and the companion. As the sdB sits on the EHB the most likely solution is a core-He burning object with a mass close to the canonical mass for the He flash of $0.47 \,\rm M_\odot$. Population synthesis models \citep{han02,han03} predict a mass range of $M_{\rm sdB}=0.37-0.48\,\rm M_\odot$, which is confirmed by asteroseismological measurements \citep{fontaine12}. A more massive ($2-3\,\rm M_\odot$) progenitor star would ignite the He core under non-degenerate conditions and lower masses down to $0.3\,\rm M_\odot$ are possible. Due to the shorter lifetime of the progenitors such lower mass hot subdwarfs would also be younger. Higher masses for the sdB were ruled out as contemporary theory did not predict that. By a combined analysis of the spectrum and the light curve the companion was derived to have a mass of $0.068\pm0.003\,\rm M_\odot$. However, the derived companion radius for this solution was significantly larger than predicted by theory.
The second solution that was consistent with the atmospheric parameters was a post-RGB star with an even lower mass of only $0.25\,\rm M_\odot$. Such an object can be formed whenever the evolution of the star on the RGB is interrupted due to the ejection of a common envelope before the stellar core mass reaches the mass, which is required for helium ignition. Those post-RGB stars, also called pre-helium white dwarfs, cross the EHB and evolve directly to white dwarfs.
In this case the companion was determined to have a mass of $0.045\pm0.003\,\rm M_\odot$ and the radius was perfectly consistent with theoretical predictions.
The discovery of J08205+0008 was followed by the discovery of two more eclipsing systems with brown dwarf (BD) companions, J162256+473051 \citep{schaffenroth14} and V2008-1753 \citep{schaffenroth15}, both with periods of less than 2 hours. Two non-eclipsing systems were also discovered by \citet{schaffenroth14a}, and a subsequent analysis of a larger population of 26 candidate binary systems by \citet{schaffenroth18} suggests that the fraction of sdB stars with close substellar companions is as high as 3 per cent, much higher than the $0.5\pm0.3$ per cent that is estimated for brown dwarf companions to white dwarfs (e.g. \citealt{steele11}). Seven of the nine known white dwarf-brown dwarf systems have primary masses within the mass range for a He-core burning hot subdwarf and might therefore have evolved through this phase before.
In this paper, we present new phase-resolved spectra of J08205+0008 obtained with ESO-VLT/UVES and XSHOOTER and high cadence light curves with ESO-NTT/ULTRACAM. Combining these datasets, we have refined the radial velocity solution and light curve fit. We performed an in-depth analysis of the sdB atmosphere and a fit of the spectral energy distribution using the ULTRACAM secondary eclipse measurements to better constrain the radius and mass of the sdB primary and the companion. We also present our photometric campaign using the SAAO/1m-telescope and BUSCA mounted at the Calar Alto/2.2m telescope which has been underway for more than 10 years now, and which has allowed us to derive variations of the orbital period.
\section{Spectroscopic and photometric data}
\subsection{UVES spectroscopy}
We obtained time-resolved, high resolution ($R\simeq40\,000$) spectroscopy of J08205+0008 with ESO-VLT/UVES \citep{dekker04} on the night of 2011-04-05 as part of program 087.D-0185(A). In total 33 single spectra with exposure times of $300\,{\rm s}$ were taken consecutively to cover the whole orbit of the binary. We used the 1" slit in seeing of $\sim$ 1" and airmass ranging from 1.1. to 1.5. The spectra were taken using cross dispersers CD\#2 and CD\#3 on the blue and red chips respectively to cover a wavelength range from \SI{3300}{\angstrom} to \SI{6600}{\angstrom} with two small gaps ($\simeq$ \SI{100}{\angstrom}) at \SI{4600}{\angstrom} and \SI{5600}{\angstrom}.
The data reduction was done with the UVES reduction pipeline in the \textsc{midas} package \citep{midas}. In order to ensure an accurate normalisation of the spectra, two spectra of the DQ type white dwarf WD\,0806$-$661 were also taken \citep{subasavage09}. Since the optical spectrum of this carbon-rich white dwarf is featureless, we divided our data by the co-added and smoothed spectrum of this star.
The individual spectra of J08205+0008 were then radial velocity corrected using the derived radial velocity of the individual spectra as described in Sect. \ref{rvs} and co-added for the atmospheric analysis.
In this way, we increased the signal-to-noise ratio to S/N\,$\sim$\,90, which was essential for the subsequent quantitative analysis.
\subsection{XSHOOTER spectroscopy}
We obtained time resolved spectra of J08205+0008 with ESO-VLT/XSHOOTER \citep{vernet11} as part of programme 098.C-0754(A). The data were observed on the night of 2017-02-17 with 300~s exposure times in nod mode and in seeing of $0.5-0.8$". We obtained 24 spectra covering the whole orbital phase (see Fig. \ref{change of atmospheric parameters vs. orbital phase}) in each of the UVB ($R\sim$\,5400), VIS ($R\sim$\,8900) and NIR ($R\sim$\,5600) arms with the $0.9-1.0$" slits. The spectra were reduced using the ESO \textsc{reflex} package \citep{reflex} and the specific XSHOOTER routines in nod mode for the NIR arm, and in stare mode for the UVB and VIS arms.
To correct the astronomical observations for atmospheric absorption features in the VIS and NIR arms, we did not require any observations of telluric standard stars, as we used the \texttt{molecfit} software, which is based on fitting synthetic transmission spectra calculated by a radiative transfer code to the astronomical data \citep{2015A&A...576A..77S, 2015A&A...576A..78K}. The parameter set-up (fitted molecules, relative molecular column densities, degree of polynomial for the continuum fit, etc.) for the telluric absorption correction evaluation of the NIR-arm spectra were used according to Table 3 of \citet{2015A&A...576A..78K}. Unfortunately, the NIR arm spectra could not be used after the telluric corrections since the signal-to-noise (S/N) ratio and the fluxes are too low. Figure \ref{VIS X-Shooter arm before and after telluric correction with molecfit} shows an example comparison between the original and the telluric absorption corrected XSHOOTER VIS arm spectra. The quality of the telluric correction is sufficient to allow us to make use of the hydrogen Paschen series for the quantitative spectral analysis.
Accurate radial velocity measurements for the single XSHOOTER spectra were performed within the analysis program SPAS \citep{2009PhDT.......273H}, whereby selected sharp metal lines listed in Table \ref{list of lines detected} were used. We used a combination of Lorentzian, Gaussian and straight line (in order to model the slope of the continuum) function to fit the line profiles of the selected absorption lines.
After having corrected all single spectra by the averaged radial velocities, a co-added spectrum was created in order to achieve S/N\,$\sim 460/260$ in the UVB and VIS channels, respectively.
The co-added spectrum then was normalized also within SPAS. Numerous anchor points were set where the stellar continuum to be normalized was assumed. In this way, the continuum was approximated by a spline function. To obtain the normalized spectrum, the original spectrum was divided by the spline.
\subsection{ULTRACAM photometry}\label{ULTRACAM}
Light curves in the SDSS $u'g'r'$ filters were obtained simultaneously using the ULTRACAM instrument \citep{dhillon} on the 3.5m-ESO-NTT at La Silla. The photometry was taken on the night of 2017-03-19 with airmass $1.15-1.28$ as part of programme 098.D-679 (PI; Schaffenroth). The data were taken in full frame mode with 1$\times$1 binning and the slow readout speed with exposure times of 5.75~s resulting in 1755 frames obtained over the full orbit of the system. The dead-time between each exposure was only 25 msec. We reduced the data using the HiperCam pipeline (\url{http://deneb.astro.warwick.ac.uk/phsaap/hipercam/docs/html}). The flux of the sources was determined using aperture photometry with an aperture scaled variably according to the full width at half-maximum. The flux relative to a comparison star within the field of view (08:20:51.941 +00:08:21.64) was determined to account for any variations in observing conditions. This reference star has SDSS magnitudes of $u'$=15.014$\pm$0.004, $g'$=13.868$\pm$0.003, $r'$=13.552$\pm$0.003 which were used to provide an absolute calibration for the light curve.
\subsection{SAAO photometry}\label{saao}
All the photometry was obtained on the 1m (Elizabeth) telescope at the Sutherland
site of the South African Astronomical Observatory (SAAO). Nearly all observations
were made with the STE3 CCD, except for the last two (Table H1), which were made
with the STE4 camera. The two cameras are very similar with the only difference being the pixel size as the STE3 is $512\times512$ pixels in size and the STE4 is $1024\times1024$. We used a $2\times2$ pre-binned mode for each CCD resulting in a read-out time of around 5 and 20s, respectively, so that
with typical exposure times around 10-12s, the time resolution of STE4 is only about
half as good as STE3. Data reduction and eclipse analysis were carried out as outlined
in \citep{kilkenny11}; in the case of J08205+008, there are several useful comparison
stars, even in the STE3 field, and - given that efforts were made to observe eclipses near the meridian - usually there were no obvious "drifts" caused by differential
extinction effects. In the few cases where such trends were seen, these were removed
with a linear fit to the data from just before ingress and just after egress. The stability of the
procedures (and the SAAO time system over a long time base) is demonstrated by the
constant-period system AA Dor \citep[Fig.1 of ][]{kilkenny14} and by the intercomparisons in Fig. 8 of \citet{baran18}, for example.
\subsection{BUSCA photometry}\label{calaralto}
Photometric follow-up data were also taken with the Bonn University
Simultaneous CAmera \cite[BUSCA; see][]{busca}, which is mounted to the
2.2 m-telescope located at the Calar Alto Observatory in Spain.
This instrument observes in four bands simultaneously giving a very accurate eclipse measurement and good estimate of the errors. The four different bands we used in our observation are given solely by the intrinsic transmission curve given by the
beam splitters (UB, BB, RB, IB, \url{http://www.caha.es/CAHA/Instruments/BUSCA/bands.txt}) and the efficiency of the CCDs, as no filters where used to ensure that all the visible light is used most efficiently.
The data were taken during one run on 25 Feb 2011 and 1 Mar 2011. We used an exposure time of 30~s. Small windows were defined around the target and four comparison stars to decrease the read-out time from 2 min to 15 s. As comparison stars we used stars with similar magnitudes ($\Delta m< 2$mag) in all SDSS bands from $u$ to $z$, which have been pre-selected using the SDSS DR 9 skyserver (\url{http://skyserver.sdss.org/dr9/en/}). The data were reduced using IRAF\footnote{http://iraf.noao.edu/}; a standard CCD reduction was performed using the IRAF tools for bias- and flatfield-correction. Then the light curves of the target and the comparison stars were extracted using the aperture photometry
package of DAOPHOT. The final light was constructed by dividing the light curve of the target by the light curves of the comparison stars.
\section{Analysis}
\subsection{The hybrid LTE/NLTE approach and spectroscopic analysis}\label{The spectroscopic analysis technique}
Both the co-added UVES and XSHOOTER (UVB and VIS arm) spectra were analyzed using the same hybrid local thermodynamic equilibrium (LTE)/non-LTE (NLTE) model atmospheric approach. This approach has been successfully used to analyze B-type stars (see, for instance, \citealt{2006BaltA..15..107P, 2006A&A...445.1099P, 2011JPhCS.328a2015P}; \citealt{2007A&A...467..295N, Nieva_2008}) and is based on the three generic codes \textsc{atlas12} \citep{1996ASPC..108..160K}, \textsc{detail}, and \textsc{surface} (\citealt{1981PhDT.......113G}; \citealt{Butler_1985}, extended and updated).
Based on the mean metallicity for hot subdwarf B stars according to \citet{naslim13}, metal-rich and line-blanketed, plane-parallel and chemically homogeneous model atmospheres in hydrostatic and radiative equilibrium were computed in LTE within \textsc{atlas12}.
Occupation number densities in NLTE for hydrogen, helium, and for selected metals (see Table \ref{summary of model atoms used for the hybrid LTE/NLTE approach}) were computed with \textsc{detail} by solving the coupled radiative transfer and statistical equilibrium equations. The emergent flux spectrum was synthesized afterwards within \textsc{surface}, making use of realistic line-broadening data.
Recent improvements to all three codes \citep[see][for details]{2018A&A...615L...5I} with regard to NLTE effects on the atmospheric structure as well as the implementation of the occupation probability formalism \citep{1994A&A...282..151H} for H\,{\sc i} and He\,{\sc ii} and new Stark broadening tables for H \citep{2009ApJ...696.1755T} and He\,{\sc i} \citep{1997ApJS..108..559B} are considered as well. For applications of these models to sdB stars see \citet{schneider18}.
We included spectral lines of H and \ion{He}{i}, and in addition, various metals in order to precisely measure the projected rotational velocity ($v\sin i$), radial velocity ($v_{\rm rad}$), and chemical abundances of J08205+0008. The calculation of the individual model spectra is presented in detail in \citet{2014A&A...565A..63I}.
In Table \ref{Hybrid LTE/NLTE model grid used for the quantitative spectral analysis of SDSS J08205+0008}, the covered effective temperatures, surface gravities, helium and metal abundances for the hybrid LTE/NLTE model grid used are listed.
The quantitative spectral analysis followed the methodology outlined in detail in \citet{2014A&A...565A..63I}, that is, the entire useful spectrum and all 15 free parameters ($T_{\rm eff}$, $\log g$, $v_{\rm rad}$, $v\sin i$, $\log{n(\text{He})}:=\log{\left[\frac{\text{N(He)}}{\text{N(all elements)}}\right]}$, plus abundances of all metals listed in Table \ref{summary of model atoms used for the hybrid LTE/NLTE approach}) were simultaneously fitted using standard $\chi^2$ minimization techniques. Macroturbulence $\zeta$ and microturbulence $\xi$ were fixed to zero because there is no indication for additional line-broadening due to these effects in sdB stars \citep[see, for instance,][]{geier:2012,schneider18}.
\begin{table}
\caption{Metal abundances of J08205+0008 derived from XSHOOTER and UVES.$^\dagger$}\label{abundance table}
\centering
\begin{tabular}{lll}
\hline\hline
Parameter & XSHOOTER & UVES\\
\hline
$\log{n(\text{C})}$ & $-4.38\pm0.05$ & $-4.39^{+0.04}_{-0.03}$\\
$\log{n(\text{N})}$ & $-4.00^{+0.03}_{-0.02}$ & $-3.98\pm0.03$\\
$\log{n(\text{O})}$ & $-4.01^{+0.05}_{-0.06}$ & $-3.86^{+0.07}_{-0.06}$\\
$\log{n(\text{Ne})}$ & $\leq -6.00$ & $\leq -6.00$\\
$\log{n(\text{Mg})}$ & $-4.98^{+0.05}_{-0.04}$ & $-5.03\pm0.05$\\
$\log{n(\text{Al})}$ & $-6.20\pm0.03$ & $\leq -6.00$\\
$\log{n(\text{Si})}$ & $-5.13\pm0.04$ & $-5.17^{+0.07}_{-0.08}$\\
$\log{n(\text{S})}$ & $-5.31^{+0.11}_{-0.10}$ & $-5.12^{+0.06}_{-0.08}$\\
$\log{n(\text{Ar})}$ & $-5.54^{+0.15}_{-0.27}$ & $-5.32^{+0.19}_{-0.23}$\\
$\log{n(\text{Fe})}$ & $-4.39\pm0.04$ & $-4.41^{+0.04}_{-0.05}$\\
\hline
\multicolumn{3}{l}{$\dagger$: Including 1$\sigma$ statistical and systematic errors.}\\
\multicolumn{3}{l}{$\log{n(\text{X})}:=\log{\left[\frac{\text{N(X)}}{\text{N(all elements)}}\right]}$}
\end{tabular}
\end{table}\noindent
\subsection{Effective temperature, surface gravity, helium content and metal abundances} \label{Effective temperature, surface gravity, and helium content}
The excellent match of the global best fit model spectrum to the observed one is demonstrated in Fig. \ref{XSHOOTER hydrogen and helium lines 1} for selected spectral ranges in the co-added XSHOOTER spectrum of J08205+0008 (UVB + VIS arm).
The wide spectral range covered by the XSHOOTER spectra allowed, besides the typical hydrogen Balmer series and prominent $\ion{He}{i}$ lines in the optical, Paschen lines to be included in the fit, which provides additional information that previously could not be used in sdB spectral analysis, but provides important consistency checks.\\
In the framework of our spectral analysis, we also tested for variations of the atmospheric parameters over the orbital phase as seen in other reflection effect systems \citep[e.g.][]{heber04,schaffenroth13}. As expected, due to the relatively weak reflection effect of less than 5\%, the variations were within the total uncertainties given in the following and can therefore be neglected (see also Fig. \ref{change of atmospheric parameters vs. orbital phase} for details).
\begin{figure*}
\begin{center}
\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.49]{SDSS0820_hydrogen_and_helium_line_profiles_1_v2.pdf}
\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.49]{SDSS0820_hydrogen_and_helium_line_profiles_2_v2.pdf}
\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.49]{SDSS0820_hydrogen_and_helium_line_profiles_3_v2.pdf}
\caption{Comparison between observation (solid black line) and global best fit (solid red line) for selected spectral ranges in the co-added XSHOOTER spectrum of J08205+0008. Prominent hydrogen and $\ion{He}{i}$ lines are marked by blue labels and the residuals for each spectral range are shown in the bottom panels, whereby the dashed horizontal lines mark mark deviations in terms of $\pm1\sigma$, i.e., values of $\chi=\pm1$ (0.2\% in UVB and 0.4\% in VIS, respectively).} Additional absorption lines are caused by metals (see Fig. \ref{SDSS0820_metal_line_profiles}). Spectral regions, which have been excluded from the fit, are marked in grey (observation) and dark red (model), respectively. Since the range between $\ion{H}{i}$ \SI{9230}{\angstrom} and $\ion{H}{i}$ \SI{9546}{\angstrom} strongly suffers from telluric lines (even after the telluric correction with \texttt{molecfit}), it is excluded from the figure.
\label{XSHOOTER hydrogen and helium lines 1}
\end{center}
\end{figure*}
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{J0820_kiel.pdf}
\caption
$T_{\text{eff}}-\log{(g)}$ diagram for J08205+0008. While the blue square represents the UVES solution, the red square results from XSHOOTER. The grey square marks the LTE solution of \citet{geier11c}. The zero-age (ZAEHB) and terminal-age horizontal branch (TAEHB) for a canonical mass sdB are shown in grey as well as evolutionary tracks for a canonical mass sdB with different envelope masses from \citet{1993ApJ...419..596D} with black dotted lines. Additionally we show evolutionary tracks with solar metallicity for different sdB masses with hydrogen layers of $0.005\rm\,M_\odot$, according to \citet{han02} to show the mass dependence of the EHB. The error bars include 1$\sigma$ statistical and systematic uncertainties as presented in the text (see Sect. \ref{Effective temperature, surface gravity, and helium content} for details).}
\label{Kiel diagram}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.45]{NLTE_abundances_plot_v2.pdf}
\caption{The chemical abundance pattern of J08205+0008 (red: XSHOOTER, blue: UVES) relative to solar abundances of \citet{asplund09}, represented by the black horizontal line. The orange solid line represents the mean abundances for hot subdwarf B stars according to \citet{naslim13} used as the metallicity for our quantitative spectral analysis. Upper limits are marked with downward arrows and $\left[\frac{\text{N}(\text{X})}{\text{N}(\text{total})}\right]:=\log_{10}{
\left\{\frac{\text{N}(\text{X})}{\text{N}(\text{total})}\right\}}-\log_{10}{\left\{\frac{\text{N}(\text{X(solar)})}{\text{N}(\text{total})}\right\}}$.}
\label{abundance}
\end{center}
\end{figure}
\begin{figure*}
\begin{center}
\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.5]{SDSS0820_metal_line_profiles_1.pdf}
\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.5]{SDSS0820_metal_line_profiles_2.pdf}
\caption{Selected metal lines in the co-added XSHOOTER spectrum of J08205+0008. The observed spectrum (solid black line) and the best fit (solid red line) are shown. Solid blue vertical lines mark the central wavelength positions and the ionization stages of the individual metal lines according to Table \ref{list of lines detected}.}
\label{SDSS0820_metal_line_profiles}
\end{center}
\end{figure*}
The resulting effective temperatures, surface gravities, and helium abundances derived from XSHOOTER and UVES are listed in Table \ref{tab:par}. The results include 1$\sigma$ statistical errors and systematic uncertainties according to the detailed study of \citet{2005A&A...430..223L}, which has been conducted in the framework of the ESO Supernova Ia Progenitor Survey. For stars with two exposures or more, \citet{2005A&A...430..223L} determined a systematic uncertainty of $\pm$374\,K for $T_{\text{eff}}$, $\pm$\,0.049\,dex for $\log{(g)}$, and $\pm$\,0.044\,dex for $\log{n(\text{He})}$ (see Table 2 in \citealt{2005A&A...430..223L} for details).
Figure \ref{Kiel diagram} shows the $T_{\text{eff}}-\log{(g)}$ diagram, where we compare the UVES and XSHOOTER results to predictions of evolutionary models for the horizontal branch for a canonical mass sdB with different envelope masses from \citet{1993ApJ...419..596D}, as well as evolutionary tracks
assuming solar metallicity and masses of $0.50$\,M$_{\rm \odot}$ and $0.55$\,M$_{\rm \odot}$ \citep{han02}. With $T_{\text{eff}}=26\,000\pm400$\,\si{\kelvin} and $\log{(g)}=5.54\pm0.05$ (XSHOOTER, statistical and systematic errors) and $T_{\text{eff}}=25\,600\pm400$\,\si{\kelvin} and $\log{(g)}=5.51\pm0.05$ (UVES, statistical and systematic errors), J08205+0008 lies within the EHB, as expected. Our final result ($T_{\text{eff}}=25\,800\pm290$\,\si{\kelvin}, $\log{(g)}=5.52\pm0.04$), the weighted average of the XSHOOTER and UVES parameters, is also in good agreement with the LTE results of \citet{geier11c}, which are $T_{\text{eff}}=26\,700\pm1000$\,\si{\kelvin} and $\log{(g)}=5.48\pm0.10$, respectively.
The determined helium content of J08205+0008 is $\log{n(\text{He})}=-2.06\pm0.05$ (XSHOOTER, statistical and systematic errors) and $\log{n(\text{He})}=-2.07\pm0.05$ (UVES, statistical and systematic errors), hence clearly subsolar (see \citealt{asplund09} for details).
The final helium abundance ($\log{n(\text{He})}=-2.07\pm0.04$), the weighted average of XSHOOTER and UVES, therefore is comparable with \citet{geier11c}, who measured $\log{n(\text{He})}=-2.00\pm0.07$, and with the mean helium abundance for sdB stars from \citet{naslim13}, which is $\log{n(\text{He})}=-2.34$ (see also Fig. \ref{abundance}).
Moreover, it was possible to identify metals of various different ionization stages within the spectra (see Table \ref{list of lines detected} and Fig. \ref{SDSS0820_metal_line_profiles}) and to measure their abundances. Elements found in more than one ionization stage are oxygen ($\ion{O}{i/ii}$), silicon ($\ion{Si}{ii/iii}$), and sulfur ($\ion{S}{ii/iii}$), whereas carbon ($\ion{C}{ii}$), nitrogen ($\ion{N}{ii}$), magnesium ($\ion{Mg}{ii}$), aluminum ($\ion{Al}{iii}$), argon ($\ion{Ar}{ii}$), and iron ($\ion{Fe}{iii}$) are only detected in a single stage. We used the model grid in Table \ref{Hybrid LTE/NLTE model grid used for the quantitative spectral analysis of SDSS J08205+0008} to measure the individual metal abundances in both the co-added XSHOOTER and the UVES spectrum. We were able to fit the metal lines belonging to different ionization stages of the same elements similarly well (see Fig. \ref{SDSS0820_metal_line_profiles}). The corresponding ionization equilibria additionally constrained the effective temperature.\\
All metal abundances together with their total uncertainties are listed in Table 1. Systematic uncertainties were derived according to the methodology presented in detail in \citet{2014A&A...565A..63I} and cover the systematic uncertainties in effective temperature and surface gravity as described earlier.
The results of XSHOOTER and UVES are in good agreement, except for the abundances of oxygen, sulfur, and argon, where differences of $0.15$\,dex, $0.19$\,dex, and $0.22$\,dex, respectively, are measured. However, on average these metals also have the largest uncertainties, in particular argon, such that the abundances nearly overlap if the corresponding uncertainties are taken into account. According to Fig. \ref{abundance}, J08205+0008 is underabundant in carbon and oxygen, but overabundant in nitrogen compared to solar (\citealt{asplund09}), showing the prominent CNO signature as a remnant of the star's hydrogen core-burning through the CNO cycle. Aluminum and the alpha elements (neon, magnesium, silicon, and sulfur) are underabundant compared to solar. With the exception of neon, which is not present, the chemical abundance pattern of J08205+0008 generally follows the metallicity trend of hot subdwarf B stars (\citealt{naslim13}), even leading to a slight enrichment in argon and iron compared to solar. The latter may be explained by radiative levitation, which occurs in the context of atomic transport, that is, diffusion processes in the stellar atmosphere of hot subdwarf stars (\citealt{Greenstein_1967}; see \citealt{Michaud_Atomic_Diffusion_in_Stars_2015} for a detailed review).
Due to the high resolution of the UVES (and XSHOOTER) spectra, we were also able to measure the projected rotational velocity of J08205+0008 from the broadening of the spectral lines, in particular from the sharp metal lines, to $v\sin{i}=66.0\pm0.1\,{\rm km\,s^{-1}}$ (UVES, 1$\sigma$ statistical errors only) and $v\sin{i}=65.8\pm0.1\,{\rm km\,s^{-1}}$ (XSHOOTER, 1$\sigma$ statistical errors only)
\subsection{Search for chemical signatures of the companion}
Although HW\,Vir type systems are known to be single-lined, traces of the irradiated and heated hemisphere of the cool companion have been found in some cases. \citet{wood99} discovered the H$\alpha$ absorption component of the companion in the prototype system HW\,Vir \citep[see also][]{edelmann08}.
Metal lines in emission were found in the spectra of the hot sdOB star AA\,Dor by \citet{vuckovic16} moving in antiphase to the spectrum of the hot sdOB star indicating an origin near the surface of the companion. After the removal of the contribution of the hot subdwarf primary, which is dominating the spectrum, the residual
spectra showed more than 100 shallow emission lines originating from the heated side of the secondary, which show their maximum intensity close to the phases
around the secondary eclipse. They analysed the residual spectrum in order to model the irradiation of the low-mass companion by the hot subdwarf star.
The emission lines of the heated side of the secondary star allowed them to determine the radial velocity semi-amplitude of the centre-of-light. After the correction to the centre-of-mass of the secondary they could derive accurate masses of both components of the AA Dor system, which is consistent with a canonical sdB mass of $0.46\,\rm M_\odot$ and a companion of $0.079\pm0.002\rm\,M_\odot$ very close to the hydrogen burning limit. They also computed a first generation atmosphere model
of the low mass secondary including irradiation effects.
J08205+0008 is significantly fainter and cooler than AA Dor but with a much shorter period.
We searched the XSHOOTER spectra for signs of the low-mass companion of J08205+0008.
This was done by subtracting the spectrum in the secondary minimum where the companion is eclipsed from the spectra before and after the secondary eclipse where most of the heated atmosphere of the companion is visible.
However, no emission or absorption lines from the companion were detected (see Fig. \ref{balmer_companion_uvb} and \ref{balmer_companion_vis}). Also, in the XSHOOTER NIR arm spectra, no emission lines could be found.
\subsection{Photometry: Angular diameter and interstellar reddening}\label{Photometric analysis}\label{SED fitting}
The angular diameter of a star is an important quantity, because it
allows the stellar radius to be determined, if the distance is known e.g. from
trigonometric parallax. The angular diameter can be determined by
comparing observed photometric magnitudes to those calculated from
model atmospheres for the stellar surface.
Because of contamination by the reflection effect the apparent magnitudes of the hot subdwarf can be measured only during
the secondary eclipse, where the companion is completely eclipsed by the larger
subdwarf. We performed a least squares fit to the flat bottom of the secondary eclipse in the ULTRACAM light curves to
determine the apparent magnitudes and
derived $u'=14.926\pm$ 0.009mag, $g'=15.025\pm$ 0.004mag, and $r'=15.450\pm$ 0.011mag (1$\sigma$ statistical errors).
Because the star lies at low Galactic latitude (b=19$^\circ$) interstellar reddening is expected to be significant.
Therefore, both the angular diameter and the interstellar colour
excess have to be determined simultaneously.
We used the reddening law of \citet{2019ApJ...886..108F} and matched a synthetic flux distribution calculated from the same grid of model
atmospheres that where also used in the quantitative spectral analysis (see Sect. \ref{The spectroscopic analysis technique}) to the observed magnitudes as
described in \citet{2018OAst...27...35H}. The $\chi^2$ based fitting routine uses two free parameters: the angular diameter $\theta$,
which shifts the fluxes up and down according to $f(\lambda)=\theta^2 F(\lambda)/4$, where f($\lambda$) is the observed flux at the detector
position and $F(\lambda)$ is the synthetic model flux at the stellar surface, and the color excess
\footnote{\citet{2019ApJ...886..108F} use $E(44-55)$, the monochromatic equivalent of the usual $E(B-V)$ in the Johnson system,
using the wavelengths $\lambda$ =4400\,\AA and 5000\,\AA, respectively. In fact, $E(44-55)$ is identical to $E(B-V)$
for high effective temperatures as determined for J08205+0008.}.
The final atmospheric parameters and their respective uncertainties derived from the quantitative spectral analysis (see Sect. \ref{Effective temperature, surface gravity, and helium content})
result in an angular diameter of $\theta= 6.22\, (\pm 0.15) \cdot 10^{-12}\,\rm rad$ and an interstellar reddening of
$E(B-V)=0.041 \pm 0.013$ mag.
The latter is consistent with values from reddening maps of \citet{1998ApJ...500..525S} and \citet{2011ApJ...737..103S}: 0.039 mag and 0.034 mag,
respectively.
In addition, ample photometric measurements of J08205+0008 are available in different filter systems, covering the spectral range all the way
from the ultraviolet (GALEX) through the optical (e.g. SDSS) to the infrared (2MASS; UKIDDS; WISE). However, those measurements are mostly averages of
observations taken at multiple epochs or single epoch measurements at unknown orbital phase.
Therefore, those measurements do not allow us to determine the angular diameter of the sdB because of the contamination by light from the heated hemisphere of the companion. However, an average spectral energy distribution of the system
can be derived. This allows us to redetermine the interstellar reddening and to search for an infrared excess caused by light from the
cool companion.
The same fitting technique is used in the analysis of the SED as
described above for the analysis of the ULTRACAM magnitudes. Besides
the sdB grid, a grid of synthetic spectra of cool stars \citep[$2300\,{\rm K} \le T_{\rm eff} \le 15 000\,\rm K$,][]{2013A&A...553A...6H} is used.
In addition to the angular diameter and reddening parameter, the temperature of the cool companion as well as the surface ratio are
free parameters in the fit. The fit results in $E(B-V) = 0.040 \pm
0.010$ mag, which is fully consistent with the one derived from the ULTRACAM photometry as well as with the reddening map.
The apparent angular diameter is larger than that from ULTRACAM photometry by 2.8\%, which is caused by the contamination by
light from the companion's heated hemisphere. The effective temperature of the companion is unconstrained and the best match is achieved for the surface ratio of zero, which means there is no signature from the
cool companion.
In a final step we allow the effective temperature of the sdB to vary and determine it along with the angular diameter and the interstellar reddening, which results in $T_{\rm eff}$= 26900$^{+1400}_{-1500}$\,K in agreement with the spectroscopic result.
\begin{figure}
\centering
\includegraphics[width=1.05\columnwidth]{photometry_SED.pdf}
\caption{\label{fig:photometry_sed}Comparison of synthetic and
observed photometry: \textit{Top panel:} Spectral
energy distribution: Filter-averaged fluxes converted from
observed magnitudes are shown in different colours.
The respective full width at tenth maximum are shown as dashed
horizontal lines. The best-fitting model, degraded to a
spectral resolution of 6\,{\tiny\AA} is plotted in gray.
In order to reduce the steep SED slope the
flux is multiplied by the wavelength cubed.
\textit{Bottom panel:} Difference between synthetic and observed magnitudes
divided by the corresponding uncertainties (residual $\chi$).
The following color code is used for the different photometric
systems: GALEX
\citep[violet,][]{2017yCat.2335....0B}; SDSS
\citep[golden,][]{2015ApJS..219...12A}; Pan-STARRS1
\citep[dark red,][]{2017yCat.2349....0C}; Johnson
\citep[blue,][]{2015AAS...22533616H};
{\it Gaia}
\citep[cyan,][with corrections and
calibrations from \citet{2018A&A...619A.180M}]{2018A&A...616A...4E}; 2MASS
\citep[red,][]{2003yCat.2246....0C}; UKIDSS
\citep[pink,][]{2007MNRAS.379.1599L};
WISE \citep[magenta,][]{2014yCat.2328....0C,2019ApJS..240...30S}.}
\end{figure}
\subsection{Stellar radius, mass and luminosity}\label{Stellar radius, mass and luminosity}
Since Gaia data release 2 \citep[DR2;][]{2018A&A...616A...1G}, trigonometric parallaxes are available for a
large sample of hot subdwarf stars, including J08205+0008
for which 10\% precision has been reached. We corrected for the Gaia DR2 parallax zero point offset of $-$0.029 mas \citep{2018A&A...616A...2L}.
Combining the parallax measurement
with the results from our quantitative spectral analysis ($\log{g}$ and $T_{\rm eff}$) and with the angular diameter $\theta$ derived
from ULTRACAM photometry, allows for the determination of the mass of the sdB primary in J08205+0008 via:
\begin{equation}
M = \frac{g \theta^2}{4 G \varpi^2}\label{mass}
\end{equation}
The respective uncertainties of the stellar parameters are derived by Monte Carlo error propagation. The uncertainties are dominated by the error of the parallax measurement. Results are summarized in Table \ref{tab:par}.
Using the gravity and effective temperature derived by the spectroscopic analysis, the mass for the
sdB is $M= 0.48^{+0.12}_{-0.09}$ M$_\odot$ and its luminosity is $L=16^{+3.6}_{-2.8}$ L$_\odot$ in agreement with canonical models for EHB stars \citep[see Fig. 13][]{1993ApJ...419..596D}.
The radius of the sdB is calculated by the angular diameter and the parallax to
$R = 0.200^{+0.021}_{-0.018}$ R$_\odot$.
\subsection{Radial velocity curve and orbital parameters}\label{rvs}
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{rv_j0820.pdf}
\caption{Radial velocity of J08205+0008 folded on the orbital period.
The residuals are shown together with a prediction of the Rossiter-McLaughlin effect using the parameters derived in this paper in blue and a model with a higher rotational velocity assuming bound rotation in green. The radial velocities were determined from spectra obtained with XSHOOTER (red circles), UVES (black triangles), EFOSC2 (black circles), and SDSS (black rectangles). The EFOSC2 and SDSS RVs have been corrected by a systematic shift (see text for details).}
\label{rv}
\end{center}
\end{figure}
The radial velocities of the individual XSHOOTER spectra were measured by fitting all spectral features simultaneously to synthetic models as described in Sect.~\ref{The spectroscopic analysis technique}.
Due to lower S/N of the individual UVES spectra, which were observed in poor conditions, only the most prominent features in the spectra are suitable for measuring the Doppler shifts. After excluding very poor quality spectra, radial velocities of the remaining 28 spectra were measured using the {\sc fitsb2} routine \citep{napiwotzki04b} by fitting a set of different mathematical functions to the hydrogen Balmer lines as well as He\,{\sc i} lines. The continuum is fitted by a polynomial, and the line wings and line core by a Lorentzian and a Gaussian function, respectively. The barycentrically corrected RVs together with formal $1\sigma$-errors are summarized in Table~\ref{RVs}.
The orbital parameters $T_{\rm 0}$, period $P$, system velocity $\gamma$, and RV-semiamplitude $K$ as well as their uncertainties were derived with the same method described in \citet{geier11b}. To estimate
the contribution of systematic effects to the total error budget additional to the statistic errors determined by the {\sc fitsb2} routine, we normalised the $\chi^{2}$ of the most probable solution by adding systematic errors to each data point $e_{\rm norm}$ until the reduced $\chi^{2}$ reached $\simeq1.0$.
Combining the UVES and XSHOOTER RVs we derived $T_{0}=57801.54954\pm0.00024\,{\rm d}$, $P=0.096241\pm 0.000003\,{\rm d}$, $K=47.9\pm0.4$\,\si{\kilo\metre\per\second} and the system velocity $\gamma=26.5\pm0.4$\,\si{\kilo\metre\per\second}. No significant systematic shift was detected between the two datasets and the systematic error added in quadrature was therefore very small $e_{\rm norm}=2.0$\,\si{\kilo\metre\per\second}. The gravitational redshift is significant at $1.6_{+0.05}^{-0.02}$\,\si{\kilo\metre\per\second} and might be important if the orbit of the companion could be measured by future high resolution measurements \citep[see, e.g.,][]{vos13}.
To improve the accuracy of the orbital parameters even more we then tried to combine them with the RV dataset from \citet{geier11c}, medium-resolution spectra taken with ESO-NTT/EFOSC2 and SDSS. A significant, but constant systematic shift of $+17.4$\,\si{\kilo\metre\per\second} was detected between the UVES+XSHOOTER and the SDSS+EFOSC2 datasets. Such zero-point shifts are common between low- or medium-resolution spectrographs. It is quite remarkable that both medium-resolution datasets behave in the same way. However, since the shift is of the same order as the statistical uncertainties of the EFOSC2 and SDSS individual RVs we refrain from interpreting it as real.
Adopting a systematic correction of $+17.4$\,\si{\kilo\metre\per\second} to the SDSS+EFOSC2 dataset, we combined it with the UVES+XSHOOTER dataset and derived $T_{0}\,(\rm BJD_{TDB})=2457801.59769\pm0.00023\,{\rm d}$, $P=0.09624077\pm 0.00000001\,{\rm d}$, which is in perfect agreement with the photometric ephemeris, $K=47.8\pm0.4$\,\si{\kilo\metre\per\second} and $\gamma=26.6\pm0.4$\,\si{\kilo\metre\per\second}. This orbital solution is consistent with the solution from the XSHOOTER+UVES datasets alone. Due to the larger uncertainties of the SDSS+EFOSC2 RVs, the uncertainties of $\gamma$ and $K$ did not become smaller. The uncertainty of the orbital period on the other hand improved by two orders of magnitude due to the long timebase of 11 years between the individual epochs. Although this is still two orders of magnitude larger than the uncertainty derived from the light curve (see Sect.~\ref{timing}), the consistency with the light curve solution is remarkable. The RV curve for the combined solution phased to the orbital period is given in Fig.~\ref{rv}. Around phase 0 the Rossiter-McLaughin effect \citep{rossiter,mclaughlin} is visible. This effect is a RV deviation that occurs as parts of a rotating star are blocked out during the transit of the companion. The effect depends on the radius ratio and the rotational velocity of the primary. We can derive both parameters much more precisely with the spectroscopic and photometric analysis, but we plotted a model of this effect using our system parameters on the residuals of the radial velocity curve to show that is consistent.
Except for the corrected system velocity, the revised orbital parameters of J08205+0008 are consistent with those determined by \citet{geier11c} ($P=0.096\pm 0.001\,{\rm d}$, $K=47.4\pm1.9$\,\si{\kilo\metre\per\second}), but much more precise.
\subsection{Eclipse timing}\label{timing}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{o-c_J0820_res}
\caption{(O--C) diagram for J08205+0008 using eclipse times observed with Merope (red squares), BUSCA (blue diamonds), ULTRACAM (green triangles) and the SAAO-1m/1.9m telescope (black circles). The solid line represents a fit of a parabola to account for the period change of the orbital period. The derived quadratic term is given in the legend. The parameters of the fit are provided in the legend. In the lower panel the residuals between the observations and the best fit are shown.}
\label{o-c}
\end{figure}
Since the discovery that J08205+00008 is an eclipsing binary in November 2009, we have monitored the system regularly using BUSCA mounted at the 2.2m-telescope in Calar Alto, Spain, ULTRACAM and the 1m in Sutherland Observatory (SAAO), South Africa. Such studies have been performed for several post-common envelope systems with sdB or white dwarf (WD) primaries and M dwarf companions \citep[see][for a summary]{lohr14}. In many of those systems period changes have been found
The most convenient way to reveal period changes is to construct an observed minus calculated (O--C) diagram. Thereby we compare the observed mid-eclipse times (O) with the expected mid-eclipse times (C) assuming a fixed orbital period $P_0$ and using the mid-eclipse time for the first epoch $T_0$. Following \citet{kepler91}, if we expand the observed mid-eclipse of the Eth eclipse ($T_E$ with $E=t/P$) in a Taylor serie
, we get the (O--C) equation:
\begin{equation}
\mathrm{O-C}=\Delta T_0+\frac{\Delta P_0}{P_0}t+\frac{1}{2}\frac{\dot{P}}{P_0}t^2+...
\end{equation}
This means that with a quadratic fit to the O--C data we can derive the ephemeris $T_0$, $P$, and $\dot{P}$ in $BJD_{TDB}$.
Together with the discovery data observed with Merope at the Mercator telescope on La Palma \citep{geier11c} it was possible to determine timings of the primary eclipse over more than 10 years, as described in Sect. \ref{saao} and \ref{calaralto}.
All measured mid-eclipse times can be found in Table \ref{ecl_time}.
We used all eclipse timings to construct an O--C diagram, which is shown in Fig. \ref{o-c}. We used the ephemeris given in \citet{geier11c} as a starting value to find the eclipse numbers of each measured eclipse time and detrended the O-C diagram by varying the orbital period until no linear trend was visible to improve the determination of the orbital period. During the first $7-8$ years of observations, the ephemeris appeared to be linear. This was also found by \citet{pulley:18}. As their data show a large scatter, we do not use it in our analysis. However, in the last two years a strong quadratic effect was revealed. The most plausible explanation is a decrease in the orbital period of the system.
This enabled us to derive an improved ephemeris for J08205+0008:
\begin{eqnarray*}
T_0&=&2455165.709211(1)\\
P&=&0.09624073885(5)\rm\,d\\
\dot{P}&=&-3.2(8)\cdot 10^{-12}\,\rm dd^{-1}
\end{eqnarray*}
\subsection{Light curve modeling}
With the new very high quality ULTRACAM $u'g'r'$ light curves we repeated the light curve analysis of \citep{geier11c} obtaining a solution with much smaller errors.
For the modeling of the light curve we used {\sc lcurve}, a code written to model detached and accreting binaries containing a white dwarf \citep[for details, see][]{copperwheat10}. It has been used to analyse several detached white dwarf-M dwarf binaries \citep[e.g.,][]{parsons_nnser}. Those systems show very similar light curves with very deep, narrow eclipses and a prominent reflection effect, if the primary is a hot white dwarf. Therefore, {\sc lcurve} is ideally suited for our purpose.
The code calculates monochromatic light curves by subdividing each star into small elements with a geometry fixed by its radius as measured along the line from the center of one star towards the center of the companion. The flux of the visible elements is always summed up to get the flux at a certain phase. A number of different effects that are observed in compact and normal stars are considered, e.g. Roche distortions observed when a star is distorted from the tidal influence of a massive, close companion, as well as limb-darkening and gravitational darkening. Moreover, lensing and Doppler beaming, which are important for very compact objects with close companions, can be included. The Roemer delay, which is a light travel-time effect leading to a shift between primary and secondary eclipse times due to stars of different mass orbiting each other and changing their distance to us, and asynchronous orbits can be considered. The latter effects are not visible in our light curves and can hence be neglected in our case.
\begin{figure}
\centering
\includegraphics[width=1.05\linewidth]{lc.pdf}
\caption{ULTRACAM $u'g'r'$ light curves of J08205+0008 together with the best fit of the most consistent solution. The light curves in the different filters have been shifted for better visualisation. The lower panel shows the residuals. The deviation of the light curves from the best fit is probably due to the fact that the comparison stars cannot completely correct for atmospheric effects due to the different colour and the crude reflection effect model used in the analysis is insufficient to correctly describe the shape of the reflection effect.}
\label{lc_0820}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=1.0\linewidth]{mcmc_g_newer.pdf}
\caption{MCMC calculations showing the distributions of the parameter of the analysis of the ULTRACAM g'-band light curve.}
\label{mcmc_r}
\end{figure*}
As we have a prominent reflection effect it is very important to model this effect as accurately as possible. The reflection effect, better called the irradiation effect, results from the huge difference in temperature between the two stars, together with their small separation.
The (most likely) tidally locked companion is heated up on the side facing the hot subdwarf because of the strong irradiation by the hot primary. Therefore, the contribution of the companion to the total flux of the system varies with phase and increases as more of the heated side is visible to the observer. We use a quite simple model, which calculates the fluxes from the temperatures of both companions using a black body approximation. The irradiation is approximated by assigning a new temperature to the heated side of the companion
\begin{equation}
\sigma T'^4_{\rm sec}=\sigma T^4_{\rm sec}+F_{\rm irr}=\sigma T^4_{\rm sec}\left[1+\alpha\left(\frac{T_{\rm prim}}{T^{\rm sec}}\right)^4\left(\frac{R_{\rm prim}}{a}\right)^2\right],
\end{equation}
with $\alpha$ being the albedo of the companion and $F_{\rm irr}$ the irradiating flux, accounting for the angle of incidence and distance from the hot subdwarf. The irradiated side is heated up to a temperature of $13\,000-15\,000$ K similar to HW Vir \citep{kiss00}, which is slightly hotter but has a longer period. Hence, the amplitude of the effect is increasing from blue to red as can be seen in Fig. \ref{lc_0820}, as the sdB is getting fainter compared to the companion in the red. If the irradiation effect is very strong, the description given above might not be sufficient, as the back of the irradiated star is completely unaffected in this description, but heat transport could heat it up, increasing the luminosity of unirradiated parts as well. This is not considered in our simple model.
As the light curve model contains many parameters, not all of them independent, we fixed as many parameters as possible (see Table \ref{param}). The temperature of the sdB was fixed to the temperature determined from the spectroscopic fit. We used the values determined by the coadded XSHOOTER spectra, as they have higher signal-to-noise.
The gravitational limb darkening coefficients were fixed to the values expected for a radiative atmosphere for the primary \citep{von_zeipel} and a convective atmosphere for the secondary \citep{lucy} using a blackbody approximation to calculate the resulting intensities. For the limb darkening of the primary we adopted a quadratic limb darkening law using the tables by \citet{claret}. As the tables include only surface gravities up to $\log{g}=5$ we used the values closest to the parameters derived by the spectroscopic analysis.
As it is a well-separated binary, the two stars are approximately spherical, which means the light curve is not sensitive to the mass ratio. Therefore, we computed solutions with different, fixed mass ratios. To localize the best set of parameters we used a \textsc{simplex} algorithm \citep{press92} varying the inclination, the radii, the temperature of the companion, the albedo of the companion (absorb), the limb darkening of the companion, and the time of the primary eclipse to derive additional mid-eclipse times. Moreover, we also allowed for corrections of a linear trend, which is often seen in the observations of hot stars, as the comparison stars are often redder and so the correction for the air mass is often insufficient. This is given by the parameter "slope". The model of the best fit is shown in Fig. \ref{lc_0820} together with the observations and the residuals.
To get an idea about the degeneracy of parameters used in the light curve solutions, as well as an estimation of the errors of the parameters we performed Markov-Chain Monte-
Carlo (MCMC) computations with \textsc{emcee} \citep{emcee} using the best solution we obtained with the \textsc{simplex} algorithm as a starting value varying the radii, the inclination, the temperature of the companion as well as the albedo of the companion. As a prior we constrained the temperature of the cool side of the companion to $3000\pm500$ K. Due to the large luminosity difference between the stars the temperature of the companion is not significantly constrained by the light curve. The computations were done for all three light curves separately.
For the visualisation we used the python package \textsc{corner} \citep[see Fig. \ref{mcmc_r}]{corner}. The results of the MCMC computations of the light curves of all three filters agree within the error (see Table \ref{param}). A clear correlation between both radii and the inclination is visible as well as a weak correlation of the albedo of the companion (absorb) and the inclination. This results from the fact that the companion is only visible in the combined flux due to the reflection effect and the eclipses and the amplitude of the reflection effect depends on the inclination, the radii, the separation, the albedo and the temperatures. Looking at the $\chi^2$ of the temperature of the companion we see that all temperatures give equally good solutions showing that the temperature can indeed not be derived from the light curve fit. The albedo we derived has, moreover, a value > 1, which has been found in other HW Vir systems as well and is due to the simplistic modeling of the reflection effect. The reason for the different distribution in the inclination is not clear to us. However, it is not seen in the other bands. It might be related to the insufficient correction of atmospheric effects by the comparison stars.
\begin{table}\caption{Parameters of the light curve fit of the ULTRACAM u'g'r' band light curves}\label{param}
\begin{tabular}{llll}
\hline\hline
band & u'&g'&r'\\\hline
\multicolumn{4}{c}{Fixed Parameters}\\\hline
q&\multicolumn{3}{c}{0.147}\\
$P$&\multicolumn{3}{c}{0.09624073885}\\
$T_{\rm eff,sdB}$&\multicolumn{3}{c}{25800}\\
$x_{1,1}$&0.1305&0.1004&0.0788\\
$x_{1,2}$&0.2608&0.2734&0.2281\\
$g_1$&\multicolumn{3}{c}{0.25}\\
$g_2$&\multicolumn{3}{c}{0.08}\\
\hline
\multicolumn{4}{c}{Fitted parameters}\\\hline
$i$&$85.3\pm0.6$&$85.6\pm0.2$&$85.4\pm0.3$\\
$r_1/a$&$0.2772\pm0.0029$&$0.2734\pm0.0010$&$0.2748\pm0.0014$\\
$r_2/a$&$0.1322\pm0.0018$&$0.1297\pm0.0006$&$0.1304\pm0.0008$\\
$T_{\rm eff,comp}$&$3000\pm500$&$2900\pm500$&$3200\pm560$\\
absorb&$1.54\pm0.08$&$1.58\pm0.03$&$2.08\pm0.05$\\
$x_2$&0.70&0.78&0.84\\
$T_0$ [MJD]&57832.0355&57832.0354&57832.0354\\
slope&-0.000968&-0.002377&0.00013417\\
$\frac{L_1}{L_1+L_2}$&0.992578&0.98735&0.97592\\
\hline
\end{tabular}
\end{table}
\subsection{Absolute parameters of J08205+00008}\label{Absolute parameters of J08205+00008}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{sdB_python.pdf}
\caption{Mass of the sdB versus the photometric $\log g$ for J08205+0008 for different mass ratios from $0.11-0.20$ in steps of 0.01 (red solid line). The parameters were derived by combining the results from the analysis of the light curves and radial velocity curve. The grey area marks the spectroscopic $\log{g}$ that was derived from the spectroscopic analysis.
The blue dashed lines indicate the $\log g$ derived by the radius from the SED fitting and the {\it Gaia} distance for different sdB masses.
The red area marks the mass range for the sdB for which we get a consistent solution by combining all different methods. The red vertical line represents the solution for a canonical mass sdB.}
\label{sdb}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{brown_dwarf_python.pdf}
\caption{Comparison of theoretical mass-radius relations of low-mass stars \citep{baraffe,baraffe_2} to results from the light curve analysis of J08205+0008. We used tracks for different ages of 1 Gyr (dashed), 5 Gyr (dotted-dashed) and 10 Gyr (dotted). Each red square together with the errors represents a solution from the light curve analysis for a different mass ratio ($q = 0.11-0.20$ in steps of 0.01). The red vertical line represents the solution for a canonical mass sdB. The red area marks the mass range of the companion corresponding to the mass range we derived for the sdB.}
\label{bd}
\end{figure}
As explained before, we calculated solutions for different mass ratios ($q=0.11-0.20$). We obtain equally good $\chi^2$ for all solutions, showing that the mass ratio cannot be constrained by the light curve fit as expected. Hence, the mass ratio needs to be constrained differently.
However, the separation, which can be calculated from the mass ratio, period, semi-amplitude of the radial-velocity curve and the inclination, is different for each mass ratio.
The masses of both companions can then be calculated from the mass function. From the relative radii derived from the light curve fit together with the separation, the absolute radii can be calculated. This results in different radii and masses for each mass ratio.
As stated before, the previous analysis of \citet{geier11c} resulted in two possible solutions: A post-RGB star with a mass of 0.25 $\rm M_\odot$ and a core helium-burning star on the extreme horizontal branch with a mass of $\rm 0.47\,M_\odot$.
From the analysis of the photometry together with the Gaia magnitudes (see Sect. \ref{Stellar radius, mass and luminosity}) we get an additional good constraint on the radius of the sdB. Moreover, the surface gravity was derived from the fit to the spectrum. This can be compared to the mass and radius of the sdB (and a photometric $\log g$: $g=GM/R^2$) derived in the combined analysis of radial velocity curve and light curve. This is shown in Fig. \ref{sdb}. We obtain a good agreement for of all three methods (spectroscopic, photometric, parallax-based) for an sdB mass between 0.39-0.60 $\rm M_\odot$. This means that we can exclude the post-RGB solution. The position of J0820 in the $T_{\rm eff}-\log{g}$ diagram, which is shown in Fig. \ref{Kiel diagram}, gives us another constraint on the sdB mass. By comparing the atmospheric parameters of J08205+0008 to theoretical evolutionary tracks calculated by \citet{han02} it is evident that the position is not consistent with sdB masses larger than $\sim 0.50\,\rm M_\odot$, which we, therefore, assume as the maximum possible mass for the sdB.
Accordingly, we conclude that the solution that is most consistent with all different analysis methods is an sdB mass close to the canonical mass ($0.39-0.50\,\rm M_\odot$). For this solution we have an excellent agreement of the parallax radius with the photometric radius only, if the parallax offset of $-0.029$ mas suggested by \citet{2018A&A...616A...2L} is used. Otherwise the parallax-based radius is too large. The companion has a mass of $0.061-0.71\,\rm M_\odot$, which is just below the limit for hydrogen-burning. Our final results can be found in Table \ref{tab:par}. The mass of the companion is below the hydrogen burning limit and the companion is hence most likely a massive brown dwarf.
We also investigated the mass and radius of the companion and compared it to theoretical calculations by \citet{baraffe} and \citet{baraffe_2} as shown in Fig.~\ref{bd}. It is usually assumed that the progenitor of the sdB was a star with about $1-2\,\rm M_\odot$ \citep{heber09,heber16}. Therefore, we expect that the system is already quite old (5-10 Gyrs). For the solutions in our allowed mass range the measured radius of the companion is about 20\% larger than expected from theoretical calculations. Such an effect, called inflation, has been observed in different binaries and also planetary systems with very close Jupiter-like planets. A detailed discussion will be given later. This effect has already been observed in other hot subdwarf close binary systems \citep[e.g.][]{schaffenroth15}.
However, if the system would still be quite young with an age of about 1 Gyr, the companion would not be inflated. We performed a kinematic analysis to determine the Galactic population of J08205+0008. As seen in Fig. \ref{toomre} the sdB binary belongs to the thin disk where star formation is still ongoing and could therefore indeed be as young as 1 Gyr, if the progenitor was a $2\,\rm M_\odot$ star. About half of the sdO/Bs at larger distances from the Galactic plane (0.5 kpc) are found in the thin disk \citep{martin17}. However, it is unclear whether a brown dwarf companion can eject the evelope from such a massive 2 $\rm M_{\odot}$ star. Hydrodynamical simulations performed by \citet{kramer20} indicate that a BD companion of $\sim0.05-0.08\,\rm M_\odot$ might just be able to eject the CE of a lower mass ($1\,\rm M_\odot$) red giant.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{{SDSSJ082053.53+000843.4_Toomre_AS}.pdf}
\caption{Toomre diagram of J08205+0008: the quantity $V$ is the velocity in direction of Galactic rotation, $U$ towards the Galactic center, and $W$ perpendicular to the Galactic plane. The two dashed ellipses mark boundaries for the thin (85\,km\,s$^{-1}$) and thick disk (180\,km\,s$^{-1}$) following Fuhrmann (2004). The red cross marks J08205+0008, the yellow circled dot the Sun, and the black plus the local standard of rest. The location of J08205+0008 in this diagram clearly hints at a thin disk membership. }
\label{toomre}
\end{figure}
\begin{table}
\caption{Parameters of J08205+0008.
}\label{tab:par}
\vspace{0.5cm}
\begin{tabular}{lll}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{3}{l}{SPECTROSCOPIC PARAMETERS}\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$\gamma$& [${\rm km\,s^{-1}}$] & {$26.5\pm0.4$} \\
$K_1$ & [${\rm km\,s^{-1}}$] & {$47.8\pm0.4$} \\
$f(M)$& [$M_{\rm \odot}$] & {$0.0011\pm0.0001$} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$T_{\rm eff,sdB}$ & [K] & {$25800\pm290^\ast$} \\
$\log{g,sdB}$ & & {$5.52\pm0.04^\ast$} \\
$\log{n(\text{He})}$ & & {$-2.07\pm0.04^\ast$} \\
$v\sin{i}$ & [${\rm km\,s^{-1}}$] & $65.9\pm0.1^\dagger$\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$a$ &[$\rm R_\odot$]&$0.71\pm0.02$\\
$M_{\rm 1}$&[$M_{\rm \odot}$]& $0.39 - 0.50$ \\
$M_{\rm 2}$ & [$M_{\rm \odot}$] & $0.061-0.071$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{3}{l}{PHOTOMETRIC PARAMETERS}\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$T_0$ & [BJD$_{TDB}$]&2455165.709211(1)\\
$P$&[d]&$0.09624073885(5)$\\
$\dot{P}$&dd$^{-1}$&$-3.2(8)\cdot 10^{-12}$\\
$i$&[$^\circ$]&$85.6\pm0.3$\\
$R_{\rm 1}$ & [$R_{\rm \odot}$] & $0.194\pm0.008$ \\
$R_{\rm 2}$ & [$R_{\rm \odot}$] & $0.092\pm 0.005$ \\
$\log g$ & &$5.52\pm0.03$\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\multicolumn{3}{l}{SED FITTING}\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$\varpi_{\text{Gaia}}$ & [mas] & $0.6899\pm0.0632^\dagger$ \\
$E(B-V)$ & [mag] & $0.040\pm0.010^\dagger$ \\
$\theta$ & [$10^{-12}$\,rad] & $6.22\pm0.15^\ast$ \\
$R_{\text{Gaia}}$ & [$R_{\odot}$] & 0.200$^{+0.021\ast}_{-0.018}$ \\
$M_{\text{Gaia}}$ & [$M_{\odot}$] & 0.48$^{+0.12\ast}_{-0.09}$ \\
$\log{(L_{\text{Gaia}}/L_{\odot})}$ & & 16$^{+3.6\ast}_{-2.8}$ \\
\noalign{\smallskip}
\hline
\multicolumn{3}{l}{Gaia: Based on measured \textit{Gaia} parallax, but applying a zero}\\ \multicolumn{3}{l}{point offset of $-0.029$\,mas (see Sect. \ref{Stellar radius, mass and luminosity} for details).}\\
\multicolumn{3}{l}{$\dagger$: 1$\sigma$ statistical errors only.}\\
\multicolumn{3}{l}{$\ast$: Listed uncertainties result from statistical and systematic}\\ \multicolumn{3}{l}{errors (see Sects. \ref{Effective temperature, surface gravity, and helium content} and \ref{SED fitting} for details).}\\
\end{tabular}
\end{table}
\section{Discussion}
\subsection{Tidal synchronisation of sdB+dM binaries} \label{syncro}
In close binaries, the rotation of the components is often assumed to be synchronised to their orbital motion. In this case the projected rotational velocity can be used to put tighter constraints on the companion mass. \citet{geier10b} found that assuming tidal synchronisation of the subdwarf primaries in sdB binaries with orbital periods of less than $\simeq1.2\,{\rm d}$ leads to consistent results in most cases. In particular, all the HW\,Vir type systems analysed in the \citet{geier10b} study turned out to be synchronised.
In contrast to this, the projected rotational velocity of J08205+0008 is much smaller than is required for tidal synchronisation. We can calculate the expected rotational velocity ($v_{\rm rot}$) using the inclination ($i$), rotational period ($P_{\rm rot}$) and the radius of the primary ($R_1$) from the light curve analysis if we assume the system is synchronised:
\begin{equation}
P_{\rm rot, 1}=\frac{2\pi R_1}{v_{\rm rot}}\equiv P_{\rm orb} \rightarrow v_{\rm synchro} \sin i=\frac{2\pi R_1\sin i}{P_{\rm orb}}.
\end{equation}
Due to the short period of this binary, the sdB should spin with $v_{\rm syncro}\simeq102\,{\rm km\,s^{-1}}$ similar to the other known systems \citep[see][and references therein]{geier10b}.
Other observational results in recent years also indicate that tidal synchronisation of the sdB primary in close sdB+dM binaries is not always established in contrast to the assumption made by \citet{geier10b}. New theoretical models for tidal synchronisation \citep{preece:18,preece:19} even predict that none of the hot subdwarfs in close binaries should rotate synchronously with the orbital period.
From the observational point of view, the situation appears to be rather complicated.
\citet{geier10b} found the projected rotational velocities of the two short-period ($P=0.1-0.12\,{\rm d}$) HW\,Vir systems HS\,0705+6700 and the prototype HW\,Vir to be consistent with synchronisation. \citet{charpinet08} used the splitting of the pulsation modes to derive the rotation period of the pulsating sdB in the HW\,Vir-type binary PG\,1336$-$018 and found it to be consistent with synchronised rotation. This was later confirmed by the measurement of the rotational broadening \citep{geier10b}.
However, the other two sdBs with brown dwarf companions J162256+473051 and V2008-1753 \citep{schaffenroth14,schaffenroth15} have even shorter periods of only 0.07 d and both show sub-synchronous rotation with 0.6 and 0.75 of the orbital period, respectively, just like J0820+0008. AA Dor on the other hand, which has a companion very close to the hydrogen burning limit and a longer period of 0.25 d, seems to be synchronised \citep[and references therein]{vuckovic16}, but it has already evolved beyond the EHB and is therefore older and has had more time to synchronise.
\citet{pablo11} and \citet{pablo12} studied three pulsating sdBs in reflection effect sdB+dM binaries with longer periods and again used the splitting of the pulsation modes to derive their rotation periods ($P\simeq0.39-0.44\,{\rm d}$). All three sdBs rotate much slower than synchronised. But also in this period range the situation is not clear, since a full asteroseismic analysis of the sdB+dM binary Feige\,48 ($P\simeq0.38\,{\rm d}$) is consistent with synchronised rotation.
Since synchronisation timescales of any kind \citep{geier10b} scale dominantely with the orbital period of the close binary, these results seem puzzling. Especially since the other relevant parameters such as mass and structure of the primary or companion mass are all very similar in sdB+dM binaries. They all consist of core-helium burning stars with masses of $\sim0.5\,M_{\odot}$ and low-mass companions with masses of $\sim0.1\,M_{\odot}$. And yet 5 of the analysed systems appear to be synchronised, while 6 rotate slower than synchronised without any significant dependence on companion mass or orbital period.
This fraction, which is of course biased by complicated selection effects, might be an observational indication that the synchronisation timescales of such binaries are of the same order as the evolutionary timescales
It has to be pointed out that although evolutionary tracks of EHB stars exist, the accuracy of the derived observational parameters (usually $T_{\rm eff}$ and $\log{g}$) is not high enough to determine their evolutionary age on the EHB by comparison with those tracks as accurate as it can be done for other types of stars (see Fig.~\ref{Kiel diagram}). As shown in Fig. \ref{Kiel diagram}, the position of the EHB is also dependent on the core and envelope mass and so it is not possible to find a unique track to a certain position in the $T_{\rm eff}-\log{g}$ diagram and in most sdB systems the mass of the sdB is not constrained accurately enough.
\citet{2005A&A...430..223L} showed that sdB stars move at linear speed over the EHB and so the distance from the zero-age extreme horizontal branch (ZAEHB) represents how much time the star already spent on the EHB.
If we look at the position of the non-synchronised against the position of the synchronised systems in the $T_{\rm eff}-\log{g}$ diagram (Fig. \ref{Kiel diagram synchro}), it is obvious that all the systems, which are known to be synchronised, appear to be older. There also seems to be a trend that systems with a higher ratio of rotational to orbital velocity are further away from the ZAEHB. This means that the fraction of rotational to orbital period might even allow an age estimate of the sdB.
The fact that the only post-EHB HW\,Vir system with a candidate substellar companion in our small sample (AA\,Dor) appears to be synchronised, while all the other HW\,Vir stars with very low-mass companions and shorter periods are not, fits quite well in this scenario.
This could be a hint to the fact that for sdB+dM systems the synchronisation timescales are comparable to or even smaller than the lifetime on the EHB.
Hot subdwarfs spend $\sim100\,{\rm Myrs}$ on the EHB before they evolve to the post-EHB stage lasting $\sim10\,{\rm Myrs}$. So we would expect typical synchronisation timescales to be of the order of a few tens of millions of years, as we see both synchronised and unsynchronised systems.
\begin{figure}
\begin{center}
\includegraphics[width=\linewidth]{J0820_synchro.pdf}
\caption{$T_{\text{eff}}-\log{(g)}$ diagram for the sdB+dM systems with known rotational periods mentioned in Sect. \ref{syncro}. The filled symbols represent synchronized systems, the open symbols, systems which are known to be non-synchronised. The square marks the position of J08205+0008. The sizes of the symbols scale with the orbital period, with longer periods having larger symbols. Plotted error bars are the estimated parameter variations due to the reflection effect, as found e.g. in \citet{schaffenroth13}. The zero-age (ZAEHB) and terminal-age extreme horizontal branch (TAEHB) for a canonical mass sdB as well as evolutionary tracks for a canonical mass sdB with different envelope masses from \citet{1993ApJ...419..596D} are also shown.}
\label{Kiel diagram synchro}
\end{center}
\end{figure}
\subsection{A new explanation for the period decrease}
There are different mechanisms of angular momentum loss in close binaries leading to a period decrease: gravitational waves, mass transfer (which can be excluded in a detached binary), or magnetic braking \citep[see][]{quian:08}.
Here, we propose that tidal synchronisation can also be an additional mechanism to decrease the orbital period of a binary.
From the rotational broadening of the stellar lines (see Sect. \ref{syncro}) we derived the rotational velocity of the subdwarf to be about half of what would be expected from the sdB being synchronised to the orbital period of the system. This means that the sdB is currently spun up by tidal forces until synchronisation is reached causing an increase in the rotational velocity.
As the mass of the companion is much smaller than the mass of the sdB, we assume synchronisation for the companion.
The total angular momentum of the binary system is given by the orbital angular momentum $J_{\rm orb}$ and the sum of the rotational angular momentum of the primary and secondary star $I_{\rm spin,1/2}$, with $\omega$ being the orbital angular velocity and $\Omega_i$ the rotational, angular velocity
\begin{align}
J_{\rm tot}&=J_{\rm orb}+\sum_{i=1}^{2}I_{\rm spin,i} \\
J_{\rm orb}&=(m_1a_1^2+m_2a_2^2)\,\omega = \frac{m_1m_2}{m_1+m_2}a^2\omega\\%\frac{2\pi}{P_{\rm orb}} \\
a^2&=\left(\frac{G(m1+m2)}{\omega^2}\right)^{2/3}\\%\left(\frac{G(m_1+m_2)}{4\pi^2}\right)^{2/3}P_{\rm orb}^{4/3} \\
I_{\rm spin,i}&=k_r^2M_iR_i^2\Omega_
\end{align}
with $k_r^2$ the radius of gyration of the star. It refers to the distribution of the components of an object around its rotational axis. It is defined as $k_r^2=I/MR^2$, where $I$ is the moment of inertia of the star. \citet{geier10b} used a value of 0.04 derived from sdB models, which we adopt.
For now we neglect angular momentum loss due to gravitational waves and magnetic braking.
If we assume that the companion is already synchronised and its rotational velocity stays constant ($\frac{\rm d\Omega_{2}}{\mathrm{d}t}=0$) and that the masses and radii do not change, as we do not expect any mass transfer after the common envelope phase, we obtain
\begin{equation}
\frac{dJ_{\rm tot}}{dt}=p_1\frac{\mathrm{d}\omega^{-1/3}}{\mathrm{d}t}+p_2\frac{\rm d\Omega_1}{\mathrm{d}t}=-p_1\frac{\dot{\omega}}{3\omega^{4/3}}+p_2\dot{\Omega}=0
\end{equation}
with
\begin{equation}
p_1=\frac{m_1m_2G^{2/3}}{(m_1+m_2)^{1/3}}
\end{equation}
and
\begin{equation}
p_2=k_r^2m_1R_1^2
\end{equation}
This shows that from an increase in the rotational velocity of the primary, which is expected from tidal synchronisation, we expect an increase of the orbital velocity, which we observe in the case of J08205+0008.
We can now calculate the current change of orbital velocity:
\begin{equation}
\dot{\Omega}_1=\frac{p_1}{3p_2}\frac{\dot{\omega}}{\omega^{4/3}}=\frac{m_2G^{2/3}}{3k_r^2R_1^2(m_1+m_2)^{1/3}}\frac{\dot{\omega}}{\omega^{4/3}}
\end{equation}
From this equation we can clearly see that rotational velocity change depends on the masses of both stars, the radius of the primary, the orbital velocity change and the current orbital velocity. An increasing rotational velocity causes an increasing orbital velocity and hence a period decrease.
\subsection{Synchronisation timescale}
If we assume that the observed period decrease is only due to the rotational velocity change, we can calculate the rate of the rotational velocity change and the timescale until synchronisation is reached.
According to \citet{preece:18}, the change of rotational angular velocity is given by
\begin{equation}
\frac{\mathrm{d}\Omega}{\mathrm{d}t}=\frac{\omega}{\tau_{\rm tide}}\left(1-\frac{\Omega}{\omega}\right)\frac{M_2}{M_1+M_2}\frac{a^2}{R^2k_r^2}\propto \left(1-\frac{\Omega}{\omega}\right)\label{eq}
\end{equation}
where $\tau_{\rm tide}$ is the tidal time-scale depending on the density, radius and mass of the star and the viscous time-scale of the convective region. The current position of J08205+0008 on the $T_{\rm eff}-\log g$ diagram and the mass we derived from our analysis suggest that the sdB is currently in the evolutionary phase of helium-burning. The lifetime of this phase is approximately 100 Myrs. So we do not expect the structure of the star to change significantly in the next few Myr. Because the moment of inertia of an sdB star is small compared to that of the binary orbit, the change in separation and angular velocity can be neglected.
Therefore, we can calculate the timescale until synchronisation is reached using the equation given in \citet{zahn:89}:
\begin{equation}
\frac{1}{T_{\rm sync}}=-\frac{1}{\Omega_1-\omega} \frac{\mathrm{d}\Omega_1}{\mathrm{d}t}
\end{equation}
Using our equation (\ref{eq}) and calculating and substituting the angular velocities by the periods we derive an expression for the synchronisation time scale:
\begin{equation}
T_{\rm sync}=\left(1-\frac{2\pi R_1\sin i}{P_{\rm orb}v\sin i}\right)\frac{P_{\rm orb}^{2/3}v\sin i}{\dot{P}_{\rm orb}\sin i}\frac{3(2\pi)^{1/3}k_r^2R_1(m_1+m_2)^{1/3}}{m_2G^{2/3}}
\end{equation}
Using the orbital period, the masses, radii and inclination from our analysis, we calculate a synchronisation time $T_{\rm sync}$ of $2.1\pm0.1$ Myrs, well within the lifetime of a helium burning object on the extreme horizontal branch. The orbital period will change by about 200 s (3.5\%) in this 2 Myrs, which means a change in the separation of only 0.01 $\rm R_\odot$, which shows that our assumption of a negligible change in separation is valid. If we assume that the rotation after the common envelope phase was close to zero, the total timescale until the system reaches synchronisation is about 4 Myrs. This assumption is plausible as most red giant progenitors rotate slowly and the common envelope phase is very short-lived and so no change of the rotation is expected.
This means that this effect could significantly add to the observed period decrease. The fact that the synchronised systems appear to be older than the non-synchronised ones confirms that the synchronisation timescale is of the expected order of magnitude and it is possible that we might indeed measure the synchronisation timescale.
As mentioned before \citet{preece:18} predict that the synchronisation timescales are much longer than the lifetime on the EHB and that none of the HW Vir systems should be synchronised.
\citet{preece:19} investigated also the special case of NY Vir, which was determined to be synchronised from spectroscopy and asteroseismolgy, and came to the conclusion that they cannot explain, why it is synchronised. They proposed that maybe the outer layers of the sdB were synchronised during the common envelope phase. However, observations show that synchronised sdB+dM systems are not rare, but that synchronisation occurs most likely during the phase of helium-burning,
which shows that synchronisation theory is not yet able to predict accurate synchronization time scales
\subsection{Orbital period variations in HW Vir systems}
As mentioned before, there are several mechanisms that can explain period changes in HW Vir systems.
The period change due to gravitational waves is usually very small in HW Vir systems and would only be observable after observations for many decades \citep[e.g.][]{kilkenny14}. Using the equation given in \citet{kupfer20} with the system parameters derived in this paper, we predict an orbital period decay due to gravitational waves of $\dot{P}=4.5\cdot10^{-14}\,\rm ss^{-1}$.
The observed change in orbital period is hence about 100 times higher than expected by an orbital decay due to gravitational waves
HW Vir and NY Vir have also been observed to show a period decrease of the same order of magnitude \citep{quian:08,kilkenny14} but have been found to rotate (nearly) synchronously.
Both also show additionally to the period decrease a long-period sinusoidal signal \citep{Lee:09,Lee:14}. These additional variations in the O--C diagram have been interpreted as caused by circumbinary planets in both cases, however the solutions were not confirmed with observations of longer baselines. Observations of more than one orbital period of the planet would be necessary to confirm it. The period decrease was explained to be caused by angular momentum loss due to magnetic stellar wind braking.
Following the approach of \citet{qian:07} we calculated the relation between the mass-loss rate and the Alfv\'{e}n radius that would be required to account for the period decrease in J08205+0008 due to magnetic braking. This is shown in Fig. \ref{mag_braking}. Using the tidally enhanced mass-loss rate of \citet{tout:88} we derive that an Alfv\'{e}n radius of $75\,\rm R_\odot$ would be required to cause the period decrease we measure, much larger than the Alfv\'{e}n radius of the Sun. This shows that, as expected, the effect of magnetic braking in a late M dwarf or massive brown dwarf is very small at best and cannot explain the period decrease we derive.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{mag_braking.pdf}
\caption{Correlation between the Alfv\'{e}n radius and the mass loss rate
for the companion of J08205+0008. The red dashed line marks the Alfv\'{e}n radius for the Sun, the blue dotted line indicates the tidally enhanced mass-loss rate determined using the parameters of the sdB using the formula of \citet{tout:88}.}
\label{mag_braking}
\end{figure}
\citet{bours:16} made a study of close white dwarf binaries and observed that the amplitude of eclipse arrival time variations in K dwarf and early M dwarf companions is much larger than in late M dwarf, brown dwarf or white dwarf companions, which do not show significant orbital period variations. They concluded that these findings are in agreement with the so-called Applegate mechanism, which proposes that variability in the binary orbits can be driven by magnetic cycles in the secondary stars. In all published HW Vir systems with a longer observational baseline of several years quite large period variations on the order of minutes have been detected \citep[see][for an overview]{zorotovic:13, pulley:18}, with the exception of AA Dor \citep{kilkenny14}, which still shows no sign of period variations after a baseline of about 40 years. Also the orbital period decrease in J08205+0008 is on the order of seconds and has only been found after 10 years of observation and no additional sinusoidal signals have been found as seen in many of the other systems. This confirms that the findings of \citet{bours:16} apply to close hot subdwarf binaries with cool companions. The fact that the synchronised HW Vir system AA Dor does not show any period variations also confirms our theory that the period variations in HW Vir systems with companions close to the hydrogen-burning limit might be caused by tidal synchronisation. In higher-mass M dwarf companions the larger period variations are likely caused by the Applegate mechanism and the period decrease can be caused dominantly by magnetic braking and additionally tidal synchronisation.
It seems that orbital period changes in HW Vir systems are still poorly understood and have also not been studied observationally to the full extent. More observations over long time spans of synchronised and non-synchronised short-period sdB binaries with companions of different masses will be necessary to understand synchronisation and orbital period changes of hot subdwarf binaries. Most likely it cannot be explained with just one effect and is likely an interplay of different effects.
\subsection{Inflation of brown dwarfs and low-mass M dwarfs in eclipsing WD or sdB binaries}
Close brown dwarf companions that eclipse main sequence stars are rare, with only 23 known to date \citep{carmichael20}. Consequently, brown dwarf companions to the evolved form of these systems are much rarer with only three (including J08205+0008) known to eclipse hot subdwarfs, and three known to eclipse white dwarfs. These evolved systems are old ($>$ 1 Gyr), and the brown dwarfs are massive, and hence not expected to be inflated \citep{thorngren18}.
Surprisingly, of the three hot subdwarfs with brown dwarf companions, J08205+0008 is the one that receives the least irradiation - almost half that received by V2008-1753 and SDSSJ162256.66+473051.1, both of which have hotter primaries (32000~K, 29000~K) and shorter periods ($\sim$1.6~hr) than J08205+0008. This suggests that more irradiation, and more irradiation at shorter wavelengths does not equate to a higher level of inflation of a brown dwarf. Indeed this finding is consistent with that for brown dwarfs irradiated by white dwarfs, where the most irradiated object with a measured radius is SDSS J1205-0242B, in a 71.2 min orbit around a 23681~K white dwarf and yet the brown dwarf is not inflated \citep{parsons17}. The brown dwarf in this system only receives a hundredth of the irradiation that J08205+0008 does. However, WD1032+011, an old white dwarf ($T_{\rm eff} \sim$ 10000~K) with a high mass brown dwarf companion (0.0665 M$_{\odot}$) does appear to be inflated \citep{casewell:2020}. As can be seen from Figure \ref{MR}, the majority of the low mass brown dwarfs (M$<$35 M$_{\rm Jup}$) are inflated, irrelevant of how much irradiation they receive. For the few old (5-10 Gyr), higher mass inflated brown dwarfs, the mechanism leading to the observed inflation is not yet understood.
\begin{figure*}
\centering
\includegraphics[width=1.05\linewidth]{sds_mr_jul31.pdf}
\caption{All known eclipsing binary white dwarfs with detached brown dwarf (triangles: \citealt{parsons17, littlefair14}) and late M dwarf companions (triangles) from \citet{parsons18}, hot subdwarfs with eclipsing brown dwarf companions (circles: \citealt{schaffenroth14a, schaffenroth15}) and all known eclipsing brown dwarf companions to main sequence stars (+: \citealt{carmichael20}). J08205+0008 is plotted as the filled square. The colour is proportional to the effective temperature of the primary in each system and the coloured circle size is proportional to the amount of total incident radiation the secondary receives. Also shown are the Sonora Bobcat brown dwarf evolutionary models of \citet{bobcat} for solar and sub-solar metallicity and the NextGen models \citep{baraffe97}.}
\label{MR}
\end{figure*}
\subsection{Previous and future evolution of the system}
As stated before, stars with a cool, low-mass companion sitting on the EHB are thought to have formed by a common-envelope phase from a progenitor of up to two solar mass on the RGB. Due to the large mass ratio only unstable mass transfer is possible. If the mass transfer happened at the tip of the RGB, a core-helium burning object with about $0.5\,\rm M_\odot$ will be formed. If the mass transfer happened earlier then the core of the progenitor has not enough mass to start He-core burning and the pre-He WD will move to the WD cooling track crossing the EHB. Our analysis of J08205+0008 showed that a low-mass solution ($0.25\,M_\odot$, as discussed previously) can be excluded and that the primary star is indeed currently a core He-burning object.
\citet{kupfer15} calculated the evolution of J08205+0008 and considering only angular momentum loss due to gravitational waves and found that the companion will fill its Roche lobe in about 2.2 Gyrs and mass transfer is expected to start forming a cataclysmic variable.
We detected a significantly higher orbital period decrease in this system than expected from gravitational waves. Up to now, we could not detect any change in the rate of this period decrease. If we assume that the orbital period change is due to rotational period change
until synchronisation is reached and afterwards the period decrease will be solely due to gravitational waves,
we can calculate when the companion will fill its Roche lobe and accretion to the primary will start. To calculate the Roche radius the equation derived in \citet{eggleton83} was used:
\begin{equation}
R_L=\frac{0.49q^{2/3}}{0.6q^{2/3}+\ln(1+q^{1/3})}a
\end{equation}
Using the values derived in our analysis we calculate that the Roche lobe of the companion will be filled at a system separation of 0.410 $\rm R_\odot$, 56\% of the current separation, which is reached at a period of 3525 s.
From this we calculate a time scale of 1.8 Gyrs until the Roche lobe will be filled.
Systems with a mass ratio $q=M_2/M_1<2/3$, with $M_1$ being the mass of the accretor, are assumed to be able to undergo stable mass transfer. Our system has a mass ratio of $0.147\ll2/3$.
The subdwarf will already have evolved to a white dwarf and a cataclysmic variable will be formed. It is expected that the period of an accreting binary with a hydrogen-rich donor star will decrease until a minimum period of $\simeq70$ min is reached at a companion mass around $0.06\,M_\odot$ and the period will increase again afterwards \citep{nelson18}. Such systems are called period bouncers. Our system comes into contact already close to the minimum period and should hence increase the period when the mass transfer starts.
The future of the system depends completely on the period evolution. A longer baseline of observations of this system is necessary to confirm that the period decrease is indeed stable and caused by the tidal synchronisation.
\section{Conclusion and summary}
The analysis of J08205+0008 with higher quality data from ESO-VLT/XSHOOTER, ESO-VLT/UVES and ESO-NTT/ULTRACAM allowed us to constrain the masses of the sdB and the companion much better by combining the analysis of the radial velocity curve and the light curve. We determine an sdB mass of $0.39-0.50\,\rm M_\odot$ consistent with the canonical mass and a companion mass of $0.061-0.071\,\rm M_\odot$ close to the hydrogen burning limit. Therefore, we confirm that the companion is likely be a massive brown dwarf.
The atmospheric parameters and abundances show that J08205+0008 is a typical sdB and comparison with stellar evolution tracks suggest that the mass has to be less than $0.50\,M_\odot$ consistent with our solution and also the mass derived by a spectrophotometric method using Gaia parallaxes and the SED derived in the secondary eclipse, where the companion is not visible.
If the sdB evolved from a $1\,\rm M_\odot$ star, the age of the system is expected to be around 10\, Gyrs. In this case the radius of the brown dwarf companion is about 20\% inflated compared to theoretical calculations. Such an inflation is observed in several sdB/WD+dM/BD systems but not understood yet. However, the inflation seems not to be caused by the strong irradiation.
The sdB binary belongs to the thin disk, as do about half of the sdB at this distance from the Galacic plane. This means that they also could be young, if they have evolved from a more massive progenitor. Then we get a consistent solution without requiring inflation of the companion. However, a brown dwarf companion might not be able to remove the envelope of a more massive progenitor.
We detected a significant period decrease in J0820+0008. This can be explained by the spin-up of the sdB due to tidal sychronisation. We calculated the synchronisation timescale to 4 Myrs well within the lifetime on the EHB. The investigation of the parameters of all known Vir systems with rotational periods (see Sect. \ref{syncro}) shows that the synchronised systems tend to be older, showing that the synchronisation timescale seems to be comparable but smaller than the lifetime on the EHB in contrast to current synchronisation theories.
By investigating the known orbital period variations in HW Vir systems, we can confirm the findings by \citet{bours:16} that period variations in systems with higher mass M dwarf companions seem to be larger. Hence, we conclude that the large period variations in those systems are likely caused by the Applegate mechanism and the observed period decreases dominantly by magnetic braking. In lower-mass companions close to the hydrogen-burning limit, on the other hand, tidal synchronisation spinning up the sdB could be responsible for the period decrease, allowing us to derive a synchronisation timescale.
The results of our analysis are limited by the precision of the available trigonometric parallax. As the Gaia mission proceeds, the precision and accuracy of the trigonometric parallax will improve, which will narrow down the uncertainties of the stellar parameters.
A very important goal is to detect spectral signatures from the companion and to measure the radial velocity curve of the companion. We failed to do so, because the infrared spectra at hand are of insufficient quality. The future IR instrumentation on larger telescopes, such as the ESO-ELT, will be needed. A high precision measurement of the radial velocity curves of both components will then allow us to derive an additional constraint on mass and radius from the difference of the stars' gravitational redshifts \citep{vos13}.
Such measurements will give an independent determination of the nature of the companion and will help to test evolutionary models for low mass star near the hydrogen burning limit via the mass-radius relation.
The combination of many different methods allowed us to constrain the masses of both components much better without having to assume a canonical mass for the sdB. This is only the fourth HW Vir system for which this is possible.
\section*{Acknowledgements}
D.S. is supported by the Deutsche Forschungsgemeinschaft (DFG) under grant HE 1356/70-1 and IR190/1-1.
V.S. is supported by the Deutsche Forschungsgemeinschaft, DFG through grant GE 2506/9-1. S.L.C. is supported by an STFC Ernest Rutherford Fellowship ST/R003726/1.
DK thanks the SAAO for generous allocations of telescope time and the National Research Foundation of South Africa and the University of the Western Cape for financial support. VSD, SPL and ULTRACAM are supported by the STFC.
We thank J. E. Davis for the development of the \texttt{slxfig} module, which has been used to prepare figures in this work. \texttt{matplotlib} \citep{2007CSE.....9...90H} and \texttt{NumPy} \citep{2011CSE....13b..22V} were used in order to prepare figures in this work. This work has made use of data from the European Space Agency (ESA) mission
{\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia}
Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement. Based on observations at the Cerro Paranal Observatory of the European Southern Observatory (ESO) in Chile under the program IDs 087.D-0185(A), and 098.C-0754(A). Based on observations at the La Silla Observatory of the European Southern Observatory (ESO) in Chile under the program IDs 082.D-0649(A), 084.D-0348(A), and 098.D-679.
This paper uses observations made at the South African Astronomical Observatory.
We made extensive use of NASAs Astrophysics Data System Abstract Service (ADS) and the SIMBAD and VizieR database, operated at CDS, Strasbourg, France.
\section*{Data availability statement}
Most data are incorporated into the article and its online supplementary material. All other data are available on request.
\bibliographystyle{mnras}
|
1,116,691,497,802 | arxiv | \section{Conclusion}
\label{sec:conclusion}
This paper describes {{\scshape GeoReach}} a novel approach that evaluates graph reachability queries and spatial range predicates side-by-side. {{\scshape GeoReach}} extends the functionality of a given graph database management system with light-weight spatial indexing entries to efficiently prune the graph traversal based on spatial constraints. {{\scshape GeoReach}} allows users to tune the system performance to achieve both efficiency and scalability. Based on extensive experiments, we show that {{\scshape GeoReach}} can be scalable and query-efficient than existing spatial and reachability indexing approaches in relatively sparse graphs. Even in rather dense graphs, our approach can outperform existing approaches in storage overhead and initialization time and still achieves faster query response time. In the future, we plan to study we plan to study the extensibility of {{\scshape GeoReach}} to support different spatial predicates. Furthermore, we aim to extend the framework to support a distributed system environment. Last but not least, we also plan to study the applicability of {{\scshape GeoReach}} to various application domains including: Spatial Influence Maximization, Location and Social-Aware Recommendation, and Location-Aware Citation Network Analysis.
\section{Experimental Evaluation}
\label{sec:experiment}
In this section, we present a comprehensive experimental evaluation of {{\scshape GeoReach}} performance. We compare the following approaches: {GeoMT0}, {GeoMT2}, {GeoMT3}, {GeoP}, {GeoRMBR} and {SpaReach}. {GeoMT0}, {GeoMT2} and {GeoMT3} are approaches that store only {ReachGrid} by setting {\tt MAX\_REACH\_GRIDS} to the total number of grids in the space and {\tt MAX\_RMBR} to A where A represent the area of the whole 2D space. Their difference lies in the value of {\tt MERGE\_COUNT}. {GeoMT0} is an approach where {\tt MERGE\_COUNT} is 0. In such approach, no higher layer grids are merged. {\tt MERGE\_COUNT} is set to 2 and 3 respectively in {GeoMT2} and {GeoMT3}. {GeoP} is an approach in which {\tt MERGE\_COUNT} = 0, {\tt MAX\_REACH\_GRIDS} = 200 and {\tt MAX\_RMBR} = A. In such approach, reachable grids in {ReachGrid} will not be merged. If the number of reachable grids of {ReachGrid}($v$) is larger than 200 then $v$ will be degraded to an R-vertex. Since {\tt MAX\_RMBR} = A, there will be no B-vertex. In {GeoRMBR}, {\tt MAX\_REACH\_GRIDS} = 0, {\tt MAX\_RMBR} = A, hence only {RMBR} s are stored. In all {ReachGrid} related approaches, the total space is split into $128\times 128$ pieces in the highest grid layer. {SpaReach} approach is implemented with both spatial index and reachability index. Graph structure is stored in Neo4j graph database. Reachability index is stored as attributes of each graph vertex in Neo4j database. Reachability index we use is proposed in~\cite{YAI+2013}. Spatial index used {SpaReach} approaches is implemented by gist index in postgresql. To integrate Neo4j and postgresql databases, for each vertex in the graph, we assign it an id to uniquely identify it.
{\bf Experimental Environment.} The source code for evaluating query response time is implemented in Java and compiled with java-7-openjdk-amd64. Source codes of index construction are implemented in c++ and complied using g++ 4.8.4. Gist index is constructed automatically by using command line in Postgresql shell. All evaluation experiments are run on a computer with an 3.60GHz CPU, 32GB RAM running Ubuntu 14.04 Linux OS.
\begin{table}[t]
\small
\centering
\caption{\small \label{real_datasets} Graph Datasets (K = $10^3$)}
\begin{tabular}{|l|c|c|c|c|}
\hline {\bf Dataset} &$|V|$&$|E|$&$d_{avg}$&$l$\\
\hline citeseerx~~&6540K&15011K&2.30&59\\
\hline go-uniprot~~&6968K&34770K&4.99&21\\
\hline patent&3775K~~&16519K&4.38&32\\
\hline uniprot22m~~&1595K&1595K&1.00&4\\
\hline uniprot100m~~&16087K&16087K&1.00&9\\
\hline uniprot150m~~&25038K&25038K&1.00&10\\
\hline
\end{tabular}
\end{table}
{\bf Datasets.} We evaluate the performance of our methods using six real datasets~\cite{CHS+2013,YAI+2013} (see Table~\ref{real_datasets}). Number of vertices and edges are listed in column $|V|$ and $|E|$. Column $d_{avg}$ and $l$ are average degree of vertices and length of the longest path in the graph, respectively. Citeseerx and patent are real life citation graphs extracted from CiteSeerx\footnote{\scriptsize http://citeseer.ist.psu.edu/} and US patents\footnote{\scriptsize http://snap.stanford.edu/data/}~\cite{YAI+2013}. Go-uniprot is a graph generated from Gene Ontology and annotation files from Uniprot\footnote{\scriptsize http://www.uniprot.org/}~\cite{YAI+2013}. Uniprot22m, uniprot100m and uniprot150m are RDF graphs from UniProt database~\cite{YAI+2013}. The aforementioned datasets represent graphs that possess no spatial attributes. For each graph, we simulate spatial data by assigning a spatial location to a subset of the graph vertices. During the experiments, we change the ratio of spatial vertices to the total number of vertices from 20\% to 80\%. During the experiments, we vary the spatial distribution to be: uniform, zipf, and clustered distributions. Unless mentioned otherwise, the number of spatial clusters is set to 4 by default.
\subsection{Query Response Time}
In this section, we fist compare the query response time performance of {SpaReach} to our {GeoP} approach. Afterwards, we change tunable parameters in {{\scshape GeoReach}} to evaluate influence of these thresholds. For each dataset, we change the spatial selectivity of the input query rectangle from 0.0001 to 0.1. For each query spatial selectivity, we randomly generate 500 queries by randomly selecting 500 random vertices and 500 random spatial locations of the query rectangle. The reported query response time is calculated as the average time taken to answer the 500 queries.
\begin{figure*}
\centering
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{uni150m_20}
\caption{uniprot150m}
\label{fig:query_time_uniprot150m}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{Patents_20}
\caption{patent}
\label{fig:query_time_Patents}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{go_uniprot_20}
\caption{go-uniprot}
\label{fig:query_time_go_uniprot}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{citeseerx_20}
\caption{citeseerx}
\label{fig:query_time_citeseerx}
\end{subfigure}
\caption{\small Query response time (80\% spatial vertex ratio, randomly-distributed spatial data, and spatial selectivity ranging from 0.0001 to 0.1)}
\label{fig:query_time}
\end{figure*}
Figure~\ref{fig:query_time} depicts the query response time of {GeoP} and {SpaReach} on four datasets. 80\% of vertices in the graph are spatial and they are randomly-distributed in space. For brevity, we omit the results of the other two datasets, i.e., {uniprot22m} and {uniprot100m}, since they have almost the same graph structure and exihibit the same performance. As it turns out In Figure~\ref{fig:query_time}, {GeoP} outperforms {SpaReach} for any query spatial selectivity in {uniprot150m}, {go-uniprot} and {citeseerx}. For these datasets, {SpaReach} approach cost more time when query selectivity increases. When we increasing the query range size, the range query step tends to return a larger number of spatial vertices. Hence, the graph reachability checking step has to check more spatial vertices. Figure~\ref{fig:query_time_go_uniprot} and~\ref{fig:query_time_citeseerx} show similar experiment results. In conclusion, {GeoP} is much more query-efficient in relatively sparse graphs. Patent dataset is the densest graph with richer reachability information. Figure~\ref{fig:query_time_Patents} indicates that even when spatial selectivity set to 0.0001, {GeoP} can achieve almost the same performance as {SpaReach}. When spatial selectivity increases, {GeoP} outperforms {SpaReach} again. In a denser graph, the performance difference between the two approaches is smaller than in sparse graphs especially when the spatial selectivity is low.
\iffalse
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.44\textwidth]{figures/query_time/uniprotenc150m_20_ours}
\caption{\small Query Reponse Time in {\bf uniprot150m}, 80\% spatial vertex ratio, and spatial selectivity ranging from 0.0001 to 0.1}
\label{fig:query_time_uniprotec150_ours}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.44\textwidth]{figures/query_time/Patent_20_ours}
\caption{\small Query Response Time in {\bf Patents}, 80\% spatial vertex ratio, and spatial selectivity ranging from 0.0001 to 0.1}
\label{fig:query_time_patent_ours}
\end{center}
\end{figure}
\fi
\begin{table*}[t]
\small
\captionsetup{font = small}
\caption{\small \label{query_time_ours} Query Response Time in three datasets, 80\% spatial vertex ratio, and spatial selectivity ranging from 0.0001 to 0.1}
\centering
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
&\multicolumn{5}{|c|}{uniprot150m}&\multicolumn{5}{|c|}{patent}&\multicolumn{5}{|c|}{citeseerx}\\
\hline Selectivity&MT0&MT2&MT3&GeoP&RMBR&MT0&MT2&MT3&GeoP&RMBR&MT0&MT2&MT3&GeoP&RMBR\\
\hline 0.0001&68&68&67&66&66&643&762&741&1570&2991&202&212&203&210&234\\
\hline 0.001&65&77&78&66&65&168&258&185&559&1965&34&460&471&207&215\\
\hline 0.01&66&66&65&65&65&87&143&98&217&915&32&408&410&189&200\\
\hline 0.1&69&65&65&75&66&51&108&59&155&348&33&399&399&160&183\\
\hline
\end{tabular}
\end{table*}
Table~\ref{query_time_ours} compares the query response time of all our approaches for the {uniprot150m}, {patent} {go-uniprot} and {citeseerx} datasets with randomly distributed spatial vertices and spatial ratio of 80\%. In {uniprot150m}, all our approaches almost have the same performance. The same pattern happens with the {uniprot22m}, {uniprot100m} and {go-uniprot} datasets. So we use {uniprot150m} as a representative.
For the {patent} graph with random-distributed spatial vertices and spatial ratio of 20\%, query efficiency difference can be easily caught. {GeoMT0} keeps information of exact reachable grids of every vertex which brings us fast query speed, but also the highest storage overhead. {RMBR} stores general spatial boundary of reachable vertices which is the most scalable. However, such approach spend the most time in answering the query. Since {GeoMT3} is an approach that {\tt MERGE\_COUNT} is set to 3, just few grids in {GeoMT3} are merged. As a result, its query time is merely little bit longer than {GeoMT0}. There are more grids getting merged in {GeoMT2} than in {GeoMT3}. Inaccuracy caused by more integration lowers efficiency of {GeoMT3} in query. {GeoP} is combination of {ReachGrid} and {RMBR}. Its query efficiency is lower than {GeoMT0} and better than {GeoRMBR}. In this case, {GeoMT2} outperforms {GeoP}. But it is not always the case. By tuning {\tt MAX\_REACH\_GRIDS} to a larger number, {GeoP} can be more efficient in query.
In {citeseerx}, {GeoMT0} keeps the best performance as expected. Performance of {GeoP} is in between {GeoMT0} and {GeoRMBR} as what is shown in patent. But {GeoMT2} and {GeoMT3} reveal almost the same efficiency and they are worse than {GeoRMBR}. Distinct polarized graph structure accounts for the abnormal appearance. In {citeseerx}, all vertices can be divided into two groups. One group consists of vertices that cannot reach any vertex. The other group contains a what we call center vertex. The center vertex has huge number of out-edge neighbor vertices and is connected by huge number of vertices as well. Because the center vertex can reach that many vertices, it can reach nearly all grid cells in space. As a result, vertices that can reach the center vertex can also reach all grid cells in space. So no matter what value is {\tt MAX\_REACH\_GRIDS}, reachable grids in {ReachGrid} of these vertices will be merged into only one grid in a lower layer until to the bottom layer which is the whole space. Then such {ReachGrid} can merely function as a {GeoB} which owns poorer locality than {RMBR}.
\subsection{Storage Overhead}
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{uniprotenc_150_index_size}
\caption{uniprot150m}
\label{fig:index_size_uni150}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{Patent_index_size_random}
\caption{patent}
\label{fig:index_size_patent_random}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{go_uniprot_index_size}
\caption{go-uniprot}
\label{fig:index_size_go_uniprot}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{citeseerx_index_size}
\caption{citeseerx}
\label{fig:index_size_citeseerx}
\end{subfigure}
\caption{\small Storage Overhead (Randomly distributed, spatial vertex ratio from 0.8 to 0.2)}
\label{fig:index_size}
\end{figure*}
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{uni150_index_size_vary_distribution}
\caption{uniprot150m}
\label{fig:index_size_uni150_vary_distribution}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{Patent_index_size_vary_distribution}
\caption{patent}
\label{fig:index_size_patent_vary_distribution}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{go_uniprot_index_size_vary_distribution}
\caption{go-uniprot}
\label{fig:index_size_go_uniprot_vary_distribution}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{citeseerx_index_size_vary_distribution}
\caption{citeseerx}
\label{fig:index_size_citeseerx_vary_distribution}
\end{subfigure}
\caption{\small Storage Overhead for varying spatial data distribution (randomly, cluster and zipf distributed) and 0.8 spatial vertex ratio)}
\label{fig:index_size_vary_distribution}
\end{figure*}
Figure~\ref{fig:index_size_uni150} gives the storage overhead of all approaches for the {uniprot150m} dataset. In this experiment, the spatial vertices are randomly distributed in space. Since {uniprot22m} and {uniprot100m} share the same pattern with {uniprot150m} (even spatial distribution of vertices varies), they are not shown in the figure. The experiments show that {{\scshape GeoReach}} and all its variants require less storage overhead than {SpaReach} because of the additional overhead introduced by the spatial index. When there are less spatial vertices, {SpaReach} obviously occupies less space because size of spatial index lessens. However, {SpaReach} always requires more storage than any other approaches. Storage overhead of {{\scshape GeoReach}} approaches shows a two-stages pattern which means it is either very high (ratio = 0.8, 0.6 and 0.4) or very low (ratio = 0.2). The reason is as follows. These graphs are sparse and almost all vertices reach the same vertex. This vertex cannot reach any other vertex. Let us call it an end vertex. If the end vertex is a spatial vertex, then all vertices that can reach the end vertex will keep their spatial reachability information (no matter what category they are) in storage. But if it is not, majority of vertices will store nothing for spatial reachability information. {GeoMT0} and {GeoP} are of almost the same index size because of sparsity and end-point phenomenon in these graphs. Such characteristic causes that almost each vertex can just reach only one grid which makes {\tt MAX\_REACH\_GRIDS} invalid in approach {GeoMT0} (number of reachable grids is always less than {\tt MAX\_REACH\_GRIDS}) which makes {GeoMT0} and {GeoP} have nearly the same size. For similar reason, {\tt MERGE\_COUNT} becomes invalid in these datasets which makes {GeoMT2} and {GeoMT3} share the same index size with {GeoMT0} and {GeoP}. We also find out that index size of {GeoRMBR} is slightly larger than {GeoMT0} approaches. Intuitively, {RMBR} should be more scalable than {ReachGrid}. But most of the vertices in these three graphs can reach only one grid. In {GeoRMBR}, for each vertex that have reachable spatial vertices, we assign an {RMBR} which will be stored as coordinates of {RMBR}'s top-left and lower-right points. It is more scalable to store one grid id than two coordinates. So when a graph is highly sparse, index size of {GeoMT0} is possible to be less than {GeoRMBR}.
Figure~\ref{fig:index_size_go_uniprot} shows that in {go-uniprot} all {{\scshape GeoReach}} approaches performs better than {SpaReach}. When we compare all the {{\scshape GeoReach}} approaches, {GeoMT0}, {GeoMT2} and {GeoMT3} lead to almost the same storage overhead. That happens due to the fact that {go-uniprot} is a very sparse graph. A vertex can only reach few grids in the whole space. Grid cells in {ReachGrid} can hardly be spatially adjacent to each other which causes no integration. The graph sparsity makes the number of reachable grids in {ReachGrid} always less than {\tt MAX\_REACH\_GRIDS} which leads to less R-vertices and more G-vertices. In consequence, {go-uniprot}, {GeoMT0}, {GeoMT2}, {GeoMT3} and {GeoP} lead to the same storage overhead. It is rational that {GeoRMBR} requires the least storage because {RMBR} occupies less storage than {ReachGrid}.
When graphs are denser, results become more complex. Figure~\ref{fig:index_size_patent_random} shows index size of different approaches in patent dataset with randomly-distributed spatial vertices. {GeoRMBR} and {GeoP}, take the first and the second least storage and are far less than other approaches because both of them use {RMBR} which is more scalable. {GeoMT0} takes the most storage in all spatial ratios for that {ReachGrid} takes high storage overhead. {GeoMT2} and {GeoMT3} require less storage than {GeoMT0} because spatially-adjacent reachable grids in {GeoMT0} are merged which brings us scalability. {GeoMT3} are more scalable than {GeoMT2} because {\tt MERGE\_COUNT} in {GeoMT2} is 2 which causes more integration. There are three approaches, {GeoMT3}, {GeoP} and {GeoRMBR}, that outperform {SpaReach} approach. By tuning parameters in {{\scshape GeoReach}}, we are able achieve different performance in storage overhead and can also outperform {SpaReach}.
Figure~\ref{fig:index_size_citeseerx} depicts index size of all approaches in {citeseerx} with randomly distributed spatial vertices. Spatial vertices ratio ranges from 0.8 to 0.2. All {{\scshape GeoReach}} approaches outperform {SpaReach} except for one outlier when spatial vertices ratio is 0.2. {GeoMT0} consumes huge storage. This is caused by the center vertex which is above-mentioned. Recall that large proportion of {ReachGrid} contains almost all grids in space. After bitmap compression, it will cause low storage overhead. This is why when spatial vertices ratio is 0.8, 0.6 and 0.4, {GeoMT0} consumes small size of index. When the ratio is 0.2, there are less spatial vertices. Although graph structure does not change, the center vertex reach less spatial vertices and less grids. Then the bitmap compression brings no advantage in storage overhead.
Figure~\ref{fig:index_size_vary_distribution} shows the impact of spatial data distribution on the storage cost. {GeoMT0}, {GeoMT2} and {GeoMT3} are all {ReachGrid}-based approaches. Spatial data distribution of vertices influences all approaches the same way. For all datasets, {SpaReach} is not influenced by the spatial data distribution. {SpaReach} consists of two sections: (1)~The reachability index size is determined by graph structure and (2)~The spatial index size is directly determined by number of spatial vertices. Hence, {SpaReach} exhibits the same storage overhead for different spatial data distributions. When spatial vertices distribution varies, {GeoRMBR} also keeps stable storage overhead. This is due to the fact that the storage overhead for each {RMBR} is a constant and the number of stored {RMBR} s is determined by the graph structure and spatial vertices ratio, and not by the spatial vertices distribution. Spatial data distribution can only influence the shape of each {RMBR}.
Figure~\ref{fig:index_size_uni150_vary_distribution} shows that each approach in {{\scshape GeoReach}} keeps the same storage overhead under different distributions in {uniprot150m}. As mentioned before, {GeoMT0}, {GeoMT2}, {GeoMT3} and {GeoP} actually represent the same data structure since there is only a single reachable grid in {ReachGrid}. When there is only one grid reachable, varying the spatial distribution becomes invalid for all approaches which use {ReachGrid}.
Figure~\ref{fig:index_size_patent_vary_distribution} and~\ref{fig:index_size_go_uniprot_vary_distribution} shows that the storage overhead introduced by {ReachGrid}-based approaches decreases when spatial vertices become more congested. Randomly distributed spatial data is the least congested while zipf distributed is the most. The number of reachable spatial vertices from each vertex do not change but these reachable spatial vertices become more concentrated in space. This leads to less reachable grids in {ReachGrid}.
Figure~\ref{fig:index_size_citeseerx_vary_distribution} shows that when spatial vertices are more congested, {ReachGrid} based approaches, i.e., {GeoMT0}, {GeoMT2} and {GeoMT3}, tend to be less scalable. Recall that {citeseerx} dataset is a polarized graph with a center vertex. One group contains vertices that can reach huge number of vertices (about 200,000) due to the center vertex. When spatial vertices are more concentrated and that will lead to more storage overhead.
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{Uniprotenc_150m_random}
\caption{uniprot150}
\label{fig:ini_time_uniprotenc_150}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{Patent_random}
\caption{patent}
\label{fig:ini_time_Patent}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{go_uniprot_random}
\caption{go-uniprot}
\label{fig:ini_time_go_uniprot}
\end{subfigure}
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{citeseerx_random}
\caption{citeseerx}
\label{fig:ini_time_citeseerx}
\end{subfigure}
\caption{\small Initialization time (Randomly distributed, spatial vertex ratio from 0.8 to 0.2)}\label{fig:ini_time}
\end{figure*}
\subsection{Initialization time}
In this section, we evaluate the index initialization time for all considered approaches. For brevity, we only show the performance results for four datasets, {uniprot150m}, {patent}, {go-uniprot} and {citeseerx}, since {uniprot22m}, {uniprot100m} and {uniprot150m} datasets exhibit the same performance. Figure~\ref{fig:ini_time_uniprotenc_150} shows that {SpaReach} requires much more construction time than the other approaches under all spatial ratios. Although these graphs are sparse, they contain large number of vertices. This characteristic causes huge overhead in constructing a spatial index which dominates the initialization time in {SpaReach}. Hence, {SpaReach} takes much more time than all other approaches. However, the {SpaReach} initialization time decreases when decreasing the number spatial vertices since the spatial index building step deals with less spatial vertices in such case. However, {SpaReach} remains the worst even when the spatial vertex ratio is set to 20\%.
Figures~\ref{fig:ini_time_Patent} and~\ref{fig:ini_time_citeseerx} gives the initialization time for both the {patent} and {citeseerx} datasets, respectively. {GeoRMBR} takes significantly less initialization time compared to all other approaches. {GeoP} takes less time than the rest of approaches because it is {ReachGrid} of partial vertices whose number of reachable grids are less than {\tt MAX\_REACH\_GRIDS} that are calculated. In most cases, {GeoMT0} can achieve almost equal or better performance compared to {SpaReach} while {GeoMT2} and {GeoMT3} requires more time due to the integration of adjacent reachable grids. To sum up, {GeoRMBR} and {GeoP} perform much better than {SpaReach} in initialization even in very dense graphs. {GeoMT0} can keep almost the same performance with {SpaReach} approach.
Figure~\ref{fig:ini_time_go_uniprot} shows the initialization time for all six approaches on the {go-uniprot} dataset. Both {RMBR} approaches, i.e., {GeoRMBR} and {GeoP}, still outperform {SpaReach}. This is due to the fact that a spatial index constitutes a high proportion of {SpaReach} initialization time. As opposed to the {uniprot150m} case, the smaller performance gap between initializing {GeoRMBR} and {SpaReach} in {go-uniprot}.is explained as follows. The size of go-uniprot is far less than {uniprot150m} which decreases the spatial index initialization cost. As a result, the index construction time in {SpaReach} is less than that in {uniprot150m}. Since this graph has more reachability information, all {{\scshape GeoReach}} approaches require more time than in {uniprot150m}. It is conjunction of {{\scshape GeoReach}} and {SpaReach} index size changes that causes the smaller gap.
\subsection{Data Structure}
\label{sec:index_overview}
In this section, we explain how {{\scshape GeoReach}} augments a graph structure with spatial indexing entries to form what we call SP\lowercase{atially}-A\lowercase{ugmented} Graph ({{SPA-Graph}}). To be generic, {\scshape GeoReach} stores the newly added spatial indexing entries the same way other properties are stored in a graph database system. The structure of a {{SPA-Graph}} is similar to that of the original graph except that each vertex $v \in V$ in a {{SPA-Graph}} $G=\{V,E\}$ stores spatial reachability information. A {{SPA-Graph}} has three different types of vertices, described as follows:
\begin{myitem}
\item {\bf B-Vertex:} a B-Vertex $v$ ($v \in V$) stores an extra bit (i.e., boolean), called Spatial Reachability Bit (abbr. GeoB) that determines whether $v$ can reach any spatial vertex ($u \in V_S$) in the graph. {GeoB} of a vertex $v$ is set to 1 (i.e., true) in case $v$ can reach at least one spatial vertex in the graph and reset to 0 (i.e., false) otherwise.
\item {\bf R-Vertex:} an R-Vertex $v$ ($v \in V$) stores an additional attribute, namely Reachability Minimum Bounding Rectangle (abbr. RMBR($v$)). RMBR($v$) represents the minimum bounding rectangle {MBR}($S$) (represented by a top-left and a lower-right corner point) that encloses all spatial polygons which represent all spatial vertices $S$ that are reachable from vertex $v$ ({RMBR}($v$) = {MBR}($RF_S(v)$), $RF_S(v)$ = $\{u|v \leadsto u, u \in V_S\}$).
\item {\bf G-Vertex:} a G-Vertex $v$ stores a list of spatial grid cells, called the reachability grid list (abbr. {ReachGrid}($v$)). Each grid cell $C$ in {ReachGrid}($v$) belongs to a hierarchical grid data structure that splits the total physical space into $n$ spatial grid cells. Each spatial vertex $u \in V_S$ will be assigned a unique cell ID $(k\in [1,n])$ in case $u$ is located within the extents of cell $k$, noted as {Grid}$(u)=k$. Each cell $C \in$ {ReachGrid}($v$) contains at least one spatial vertex that is reachable from $v$ ({ReachGrid}($v$) = $\cup~${Grid}($u$), $\{u|v\leadsto u,~{u \in {V_S}}\}$).
\end{myitem}
\begin{lemma}
\label{lemma:RMBR}
Let $v$ ($v\in V$) be a vertex in a {{SPA-Graph}} $G=\{V,E\}$ and $V_{v}^{out}$ be the set of vertices that can be reached via a direct edge from a vertex $v$. The reachability minimum bounding rectangle of $v$ ({RMBR}($v$)) is equivalent to the minimum bounding rectangle that encloses all its out-edge neighbors $V_{v}^{out}$ and their reachability minimum bounding rectangles. {\small {RMBR}($v$) = {MBR}$_{v'\in V_{v}^{out}}$({RMBR}($v'$), $v'.spatial$)}.
\end{lemma}
\begin{IEEEproof}
Based on the reachability definition, the set of reachable vertices $RF(v)$ from a vertex $v$ is equal to the union of the set of vertices that is reached from $v$ via a direct edge ($V_{v}^{out}$) and all vertices that are reached from each vertex {\small$v^{'} \in V_{v}^{out}$}. Hence, the set ($RF_S(v)$) of reachable spatial vertices from $v$ is given in Equation~\ref{eq:reachablespatialvertices}.
\begin{small}
\begin{equation}
\label{eq:reachablespatialvertices}
RF_S(v) = \bigcup\limits_{v^{'}\in V_{v}^{out}} (v' \cup RF_S(v^{'}))
\end{equation}
\end{small}
And since {\small {RMBR}($v$) = {MBR}($RF_S(v)$)}, then the the reachability minimum bounding rectangle of $v$ is as follows:
\begin{small}
\begin{equation}
\begin{split}
{RMBR}(v) & = {MBR}(\bigcup\limits_{v^{'}\in V_{v}^{out}} (v' \cup RF_S(v^{'})))) \\
& = {MBR}_{v'\in V_{v}^{out}}(RMBR(v'), v'.spatial)
\end{split}
\end{equation}
\end{small}
That concludes the proof.
\end{IEEEproof}
\begin{lemma}
\label{lemma:ReachGrid}
The set of reachable spatial grid cells from a given vertex $v$ is equal to the union of all spatial grid cells reached from its all its out-edge neighbors and grid cells that contain the spatial neighbors
\begin{small}
\begin{equation}
ReachGrid(v) = \bigcup\limits_{v'\in V_v^{out}}(ReachGrid(v') \cup Grid(v'))
\end{equation}
\end{small}
\end{lemma}
\begin{IEEEproof}
\label{lemma:RMBR}
Similar to that of Lemma~\ref{lemma:RMBR}.
\end{IEEEproof}
{\bf Example.} Figure~\ref{fig:graph_overview} gives an example of a {{SPA-Graph}}. {GeoB} of vertex $b$ is set to $1$ (true) since $b$ can reach three spatial vertices $e$, $f$ and $h$. {GeoB} for $d$ is $0$ since $d$ cannot reach any spatial vertex in the graph. Figure~\ref{fig:graph_overview} also gives an example of a Reachability Minimum Bounding Rectangle {RMBR} of vertex $j$ (i.e., {RMBR}($j$)). All reachable spatial vertices from $j$ are $g$, $i$, $h$ and $f$. Figure~\ref{fig:graph_overview} also depicts an example of {ReachGrid}. There are three layers of grids, denoted as $L_0$, $L_1$, $L_2$ from top to bottom. The uppermost layer $L_0$ is split into $4\times4$ grid cells; each cell is assigned a unique id from $1$ to $16$. We denote grid cell with id $1$ as $G_1$ for brevity. The middle layer gird $L_1$ is split into four cells $G_{17}$ to $G_{20}$. Each cell in $L_1$ covers four times larger space than each cell in $L_0$. $G_{17}$ in $L_1$ covers exactly the same area of $G_1$, $G_2$, $G_5$, $G_6$ in $L_0$. The bottom layer $L_2$ contains only a single grid cell which covers all four grids in $L_1$ and represents the whole physical space. All spatial vertices reachable from vertex $a$ are located in $G_2$, $G_7$, $G_9$, $G_{12}$ and $G_{14}$, respectively. Hence, {ReachGrid}($a$) can be $\{2,7,9,12,14\}$. Notice that vertex $e$ and $f$ are both located in $G_9$ and $G_{14}$ covered by $G_{19}$ in {ReachGrid}($a$) can be replaced by $G_{19}$. Then, {ReachGrid}($a$) = $\{2,7,12,19\}$. In fact, there exist more options to represent {ReachGrid}($a$), such as $\{17,18,19,20\}$ or $\{21\}$ by merging into only a single grid cell in $L_2$. When we look into {ReachGrid} of connected vertices, for instance $g$, {ReachGrid}($g$) is $\{12, 14\}$ and {ReachGrid}($i$) is $\{14\}$. It is easy to verify that {ReachGrid}($g$) is {ReachGrid}($i$)$\cup${Grid}($i.spatial$), which accords with lemma~\ref{lemma:ReachGrid}.
{\bf {SPA-Graph} Intuition.} The main idea behind the {{SPA-Graph}} is to leverage the spatial reachability bit, reachability minimum bounding rectangle and reachability grid list stored in a B-Vertex, R-Vertex or a G-Vertex to prune graph paths that are guaranteed (or not) to satisfy both the spatial range predicate and the reachability condition. That way, {{\scshape GeoReach}} cuts down the number of traversed graph vertices and edges and hence significantly reduce the overall latency of a {\small \sffamily RangeReach} query.
\section{Initialization \& Maintenance}
\section{Initialization \& Maintenance}
\label{sec:initialization}
\begin{algorithm}[t]
\caption{{{{\scshape GeoReach}}} Initialization Algorithm}
\begin{scriptsize}
\label{alg:GeoReach_construct}
\begin{algorithmic}[1]
\STATE {\bf Function} \textsc{Initialize}(Graph $G=\{V,E\}$)
\STATE {/*PHASE I: SPA-Graph Vertex Initialization */}
\FOR {{\bf each} Vertex $v$ $\in$ $V$ according their sequence in topology}
\STATE {{\bf InitializeVertex}($G$, $v$, {\tt MAX\_REACH\_GRIDS}, {\tt MAX\_RMBR})}
\ENDFOR
\STATE {/* PHASE II: Reachable Grid Cells Merging */}
\label{line:phaseII:strat}
\FOR {{\bf each} G-vertex $v$}
\FOR {{\bf each} layer $L_i$ from $L_1$ to $L_{bottom}$}
\FOR {{\bf each} grid cell $G_i$ in $L_i$}
\IF {Number of reachable grids in corresponding region in $L_{i-1}$ is larger than {\tt MERGE\_COUNT}}
\STATE {Add $G_i$ in $L_i$ into {ReachGrid}($v$)}
\STATE {Remove reachable grid cells that are covered by $G_i$ in higher layers}
\ENDIF
\ENDFOR
\ENDFOR
\ENDFOR
\label{line:phaseII:end}
\end{algorithmic}
\end{scriptsize}
\end{algorithm}
This section describes the {{SPA-Graph}} initialization algorithm. The {{\scshape GeoReach}} initialization algorithm (Pseudocode is given in Algorithm~\ref{alg:GeoReach_construct}) takes as input a graph Graph $G=\{V,E\}$ and runs in two main phases: {\em (1)~Phase~I: SPA-Graph Vertex Type Initialization:} this phase leverages the tradeoff between query response time and storage overhead explained in Section~\ref{sec:spagraphanalysis} to determine the type of each vertex. {\em (2)~Phase~II: Reachable Grid Cells Merging}: This step further reduces the storage overhead of each G-Vertex in the SPA-Graph by merging a set of grid cells into a single grid cell. Details of each phase are described in Section~\ref{subsec:vetextypeinit} and~\ref{subsec:cellmerging}
\subsection{SPA-Graph Vertex Type Initialization}
\label{subsec:vetextypeinit}
\begin{algorithm}[t]
\caption{SPA-Graph Vertex Initialization Algorithm}
\begin{scriptsize}
\label{alg:GeoReach_Vertex_Init}
\begin{algorithmic}[1]
\STATE {\bf Function} \textsc{InitializeVertex}(Graph $G=\{V,E\}$, Vertex $v$)
\STATE {{\em Type} $\leftarrow$ {\bf InitializeType(v)}}
\SWITCH {\emph{Type}}
\CASE{B-vertex}
\STATE {Set $v$ B-vertex and {GeoB}($v$) = \emph{true}}
\end{ALC@g}
\CASE{G-vertex}
\STATE {{ReachGrid}($v$) $\leftarrow$ $\emptyset$}
\FOR {{\bf each} Vertex $v{'}$ $\in$ $V_{v}^{out}$}
\STATE {\textbf{Maintain-GVertex}($v$, $v'$)}
\IF {Number of grids in {ReachGrid}($v$) > {\tt MAX\_REACH\_GRIDS}}
\STATE {Set $v$ R-vertex} and {\bf break}
\ENDIF
\ENDFOR
\STATE {\emph{Type} $\leftarrow$ R-vertex}
\IF {Number of grids in {ReachGrid}($v$) = 0}
\STATE {Set $v$ B-vertex, {GeoB}($v$)} $\leftarrow$ \emph{false} and {\textbf{break}}
\ENDIF
\end{ALC@g}
\CASE{R-vertex}
\STATE {{RMBR}($v$) $\leftarrow$ $\emptyset$}
\FOR {{\bf each} Vertex $v{'}$ $\in$ $V_{v}^{out}$}
\STATE {\textbf{Maintain-RVertex}($v$, $v'$)}
\IF {Area({RMBR}($v$)) > {\tt MAX\_RMBR}}
\STATE {Set $v$ B-vertex, {GeoB}($v$) $\leftarrow$ \emph{true}} and {{\bf break}}
\ENDIF
\ENDFOR
\end{ALC@g}
\STATE \textbf{end switch}
\end{algorithmic}
\end{scriptsize}
\end{algorithm}
To determine the type of each vertex, the initialization algorithm takes into account the following system parameters:
\begin{myitem}
\item {\bf {\tt MAX\_RMBR}}: This parameter represents a threshold that limits space area of each {RMBR}. If a vertex $v$ is an R-vertex, area of {RMBR}($v$) cannot be larger than {\tt MAX\_RMBR}. Otherwise, $v$ will be degraded to a B-vertex.
\item {\bf {\tt MAX\_REACH\_GRIDS}}: This parameter sets up the maximum number of grid cells in each {ReachGrid}. If a vertex $v$ is a G-vertex, number of grid cells in {ReachGrid}($v$) cannot exceed {\tt MAX\_REACH\_GRIDS}. Otherwise, $v$ will be degraded to an R-vertex.
\end{myitem}
Algorithm~\ref{alg:GeoReach_Vertex_Init} gives the pseudocode of the vertex initialization algorithm. Vertices are processed based on their topological sequence in the graph. For each vertex, the algorithm first determines the initial vertex type using the {InitializeType} function (pseudocode omitted for brevity). For a vertex $v$, categories of vertex $v'$ ($\{v'|~v'\in V_{v}^{out}\}$) will be checked. If there is any B-vertex $v'$ with {GeoB}($v'$) = \emph{true}, $v$ is directly initialized to a B-vertex with {GeoB}($v$) = \emph{true}. Otherwise, if there is any R-vertex, the function will return an R-vertex type, which means that $v$ is initialized to R-vertex. If either of the above happens, the function returns G-vertex type. Based on the initial vertex type, the algorithm may encounter one of the following three cases:
{\bf Case~I (B-vertex):} The algorithm directly sets $v$ as a B-vertex and {GeoB}($v$) = \emph{true} because there must exist one out-edge neighbor $v'$ of $v$ such that {GeoB}($v'$) = \emph{true}.
{\bf Case~III (R-vertex):} For each $v'$ ($v' \in V_v^{out}$), the algorithm calls the {Maintain-RVertex} algorithm. Algorithm~\ref{alg:Ini_R} shows the pseudocode of the {Maintain-RVertex} algorithm. {Maintain-RVertex} aggregates {RMBR} information. After each aggregation step, area of {RMBR}($v$) will be compared with {\tt MAX\_RMBR}: In case the area of {RMBR}($v$) is larger than {\tt MAX\_RMBR}, the algorithm sets $v$ to be a B-vertex with a true GeoB value and terminates. When $v'$ is either a G-vertex or an R-vertex, the algorithm uses the new bounding rectangle returned from {MBR}({RMBR}($v$), {RMBR}($v'$), $v'.spatial$) to update the current {RMBR}($v$). The algorithm calculates the {RMBR} of a G-vertex in case~III. In case $v'$ is a B-vertex, {GeoB}($v'$) must be reset to false. The algorithm then updates {RMBR}($v$) to {MBR}({RMBR}($v$), $v.spatial$).
\begin{algorithm}[t]
\caption{Maintain R-vertex}
\begin{scriptsize}
\label{alg:Ini_R}
\begin{algorithmic}[1]
\STATE {\bf Function} \textsc{Maintain-RVertex}(From-side vertex $v$, To-side vertex $v'$)
\SWITCH {Type of $v'$}
\CASE {B-vertex}
\IF {{GeoB}($v'$) = true}
\STATE {Set $v'$ B-vertex}~and~{{GeoB}($v$) $\leftarrow$ \emph{true}}
\ELSIF {{RMBR}($v$) fully contains {MBR}($v'.spatial$)}
\RETURN {\emph{false}}
\ELSE
\STATE {{RMBR}($v$) $\leftarrow$ {MBR}({RMBR}($v$), $v'.spatial$)}
\ENDIF
\end{ALC@g}
\CASE {R-vertex}
\IF {{RMBR}($v$) fully contains {MBR}({RMBR}($v'$), $v'.spatial$)}
\RETURN {\emph{false}}
\ELSE
\STATE {{RMBR}($v$) $\leftarrow$ {MBR}({RMBR}($v$), {RMBR}($v'$), $v'.spatial$)}
\ENDIF
\end{ALC@g}
\CASE {G-vertex}
\IF {{RMBR}($v$) fully contains {MBR}({RMBR}($v'$), $v'.spatial$)}
\RETURN {\emph{false}}
\ELSE
\STATE {{RMBR}($v$) $\leftarrow$ {MBR}({RMBR}($v$), {RMBR}($v'$), $v'.spatial$)}
\ENDIF
\end{ALC@g}
\STATE \textbf{end switch}
\RETURN {\emph{true}}
\end{algorithmic}
\end{scriptsize}
\end{algorithm}
{\bf Case~II (G-vertex):} For each vertex $v'$ ($v' \in V_v^{out}$), {Maintain-GVertex} (pseudocode omitted for the sake of space) is invoked to calculate the {ReachGrid} of $v'$.
In case $v'$ is a B-vertex with {GeoB}($v'$) = \emph{false} and $v'$ is a spatial vertex, the grid cell that contains the location of $v'$ will be added into {ReachGrid}($v$). If $v'$ is a G-vertex, all grid cells in {ReachGrid}($v'$) and {Grid}($v'.spatial$) will be added into {ReachGrid}($v$). It does not matter whether $v'$ is a spatial vertex or not. If $v'$ is not a spatial vertex, {Grid}($v'.spatial$) is $\emptyset$. After accumulating information from each neighbor $v'$, the algorithm changes the type of $v$ to R-vertex immediately in case the number of reachable grid cells in {ReachGrid}($v$) is larger than {\tt MAX\_REACH\_GRIDS}. Therefore, the algorithm sets the \emph{Type} to R-vertex since {RMBR}($v$) should be calculated for possible future usage, e.g. {RMBR} of in-edge neighbors of $v$(it will be shown in R-vertex case).
{\bf Example.} Figure~\ref{fig:graph_overview} depicts a {{SPA-Graph}} with {\tt MAX\_RMBR} = 0.8A and {\tt MAX\_REACH\_GRIDS} = 4, where A is area of the whole space. Each vertex is attached with some information and affiliated to one category of {{{\scshape GeoReach}}} index. Their affiliations are listed in the figure. It is obvious that those vertices which cannot reach any spatial vertices will be stored as B-vertex and have a false boolean {GeoB} value to represent such condition. Vertices $d$, $f$, $h$, $i$, $j$ and $k$ are assigned a false value. Other vertices are G-vertex initially. {ReachGrid}($a$) = $\{2,7,9,12,14\}$, {ReachGrid}($b$) = $\{2,9,14\}$, {ReachGrid}($c$) = $\{12, 14\}$, {ReachGrid}($e$) = $\{14\}$, {ReachGrid}($g$) = $\{12,14\}$, {ReachGrid}($i$) = $\{14\}$, {ReachGrid}($j$) = $\{2,7,12,14\}$ and {ReachGrid}($l$) = $\{2\}$. Because of {\tt MAX\_REACH\_GRIDS}, some of them will be degraded to an R-vertex. Number of reachable grids in {ReachGrid}($a$) and {ReachGrid}($j$) are 4 and 5, respectively. Both of them are larger than or equal to {\tt MERGE\_COUNT}. They will be degraded to R-vertex first. Then area of their {RMBR} are compared with {\tt MAX\_RMBR}. Area of {RMBR}($a$) is apparently over 80\% of the total space area. According to {\tt MAX\_RMBR}, $a$ is stored as a B-vertex with a true value while $j$ is stored as an R-vertex with an {RMBR}.
\subsection{Reachable Grid Cells Merging}
\label{subsec:cellmerging}
After the type of each vertex is decided, the initialization algorithm performs the reachable grid cells merging phase (lines~\ref{line:phaseII:strat} to ~\ref{line:phaseII:end} in Algorithm~\ref{alg:GeoReach_construct}). In this phase, the algorithm merges adjacent grid cells to reduce the overall storage overhead of each G-Vertex. To achieve that, the algorithm assumes a system parameter, namely {\bf {\tt MERGE\_COUNT}}. This parameter determines how {{\scshape GeoReach}} merges spatially adjacent grid cells according to {\tt MERGE\_COUNT}. In each spatial region with four grid cells, the number of reachable grid cells should not be less than {\tt MERGE\_COUNT}. Otherwise, we merge the four grid cells into a single grid cell in the lower layer.
For each G-vertex $v$, all grid cells in grid cell layers $L_1$ to $L_{bottom}$ are checked. When a grid cell $G_i$ in $L_i$ is processed, four grid cells in $L_{i-1}$ that cover the same space with $G_i$ will be accessed. If number of reachable grid cells is larger than or equal to {\tt MERGE\_COUNT}, $G_i$ should be added in {ReachGrid}($v$) first. Then all grid cells covered by $G_i$ in layers from $L_0$ to $L_{i-1}$ should be removed. In order to achieve that, a recursive approach is implemented as follows. For each grid cell in $L_{i-1}$ that is reachable from $v$, the algorithm directly remove it from {ReachGrid}($v$). The removal stops at this grid in this layer. No recursive checking is required on grid cells in higher layers for which the space is covered by the reachable grid cell. Since all those reachable grid cells have been removed already. For those grid cells that are not reachable from $v$, the algorithm cannot assure that they do not cover some reachable grids in a higher layer. Hence, the recursive removal is invoked until the algorithm reaches the highest layer or other reachable grid cells are visited.
The {{SPA-Graph}} in Figure~\ref{fig:graph_overview} has a {\tt MERGE\_COUNT} set to 2. There is no merging in $e$, $i$ and $l$ because their {ReachGrid}s contain only one grid. The rest are $b$, $c$ and $g$. In {ReachGrid}($b$), for each grid in $L_1$, we make the {\tt MERGE\_COUNT} checking. $G_{17}$ covers four grids $G_1$, $G_2$, $G_5$ and $G_6$ in $L_0$. In such four-grids region, only $G_2$ is reachable from $b$. The merging will not happen in $G_{17}$. It is the same case in $G_{18}$ and $G_{20}$. However, there are two grids, $G_9$ and $G_{14}$ covered by $G_{19}$ in $L_1$. As a result, the two grids in $L_0$ will be removed from {ReachGrid}($b$) with $G_{19}$ being added instead. For the grid $G_{21}$ in $L_2$, the same checking in $L_1$ will be performed. Since, only $G_{19}$ is reachable, no merging happens. Finally, {ReachGrid}($b$) = $\{2,19\}$. Similarly, we can have {ReachGrid}($c$) = $\{12,14\}$ and {ReachGrid}($g$) = $\{12,14\}$ where no merging occurs.
\begin{algorithm}[t]
\caption{Maintain B-vertex}
\begin{scriptsize}
\label{alg:Ini_B}
\begin{algorithmic}[1]
\STATE {\bf Function} \textsc{Maintain-BVertex}(From-side vertex $v$, To-side vertex $v'$)
\IF {{GeoB}($v$) = \emph{true}}
\STATE {{\bf return} \emph{false}}
\ELSE
\SWITCH {Type of $v'$}
\CASE {B-vertex}
\IF {{GeoB}($v'$) = \emph{true}}
\STATE {{GeoB}($v$) $\leftarrow$ \emph{true}}
\ELSIF {$v'.spatial$ $\neq$ NULL}
\STATE {{ReachGrid}($v$) $\leftarrow$ {Grid}($v'.spatial$)}
\ELSE
\RETURN {\emph{false}}
\ENDIF
\end{ALC@g}
\CASE {R-vertex}
\STATE {{RMBR}($v$) $\leftarrow$ {MBR}({RMBR}($v'$), $v'.spatial$)}
\end{ALC@g}
\CASE {G-vertex}
\STATE {{ReachGrid}($v$) $\leftarrow$ {ReachGrid}($v'$)$\cup${Grid}($v'.spatial$)}
\end{ALC@g}
\STATE \textbf{end switch}
\ENDIF
\RETURN {\emph{true}}
\end{algorithmic}
\end{scriptsize}
\end{algorithm}
\section{Introduction}
\label{sec:introduction}
Graphs are widely used to model data in many application domains, including social networking, citation network analysis, studying biological function of genes, and brain simulation. A graph contains a set of vertices and a set of edges that connect these vertices. Each graph vertex or edge may possess a set of properties ({\em aka.} attributes). Thanks to the wide spread use of GPS-enabled devices, many applications assign a spatial attribute to a vertex (e.g., geo-tagged social media). Figure~\ref{fig:graph_example} depicts an example of a social graph that has two types of vertices: {\tt Person} and {\tt Venue} and two types of edges: {\tt Follow} and {\tt Like}. Vertices with type {\tt Person} have two properties (i.e., attributes): name and age. Vertices with type {\tt Venue} have two properties: name and {\em spatial} location. A spatial location attribute represents the spatial location of the entity (i.e., Venue) represented by such vertex. In Figure~\ref{fig:graph_example}, vertices $\{e,f,g,h,i\}$ are spatial vertices which represent venues.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.44\textwidth]{graph_overview_introduction}
\caption{Location-Aware Social Graph}
\label{fig:graph_example}
\end{center}
\end{figure}
Graph Database Management Systems (GDBMSs) emerged as a prominent NoSQL approach to store, query, and analyze graph data~\cite{GLG12,CHW+2013,SEH+13,SEH+12,SWL13}. Using a GDBMS, users can pose {\em reachability analysis} queries like: (i)~Find out whether two vertices in the graph are reachable, e.g., Are Alice (vertex $a$) and Katharine (vertex $l$) reachable in the social graph given in Figure~\ref{fig:graph_example}. (ii)~Search for graph paths that match a given regular language expression representing predicates on graph elements, e.g., Find all venues that Alice's Followees and/or her Followees' Followees also liked. Similarly, users may issue a {\em Reachability Query with Spatial Range Predicate} ({\em abbr.} {\small\sffamily RangeReach}). A {\small\sffamily RangeReach} query takes as input a graph vertex $v$ and a spatial range $R$ and returns true only if $v$ can reach any spatial vertex (that possesses a spatial attribute) which lies within the extent of $R$ (formal definition is given in Section~\ref{sec:preliminaries}). An example of a {\small\sffamily RangeReach} query is: Find out whether Alice can reach any of the Venues {\em located within the geographical area of Arizona State University} (depicted as a dotted red rectangle $R$ in Figure~\ref{fig:graph_example}). As given in Figure~\ref{fig:graph_example}, The answer to this query is true since Alice can reach Sushi 101 (vertex $g$) which is located within $R$. Another query example is to find out whether Katharine can reach any of the venues located within $R$. The answer to this query is false due to the fact that the only venue reachable from Katharine, Subway (vertex $h$), is not located within $R$.
There are several straightforward approaches to execute a {\small\sffamily RangeReach} query: (1)~{\em Traversal Approach:}~The naive approach traverses the graph, checks whether each visited vertex is a spatial vertex and returns true as the answer if the vertex's spatial attribute lies within the input query range $R$. This approach yields no storage/maintenance overhead since no pre-computed data structure is maintained. However, the Traversal approach may lead to high query response time since the algorithm may traverse the whole graph to answer the query.
(2)~{\em Transitive Closure (TC) Approach:} this approach leverages the pre-computed transitive closure~\cite{SAB+2013} of the graph to retrieve all vertices that are reachable from $v$ and returns true if at least one spatial vertex (located in the spatial range $R$) that is reachable from $v$. The TC approach achieves the lowest query response time, however it needs to pre-compute (and maintain) the graph transitive closure which is deemed notoriously infeasible especially for large-scale graphs.
(3)~{\em Spatial-Reachability Indexing (SpaReach) Approach:} uses a spatial index~\cite{BKS+90,Sam90a} to locate all spatial vertices $V_R$ that lie within the spatial range $R$ and then uses a reachability index~\cite{ZLW+2014} to find out whether $v$ can reach any vertex in $V_R$. SpaReach achieves better query response time than the Traversal approach but it still needs to necessarily probe the reachability index for spatial vertices that may never be reached from the $v$. Moreover, SpaReach has to store and maintain two index structures which may preclude the system scalability.
In this paper, we propose {{\scshape GeoReach}}, a scalable and time-efficient approach that answers graph reachability queries with spatial range predicates ({\small\sffamily RangeReach}). {{\scshape GeoReach}} is equipped with a light-weight data structure, namely {{SPA-Graph}}, that augments the underlying graph data with spatial indexing directories. When a {\small\sffamily RangeReach} query is issued, the system employs a pruned-graph traversal approach. As opposed to the SpaReach approach, {{\scshape GeoReach}} leverages the Spa-Graph' s auxiliary spatial indexing information to alternate between spatial filtering and graph traversal and early prunes those graph paths that are guaranteed: (a)~not to reach any spatial vertex or (b)~to only reach spatial vertices that outside the input spatial range query. As opposed to the TC and SpaReach approaches, {{\scshape GeoReach}} decides the amount of spatial indexing entries (attached to the graph) that strikes a balance between query processing efficiency on one hand and scalability (in terms of storage overhead) on the other hand. In summary, the main contributions of this paper are as follows:
\begin{itemize}
\item To the best of the authors' knowledge, the paper is the first that formally motivates and defines {\small\sffamily RangeReach}, a novel graph query that enriches classic graph reachability analysis queries with spatial range predicates. {\small\sffamily RangeReach} finds out whether an input graph vertex can reach any spatial vertex that lies within an input spatial range.
\item The paper proposes {{\scshape GeoReach}} a generic approach that adds spatial data awareness to an existing GDBMS. {{\scshape GeoReach}} allows efficient execution of {\small\sffamily RangeReach} queries issued on a GDBMS, yet without compromising a lot on the overall system scalability (measured in terms of storage size and initialization/maintenance time).
\item The paper experimentally evaluates {{\scshape GeoReach}}~\footnote{\scriptsize https://github.com/DataSystemsLab/GeoGraphDB--Neo4j} using real graph datasets based on a system implementation inside Neo4j (an open source graph database system). The experiments show that {{\scshape GeoReach}} exhibits up to two orders of magnitude better query response time and occupies up to four times less storage than the state-of-the-art spatial and reachability indexing approaches.
\end{itemize}
The rest of the paper is organized as follows: Section~\ref{sec:preliminaries} lays out the preliminary background and related work. The SPA-Graph data structure, {{\scshape GeoReach}} query processing, initialization and maintenance algorithms are explained in Sections~\ref{sec:georeachoverview} to~\ref{sec:initialization}. Section~\ref{sec:experiment} experimentally evaluates the performance of {{\scshape GeoReach}}. Finally, Section~\ref{sec:conclusion} concludes the paper.
\subsection{SPA-Graph Maintenance}
\label{sec:maintenance}
When the structure of a graph is updated, i.e., adding or deleting edges and/or vertices, {\scshape GeoReach} needs to maintain the SPA-Graph structure accordingly. Moreover, when the spatial attribute of a vertex changes, {\scshape GeoReach} may need to maintain the RMBR and/or ReachGrid properties of that vertex and other connected vertices as well. As a matter of fact, all graph updates can be simulated as a combination of adding and/or deleting a set of edges.
{\bf Adding an edge.} When an edge is added to the graph, the directly-influenced vertices are those that are connected to another vertex by the newly added edge. The spatial reachability information of the to-side vertex will not be influenced by the new edge. Based upon Lemmas~\ref{lemma:RMBR} and~\ref{lemma:ReachGrid}, the spatial reachability information, i.e., RMBR or ReachGrid, of the to-side vertex should be modified based on the the from-side vertex. On the other hand, the from-side vertex may remain the same or change. In the former case, there is no recursive updates required for the in-edge neighbors of the from-side vertex. Otherwise, the recursive updates are performed in the reverse direction until no change occurs or there is no more in-edge neighbor. A queue $Q$ will be exploited to track the updated vertices. When $Q$ is not empty, which means there are still some in-edge neighbors waiting for updates, the algorithm retrieves the next vertex in the queue. For such vertex, all its in-edge neighbors are updated by using the reachability information stored on this vertex. Updated neighbors will then be pushed into the queue. The algorithm halts when the queue is empty. Depending on category of the from-side vertex, corresponding maintenance functions, including {Maintain-BVertex}, {Maintain-RVertex} and {Maintain-GVertex} are used to update the newly added spatial reachability information.
Algorithm~\ref{alg:Ini_B} is used when the from-side vertex is a B-vertex.
In algorithm~\ref{alg:Ini_B}, if the from-side vertex $v$ is already a B-vertex with {GeoB}($v$) = \emph{true}. The added edge will never cause any change on $v$. Hence a false value is returned. In case {GeoB}($v$) = \emph{false}, the algorithm considers type of the to-side vertex $v'$.
\begin{myitem}
\item {\bf B-vertex.} If {GeoB}($v'$) = \emph{true}, it is no doubt that {GeoB}($v$) will be set to true and a true value will be returned. Otherwise, the algorithm checks whether $v'$ is spatial. If it is, {ReachGrid}($v$) is updated with {Grid}($v'spatial$). Otherwise, the algorithm returns false because $v$ is not changed.
\item {\bf R-vertex.} In such case, it is certain that $v$ will be updated to an R-vertex. The algorithm merely updates {RMBR}($v$) with {MBR}({RMBR}($v'$), $v'.spatial$).
\item {\bf G-vertex.} It is similar to the R-vertex case. Type of $v'$ can decide that $v$ should be a G-vertex and the algorithm updates {ReachGrid}($v$) with {ReachGrid}($v'$)$\cup${Grid}($v'.spatial$)
\end{myitem}
{Maintain-BVertex} and {Maintain-RVertex} are what we use in the initialization. However, there is a new condition that should be taken into consideration. When the from-side vertex $v$ is an R-vertex and the to-side vertex $v'$ is a G-vertex, the algorithm needs to update the {RMBR}($v$) with {ReachGrid}($v'$). Under such circumstance, first a dummy {RMBR}($v'$) will be constructed using {ReachGrid}($v'$). Although it is not the exact {RMBR} of $v'$, it is still precise. Error of the width and height will not be greater than size of a grid cell.
No matter what function is invoked to update the from-side vertex, {{\scshape GeoReach}} takes into account the system parameters {\tt MAX\_RMBR} and {\tt MAX\_REACH\_GRIDS} are checked on {RMBR} and {ReachGrid}, respectively.
{\bf Deleting an edge.} When an edge is removed, the to-side vertex will be not impacted by the deleting which is the same with adding an edge. To maintain the correctness of spatial reachability information stored on the from-side vertex, the only way is to reinitialize its spatial reachability information according to all its current out-edge neighbors. If its structure is different from the original state due to the deleting, the structure of all its in-edge neighbors will be rebuilt recursively. A queue $Q$ is used to keep track of the changed vertices. The way {\scshape GeoReach} maintains the queue and the operations on each vertex in the queue are similar to the AddEdge procedure. Maintenance cost of deleting an edge will be $O(kn^3)$ because the whole {{\scshape GeoReach}} index may be reinitialized.
\subsection{Query Processing}
\label{sec:query}
\begin{algorithm}[t]
\caption{Reachability Query with Spatial Range Predicate}
\begin{scriptsize}
\label{alg:spa_reach_query}
\begin{algorithmic}[1]
\STATE {\bf Function} \textsc{{RangeReach}}($v$, $R$)
\STATE {{\bf if} v is a spatial vertex {\bf and} $v.spatial$ Lie In $R$ {\bf then} {\emph{return true}}}
\STATE {Terminate $\leftarrow$ \emph{true}}
\IF {$v$ is a B-vertex}
\STATE {{\bf if} {GeoB}($v$) = true {\bf then} Terminate $\leftarrow$ \emph{false}}
\ELSIF {$v$ is a R-vertex}
\STATE {{\bf if} $R$ full contains {RMBR}($v$) {\bf then} \emph{return true}}
\STATE {{\bf if} $R$ no overlap with {RMBR}($v$) {\bf then} \emph{return false}}
\STATE {Terminate $\leftarrow$ \emph{false}}
\ELSIF {$v$ is a G-vertex}
\FOR {{\em each} grid $G_i\in$ {ReachGrid}($v$)}
\STATE {{\bf if} $R$ fully contains $G_i$ {\bf then} \emph{return true}}
\STATE {$G_i$ partially overlaps with $R$ {\bf then} Terminate $\leftarrow$ false}
\ENDFOR
\ENDIF
\IF {Terminate = \emph{false}}
\FOR {{\em each} vertex $v'\in V_{v}^{out}$}
\STATE {{\bf if} \textsc{{RangeReach}}($v'$, $R$) = \emph{true} {\bf then} \emph{return true}}
\ENDFOR
\ENDIF
\STATE {{\em return false}}
\end{algorithmic}
\end{scriptsize}
\label{algo:query}
\end{algorithm}
This section explains the {\small\sffamily RangeReach} query processing algorithm. The main objective is to visit as less graph vertices and edges as possible to reduce the overall query latency. The query processing algorithm accelerates the SPA-Graph traversal procedure by pruning those graph paths that are guaranteed (or not) to satisfy the spatial reachability constraint. Algorithm~\ref{algo:query} gives pseudocode for query processing. The algorithm takes as input a graph vertex $v$ and query rectangle $R$. It then starts traversing the graph starting from $v$. For each visited vertex $v$, three cases might happen, explained as follows:
{\bf Case~I (B-vertex):} In case {GeoB} is false, a B-vertex cannot reach any spatial vertex and hence the algorithm stops traversing all graph paths after this vertex. Otherwise, further traversal from current B-vertex is required when {GeoB} value is true. Line 4 to 5 in algorithm~\ref{algo:query} is for processing such case.
{\bf Case~II (R-vertex):} For a visited R-vertex $u$, there are three conditions that may happen (see figure~\ref{fig:threecondition}). They are the case from line 6 to 9 in algorithm~\ref{algo:query}:
\begin{myitem}
\item {\bf Case~II.A}: {RMBR}($u$) lies within the query rectangle (see Figure~\ref{fig:threecondition_lie_in}). In such case, the algorithm terminates and returns {\bf true} as the answer to the query since there must exist at least a spatial vertex that is reachable from $v$.
\item {\bf Case~II.B}: The spatial query region $R$ does not overlap with {RMBR}{($u$)} (see Figure~\ref{fig:threecondition_no_overlap}). Since all reachable spatial vertices of $u$ must lie inside {RMBR}($u$), there is no reachable vertex can be located in the query rectangle. As a result, graph paths originating at $u$ can be pruned.
\item {\bf Case~III.C}: {RMBR}{($u$)} is partially covered by the query rectangle (see Figure~\ref{fig:threecondition_partially_cover}). In this case, the algorithm keeps traversing the graph by fetching the set of vertices $V_{v}^{out}$ that can be reached via a direct edge from $v$.
\end{myitem}
\begin{figure}
\centering
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{threecondition_no_overlap}
\caption{No Overlap}
\label{fig:threecondition_no_overlap}
\end{subfigure}
\hspace{-5mm}
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=0.7\linewidth]{threecondition_lie_in}
\caption{Lie In}
\label{fig:threecondition_lie_in}
\end{subfigure}
~
\begin{subfigure}[t]{0.46\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{threecondition_partially_cover}
\caption{Partially Covered By}
\label{fig:threecondition_partially_cover}
\end{subfigure}
\caption{\small Relationships between {RMBR} and a query rectangle}
\label{fig:threecondition}
\end{figure}
{\bf Case~III (G-vertex):} For a G-vertex $u$, it store many reachable grids from $u$. Actually, it can be regarded as many smaller {RMBR}s. So three cases may also happen. Algorithm~\ref{algo:query} line 13 to 18 is for such case. Three cases will happen are explained as follows:
\begin{myitem}
\item {\bf Case~III.A}: The query rectangle $R$ fully contains any grid cell in {ReachGrid}{($u$)}. In such case, the algorithms terminates and returns {\em true} as the query answer.
\item {\bf Case~III.B}: The query rectangle have no overlap with all grids in {ReachGrid}{($u$)}. This case means that $v$ cannot reach any grids overlapped with $R$. Then we never traverse from $v$ and this search branch is pruned.
\item {\bf Case~III.C}: If the query rectangle fully contains none of the reachable grid and partially overlap with any reachable grid, it corresponds to Partially Covered By case for {RMBR}. So further traversal is performed.
\end{myitem}
Figure~\ref{fig:graph_overview} gives an example of {\small\sffamily RangeReach} that finds whether vertex $a$ can reach query rectangle $Q$ (the shaded one in figure~\ref{fig:graph_overview}). At the beginning of the traversal, the algorithm checks the category of $a$. In case, It is a B-vertex and its {GeoB} value is true, the algorithm recursively traverses out-edge neighbors of $a$ and perform recursive checking. Therefore, the algorithm retrieves vertices $b$, $c$, $d$ and $j$. For vertex $b$, it is a G-vertex and its reachable grids are $G_2$ and $G_{19}$. $G_{19}$ cover the range of four grids in $L_0$. They are $G_9$, $G_{10}$, $G_{13}$ and $G_{14}$. The spatial range is merely partially covered by $Q$ ({\bf Case~III.C}), hence it is possible for $b$ to reach $Q$. We cannot make an assured decision in this step so $b$ is recorded for future traversal. Another neighbor is $c$. {ReachGrid}($c$) is $\{12,14\}$ which means that $G_{12}$ and $G_{14}$ are reachable from $c$. $G_{14}$ lies in $Q$ ({\bf Case~III.A}). In such case, since $a\leadsto c$, we can conclude that $a\leadsto R$. The algorithm then halts the graph traversal at this step and returns true as the query answer.
\section{Preliminaries and Background}
\label{sec:preliminaries}
This section highlights the necessary background and related research work. Table~\ref{table:mathnotations} summarizes the main notations in the paper.
\subsection{Preliminaries}
\begin{table} [t]
\centering
\begin{scriptsize}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{ c | p{6cm} }
\hline
{\bf Notation} & {\bf Description} \tabularnewline
\hline
$G=\{V,E\}$ & A graph $G$ with a set of vertices $V$ and set of edges $E$ \tabularnewline
\hline
$V_{v}^{out}$ &
The set of vertices that can be reached via a direct edge from a vertex $v$ \tabularnewline
\hline
$V_{v}^{in}$ &
The set of vertices that can reach (via a direct edge) vertex $v$ \tabularnewline
\hline
$RF(v)$ &
The set of vertices that are reachable from (via any number of edges) vertex $v$ \tabularnewline
\hline
$V_S$ &
The set of spatial vertices in $G$ such that $V_{S} \subseteq V$ \tabularnewline
\hline
$RF_S(v)$ &
The set of spatial vertices that are reachable from (via any number of edges) vertex $v$ \tabularnewline
\hline
$n$ &
The cardinality of V ($n=|V|$); the number of vertices in $G$ \tabularnewline
\hline
$m$ &
The cardinality of E ($m=|E|$); the number of edges in $G$ \tabularnewline
\hline
$v_1 \leadsto v_2$ &
$v_2$ is reachable from $v_1$ via connected path in $G$ (such that both $v_1$ and $v_2$ $\in$ $V$) \tabularnewline
\hline
{MBR}($P$) &
Minimum bounding rectangle of a set of spatial polygons $P$ (e.g., points, rectangles) \tabularnewline
\hline
\end{tabular}
\end{scriptsize}
\caption{Notations.}
\label{table:mathnotations}
\end{table}
{\bf Graph Data.} {{\scshape GeoReach}} deals with a directed property graph $G=(V,E)$ where (1)~$V$ is a set of vertices such that each vertex has a set of properties (attributes) and (2)~$E$ is a set of edges in which every edge can be represented as a tuple of two vertices $v_1$ and $v_2$ ($v_1,v_2\in V$). The set of spatial vertices $V_S \subseteq V$ such that each $v \in V_{S}$ has a spatial attribute (property) $v.spatial$. The spatial attribute $v.spatial$ may be a geometrical point, rectangle, or a polygon. For ease of presentation, we assume that a spatial attribute of spatial vertex is represented by a point. Figure~\ref{fig:graph_example} depicts an example of a directed property graph. Spatial Vertices $V_S$ are represented by black colored circles and are located in a two-dimensional planer space while white colored circles represent regular vertices that do not possess a spatial attribute. Arrows indicate directions of edges in the graph.
\noindent {\bf Graph Reachability ($v_1 \leadsto v_2$).} Given two vertices $v_1$ and $v_2$ in a graph $G$, $v_1$ can reach $v_2$ ($v_1 \leadsto v_2$) or in other words $v_2$ is reachable from $v_1$ if and only if there is at least one graph path from $v_1$ to $v_2$. For example, in Figure~\ref{fig:graph_example}, vertex $a$ can reach vertex $f$ through the graph path {\small \tt a->c->i->f} so it can be represented as $a \leadsto f$. On the other hand, $c$ cannot reach $h$.
\noindent {\bf Reachability with Spatial Range Predicate ({\small\sffamily RangeReach}).} {\small\sffamily RangeReach} queries find whether a graph vertex can reach a specific spatial region (range) $R$. Given a vertex $v \in V$ in a Graph $G$ and a spatial range $R$, {\small\sffamily RangeReach} can be described as follows:
\begin{small}
\begin{equation}
\emph{\small \sffamily RangeReach(v, R)} = \left\{ \,
\begin{IEEEeqnarraybox}[][c]{l?s}
\IEEEstrut
{true} & if $\exists$~$v'$ such that\\
~ & ~(1)~$v' \in V_{S}$\\
~ & ~(2)~$v'.spatial$ lies within $R$\\
~ & ~(3)~$v \leadsto v'$\\
{false} & Otherwise.
\IEEEstrut
\end{IEEEeqnarraybox}
\right.
\label{eq:rangereach}
\end{equation}
\end{small}
As given in Equation~\ref{eq:rangereach}, if any spatial vertex $v' \in V_{S}$ that lies within the extent of the spatial range $R$ is reachable from the input vertex $v$, then {\small\sffamily RangeReach}($v$, $R$) returns true (i.e., $v \leadsto R$). For example, in Figure~\ref{fig:graph_example}, {\small\sffamily RangeReach}($a$, $R$) = \emph{true} since $a$ can reach at least one spatial vertex $f$ in $R$. However, {\small\sffamily RangeReach}($l$, $R$) = \emph{false} since $l$ can merely reach a spatial vertex $h$ which is not located in $R$. Vertex $d$ cannot reach $R$ since it cannot reach any vertex.
\iffalse
\begin{figure}[t]
\centering
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/graph_background}
\caption{Geo-RMBR}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/GeoReachGrid-M}
\caption{Geo-Grid}
\end{subfigure}
~
\begin{subfigure}[t]{0.44\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{figures/graph_overview}
\caption{Graph Overview}
\end{subfigure}
\caption{Index Structure}\label{fig:index_structure}
\end{figure}
\fi
\subsection{Related Work}
This section presents previous work on reachability indexes, spatial indexes, and straightforward solutions to processing graph reachability queries with spatial range predicates ({\small\sffamily RangeReach}).
{\bf Reachability Index.} Existing solutions to processing graph reachability queries ($u \leadsto v$) can be divided into three categories~\cite{ZLW+2014}: (1)~Pruned Graph Traversal~\cite{CGK+2005,TSL+2007,YCZ+2010}: These approaches pre-compute some auxiliary reachability information offline. When a query is issued, the query processing algorithm traverses the graph using a classic traversal algorithm, e.g., Depth First Search (DFS) or Breadth First Search (BFS), and leverages the pre-computed reachability information to prune the search space. (2)~Transitive closure retrieval~\cite{ABJ+1989,CC+2008,J+1990,JXR+2008,SAB+2013,SSM+2011,WHY+2006}: this approach pre-computes the transitive closure of a graph offline and compresses it to reduce its storage footprint. When a query $u \leadsto v$ is posed, the transitive closure of the source vertex $u$ is fetched and decomposed. Then the query processing algorithm checks whether the terminal vertex $v$ lies in the transitive closure of $u$. and (3)~Two-Hop label matching~\cite{CPC+2010,CHW+2013,CSC+2012,CYL+2008,CHK+2003,STW+2004}: The two-hop label matching approach assigns each vertex $v$ in the graph an out-label set $L_{out}(v)$ and an in-label set $L_{in}(v)$. When a reachability query is answered, the algorithm decides that $u \leadsto v$ if and only if $L_{out}(v)\cap L_{in}(v)\not\neq\varnothing$. Since the two label sets do not contain all in and out vertices, size of the reachability index reduces.
{\bf Spatial Index.} A spatial index ~\cite{RSV02,Sam06,SC03} is used for efficient retrieval of either multi-dimensional objects (e.g., $\langle$$x$,$y$$\rangle$ coordinates of an object location) or objects with spatial extents, e.g., polygon areas represented by their minimum boundary rectangles (MBR). Spatial index structures can be broadly classified to hierarchical (i.e., tree-based) and non-hierarchical index structures. Hierarchical tree-based spatial index structures can be classified into another two broad categories: (a)~the class of \emph{data-partitioning trees}, also known as the class of Grow-and-Post trees~\cite{Lom91}, which refers to the class of hierarchical data structures that basically extend the B-tree index structure~\cite{BM72,Com79} to support multi-dimensional and spatial objects. The main idea is to recursively partition the spatial data based on a spatial proximity clustering, which means that the spatial clusters may overlap. Examples of spatial index structures in this category include R-tree~\cite{Gut84} and R*-tree~\cite{BKS+90}. (b)~the class of \emph{space-partitioning trees} that refers to the class of hierarchical data structures that recursively decomposes the space into disjoint partitions. Examples of spatial index structures in this category include the Quad-tree~\cite{FB74} and k-d tree~\cite{Ben75}.
{\bf Spatial Data in Graphs.} Some existing graph database systems, e.g., Neo4j, allow users to define spatial properties on graph elements. However, these systems do not provide native support for {\small\sffamily RangeReach} queries. Hence, users need to create both a spatial index and a reachability index to efficiently answer a {\small\sffamily RangeReach} queries (drawbacks of this approach are given in the following section). On the other hand, existing research work~\cite{LMB+14} extends the RDF data with spatial data to support RDF queries with spatial predicates (including range and spatial join). However, such technique is limited to RDF and not general graph databases. It also does not provide an efficient solution to handle reachability queries.
\subsection{Straightforward Solutions}
There are three main straightforward approaches to process a {\small\sffamily RangeReach} query, described as follows:
{\bf Approach~I: Graph Traversal.} This approach executes a spatial reachability query using a classical graph traversal algorithm like DFS (Depth First Search) or BFS (Breadth First Search). When {\small\sffamily RangeReach}($v$, $R$) is invoked, the system traverses the graph from the starting vertex $v$. For each visited vertex, the algorithm checks whether it is a spatial vertex and returns true as the query answer if the vertex's location lies within the input query range $R$ because the requirement of spatial reachability is satisfied and hence $v \leadsto R$. Otherwise, the algorithm keeps traversing the graph. If all vertices that $v$ can reach do not lie in $R$, that means $v$ cannot reach $R$.
{\bf Approach~II: Transitive Closure (TC).} This approach pre-computes the transitive closure of the graph and stores it as an adjacency matrix in the database. Transitive closure of a graph stores the connectivity component of the graph which can be used to answer reachability query in constant time. Since the final result will be determined by spatial vertices, only spatial vertices are stored. When {\small\sffamily RangeReach}($v$, $R$) is invoked, the system retrieves all spatial vertices that are reachable from $v$ by means of the transitive closure. The system then returns true if at least one spatial vertex that is reachable from $v$ is also located in the spatial range $R$.
{\bf Approach~III: SpaReach.} This approach constructs two indexes a-priori: (1)~A Spatial Index: that indexes all spatial vertices in the graph and (2)~A Reachability Index: that indexes the reachability information of all vertices in the graph. When a {\small\sffamily RangeReach} query is issued, the system first takes advantage of the spatial index to locate all spatial vertices $V_R$ that lie within the spatial range $R$. For each vertex $v' \in V_R$, a reachability query against the reachability index is issued to test whether $v$ can reach $v'$. For example, to answer {\small\sffamily RangeReach}($a$, $R$) in Figure~\ref{fig:graph_overview}, spatial index is exploited first to retrieve all spatial vertices that are located in $R$. From the range query result, it can be known that $g$, $i$ and $f$ are located in rectangle $R$. Then graph reachability index is accessed to determine whether $a$ can reach any located-in vertex. Hence, it is obvious {\small\sffamily RangeReach}($a$, $R$) = \emph{true} by using this approach.
{\bf Critique.}
The Graph Traversal approach yields no storage/maintenance overhead since no pre-computed data structure is maintained. However, the traversal approach may lead to high query response time ($O(m)$ where $m$ is the number of edges in the graph) since the algorithm may traverse the whole graph to answer the query.
The TC approach needs to pre-compute (and maintain) the graph transitive closure which is deemed notoriously infeasible especially for large-scale graphs. The transitive closure computation is $O(kn^3)$ or $O(nm)$ and the TC storage overhead is $O(kn^2)$ where $n$ is total number of vertices and $k$ is the ratio of spatial vertices to the total number of vertices in the graph. To answer a {\small\sffamily RangeReach} query, the TC approach takes $O(kn)$ time since it checks whether each reachable spatial vertex in the transitive closure is located within the query rectangle.
On the other hand, SpaReach builds a reachability index, which is a time-consuming step, in $O(n^3)$~\cite{WHY+2006} time. The storage overhead of a spatial index is $O(n)$ and that of a reachability index is $O(nm^{1/2})$. To store the two indices, the overall storage overhead is $O(nm^{1/2})$. Storage cost of this approach is far less than TC approach but still not small enough to accommodate large-scale graphs. The query time complexity of a spatial index is $O(kn)$ while that of reachability index is $m^{1/2}$. But for a graph reachability query, checking is demanded for each spatial vertex in the result set generated by the range query. Hence, cost of second step reachability query is $O(knm^{1/2})$. The total cost should be $O(knm^{1/2})$. Query performance of Spa-Reach is highly impacted by the size of the query rectangle since the query rectangle determines how many spatial vertices are located in the region. In Figure~\ref{fig:graph_example}, query rectangle $R$ overlaps with three spatial vertices. For example, to answer {\small\sffamily RangeReach}($l$, $R$), all three vertices $\{f,g,i\}$ will be checked against the reachability index to decide whether any of them is reachable from $l$ and in fact neither of them is reachable. In a large graph, a query rectangle will possibly contain a large number of vertices. That will definitely lead to high unreasonable high query response time.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{graph_overview}
\caption{{{SPA-Graph}} Overview}
\label{fig:graph_overview}
\end{figure}
\section{Our Approach: GeoReach}
\label{sec:georeachoverview}
In this section, we give an overview of {{\scshape GeoReach}} an efficient and scalable approach for executing graph reachability queries with spatial range predicates.
\input{IndexStructure}
\input{QueryProcessing}
\input{Trade-off}
\input{Initialization}
\input{Maintenance}
\section{SPA-Graph Analysis}
\label{sec:spagraphanalysis}
This section analyzes each SPA-Graph vertex type rom two perspectives: (1)~Storage Overhead: the amount of storage overhead that each vertex type adds to the system (2)~Pruning Power: the probability that the query processing algorithm terminates when a vertex of such type is visited during the graph traversal.
{\bf B-vertex.} When visiting a B-Vertex, in case {GeoB} is false, the query processing algorithm prunes all subsequent graph paths originated at such vertex. That is due to the fact that such vertex cannot reach any spatial vertex in the graph. Otherwise, the query processing algorithm continues traversing the graph. As a result, pruned power of a B-vertex lies in the condition that {GeoB} is false. For a given graph, number of vertices that can reach any space is a certain value. So probability that a vertex can reach any spatial vertex is denoted as $P_{true}$. This is also the probability of a B-vertex whose {GeoB} value is true. Probability of a B-vertex whose {GeoB} value is false, denoted as $P_{false}$, will be $1-P_{true}$. To sum up, pruned power of a B-vertex is $1-P_{true}$ or $P_{false}$
\begin{figure}[t]
\centering
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{CostAnalysis_R-vertex_no_overlap}
\caption{{}}
\label{fig:cost_analysis_R_vertex_no_overlap}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{CostAnalysis_R-vertex_no_overlap_general}
\caption{{}}
\label{fig:cost_analysis_R_vertex_no_overlap_general}
\end{subfigure}
~
\begin{subfigure}[t]{0.23\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{CostAnalysis_R-vertex_lie_in}
\caption{{}}
\label{fig:cost_analysis_R_vertex_lie_in}
\end{subfigure}
\caption{R-vertex Pruning Power}\label{fig:cost_analysis_R-vertex}
\end{figure}
{\bf R-vertex.} When an R-vertex is visited, the condition whether the vertex can reach any space still exists. If the R-vertex cannot reach any space, we assign the R-vertex a specific value to represent it(e.g. set coordinates of {RMBR}'s bottom-left point bigger than that of the top-right point). In this case, pruned power of a R-vertex will be the same with a B-vertex, which is $P_{false}$. Otherwise, when the R-vertex can reach some space, it will be more complex. Because information of {RMBR} and query rectangle $R$ have some impact on the pruned power of this R-vertex. The algorithm stops traversing the graph in both the {\em No Overlap} and {\em Lie In} cases depicted in Figures~\ref{fig:threecondition_no_overlap} and~\ref{fig:threecondition_lie_in}. Figure~\ref{fig:cost_analysis_R-vertex} shows the two cases that R-vertex will stop the traversal. In Figure~\ref{fig:cost_analysis_R-vertex}, width and height of the total 2D space are denoted as $A$ and $B$. Assume that the query rectangle can be located anywhere in the space with equal probability. We use $(x_1, y_1)$ and $(x_2, y_2)$ to represent the {RMBR}'s top-left corner and lower-right point coordinates, respectively. Then all possible areas where top-left vertex of query rectangle $Q$ should be part of the total space, denoted as $I$ (see the shadowed area in the figure. Its area is determined by size of query rectangle. Denote width and height of $Q$ are $e$ and $f$, then area of $I$, $A_I=(A-e)\times (B-f)$.
First, we estimate probability of {\em No Overlap} case. Figure~\ref{fig:cost_analysis_R_vertex_no_overlap} shows one case of {\em No Overlap}. If the query rectangle $Q$ do not overlap with {RMBR}, top-left vertex of $Q$ must lie outside rectangle $R$ which is forms the overlap region (drawn with solid line in Figure~\ref{fig:cost_analysis_R_vertex_no_overlap_general}). Area of $R$ (denoted as $A_R$) is obviously determined by the {RMBR} location and size of $Q$. It can be easily observed that $A_R = (x_2 - (x_1 - e))\times (y_2 - (y_1 - f))$. Another possible case is demonstrated in Figure~\ref{fig:cost_analysis_R_vertex_no_overlap_general}. In such case, if we calculate $R$ in the same way, range of $R$ will exceeds area of $I$ which contains all possible locations. As a result, $A_R = A_I$ in this case. As we can see, area of overlap region is determined by the range of $R$ and $I$ altogether. Then we can have a general representation of the overlap area $A_{Overlap} = (min(A-e, x_2) - max(0, x_1-e))\times (min(B-f, y_2) - max(0, x_2 - f)$. The {\em No Overlap} area is $A_I - A_{Overlap}$ and the probability of having a {\em No Overlap} case is calculated as follows:
\begin{equation}
P_{NoOverlap} = \frac{A_I - A_{Overlap}}{A_I} = 1 - \frac{A_{Overlap}}{A_I}.
\end{equation}
Figure~\ref{fig:cost_analysis_R_vertex_lie_in} depicts the {\em Lie In} case. When top-left vertex of $Q$ lies in region $R$, then such Lie In case will happen. To ensure that $R$ exists, it is necessary that $e>(x_2-x_1)$ and $f>(y_2-y_1)$. If it is not, then probability of such case must be 0. If this requirement is satisfied, then $A_R = (x_1 - (x_2 - e))\times (y_1 - (y_2 - f))$. Recall what is met in the above-mentioned case, $R$ may exceed the area of $I$. Similarly, more general area should be $A_R = (min(A-e, x_1) - max(0, x_1 - (x_2 - e)))\times (min(B-f, y_1) - max(0, y_1 - (y_2 - f)))$. Probability of such case should be $\frac{A_R}{A_I}$. To sum up, we have
\begin{equation}
P_{LieIn} = \begin{cases}
\frac{A_R}{A_I}&\mbox{$e>(x_2-x_1)$ and $f>(y_2-y_1)$}\\
0&\mbox{else}
\end{cases}
\end{equation}
After we sum up all conditional probabilities based on $P_{true}$ and $P_{false}$, pruning power of an R-vertex is equal to $(P_{\small NoOverlap}+P_{\small LieIn})\times P_{true} + P_{false}$.
Evidently, the pruning power of an R-vertex is more powerful than a B-vertex. When the storage overhead of an R-vertex is considered, coordinates of {RMBR}'s top-left and lower-right vertices should be stored. Thus its storage will be at least four bytes depending on the spatial data precision. That means the storage overhead of a G-Vertex is always higher than that of a B-Vertex.
{\bf G-vertex.}
For a high resolution grid, it is of no doubt that a G-vertex possesses a high pruning power. However, this comes at the cost of higher storage overhead because more grid cells occupies more space. When a G-vertex is compared with an R-vertex, the area of an R-vertex is much larger than a grid. In this case, an R-vertex can be seen as a a simplified G-vertex for which the grid cell size is equal to that of {RMBR}. One extreme case of R-vertex is that the vertex can reach only one spatial vertex. In such case, {RMBR} is location of the reachable spatial vertex. Such R-vertex can still be counted as a G-vertex whose grid size $x\rightarrow 0$. According the rule, it should be with higher storage overhead and more accuracy. Actually, storing it as a G-vertex will cost an integer while any R-vertex requires storage for four float or even double number.
|
1,116,691,497,803 | arxiv | \section{Introduction}
\subsection{Motivation}
The glass transition, an essentially dynamic phenomenon,
can be described as the slowing down and eventual freezing of
$\alpha$ relaxation.
According to mode-coupling (MC) theory \cite{Got91,x15},
the physical origin of this process must be sought on a picosecond scale
where long-ranged transport starts to evolve
from vibrational short-time dynamics.
In the long-time limit, MC theory reproduces the well-established
phenomenology of $\alpha$ relaxation.
New results are obtained for shorter times.
In particular, the theory predicts a change of transport mechanism
around a cross-over temperature~$T_{\rm c}$,
located in the moderately viscous liquid phase well above the
conventional glass transition temperature~$T_g$.
On cooling towards $T_{\rm c}$, particles spend more and more time
being trapped in transient cages;
this process, labelled fast $\beta$ relaxation,
is predicted to obey remarkably universal scaling.
In a couple of structural glass formers,
MC predictions have been confirmed \cite{GoSj92,Got99}
primarily by different scattering techniques \cite{CuLD95,PeWu95}.
More recently, the GHz--THz dynamics also became accessible
by dielectric spectroscopy \cite{LuPD96b,LuPL97}.
Although results are in accord with the MC scenario,
in several cases the data did not fall together
with dynamic susceptibilities from scattering experiments \cite{LuPD97b,ScLB98}
which is in conflict with the asymptotic factorisation property
of MC $\beta$ relaxation.
Recent theoretical developments suggest explanations on the basis of
corrections to scaling \cite{FrFG97b,FuGM98}
and orientational degrees of freedom \cite{ScSc97,FrFG97c,FaSS98}.
For experimental tests,
more detailed comparisons of different observables are needed.
In this context,
we performed incoherent neutron and depolarised light scattering experiments
on propylene carbonate (PC, 4-methyl-1,3-dioxolan-2-on, C$_4$O$_2$H$_6$),
a fragile glass former ($T_g=160\,$K) with low molecular weight ($M=102.1$)
which has already been studied by various experimental techniques.
A synopsis of available data shall be given in subsection I.C
after collecting the essential results of MC theory in I.B.
\subsection{The Mode-Coupling Cross-Over}
In its simplest (``idealised'') formulation,
MC theory describes an ergodic-to-nonergodic transition at $T_{\rm c}$.
On the low-temperature side,
the onset of fast $\beta$ relaxation leads
to an anomalous decrease of the Debye-Waller or Lamb-M\"o\ss bauer factor,
\begin{equation}\label{Esqrt}
f_q = f_q^c + h_q |\sigma|^{1/2} {\,,~}\sigma>0\,,
\end{equation}
with a reduced temperature $\sigma=(T_{\rm c}-T)/T_{\rm c}$.
On the high-temperature side,
the time constant of $\alpha$ relaxation diverges
with a fractal exponent~$\gamma$,
\begin{equation}\label{Etaua}
\tau \propto |\sigma|^{-\gamma} {\,,~}\sigma<0\,.
\end{equation}
Such a transition has actually been observed in a colloidal suspension
\cite{Meg95}.
In a structural glass former,
the singularities
(\ref{Esqrt}) and~(\ref{Etaua}) are smeared out
because activated hopping processes restore ergodicity \cite{GoSj87b}.
Under these limitations,
integral quantity like $f_q$ or~$\tau_\alpha$
do not allow for decisive tests of theory.
One rather has to study the full dynamics,
as represented by the dynamic susceptibility $\chi''_q(\omega)$,
or any other dynamic variable coupling to it.
Stronger predictions are made for the fast $\beta$ regime:
around the minimum between $\alpha$ peak and vibrational excitations,
in a temperature range close enough to, but sufficiently above $T_{\rm c}$,
any susceptibility is expected to reach the same asymptotic limit
\begin{equation}\label{EminL}
\chi''(\omega) = \chi_\sigma g_\lambda(\omega/\omega_\sigma)
\end{equation}
where the scaling function $g_\lambda$ is fully determined by
one single parameter~$\lambda$ \cite{Got90}.
Amplitude and frequency scale of (\ref{EminL}) should become singular
on cooling towards $T_{\rm c}$,
\begin{equation}\label{Epars}
\chi_\sigma \propto |\sigma|^{1/2}\,{\rm,~and~}
\omega_\sigma \propto |\sigma|^{1/2a} {\,,~}\sigma<0
\end{equation}
where the exponent~$a$, just as $\gamma$ in (\ref{Etaua}),
is determined by~$\lambda$.
\subsection{Previous Studies of Propylene Carbonate}
Table~1 summarises previous studies of fast dynamics in PC.
All authors reported at least partial accord with MC predictions.
However,
not all MC interpretations were consistent with each other.
It is an essential result of idealised MC theory
that in the asymptotic $\beta$ regime all
dynamic observables show the same spectral distribution and
the same temperature dependence.
Therefore,
one material is characterised by just one~$\lambda$ and one~$T_{\rm c}$.
Some of the $T_{\rm c}$ reported for PC
are therefore incompatible with MC theory.
As with other materials,
early scattering experiments \cite{BoHo91,ElBT92} concentrated
on the square-root singularity of~$f_q$.
As in other materials,
this singularity remains elusive:
in Brillouin scattering,
data fitting depends on uncontrolled approximations
for the memory function \cite{DuLC94}.
Similarly,
in neutron scattering a determination of $T_{\rm c}$ from (\ref{Esqrt})
works at best if the full lineshape on the $\sigma>0$ side
is known from high-resolution spectroscopy \cite{BaFK89b}.
Therefore, the isolated results $T_{\rm c}=210$ or 270\,K
can be discarded from further consideration.
In comparison,
the determination of $T_{\rm c}$ from viscosity or $\alpha$-relaxation data
works better.
However,
available data do not allow for an independent determination of~$\lambda$,
and even for one given value $\lambda=0.70$, results vary
between $T_{\rm c}=180$ and 196\,K.
A dynamic susceptibility has been measured first
by depolarised light scattering \cite{DuLC94},
yielding $\lambda=0.78$ and $T_{\rm c}=187$.
These values have shown to be consistent with
dielectric loss spectroscopy \cite{LuPD97b,LuPD96c,ScLB99}.
However, a time-domain optical measurement of the solvation response
of a solute molecule found a significantly lower~$T_{\rm c}$
if the light scattering value of $\lambda=0.78$ was assumed
\cite{MaBB95,MaBB96}.
In the following section II,
we study the fast relaxation regime by neutron scattering.
The $\lambda$ and $T_{\rm c}$ we obtain
differ significantly from results of the other dynamic measurements
which motivates us to remeasure some light scattering spectra (sect.~III)
and to reanalyse dielectric-loss and solvation-response data (sect.~IV).
Only then, all susceptibilities will be compared in section~V.
\section{Neutron Scattering}
\subsection{Experiments}
Inelastic, incoherent neutron scattering has been measured
on the time-of-flight spectrometers Mib\'emol at the
Laboratoire L\'eon Brillouin, Saclay,
and {\sc Neat} at the Hahn-Meitner-Institut, Berlin.
On Mib\'emol, the counter-rotating choppers were operated with 10\,000~rpm.
With an incident neutron wavelength of $\lambda_{\rm i}=8.5\,$\AA,
we obtained a resolution (fwhm) of $6.5-9.5\,$GHz, depending on angle.
At the Berlin reactor, the flux delivered by the undermoderated cold source
decreases rather fast for wavelengths beyond 5 or 6\,\AA;
therefore $\lambda_{\rm i}=5.5\,$\AA\ was chosen.
With counter-rotating choppers at 20\,000~rpm,
we achieved nevertheless a resolution of $11-13.5\,$GHz.
Total count rates were of the same order for both instruments;
a precise comparison cannot be made
because different sample geometries were used.
For the Mib\'emol experiment,
we used an Al hollow cylinder \cite{Wut99}
with 30\,mm outer diameter and
a sample layer of about 0.1\,mm thickness.
In this container,
the sample crystallised partially during a 190\,K scan;
at 175\,K,
as far as one can tell from the elastic structure factor of a
predominantly incoherent scatterer,
complete or nearly complete crystallisation occured within 2 hours.
On the other hand, after rapidly cooling from 260\,K,
it was possible to perform a 2\,K scan in the fully amorphous state.
On {\sc Neat}, we tried to remeasure the dynamics of the supercooled state
below 200\,K.
To this purpose,
we filled the sample into nearly 200 thin capillaries
(soda lime, inner diameter 0.2\,mm, Hilgenberg).
As in other liquids,
this packaging proved highly successful;
comparison to a vanadium scattering and visual inspection showed that
down to 168\,K no crystallisation occured.
Because the detectors are located at only 250\,cm from the sample,
the resolution of {\sc Neat} is very sensitive to flightpath differences.
We therefore abstained from using a hollow cylindrical geometry;
instead, we placed the capillaries in a rectangular holder
with a 30\,mm base length
which was then mounted at $45^\circ$ with respect to the incoming beam.
For both experiments,
the sample material, 1,2-propylene carbonate (99.7\,\%, Sigma--Aldrich),
was loaded under an inert gas.
After converting the raw data to $S(2\theta,\omega)$,
they were binned in about 30 angular groups;
a non-equidistant frequency binning was imposed by
requiring statistic fluctuations to fall below a given mark.
Only then was the container scattering subtracted.
The absolute intensity scale was taken from the elastic scattering
of the Mib\'emol 2\,K scan;
for {\sc Neat}, detectors were first calibrated to vanadium
before the overall scale was fit to Mib\'emol at 210\,K.
The scattering law $S(2\theta,\omega)$ depends still on $\lambda_{\rm i}$.
Only after interpolation to constant wavenumbers~$q$
one obtains spectra $S(q,\omega)$
which are independent of the kinematics of scattering,
and only then a direct comparison between our two experiments becomes possible.
This comparison is performed explicitely at 207/210\,K in Fig.~\ref{Fnsqw}.
In the quasielastic scattering range, up to some 100\,GHz,
the accord between Mib\'emol and {\sc Neat} is excellent.
\subsection{The Factorisation Property}
For analysing the data, especially in the quasielastic range,
it is advantageous to visualise the scattering law as a susceptibility
\begin{equation}\label{Etriv}
\chi''_q(\omega) = S(q,\omega) / n(\omega)
\end{equation}
with the Bose factor $n(\omega)= {(\exp(\hbar\omega/k_{\rm B}T)-1)}^{-1}$.
Around the susceptibility minimum,
mode coupling theory predicts
that the $\beta$ line shape is asymptotically
the same for all experimental observables
which couple to density fluctuations;
thus {\it a fortiori}\/
it must be the same for incoherent neutron susceptibilities
measured at different wavenumbers~$q$.
We therefore expect $\chi''_q(\omega)$ to factorise into
an $\omega$-independent amplitude
and a $q$-independent spectral function,
\begin{equation}\label{Efact}
\chi''_q(\omega) = h_q \chi''(\omega)\;.
\end{equation}
On different theoretical grounds,
such a factorisation is also expected for incoherent scattering
from harmonic vibrations in lowest order of mass expansion
\cite{Pla54,WuKB93}.
There are several ways to test (\ref{Efact}) and to determine $h_q$:
{\it e.\,g.}, one may iteratively construct a model for $\chi''(\omega)$
and fit it to the individual $\chi''_q(\omega)$ data sets.
More simply, $h_q$ can be calculated from a least-squares match between
neighbouring $q$ cuts \cite{WuKB93}.
For the present data,
we find that $h_q$ does not depend on the chosen procedure,
nor does it vary with the frequency subrange from which it is determined.
A first surprising result is then
the strictly linear wavenumber dependence of $h_q$
(Fig.~\ref{Ffact1}a).
Within MC theory, $q$ dependences can be calculated
only if a specific microscopic structure is put in.
Within harmonic theory, however,
there is a clear expectation that $h_q\propto q^2$.
Astonishingly enough,
this $q^2$ dependence is never seen.
Instead, in at least two other molecular systems
a linear $h_q\propto q$ dependence is found
though the temperature dependence of THz modes
indicates pure harmonic behaviour \cite{KiBD92,x34}.
From other time-of-flight studies \cite{WuKB93,CuDo90} we suspect
and a simulation \cite{x41} confirms
that this is mainly a multiple-scattering effect,
Anyway, the linearity is so accurate
that in our analysis we shall use (\ref{Efact}) with
$h_q=q/q_0$ (with an arbitrary normalisation $q_0=1\,$\AA$^{-1}$)
instead of employing the empirical values.
Fig.~\ref{Fnsu} shows that
the factorisation holds over the full experimental data range,
at least from 30 to 2500\,GHz,
except for the onset of $\alpha$ relaxation at the highest temperature
(which is associated with long-ranged transport
and depends therefore strongly on~$q$).
This allows us to condense our huge two-dimensional data sets into
$q$-independent functions $\chi''(\omega)$ with much improved statistics;
only on this basis is it possible to proceed with a
quantitative analysis of noisy data accumulated in relatively short scans.
Before proceeding with the analysis
we have to note that the $\chi''(\omega)$ determined by this method
and shown for 207/210\,K in Fig.~\ref{Ffact1}b
reveals considerable discrepancies
between Mib\'emol and {\sc Neat}
at frequencies above about 400\,GHz.
Although these deviations passed unnoticed on the double logarithmic scale
of Fig.~\ref{Fnsqw}b, they are already present in the raw data.
While we can exclude container scattering and dark counts as possible causes,
two other frequency-dependent effects are likely to contribute:
multiple scattering \cite{x41}
and an inaccurate detector efficiency correction \cite{x42}.
Multiple scattering cannot be corrected for
unless a comprehensive theoretical model of $S(q,\omega)$ is used as input;
therefore,
it presents a fundamental limitation to the determination of spectral shapes
by quasielastic neutron scattering.
\subsection{\boldmath Master Curves for $\beta$ Relaxation}
The $q$-independent $\chi''(\omega)$ can now be used
to test the scaling form (\ref{EminL}).
We use fits with a fixed~$\lambda$ to determine
$\chi_\sigma$ and $\omega_\sigma$.
With these values,
a master curve $\chi''(\omega/\omega_\sigma)/\chi_\sigma$ is constructed,
and from the master curve the scaling range is read off
which is then used for improved fits to the original data.
Figure~\ref{Fnsufi} shows fits to the original
$\chi''(\omega)$ for $\lambda=0.72$,
and Figure~\ref{F3nma} shows master curves for three $\lambda$'s.
In principle,
a self-consistent $\lambda$ can be determined in an iterative procedure
from free fits to the master curves.
For our $220-260\,$K data, $\lambda$ tends towards values around 0.69
(Figure~\ref{F3nma}a),
but the convergence is erratic,
and the outcome depends on the subjective decision of
which points to include in the master curve.
With the data on hand,
a more restrictive determination of $\lambda$
is possible from the temperature dependence~(\ref{Epars}).
The insets in Figure~\ref{F3nma} show that
a consistent linear behaviour
${\chi_\sigma}^2\propto{\omega_\sigma}^{2a}\propto\sigma$
is found only for $\lambda=0.72$ and $T_{\rm c}=182\,$K;
the exponent $a=0.32$, which corresponds to $\lambda=0.72$,
is confirmed by cross-checking
$\ln \chi_\sigma$ {\it vs.} $\ln \omega_\sigma$
(inset of Figure~\ref{Fnsufi}).
Figure~\ref{F3nma}c demonstrates that the value $\lambda=0.78$
suggested by preceeding light scattering and dielectric loss measurements
do not give a good description of the neutron scattering data:
the master curve is of poorer quality than for $\lambda=0.69$ or 0.72,
especially at frequencies around and below the minimum;
free fits with $g_\lambda$ show a clear trend towards smaller values of
$\lambda$;
and the scales $\chi_\sigma$ and $\omega_\sigma$
do not consistently follow~(\ref{Epars}).
This discrepancy motivates us to remeasure some light scattering spectra
around the susceptibility minimum.
\section{Light Scattering}
\subsection{Previous Studies}
Light scattering, just as neutron scattering,
comprises coherent and incoherent contributions.
Since the wavelength of visible light is much longer than molecular dimensions,
coherent scattering arises only from sound modes,
giving rise to discrete Brillouin lines.
Incoherent scattering, on the other hand,
sees local motion and yields a continuous spectrum.
In a very first approximation, this spectrum can be interpreted
as if it were a $q$ averaged neutron scattering law;
microscopic models suggest that the scattering mechanism involves
four-point density correlations and/or rotational motion.
Brillouin scattering
yields the velocity and damping constant of sound waves.
In principle \cite{FuGL90},
the sound dispersion through the glass transition reveals
the strength of the $\alpha$ relaxation so that it is possible
to read off the Debye-Waller factor~$f_q$ from fits to the Brillouin lines.
In practice,
this led to an inconsistent estimate of~$T_{\rm c}$ in PC \cite{ElBT92};
for a reliable determination of $f_q(T)$,
one should not only determine one limiting sound velocity
by ultrasonic experiments,
but also provide an independently determined memory function
as input \cite{DuLC94}.
More direct information on the microscopic dynamics is obtained from
the incoherent continuous spectra.
In order to suppress the much stronger Brillouin lines,
these spectra are preferentially gathered
in depolarised (VH) back-scattering geometry.
In an extensive study of PC,
besides a thorough discussion of Brillouin scattering,
Du {\it et al.} \cite{DuLC94}
have also measured VH spectra over a wide frequency band,
made accessible by combining
a tandem interferometer with a grating monochromator.
The measurements extended over a large temperature range,
including $\alpha$ relaxation as well as hopping processes below $T_{\rm c}$,
and were analysed within
both the ``idealised'' and ``extended'' version of MC theory.
While these measurements gave a broad overview
of the dynamics of glass-forming PC,
they did not yield spectral lineshapes with the precision we need now
for a quantitative comparison with neutron scattering results.
In particular,
the exponent parameter $\lambda=0.78$,
obtained within idealised theory from a global fit to $T>T_{\rm c}$ data,
was given with a relatively large error range of $\pm0.05$.
Furthermore,
as in other broad band light scattering studies performed
until quite recently,
the tandem interferometer was used in series with an insufficient bandpass
which did not fully suppress higher-order transmissions of the interferometer
\cite{SuWN98,BaLS99}.
For these reasons, we remeasured some depolarised spectra,
concentrating on the temperature and frequency range
of the asymptotic $\beta$ regime.
\subsection{Experimental Set-up}
Experiments were performed in Garching on
a Fabry-Perot six-pass tandem interferometer.
The instrument, bought from J.~R.\ Sandercock,
has been modified in several details to allow for stable operation and
high contrast.
The six-pass optics has been placed in a thermally isolating housing,
and the scanning stage is actively temperature stabilised.
The analogue-electronic stabilisation of the interferometer piezos
is replaced by computer control.
Entrance and exit pinholes are spatially separated from the six-pass optics,
so that the most critical alignements can be done without disturbing the
interferometer operation.
By placing additional masks in the six-pass optics,
in particular on mirror surfaces,
the cross-talk between different passes could be reduced by several orders,
and a straylight rejection of better than $10^{11}$ was achieved.
Depending on the free spectral range,
the instrument is used in series with an interference filter of
0.15 or 1\,GHz band width (Andover);
these filters are placed in a special housing
with active temperature stabilisation.
Furthermore, to account for long-term drift,
the instrument function is redetermined periodically
by automatic white-light scans.
Although the spectrometer guaranteed excellent straylight rejection,
special care was taken to prevent direct or diffuse reflections of laser light
from entering the instrument.
Therefore, instead of 180$^\circ$ back-scattering,
a 169$^\circ$ VH geometry was chosen.
From the intensity
transmitted through the ``ghosts'' of the tandem instrument,
we conclude that the straylight was about $10^5$ times weaker than
the inelastic scattering from PC,
and completely negligible
compared to the detector's dark count rate of about 2.5\,sec$^{-1}$.
After subtraction of these dark counts,
and normalisation to the corresponding white light scans,
the resulting spectra showed excellent detailed balance symmetry.
The sample material was from the same source
as for the neutron scattering experiment,
and was vacuum sealed in a Duran cell.
\subsection{\boldmath Susceptibilities Around $\omega_\sigma$}
For a precise determination of the spectral lineshape,
subsequent measurements over different spectral ranges
were performed after stabilising the temperature over night.
The most restrictive determination of the exponent parameter was possible
at $T=216\,$K.
As shown in Fig.~\ref{Fl1su},
the matching of the three overlapping spectral ranges is excellent.
Except for the leaking VV Brillouin mode,
the VH susceptibility is described over more than two decades by the
mode coupling asymptote $g_\lambda$.
Fits yield $\lambda=0.72\pm0.01$,
which is confirmed by measurements at other temperatures
as well as by a bulk of earlier experiments
we had performed under less ideal experimental conditions.
This result is at the margin of the
error range in the literature value $0.78\pm0.05$ \cite{DuLC94};
the figure shows that 0.78 itself is clearly incompatible with our present data.
Secondly,
for two different spectral ranges we measured temperature series.
Fig.~\ref{FlTsu} shows some of the composite susceptibilities.
From fits with fixed $\lambda=0.72$,
the frequencies $\omega_\sigma$ were obtained.
Then, the individual susceptibilities,
measured around interferometer mirror spacings $z_0=0.8$ and 2.4\,mm
were fitted with fixed $\lambda$ and $\omega_\sigma$
so that we obtained two independent data sets for the amplitudes $\chi_\sigma$.
The temperature dependence of $\omega_\sigma$ is shown in Fig.~\ref{Flpar}a.
Data between 190 and 230\,K extrapolate to the same $T_{\rm c}=182\pm1$\,K
as found from neutron scattering.
Extending the fit range to higher temperatures leads to a $T_{\rm c}$
which is about 2\,K lower.
Again, these results are marginally compatible with $T_{\rm c}=187\pm5$\,K
from the earlier light scattering study \cite{DuLC94}.
Measuring amplitudes in light scattering is difficult,
and the prediction (\ref{Epars}) can be verified only
over a reduced temperature range.
Results are visualised best in a logarithmic plot of $\chi_\sigma$
{\it vs.} $T-T_{\rm c}$ (Fig.~\ref{Flpar}b)
which suggests there are two different regions
in which $\chi_\sigma$ is proportional to $|\sigma|^{1/2}$,
separated by some step.
Without further experiments, we must leave open whether this step
comes from the sample or presents an experimental artifact,
due for instance to distortions of the optical paths.
We note, however, that after 30 hours and a full temperature cycle,
we were able to reproduce an amplitude $\chi_\sigma$ within less than 1\,\%.
Thus, irregularities in the amplitude are due not to a drift in time,
but mainly to reversible effects of temperature variation.
Putting aside the amplitude problem,
our results are in excellent accord with the neutron scattering data
of sect.~II.
This motivates us
to reconsider dielectric loss data
which have been analysed previously with $\lambda=0.78$.
\section{Other Spectroscopies}
\subsection{Dielectric Loss}
Dielectric spectroscopy on PC has been described recently
\cite{LuPD97b,LuPD96c,ScLB99}.
The measurements extended over a wide range of temperatures and many decades
in frequency,
and the analysis has addressed different issues
which are currently debated in the context of glass transition dynamics.
Within the GHz--THz range,
the dielectric susceptibility~$\epsilon''(\omega)$ passes through a minimum
as suggested by MC theory.
Fits with Eq.~(\ref{EminL}), approximated as a sum of two power laws \cite{x39},
gave $\lambda=0.78$,
and from the temperature dependence of frequency and amplitude (\ref{Epars})
$T_{\rm c}\simeq187\,$K was found.
Though these values had strong support in the existing literature,
they differ from our neutron and light scattering results.
It is therefore interesting to ask whether the dielectric loss would
also allow for $\lambda=0.72$.
Since the available dielectric data are even noisier
than the neutron susceptibilities,
there is no unique way to determine the scaling range within which
Eq.~(\ref{EminL}) applies;
therefore, any value of $\lambda$ depends on the choice of the fit range.
In the previous analysis \cite{ScLB99},
fits were applied between about 1 and 600\,GHz.
For temperatures around 200\,K this range covered both sides
of the minimum equally well.
When $\lambda=0.72$ is imposed in an iterative master curve construction,
the scaling range evades towards lower frequencies,
as can be seen from the fits in Fig.~\ref{Fdie72}.
For direct comparison,
Fig.~\ref{Fd2ma} shows master curves constructed with
two different values of $\lambda$.
In the upper curve, with $\lambda=0.78$,
the measured susceptibilities have been
rescaled with exactly the $\omega_\sigma$ and $\epsilon_\sigma$
shown in Fig.~8 of Ref.~\onlinecite{ScLB99}.
The rescaling works particularly well on the high-frequency side of the minimum
and shows nicely the $\omega^a$ limit which has remained elusive
in so many other experimental investigations.
The lower curve,
constructed with $\lambda=0.72$, %
shows that the scaling property is completely lost above $\omega_\sigma$,
whereas it has significantly improved in the low-frequency wing.
Furthermore,
the $\omega^{-b}$ limit of $g_\lambda$ describes the data
nearly up to the $\alpha$-relaxation maximum
where they almost coincide,
except for the lowest two temperatures.
This unexpected observation motivates a new master curve construction
which is quite common in the conventional analysis of $\alpha$ relaxation,
but which has never before been applied to MC $\beta$ relaxation:
in Fig.~\ref{Fama}, the dielectric data are shown with original amplitude,
rescaled only in frequency
by the $\alpha$ maximum frequency $\omega_{\rm max}$.
As usual for $\alpha$ relaxation
(and in accord with the second scaling law of MC theory)
the amplitude and the line shape of the $\alpha$ peak
are temperature independent.
But here,
the scaling behaviour extends far down into the high-frequency wing ---
in fact, it extends as far as the data have been measured,
except again the two lowest temperatures, 193 and 203\,K.
However, a master curve which extends from the maximum up to the minimum
of a dynamic susceptibility cannot simultaneously obey the first and the
second scaling law of MC: we expect
\begin{equation}
\omega_{\rm max}/\omega_{\rm min}\propto {|\sigma|}^{1/2b}\,,
\end{equation}
and, even more elementary, $\epsilon_{\rm min}\propto{|\sigma|}^{1/2}$
whereas $\epsilon_{\rm max}$ remains constant.
Since there is no doubt about the scaling of $\alpha$ relaxation,
we are bound to conclude that the dielectric data do not reach
the asymptotic regime of fast $\beta$ relaxation,
except in a rather small temperature range that extends
at best to about 210\,K.
For higher temperatures,
the high-frequency wing of $\alpha$ relaxation,
though technically describable by $g_\lambda$,
does not represent the first scaling law limit.
This conclusion is independent of any fitting,
and does in particular not depend upon an imposed value of~$\lambda$.
\subsection{Solvation Dynamics}
The solvation response of $s$-tetrazine has been measured in PC
from 1.5 to 100\,ps \cite{MaBB95}.
A MC analysis in the time domain identified both $\alpha$- and $\beta$-scaling
regions in the dynamics \cite{MaBB96}.
A unified analysis of both regions was consistent with MC theory for
a range of $\lambda$--$T_{\rm c}$ pairs.
In the original publication, $\lambda=0.78$ was fixed,
in accord with the light scattering analysis of Du {\it et al.} \cite{DuLC94}.
However, this value of $\lambda$ yielded
a cross-over temperature $T_{\rm c}=176\,$K
substantially below the value obtained by Du {\it et al.} ($T_{\rm c}=187\,$K).
The solvation data have been reanalysed by the same method used
in Ref.~\cite{MaBB96}, but using $\lambda=0.72$.
The analysis of the $\alpha$-scaling region is unchanged.
Figure~\ref{Fholes} shows the new beta-scaling plot
and the fit to $g_\lambda(t/t_\sigma)$,
both of which are as good as in the previous analysis.
The inset shows a temperature scaling plot
of $t_\sigma$ and $\tau_\alpha$.
As expected, the scaling law deteriorates at temperatures far above $T_{\rm c}$.
Depending on the range of temperatures fit,
acceptable values of $T_{\rm c}$ lie in the range 178--182\,K.
Thus, the solvation response is consistent
with both the $\lambda$ and $T_{\rm c}$ obtained in Sects.~II and III
by neutron and light scattering.
\section{Comparison of Dynamic Observables}
\subsection{Direct Comparison of Susceptibilities}
Neutron scattering,
light scattering with or without proper bandpass,
dielectric loss:
each experiment, taken alone,
seemed in full accord with the asymptotic predictions of MC theory.
Taken together, the situation becomes more complicated.
In the fast $\beta$ regime,
any observable that couples to density fluctuations
is expected to tend towards the same asymptotic limit~(\ref{EminL}).
For neutron scattering,
this prediction takes the form of a $q$,$\omega$ factorisation,
and is confirmed, in PC as in other materials, over a wide frequency range.
However, it breaks down completely when depolarised light scattering
or dielectric loss are included.
In Fig.~\ref{F3sus3},
dynamic susceptibilities from
neutron scattering, light scattering \cite{x40}, and dielectric loss
are shown for direct comparison
on an arbitrary intensity scale, but in absolute frequency units.
The result is in flagrant contradiction to any factorisation property:
there is no one temperature for which the measured
$\epsilon''(\omega)$ and $\chi''(\omega)$ fall together;
in particular, their minimum frequencies differ systematically.
On a first sight,
this outcome is a bad surprise,
and could make us doubt whether
fits of individual data sets with the asymptotic laws
(\ref{EminL}) and~(\ref{Epars}) are meaningful at all.
On closer examination,
the discrepancies between the three data sets can be traced back
to two major differences:
to the individually different temperature ranges
within which the $\beta$ asymptote applies,
and to a systematic shift in the frequency scale $\omega_\sigma$.
\subsection{\boldmath Temperature Range of $\beta$ Relaxation}
Within idealised MC theory,
all dynamic observables converge towards the same scaling limit
(\ref{EminL}) and~(\ref{Epars}),
characterised by just one lineshape parameter~$\lambda$ and one
frequency scale~$\omega_\sigma$.
This universality, however, is restricted to the lowest-order asymptote
and does not imply a universal radius of convergence:
the next-to-leading-order corrections already depend
on the microscopic coupling \cite{FrFG97b}
so that different observables may reach the asymptote
at different temperatures and frequencies.
Nevertheless, for neutron and light scattering,
as well as for solvation response,
(\ref{EminL}) and~(\ref{Epars}) hold over rather large temperature intervals.
In neutron scattering, (\ref{Epars}) holds best between 220 and 260\,K;
at 210\,K, the susceptibility minimum approaches the instrumental
resolution function, and the signal becomes very weak:
these technical limitations prevent us from following $S(q,\omega)$
closer to $T_{\rm c}$.
For light scattering, Fig.~\ref{Flpar} confirms (\ref{Epars})
with reasonable precision up to nearly 1.5\,$T_{\rm c}$.
On the other hand,
the $\alpha$ master curve in Fig.~\ref{Fama} suggested
that the range of $\beta$ scaling is rather small for dielectric loss.
This is corroborated by Fig.~\ref{F3ws} in which the $\omega_\sigma$
from fits with fixed $\lambda=0.72$ are compiled.
For the lowest temperatures 193--213\,K,
the minima of $\epsilon''(\omega)$ coincide with the light scattering data,
whereas for higher temperatures
the dielectric $\omega_\sigma$ cross over
to a much steeper temperature dependence characteristic for $\alpha$ relaxation
(actually even steeper than the MC prediction
${|\sigma|}^{1/2a+1/2b}$).
\subsection{\boldmath Frequency Scales $\omega_\sigma$}
As Fig.~\ref{F3ws} shows,
the $\omega_\sigma$ from neutron scattering fall about 35\,\%
below the $\omega_\sigma$ from light scattering.
The good agreement of $\omega_\sigma$ from neutron scattering
with ${t_\sigma}^{-1}$ from solvation dynamics
excludes the possibility that the discrepancy
between neutron and light scattering lies only in
multiple scattering or other technical shortcomings of neutron scattering.
{\it Ergo}, at least one of the two scattering techniques does not see
the true $\beta$ asymptote of MC theory.
On the other hand,
the observed scaling,
the quality of the $g_\lambda$ fits, and
the accord of the $\lambda$ and $T_{\rm c}$
call for a MC interpretation,
and indicate a transient behaviour that is closely coupled
to the universal $\beta$ asymptote.
\section{Discussion}
\subsection{\boldmath Strength of $\alpha$ Relaxation}
Although there is no universal criterion for the validity of $\beta$ scaling,
an upper limit may be given:
it is plausible that the asymptote~(\ref{EminL})
which is based on an expansion around the susceptibility minimum
will no longer apply when the minimum ceases to exist.
This happens when the height of the minimum, $\chi_\sigma$,
becomes comparable to the maximum of $\chi''(\omega)$ in the vibrational band.
This condition also determines the highest temperature for which
the proportionality~(\ref{Epars}) can hold.
On the other hand, the temperature range over which~(\ref{Epars}) holds
can be related to the strength of the $\alpha$ peak.
To explain this relation, we refer back to
the $\alpha$ master plot in Fig.~\ref{Fama}.
Besides the dielectric $\epsilon''(\omega/\omega_{\rm max})$,
the figure also shows
$\chi''_{\rm VH}(\omega/\omega_{\rm max})$ from depolarised light scattering.
For both data sets,
the same frequency scale $\omega_{\rm max}$ has been applied,
as determined from the maximum position of the dielectric $\alpha$ peak.
The intensity scale of the light scattering data is arbitrary
and has been adjusted by a global factor so that the maximum
of the microscopic excitations has about the same height as for the
dielectric data which are measured in absolute units.
Compared to the high-frequency maximum,
the $\alpha$ peak is about four times weaker in light scattering
than in dielectric loss.
In the high-frequency wing, at, say, $\omega\simeq30\omega_{\rm max}$,
the ratio is reduced to a factor of about 2.
This reduction is due to different positions of the $\alpha$ maxima,
and to different slopes of the wings.
At still higher frequencies, the susceptibilities cross over towards
the minimum.
For the lowest temperatures shown in the figure,
the ratio of~2 between dielectric and light scattering data
survives in this frequency range.
Thus, for a low-temperature value of~$\sigma$,
the $\beta$ amplitude~$\chi_\sigma$ for light scattering
is about two times smaller than for dielectric loss.
On the other,
from the condition stated above,
and because we have rescaled the susceptibilities to about
the same values at high frequencies,
the temperature range of $\beta$ scaling ends for both techniques
at about the same absolute value of $\chi_\sigma$.
Using the proportionality~$\chi_\sigma\propto{|\sigma|}^{1/2}$,
the dielectric data reach this limit after a 4 times smaller
temperature change than the light scattering data do.
Thus, the strength of $\alpha$ relaxation can explain
why the $\beta$ scaling regime extends over only 30\,K in dielectric loss,
compared to 90\,K or more in light scattering.
\subsection{\boldmath Frequency Range of $\beta$ Relaxation}
One can imagine many experimental imperfections that distort
spectra measured in a scattering experiment.
For instance,
multiple scattering could overlay $S(q,\omega)$ with a convolution with
itself.
But any such distortions
would affect the wings of the susceptibility much more than
the region around the minimum.
In particular, convolutions lead to similar corrections
as intrinsic next-to-leading-order terms.
Therefore,
it is not easy to explain the discrepancy observed in the $\omega_\sigma$.
We note, however, that the $g_\lambda$ fits to the neutron scattering data
in Fig.~\ref{Fnsufi} extend up to frequencies between 200 and 400\,GHz,
whereas in light scattering, the fitted $g_\lambda$ already deviate from the
measured susceptibilities a little above 100\,GHz.
It is therefore quite possible
that our neutron scattering analysis sees a preasymptotic transient
rather than the true $g_\lambda$.
This would ressemble the situation in glycerol \cite{WuHL94},
where the $\chi''(\omega)$ allowed for the construction of a master curve
and followed some MC predictions,
although the full $\beta$ asymptote was not reached.
Numeric solutions of a simple two-correlator model showed
how such a scenario can arise from a MC ansatz \cite{FrGM97}.
Similar fits, with one slave correlator for each observable,
also work for propylene carbonate \cite{GoVo00}.
\subsection{Current Theoretical Developments}
Some of the questions raised by our experiments are
also addressed by recent numerical solutions of MC equations
and molecular-dynamics simulations.
A MC analysis of the hard-sphere liquid as a function of wavenumber
has demonstrated that corrections to scaling are of differing importance
for different observables.
Analytic expansions have shown that corrections to the $\alpha$ process are
of higher order than corrections to the $\beta$ asymptote:
this may explain why $\alpha$ scaling holds over a wider range
than the factorisation property of $\beta$ relaxation \cite{FrFG97b}.
In a molecular-dynamics simulation
of a liquid made of rigid diatomic molecules \cite{KaKS98b},
orientational correlation functions have been analysed
for different angular momenta $l=0,1,2,\ldots$
All correlators were found to fulfil the MC factorisation ---
with the pronounced exception of $l=1$ for which the position of the
susceptibility minimum is shifted by a factor of~10 [{\it loc.\ cit.}, Fig.~8].
The authors attribute this peculiar behaviour to $180^\circ$ jumps,
which is not necessarily the right explanation for similar behaviour
in more complex fluids.
A scenario with an underlying type A transition
for odd $l$ has been proposed recently \cite{Sin99}.
It has been noted before that the dielectric response couples only weakly
to vibrational excitations \cite{LuPD96b},
and the peculiar strength of the $\alpha$ peak in the $l=1$ correlator
has also been found in an analytic extension of MC theory
to non-spherical particles \cite{ScSc97}.
A decomposition according to angular momenta might also explain
the excellent accord between neutron scattering and solvation response
found in Fig.~\ref{F3ws},
since both techniques see $l=0$
whereas light scattering might be dominated by $l=2$ \cite{LeDG97}.
\section*{Acknowledgements}
We thank H.~Z.~Cummins, M.~Diehl, J.~K.~Kr\"uger, H.~Leyser,
J.~R.\ Sandercock, A.~P.\ Sokolov, and
J.~Wiedersich for invaluable advice in setting up our Fabry-Perot spectrometer.
We are grateful to W.~Petry for continuous support,
and we thank him as well as
M.~Fuchs, W.~G\"otze, A.~P.~Singh and T.~Voigtmann for fruitful discussions.
We acknowledge financial aid by
the Bundes\-mini\-sterium f\"ur Bildung, Wissenschaft, Forschung und Technologie
through Verbundprojekte 03{\sc pe}4{\sc tum}9 and
03{\sc lo}5{\sc au\footnotesize 2}8
and through contract no.\ 13{\sc n}6917,
by the Deutsche Forschungsgemeinschaft under grant no.\ {\sc Lo}264/8--1,
by the European Commission
through Human Capital and Mobility Program ERB CHGECT 920001,
and by the National Science Foundation under {\sc Che9809719}.
The Laboratoire L\'eon Brillouin is a laboratoire commun CEA -- CNRS.
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\narrowtext
\begin{table}
\caption
{Exponent parameter~$\lambda$ and cross-over temperature~$T_{\rm c}$
of propylene carbonate as determined by different experimental techniques.}
\label{LitTab}
\tablenotetext[1]{This column indicates whether
the MC fit was based on the Debye-Waller factor $f_q$ (\ref{Esqrt}),
the $\alpha$ time scale $\tau_\alpha$ (\ref{Etaua}),
or the $\beta$ relaxation parameters $\omega_\sigma$ and $\chi_\sigma$
(\ref{Epars}). The subscript $\sigma_\pm$ indicates that $T<T_{\rm c}$ data
have also been used.}
\tablenotetext[2]{Used as input.}
\tablenotetext[3]{Determination of MC parameters judged unreliable
by the original authors.}
\tablenotetext[4]{The value $179\pm2$\,K obtained
in an alternative analysis using extended MC theory
is not directly comparable to the $T_{\rm c}$'s from idealised theory.}
\tablenotetext[5]{Transient hole burning on $s$-tetrazine
in propylene carbonate. Original analysis.}
\tablenotetext[6]{Original data from Ref.~\cite{BoHu85}.}
\begin{tabular}[th]{lllll}
method & param.\tablenotemark[1] & $\lambda$ & $T_{\rm c}\;$(K) & reference \\
\hline
neutron scattering & $f_q$ & -- & 210 &
\protect\onlinecite{BoHo91} \\
Brillouin scattering & $f_q$ & -- & $270\pm5$ &
\protect\onlinecite{ElBT92} \\
Brillouin scattering & $f_q$ &
--\tablenotemark[3] &
--\tablenotemark[3] &
\protect\onlinecite{DuLC94} \\
viscosity\tablenotemark[6] & $\tau_\alpha$ & 0.70 & 196 &
\protect\onlinecite{BoET90} \\
{\it same data}
& $\tau_\alpha$ & 0.78\tablenotemark[2] & $188\pm3$ &
\protect\onlinecite{DuLC94} \\
neutron scattering & $\tau_\alpha$ & 0.70\tablenotemark[2]& 180/188 &
\protect\onlinecite{BoET90,BoHo91} \\
dielectric loss & $\tau_\alpha$ & 0.78\tablenotemark[2] & 187\tablenotemark[2] &
\protect\onlinecite{ScLB99} \\
VH light scattering & $\omega_{\sigma_\pm}$, $\chi_{\sigma_\pm}$
& 0.78 & $187\pm5$\tablenotemark[4] &
\protect\onlinecite{DuLC94} \\
dielectric loss & $\omega_\sigma$, $\chi_\sigma$ & 0.78 & 187 &
\protect\onlinecite{LuPD97b,LuPD96c,ScLB99} \\
solvation response\tablenotemark[5]
& $t_\sigma$, $\tau_\alpha$ & 0.78\tablenotemark[2] & 176 &
\protect\onlinecite{MaBB96} \\
{\it same data}
& $t_\sigma$, $\tau_\alpha$ & 0.72\tablenotemark[2] & $180\pm2$ &
{\it this work} \\
neutron scattering & $\omega_\sigma$, $\chi_\sigma$ & $0.72$ &
$182$ & {\it this work} \\
VH light scattering & $\omega_\sigma$, $\chi_\sigma$
& 0.72 & $182$ & {\it this work} \\
\end{tabular}
\end{table}
\begin{figure}\caption
{Incoherent neutron scattering law $S(q,\omega)$ of propylene carbonate
at $T=207-210\,$K and $q=1.2\,$\AA$^{-1}$,
measured on two time-of-flight spectrometers
under different experimental conditions:
Mib\'emol ($\lambda_{\rm i}=8.5\,$\AA, circular Al container)
and {\sc Neat} ($\lambda_{\rm i}=5.5\,$\AA,
linear arrangement of glass capillaries).
Figure (b) repeats the data of (a) on a double logarithmic scale
and compares them to the measured resolution functions (lines).
Above 20\,GHz, the quasielastic spectra agree within about statistical error.}
\label{Fnsqw}
\end{figure}
\begin{figure}\caption
{Decomposition of $\chi''_q(\omega)$
according to the factorisation~(\protect\ref{Efact})
for Mib\'emol (210\,K) and {\sc Neat} (207\,K) data.
(a) The amplitude $h_q$ has been determined by least-squares matching
between 50 and 2500\,GHz.
Other methods yield almost identical results. ---
(b) The spectral function $\chi''(\omega)$
is obtained after dividing measured $\chi''_q(\omega)$ by $h_q$.
At low frequencies, the accord between both experiments is excellent;
however, at high wavenumbers and especially at high frequencies,
deviations are strong and systematic.
Multiple scattering
and imperfect correction for energy-dependent detector efficiency
are probably the main causes for distortions of the spectral line shape.}
\label{Ffact1}
\end{figure}
\begin{figure}\caption
{Rescaled susceptibilities $\chi''_q(\omega)/h_q$.
The resolution-broadened elastic line has been cut off,
except at 285\,K where the $q$-dependent $\alpha$ relaxation
can be fully resolved.
The factorisation (\ref{Efact}) is confirmed for both
the mode-coupling $\beta$ regime and the vibrational spectra in the THz region.}
\label{Fnsu}
\end{figure}
\begin{figure}\caption
{Susceptibilities $\chi''(\omega)$,
obtained as $q$-independent averages $\chi''_q(\omega)/h_q$
from Mib\'emol data,
for temperatures varying in 10\,K steps from 210 to 260\,K.
Solid lines are fits
with the mode coupling asymptote $g_\lambda(\omega/\omega_\sigma)$
(\protect\ref{EminL}) with $\lambda=0.72$ fixed.
The fit ranges, indicated by full symbols,
have been determined in a self-consistent iteration
from Fig.~\protect\ref{F3nma};
at low frequencies, they are limited by resolution or $\alpha$ relaxation,
at high frequencies by the low-lying vibrational peak. ---
In the inset, the logarithms of the fit parameters
$\chi_\sigma$ and $\omega_\sigma$ are plotted against each other.
The straight line shows the slope $a=0.318$ expected from theory.}
\label{Fnsufi}
\end{figure}
\begin{figure}\caption
{Susceptibility master curves
$\chi''(\omega) / \chi_\sigma$ vs. $\omega/\omega_\sigma$ from the
data set of Fig.~\protect\ref{Fnsufi}.
Fitting the same original data with different values of~$\lambda$
yields different scales $\chi_\sigma$ and $\omega_\sigma$
which lead then to distinctly differing master curves.
The temperature dependence~(\protect\ref{Epars})
of $\chi_\sigma$ ($\square$)
and $\omega_\sigma$ ($\blacklozenge$) is tested in the insets.
---
(a) Iterative rescaling yields a $\lambda$ of about 0.69
(but unstable and depending much upon idiosyncrasies of the fitting procedure).
The temperature dependence of $\chi_\sigma$ and $\omega_\sigma$
is in conflict with~(\protect\ref{Epars}). ---
(b) For $\lambda\simeq0.72$,
the rescaling is as good as for 0.69,
and $\chi_\sigma$ and $\omega_\sigma$ extrapolate from 220--260\,K
to a consistent~$T_{\rm c}=182\,$K. ---
(c) With $\lambda=0.78$ as suggested by part of the literature,
the master curve is of poorer quality,
fits with $g_\lambda$ show that the imposed value of $\lambda$ is
not self consistent,
and $\chi_\sigma$ and $\omega_\sigma$ are again
in conflict with~(\protect\ref{Epars}).}
\label{F3nma}
\end{figure}
\begin{figure}\caption
{Susceptibility $\chi_{\rm VH}(\omega)$ from depolarised light scattering
from propylene carbonate at 216\,K,
measured at the Fabry-Perot-Sandercock tandem interferometer at Garching,
with $\lambda_{\rm i}=514.5$\,nm and
with mirror spacings $z_0=0.8,2.4$ and 7.2\,mm.
The matching of the three data sets is excellent and would extent
even further without the VV Brillouin mode leaking around 10\,GHz.
The solid lines is a fit with the mode-coupling asymptote
$\chi_\sigma g_\lambda(\omega/\omega_\sigma)$ with $\lambda=0.72$.
The dotted line corresponds to the mean literature value $\lambda=0.78$.}
\label{Fl1su}
\end{figure}
\begin{figure}\caption
{Depolarised light scattering susceptibilities,
combined from two spectral ranges ($z_0=0.8$ and 2.4\,mm),
as function of temperature.
Solid lines are mode-coupling fits with fixed $\lambda=0.72$.
Open symbols indicate regions which had to be excluded from the fits:
above about 80\,GHz the cross-over to the non-universal vibrational spectrum;
around 8--15\,GHz the Brillouin line,
and at the highest temperature
the beginning of $\alpha$-relaxation.}
\label{FlTsu}
\end{figure}
\begin{figure}\caption
{Scaling parameters obtained
from mode-coupling fits.
The rec\-ti\-fied plot (a) confirms that $\omega_\sigma$
evolves with ${|\sigma|}^{1/{2a}}$.
When fitting only the full symbols, $T_{\rm c}=182\pm1\,$K is obtained in
full accord with the neutron scattering result from
Fig.~\protect\ref{F3nma}. When extending the temperature range,
$T_{\rm c}$ tends to decrease by a few K. ---
The amplitudes,
determined in\-de\-pen\-dent\-ly for two spectral ranges,
are ad\-van\-ta\-geous\-ly represented (b) in a logarithmic plot
{\it vs.} $T-T_{\rm c}$ with $T_{\rm c}=182\,$K as determined above.
The solid line shows the slope 1/2 expected from theory.}
\label{Flpar}
\end{figure}
\begin{figure}\caption
{Dielectric loss data as published in
Refs.~\protect\cite{LuPD97b,LuPD96c,ScLB99}.
Solid lines are MC fits (\ref{EminL}) with fixed $\lambda=0.72$.
This figure can be compared directly
to Fig.~7 in Refs.~\protect\cite{ScLB99}
where the same data are fitted with $\lambda=0.78$.}
\label{Fdie72}
\end{figure}
\begin{figure}\caption
{Dielectric loss data \protect\cite{LuPD97b,LuPD96c,ScLB99},
rescaled to two versions of a master function
$\epsilon''(\omega/\omega_\sigma)/\epsilon_\sigma$.
In the upper curve, the same $\omega_\sigma$ and $\epsilon_\sigma$
are used as determined in Ref.~\protect\onlinecite{ScLB99}
for $\lambda=0.78$.
For the lower curve, $\lambda=0.72$ was imposed which shifts the scaling
range towards the low-frequency wing of the susceptibility minimum.}
\label{Fd2ma}
\end{figure}
\begin{figure}\caption
{In this master plot,
the frequency scale is taken from the position of the $\alpha$ relaxation peak
in dielectric loss, and applied to both
dielectric loss $\epsilon''(\omega)$
(same data as in~\protect\cite{ScLB99})
and depolarised light scattering $\chi_{\rm VH}''(\omega)$
(scanned from Ref.~\protect\cite{DuLC94}).
While the $\epsilon''(\omega)$ are shown in absolute units,
the amplitude of $\chi_{\rm VH}''(\omega)$ is arbitrarily rescaled
by a global factor
so that the microscopic peaks culminate at about equal height. ---
This scaling representation elucidates
how a strong $\alpha$ peak reduces the temperature range
for which a susceptibility minimum can be observed.}
\label{Fama}
\end{figure}
\begin{figure}\caption
{Solvation response of $s$-tetrazine in propylene carbonate as
measured by transient hole burning.
Same data as in Fig.~6 of \protect\cite{MaBB96},
but now reduced with $\lambda=0.72$, $f_c=0.56$.
The solid curve is the scaling function $g_\lambda(t/t_\sigma)$.
The deviations at long times are attributed to $\alpha$ relaxation. ---
The inset ({\it cf.} Fig.~7 of \protect\cite{MaBB96})
shows the temperature scaling of $t_\sigma$ ($t_\beta$ in \protect\cite{MaBB96})
and $\tau_\alpha$ (both in ps),
using the exponents
$2a=0.636$ and $\gamma=2.395$ that correspond to $\lambda=0.72$.
The fits shown extrapolate to a common $T_{\rm c}=182\,$K in accord
with neutron scattering measurements.
(Good fits can be obtained over the range 178--182\,K).}
\label{Fholes}
\end{figure}
\begin{figure}\caption
{Direct comparison of susceptibilities
from neutron scattering [Mib\'emol, this work],
depolarised light scattering \protect\cite{DuLC94},
and dielectric loss \protect\cite{LuPD97b,LuPD96c,ScLB99}.
The absolute scale is chosen arbitrarily (but temperature-independent)
to make the data approximately coincide in the low-temperature,
low-frequency corner
where they possibly reach a common asymptotic regime.}
\label{F3sus3}
\end{figure}
\begin{figure}\caption
{Frequencies $\omega_\sigma$,
determined from mode-coupling fits with the same $\lambda=0.72$,
for neutron scattering, light scattering,
dielectric loss, and solvation dynamics
(from the time-domain analysis converted to $\omega_\sigma = 1/t_\sigma$).
Temperatures are shown as $T-T_{\rm c}$ with $T_{\rm c}=182\,$K.
The solid line shows the slope $1/2a=1.572$ expected for the given value
of~$\lambda$;
the dotted line, fitted to some dielectric data points,
has a slope of~3.1 which is even steeper than $1/2a+1/2b=2.4$ expected
for the $\alpha$-relaxation limit.}
\label{F3ws}
\end{figure}
\end{multicols}
\end{document}
|
1,116,691,497,804 | arxiv |
\section{Introduction}
Outcome prediction is based on discriminant analysis,
which is the use of known classifications of some observations to classify others \cite{Hastie1996}. The aim is to use some observations in the data set under study as the training set, to automatically discover and learn the rules linking an observation to its class. These rules can then be applied to the remaining observations in the test or validation set to assign them to classes. Discriminant analysis methods are often probabilistic, or model-based, where the observations in each class are assumed to be generated by a distribution specific to that class \cite{Hastie1996}.
Fraley and Raftery {\cite{Fraley2002} developed a popular model-based discriminative analysis approach to outcome prediction that has been used in a variety of contexts \cite{Dean2006, Fraley2007, Iverson2009, Murphy2010, Andrews2012}. This involves producing Bayesian posterior probabilities of belonging to a class, or class-conditional probabilities, by using model-based clustering to fit a Gaussian mixture model (GMM) to each class in the training set, and estimating the optimal parameters
whilst ignoring the uncertainty in these estimates. These are then used to calculate the class-conditional probabilities of the remaining observations.
Using leave-one-out cross-validation (LOOCV) \cite{Lachenbruch1968, Stone1974, Meek2001}, we found that whilst their approach produces accurate and efficient results for data sets consisting of a limited number of samples with low dimensionality, a number of challenges arise for data sets with large dimensions. These are data overfitting \cite{Clarke2008, Michiels2011}, and computational inability \cite{Clarke2008}. These factors combined produce outcome predictions of limited reliability if any at all. Each parameter contributes a $d$-dimensional Bayesian integral to the predictive distribution, which is at best computationally expensive to estimate numerically. Instead, at the expense of ignoring the uncertainty in the parameters, they are estimated by the most probable values using the Expectation-Maximisation algorithm (EM) \cite{Dempster1977, Fraley2002}. Yet this is also computationally expensive in high-dimensions \cite{Fraley2002, Bouveyron2007, McNicholas2011}. Several dimension-reduction approaches such as principal component analysis (PCA) \cite{Joliffe2002, Fraley2002, Mclustversion4}, subspace clustering \cite{Scott1983, Bouveyron2007}, and/or using constrained and parsimonious models \cite{McNicholas2011}, have been suggested in literature. The first approach runs the risk of compromising the prediction accuracy through loss of information found in higher dimensions \cite{Chang1983, Bouveyron2014}. The second assumes that high-dimensional data live in subspaces with dimensionality less than $d$ \cite{Scott1983, Bouveyron2007, Bishop2006} but relies on finding a good classifier in high dimensions \cite{Bishop2006, Bouveyron2014}, and the third approach is a trade-off between perfect modelling and what can be correctly estimated in practice. The latter two have been combined to reduce the number of parameters and dimensions for EM to consider \cite{Bouveyron2007, Berge2011, Bouveyron2014}. However underestimated parameter uncertainty remains.
The solution to these challenges will involve some form of dimension reduction without underestimating parameter uncertainty or loss of information in the data. In this paper we show that it is possible to do so by solving the $d$-dimensional Bayesian integrals analytically, and propose a Bayesian outcome prediction method that can be applied to any arbitrary number of dimensions $d$ without any approximations at parameter level. In Section $2$, we briefly describe the method developed by Fraley and Raftery \cite{Fraley2002}, and in Section 3 we describe our Bayesian outcome prediction method. In Section $4$, we apply our method to simulated examples, and show how the curse of dimensionality \cite{Bellman1957} is lifted in terms of CPU and memory. In Section $5$, we apply our method to a real biomedical example. We conclude with a discussion in Section $6$.
\section{Model-based discriminant analysis}
Fraley and Raftery \cite{Fraley2002} developed a model-based approach to discriminant analysis which is implemented in the R software environment \cite{R2013} using the reference package \textit{mclust} \cite{Fraley2002, Mclustversion4}. In this section, we briefly describe their method.
Many methods have been proposed for discriminant analysis, and it is applicable in a wide variety of settings \cite{Ripley1996, Duda2001, Mclachlan2003}. Discriminant analysis methods are often probabilistic, based on the assumption that observations in each class are generated by a distribution specific to that class. The number of classes $ J$ is assumed to be known. We typically have a multi-dimensional data set $D$ consisting of $N$ observations, and their corresponding class labels ${\sigma} \in \{0, \ldots, (J-1)\}$ that indicate which class each observation belongs to: i.e. $~D=\{({\ensuremath{\mathbf{m}} x}_{1},\sigma_{1}),\ldots,({\ensuremath{\mathbf{m}} x}_{N},\sigma_{N})\}$, consisting for each individual $i$, of a $d$-dimensional covariate vector ${\ensuremath{\mathbf{m}} x}_{i}$, and the class $\sigma_{i}$ to which that individual belongs to. The data set is then split into a training and test set. Each pair $(\ensuremath{\mathbf{m}} x_{i},\sigma_{i})$ is assumed to be is drawn independently, from an unknown joint probability distribution $p(\ensuremath{\mathbf{m}} x,\sigma)$, that describes the data set being studied. In biomedical data sets, we typically have $d \gg N$ \cite{Clarke2008, Michiels2011, Wang2008, Rinaldis2013, Naturebreast2012, Natureovarian2011, CCLE2012}. If we take $p(\ensuremath{\mathbf{m}} x_{i}|\sigma)$ to be the probability distribution of the observations belonging to class $\sigma$, and $p(\sigma)$ as the proportion of observations that belong to that class, or class-imbalance parameter, then according to Bayes' theorem, the posterior probability that an observation $\ensuremath{\mathbf{m}} x_{i}$ belongs to class $\sigma$, or class-conditional probability, is
\begin{equation}
p(\sigma|\ensuremath{\mathbf{m}} x_{i}) = \frac{p(\ensuremath{\mathbf{m}} x_{i}|\sigma)~p(\sigma)}{ \sum^{J-1}_{\sigma=0} p(\ensuremath{\mathbf{m}} x_{i}|\sigma)~p(\sigma)}
\label{eqn:bayes_theorem}
\end{equation}
In probabilistic discriminant analysis, a model is fit to each class in the training set, and observations in the test set are assigned to the class corresponding to the model in which they have the highest class-conditional probability. Most common model-based discriminant analysis methods are based on the assumption that the observations in each class are multivariate Gaussian \cite{Dasgupta1998, Fraley1998}. GMMs are also commonly used \cite{Hastie1996, Fraley2002, McNicholas2011, Bouveyron2014}.
Fraley and Raftery \cite{Fraley2002} fit one GMM to each class in the training set using model-based clustering, and applied the optimal model to the test set. This withdrew the need to know the number of classes, and covariance matrix structure a priori by allowing the data in the training set to determine the number of classes in the data set. The procedure is called \textit{mclustDA} \cite{Fraley2002, Mclustversion4}, and can be summarised as follows: Set the number of GMM components, and type(s) of covariance matrix structure to consider. Apply model-based hierarchical clustering \cite{Fraley2002, Banfield1993, Dasgupta1998, Fraley1998} to the training set to provide initial, suboptimal partitions. These provide initial values for the parameters and class-conditional probabilities, which are used to initialise the Expectation-Maximisation algorithm (EM) \cite{Dempster1977, Fraley1998, Dasgupta1998}. EM
is applied to find the optimal parameters for each GMM, assigning a class-conditional probability to each data point. The Bayesian Information Criterion (BIC) \cite{Schwarz1978, Dasgupta1998, Fraley1998, Fraley2002} is then computed for each GMM, with the one with the highest BIC estimated to be the optimal model. Degenerate solutions are alleviated by using the posterior mode (MAP) instead of the maximum likelihood estimates (MLE) for the EM and BIC calculations \cite{Fraley2007}. The optimal model is then applied to the test set, and the class-conditional probabilities are calculated using (\ref{eqn:bayes_theorem}). An observation in the test set is assigned to the class corresponding to the highest class-conditional probability. It has been suggested that \textit{mclustDA} may not work well or be insufficient for high-dimensional data due to computational inability and/or overfitting \cite{Fraley2002, Wehrens2004, Fraley2005, McNicholas2011, Mclustversion4}. A suggested solution is to restrict the parameters of the model \cite{McNicholas2011, Mclustversion4} or apply a dimension-reduction technique such as PCA \cite{Mclustversion4}. We apply the former at the risk of fitting an inadequate model to the data rather than risk losing information in the data by using an a priori dimension reduction technique.
We fit one isotropic Gaussian distribution to each class such that ${\ensuremath{\mathbf{m}} x}|\sigma \sim \mathcal{N}({\ensuremath{\mathbf{m}} \mu}_ {\sigma}, \lambda^{2}_ {\sigma} \, \mathbb{I}_{d\times d})$ where ${\ensuremath{\mathbf{m}} \mu}_ {\sigma}$ is a $d \times 1$ mean vector, and $\lambda^{2}_ {\sigma}$ is the variance of the distribution for class $\sigma$. This is achieved by choosing \textbf{EII} (Equal (\textbf{E}) volume, Identity Matrix (\textbf{I}) shape and Identity Matrix (\textbf{I}) orientation) as the covariance structure, \textit{mclustDA} as the model type, \textit{defaultPrior} as the prior, and setting G (the number of Gaussians to be considered per class) to $1$ \cite{Fraley2002, Mclustversion4}.
Using LOOCV, we found that for data sets consisting of a limited number of samples ($N=100$) with $d=3, 10, 100, 1000$, \textit{mclustDA} produces predictions in a reasonably quick time with little computational expense. For data sets consisting of a large number of samples with low $d$, \textit{mclustDA} was computationally expensive due to the large amount of time taken by model-based hierarchical clustering \cite{Fraley2002, Wehrens2004, Fraley2005}. Taking a random subset of the data set as the training set, and classifying the remaining observations (in reasonably sized blocks) \cite{Banfield1993, Fraley1998, Fraley2002, Mclustversion4} is more efficient, but can lead to decreasing accuracy \cite{Meek2001, Wehrens2004, Fraley2005}. A more accurate incremental approach has been developed but only for data sets whose storage and memory requirements are not too large to be handled in whole by the \textit{mclust} software package \cite{Wehrens2004, Fraley2005}.
For large $d$ (i.e. $d=10,000$), \textit{mclustDA} could not handle the large dimensionality of the data set, and failed to produce any results.
We surmise that for a large number of samples, it would be extremely expensive in terms of CPU power and memory to produce any results given the combined computational expense of model-based hierarchical clustering, and EM. For a limited number of samples, we would be overfitting heavily. We describe a method that both overcomes this computational inability, and reduce overfitting without sacrificing any information contained in the data set in Section $3$.
\section{Bayesian binary outcome prediction for any number dimensions}
In this section, we present a Bayesian outcome prediction method that can be applied to $2$ homogeneous classes, and any number of dimensions. We reduce the dimension of numerical integrals from $2d$ dimensions to $4$, for any $d$ (FD). For large $d$, this is reduced further to $3$ (LD), and we obtain a simple outcome prediction formula without integrals in leading order for very large $d$ (LOLD).
We fit one isotropic Gaussian distribution to each class such that ${\ensuremath{\mathbf{m}} x}|\sigma \sim \mathcal{N}({\ensuremath{\mathbf{m}} \mu}_ {\sigma}, \lambda^{2}_ {\sigma}\, \mathbb{I}_{d\times d})$ where ${\ensuremath{\mathbf{m}} \mu}_ {\sigma}$ is a $d \times 1$ mean vector, and $\lambda^{2}_ {\sigma}$ is the variance of the distribution for class $\sigma$. The parameter prior with respect to the mean is set to be ${\ensuremath{\mathbf{m}} \mu}_ {\sigma} \sim \mathcal{N}(0, \alpha^{-1}_ {\sigma}\,\mathbb{I}_{d\times d})$, where $\alpha_ {\sigma}$ is the precision of the distribution, and use $\delta(\lambda_ {\sigma} - \psi_ {\sigma})$ to fix $\lambda_ {\sigma}$, and treat it as hyperparameter. We only take the uncertainty at parameter level into account. If we know precisely or wish to control the class-imbalance in the data set, we can fix $p(\sigma)$ to our chosen value $\sigma^{\prime}_ {\sigma}$. In the absence of any information about the class-imbalance in the data set, we take this to be the class-imbalance in the training set $\bar{\sigma}_ {\sigma} = {N_ {\sigma}}/{N}$ (the maximum entropy distribution for $p(\sigma)$), where $N_ {\sigma}$ is the number of observations in class $\sigma$ in the training set \cite{Bishop2006}. Proceeding with $\sigma=1$ ($p(0|{\ensuremath{\mathbf{m}} x}^{\star}, D)$ can be calculated by subtracting $p(1|{\ensuremath{\mathbf{m}} x}^{\star}, D)$ from $1$), the predictive distribution for a new observation ${\ensuremath{\mathbf{m}} x}^{\star}$ is
\begin{align*}
p(1|{\ensuremath{\mathbf{m}} x}^{\star}, D) &= \int^{\infty}_{-\infty} \int^{\infty}_{-\infty}\mathrm{d} {\ensuremath{\mathbf{m}} \mu}_{0} \mathrm{d} {\ensuremath{\mathbf{m}} \mu}_{1}\,p({\ensuremath{\mathbf{m}} \mu}_{0}|D, \hat{\psi}_{0}, \hat{\alpha}_{0})\,p({\ensuremath{\mathbf{m}} \mu}_{1}|D, \hat{\psi}_{1}, \hat{\alpha}_{1})\,\frac{p({\ensuremath{\mathbf{m}} x}^{\star}|1, {\ensuremath{\mathbf{m}} \mu}_{1}, \hat{\psi}_{1})\,\bar{\sigma}_{1}}{p({\ensuremath{\mathbf{m}} x}^{\star}|0, {\ensuremath{\mathbf{m}} \mu}_{0}, \hat{\psi}_{0})\,\bar{\sigma}_{0}\,+\,p({\ensuremath{\mathbf{m}} x}^{\star}|1, {\ensuremath{\mathbf{m}} \mu}_{1}, \hat{\psi}_{1})\,\bar{\sigma}_{1}}
\\
\\
&\propto \int^{\infty}_{-\infty}\int^{\infty}_{-\infty} \mathrm{d} {\ensuremath{\mathbf{m}} \mu}_{0} \mathrm{d} {\ensuremath{\mathbf{m}} \mu}_{1}\,\mathrm{e}^{-\frac{1}{2\hat{s}^{2}_{0}}({\ensuremath{\mathbf{m}} \mu}_{0} -(\frac{\hat{s}_{0}}{\hat{\psi}_{0}})^{2}\sum_{i=1}^{N} \delta_{\sigma_{i}, 0} \,{\ensuremath{\mathbf{m}} x}_{i})^{2} -\frac{1}{2\hat{s}^{2}_{1}}(\ensuremath{\mathbf{m}}{\mu}_{1}-(\frac{\hat{s}_{1}}{\hat{\psi}_{1}})^{2} \sum_{i=1}^{N} \delta_{\sigma_{i}, 1} \,\ensuremath{\mathbf{m}} x_{i})^{2}} \Big[1+\frac{p({\ensuremath{\mathbf{m}} x}^{\star}|0, {\ensuremath{\mathbf{m}} \mu}_{0}, \hat{\psi}_{0})\bar{\sigma}_{0}}{p({\ensuremath{\mathbf{m}} x}^{\star}|1, {\ensuremath{\mathbf{m}} \mu}_{1}, \hat{\psi}_{1})\bar{\sigma}_{1}}\Big]^{-1}
\\
\\
&\text{with} \quad \hat{s}^{-2}_{0} = \hat{\alpha}_{0} + \hat{\psi}^{-2}_{0} \sum^{N}_{i=1} \delta_{\sigma_{i}, 0}\quad\quad \hat{s}^{-2}_{1} = \hat{\alpha}_{1} + \hat{\psi}^{-2}_{1} \sum^{N}_{i=1} \delta_{\sigma_{i}, 1}
\end{align*}
Transforming the integration variables using
\begin{align*}
{\ensuremath{\mathbf{m}} \mu}_{0} &= (\frac{\hat{s}_{0}}{\hat{\psi}_{0}})^{2} \sum_{i=1}^{N} \delta_{\sigma_{i}, 0} \,{\ensuremath{\mathbf{m}} x}_{i} + \hat{s}_0 {\ensuremath{\mathbf{m}} y_{0}}\quad \quad
{\ensuremath{\mathbf{m}} \mu}_{1} = (\frac{\hat{s}_{1}}{\hat{\psi}_{1}})^{2} \sum_{i=1}^{N} \delta_{\sigma_{i}, 1} \,{\ensuremath{\mathbf{m}} x}_{i} + \hat{s}_1 {\ensuremath{\mathbf{m}} y_{1}}
\\
\\
&\text{where} \quad {\ensuremath{\mathbf{m}} y_{0}} \sim \mathcal{N}(0,\mathbb{I}_{d\times d}) \quad \quad {\ensuremath{\mathbf{m}} y_{1}} \sim \mathcal{N}(0,\mathbb{I}_{d\times d})
\end{align*}
gives
\begin{align*}
p(1|{\ensuremath{\mathbf{m}} x}^{\star}, D)
&= \frac{\bar{\sigma}_{1}\hat{\psi}^{d}_{0}}{\bar{\sigma}_{0}\hat{\psi}^{d}_{1}}\,\int^{\infty}_{-\infty}\int^{\infty}_{-\infty} \,
\Big[\frac{\bar{\sigma}_{1}\hat{\psi}^{d}_{0}}{\bar{\sigma}_{0}\hat{\psi}^{d}_{1}}+\mathrm{e}^{\frac{1}{2}(\frac{\hat{s}_1}{\hat{\psi}_{1}})^{2}\Big(\frac{{\ensuremath{\mathbf{m}} x}^{\star}}{\hat{s}_{1}} - \frac{\hat{s}_{1}}{\hat{\psi}^{2}_{1}} \sum^{N}_{i=1}\delta_{\sigma_{i}, 1}~{\ensuremath{\mathbf{m}} x}_{i} - {\ensuremath{\mathbf{m}} y_{1}}\Big)^{2}- \frac{1}{2}(\frac{\hat{s}_0}{\hat{\psi}_{0}})^{2}\Big(\frac{{\ensuremath{\mathbf{m}} x}^{\star}}{\hat{s}_{0}} - \frac{\hat{s}_{0}}{\hat{\psi}^{2}_{0}} \sum^{N}_{i=1}\delta_{\sigma_{i}, 0}~{\ensuremath{\mathbf{m}} x}_{i} - {\ensuremath{\mathbf{m}} y_{0}}\Big)^{2}}\Big]^{-1}
\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \mathcal{D}{\ensuremath{\mathbf{m}} y_{1}}\,\mathcal{D}{\ensuremath{\mathbf{m}} y_{0}}
\end{align*}
\\
This is a $2d$-dimensional integral of the following form
\begin{equation}
p(1|{\ensuremath{\mathbf{m}} x}^{\star},D) =\int^{\infty}_{-\infty}\int^{\infty}_{-\infty} \frac{1}{1 + \mathrm{e}^{a_{1}({\ensuremath{\mathbf{m}} y_{1}}-\tilde{{\ensuremath{\mathbf{m}} y}}_{1})^{2} -a_{0}({\ensuremath{\mathbf{m}} y_{0}}-\tilde{{\ensuremath{\mathbf{m}} y}}_{0})^{2}-d\log(C_{01}) + \log(E_{01})}}\,\mathcal{D} {\ensuremath{\mathbf{m}} y_{1}} \,\mathcal{D} {\ensuremath{\mathbf{m}} y_{0}}
\label{eqn:predictive_distribution_simplified}
\end{equation}
with
\begin{equation*}
a_{1} = \frac{1}{2}(\frac{\hat{s}_{1}}{\hat{\psi}_{1}})^{2}\quad a_{0} = \frac{1}{2}(\frac{\hat{s}_{0}}{\hat{\psi}_{0}})^{2}\quad
\tilde{{\ensuremath{\mathbf{m}} y}}_{1} = \frac{{\ensuremath{\mathbf{m}} x}^{\star}}{\hat{s}_{1}} - \frac{\hat{s}_{1}}{\hat{\psi}^{2}_{1}} \sum^{N}_{i=1}\delta_{\sigma_{i},1}~{\ensuremath{\mathbf{m}} x}_{i}\quad \tilde{{\ensuremath{\mathbf{m}} y}}_{0} = \frac{{\ensuremath{\mathbf{m}} x}^{\star}}{\hat{s}_{0}} - \frac{\hat{s}_{0}}{\hat{\psi}^{2}_{0}} \sum^{N}_{i=1}\delta_{\sigma_{i},0}~{\ensuremath{\mathbf{m}} x}_{i}\quad
C_{01} = \frac{\hat{\psi}_{0}}{\hat{\psi}_{1}}\quad E_{01} = \frac{\bar{\sigma}_{0}}{\bar{\sigma}_{1}}
\end{equation*}
The MAP estimates for the hyperparameters $\psi_ {\sigma}$ and $\alpha_ {\sigma}$ are found by minimising the log-likelihood in the usual way.
We choose a flat prior for $\psi_ {\sigma}$. The only information we have about $\alpha_{\sigma}$ is that it is the precision of the mean ${\ensuremath{\mathbf{m}} \mu}_{\sigma}$, and $\geq 0$. Therefore we can choose an entropic prior \cite{Rodriguez2002, Caticha2004, Neumann2007} for $\alpha_{\sigma}$. This also has the advantage of rounding off the Bayesian hierarchy at hyperparameter level. The optimal hyperparameters used in (\ref{eqn:predictive_distribution_simplified}) are:
\begin{equation*}
\hat{\alpha}_ {\sigma} = 0 \quad \hat{\psi}^{2}_ {\sigma} =
\frac{N_ {\sigma}{\ensuremath{\mathbf{m}} \Sigma}_ {\sigma}^{2}}{(N_ {\sigma}-1)d}
\end{equation*}
where
\begin{equation*}
N_{\sigma} =\sum^{N}_{i=1}\delta_{\sigma_{i}, \sigma}\quad \langle {\ensuremath{\mathbf{m}} x} \rangle_{\sigma} = \frac{1}{N_ {\sigma}} \sum^{N}_{i=1}\delta_{\sigma_{i}, \sigma}~{\ensuremath{\mathbf{m}} x}_{i}\quad
\langle {\ensuremath{\mathbf{m}} x}^{2} \rangle_{\sigma} =\frac{1}{N_ {\sigma}} \sum^{N}_{i=1}\delta_{\sigma_{i}, \sigma}~{\ensuremath{\mathbf{m}} x}^{2}_{i}\quad {\ensuremath{\mathbf{m}} \Sigma}_ {\sigma}^{2} = \langle {\ensuremath{\mathbf{m}} x}^{2} \rangle_ {\sigma} - \langle {\ensuremath{\mathbf{m}} x} \rangle_ {\sigma}^{2}
\end{equation*}
\subsection*{Evaluating integral $p(1|{\bf x^{\star}},D)$ for finite d}
We rotate the basis for the vectors ${\ensuremath{\mathbf{m}} y_{1}}$ and ${\ensuremath{\mathbf{m}} y_{0}}$ such that the first unit vector in the space of ${\ensuremath{\mathbf{m}} y_{1}}$ ($y_{11}$) points towards $\tilde{{\ensuremath{\mathbf{m}} y}}_{1}$, and the first unit vector in the space of ${\ensuremath{\mathbf{m}} y_{0}}$ ($y_{01}$) points in the direction of $\tilde{{\ensuremath{\mathbf{m}} y}}_{0}$. This results in a formula in terms of the absolute values $\tilde{y}_{1} = |\tilde{{\ensuremath{\mathbf{m}} y}}_{1}|$, and $\tilde{{y_{0}}} = |\tilde{{\ensuremath{\mathbf{m}} y_{0}}}|$ alone:
\begin{align}
p(1|{\ensuremath{\mathbf{m}} x}^{\star},D)
&= \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty} W(Y_{1})\, W(Y_{0})\frac{1}{1 + \mathrm{e}^{a_{1}Y_{1} - a_{0}Y_{0} + a_{1} (y_{11}-\tilde{y}_{1})^{2} - a_{0} (y_{01}-\tilde{y}_{0})^{2}-d \log(C) - \log(E)}}\nonumber
\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mathrm{d} Y_{1}\, \mathrm{d} Y_{0}\,\mathcal{D} y_{11}\,\mathcal{D}y_{01}
\label{enq:4d_prediction_formula}
\\\nonumber
\text{where, for} \,\, d \geq 3,\nonumber
\end{align}
\begin{equation}
W(Y_{\sigma}) = \int^{\infty}_{-\infty} \mathcal{D}{y_{2}}\ldots \mathcal{D}{y_{d}}~\delta[Y_{\sigma}-\sum^{d}_{i=2} y_{i}^{2}] = \int_{-\infty}^{\infty} \frac{\mathrm{d} v}{2\pi}~\frac{\mathrm{e}^{ivY_{\sigma}}}{(1+2iv)^{\frac{d-1}{2}}} = \frac{2^{-\frac{(d-1)}{2}}}{\Gamma{(\frac{d-1}{2})}}~\theta(Y_{\sigma})~~\mathrm{e}^{[-\frac{Y_{\sigma}}{2} ~+~ (\frac{d-3}{2})\log(Y_{\sigma})]}
\label{eqn:W(Y_sigma)}
\end{equation}
Substituting (\ref{eqn:W(Y_sigma)}) into (\ref{enq:4d_prediction_formula}) gives
\begin{align}
p(1|{\ensuremath{\mathbf{m}} x}^{\star},D) &= \frac{2^{-(d-1)}}{\Big(\Gamma{(\frac{d-1}{2})}\Big)^2}~\int^{\infty}_{-\infty} \int^{\infty}_{-\infty} \int^{\infty}_{0} \int^{\infty}_{0} \frac{e^{-\frac{1}{2}(Y_{1}+Y_{0})\,+\,(\frac{d-3}{2})\log(Y_{1}Y_{0})}}{1 +\mathrm{e}^{a_{1}Y_{1} - a_{0}Y_{0} + a_{1} (y_{11}-\tilde{y}_{1})^{2} - a_{0} (y_{01}-\tilde{y}_{0})^{2}-d \log(C) - \log(E)}}\nonumber
\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \mathrm{d} Y_{1}\, \mathrm{d} Y_{0}\,\mathcal{D} y_{11}\,\mathcal{D}y_{01}
\label{eqn:4d_prediction_formula_simplified}
\end{align}\\
where (\ref{eqn:4d_prediction_formula_simplified}) is $4$-dimensional integral. Thus for $d \geq 3$ we have shown that solving the $d$-dimensional Bayesian integrals analytically has simplified our predictive distribution to a $4$-dimensional integral (FD).
\begin{figure}[t]
\centering
\includegraphics[height=6cm, width=12cm, scale=0.75]{synthetic_100N_s0_dv_av10_entropic_all_bayesian_mclust_ROC_methods_summary_plot.ps}
\caption{ROC curves for the simulated data sets using the different versions of our Bayesian method for $d=3$ (solid line), $d=10$ (dashed line), and $d=100$ (dotted line).}
\label{fig:roc_curves_all_versions_comparison_simulated_c2_data}
\end{figure}
\subsection*{Evaluating integral $p(1|{\bf x^{\star}},D)$ for large d}
For large $d$, we can use the central limit theorem to write $Y_{\sigma} = (d-1) + Y_{\sigma}^{\star}\sqrt{2(d-1)}$, where $Y_{\sigma}^{\star} \sim \mathcal{N}(0,1)$. This gives
\begin{equation}
W(Y_{\sigma}) = \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} \mathrm{d} Y_{\sigma}^{\star}\, \mathcal{D}Y_{\sigma}^{\star}~\delta \Big[Y_{\sigma} - \Big((d-1) + Y_{\sigma}^{\star}\sqrt{2(d-1)}\Big)\Big]
\label{eqn:W(Y_sigma)_central_limit_theorem}
\end{equation}
Substituting (\ref{eqn:W(Y_sigma)_central_limit_theorem}) into (\ref{enq:4d_prediction_formula}) gives
\begin{align}
p(1|{\ensuremath{\mathbf{m}} x}^{\star},D) &= \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} \frac{1}{1 + \mathrm{e}^{(a_{1}-a_{0})(d-1) + \sqrt{2(d-1)}\sqrt{a_{1}^2+a_{0}^2}w} \, \mathrm{e}^{a_{1}(y_{11}-\tilde{y}_{1})^{2} - a_{0}(y_{01}-\tilde{y}_{0})^{2}-d\log(C) - \log(E)}}\nonumber
\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mathcal{D} w \,\mathcal{D} y_{11} \,\mathcal{D} y_{01}
\label{eqn:3d_prediction_formula}
\end{align}
where we have combined the sum $a_{1}Y_{1}^{\star}-a_{0}Y_{0}^{\star}$ into a single Gaussian variable $w$, with variance $a_{1}^2+a_{0}^{2}$. Thus for large $d$ we reduce the $4$-dimensional integral to a $3$-dimensional one (LD).
\begin{figure}[t]
\centering
\includegraphics[height=6cm, width=12cm, scale=0.75]{synthetic_100N_s0_dv_av10_entropic_bayesian_leading_order_large_D_mclust_ROC_plot.ps}
\caption{ROC curves for the simulated data sets $d=3,10,100,1000$, using LOLD (solid line) and \textit{mclustDA} (dashed line).}
\label{fig:roc_curves_LOLD_mclust_simulated_c2_data}
\end{figure}
\subsection*{Evaluating integral $p(1|{\bf x^{\star}},D)$ for leading order large d}
If we only take the leading order terms (the other terms become negligible as $d$ gets large), then (\ref{eqn:3d_prediction_formula}) becomes a simple formula
\begin{equation*}
p(1|{\ensuremath{\mathbf{m}} x}^{\star},D) = (1+\mathrm{e}^{\,d\,\Lambda_{10}})^{-1} ,\quad\quad \text{with} \quad \Lambda_{10} = a_{1}-a_{0} - \log(C) + \frac{a_{1}\tilde{y}_{1}^{2}}{d}-\frac{a_{0}\tilde{y}_{0}^{2}}{d}
\end{equation*}
Thus we have
\begin{equation*}
p(1|{\ensuremath{\mathbf{m}} x}^{\star},D)=
\begin{cases} 0 \,+\, \mathcal{O}(\mathrm{e}^{-d\,\Lambda_{10}}) & \text{if} \quad\Lambda_{10}>0
\\
1 \,-\,\mathcal{O}(\mathrm{e}^{-d\,|\Lambda_{10}|}) & \text{if} \quad \Lambda_{10}<0
\end{cases}
\end{equation*}
which gives
\begin{equation}
p(1|{\ensuremath{\mathbf{m}} x}^{\star},D) = \theta(-\Lambda_{10}) \,\pm\, \mathcal{O}(\mathrm{e}^{-d\,|\Lambda_{10}|})
\label{eqn:simple_predictive_equation}
\end{equation}
Thus we have shown that for leading order large $d$ the $2d$-dimensional predictive distribution simplifies to a simple formula (LOLD) in (\ref{eqn:simple_predictive_equation}).
\section{Simulation study}
In this section, we apply our proposed Bayesian method, and \textit{mclustDA} to simulated data sets.
In order to compare our proposed Bayesian method to \textit{mclustDA}, we simulated data sets consisting of $N=100$ observations, and their corresponding classifications. The model complexity (the number of unknown parameters) was varied by increasing the dimension of each data set, from the lowest dimension that our method can be applied to, $d=3$, to $d=10, 100, 1000,10000$. The last dimension cannot be handled by \textit{mclustDA}, and is used to illustrate the power of our Bayesian method in very large dimensions that are typical of large biomedical data sets \cite{Clarke2008, Michiels2011, Wang2008, Rinaldis2013, Naturebreast2012, Natureovarian2011, CCLE2012}.
\begin{figure}[t]
\centering
\includegraphics[height=6cm, width=12cm, scale=0.75]{synthetic_100N_c2_s0_dv_av10_entropic_leading_order_large_D_mclust_error_plot.ps}
\caption{Error curves for the simulated data sets with $d=3,10,100,1000, 10000$ at a decision threshold of $0.5$, using LOLD (solid line), and \textit{mclustDA} (dashed line). The performance improves in high dimensions as the peaks (mass) of the Gaussian distributions move away from each other whilst the width of each distribution remains constant \cite{Bishop2006}.}
\label{fig:error_curves_simulated_c2_data}
\end{figure}
We simulated data sets whose classes had the same mean vector in order to test the predictive and classification power of our methods, since well-separated classes are easier to classify particularly in higher dimensions \cite{Scott1983, Bishop2006}. The variances of the classes were set to be weakly different values, in order to allow one class to nest inside the space of the other. This allowed the classes to become more separated as the dimension increased, since the location of the peaks (mass) of the Gaussian distributions move away from each other whilst the width of each distribution increases much slower (until it becomes constant) as the dimension increases \cite{Bishop2006}. In contrast, where the mean vectors and variances are the same, the classes are indistinguishable \cite{Bishop2006}.
For each dimension, two classes, $\sigma = \{0,1\}$, of observations were simulated according to ${\ensuremath{\mathbf{m}} x}|\sigma \sim \mathcal{N}({\ensuremath{\mathbf{m}} \mu}_ {\sigma}, \lambda^{2}_ {\sigma} \, \mathbb{I}_{d\times d})$, with $\mu_{0} = \mu_{1} = 0$, $\lambda_{0} = 0.24$, and $\lambda_{1} = 0.28$. The number of observations in each class was kept the same ($N_{0}=N_{1}=50$). Classifications, $\sigma=\{0,1\}$, were generated for the number of observations in each class respectively.
The model parameters, class-imbalance parameters, and number of observations for each dimension are summarised in Table \ref{table:simulated_data_sets}.
\begin{table}[h]
\centering
\caption{Parameters of the simulated isotropic Gaussian data sets consisting of two distinct classes.}
\vspace*{2mm}
\begin{tabular}{|cccccccccc|}
\toprule
& $N$ &$N_{0}$ &$N_{1}$ &$\ensuremath{\mathbf{m}} \mu_{0}$ & $\ensuremath{\mathbf{m}} \mu_{1}$ & $\lambda_{0}$ & $\lambda_{1}$ & $\bar{\sigma}_{0}$ &$\bar{\sigma}_{1}$ \\
\midrule
& $100$ &$50$ &$50$ & $0$ & $0$ & $0.24$ & $0.28$ &$0.5$ &$0.5$\\
\bottomrule
\end{tabular}
\label{table:simulated_data_sets}
\end{table}
In order to determine when to use the different versions (FD, LD, LOLD) of our method, we applied all three versions to the simulated data sets with relatively low dimensions ($d=3,10$). Only LD and LOLD were applied to the simulated data sets with $d=100$. \textit{mclusDA} was applied as a reference.
The results using a threshold of $0.5$ are shown in Table \ref{table:types_of_bayesian_method_comparison}. The receiver operating characteristic (ROC) curves \cite{Gerds2008} for all the versions and dimensions are displayed in Figure \ref{fig:roc_curves_all_versions_comparison_simulated_c2_data}.
We observe that LOLD preforms best which suggests that the leading order (constant) $\mathcal{O}(d)$ terms dominate the integrated terms which are $\mathcal{O}(1)$. Therefore we continue our analysis using LOLD.
The ROC curves for both methods in the different dimensions are shown in Figure \ref{fig:roc_curves_LOLD_mclust_simulated_c2_data}. As expected, neither LOLD nor \textit{mclustDA} perform well for $d=3$ as they cannot distinguish between the classes. Both methods improve as the dimension of the data set increases given that it lives in subspaces with an effective dimensionality less than $d$ \cite{Bishop2006, Scott1983}, although LOLD outperforms \textit{mclustDA} on average in all dimensions. For $d=10000$, LOLD performs with $99$ per cent accuracy.
Using a classification threshold of $0.5$, Figure \ref{fig:error_curves_simulated_c2_data} illustrates the average training and test error for both methods in different dimensions. We notice that for LOLD overfitting is generally reduced as the number of dimensions increases until it ceases for $d=1000$. For \textit{mclustDA}, overfitting is reduced at a slower rate but remains for $d=1000$. Therefore we have demonstrated that on average, LOLD performs as well as or better than \textit{mclustDA}, and reduces overfitting until it ceases for $d=1000$.
\begin{table}[h]
\centering
\caption{Average performance comparison for the different methods.}
\vspace*{2mm}
\begin{tabular}{|c||cccc|}
\toprule
$d$ &Method &Threshold &Accuracy &Computation time (mins) \\
\midrule
& \textit{mclustDA} &$0.5$ &$0.52$ & $0.01$\\[0.5ex]
\raisebox{-1.5ex}{$3$} & FD &$0.5$ &$0.52$ & 32551.78\\[0.5ex]
& LD &$0.5$ &$0.52$ & 2214.55\\[0.5ex]
& LOLD &$0.5$ &$0.54$ & 0.00\\[1ex]
\hline
\hline
\\[-2ex]
& \textit{mclustDA} &$0.5$ &$0.60$ & $0.01$\\[0.5ex]
\raisebox{-1.5ex}{10} & FD &$0.5$ &$0.61$ & $57295.17$\\[0.5ex]
& LD &$0.5$ &$0.61$ & $3897.88$\\[0.5ex]
& LOLD &$0.5$ &$0.61$ & $0.00$\\[1ex]
\hline
\hline
\\[-2ex]
& \textit{mclustDA} &$0.5$ &$0.74$ & $0.09$\\[0.5ex]
\raisebox{-0.5ex}{100}
& LD &$0.5$ &$0.76$ & $4263.10$\\[0.5ex]
& LOLD &$0.5$ &$0.76$ & $0.00$
\\[1ex]
\hline
\hline
\\[-2ex]
& \textit{mclustDA} &$0.5$ &$0.94$ & $9.14$\\[0.5ex]
\raisebox{1ex}{1000}
& LOLD &$0.5$ &$0.99$ & $0.00$
\\[1ex]
\hline
\hline
\\[-2ex]
{10000}
& LOLD &$0.5$ &$0.99$ & $0.07$
\\[1ex]
\bottomrule
\end{tabular}
\label{table:types_of_bayesian_method_comparison}
\end{table}
\section{Application to triple-negative breast cancer data}
In this section, we apply our proposed Bayesian method, and \textit{mclustDA} to a real example. The gene expression data set for triple-negative breast cancer patients (\textit{exprset2}) from \textit{de Rinaldis et al.} \cite{Rinaldis2013} has been used.
The data set consists of $N=176$ patients,
who were treated at Guy's and St Thomas's Hospitals, London, UK between 1979 and 2001, and had at least $5.5$ years follow-up. Each patient had $d=22035$ gene expressions recorded. Patients with missing data or who were lost to follow up were excluded from our analysis. This yielded $N=165$ patients to study.
Patients who survived for an interval of at least $5$ years from initial diagnosis were designated as class $\sigma=1$, the good prognosis group: GP($1$). Patients who died from breast cancer within $5$ years were designated as class $\sigma=0$, the poor prognosis group: GP($0$). The aim was to use the patients' gene expression data to predict which prognosis group they belong to. The total number of patients, the number of patients in each class, and label-imbalance parameters are summarised in Table \ref{table:exprset2_data_set}.
\begin{table}[h]
\centering
\caption{\textit{exprset2} data set.}
\vspace*{2mm}
\begin{tabular}{|cccccccc|}
\toprule
$d$ &GP($0$) &GP($1$) &$N$ &$N_{0}$ &$N_{1}$ & $\bar{\sigma}_{0}$ &$\bar{\sigma}_{1}$ \\
\midrule
$22035$ & $<5 yrs$ & $\geq 5 yrs$ &$165$ &$41$ & $124$ &$0.25$ &$0.75$\\
\bottomrule
\end{tabular}
\label{table:exprset2_data_set}
\end{table}
In order to apply \textit{mclustDA} to the \textit{exprset2} data set, we reduced the number of dimensions a priori since \textit{mclustDA} could not handle such large dimensions. Based on the hypothesis that genes with greater correlation with outcome are likely candidates for reporting prognosis \cite{Veer2002, Vijver2002}, the correlation coefficient of the expression for each gene, and outcome was calculated. The genes were then ranked-ordered according to the magnitude of the correlation coefficient \cite{Veer2002, Vijver2002}. The highest ranked $100$, $1000$ and $10000$ genes were chosen. \textit{mclustDA} was applied to the first two reduced \textit{exprset2} data sets consisting of $N=165$ patients and $d=100$ genes, and $N=165$ patients and $d=1000$ genes respectively. LOLD was applied to all three reduced data sets, and the full data set. The results using a decision threshold of $0.5$ are shown in Table \ref{table:exprset2_data_results}. LOLD performed as well as \textit{mclustDA}, and was more efficient for $d=100$. For $d=1000$, LOLD was both more accurate and efficient than \textit{mclustDA}. In higher dimensions, overfitting causes the predictive accuracy of LOLD to decrease. Using a classification threshold of $0.5$, Figure \ref{fig:error_curves_exprset2_c2_data} illustrates the training, and test error for both methods. Compared to \textit{mclustDA}, LOLD reduces overfititng for the two lower dimensions. For the two higher dimensions, LOLD is clearly overfitting although the optimal predictive accuracy only decreases by 6 per cent.
\begin{table}[h]
\centering
\caption{\textit{exprset2} data set results.}
\vspace*{2mm}
\begin{tabular}{|c||cccc|}
\toprule
Method &$d$ &Threshold &Accuracy &Computation time (mins) \\
\midrule
& $100$ &$0.5$ &$0.76$ & $0.09$\\[-1ex]
\raisebox{1.5ex}{\textit{mclustDA}} & $1000$ &$0.5$ &$0.75$ & $11.36$\\[2ex]
\hline
\hline
\\[-2ex]
& $100$ &$0.5$ &$0.76$ & $0.00$\\[0.5ex]
\raisebox{-1.5ex}{LOLD} & $1000$ &$0.5$ &$0.76$ & $0.03$\\[0.5ex]
& $10000$ &$0.5$ &$0.74$ & 0.40\\[0.5ex]
& $22035$ &$0.5$ &$0.70$ & $1.67$
\\[1ex]
\bottomrule
\end{tabular}
\label{table:exprset2_data_results}
\end{table}
Additionally, we now know a posteriori that the predictive information lies in the first $1000$ rank-ordered genes for this data set. This would not have been possible with \textit{mclustDA} as it was less accurate for these $1000$ genes, and could not handle the first $10000$ rank-ordered genes. This information may have been lost using a dimension reduction technique a priori, since it would require a user-defined arbitrary cut-off which may have been less than the first $1000$ rank-ordered genes.
\begin{figure}[t]
\centering
\includegraphics[height=6cm, width=12cm, scale=0.75]{exprset2_DIAG_to_DEATH_2_entropic_leading_order_large_D_mclust_error_plot.ps}
\caption{Error curves for the exprset2 data sets $d=100,1000,10000, 22035$ at a decision threshold of $0.5$, using LOLD (solid line), and \textit{mclustDA} (dashed line).}
\label{fig:error_curves_exprset2_c2_data}
\end{figure}
\section{Discussion}
We have proposed a Bayesian binary outcome prediction method that can be applied to any arbitrary number of dimensions without any approximations at parameter level. Our method overcomes the computational inability associated with other methods in large dimensions without sacrificing any information from the data set. Although three prediction formulas, (FD, LD, LOLD) have been developed, we have shown that it suffices to use the most efficient version, LOLD, for any arbitrary dimension, without compromising predictive performance.
We compared our method to \textit{mclusDA} \cite{Fraley2002, Mclustversion4} due to its Bayesian approach, and wide application \cite{Dean2006, Fraley2007, Iverson2009, Murphy2010, Andrews2012}. For data sets where the methods produce the same results, our method is more efficient. For high-dimensional data sets, our Bayesian method is superior in both accuracy and efficiency. Our method also succeeds in reducing overfitting although it remained in very high-dimensions.
Our method can be seen as an extension for \textit{mclustDA}, in a homogeneous binary class setting, where we take the uncertainty at parameter level into account instead of estimating the Bayesian integrals using EM. However the methods differ in their application to the training set. We use a supervised method by applying our Bayesian method to training set using both the training data and outcome information whilst \textit{mclustDA} applies an unsupervised model-based clustering algorithm to the training data only. A limitation of our approach is that we must set the number of classes in the data set, and select the optimal model a priori, whilst \textit{mclustDA} calculates this from the data.
We also limited our approach to only two homogeneous classes. A natural extension would be to expand our approach to consider more than two classes. Our method could also extend to heterogeneous classes. The would entail fitting a more complex GMM to each class. We would also need to consider Bayesian model selection in order to choose the optimal parametrisation for each GMM. The uncertainty in model selection, and at hyperparameter level could also be incorporated into our frame work. However the risk of heavily overfitting remains for all these ideas, as the number of unknown parameters, and hyperparameters will increase considerably.
Finally, we have only compared our method to the \textit{mclustDA}. Many other model-based discriminant analysis methods exist that overcome computational inability in high-dimension \cite{McNicholas2011, Bouveyron2014}. These include using variable selection \cite{Raftery2006, Murphy2010}, and combining subspace clustering with constrained and parsimonious models \cite{Bouveyron2007, Bouveyron2014}. However, these methods may still suffer from loss of information, and/or underestimated uncertainty at parameter level. Overfitting remains where the imbalance between the number of observations and dimensions remains high, even after reducing the dimensionality of the data set. The method we propose overcomes these problems by reducing the number of dimensions analytically, whilst also taking the uncertainty at parameter level into account. No information is lost during this process, and we postpone the use of computer resources until the resulting formulae can no longer be simplified.
\newpage
\bibliographystyle{wileyj}
|
1,116,691,497,805 | arxiv | \section{Introduction}
\subsection{Problems causing confusion in understanding neutrino oscillations}
Although an extensive review of neutrino oscillations does not mention time
at all\cite{review} and bypasses all confusion,
the continuing argument about the roles of energy, momentum and time in neutrino
oscillations\cite{neusb} still causes confusion\cite{cutapp}. The ``equal
energy assumption" of one paper\cite{Okun} was criticized by another
\cite{Giunti} while shown to arise naturally from the interaction of the
neutrino with its environment\cite{Leo}. This interaction is still often
ignored\cite{Giunti2,Giunti3} and the controversy
continues\cite{OkunA,OkunB,OkunC}.
Why are different mass states coherent? What is the correct formula for the
oscillation phase? How can textbook formulas for oscillations in time describe
experiments which never measure time? How can we treat the different velocities
and different transit times of different mass eigenstates and avoid incorrect
factors of two? How can textbook forumulas which describe coherence between
energy states be justified when Stodolsky's theorem states there is no
coherence between different energies? Is covariant relativistic quantum field
theory necessary to describe neutrino oscillations? How important is the
detector, which is at rest in the laboratory and cannot be Lorentz tranformed
to other frames?
These questions are answered by a simple rigorous calculation which includes
the quantum fluctuations in the position of the detector and in the transit
time between source and detector. The commonly used standard formula for
neutrino oscillation phases is confirmed. An ``ideal" detector
which measures precisely the energy and momentum of the neutrino destroys all
phases in the initial wave packet and cannot observe oscillations. A realistic
detector preserves the phase differences between neutrinos having the same
energy and different momenta and confirms the standard formula. Whether phase
differences between neutrinos with different energies are observable or
destroyed by the detector is irrelevant.
The confusion begins with the textbooks
that claim that a neutrino in a mixture of mass eigenstates
with different energies will oscillate in time.
Nobody measures time. What is measured is distance. Confusion arises in
converting the period of oscillation in time to a wave length in distance,
combined with the observation that neutrinos with different energies
travel with different velocities and arrive at the detector at different
times. How to treat this is clear to anyone who understands first year
quantum mechanics and wave-particle duality. Confusion arises because too many
papers on neutrino
oscillations are written by people who don't and double counting leads to
the factor of two.
A second source of confusion is that neutrino oscillations are observable
only if the neutrino detector is able to detect the relative phase of
components in an incident wave packet. Leo Stodolsky\cite{Leo} has pointed out
that any stationary detector is described by a density matrix that is diagonal
in energy and that phases between incident neutrino amplitudes with different
energies are not observable. Only interference between components with the
SAME ENERGY and different momenta are observable. But since this point is
continuously ignored in the literature and in conferences it is spelled out
clearly and unambiguously in the present paper. It may be only first year
quantum mechanics, but unfortunately it seems to be needed.
These problems are illustrated in the section on neutrino mass, mixing and
flavor change in the review of particle physics\cite{PDG} by the Particle Data
Group.
The section contains this caveat ``The quantum mechanics leading to the same
result is somewhat subtle. To do justice to the physics requires a more refined
treament than the one we have given. Sophisticated treatments continue to yield
new insights". Unfortunately most readers ignore the caveat and a literal
interpretation of the text contributes to the confusion.
The time-dependent neutrino wave function described by eq. (13.4) has a
``Lorentz-invariant phase factor". The text does not note that this phase
factor is not gauge invariant and is not observable; only phase differences are
observable. Although first year quantum mechanics tells us that the overall
phase of a wave function is not observable, many readers fall into this trap
and take the phase seriously.
The text considers a neutrino produced with momentum p; all components
have same momentum. The time-dependent phase is replaced by a
distance-dependent phase by noting that the neutrino is extremely
relativistic and sets $t\approx L$. However, this is not strictly true for
finite mass neutrinos. Neutrinos with the same momentum and different
masses have different velocities and travel the distance $L$ in sufficiently
different times to cause confusion.
This has recently been pointed out to the Particle Data Group by a reader who
noted that the exact relation between $t$ and $L$ reads $L = t\cdot v$, with
$v= p/E$. Substituting $L/v = L\cdot E/p$ into the ``Lorentz-invariant phase
factor" he gets $E\cdot t - p\cdot L = E^2\cdot L/p - p^2\cdot L/p = (E^2 -
p^2)\cdot L/p = m^2\cdot L/p$ which differs from the standard result by a
factor of 2.
The text then examines the relative phases of the mass eigenstates at the
distance $L$ and converts the state to a linear combination of flavor
eigenstates. This assumes that the states with different energies are coherent
and that their relative phase is observable. But Stodolsky has shown\cite{Leo}
that neutrinos with different energies lose all coherence in the interaction
with the detector and their relative phase is not observable.
Our purpose here is to present a coherent treatment that can be
easily understood by physicists who are unaware of the ``somewhat subtle
quantum mechanics" needed to understand neutrino oscillations.
If the interaction with the environment is ignored, the problem reduces to the
propagation of a single noninteracting Dirac fermion, easily solved without
relativistic field theory. In a real experiment the neutrino passes the
detector as a wave with a finite length in time and can interact with the
detector any time during this finite interval. These quantum fluctuations in
transit time affect the relative phase observed at the detector between two
neutrino waves. Confusion and erroneous factors of two arise in incorrectly
treating this transit time as a well-defined classical variable.
We present here a rigorous treatment showing that the standard relation between
phases of amplitudes having different masses always holds irrespective of
whether states with equal energy, equal momentum or any other combination are
used. Effects of transit time variations are shown to be negligible.
The observability of these phases depends on the detector. An ideal detector;
e.g. a nucleon at rest, measures the neutrino mass and destroys all coherence
and relative phases. This ``missing mass" experiment has no oscillations.
More realistic detectors preserve the relative phases of amplitudes having
the same energy and different momenta; the detecting nucleon is confined to a
region of space small compared to the wave length of the observed neutrino
oscillations. The relative amplitudes for final states containing an $e$,
$\mu$ or $\tau$ are thus determined for each energy. Since these relative
magnitudes do not change appreciably with energy in realistic experiments, the
flavor output of the detector is determined without needing Stodolsky's
stationary environment assumption\cite{Leo}. Even if a judicious time
measurement preserves some coherence between different energies, the flavor
output is unchanged.
\subsection{The basic problem}
The source of the confusion is easily seen by examining the state of a
neutrino wave packet incident on a detector: a mixture of states having
different masses, energies and momenta. The phase change in the wave function
of a component of the wave packet in traveling a distance $x$ from the source
to the detector in a time $t$ is
\beq{phase}
\phi(x,t) = px -Et
\end{equation}
This phase by itself is unobservable, like any overall phase of a wave
function. It is not gauge invariant since a gauge transformation can multiply
the wave function by an arbitrary function of x.
What is gauge invariant and observable is
the phase difference between any two components of the wave function having
masses $m_1$ and $m_2$
\beq{shift}
\delta \phi(12) = \phi_1(x,t) - \phi_2(x,t) = (p_1 -p_2)x -(E_1 -E_2)t =
{{p_1^2-p_2^2}\over{p_1+ p_2}} \cdot x - {{E_1^2-E_2^2}\over{E_1+ E_2}} \cdot t
\end{equation}
where $p_1$, $p_2$, $E_1$ and $E_2$ are respectively the momenta and energies
of these two components in the wave packet.
To relate this result (\ref{shift}) to the phase difference actually observed
in a detector, we set $x$ equal to the distance between the source and the
detector. The problem arises in choosing a value for t. We parametrize the
problem by setting $t$ equal to the time required to traverse the distance x
using an average group velocity $\bar v$ for the two mass eigenstates, with a
correction $\delta t$ to describe any discrepancies, including quantum
fluctuations.
\beq{velocity2}
\bar v \equiv {{p_1+ p_2}\over{E_1+ E_2}}; ~ ~ ~
t= {x \over \bar v} + \delta t = {{E_1+ E_2}\over{p_1+ p_2}}\cdot x +\delta t
\end{equation}
Thus
\beq{shiftb}
\delta \phi(12) =
{{p_1^2-p_2^2}\over{p_1+ p_2}} \cdot x - {{E_1^2-E_2^2}\over{E_1+ E_2}} \cdot t
= {{m_2^2-m_1^2}\over{p_1+ p_2}} \cdot x - {{E_1^2-E_2^2}\over{E_1+ E_2}}
\cdot \delta t
\end{equation}
This gives the standard formula for the phase shift independent of the momentum
and energy values of the two interfering waves with a correction term which
vanishes when the two waves have the same energy. The physics here is evident.
The relative phase at the same point in space between two amplitudes with
different energies is proportional to the product of the energy difference
and time and sensitive to the exact time when the neutrino is detected. The
$\delta t$ correction is appreciable for the understanding of time behavior.
But a quantitative analysis given below shows the correction to the formula
(\ref{shiftb}) to be negligible. This explains why treatments using waves with
the same momentum and different energies give the same answer as those using
the same energy and different momenta.
The time behavior has caused much confusion in the community and in the
literature. We therefore we investigate in detail its physical meaning and
quantum fluctuations.
\section{The elementary quantum mechanics of space and time}
Neutrinos arrive at a detector at distance $x$ from
the source as a coherent linear combination of states having different masses,
energies and momenta. Since the source and absorber are quantum
systems, quantum fluctuations arise in both the distance $x$ and in the time
interval $t$ between the emission from the source and its observation at the
detector. The distance fluctuations can be shown to be small in any experiment
where the positions of the source and absorber are known to a precision much
smaller than the wave length of the oscillation to be measured. The
nature of the quantum fluctuations in the time interval $t$ are not generally
well understood and are the source of much confusion.
The neutrino wave function at the detector defines a probability
amplitude for its detection at any time during the arrival of the
wave packet. The value of the transit time $t$ is thus not
precisely defined and subject to the usual quantum fluctuations of
any variable described by a quantum-mechanical probability amplitude.
The waves for different mass eigenstates travel with different group
velocities, and their centers may arrive at the detector at different times.
But the detector does not measure the arrival time of a wave packet center.
Each event has a single value of the time $t$. The flavor output of the
detector is determined by the relative phases of the different components of
the wave packet at that time. The quantum fluctuations in time can be observed
if there is a measurement of the time spectrum of events.
We now define the quantities necessary to
evaluate the phase shifts in eq.(\ref{shiftb}).
The time variable $t$ to use in eq. (\ref{shift}) is defined as the
interval between the time when the neutrino is detected and a time at the
source when the neutrino is in a flavor eigenstate. This is then subject to the
uncertainty denoted by $\delta t$. The neutrino flavor at the source can be
measured by placing a detector a short distance from the source. This was in
fact the first experiment that showed that there were different flavors of
neutrinos.
We now check the effect of $\delta t$ for the case of equal momenta
$p_1 = p_2 \equiv p$ in (eq.\ref{shiftb}).
\beq{shiftb3}
\delta \phi(12) = {{m_2^2-m_1^2}\over{p_1+ p_2}} \cdot \left( x +
{{E_1^2-E_2^2}\over{m_1^2-m_2^2}}\cdot \bar v \cdot \delta t\right) =
{{m_2^2-m_1^2}\over{2p}} \cdot x
\cdot \left(1 + {{\bar v \cdot \delta t}\over x } \right)
\end{equation}
Since the spatial size of the wave packet, $ \bar v \cdot \delta t$ is much
smaller than the distance x between source and detector the neglect of $\delta
t$ in eqs. (\ref{shiftb}) and (\ref{shiftb3}) is clearly justified. But $\delta
t$ is also seen to be much larger than the time separation $\delta
t_{wc}$
between the centers of the two mass eigenstate wave packets, which have moved
through the distance $x$ with group velocities $v_1 = p/E_1$ and $v_2 = p/E_2$
respectively.
\beq{shiftb4}
\bar v \cdot \delta t_{wc} =
\bar v \cdot \left({x \over v _1 }- {x \over v _2 }\right)
\approx {{m_2^2-m_1^2}\over{2p^2}} \cdot x << \bar v \cdot \delta t << x
\end{equation}
The transit time fluctuations $\delta t$ are seen to be much larger than
$\delta t_{wc}$ but sufficiently small so that the correction to eq.
(\ref{shiftb}) is negligible.
The time difference $\delta t_{wc}$ is automatically included in the above
calculation and needs no further correction. Unnecessary additional corrections
motivated by the erroneous classical picure of a particle traveling on a
classical path with a well defined velocity has led to the spurious factors of
2.
\section {What is observable}
Whether the phase (\ref{shiftb}) is experimentally observable depends
upon the detector.
\subsection {An ``ideal" detector}
Consider an ``ideal" detector which is a nucleon at rest. It absorbs
the incident neutrino and emits a lepton. Since energy and momentum are
conserved, the energy and momentum transfers from the neutrino to the detector
denoted by $\delta E$ and $\delta p$ are equal respectively to the energy and
momentum of the incident neutrino, denoted by $E_\nu$ and $p_\nu$
\beq{deltaE}
\delta E = T_N + E_{lepton}=E_\nu; ~ ~ ~
\delta p =p_N + p_{lepton} = p_\nu
\end{equation}
where $T_N$ and $p_N$ are the kinetic energy and momentum of the final nucleon,
and $E_{lepton}$ and $p_{lepton}$ are the total energy and momentum of the
emitted lepton.
This ``missing mass experiment" determines the neutrino mass
\beq{mnu}
m_\nu=\sqrt{E_\nu^2 - p_\nu^2}
\end{equation}
Since the detector determines the neutrino mass, waves arriving at the detector
with different masses are not coherent. All phase information is
destroyed in this ``ideal" detector.
\subsection {A realistic localized detector}
Realistic detectors are not ideal. The localization of the detector
nucleon in a small region of space makes its momentum uncertain. There is no
comparable localization in time.
This asymmetry between energy and momentum in the initial detector state
destroys the apparent symmetry between energy and momentum noted in all
covariant descriptions of neutrino oscillations. A fully covariant description
of any experiment which can be used also to consider detectors moving with
relativistic velocities is not feasible at present. A covariant description
which neglects the quantum mechanics of the neutrino-detector interaction is
neglecting essential physics of all realistic oscillation experiments.
The following section gives a rigorous treatment of the handwaving argument
using the uncertainty principle to relate the momentum spread of the wave
function to the size of the detector. The phase between components of the
neutrino wave with the same energy and different masses and momenta is shown to
be observable if quantum fluctuations in the position of the
detector nucleon are very small in comparison with the wave length of the
neutrino oscillations in space produced by the mass difference between neutrino
mass eigenstates.
The simple hand-waving argument for this physics states that the uncertainty
principle and the localization in space of the detector nucleon that absorbs
the neutrino prevents the detector from knowing the difference between
components of the incident neutrino wave packet with slightly different momenta
and the same energy.
\section {A detailed rigorous calculation of the detection process}
The rigorous quantum-mechanical argument notes that the product $\delta p
\cdot \delta x$ of the quantum fluctuations in the position of the detector
nucleon $\delta x$ and the range of momenta $\delta p$ in relevant neutrino
states having the same energy is a small quantity. Taking the leading terms in
the expansion of the transition matrix elements in powers of $\delta p \cdot
\delta x$ gives the result that the flavor spectrum of the charged leptons
emitted from the detector at a given energy is determined by the relative phase
of the components of the incident neutrino wave packet having the same energy
and different momenta. Whether coherence between amplitudes with different
energies is destroyed, as required by Stodolsky's theorem, or can be preserved by
judicious time measurements is irrelevant to this result.
The detection of the neutrino is a weak interaction described in first order
perturbation theory by transition matrix elements between the initial state of
the neutrino-detector system before the interaction and all possible final
states.
Consider the transition matrix element between an initial state $\ket{i(E)}$
with energy $E$ of the entire neutrino - detector system and a final state
$\ket{f(E)}$ of the system of a charged muon and the detector with the same
energy E, where a neutrino $\nu_k$ with energy, mass and momentum $E_\nu$,
$m_k$ and $\vec P_o + \vec {\delta P_k}$ is detected via the transition
\beq{weak1}
\nu_k + p \rightarrow \mu^+ + n
\end{equation}
occurring on a proton in the detector. We express the neutrino momentum
as the sum of the mean momentum $\vec P_o$ of all the neutrinos
with energy
$E_\nu$ and the difference $\vec {\delta P_k}$ between the momentum of each
mass eigenstate and the mean momentum,
The transition matrix element depends upon the individual mass eigenstates $k$
only in the momentum difference $\vec {\delta P_k}$ and a factor $c_{k\mu}$ for
each mass eigenstate which is a function of neutrino mixing angles. $c_{k\mu}$
describes the transition amplitude for this mass eigenstate to produce a muon
when it reaches the detector, multiplied by a phase given by the generalization
to the case of three neutrinos of eq. (\ref{shiftb}) with $E_1=E_2$ The
transition matrix element can thus be written in a factorized form with one
factor $T_o$ independent of the mass $m_k$ of the neutrino eigenstate and a
factor depending on $m_k$.
\beq{weak2} \bra{f(E)}T \ket{i(E)} = \sum_k
\bra{f(E)}T_o \cdot c_{k\mu} e^{i\vec {\delta P_k} \cdot \vec X} \ket{i(E)}
\end{equation}
where $\vec X$ denotes the co-ordinate of the nucleon that absorbs the
neutrino. Then if the product $\vec {\delta P_k} \cdot \vec X$ of the momentum
spread in the neutrino wave packet and the fluctuations in the position of the
detector nucleon is small, the exponential can be expanded and approximated
by the leading term
\beq{weak3}
\bra{f(E)}T \ket{i(E)} =
\sum_k \bra{f(E)}T_o \cdot c_{k\mu} e^{i\vec {\delta P_k} \cdot \vec X} \ket{i(E)}
\approx
\sum_k \bra{f(E)}T_o \cdot c_{k\mu} \ket{i(E)}
\end{equation}
The transition matrix element for the probability that a muon is observed at
the detector is thus proportional to the coherent sum of the amplitudes
$c_{k\mu}$ for neutrino components with the same energy and different masses
and momenta to produce a muon at the detector. A similar result is obtained
for the probability of observing each other flavor. The final result is
obtained by summing the contributions over all the energies in the incident
neutrino wave packet. But as long as the flavor output for each energy is
essentially unchanged over the energy region in the wave packet, the flavor
ouput is already determined for each energy, and is independent of any
coherence or incoherence between components with different energies.
For the case of two neutrinos with energy $E$ and mass eigenstates $m_1$ and
$m_2$ the relative phase of the two neutrino waves at a distance $x$ is given
by eq. (\ref{shiftb}) with $E_1=E_2$
\beq{WW3a}
\phi^E_m(x)= (p_1 - p_2)\cdot x =
{{(p_1^2 - p_2^2)}\over{(p_1 + p_2)}}\cdot x =
{{m_2^2-m_1^2}\over{2 \bar p}} \cdot x
\end{equation}
The flavor output of the detector is thus seen to be determined by the
interference between components in the neutrino wave paclet with the same
energy and different masses and momenta. All the relevant physics is
in the initial state of the nucleon in the detector that detects the
neutrino and emits a charged lepton, together with the relative phases of
the components of the incident neutrino wave packet with the same energy.
This result (\ref{weak3}-\ref{WW3a}) is completely independent of the neutrino
source and in particular completely independent of whether the source satisfies
Stodolsky's stationarity condition\cite{Leo}. No subsequent time measurements
or additional final state interactions that mix energies can change this flavor
output result.
\section{Conclusions}
\subsection{Problems in calculating phase differences}
\begin{enumerate}
\item A reliable calculation of neutrino oscillations that avoids confusion
requires an understanding of the effects of variations in the transit time of a
neutrino between source and detector and its quantum fluctuations.
\item When the small quantum fluctuations in time are neglected, the phase
difference between two components of a narrow wave packet with different
masses depends only on their squared mass difference and the mean momentum in
the wave packet and is otherwise independent of their momenta and energies.
\item The value of this phase difference is given by the standard formula used
in all analyses of neutrino oscillations.
\end{enumerate}
\subsection{Problems in observing phase differences}
\begin{enumerate}
\item The observability of
this phase difference depends upon the quantum mechanics of the detector.
\item
All coherence is destroyed in an ``ideal" detector, which precisely measures
the momentum and energy of the neutrino.
\item A realistic detector preserves
the relative phase between states having the same energy and different momenta.
This is sufficient to uniquely determine the flavor output of the detector and
show that it satisfies the standard formula for neutrino oscillations.
\item
Whether the coherence between states having different energies is destroyed, as
required by Stodolsky's stationarity theorem\cite{Leo}, or can somehow be
preserved by judicious time measurements is irrelevant to the flavor output of
the detector.
\item The standard formula used in all analyses of neutrino
oscillations can be rigorously justified. All sources of confusion are resolved
by using states with the same energy and different momenta to calculate
neutrino oscillations.
\end{enumerate}
\subsection{Some concluding remarks}
The initial uncertainty in the momentum of the detector nucleon in a localized
detector destroys all memory of the initial neutrino momentum and of the
initial neutrino mass after the neutrino has been absorbed. The hand-waving
justification of the result (\ref{weak3}) uses the uncertainty principle to
say that if we know where the detector is we don't know its momentum and can't
use momentum conservation to determine the mass of the incident neutrino. The
above rigorous justification shows full interference between the contributions
from different neutrino momentum states with the same energy as long as the
product of the momentum difference and the quantum fluctuations in the initial
position of the detector nucleon is negligibly small in the initial detector
state.
This treatment of the neutrino detector is sufficient to determine the flavor
output of any experiment in which the incident neutrino wave packet is the same
well defined linear combination of mass eigenstates throughout the whole wave
packet.
Time measurements and all possible coherence between amplitudes
from components of the incident neutrino wave packet with different energies
are not considered and seen to be unnecessary.
There may be fancy time measurements which
can introduce such coherence. But the coherence between incident neutrino
states with the same energy and different momentum already determines the
flavor output of the detector for each incident neutrino energy and cannot be
destroyed by time measurements.
The question remains of possible variation of detector flavor output
as a function of energy. As long as the wave packet is sufficiently
narrow in momentum this flavor output does not change appreciably over the
relevant energy range in the wave packet and the standard neutrino oscillation
formulas are valid. When the flavor output varies widely as a function of
energy, oscillations are no longer observed. This can be seen in the case of
neutrinos traveling large distances with many oscillation wave lengths, as in
neutrinos arriving from the sun or a supernova. Here the neutrino wave packet
can separate into components with different mass eigenstates, traveling with
different velocities and reaching the detector at measurably different times.
All this time variation appears simply\cite{Leo} in the energy spectrum, which
is the fourier transform of the time behavior.
\section{acknowledgments}
It is a pleasure to thank Terry Goldman, Boris Kayser, Lev Okun, Georg Raffelt,
Alexei Smirnoff, Leo Stodolsky and Lincoln Wolfenstein for helpful discussions
and comments as well as to thank the organizers and participants in the
Ringberg CUTAPP 2005 (Leofest) workshop\cite{cutapp} for many clarifying
interactions. This work is supported in part by the U.S. Department of Energy,
Division of High Energy Physics, Contract W-31-109-ENG-38.
\catcode`\@=11
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|
1,116,691,497,806 | arxiv | \section{Introduction and preliminaries}
For our general setting we consider the problem of finding $u$ in an equation of the form
\begin{equation}\label{eq:prob-inv}
Tu=v,
\end{equation}
where $T:\X\rightarrow \Y$ is a bounded linear operator between two infinite dimensional Hilbert spaces $\X$ and
$\Y$, the range of $T$ is non-closed and $v$ is the data, which is supposed to be known, perhaps with a certain
degree of error. In the sequel and unless otherwise specified, the space $\X$ will be $L^2(\Omega)$ where $\Omega\subset\mathbb{R}^n$ is a bounded open convex set with Lipschitz boundary.
It is well known that under these hypotheses problem (\ref{eq:prob-inv}) is ill-posed in the sense of Hadamard (\cite{ref:Hadamard-1902}) and it must
be regularized before any attempt to approximate its solutions is made (\cite{refb:Engl-Hanke-96}). The most usual way of
regularizing a problem is by means of the use of the \textit{Tikhonov-Phillips regularization method} whose general
formulation can be given within the context of an unconstrained optimization problem. In fact, given an appropriate
penalizer $W(u)$ with domain $\mathcal D\subset \X$, the regularized solution obtained by the Tikhonov-Phillips
method and such a penalizer, is the minimizer $u_\alpha$, over $\mathcal D$, of the functional
\begin{equation}\label{eq:formulacion W}
J_{\alpha,W}(u)= \norm{Tu-v}^2+\alpha W(u),
\end{equation}
where $\alpha$ is a positive constant called regularization parameter. For general penalizers $W$, sufficient
conditions guaranteeing existence, uniqueness and weak and strong stability of the minimizers under different types
of perturbations where found in \cite{ref:Mazzieri-Spies-Temperini-JMAA-2012}.
Each choice of an appropriate penalizer $W$ originates a different regularization method producing a particular
regularized solution possessing particular properties. Thus, for instance, the choice of $W(u)=\norm{u}^2_{\scriptscriptstyle L^2(\Omega)}$ gives
raise to the classical Tikhonov-Phillips method of order zero producing always smooth regularized approximations which approximate, as $\alpha\to 0^+$, the best approximate solution (i.e. the least squares solution of minimum norm) of problem (\ref{eq:prob-inv}) (see \cite{refb:Engl-Hanke-96}) while for $W(u)=\norm{|\nabla u|\,}^2_{\scriptscriptstyle L^2(\Omega)}$ the order-one Tikhonov-Phillips method is obtained. Similarly, the choice of $W(u)=\norm{u}_{\scriptscriptstyle \text{BV}(\Omega)}$ (where $\norm{\cdot}_{\scriptscriptstyle \text{BV}}$ denotes the total variation norm) or $W(u)=\norm{|\nabla u|\,}_{\scriptscriptstyle L^1(\Omega)}$, result in the so called ``bounded variation regularization methods'' (\cite{ref:Acar-Vogel-94}, \cite{ref:Rudin-Osher-Fatemi-1992}). The use of these penalizers is appropriate when preserving discontinuities or edges is an important matter. The method, however, has as a drawback that it tends to produce piecewise constant approximations and therefore, it will
most likely be inappropriate in regions where the exact solution is smooth (\cite{ref:Chambolle-Lions-97}) producing the so called
``staircasing effect".
In certain types of problems, particularly in those in which it is known that the regularity of the exact solution
is heterogeneous and/or anisotropic, it is reasonable to think that using and spatially adapting two or more penalizers
of different nature could be more convenient. During the last 15 years several regularization methods have been
developed in light of this reasoning. Thus, for instance, in 1997 Blomgren \textit{et al.} (\cite{ref:Blomgren-etal-1997}) proposed
the use of the following penalizer, by using variable $L^p$ spaces:
\begin{equation} \label{eq:penalizante Blomgren}
W(u)=\int_\Omega |\nabla u|^{p(|\nabla u|)} dx,
\end{equation}
where $\underset{u\rightarrow 0^+}{\lim}p(u)=2$, $\underset{u\rightarrow \infty}{\lim}p(u)=1$ and $p$ is a decreasing
function. Thus, in regions where the modulus of the gradient of $u$ is small the penalizer is approximately equal
to $\norm{\abs{\nabla u}}_{L^2(\Omega)}^2$ corresponding to a Tikhonov-Phillips method of order one (appropriate for
restoration in smooth regions). On the other hand, when the modulus of the gradient of $u$ is large, the
penalizer resembles the bounded variation seminorm $\norm{\abs{\nabla u}}_{L^1(\Omega)}$, whose use, as mentioned
earlier, is highly appropriate for border detection purposes. Although this model for $W$ is quite reasonable,
proving basic properties of the corresponding generalized Tikhonov-Phillips functional turns out to be quite
difficult. A different way of combining these two methods was proposed by Chambolle and Lions (\cite{ref:Chambolle-Lions-97}). They
suggested the use of a thresholded penalizer of the form:
\begin{equation*}
W_\beta(u)=\int_{|\nabla u|\leq \beta}\left|\nabla u\right|^{2} dx +\int_{|\nabla u|> \beta}\left|\nabla u\right|
dx,
\end{equation*}
where $\beta>0$ is a prescribed threshold parameter. Thus, in regions where borders are more likely to be present ($|\nabla u|>
\beta$), penalization is made with the bounded variation seminorm while a standard order-one Tikhonov-Phillips
method is used otherwise. This model was shown to be successful in restoring images possessing regions with
homogeneous intensity separated by borders. However, in the case of images with non-uniform or highly degraded
intensities, the model is extremely sensitive to the choice of the threshold parameter $\beta$. More recently, penalizers of
the form
\begin{equation}
W(u)=\int_\Omega |\nabla u|^{p(x)} dx,
\end{equation}
for certain functions $p$ with range in $[1,2]$, were studied in \cite{ref:Chen-Levine-Rao-2006} and \cite{ref:Li-Li-Pi-2010}. It is timely to point
out here that all previously mentioned results work only for the case of denoising, i.e. for the case $T=id$.
In this work we propose the use of a model for general restoration problems, which combines, in an appropriate way,
the penalizers corresponding to a zero-order Tikhonov-Phillips method and the bounded variation seminorm. Although
several mathematical issues for this model still remain open, its use in some signal and image restoration problems has already proved
to be very promising. The purpose of this article is to introduce the model, show mathematical results
regarding the existence of the corresponding regularized solutions, and present some results of its application to
signal restoration.
The following Theorem, whose proof can be found in \cite{ref:Acar-Vogel-94} (Theorem 3.1),
guarantees the well-posedness of the unconstrained minimization problem
\begin{equation}\label{eq: funcional AV}
u^\ast\;=\;\underset{u\in L^p(\Omega)}{\rm argmin}\; J(u).
\end{equation}
\begin{thm}\label{teo:existencia AV}
Let $J$ be a $BV$-coercive functional defined on $L^p(\Omega)$. If $1\leq p<\frac{n}{n-1}$ and $J$ is lower
semicontinuous, then problem (\ref{eq: funcional AV}) has a solution. If $p=\frac{n}{n-1}$, $n\geq 2$ and in
addition $J$ is weakly lower semicontinuous, then a solutions also exists. In either case, the solution is unique
if $J$ is strictly convex.
\end{thm}
The following theorem, whose proof can also be found in \cite{ref:Acar-Vogel-94} (Theorem 4.1), is very important for the
existence and uniqueness of minimizers of functionals of the form
\begin{equation}\label{eq:funcional-JOTA}
J(u)\;=\;\|Tu-v\|^2+ \alpha J_{\s 0}(u),
\end{equation}
where $\alpha>0$ and $J_{\s 0}(u)$ denotes the bounded variation seminorm given by
\begin{equation}\label{eq:funcional-JOTA-cero}
J_{\s 0}(u)=\underset{\vec\nu\in\mathcal V}{\sup}\int_{\Omega}-u\, \text{div}\vec\nu\, dx,
\end{equation}
with $\mathcal V\doteq\{\vec\nu: \Omega \rightarrow\mathbb R^n \text{ such that } \vec \nu\in C^{\s 1}_{\s
0}(\Omega) \text{ and } |\vec\nu (x)|\leq 1 \,\forall\, x\in\Omega\}.$
\begin{thm}\label{teo:existencia-JOTA} Suppose that $p$ satisfies the restrictions of Theorem \ref{teo:existencia
AV} and $T\chig_\Omega\ne 0$. Then $J(\cdot)$ defined by (\ref{eq:funcional-JOTA}) is $BV$-coercive.
\end{thm}
Note here that (\ref{eq:funcional-JOTA}) is a particular case of (\ref{eq:formulacion W}) with $W(u)=J_{\s 0}(u)$. The
following theorem, whose proof can be found in \cite{ref:Mazzieri-Spies-Temperini-JMAA-2012}, gives conditions
guaranteeing existence and uniqueness of minimizers of (\ref{eq:formulacion W}) for general penalizers $W(u)$. This
theorem will also be very important for our main results in the next section.
\begin{thm}\label{teo:existencia JMAA} Let $\X,\,\Y$ be normed vector spaces,
$T\in\mathcal L(\X,\Y)$, $v\in\Y,$ $\D\subset \X$ a convex set and
$W:\mathcal{D}\longrightarrow \mathbb R$ a functional bounded from
below, $W$-subsequentially weakly lower semicontinuous, and such
that $W$-bounded sets are relatively weakly compact in $\X$. More
precisely, suppose that $W$ satisfies the following hypotheses:
\begin{itemize}
\item \textit{(H1): }$\exists \,\,\gamma\geq 0$ such that $W(u)\geq
-\gamma\hspace{0.3cm} \forall\,u\in\mathcal D$.
\item \textit{(H2):\;}for every $W$-bounded sequence $\{u_n\}\subset\D$ such that
$u_{n}\overset{w}{\rightarrow} u\in\D,$ there exists a subsequence
$\{u_{n_j}\}\subset \{u_n\}$ such that $
W(u)\leq\liminf_{j\rightarrow\infty}W(u_{n_j})$.
\item \textit{(H3):} for every $W$-bounded sequence
$\{u_n\} \subset \mathcal D$ there exist a subsequence
$\{u_{n_j}\}\subset \{u_n\}$ and $u\in \D$ such that
$u_{n_j}\overset{w}{\rightarrow} u$.
\end{itemize}
Then the functional $J_{W,\alpha}(u)\doteq \|Tu-v\|^2+\alpha W(u)$ has a global minimizer on $\mathcal{D}$. If moreover $W$ is
convex and $T$ is injective or if $W$ is strictly convex, then such a minimizer is unique.
\end{thm}
\begin{proof} See Theorem 2.5 in \cite{ref:Mazzieri-Spies-Temperini-JMAA-2012}.
\end{proof}
\section{Main results}
In this section we will state and prove our main results concerning existence and uniqueness of minimizers of particular
generalized Tikhonov-Phillips functionals with combined spatially-varying $L^2$-BV penalizers. In what follows $\mathcal{M}(\Omega)$ shall denote the set of all real valued measurable functions defined on $\Omega$ and $\widehat{\mathcal{M}}(\Omega)$ the subset of $\mathcal{M}(\Omega)$ formed by those functions with values in $[0,1]$.
\begin{defn} Given $\theta\in \widehat{\mathcal{M}}(\Omega)$ we define the functional $W_{\scriptscriptstyle 0, \theta}(u)$ with values on the extended reals by
\begin{equation}\label{eq:funcional W0theta}
W_{\scriptscriptstyle 0, \theta}(u)\doteq \sup_{\vec\nu\in\mathcal V_{\scriptscriptstyle \theta}}\int_{\Omega}-u\, \text{div}(\theta \vec\nu)\, dx,
\hspace{0.5cm} u \in \mathcal{M}(\Omega)
\end{equation}
where $\mathcal V_{\s \theta}\doteq\{\vec\nu: \Omega\rightarrow\mathbb R^n \text{ such that }
\theta\vec \nu\in C^{\s 1}_{\s 0}(\Omega) \text{ and } |\vec\nu (x)|\leq 1 \,\forall\, x\in\Omega\}.$
\end{defn}
\begin{lem}\label{lem: J0 gradiente}
If $u\in C^{\s 1}(\Omega)$ then $W_{\s 0,\theta}(u)=\|\theta \abs{\nabla u}\|_{L^{\s
1}(\Omega)}$.
\end{lem}
\begin{proof}
Let $u\in C^1(\Omega).$ For all $\vec\nu \in \mathcal V_{\s \theta}$ it follows easily that
\begin{align}\label{eq: desig-thetagrad}
\nonumber \ds\int_{\Omega}-u\, \text{div}(\theta \vec\nu)\, dx&=\ds\int_{\Omega}\nabla u \cdot \theta\vec \nu \,
dx - \ds\int_{\delta\Omega}(u\theta\vec\nu \cdot \vec n )\, dS \\ \nonumber
&=\int_{\Omega}\nabla u\cdot\theta\vec\nu\;dx \qquad (\text{since } \theta \vec\nu|_ {\delta\Omega}=0) \\ \nonumber
&\leq \ds\int_{\Omega}\abs{\theta\nabla u} \abs{\vec \nu}\, dx\\
&\leq \ds\int_{\Omega}\abs{\theta\nabla u} \, dx \quad (\text{since} \abs{\vec\nu(x)}\leq 1),
\end{align}
where $\vec n$ denotes the outward unit normal to $\delta\Omega$. Taking supremum over $\vec \nu \in \mathcal
V_{\s\theta}$ it follows that
$$W_{\scriptscriptstyle 0, \theta}(u) \leq \|\theta \abs{\nabla u} \|_{ L^1(\Omega)}.$$
For the opposite inequality, define $\vec\nu_\ast(x)\doteq \begin{cases}
\frac{\nabla u(x)}{\abs{\nabla u(x)}}, &\text{if }\abs{\nabla u(x)}\neq 0,\\
0, &\text{if } \abs{\nabla u(x)}= 0,
\end{cases}$. Then one has that $\abs{\vec\nu_\ast(x)}\leq 1\; \forall \, x \in \Omega$ and $\vec\nu_\ast\in C(\Omega, \mathbb{R}^n)$ since $u\in C^{\s 1}(\Omega).$ Also,
$$\ds\int_{\Omega}(\nabla u \cdot \theta\vec\nu_\ast)\, dx = \ds\int_{\Omega}\abs{\theta\nabla u} \, dx.$$ By convolving $\vec\nu_\ast$ with an appropriately chosen function $\varphi \in C^{\s \infty}_{\s 0}(\Omega, \mathbb{R}^n)$, we can obtain a function $\vec\nu \in \mathcal V_{\s \theta} \cap C^{\s \infty}_{\s 0}(\Omega, \mathbb{R}^n)$ for which the left hand side of (\ref{eq: desig-thetagrad}) is arbitrarily close to $\ds\int_{\Omega}\abs{\theta\nabla u} \, dx.$ Then taking supremum over $\vec\nu \in \mathcal V_\theta$ we have that $$W_{\scriptscriptstyle 0, \theta}(u) \geq \|\theta \abs{\nabla u} \|_{ L^1(\Omega)}.$$ Hence $W_{\scriptscriptstyle 0, \theta}(u) = \|\theta \abs{\nabla u} \|_{ L^1(\Omega)},$ as we wanted to prove.
\end{proof}
\noindent Observation: From de density of $C^{\s 1}(\Omega)$ in $ W^{\s 1,1}(\Omega)$ it follows that Lemma
\ref{lem: J0 gradiente} holds for every $ u\, \in W^{\s 1,1}(\Omega)$.
\begin{rem}
For any $\theta \in \widehat{\mathcal{M}}(\Omega)$, it follows easily that
\begin{equation}\label{eq:desigW-J}
W_{\s 0,\theta}(u)\leq J_{\s 0}(u), \quad \forall \,u \in \mathcal{M}(\Omega).
\end{equation}
In fact, for any $\vec\nu\in \mathcal V_{\s \theta}$ and for any $u \in \mathcal{M}(\Omega)$ we have that
\begin{align}\label{eq:desig-remark}
\int_{\Omega}-u\, \text{div}(\theta \vec\nu)\, dx&\leq \underset {\vec\nu \in \mathcal{V}} {\sup}\int_{\Omega}-u\, \text{div}\,\vec\nu\, dx \\ \nonumber
&=J_{\s 0}(u),
\end{align}
where inequality (\ref{eq:desig-remark}) follows from the fact that $\theta\vec\nu \in \mathcal V$ (since
$|\theta(x)|\leq 1 \;\forall x\in\Omega$). By taking supremum for $\vec\nu \in \mathcal{V}_\theta$ inequality
(\ref{eq:desigW-J}) follows.
\end{rem}
Although inequality (\ref{eq:desigW-J}) is important by itself since it relates the functionals $W_{\s 0,\theta}$ and $J_{\s 0}$,
in order to be able to use the known coercitivity properties of $J_{\s 0}$ (see \cite{ref:Acar-Vogel-94}), an inequality of the opposite type is highly desired. That is, we would like to show that, under certain conditions on $\theta(\cdot)$, there exists a constant $C=C(\theta)$ such that $W_{\s 0,\theta}(u)\geq CJ_{\s 0}(u)$ for all $u \in \mathcal{M}(\Omega)$. The following theorem provides sufficient conditions on $\theta$ assuring such an inequality.
\begin{thm}\label{teo:desig-WgJ}
Let $\theta \in \widehat{\mathcal{M}}(\Omega)$ be such that $\frac1\theta\in L^\infty(\Omega)$ and
let $J_{\s 0}$, $W_{\s 0,\theta}$ be the functionals defined in (\ref{eq:funcional-JOTA-cero}) and
(\ref{eq:funcional W0theta}), respectively. Then $J_{\s 0}(u) \le \|\frac1\theta\|_{\s L^\infty(\Omega)}\, W_{\s 0,\theta}(u)$
for all $u \in \mathcal{M}(\Omega)$.
\end{thm}
\begin{proof}
Let $u \in \mathcal{M}(\Omega)$ and $K_{\s\theta}\doteq\|\frac1\theta\|_{\s L^\infty(\Omega)}$. Then for all $\vec\nu \in \mathcal{V}$
\begin{align*}
\int_{\Omega}-u\, \text{div}\,\vec\nu\, dx&= K_{\s\theta} \int_{\Omega}-u\, \text{div}\left(
\frac{\theta\vec\nu}{K_{\s\theta}\theta}\right)\, dx \\
&\leq K_{\s\theta}\;\underset {\vec\omega \in \mathcal{V}_{\s \theta}} {\sup}\int_{\Omega}-u\,
\text{div}\,(\theta\,\vec\omega)\,
dx \\
&=K_{\s\theta}\;W{\s 0,\theta}(u),
\end{align*}
where the last inequality follows from the fact that $\frac{\vec\nu}{K_{\s\theta}\,\theta}\in \mathcal{V}_{\s \theta}$
since $K_{\s \theta}\geq 1, |K_{\s \theta} \theta(x)|\geq 1 \, \forall \, x \in\Omega$ and $\vec \nu \in \mathcal V$. Then, taking supremum for $\vec\nu\in\mathcal{V}$ we conclude that $J_{\s 0}(u)
\le K_{\s\theta}\, W_{\s 0,\theta}(u)$.
\end{proof}
The following lemma will be of fundamental importance for proving several of the upcoming results.
\begin{lem}\label{lem:wls-W}
The functional $W_{\s 0,\theta}$ defined by (\ref{eq:funcional W0theta}) is weakly lower semicontinuous
with respect to the $L^p$ topology, $\forall\, p\in[1,\infty)$.
\end{lem}
\begin{proof}
Let $p\in [1,\infty)$, $\{u_n\}\subset L^p(\Omega)$ and $u\in L^p(\Omega)$ be such that $u_n\overset{w}{\rightarrow} u$. Let $\vec{\nu}_\ast\in \mathcal V_{\s \theta}$ and $q$ the conjugate dual of $p$. Since $\theta\vec \nu_\ast\in C^{\s 1}_{\s 0}(\Omega)$, it follows that $\text{div}(\theta\vec\nu_\ast)$
is uniformly bounded on $\Omega$ and therefore, $\text{div}(\theta\vec\nu_\ast)\in L^{\infty}(\Omega)\subset L^q(\Omega).$ Then, from the weak convergence of $u_n$ it follows that $\underset{n\to \infty}{\lim} \int_{\Omega}-u_n\, \text{div}(\theta \vec\nu_\ast)\, dx = \ds \int_{\Omega}-u\, \text{div}(\theta \vec\nu_\ast)\, dx$.
Hence $\ds\int_{\Omega}-u\, \text{div}(\theta \vec\nu_\ast)\, dx=\underset{n\to \infty}{\lim} \int_{\Omega}-u_n\, \text{div}(\theta \vec\nu_\ast)\, dx\leq \liminf_{n\rightarrow\infty} \sup_{\vec \nu \in\mathcal V_{\s \theta}}\int_{\Omega}-u_n\, \text{div}(\theta \vec\nu)\, dx =\break \liminf_{n\rightarrow\infty}W_{\s 0,\theta}(u_n)$. Thus $\forall \, \vec\nu_\ast \in \mathcal V_{\s \theta}$
$$\ds\int_{\Omega}-u\, \text{div}(\theta \vec\nu_\ast)\, dx\leq \liminf_{n\rightarrow\infty}W_{\s 0,\theta}(u_n).$$
Taking supremum over all $\vec\nu_\ast \in \mathcal V_{\s \theta}$ it follows that $W_{\s 0,\theta}(u)\leq
\underset{n\rightarrow\infty}{\liminf} \,W_{\s 0,\theta}(u_n).$
\end{proof}
We are now ready to present several results on existence and uniqueness of minimizers of generalized
Tikhonov-Phillips functionals with penalizers involving spatially varying combinations of the $L^2$-norm and of the
functional $W_{\s 0,\theta}$, under different hypotheses on the function $\theta$.
\begin{thm}\label{teo:existencia_unicidad ep1-ep2}
Let $\Omega\subset\mathbb{R}^n$ be a bounded open convex set with Lipschitz boundary, $\X=L^2(\Omega)$, $\Y$ a normed
vector space, $T\in \mathcal L(\X, \Y)$, $v\in\Y$, $\alpha_{\scriptscriptstyle 1},\, \alpha_{\scriptscriptstyle 2}$
positive constants, $\theta \in \widehat{\mathcal{M}}(\Omega)$ and $F_{\s\theta}$ the functional defined by
\begin{equation}\label{eq:funcional J}
F_{\s\theta}(u)\doteq \norm{Tu-v}^2_{\Y}+\alpha_{\scriptscriptstyle 1}\|\sqrt{1-\theta}\,u\|^2_{\scriptscriptstyle
L^2(\Omega)}+\alpha_{\scriptscriptstyle 2}\,W_{\s 0, \theta}(u),\hspace{0.5cm} u\in \mathcal{D}\doteq L^2(\Omega).
\end{equation}
If there exists $\varepsilon_{\s 2}\in \mathbb{R}$, such that
$\theta(x)\leq\varepsilon_{2}<1$ for a.e. $x \in \Omega$, then the functional (\ref{eq:funcional J})
has a unique global minimizer $u^*\in L^2(\Omega)$. If moreover there exists $\varepsilon_{\s 1}\in \mathbb{R}$ such that
$0<\varepsilon_{\s 1}\leq \theta(x)$ for a.e. $x \in \Omega$, then $u^*\in BV(\Omega)$.
\end{thm}
\begin{proof}
By virtue of Theorem \ref{teo:existencia JMAA} it is sufficient to show that the functional $$W(u)\doteq \alpha_{\scriptscriptstyle 1}\|\sqrt{1-\theta}\,u\|^2_{\scriptscriptstyle L^2(\Omega)}+\alpha_{\scriptscriptstyle 2}\,W_{\s 0, \theta}(u),\, u\in L^2(\Omega)$$
satisfies hypotheses \textit{(H1)}, \textit{(H2)} and \textit{(H3)}. Clearly \textit{(H1)} holds with $\gamma=0$.
To prove (\textit{H2}) let $\{u_n\}\subset L^2(\Omega)$ such that $u_n \overset{w}{\longrightarrow} u\in L^2(\Omega)$ and $W(u_n)\leq c_1<\infty$. We want to show that $W(u)\leq \underset{n\rightarrow\infty}{\liminf} \,W(u_n).$ Since
$\sqrt{1-\theta}\in L^\infty(\Omega)$ one has $\sqrt{1-\theta}\,u_n\overset{w}{\longrightarrow} \sqrt{1-\theta}\,u$.
The condition $\theta(x)\leq \varepsilon_2<1$ for a.e. $x \in \Omega$, clearly implies that $\|\sqrt{1-\theta}\cdot\|_{\s L^2(\Omega)}$ is a norm.
Then, from the weak lower semicontinuity of $\|\sqrt{1-\theta}\cdot\|^2_{\s L^2(\Omega)},$ it follows that
\begin{equation}\label{eq:dsi norma 2}
\|\sqrt{1-\theta}\,u\|^2_{L^2(\Omega)}\leq \liminf_{n\rightarrow\infty}\|\sqrt{1-\theta}\,u_n\|^2_{L^2(\Omega)}.
\end{equation}
On the other hand, from the weak lower semicontinuity of $\W$ in $L^2(\Omega)$ (see Lemma \ref{lem:wls-W}) it follows that
\begin{equation}\label{eq:dsi J0}
\W(u)\leq\liminf_{n\rightarrow\infty}\W(u_n).
\end{equation}
From (\ref{eq:dsi norma 2}) and (\ref{eq:dsi J0}) we then conclude that
\begin{align*}
W(u)&=\alpha_{\scriptscriptstyle 1}\|\sqrt{1-\theta}\,u\|^2_{\scriptscriptstyle L^2(\Omega)}+\alpha_{\scriptscriptstyle 2}\,W_{\s 0, \theta}(u)\\
&\leq \alpha_{\scriptscriptstyle 1}\liminf_{n\rightarrow\infty}\|\sqrt{1-\theta}\,u_n\|^2_{L^2(\Omega)}+\alpha_{\scriptscriptstyle 2} \liminf_{n\rightarrow\infty}\W(u_n)\\
&\leq \liminf_{n\rightarrow\infty}\left(\alpha_{\scriptscriptstyle 1}\|\sqrt{1-\theta}\,u_n\|^2_{L^2(\Omega)}+\alpha_{\scriptscriptstyle 2} \W(u_n)\right)\\
&= \liminf_{n\rightarrow\infty}W(u_n),
\end{align*}
what proves (\textit{H2}).
To prove (\textit{H3}) let $\{u_n\}\subset L^2(\Omega)$ be such that $W(u_n)\leq c_1 <\infty,\,\,\forall \,n$. We want to show that there exist $\{u_{n_j}\}\subset\{u_n\}$ and $u\in L^{2}(\Omega)$ such that $u_{n_j}\overset{w}{\longrightarrow} u$. For this note that
\begin{align}\label{eq:desig-H3}
(1-\varepsilon_{\s 2})\|u_n\|^2_{L^2(\Omega)}\leq \|\sqrt{1-\theta}\,u_n\|^2_{L^2(\Omega)}\leq W(u_n)\leq c_1.
\end{align}
Thus $\|u_n\|_{\s L^2(\Omega)}$ is uniformly bounded and therefore there exist $\{u_{n_j}\}\subset\{u_n\}$ and $u^*\in L^2(\Omega)$ such that $u_{n_j}\overset{w}{\longrightarrow} u^*$. Hence, by Theorem \ref{teo:existencia JMAA}, the functional $F_{\s\theta}(u)$ given by (\ref{eq:funcional J}) has a global minimizer $u^* \in L^2(\Omega).$ The condition $\theta(x)\leq \varepsilon_2<1$ for a.e. $x \in \Omega$ clearly implies the strict convexity of $F_{\s \theta}$ and therefore the uniqueness of such a global minimizer.
To prove the second part of the theorem, assume further that there exists $\varepsilon_1>0$ such that $\theta(x)\geq \varepsilon_1$ for a.e. $x \in \Omega.$ Following the proof of Theorem 5.1 in \cite{ref:Mazzieri-Spies-Temperini-JMAA-2012}, it suffices to show that under this additional hypothesis the weak limit $u$ in \textit{(H3)} above belongs to $BV(\Omega).$ For this note that from (\ref{eq:desig-H3}) it follows that there exist $c_2<\infty$ such that
\begin{equation}\label{eq:desig2-H3}
\|u_n\|_{\s L^1(\Omega)}\leq c_2 \quad \forall \;n.
\end{equation}
Also, by Theorem \ref{teo:desig-WgJ} $W_{\scriptscriptstyle 0, \theta}(u)\geq \varepsilon_{\s 1}J_{\s 0}(u)\,\,\forall\, u\in \mathcal M(\Omega).$ This, together with (\ref{eq:desig2-H3}) implies that
\begin{align*}
\|u_n\|_{BV(\Omega)}= \|u_n\|_{L^1(\Omega)}+ J_{\s 0}(u_n)\leq c_2+\frac{W_{\s 0,\theta}(u_n)}{\varepsilon_1}\leq c_3<\infty\,\,\forall \,n,
\end{align*}
where the previous to last inequality follows from the uniform boundedness of $W_{\s 0,\theta}(u_n)$, which, in turn, follows from the uniform boundedness of $W(u_n)$. Hence the fact that the weak limit in (\textit{H3}) is in $BV(\Omega$) follows from the compact embedding of $BV(\Omega$) in $L^2(\Omega)$. This result is an extension of the Rellich-Kondrachov Theorem which can be found, for instance, in \cite{refb:Adams-1975} and \cite{refb:Attouch-Buttazzo-Michaille-2006}.
\end{proof}
\begin{rem}
Note that if $\theta(x)=0\; \forall\, x\in\Omega$, then $W(u)=\|u\|_{L^2(\Omega)}^2$ and $F_{\s \theta}$ as defined in (\ref{eq:funcional J}) is the
classical Tikhonov-Phillips functional of order zero. On the other hand, if $\theta(x)=1\; \forall\, x\in\Omega$
then $W(u)=J_{\s 0}(u)$ and $F_{\s \theta}$ has a global minimizer provided that $T \chig_\Omega\neq 0$. If moreover $T$ is injective then
such a global minimizer is unique. All these facts follow immediately from Theorems 3.1 and 4.1 in
\cite{ref:Acar-Vogel-94}.
\end{rem}
\begin{thm}\label{teo:L1-Linf}
Let $\Omega\subset\mathbb{R}^n$ be a bounded open convex set with Lipschitz boundary, $\X=L^2(\Omega)$, $\Y$ a normed
vector space, $T\in \mathcal L(\X, \Y)$, $v\in\Y$, $\alpha_{\scriptscriptstyle 1},\, \alpha_{\scriptscriptstyle 2}$
positive constants and $\theta \in \widehat{\mathcal{M}}(\Omega)$ such that $\frac{1}{1-\theta}\in L^{\s 1}(\Omega)$ and $\frac{1}{\theta}\in L^{\s \infty}(\Omega)$. Then the functional (\ref{eq:funcional J}) has a unique global minimizer $u^*\in BV(\Omega)$.
\end{thm}
\begin{proof}
Let us consider the functional $$W(u)\doteq \alpha_{\scriptscriptstyle
1}\|\sqrt{1-\theta}\,u\|^2_{\scriptscriptstyle L^2(\Omega)}+\alpha_{\scriptscriptstyle 2}\,W_{\s 0, \theta}(u),\,
u\in L^2(\Omega).$$
By virtue of Theorems \ref{teo:existencia JMAA} and \ref{teo:existencia_unicidad ep1-ep2} and the compact embedding of $BV(\Omega)$ in $L^2(\Omega)$, it suffices to show that $W(\cdot)$ satisfies \textit{(H1)} and \textit{(H2)} and that every $W$-bounded sequence is also $BV$-bounded. Clearly $W(\cdot)$ satisfies \textit{(H1)} with $\gamma=0$. That it satisfies \textit{(H2)} follows immediately from the fact that the condition $\frac{1}{1-\theta}\in L^{\s 1}(\Omega)$ implies that
$\|\sqrt{1-\theta}\cdot\|_{\s L^2(\Omega)}$ is a norm.
Now, let $\{u_n\}\subset L^2(\Omega)$ be a $W$-bounded sequence, i.e. such that $W(u_n)\leq c <\infty,\,\,\forall
\,n$. We will show that $\{u_n\}$ is $BV$-bounded. Since $W(u_n)$ is uniformly bounded, there exist $K<\infty$ such
that $\|\sqrt{1-\theta} \,u_n\|_{\s L^2(\Omega)}\leq K \; \forall \,n.$ From this and the fact that
$\frac{1}{1-\theta}\in L^{\s 1} (\Omega)$ it follows that
\begin{align}\label{eq:desig3-H3}\nonumber
\|u_n\|_{\s L^1(\Omega)}&=\int_\Omega \frac{1}{\sqrt{1-\theta}}\;\sqrt{1-\theta}\,|u_n|\, dx\\ \nonumber
&\leq \left(\int_\Omega \frac{1}{1-\theta}\,dx\right)^{\frac 1 2}\,\left(\int_\Omega
(1-\theta)\,u_n^2\,dx\right)^{\frac 1 2}\\ \nonumber
&=\left\|\frac{1}{1-\theta}\right \|_{\s L^1(\Omega)}^{\frac 1 2}\, \|\sqrt{1-\theta}\,u_n\|_{\s L^2(\Omega)}\\
&\leq K\,\left\|\frac{1}{1-\theta}\right \|_{\s L^1(\Omega)}^{\frac 1 2}<\infty\quad \forall\, n.
\end{align}
On the other hand from Theorem \ref{teo:desig-WgJ} $J_{\s 0}(u)\leq W_{\scriptscriptstyle 0, \theta}(u)\, \left\|\frac 1 {\theta}\right\|_{\s L^\infty(\Omega)}
\, \forall\, u\in L^{\s 2}(\Omega)$. Since $\frac 1 \theta \in L^{\s \infty}(\Omega)$ and $W_{\scriptscriptstyle 0,
\theta}(u_n)$ is uniformly bounded, it then follows that there exists $C<\infty$ such that
\begin{equation}\label{eq:desig4-H3}
J_{\s 0}(u_n)\leq C \; \forall \,n.
\end{equation}
From (\ref{eq:desig3-H3}) and (\ref{eq:desig4-H3}) it follows that
\begin{align*}
\|u_n\|_{BV(\Omega)}= \|u_n\|_{L^1(\Omega)}+ J_{\s 0}(u_n)\leq K\,\left\|\frac{1}{1-\theta}\right
\|_{\s L^1(\Omega)}^{\frac 1 2}+C<\infty\,\,\forall \, n.
\end{align*}
Hence $\{u_n\}$ is $BV$-bounded. The existence of a global minimizer of functional (\ref{eq:funcional J}) belonging to $BV(\Omega)$ then follows.
Finally note that the condition $\frac{1}{1-\theta}\in L^{\s 1}(\Omega)$ implies the strict convexity of
$F_{\s \theta}$ and therefore the uniqueness of the global minimizer.
\end{proof}
\begin{rem}
Note that the condition $\frac{1}{1-\theta}\in L^{\s 1}(\Omega)$ in Theorem \ref{teo:L1-Linf} is weaker than the condition $\theta(x)\leq\varepsilon_{2}<1$ for a.e. $x \in \Omega$ of Theorem \ref{teo:existencia_unicidad ep1-ep2}. While the latter suffices to guarantee the existence of a global minimizer in $L^{\s 2}(\Omega)$, the former does not. However this weaker condition $\frac{1}{1-\theta}\in L^{\s 1}(\Omega)$ together with the condition $\frac1\theta \in L^\infty(\Omega)$ are enough for guaranteing not only the existence of a unique global minimizer, but also the fact that such a minimizer belongs to $BV(\Omega)$.
\end{rem}
It is timely to note that in both Theorems \ref{teo:existencia_unicidad ep1-ep2} and \ref{teo:L1-Linf}, the function $\theta$ cannot assume the extreme values 0 and 1 on a set of positive measure. In some cases a pure $BV$ regularization in some regions and a pure $L^2$ regularization in others may be desired, and therefore that restraint on the function $\theta$ will turn out to be inappropriate. In the next three theorems we introduce different conditions which allow the function $\theta$ to take the extreme values on sets of positive measure.
\begin{thm}
Let $\Omega\subset\mathbb{R}^n$ be a bounded open convex set with Lipschitz boundary, $\X=L^2(\Omega)$, $\Y$ a normed
vector space, $T\in \mathcal L(\X, \Y)$, $v\in\Y$, $\alpha_{\scriptscriptstyle 1},\, \alpha_{\scriptscriptstyle 2}$
positive constants, $\theta \in \widehat{\mathcal{M}}(\Omega)$ and $\Omega_{\s 0}\doteq \{x\in\Omega \text{ such that }\theta(x)=0\}$. If
$\frac{1}{\theta}\in L^{\s \infty}(\Omega_{\s \,0}^{\s \,c})$ and $\frac{1}{1-\theta}\in L^{\s 1}(\Omega_{\s \,0}^{\s \,c})$
then functional (\ref{eq:funcional J}) has a unique global minimizer $u^*\in L^2(\Omega)\cap BV(\Omega_{\s \,0}^{\s \,c}).$
\end{thm}
\begin{proof}
Under the hypotheses of the theorem the functional $W(u)$ can be written as
\begin{equation}\label{eq:W}
W(u)=\alpha_{\s 1}\|u\|^2_{\scriptscriptstyle L^2(\Omega_{\s \,0})}+\alpha_{\s 1}\|\sqrt{1-\theta}\,u\|^2_{\s L^2(\Omega_{\s \,0}^{\s \,c})}+\alpha_{\s 2} \sup_{\vec\nu\in\mathcal V_{\s \theta}}\int_{\Omega_{\s \,0}^{\s \,c}}-u|_{\Omega_{\,0}^{\,c}}\, \text{div}(\theta \vec\nu)\, dx.
\end{equation}
Just like in Theorem \ref{teo:L1-Linf} it follows easily that $W(\cdot)$ satisfies \textit{(H1)} and \textit{(H2)}.
Let now $\{u_n\}\subset L^2(\Omega)$ be a $W$-bounded sequence. From (\ref{eq:W}) we conclude that there exist $u^*_1\in L^2(\Omega_{\s 0})$ and a subsequence $\{u_{n_j}\}\subset\{u_n\}$ such that $u_{n_j}|_{\s {\Omega_{\,0}}} \overset{w-L^2(\Omega_0)}{\longrightarrow} u^*_1$. On the other hand from the uniform boundedness of $\sup_{\vec\nu\in\mathcal V_{\s \theta}}\int_{\Omega_{\s \,0}^{\s \,c}}-u_{n_j}|_{\Omega_{\,0}^{\,c}}\, \text{div}(\theta \vec\nu)\, dx$, by using Theorem \ref{teo:desig-WgJ} with $\Omega$ replaced by $\Omega_{\s \,0}^{\s \,c}$, it follows that there exists a constant $C\leq \infty$ such that $J_{\s 0}(u_{n_j}|_{\Omega_{\,0}^{\,c}})\leq C$ for all $n_j$. Also, from (\ref{eq:W}) and the hypothesis that $\frac{1}{1-\theta}\in L^{\s 1}(\Omega_{\s \,0}^{\s \,c})$, it can be easily proved that the sequence $\{u_n\}$ is uniformly bounded in $ L^{\s 1}(\Omega_{\s \,0}^{\s \,c}).$ Hence $\left\{u_{n_j}|_{\Omega_{\,0}^{\,c}}\right\}$ is uniformly $BV$-bounded. By using the compact embedding of $BV(\Omega_{\s \,0}^{\s \,c})$ in $L^2(\Omega_{\s \,0}^{\s \,c})$ it follows that there exist a subsequence $\{u_{n_{j_k}}\}$ of $\{u_{n_j}\}$ and $u^*_{\s 2}\in BV(\Omega_{\s \,0}^{\s \,c})$ such that $u_{n_{j_k}}\overset{w-L^2(\Omega_{\s \,0}^{\s \,c})}{\longrightarrow}u^*_{\s 2}.$
Let us define now
\begin{equation*}
\hat{u}_{\s 1}(x)\doteq\left\{
\begin{array}{ll}
u^*_{\s 1}(x) , & \hbox{if $x\in \Omega_{\s \,0}$,} \\
0, & \hbox{if $x\in \Omega_{\s \,0}^{\s \,c}$,}
\end{array}
\right.
\end{equation*}
\begin{equation*}
\hat{u}_{\s 2}(x)\doteq\left\{
\begin{array}{ll}
u^*_{\s 2}(x) , & \hbox{if $x\in \Omega_{\s \,0}^{\s \,c}$,} \\
0, & \hbox{if $x\in \Omega_{\s \,0}$,}
\end{array}
\right.
\end{equation*}
and $u^*\doteq \hat{u}_{\s 1}+ \hat{u}_{\s 2}.$ Then one has that $u^*\in L^2(\Omega)$, $u^*|_{\s {\Omega_{\,0}^{\,c}}}=u_2^*\in BV(\Omega_{\s \,0}^{\s \,c})$ and $u_{n_{j_k}}\overset{w-L^2(\Omega)}{\longrightarrow}u^*.$
The existence of a global minimizer of functional (\ref{eq:funcional J}) then follows immediately from Theorem $\ref{teo:existencia JMAA}$. Uniqueness is a consequence of the fact that the hypothesis $\frac{1}{1-\theta}\in L^{\s 1}(\Omega_{\s \,0}^{\s \,c})$ implies that $\|\sqrt{1-\theta}\cdot\|_{\s L^2(\Omega_{\s \,0}^{\s \,c})}$ is a norm.
\end{proof}
\begin{thm}\label{teo omega1}
Let $n\leq 2$, $\Omega\subset\mathbb{R}^n$ be a bounded open convex set with Lipschitz boundary, $\X=L^2(\Omega)$, $\Y$ a
normed vector space, $T\in \mathcal L(\X, \Y)$, $v\in\Y$, $\alpha_{\scriptscriptstyle 1},\,
\alpha_{\scriptscriptstyle 2}$ positive constants. Let $\theta \in \widehat{\mathcal{M}}(\Omega)$ and $\Omega_{\s 1}\doteq
\{x\in\Omega \text{ such that } \theta(x)=1\}$. If $\frac{1}{\theta}\in L^{\s \infty}(\Omega_{\, \s 1}^{\,\s c})$,
$\frac{1}{1-\theta}\in L^{\s 1}(\Omega_{\,\s 1}^{\,\s c})$ and $T\chig_{\s \Omega}\neq 0$, then the
functional (\ref{eq:funcional J}) has a global minimizer $u^*\in L^2(\Omega) \cap BV(\Omega_{\s 1}^{\s c})$. If moreover $\mathcal{N}(T)$ does not contain functions vanishing on $\Omega_1$, i.e. if $Tu=0$ implies $u|_{\s \Omega_1}\ne 0$, then such a global minimizer is unique.
\end{thm}
\begin{proof}
We will prove that under the hypotheses of the theorem, the functional $F_{\s\theta}(\cdot)$ defined by (\ref{eq:funcional J}) is weakly lower semicontinuous with respect to the $L^2(\Omega)$ topology and $BV$-coercive.
First note that under the hypotheses of the theorem we can write
\begin{equation}\label{eq:F-theta-omega1}
F_{\s \theta}(u)= \|Tu-v\|^2_\Y+\alpha_{\scriptscriptstyle 1}\|\sqrt{1-\theta}\,u\|^2_{\scriptscriptstyle L^2(\Omega_{\s \, 1}^{\,c})}+\alpha_{\scriptscriptstyle 2}\,W_{\s 0, \theta}(u).
\end{equation}
Since $\frac{1}{1-\theta}\in L^{\s 1}(\Omega_{\,\s 1}^{\,\s c})$, it follows that $\|\sqrt{1-\theta}\,\cdot\|_{L^2(\Omega_1^c)}$ is a norm in $L^2(\Omega_1^c)$ and therefore it is weakly lower semicontinuous. The weak lower semicontinuity of $F_{\s\theta}(\cdot)$ then follows immediately from this fact, from Lemma \ref{lem:wls-W} and from the weak lower semicontinuity of the norm in $\Y$.
For the $BV$-coercitivity, note that
\begin{align}\label{eq:lalala}\nonumber
\|Tu-v\|^2+\alpha_2 J_{\s 0}(u) &\le \|Tu-v\|^2 +\alpha_2\left\|\frac1\theta\right\|_{\s L^\infty(\Omega_1^c)} W_{\s 0,\theta}(u) \quad(\text{from Theorem (\ref{teo:desig-WgJ})}) \\ \nonumber
&\le \|Tu-v\|^2 +\alpha_2\left\|\frac1\theta\right\|_{\s L^\infty(\Omega_1^c)} W_{\s 0,\theta}(u) +\alpha_1\|\sqrt{1-\theta}\,u\|^2_{\s L^2(\Omega_1^c)}\\
&\le \left\|\frac1\theta\right\|_{\s L^\infty(\Omega_1^c)} F_{\s \theta}(u)\qquad\qquad(\text{since } \left\|\theta^{-1}\right\|_{\s L^\infty(\Omega_1^c)}\ge 1).
\end{align}
Now, since $T\chig_{\s \Omega}\neq 0$, by Theorem \ref{teo:existencia-JOTA} the functional $J(u)\doteq\|Tu-v\|^2+\alpha_2J_{\s 0}(u)$ is $BV$-coercive on $L^2(\Omega)$. From this and inequality (\ref{eq:lalala}) it follows that $F_{\s \theta}(\cdot)$ is also $BV$-coercive.
The existence of a global minimizer $u^* \in L^{\s 2}(\Omega)$ then follows from Theorem \ref{teo:existencia AV}. Since $F_{\s \theta}(u^*)<\infty$ it follows that both $\|u^*\|_{\s L^1(\Omega_1^c)}$ and $W_{\s 0,\theta}(u^*)$ are finite. The fact that $u^*$ is of bounded variation on $\Omega_1^c$ then follows from Theorem \ref{teo:desig-WgJ}.
Finally, if $\mathcal{N}(T)$ does not contain functions vanishing on $\Omega_1$ then it follows easily that $F_{\s \theta}(u)$ is strictly convex and therefore such a global minimizer is unique.
\end{proof}
\begin{thm}
Let $\Omega\subset\mathbb{R}^n$, $n\leq 2$ be a bounded open convex set with Lipschitz boundary, $\X=L^{\s 2}(\Omega)$, $\Y$ a
normed vector space, $T\in \mathcal L(\X, \Y)$, $v\in\Y$, $\alpha_{\scriptscriptstyle 1},\,
\alpha_{\scriptscriptstyle 2}$ positive constants. Let $\theta \in \widehat{\mathcal{M}}(\Omega)$, $\Omega_{\s 0}\doteq \{x\in\Omega \text{ such that } \theta(x)=0\}$ and $\Omega_{\s 1}\doteq
\{x\in\Omega \text{ such that } \theta(x)=1\}$. If $\frac{1}{\theta}\in L^{\s \infty}(\Omega_{\, \s 0}^{\,\s c})$,
$\frac{1}{1-\theta}\in L^{\s \infty}(\Omega_{\,\s 1}^{\,\s c})$ and $\mathcal{N}(T)$ does not contain functions vanishing on $\Omega_1$, i.e. if $Tu=0$ implies $u|_{\s \Omega_1}\ne 0$, then functional (\ref{eq:funcional J}) has a unique global minimizer $u^*\in L^2(\Omega) \cap BV(\Omega_{\s 1}^{\s c}\cap \Omega_{\s 0}^{\s c})$.
\end{thm}
\begin{proof}
For the existence of a global minimizer
it is sufficient to prove that the functional $F_{\theta}$ defined by (\ref{eq:F-theta-omega1}) is weakly lower semicontinuos and $L^{\s 2}(\Omega)$-coercive. For this, note that
\begin{equation}\label{eq:F-theta-omega01}
F_{\s \theta}(u)= \|Tu-v\|^2_\Y+\alpha_{\scriptscriptstyle 1}\|\sqrt{1-\theta}\,u\|^2_{\scriptscriptstyle L^2(\Omega_{\s \, 1}^{\,c})}+\alpha_{\scriptscriptstyle 2}\,\sup_{\vec\nu\in\mathcal V_{\s \theta}}\int_{\Omega_{\s \,0}^{\s \,c}}-u\, \text{div}(\theta \vec\nu)\, dx.
\end{equation}
The weak lower semicontinuity of $F_{\s\theta}(\cdot)$ follows from Lemma \ref{lem:wls-W} and the weak lower semicontinuity of the norms in $\Y$ and $\|\sqrt{1-\theta}\,\cdot\|_{L^2(\Omega_1^c)}$.
We shall now prove that $F_{\s\theta}(\cdot)$ is $L^{\s 2}(\Omega)$-coercive. For that, assume $\{u_n\}$ is a sequence in $L^{\s 2}(\Omega)$ such that
$\|u_n\|_{\s L^{\s 2}(\Omega)}\rightarrow \infty$. Then either $\|u_n\|_{\s L^{\s 2}(\Omega_1^c)}\rightarrow \infty$ or $\|u_n\|_{\s L^{\s 2}(\Omega_1)}\rightarrow \infty$. If $\|u_n\|_{\s L^{\s 2}(\Omega_1^c)}\rightarrow \infty$, then the hypothesis $\frac{1}{1-\theta}\in L^{\s \infty}(\Omega_{\,\s 1}^{\,\s c})$ implies that $\|\sqrt{1-\theta}\,u\|^2_{\scriptscriptstyle L^2(\Omega_{\s \, 1}^{\,c})}\,\rightarrow \infty$ and therefore $F_{\s \theta}(u_n)\rightarrow \infty$. Suppose now that $\|u_n\|_{\s L^{\s 2}(\Omega_1)}\rightarrow \infty$ and without loss of generality assume that $\|u_n\|_{\s L^{\s 2}(\Omega_{\s 1}^{\s c})} \le C<\infty$. Then due to the compact embedding $BV(\Omega_{\s 1})\hookrightarrow L^{\s 2}(\Omega_{\s 1})$
it follows that $\|u_n\|_{\s BV(\Omega_1)} \rightarrow \infty$. Since $\mathcal{N}(T)$ does not contain functions vanishing on $\Omega_1$, it follows that $T\chig_{\s \Omega_1}\neq 0$. Then, by Theorem 1.2, the functional $\|Tu_n-v\|^2_\Y + \alpha_2 J_{\s 0}^{\s \Omega_1}(u_n)$ is $BV$-coercive; i.e:
\begin{equation}\label{eq:caca1}
\|Tu_n-v\|^2_\Y + \alpha_2 J_{\s 0}^{\s \Omega_1}(u_n)\rightarrow\infty.
\end{equation}
Now clearly
\begin{equation} \label{eq:caca2}
\begin{aligned}
\|Tu_n-v\|^2_\Y + \alpha_2 J_{\s 0}^{\s \Omega_1}(u_n) &\le \|Tu_n-v\|^2_\Y +\alpha_{\scriptscriptstyle 2}\,\sup_{\vec\nu\in\mathcal V_{\s \theta}}\int_{\Omega_{\s \,0}^{\s \,c}}-u_n\, \text{div}(\theta \vec\nu)\, dx \\
&\le F_{\s \theta}(u_n).
\end{aligned}
\end{equation}
From (\ref{eq:caca1}) and (\ref{eq:caca2}) it follows that $F_{\s \theta}(u_n)\rightarrow \infty$. Hence $F_{\s \theta}$ is $L^{\s 2}(\Omega)$-coercive. The existence of a global minimizer then follows.
Finally, the hypothesis that $\mathcal{N}(T)$ does not contain functions vanishing on $\Omega_1$ also implies that $F_{\s \theta}(u)$ is strictly convex and therefore such a global minimizer is unique.
\end{proof}
\section{Signal restoration with $L^2$-$BV$ regularization}
\label{sec:3}
The purpose of this section is to show some applications of the regularization method developed in the previous section consisting in the simultaneous use of penalizers of $L^2$ and of bounded-variation (BV) type to signal restoration problems.
\medskip
A basic mathematical model for signal blurring is given by convolution, as a
Fredholm integral equation of first kind:
\begin{equation}\label{eq:I}
v(t)=\int_0^1 k(t,s) u(t) ds,
\end{equation}
where $k(t,s)$ is the blurring kernel or point spread function, $u$ is the exact (original) signal and $v$ is the blurred signal. For the examples that follow we took a Gaussian blurring kernel, i.e. $k(t,s)=\frac{1}{\sqrt{2\pi}\sigma_b}\,\textrm{exp}\left(-\frac{(t-s)^2}{2\sigma_b^2}\right)$, with $\sigma_b> 0$. Equation (\ref{eq:I}) was discretized in the usual way (using collocation and quadrature), resulting in a
discrete model of the form
\begin{equation}\label{eq:II}
Af=g,
\end{equation}
where $A$ is a $(n+1) \times (n+1)$ matrix, $\,f,g \in \mathbb{R}^{n+1}$ ($f_j=u(t_j), \, g_j=v(t_j),\, t_j=\frac j n, \, 0\leq j\leq n$). We took $n=130$ and $\sigma_b=0.05$. The data $g$ was contaminated with a 1\%
zero-mean Gaussian additive noise (i.e. standard deviation equal to 1\% of the range of $g$).
\medskip
\textbf{Example 3.1.} For this example, the original signal (unknown in real life problems) and the blurred noisy signal which constitutes the data
of the inverse problem for this example are shown in Figure 1.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=.5]{solexacta_y_dato.eps}
\caption{Original signal ({\color{red}{- -}}) and blurred noisy signal ({\color{blue}{---}}).}
\label{fig:1}
\end{center}
\end{figure}
Figure 2 shows the regularized solutions obtained with the classical Tikhonov-Phillips method of order zero (left) and with penalizer associated to the bounded variation seminorm $J_{\s 0}$ (right). As expected, the regularized solution obtained with the $J_{\s 0}$ penalizer is significantly better than the one obtained with the classical Tikhonov-Phillips method near jumps and in regions where the exact solution is piecewise constant.
The opposite happens where the exact solution is smooth.
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=.48]{sol_T0.eps}\quad
\includegraphics[scale=.48]{sol_BV.eps}\\
\caption{Original signal ({\color{red}{- -}}) and regularized solutions ({\textcolor[rgb]{0.00,0.50,0.00}{---}}) obtained with Tikhonov-Phillips (left) and bounded variation seminorm (right).}
\label{fig:2}
\end{center}
\end{figure}
Figure 3 shows the regularized solution obtained with the combined $L^2-$BV method (see (\ref{eq:funcional J})). In this case the weight function $\theta(t)$ was chosen to be $\theta(t)\doteq 1$ for $t\in (0, 0.4]$ and $\theta(t)\doteq 0$ for $t\in (0.4, 1)$. Although this
choice of $\theta(t)$ is clearly based upon ``\textit{a-priori}'' information about the regularity of exact
solution, other reasonable choices of $\theta$ can be made by using only data-based information. Choosing a ``good" weighting function $\theta$ is a very important issue but we shall not discuss this matter in this article. For instance, one way of constructing a reasonable function $\theta$ is by computing the normalized (in $[0, 1]$) convolution of a Gaussian function of zero mean and standard deviation $\sigma_b$ and the modulus of the gradient of the regularized solution obtained with a pure zero-order Tikhonov-Phillips method (see Figure 4). For this weight function $\theta$, the corresponding regularized solution obtained with the combined $L^2-$BV method is shown in Figure 5. In all cases reflexive boundary conditions were used (\cite{refb:Hansen2010}) and the regularization parameters were calculated using Morozov's discrepancy principle with $\tau=1.1$ (\cite{refb:Engl-Hanke-96}).
\begin{figure}[H]
\begin{center}
\includegraphics[scale=.461]{solT0BV_thetabin.eps}
\caption{Original signal ({\color{red}{- -}}) and regularized solution ({\textcolor[rgb]{0.00,0.50,0.00}{---}}) obtained with the combined $L^2-BV$ method and binary weight function $\theta$.}
\label{fig:3}
\end{center}
\end{figure}
\begin{figure}[H]
\begin{center}
\includegraphics[scale=.5]{thetaconv.eps}
\caption{Weight function $\theta$ computed by normalizing the convolution of a Gaussian kernel and the modulus of the gradient of the regularized solution with a pure Tikhonov-Phillips method.}
\label{fig:4}
\end{center}
\end{figure}
\begin{figure}[H]
\begin{center}
\includegraphics[scale=.5]{solT0BV_thetaconv.eps}
\caption{Original signal ({\color{red}{- -}}) and regularized solution ({\textcolor[rgb]{0.00,0.50,0.00}{---}}) obtained with the combined $L^2-BV$ method and the data-based weight function $\theta$ showed in Fig. \ref{fig:4}.}
\label{fig:5}
\end{center}
\end{figure}
As it can be seen, the improvement of the result obtained with the combined $L^2-BV$ method and ``\textit{ad-hoc}'' binary function $\theta$ with respect to the pure simple methods, zero-order Tikhonov-Phillips and pure $BV$, is notorious. As previously mentioned however, in this case the construction of the function $\theta$ is based on ``\textit{a-priori}'' information about the exact solution, which most likely will not be available in concrete real life problems. Nevertheless, the regularized solution obtained with the data-based weight function $\theta$ shown in Figure \ref{fig:4} is also significantly better than those obtained with any of the single-based penalizers. This fact is clearly and objectively reflected by the Improved Signal-to-Noise Ratio (ISNR) defined as
$$ISNR=10 \log_{10}\left(\frac{\norm{f-g}^2}{\norm{f-f_\alpha}^2}
\right),$$ where $f_\alpha$ is the restored signal obtained with regularization parameter $\alpha$. For all the
previously shown restorations, the ISNR was computed in order to have a parameter for objectively measuring and comparing the
quality of the regularized solutions (see Table \ref{tab:1}).
\begin{table}[H]
\caption{ISNR's for Example 3.1.}
\label{tab:1}
\smallskip
\hskip 2cm
\begin{tabular}{|p{11cm}|p{1.5cm}|}
\hlin
\textbf{Regularization Method} & \textbf{ISNR} \\
\hlin
Tikhonov-Phillips of order zero & 2.5197\\
\hlin
Bounded variation seminorm & 4.2063\\
\hlin
Mixed $L^2-$BV method with binary $\theta$ & 5.7086\\
\hlin
Mixed $L^2-$BV method with zero-order Tikhonov-based $\theta$ & 4.4029\\
\hlin
\end{tabular}
\end{table}
\medskip
\textbf{Example 3.2.} For this example we considered a signal which is smooth in two disjoint intervals and it is piecewise constant in their complement, having three jumps. The signal was blurred and noise was added just as in the previous example. The original and blurred-noisy signal are depicted in Figure 6.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=.5]{solexacta_y_dato2.eps}
\caption{Original ({\color{red}{- -}}) and blurred-noisy ({\color{blue}{---}}) signals for Example 3.2.}
\label{fig:6}
\end{center}
\end{figure}
Figure 7 shows the restorations obtained with the classical zero-order Tikhonov-Phillips method (left) and $BV$ with penalizer $J_{\s 0}$ (right).
\begin{figure}[H]
\begin{center}
\includegraphics[scale=.48]{sol_T02.eps}\quad
\includegraphics[scale=.48]{sol_BV2.eps}\\
\caption{Original signal ({\color{red}{- -}}) and regularized solutions ({\textcolor[rgb]{0.00,0.50,0.00}{---}}) obtained with Tikhonov-Phillips (top) and bounded variation seminorm (bottom).}
\label{fig:7}
\end{center}
\end{figure}
An ad-hoc binary weight function theta for this example was defined on the interval $[0,1]$ as $\theta(t)=\chi_{\s [0.3, 0.65]}(t)$. The regularized solution obtained with this weight function and the combined $L^2-BV$ method is shown in Figure 8. Once again, the improvement with respect to any of the classical pure methods is clearly notorious.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=.5]{solT0BV_thetabin2.eps}
\caption{Original signal ({\color{red}{- -}}) and regularized solution ({\textcolor[rgb]{0.00,0.50,0.00}{---}}) obtained with the combined $L^2-BV$ method and binary function $\theta$.}
\label{fig:8}
\end{center}
\end{figure}
Here also we constructed a data based weight function $\theta$ as in Example 3.1, by convolving a Gaussian kernel with the modulus of the gradient of a a Tikhonov regularized solution and normalizing the result. This weight function $\theta$ is now depicted in Figure 9, while the corresponding restored signal is shown in Figure 10.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=.5]{thetaconv2.eps}
\caption{Tikhonov-based weight function $\theta$ for Example 3.2.}
\label{fig:9}
\end{center}
\end{figure}
\begin{figure}[H]
\begin{center}
\includegraphics[scale=.5]{solT0BV_thetaconv2.eps}
\caption{Original signal ({\color{red}{- -}}) and regularized solution ({\textcolor[rgb]{0.00,0.50,0.00}{---}}) obtained with the combined $L^2-BV$ method and function $\theta$ showed in Fig. \ref{fig:9}.}
\label{fig:10}
\end{center}
\end{figure}
In table \ref{tab:2} the values of the ISNR for the four restorations are presented. These values show once again a significant improvement of the combined method with respect to any of the pure single methods.
\begin{table}[H]
\caption{ISNR's for Example 3.2.}
\smallskip
\label{tab:2}
\hskip 2cm
\begin{tabular}{|p{11cm}|p{1.5cm}|}
\hlin
\textbf{Regularization Method} & \textbf{ISNR} \\
\hlin
Tikhonov-Phillips of order zero & 2.6008\\
\hlin
Bounded variation seminorm & 2.8448\\
\hlin
Mixed $L^2-$BV method with binary $\theta$ & 4.8969\\
\hlin
Mixed $L^2-$BV method with zero-order Tikhonov-based $\theta$ & 4.3315\\
\hlin
\end{tabular}
\end{table}
\section{Conclusions}
In this article we introduced a new generalized Tikhonov-Phillips regularization method in which the penalizer
is given by a spatially varying combination of the $L^2$ norm and of the bounded variation seminorm. For particular cases, existence and uniqueness of global minimizers of the corresponding functionals were shown. Finally, applications of the new method to signal restoration problem were shown.
Although these preliminary results are clearly quite promising, further research is needed. In particular, the choice or construction
of a weight function $\theta(t)$ in a somewhat optimal way is a matter which undoubtedly deserves much further attention and study.
Research in these directions is currently under way.
\section*{Acknowledgments}
This work was supported in part by Consejo Nacional de Investigaciones Cient\'{\i}ficas y T\'{e}cnicas, CONICET, through PIP
2010-2012 Nro. 0219, by Agencia Nacional de Promoci\'{o}n Cient\'{\i}fica y Tecnol\'{o}gica, ANPCyT, through project PICT 2008-1301, by Universidad Nacional del Litoral, through projects CAI+D 2009-PI-62-315, CAI+D PJov 2011 Nro. 50020110100055, CAI+D PI 2011 Nro. 50120110100294 and by the Air Force Office of Scientific Research, AFOSR, through Grant FA9550-10-1-0018.
\bibliographystyle{amsplain}
|
1,116,691,497,807 | arxiv | \section{Introduction}\label{sec1}
We aim to solve the equation
\begin{equation*}
Kx=y^\delta
\end{equation*}
for $K:\mathcal{X}\to \mathcal{Y}$ a compact operator between infinite-dimensional Hilbert spaces. Hereby, $y^\delta$ is a noisy measurement of the exact data $y^\dagger\in \mathcal{R}(K)$ and $x^\dagger:=K^+y^\dagger$ is the exact minimum norm solution we would like to reconstruct. We exclude the trivial case when the range of $K$ is finite-dimensional, which yields, due to compactness of $K$, that the above equation is ill-posed in the sense that the (pseudo) inverse $K^+$ is not bounded. The data is assumed to be corrupted by additive white noise $Z$, i.e., it holds that $y^\delta=y^\dagger+\delta Z$ where $Z:\mathcal{Y}\to L^2(\Omega,\mathbb{R})$ fulfills
\begin{itemize}
\item $\mathbb{E}[(Z,y)] = 0$,
\item $\mathbb{E}[(Z,y),(Z,y')] = (y,y')$,
\item $\|y'\|(Z,y) \stackrel{d}{=} \|y\| (Z,y')$
\end{itemize}
for all $y,y'\in \mathcal{Y}$. We stress that the distribution of the white noise is merely assumed to be centered with finite variance. We call $Z$ a Gaussian white noise, when $(Z,y/\|y\|)$ is standard Gaussian. The standard way to approach the problem is through regularisation, i.e., the unbounded inverse $K^+$ is replaced with a whole family of linear bounded reconstructions and then a particular element of this family is chosen dependent on the measurement such that the exact solution $x^\dagger$ is approached for vanishing noise level $\delta\to0$, see the classic monographs from Tikhononv et al. \cite{tikhonov1995numerical} and Engl et al. \cite{EnglHankeNeubauer:1996}, or the more recent ones from Lu \& Pereverzev \cite{lu2013regularization}, Ito \& Jin \cite{ItoJin:2015} and Hanke \cite{hanke2017taste}, to name a few for a general overview.
In this note a method for choosing the reconstruction is presented that does not require any additional knowledge apart from $y^\delta$, neither of the noise distribution nor of the noise level $\delta$. This class of methods is called noise level free or heuristic and has been a subject of study ever since due to its importance for practical applications, see Bauer \& Lukas \cite{bauer2011comparingparameter} or Kindermann \cite{kindermann2011convergence}.
For conceptual reasons we rely on spectral cut-off regularisation using explicitly the singular value decomposition of the operator $K$. This decomposition consists of orthonormal bases $(v_j)_{j\in\mathbb{N}}\subset \mathcal{N}(K)^\perp\subset \mathcal{X}$ and $(u_j)_{j\in\mathbb{N}}\subset \overline{\mathcal{R}(K)}\subset \mathcal{Y}$ as well as a sequence of singular values $\sigma_1\ge \sigma_2 \ge ...\searrow 0$ such that $Kv_j=\sigma_j u_j$ and $K^*u_j=\sigma_j v_j$ for all $j\in\mathbb{N}$. The spectral cut-off regularisation yields a family of approximations of $x^\dagger=K^+y^\dagger= \sum_{j=1}^\infty\frac{(y^\dagger,u_j)}{\sigma_j}u_j$ through
$$x_k^\delta:=\sum_{j=1}^k \frac{(y^\delta,u_j)}{\sigma_j}v_j.$$
From a practical point of view the desire to solve the inverse problem without knowledge of $\delta$ is perfectly understandable, however, delicate from the mathematical side due to the famous Bakunshinski veto \cite{bakushinskii1984remarks}. This result rigorously formalises the longstanding paradigm in inverse problems, namely that those are only solvable if additional information of the size of the noise is available. Precisely, it states that in the classical deterministic setting, where, different to here, $y^\delta=y^\dagger+\delta \xi_\delta$ with $\xi_\delta\in \mathcal{Y}, \|\xi_\delta\|\le 1$, it is impossible to solve the equation without knowledge of $\delta$ for arbitrary error directions $\xi_\delta$, and this is exactly the reason why noise level free methods are called 'heuristic'. Convergence results for noise level free methods in the classical setting are therefore subject to a noise restricted analysis, see Kindermann \& Neubauer \cite{kindermann2008convergence}, i.e., the additional assumption that the $\xi_\delta$ are elements of some adequate subset of $\mathcal{Y}$ has to be imposed. In the stochastic setting a possible workaround for the Bakushinskii veto is to estimate the noise level using multiple data sets $y^\delta_1, y^\delta_2,...\in \mathcal{Y}$ see Harrach et al. \cite{harrach2020beyond,10.1093/imanum/drab098} and \cite{jahn2021modified} by the author. In this article we present a method which only uses one measurement $y^\delta$.
The main difference of the white noise setting (see Bissantz et al. \cite{bissantz2007convergence} and Cavalier \cite{cavalier2011inverse}) to the classical setting of deterministic bounded noise is that the measurement $y^\delta$ is no longer an element of the Hilbert space, since its variance is unbounded $\mathbb{E}\|y^\delta-y^\dagger\|^2 =\sum_{j=1}^\infty\mathbb{E}[(y^\delta-y^\dagger,u_j)^2] = \sum_{j=1}^\infty \delta^2 = \infty$. Because of that most classic methods cannot be applied in a straight forward fashion in the white noise scenario and often methods originating from statistics, as, for example, Cross validation from Wahba \cite{wahba1977practical}, empirical risk minimisation (Cavalier et al. \cite{cavalier2002oracle}) or Lepski's method \cite{lepskii1991problem} (see Math\'{e} \& Pereverzev \cite{mathe2006regularization}) are used. Still, those methods typically require knowledge of $\delta$ or do not work for general compact operators, see Lucka et al. \cite{lucka2018risk}.
Due to the above mentioned reminiscence, regarding heuristic methods for statistical inverse problems only few results are available. An exemption is Bauer \& Rei\ss{} \cite{bauer2008regularization}, where remarkably, optimal convergence of the quasi-optimality criterion, dating back to Tikhonov \cite{tikhonov1963solution}, under Gaussian white noise in a Bayesian setup is presented.
Hereby, some drawbacks are that only a rather specific setup is covered regarding the spectral properties of the forward operator, the distribution of the white noise and the source condition. We will consider a related setup in Theorem \ref{th2} below, where we quantify the probability that the optimal error bound holds. Note that for the implementation of the method from Bauer \& Rei\ss{} additional subsampling functions have to be chosen, tailored to the type of ill-posedness and source condition, which require at least some knowledge on the underlying ground truth. The method presented in this manuscript might be a bit more user-friendly and moreover applicable in more general settings. Also, the quasi-optimality criterion has been investigated in Kindermann et al. \cite{kindermann2018quasi} for stochastic noise with finite variance, which allows to perform a classical analysis.
Another approach to handle the unboundedness of the noise, well-known from statistical inverse problems, is to smooth the data and equation. This has been done for heuristic rules under weakly bounded deterministic noise by Kindermann \& Raik \cite{kindermann2019heuristic} recently. However, presmoothing is usually not the first choice from a numerical view point since it yields a rapid increase of the degree of ill-posedness in first place.
In the recent publications \cite{jahn2021optimal, jahn2022probabilistic} by the author of this manuscript an alternative approach to handle the white noise was presented, which constitutes of using adaptively a combination of regularisation and discretisation for the discrepancy principle due to Phillips \cite{phillips1962technique} and Morozov \cite{Morozov:1966}, one of the most popular non-heuristic methods. Here we apply the very same ideas to the heuristic discrepancy principle, which was originally introduced by Hanke \& Raus \cite{hanke1996general}. The heuristic discrepancy principle for choosing $k$ is defined as follows
$$k^\delta_{\rm cHD}:=\arg\min_{k\in\mathbb{N}}\Psi(k,y^\delta) := \arg\min_{k\in\mathbb{N}} \frac{1}{\sigma_k}\sqrt{\sum_{j=k}^\infty(y^\delta,u_j)^2}.$$
Due to the unboundedness of the white noise the above is not applicable, since the right hand side is infinity for all $k\in\mathbb{N}$. We therefore introduce a discretisation parameter $m\in\mathbb{N}$ and define accordingly a discretised variant
\begin{equation}\label{eq00}
k^\delta_{\rm HD}(m):=\arg\min_{k\le \frac{m}{2}}\Psi_m(k,y^\delta)=\arg\min_{k\le \frac{m}{3}} \frac{1}{\sigma_k}\sqrt{\sum_{j=k}^m(y^\delta,u_j)^2}.
\end{equation}
The reason that we restrict to $k\le m/2$ for the minimisation instead of, say, $k\le m$ is to avoid random fluctuations of few components to dominate. The particular choice of $m/2$ as the upper bound is for convenience, other choices are possible as well (as long as it is taken care of that the randomness of single components is negligible). In order to determine the final approximation we have to choose the discretisation level $m$. For that we follow exactly the authors' paradigm from \cite{jahn2021optimal, jahn2022probabilistic} and set
\begin{equation}\label{eq0}
k^\delta_{\rm HD}:=\max_{m\in\mathbb{N}} k^\delta_{\rm HD}(m).
\end{equation}
We will formulate the main results in Section \ref{sec2} and prove them in Section \ref{sec3}. The paper closes with some numerical experiments in Section \ref{sec4}.
\section{Results}\label{sec2}
We formulate the main result, which shows convergence in probability of the method for general ill-posed $K$ and general (non-Gaussian) white noise.
\begin{theorem}\label{th1}
For any compact operator $K$ with nonclosed range and any admissible data $y^\dagger\in\mathcal{R}(K) \cup \mathcal{R}(K)^\perp$ there holds for all $\varepsilon>0$ that
\begin{equation*}
\mathbb{P}\left(\|x_{k^\delta_{\rm HD}}^\delta - K^+y^\dagger\|\le \varepsilon\right)\to 1
\end{equation*}
as $\delta\to0$, i.e., $x_{k^\delta_{\rm HD}}$ converges to $K^+y^\dagger$ in probability.
\end{theorem}
We stress that Theorem \ref{th1} explains that the Bakushinskii veto does not apply in the white noise setting in full generality since clearly no knowledge of $\delta$ is needed for the determination of the final truncation index $k^\delta_{\rm HD}$. The type of convergence in Theorem \ref{th1} is weaker then convergence in $L^2$ (called convergence in mean integrated square error in this context), which is more commonly used in statistical inverse problems. The following counter example shows that convergence of the mean integrated square error does not hold here, even if the (component) distribution of the noise is bounded.
\begin{lemma}\label{ex1}
For $x^\dagger=0$, let $\sigma_j^2=e^{-j}$ and assume that the white noise has distribution $\mathbb{P}\left((Z,y) = 0\right)=1/2$, $\mathbb{P}\left((Z,y)=\pm \sqrt{2}\right)=1/4$. Then, there holds
$$\mathbb{E}\left[\left\| x^\delta_{k^\delta_{\rm HD}}-x^\dagger\right\|^2\right] \ge \frac{1}{4\delta}\to\infty$$
as $\delta\to0$, i.e., $x^\delta_{\rm HD}$ does not converge in mean integrated square error.
\end{lemma}
It is well-known that the heuristic discrepancy principle yields only suboptimal rates in the standard setting of general source conditions. However, the following result states that it actually fulfills an oracle inequality, i.e., it attains the minimal possible error (up to a constant factor) asymptotically, in a similar setup to the one from Bauer \& Rei\ss{} \cite{bauer2008regularization}, where polynomially ill-posed operators were considered in a Bayesian-like framework. This constitutes of the assumption that the components of the unknown solution $x^\dagger$ are randomly sampled, with respective decreasing variance. For a general introduction to Bayesian inverse problems we refer to Kaipio \& Somersalo \cite{kaipio2006statistical}.
\begin{theorem}\label{th2}
Assume that there exist $q>0$ and $\eta>1$ such that $\sigma_j^2=j^{-q}$ and $(x^\dagger,v_j) = j^{-\frac{\eta}{2}} X_j$ for all $j\in\mathbb{N}$, where the $(X_j)_{j\in\mathbb{N}}$ are i.i.d. standard Gaussian. Moreover, assume that $Z$ is Gaussian white noise. Then for all $0\le \varepsilon<1/3$ and $0< \delta\le 1$
\begin{equation}\label{eqth2}
\mathbb{P}\left( \|x^\delta_{k^\delta_{\rm HD}}-x^\dagger\|\le C_{q,\eta,\varepsilon} \min_{k\in\mathbb{N}}\|x^\delta_k-x^\dagger\|\right)\ge1-4 e^{-p_{q,\eta,\varepsilon} \delta^\frac{2(1-\eta)}{(q+\eta-1)(q+\eta)}}
\end{equation}
with $p_{q,\eta,\varepsilon}$ and $C_{q,\eta,\varepsilon}$ given in \eqref{prob} and \eqref{bound} below.
\end{theorem}
Note that a smaller $\varepsilon$ yields smaller $p_{q,\eta,\varepsilon}$ and $C_{q,\eta,\varepsilon}$ in \eqref{eqth2} and hence a better bound for $\|x^\delta_{k^\delta_{\rm HD}}-x^\dagger\|$, but which holds with a smaller probability.
Theorem \ref{th1} and \ref{th2} above show that the here proposed method is reasonable and allows for regularisation without knowledge of the noise level. This will be confirmed numerically in Section \ref{sec4} below. Clearly, it would be interesting to apply the approach to other heuristic methods as, e.g., the L-curve (Hansen \cite{hansen1992analysis}) the Hanke-Raus-rule \cite{hanke1996general} or the aforementioned quasi-optimality criterion.
\section{Proofs}\label{sec3}
We start with a central proposition which controls the discretised measurement errors simultaneously.
\begin{proposition}\label{prop1}
For $\varepsilon'>0$ and $\kappa\ge \left(\frac{3}{2}+\frac{3}{2\varepsilon'}\right)^2$ there holds
\begin{align*}
&\mathbb{P}\left(\left|\sum_{j=k+1}^m(y^\delta-y^\dagger,u_j)^2-(m-k)\delta^2\right|\le \varepsilon'(m-k)\delta^2,~\forall m\ge \kappa,~k\le m/2\right)\\
\ge\qquad&1-\frac{1}{\varepsilon'}\mathbb{E}\left[\left|\frac{1}{\sqrt{\kappa}}\sum_{j=1}^{\sqrt{\kappa}}\left(\delta^{-2}(y^\delta-y^\dagger,u_j)^2-1\right)\right|\right].
\end{align*}
Moreover,
\begin{align*}
\lim_{\kappa\to \infty}\mathbb{E}\left[\left|\frac{1}{\sqrt{\kappa}}\sum_{j=1}^{\sqrt{\kappa}}\left(\delta^{-2}(y^\delta-y^\dagger,u_j)^2-1\right)\right|\right] = 0.
\end{align*}
\end{proposition}
\begin{proof}[Proof of Proposition \ref{prop1}]
We rely on Proposition 4.1 from \cite{jahn2022probabilistic}, which states that
for
$$\Omega_\kappa:=\left\{ \left|\sum_{j=1}^l(y^\delta-y^\dagger,u_j)^2-l\delta^2\right|\le \frac{\varepsilon'}{3},~\forall l\ge \sqrt{\kappa}\right\}$$
there holds
$$\mathbb{P}\left(\Omega_\kappa\right)\ge 1 - \frac{3}{\varepsilon'}\mathbb{E}\left[\left|\frac{1}{\sqrt{\kappa}}\sum_{j=1}^{\sqrt{\kappa}}\left(\delta^{-2}(y^\delta-y^\dagger,u_j)^2-1\right)\right|\right]\stackrel{(\kappa\to\infty)}{\longrightarrow} 1$$
and whose proof consists of an application of Doob's extremal theorem to the backward martingale $\left(\frac{1}{l}\sum_{j=1}^l\left((y^\delta-y^\dagger,u_j)^2\delta^{-2}-1\right)\right)_{l\in\mathbb{N}}$.
From $k\le m/2$ it follows that $m+2k\le 2m$ and $k\le m-k$. Thus
$$m+k\le 2m-k=2(m-k)+k\le 2(m-k)+(m-k)=3(m-k).
$$
We distinguish the cases $k\le \sqrt{\kappa}$ and $k>\sqrt{\kappa}$.
In the latter case we have that
\begin{align*}
&\sum_{j=k+1}^m(y^\delta-y^\dagger,u_j)^2\chi_{\Omega_\kappa}=\sum_{j=1}^m(y^\delta-y^\dagger,u_j)^2\chi_{\Omega_\kappa} -\sum_{j=1}^k(y^\delta-y^\dagger,u_j)^2\chi_{\Omega_\kappa}\\
\le &\left(1+\frac{\varepsilon'}{3}\right)m\delta^2 - \left(1-\frac{\varepsilon'}{3}\right)k\delta^2 = (m-k)\delta^2 + \frac{\varepsilon'}{3}(m+k)\delta^2\\
\le &(1+\varepsilon')(m-k)\delta^2
\end{align*}
and similarly
\begin{align*}
&\sum_{j=k+1}^m(y^\delta-y^\dagger,u_j)^2\chi_{\Omega_\kappa}=\sum_{j=1}^m(y^\delta-y^\dagger,u_j)^2\chi_{\Omega_\kappa} -\sum_{j=1}^k(y^\delta-y^\dagger,u_j)^2\chi_{\Omega_\kappa}\\
\ge &\left(1-\frac{\varepsilon'}{3}\right)m\delta^2\chi_{\Omega_\kappa} - \left(1+\frac{\varepsilon'}{3}\right)k\delta^2\chi_{\Omega_\kappa}
\ge (1-\varepsilon')(m-k)\delta^2\chi_{\Omega_\kappa}.
\end{align*}
In the former case there holds
\begin{align*}
\sum_{j=k+1}^m(y^\delta-y^\dagger,u_j)^2\chi_{\Omega_\kappa}&\le \sum_{j=1}^m(y^\delta-y^\dagger,u_j)^2\chi_{\Omega_\kappa}\le\left(1+\frac{\varepsilon'}{3}\right)m\delta^2\chi_{\Omega_\kappa}\\
= &\left(1+\frac{\varepsilon'}{3}\right)(m-k)\delta^2 +\frac{\varepsilon'}{3} k\delta^2\le (1+\varepsilon')(m-k)\delta^2,
\end{align*}
since $m-k\ge k$, and
\begin{align*}
&\sum_{j=k+1}^m(y^\delta-y^\dagger,u_j)^2\chi_{\Omega_\kappa}\ge \sum_{j=\sqrt{\kappa}+1}^m(y^\delta-y^\dagger,u_j)^2\chi_{\Omega_\kappa}\\
=&\sum_{j=1}^m(y^\delta-y^\dagger,u_j)^2\chi_{\Omega_\kappa} - \sum_{j=1}^{\sqrt{\kappa}}(y^\delta-y^\dagger,u_j)^2\chi_{\Omega_\kappa}\\
\ge &\left(1-\frac{\varepsilon'}{3}\right)m\delta^2\chi_{\Omega_\kappa} - \left(1+\frac{\varepsilon'}{3}\right)\sqrt{\kappa}\delta^2\\
\ge&\left(1-\frac{\varepsilon'}{3}\right)(m-k)\delta^2\chi_{\Omega_\kappa}-\left(1+\frac{\varepsilon'}{3}\right)\sqrt{\kappa}\delta^2\\
=&\left(1-\frac{\varepsilon'}{3}\right)(m-k)\delta^2\chi_{\Omega_\kappa} -\frac{2\varepsilon'}{3}(m-k)\delta^2 \frac{\left(1+\frac{\varepsilon'}{3}\right)\sqrt{\kappa}}{\frac{2\varepsilon'}{3}(m-k)}\\
\ge&\left(1-\frac{\varepsilon'}{3}\right)(m-k)\delta^2\chi_{\Omega_\kappa} - \frac{2\varepsilon'}{3}(m-k)\delta^2 \frac{\left(1+\frac{\varepsilon'}{3}\right)}{\frac{2\varepsilon'}{3}(\sqrt{\kappa}-1)}\ge (1-\varepsilon')(m-k)\delta^2\chi_{\Omega_\kappa},
\end{align*}
since $\kappa\ge \left(\frac{3}{2}+\frac{3}{2\varepsilon'}\right)^2$.
\end{proof}
\subsection{Proof of Theorem \ref{th1}}
We first define the 'nice' events where the error behaves regularly and thus allows for perfect control of the measurement error, subject to a small parameter $\varepsilon'>0$ and a sequence $(m^\delta)_{\delta>0}$ converging to $\infty$, which will be specified later.
\begin{equation}\label{om}
\Omega_{\delta}:=\left\{\left|\sqrt{\sum_{j=k}^m(y^\delta - y^\dagger,u_j)^2}-\sqrt{m-k}\delta\right|\le \varepsilon' \sqrt{m-k}\delta,~\forall m\ge m_\delta, k\le \frac{m}{2}\right\}.
\end{equation}
Proposition \ref{prop1} above yields $\mathbb{P}\left(\Omega_\delta\right)\to 1$ as $\delta\to 0$ which allows us to restrict our analysis to the events $\Omega_\delta$.
We start by showing that the proposed rule \eqref{eq0} is well-defined. Clearly, $k^\delta_{\rm HD}(m)$ is well-defined for all $m\in\mathbb{N}$ and the same holds true for $k^\delta_{\rm HD}$ (on $\Omega_\delta$), as follows directly from the next proposition. In fact the propsition already guarantees stability of the proposed method in that it upper bounds the regularisation parameter appropriately.
\begin{proposition}\label{prop3}
For the choice
$$m_\delta=k_{\delta,\varepsilon}:=\max\left\{k\in\mathbb{N}~:~\frac{k}{\sigma_k^2} \delta^2 \le \frac{\varepsilon^2}{4}\right\}$$
and $\varepsilon'<\frac{1}{4}$ in \eqref{om} there holds
$$k^\delta_{\rm HD}\chi_{\Omega_{\delta}} \le k_{\delta,\varepsilon}$$
for $\delta$ small enough.
\end{proposition}
\begin{proof}[Proof of Proposition \ref{prop3}]
We first observe that for $\varepsilon$ fixed
\begin{equation}\label{eq1}
k_{\delta,\varepsilon}\nearrow \infty
\end{equation}
as $\delta\to0$. By definition of $k_{\delta,\varepsilon}$, \eqref{eq1} and $\|x^\dagger\|<\infty$ we deduce that for all $k>k_{\delta,\varepsilon}$
\begin{align}\label{eq1a}
\sqrt{\sum_{j=k}^\infty (y^\dagger,u_j)^2} &\le \sigma_{k}^{\delta,\varepsilon} \sqrt{\sum_{j=k}^\infty(x^\dagger,v_j)^2} \le \sqrt{k} \delta \frac{2\sqrt{\sum_{j=k}^\infty(x^\dagger,u_j)^2}}{\varepsilon}< \frac{\sqrt{k}}{2}\delta
\end{align}
for $\delta$ sufficiently small. Now we set
$$k_\varepsilon:=\min\left\{k\in\mathbb{N}~:~ \sqrt{\sum_{j=k}^\infty (x^\dagger,v_j)^2}\le \frac{\varepsilon}{16} \right\}.$$
Since $k_\varepsilon$ is independent from $\delta$ and because of \eqref{eq1} and \eqref{eq1a}, for $\delta$ sufficiently small there holds
\begin{equation}\label{eq2}
\frac{1}{4\sqrt{2}\sigma_{k_{\delta,\varepsilon}+1}}>\frac{5}{2\sigma_{k_\varepsilon}} \quad \mbox{and}\quad \sum_{j=k}^\infty(y^\dagger,u_j)^2 \le k \delta^2,
\end{equation}
for all $k>k_{\delta,\varepsilon}$. Consequently, for all $m/2\ge k> k_2^{\delta,\varepsilon}$ the reverse triangle inequality yields
\begin{align*}
\Psi_m(k,y^\delta)\chi_{\Omega_{\delta}} &=\frac{1}{\sigma_k}\sqrt{\sum_{j=k}^m(y^\delta,u_j)^2}\chi_{\Omega_{\delta}}\\
&\ge \frac{1}{\sigma_k}\left(\sqrt{\sum_{j=k}^m(y^\delta-y^\dagger,u_j)^2}-\sqrt{\sum_{j=k}^m(y^\dagger,u_j)^2}\right)\chi_{\Omega_{\delta}}\\
&> \frac{1}{\sigma_k}\left((1-\varepsilon')\sqrt{m-k}\delta -\frac{\sqrt{k}}{2}\delta\right)\chi_{\Omega_{\delta}}\ge \frac{\sqrt{m}\delta}{\sqrt{2}\sigma_k}\left(1-\varepsilon'-\frac{1}{2}\right)\chi_{\Omega_{\delta}}\\
&\ge \frac{1}{\sigma_{k_{\delta,\varepsilon}+1}}\frac{\sqrt{m}\delta}{4\sqrt{2}}\chi_{\Omega_{\delta}},
\end{align*}
while by the triangle inequality and definition of $k_\varepsilon$
\begin{align*}
\Psi_m(k_{\varepsilon},y^\delta)&\le \frac{1}{\sigma_{k_\varepsilon}}\sqrt{\sum_{j=k_\varepsilon}^m(y^\delta,u_j)^2}\chi_{\Omega_{\delta}}\le \frac{(1+\varepsilon')\sqrt{m}\delta}{\sigma_{k_\varepsilon}}+\frac{\sqrt{\sum_{j=k_\varepsilon}^m(y^\dagger,u_j)^2}}{\sigma_{k_\varepsilon}}\\
&\le \frac{1}{\sigma_{k_\varepsilon}} \frac{5}{4}\sqrt{m}\delta + \sqrt{\sum_{j=k_\varepsilon}^\infty(x^\dagger,v_j)^2}\\
&< \frac{1}{\sigma_{k_\varepsilon}}\frac{5}{4}\sqrt{m}\delta + \frac{\varepsilon}{16}.
\end{align*}
Furthermore, by \eqref{eq2} on the one hand
\begin{align*}
\frac{\sqrt{m}\delta}{\sigma_{k_{\delta,\varepsilon}+1}4\sqrt{2}}> \frac{5\sqrt{m}\delta}{2\sigma_{k_\varepsilon}},
\end{align*}
whereas by definition of $k_{\delta,\varepsilon}$ on the other hand
\begin{align*}
\frac{\sqrt{m} \delta}{4\sqrt{2}} \frac{1}{\sigma_{k_{\delta,\varepsilon}+1}} &= \frac{1}{4\sqrt{2}} \sqrt{\frac{m}{k_{\delta,\varepsilon}+1}} \frac{\sqrt{k_{\delta,\varepsilon}+1} \delta}{\sigma_{k_{\delta,\varepsilon}+1}}\\
&\ge \frac{1}{4\sqrt{2}}\sqrt{\frac{m}{k_{\delta,\varepsilon}+1}} \frac{\varepsilon}{2}\ge \frac{\varepsilon}{8}
\end{align*}
and thus putting the preceiding two estimates together yields
\begin{align*}
\Psi_m(k,y^\delta)\chi_{\Omega_{\delta}} &> \frac{1}{\sigma_{k_{\delta,\varepsilon}+1}}\frac{\sqrt{m}\delta}{4\sqrt{2}}\chi_{\Omega_{\delta}}\\
&\ge 2\max\left(\frac{1}{\sigma_{k_\varepsilon}} \frac{5\sqrt{m}\delta}{4},\frac{\varepsilon}{16}\right)\chi_{\Omega_{\delta}}\\
&\ge \left(\frac{1}{\sigma_{k_\varepsilon}}\frac{5\sqrt{m}\delta}{4}+\frac{\varepsilon}{16}\right)\chi_{\Omega_{\delta}}
\ge \Psi_m(k_\varepsilon,y^\delta)\chi_{\Omega_{\delta}}.
\end{align*}
We conclude $k_{\rm HD}^\delta(m)\chi_{\Omega_{\delta}} \le \max(k_{\varepsilon},k_{\delta,\varepsilon})$ for all $m\ge 2k_{\delta,\varepsilon}$ and thus, for $\delta$ sufficiently small,
\begin{align*}
\max_{m\in\mathbb{N}}\arg\min_{k\le m/2}\Psi_m(k,y^\delta)\chi_{\Omega_{\delta}}&=\max\left(\max_{m\le 2k_{\delta,\varepsilon}}\arg\min_{k\le m/2}\Psi_m(k,y^\delta),\max(k_{\delta,\varepsilon},k_\varepsilon)\right)\\
&\le\max\left(k_{\delta,\varepsilon},k_\varepsilon\right)\le k_{\delta,\varepsilon},
\end{align*}
since $k_{\varepsilon}\le k_{\delta,\varepsilon}$ for $\delta$ sufficiently small, which finishes the proof of the proposition.
\end{proof}
The preceding proposition guarantees that we do not stop too late. In order to obtain convergence it needs to be assured that one also stops sufficiently late.
\begin{proposition}\label{prop4}
For $\varepsilon'>0$ and $m_\delta=m_{\delta,\eta}$ (given below) in \eqref{om} there holds
\begin{equation}
k^\delta_{\rm HD}\chi_{\Omega_{\delta}} \ge k_\varepsilon\chi_{\Omega_{\delta}}
\end{equation}
for $\delta$ sufficiently small, where
$$k_\varepsilon:=\min\left\{k\in\mathbb{N}_0~:~\sqrt{\sum_{j=k+1}^\infty(x^\dagger,v_j)^2}\le \frac{\varepsilon}{2}\right\}.$$
\end{proposition}
\begin{proof}[Proof of Proposition \ref{prop4}]
We set
$$\eta:=\frac{1}{\sigma_{1}}\sqrt{\sum_{j=k_\varepsilon}^\infty(y^\dagger,u_j)^2}.$$
Clearly, $\eta>0$ if $\|x^\dagger\|>\varepsilon/2$. This we assume in the following, since $\|x^\dagger\|\le \varepsilon/2$ would imply $k_\varepsilon=0$, which would finish the proof already. Further define
$$k_{\varepsilon,\eta}:=\min\left\{k\ge k_\varepsilon+1~:~\sqrt{\sum_{j=k}^\infty(x^\dagger,v_j)^2}\le\frac{\eta}{8}\right\}.$$
Then for
$$m_{\delta,\eta}:=\max\left\{ m\in\mathbb{N}~:~ \frac{\sqrt{m}\delta}{\sigma_{k_{\varepsilon,\eta}}}\le \frac{\eta}{8(1+\varepsilon')}\right\}$$
and $\varepsilon$ fixed there clearly hold $m_{\delta,\eta} \nearrow \infty$ as $\delta\to0$. Let $\delta$ so small that $\sigma_1^{-1}\sqrt{\sum_{j=k_\varepsilon}^{m_{\delta,\eta}}(y^\dagger,u_j)^2}\ge \eta/2$ and $m_{\delta,\eta}\ge 2 k_\varepsilon$. Now, for all $k\le k_\varepsilon$ we have
\begin{align*}
\Psi_{m_{\delta,\eta}}(k,y^\delta)\chi_{\Omega_{\delta}}&= \frac{1}{\sigma_k}\sqrt{\sum_{j=k}^{m_{\delta,\eta}}(y^\delta,u_j)^2}\chi_{\Omega_{\delta}}\ge \frac{1}{\sigma_1}\sqrt{\sum_{j=k_\varepsilon}^{m_{\delta,\eta}}(y^\delta,u_j)^2}\chi_{\Omega_{\delta}}\\
&\ge \frac{1}{\sigma_1}\sqrt{\sum_{j=k_1^\varepsilon}^{m^{\delta,\eta}}(y^\dagger,u_j)^2}\chi_{\Omega_{\delta}} - \frac{1}{\sigma_1}\sqrt{\sum_{j=k_1^\varepsilon}^{m^{\delta,\eta}}(y^\delta-y^\dagger,u_j)^2}\chi_{\Omega_{\delta}}\\
&\ge \frac{\eta}{2}\chi_{\Omega_{\delta}} - \frac{1+\varepsilon'}{\sigma_1}\sqrt{m^{\delta,\eta}}\delta\chi_{\Omega_{\delta}}\ge \frac{3\eta}{8}\chi_{\Omega_{\delta}}.
\end{align*}
On the contrary,
\begin{align*}
\Psi_{m_{\delta,\eta}}\left(k_{\varepsilon,\eta},y^\delta\right)\chi_{\Omega_{\delta}}&\le \frac{1}{\sigma_{k_{\varepsilon,\eta}}}\sqrt{\sum_{j=k_{\varepsilon,\eta}}^{m_{\delta,\eta}}(y^\dagger,u_j)^2} +\frac{1}{\sigma_{k_{\varepsilon,\eta}}}\sqrt{\sum_{j=1}^{m_{\delta,\eta}}(y^\dagger-y^\delta,u_j)^2}\chi_{\Omega_{\delta}}\\
&\le \sqrt{\sum_{j=k_{\varepsilon,\eta}}^\infty(x^\dagger,v_j)^2} + (1+\varepsilon')\frac{\sqrt{m_{\delta,\eta}}}{\sigma_{k_{\varepsilon,\eta}}}\delta \le \frac{\eta}{8}+\frac{\eta}{8} \le \frac{\eta}{4}
\end{align*}
The preceding two estimates directly imply $\Psi_{m_{\delta,\eta}}(k,y^\delta)\chi_{\Omega_{\delta}} > \Psi_{m_{\delta,\eta}}(k_{\varepsilon,\eta})\chi_{\Omega_{\delta}}$ for all $k\le k_\varepsilon$, thus since $k_\varepsilon< k_{\varepsilon,\eta}$ we deduce $k_{\rm HD}^\delta(m_{\delta,\eta})\chi_{\Omega_{\delta}} > k_\varepsilon\chi_{\Omega_{\delta}}$. Therefore in particular $k_{\rm HD}^\delta \chi_{\Omega_{\delta}} >k_\varepsilon\chi_{\Omega_{\delta}}$, and the proof is finished.
\end{proof}
We come to the main proof of Theorem \ref{th1}. Fix $\varepsilon'<\frac{1}{4}$ and $m_\delta=\max(k_{\delta,\varepsilon},m_{\delta,\eta})$ (where the arguments are from Proposition \ref{prop3} and \ref{prop4}) in \eqref{om}. We apply Proposition \ref{prop3} and \ref{prop4} to the canoncial error decomposition into a data propagation and an approximation error (known as the bias-variance decomposition) and obtain, for $\delta$ sufficiently small,
\begin{align*}
\|x^\delta_{k^\delta_{\rm HD}} - K^+y^\dagger\|\chi_{\Omega_{\delta}}&=\left\|\sum_{j=1}^{k^\delta_{\rm HD}}\frac{(y^\delta,u_j)}{\sigma_j} v_j - \sum_{j=1}^\infty(x^\dagger,v_j) v_j\right\|\chi_{\Omega_{\delta}}\\
&\le \sqrt{\sum_{j=1}^{k_{\rm HD}^\delta}\frac{(y^\delta-y^\dagger,u_j)^2}{\sigma_j^2}}\chi_{\Omega_{\delta}} + \sqrt{\sum_{j=k^\delta_{\rm HD}+1}^\infty(x^\dagger,v_j)^2}\chi_{\Omega_{\delta}}\\
&\le \frac{(1+\varepsilon')\sqrt{k_{\delta,\varepsilon}}\delta}{\sigma_{k_{\delta,\varepsilon}}}+\sqrt{\sum_{j=k_\varepsilon+1}^\infty(x^\dagger,v_j)^2}\le \frac{\varepsilon}{2}+\frac{\varepsilon}{2}\le \varepsilon.
\end{align*}
Proposition \ref{prop1} guarantees that $\mathbb{P}(\Omega_\delta)\to1$ as $\delta\to0$ and thus finishes the proof of the theorem.
\subsection{Proof of Lemma \ref{ex1}}
Let $m_\delta:=\lceil\frac{-3\log(\delta)}{1-\log(2)}\rceil+1$ and define the event
$$\Omega_\delta:=\left\{|(Z,u_{m_\delta-1})|=\sqrt{2},~(Z,u_m)=0~\forall m_\delta\le m \le 2m_\delta \right\}.$$
By independency and the distribution of the white noise it holds that $\mathbb{P}\left(\Omega_\delta\right)=(1/2)^{m_\delta+2}$. Since $y^\dagger=Kx^\dagger=0$ there holds $(y^\delta,u_j)\chi_{\Omega_{\delta}}=0$ for $m_\delta-1<k\le 2m_\delta$ and $|(y^\delta,u_{m_\delta-1})|\chi_{\Omega_{\delta}}=|(Z,u_{m_\delta-1})|\delta\chi_{\Omega_{\delta}}=\sqrt{2}\delta\chi_{\Omega_{\delta}}$. Thus we have
\begin{align*}
\Psi_{2m_\delta}(k,y^\delta)\chi_{\Omega_{\delta}} =&\frac{1}{\sigma_k}\sqrt{\sum_{j=k}^{2m_\delta}(y^\delta,u_j)^2}\chi_{\Omega_{\delta}}\\
&\ge \frac{1}{\sigma_{m_\delta-1}}|(y^\delta,u_{m_\delta-1})|\chi_{\Omega_{\delta}}=e^{-(m_\delta-1)} \sqrt{2}\chi_{\Omega_{\delta}},
\end{align*}
while
$$\Psi_{2m_\delta}(m_\delta,y^\delta)\chi_{\Omega_{\delta}}= \frac{1}{\sigma_{m_\delta}}\sqrt{\sum_{j=m_\delta}^{2m_\delta}(y^\delta,u_j)^2}\chi_{\Omega_{\delta}}=0.$$
Therefore $k^\delta_{\rm HD}\chi_{\Omega_{\delta}} \ge k^\delta_{\rm HD}(2m_\delta)\chi_{\Omega_{\delta}} \ge m_\delta \chi_{\Omega_{\delta}}$. Furthermore, for $k\ge m_\delta$,
$$\|x_k^\delta-x^\dagger\|^2\chi_{\Omega_{\delta}} =\sum_{j=1}^k\frac{(y^\delta-y^\dagger,u_j)^2}{\sigma_j^2}\chi_{\Omega_{\delta}} \ge \frac{\delta^2(Z,u_{m_\delta}-1)^2}{\sigma_{m_\delta-1}^2}\chi_{\Omega_{\delta}} =2 \delta^2e^{(m_\delta-1)}\chi_{\Omega_{\delta}}.$$
Finally, restricting the expectation to the event $\Omega_\delta$ yields
\begin{align*}
\mathbb{E}\|x^\delta_{k^\delta_{\rm HD}}-x^\dagger\|^2 &\ge \mathbb{E}\left[\|x^\delta_{k^\delta_{\rm HD}}-x^\dagger\|^2 \chi_{\Omega_{\delta}}\right] \ge 2\delta^2e^{(m_\delta-1)}\mathbb{P}\left(\Omega_\delta\right)\\
&= 2\delta^2e^{(m_\delta-1)}\left(1/2\right)^{m_\delta+2}
= \frac{\delta^2}{4} e^{(m_\delta-1)(1-\log(2))} = \frac{1}{4\delta}
\end{align*}
and the proof is finished.
\subsection{Proof of Theorem \ref{th2}}
We will need the following technical proposition similar to Proposition \ref{prop1} above.
\begin{proposition}\label{prop5}
Let $(X_j)_{j\in\mathbb{N}}$ be i.i.d. standard Gaussian. Then, for $\varepsilon>0$ there holds
$$\mathbb{P}\left(\sup_{m\ge M}\left|\frac{1}{m}\sum_{j=1}^m (X_j^2-1)\right|\ge \varepsilon\right)\le 2e^{-\frac{M}{2}\left(\varepsilon-\log(1+\varepsilon)\right)}.$$
\end{proposition}
\begin{proof}[Proof of Proposition \ref{prop5}]
We apply Doob's extremal inequality to the reverse submartingale $\left(\exp\left|\frac{1}{m}\sum_{j=1}^m(X_j^2-1)\right|\right)_{m\in\mathbb{N}}$ and obtain, for $\lambda$ to be optimised
\begin{align*}
&\mathbb{P}\left(\sup_{m\ge M}\frac{1}{m}\left|\sum_{j=1}^m(X_j^2-1)\right|>\varepsilon\right)\\
\le &\frac{\mathbb{E}\left[e^{\lambda\left|\frac{1}{M}\sum_{j=1}^M X_j^2-1\right|}\right]}{e^{\lambda\varepsilon}}\le e^{-\lambda\varepsilon}\left( \mathbb{E}\left[e^{\frac{\lambda}{M} \sum_{j=1}^M X_j^2}\right] e^{-\lambda} + \mathbb{E}\left[e^{-\frac{\lambda}{M}\sum_{j=1}^M X_j^2}\right]e^\lambda\right)\\
\le &e^{-\lambda\varepsilon}\left(\frac{e^{-\lambda}}{\left(1-\frac{2\lambda}{M}\right)^\frac{M}{2}} + \frac{e^\lambda}{\left(1+\frac{2\lambda}{M}\right)^\frac{M}{2}}\right)\le 2 e^{-(1+\varepsilon)\lambda - \frac{M}{2}\log\left(1-\frac{2\lambda}{M}\right)}\le 2 e^{-\frac{M}{2}(\varepsilon-\log(1+\varepsilon))},
\end{align*}
where we used that $\sum_{j=1}^M X_j^2$ is $\chi^2$-distributed in the third step, that the first term dominates the second in the fourth step and that the expression is minimal for $\lambda=\frac{M}{2}\frac{\varepsilon}{1+\varepsilon}$ in the fifth step.
\end{proof}
We start with the proof of the Theorem and first derive a lower bound for the minimal error. Define $k_\delta:=\lceil\delta^\frac{-2}{q+\eta}\rceil$; which is the index approximately balancing the bias and the variance of $\|x_k^\delta-x^\dagger\|$ and which will be shown to accurately estimate the $\arg\min$ of $\|x_k^\delta-x^\dagger\|$. Moreover, set $\kappa_\delta = \lceil k_\delta^\frac{\eta-1}{q+\eta-1}\rceil-1$. We perform the analysis on the following events
\begin{align*}
\Omega_\delta:&=\left\{\frac{1}{m}\left|\sum_{j=1}^m(X_j^2-1)\right|\le \varepsilon,~ \forall m\ge c_{q,\eta,\varepsilon'}\kappa_\delta\right\}\\
&\qquad\cap \left\{\frac{1}{m}\left|\sum_{j=1}^m(y^\delta-y^\dagger,u_j)^2-\delta^2\right|\le \varepsilon \delta^2,~\forall m\ge k_\delta\right\},\\
\end{align*}
where $c_{q,\eta,\varepsilon'}$ is given below.
Now, using that $(1+\varepsilon)m-(1-\varepsilon)k \le (1+3\varepsilon)(m-k)$ and $(1-\varepsilon)m-(1+\varepsilon)k\ge (1-3\varepsilon)(m-k)$ for $m\ge 2k$, we see that
for $k\le 2k_\delta$ we have
\begin{align*}
\|x_k^\delta-x^\dagger\|^2\chi_{\Omega_{\delta}} &= \sum_{j=1}^k\frac{(y^\delta-y^\dagger,u_j)^2}{\sigma_j^2}\chi_{\Omega_{\delta}} + \sum_{j=k+1}^\infty(x^\dagger,v_j)^2\chi_{\Omega_{\delta}}\\
&\ge \sum_{j=2k_\delta+1}^\infty(x^\dagger,v_j)^2\chi_{\Omega_{\delta}}\ge \sum_{l=1}^\infty \left((l+2) k_\delta\right)^{-\eta}\sum_{j=(l+1)k_\delta+1}^{(l+2)k_\delta} X_j^2\chi_{\Omega_{\delta}}\\
&\ge k_\delta^{-\eta}\sum_{l=1}^\infty(l+2)^{-\eta} (1-3\varepsilon) k_\delta\chi_{\Omega_{\delta}} \ge (1-3\varepsilon)\frac{2^{1-\eta}}{\eta-1}k_\delta^{1-\eta}\chi_{\Omega_{\delta}},
\end{align*}
while for $k\ge 2k_\delta$ we have
\begin{align*}
\|x_k^\delta-x^\dagger\|^2\chi_{\Omega_{\delta}} &= \sum_{j=1}^k\frac{(y^\delta-y^\dagger,u_j)^2}{\sigma_j^2}\chi_{\Omega_{\delta}} + \sum_{j=k+1}^\infty(x^\dagger,v_j)^2\chi_{\Omega_{\delta}}\\
&\ge \sum_{j=k_\delta+1}^{2k_\delta}\frac{(y^\delta-y^\dagger,u_j)^2}{\sigma_j^2}\chi_{\Omega_{\delta}}\ge k_\delta^q \sum_{j=k_\delta+1}^{2k_\delta}(y^\delta-y^\dagger,u_j)^2\chi_{\Omega_{\delta}}\\
&\ge (1-3\varepsilon) k_\delta^{1+q}\delta^2\chi_{\Omega_{\delta}}.
\end{align*}
By the choice of $k_\delta$ there holds $k_\delta^{1-\eta} \ge 2^{1-\eta} \delta^{2\frac{\eta-1}{\eta+q}}$ and $k_\delta^{1+q}\delta^2\ge \delta^{2\frac{\eta-1}{\eta+q}}$. Therefore
\begin{align}\label{eq3a}
\min_{k\in\mathbb{N}}\|x_k^\delta-x^\dagger\|^2\chi_{\Omega_{\delta}} \ge (1-3\varepsilon)\min\left(1,\frac{4^{1-\eta}}{\eta-1}\right) \delta^{2\frac{\eta-1}{\eta+q}}\chi_{\Omega_{\delta}}.
\end{align}
We will use in the following that for $\varepsilon<1/3$ there holds
\begin{align*}
&(1-\varepsilon')\sqrt{m-k}\chi_{\Omega_{\delta}}\le\sqrt{\sum_{j=k+1}^{m}X_j^2}\chi_{\Omega_{\delta}}\le(1+\varepsilon')\sqrt{m-k},\\
&(1-\varepsilon')\sqrt{m-k}\delta\chi_{\Omega_{\delta}}\le\sqrt{\sum_{j=k+1}^m(y^\delta-y^\dagger,u_j)^2}\chi_{\Omega_{\delta}}\le(1+\varepsilon')\sqrt{m-k}\delta,\\
&(1-\varepsilon')\sqrt{m}\delta\chi_{\Omega_{\delta}}\le\sqrt{\sum_{j=1}^m(y^\delta-y^\dagger,u_j)^2}\chi_{\Omega_{\delta}}\le(1+\varepsilon')\sqrt{m}\delta,
\end{align*}
for all $m,k\ge \kappa_\delta$ and $m\ge 2k$, where $\varepsilon':=1-\sqrt{1-3\varepsilon}.$ We only show the first of the above inequalities. We have
\begin{align*}
\sqrt{\sum_{j=k+1}^{m}X_j^2\chi_{\Omega_{\delta}}} &= \sqrt{\sum_{j=1}^{m}X_j^2 - \sum_{j=1}^{k}X_j^2}\chi_{\Omega_{\delta}}\le \sqrt{(1+\varepsilon)m -(1-\varepsilon)k}\\
&=\sqrt{1+3\varepsilon}\sqrt{m-k}
\end{align*}
and
\begin{align*}
\sqrt{\sum_{j=k+1}^{m}X_j^2}\chi_{\Omega_{\delta}} &= \sqrt{\sum_{j=1}^{m}X_j^2 - \sum_{j=1}^{k}X_j^2}\chi_{\Omega_{\delta}}\ge \sqrt{(1-\varepsilon)m -(1+\varepsilon)k}\\
&=\sqrt{1-3\varepsilon}\sqrt{m-k},
\end{align*}
and the claim follows.
We first show that we do not stop too late. Indeed, for all $m, k\ge C_{q,\eta,\varepsilon'}k_\delta$ with $C_{q,\eta,\varepsilon'}:=\max\left(\left(\frac{2(1+\varepsilon')}{1-\varepsilon'}\right)^\frac{2}{q+\eta},\left(\frac{4\sqrt{3}(1+\varepsilon')}{1-\varepsilon'}\right)^\frac{2}{q}\right)$ and $m\ge 2k$ we have
\begin{align*}
\Psi_m(k,y^\delta)\chi_{\Omega_{\delta}} &=\frac{1}{\sigma_k}\sqrt{\sum_{j=k}^m(y^\delta,u_j)^2}\chi_{\Omega_{\delta}}\\
&\ge \frac{1}{\sigma_k}\left(\sqrt{\sum_{j=k}^m(y^\delta-y^\dagger,u_j)^2}-\sqrt{\sum_{j=k}^m(y^\dagger,u_j)^2}\right)\chi_{\Omega_{\delta}}\\
&> \frac{1}{\sigma_k}\left((1-\varepsilon')\sqrt{m-k-1}\delta - k^\frac{-(q+\eta)}{2} \sqrt{\sum_{j=k}^m Z_j^2}\right)\chi_{\Omega_{\delta}}\\
&> \frac{1}{\sigma_k}\left((1-\varepsilon')\sqrt{m-k-1}\delta - k^{-\frac{q+\eta}{2}}(1+\varepsilon')\sqrt{m-k-1}\right)\chi_{\Omega_{\delta}}\\
&\ge \frac{\sqrt{m-k-1}\delta}{\sigma_k}\left((1-\varepsilon') - C_{q,\eta,\varepsilon'}^\frac{-(q+\eta)}{2}(1+\varepsilon')k_\delta^{-\frac{q+\eta}{2}}\delta^{-1}\right)\chi_{\Omega_{\delta}}\\
&=\frac{\sqrt{m-k-1}\delta}{\sigma_k}\left(1-\varepsilon'-C_{q,\eta,\varepsilon'}^\frac{-(q+\eta)}{2}(1+\varepsilon')\right)\chi_{\Omega_{\delta}}\\
&> \frac{C_{q,\eta,\varepsilon'}^\frac{q}{2}}{\sqrt{3}}\left(1-\varepsilon'-C_{q,\eta,\varepsilon'}^\frac{-(q+\eta)}{2}(1+\varepsilon')\right)\sqrt{m k_\delta^q}\delta\chi_{\Omega_{\delta}},
\end{align*}
while
\begin{align*}
&\Psi_m(k_\delta,y^\delta)\chi_{\Omega_{\delta}}\\
= &\frac{1}{\sigma_{k_\delta}}\sqrt{\sum_{j=k_\delta}^m(y^\delta,u_j)^2}\chi_{\Omega_{\delta}}\\
\le &k_\delta^\frac{q}{2}\left(\sqrt{\sum_{j=k_\delta}^m(y^\delta-y^\dagger,u_j)^2} + k_\delta^{-\frac{q+\eta}{2}} \sqrt{\sum_{j=k_\delta}^m X_j^2}\right)\chi_{\Omega_{\delta}}\\
< &k_\delta^\frac{q}{2} \left((1+\varepsilon')\sqrt{m-k_\delta-1}\delta+(1+\varepsilon')k_\delta^\frac{-(q+\eta)}{2} \sqrt{m-k_\delta-1}\right)\chi_{\Omega_{\delta}}\\
\le &2(1+\varepsilon')\sqrt{k_\delta^q m}\delta\chi_{\Omega_{\delta}}\le\frac{C_{q,\eta,\varepsilon'}}{\sqrt{3}}\left(1-\varepsilon'-C_{q,\eta,\varepsilon'}^\frac{-(q+\eta)}{2}(1+\varepsilon')\right)\sqrt{m k_\delta^q}\delta\chi_{\Omega_{\delta}}< \Psi_m(k,y^\delta)
\end{align*}
by choice of $C_{q,\eta,\varepsilon'}$ (note that $C_{q,\eta,\varepsilon'}^\frac{-(q+\eta)}{2}(1+\varepsilon')\le \frac{1-\varepsilon'}{2}$). Therefore,
\begin{equation}\label{eq3}
k^\delta_{\rm HD} \chi_{\Omega_{\delta}} \le C_{q,\eta,\varepsilon'} k_\delta.
\end{equation}
Next we show that we stop sufficiently late. First,
\begin{align*}
\Psi_{2k_\delta}(k_\delta,y^\delta)\chi_{\Omega_{\delta}}&\le \frac{1}{\sigma_{k_\delta}}\left(\sqrt{\sum_{j=k_\delta}^{2k_\delta}(y^\dagger,u_j)^2} + \sqrt{\sum_{j=k_\delta}^{2k_\delta} (y^\delta-y^\dagger,u_j)^2}\right)\chi_{\Omega_{\delta}}\\
&\le \sqrt{k_\delta^q}\left((1+\varepsilon') \sqrt{k_\delta^{-(q+\eta)} (k_\delta+1)} +(1+\varepsilon')\sqrt{k_\delta+1}\delta\right)\\
&\le 2\sqrt{2}(1+\varepsilon')\sqrt{k_\delta^{1+q}} \delta.
\end{align*}
Then, for $k\le c_{q,\eta,\varepsilon'}\kappa_\delta$ with $c_{q,\eta,\varepsilon'}:=\left(2^\frac{q+\eta}{2}\frac{3\sqrt{2}(1+\varepsilon')}{1-\varepsilon'}\right)^\frac{2}{1-q-\eta}<1$ (note that $\eta>1$)
\begin{align*}
&\Psi_{2k_\delta}(k,y^\delta)\chi_{\Omega_{\delta}}\ge \frac{1}{\sigma_k}\left(\sqrt{\sum_{j=k}^{2k_\delta}(y^\dagger,u_j)^2}-\sqrt{\sum_{j=1}^{2k_\delta}(y^\delta-y^\dagger,u_j)^2}\right)\chi_{\Omega_{\delta}}\\
\ge &\frac{1}{\sigma_k}\left( \sqrt{\left(2c_{q,\eta,\varepsilon'}\kappa_\delta\right)^{-(q+\eta)}\sum_{j=\lfloor c_{q,\eta,\varepsilon'}\kappa_\delta\rfloor+1}^{2\lfloor c_{q,\eta,\varepsilon'}\kappa_\delta\rfloor}X_j^2} - \sqrt{\sum_{j=1}^{2k_\delta}(y^\delta-y^\dagger,u_j)^2}\right)\chi_{\Omega_{\delta}}\\
&> \frac{1}{\sigma_k}\left(\left(\lfloor2c_{q,\eta,\varepsilon'}\kappa_\delta\rfloor\right)^\frac{-q-\eta}{2}(1-\varepsilon') \sqrt{\lfloor c_{q,\eta,\varepsilon'}\kappa_\delta\rfloor} - (1+\varepsilon') \sqrt{2k_\delta}\delta\right)\chi_{\Omega_{\delta}}\\
&\ge(1-\varepsilon')2^{-\frac{q+\eta}{2}}c_{q,\eta,\varepsilon'}^\frac{1-q-\eta}{2}(\kappa_\delta-1)^\frac{1-q-\eta}{2} - (1+\varepsilon') \sqrt{2k_\delta}\delta\chi_{\Omega_{\delta}}\\
&\ge \left(c_{q,\eta,\varepsilon'}^\frac{1-q-\eta}{2} 2^\frac{-q-\eta}{2}(1-\varepsilon') - (1+\varepsilon')\sqrt{2}\right) \sqrt{k_\delta^{1+q}}\delta\chi_{\Omega_{\delta}}\\
&= 2\sqrt{2}(1+\varepsilon')\sqrt{k_\delta^{1+q}}\delta\chi_{\Omega_{\delta}}
\end{align*}
by definition of $\kappa$ and choice of $c_{q,\eta,\varepsilon'}$. On the other hand, for $c_{q,\eta,\varepsilon'} \kappa_\delta\le k\le c_{q,\eta,\varepsilon'} k_\delta$ we also have
\begin{align*}
\Psi_{2k_\delta}(k,y^\delta)\chi_{\Omega_{\delta}}&\ge \frac{1}{\sigma_k}\left(\sqrt{\sum_{j=k}^{2k-1}(y^\dagger,u_j)^2}-\sqrt{\sum_{j=1}^{2k_\delta}(y^\delta-y^\dagger,u_j)^2}\right)\chi_{\Omega_{\delta}}\\
&> \frac{1}{\sigma_k}\left(\left(2k\right)^\frac{-q-\eta}{2} (1-\varepsilon')\sqrt{k}- (1+\varepsilon') \sqrt{2k_\delta}\delta\right)\chi_{\Omega_{\delta}}\\
&\ge 2^\frac{-q-\eta}{2} (1-\varepsilon')k^\frac{1-q-\eta}{2}- (1+\varepsilon') \sqrt{2k_\delta}\delta k^\frac{q}{2}\chi_{\Omega_{\delta}}\\
&\ge \left(c_{q,\eta,\varepsilon'}^\frac{1-q-\eta}{2} 2^{-q-\eta}(1-\varepsilon') - (1+\varepsilon')\sqrt{2}\right) \sqrt{k_\delta^{1+q}}\delta\\
&\ge 2\sqrt{2}(1+\varepsilon')\sqrt{k_\delta^{1+q}}\delta\chi_{\Omega_{\delta}}.
\end{align*}
Putting the last three estimates together gives $\Psi_{2k_\delta}(k,y^\delta)\chi_{\Omega_{\delta}}> \Psi_{2k_\delta}(k_\delta,y^\delta)\chi_{\Omega_{\delta}}$ for all $k\le c_{q,\eta,\varepsilon'}k_\delta$, which yields
\begin{equation}\label{eq4}
k^\delta_{\rm HD}\chi_{\Omega_{\delta}} > c_{q,\eta,\varepsilon'} k_\delta\chi_{\Omega_{\delta}}.
\end{equation}
Thus, using \eqref{eq3} and \eqref{eq4} in the data propagation and the approximation error respectively and \eqref{eq3a} for the minimal error, there holds
\begin{align*}
&\|x_{k_{\rm HD}^\delta}^\delta-x^\dagger\|\chi_{\Omega_{\delta}}\\
\le &\sqrt{\sum_{j=1}^{k_{\rm HD}^\delta}\frac{(y^\delta-y^\dagger,u_j)^2}{\sigma_j^2}}\chi_{\Omega_{\delta}} + \sqrt{\sum_{j=k^\delta_{\rm HD}+1}^\infty(x^\dagger,v_j)^2}\chi_{\Omega_{\delta}}\\
\le &(1+\varepsilon') \left(C_{q,\eta,\varepsilon'}k_\delta\right)^\frac{1+q}{2}\delta\chi_{\Omega_{\delta}} + \sqrt{\sum_{j=\lfloor c_{q,\eta,\varepsilon'}k_\delta\rfloor+1}^\infty j^{-\eta} X_j^2}\chi_{\Omega_{\delta}} \\
\le &(1+\varepsilon')\left(C_{q,\eta,\varepsilon'}k_\delta\right)^\frac{1+q}{2}\delta\chi_{\Omega_{\delta}} + \sqrt{\sum_{l=1}^\infty (lc_{q,\eta,\varepsilon'}k_\delta)^{-\eta}\sum_{j=l\lfloor c_{q,\eta,\varepsilon'}k_\delta\rfloor+1}^{(l+1)\lfloor c_{q,\eta,\varepsilon'}k_\delta\rfloor} X_j^2}\chi_{\Omega_{\delta}}\\
\le &(1+\varepsilon') \left(C_{q,\eta,\varepsilon'}k_\delta\right)^\frac{1+q}{2}\delta\chi_{\Omega_{\delta}} + \sqrt{k_\delta^{1-\eta} c_{q,\eta,\varepsilon'}^{-\eta}(1+\varepsilon')^2 \sum_{l=1}^\infty l^{-\eta}}\chi_{\Omega_{\delta}}\\
\le &(1+\varepsilon')\left(\sqrt{C_{q,\eta,\varepsilon'^{1+q}}}\chi_{\Omega_{\delta}} + c_{q,\eta,\varepsilon'}^\frac{-\eta}{2} \sqrt{\frac{\eta}{\eta-1}}\right) \delta^\frac{\eta-1}{\eta+q}\chi_{\Omega_{\delta}}\\
\le &(1+\varepsilon')\left(\sqrt{C_{q,\eta,\varepsilon'}^{1+q}} + c_{q,\eta,\varepsilon'}^\frac{-\eta}{2} \sqrt{\frac{\eta}{\eta-1}}\right)\sqrt{\max\left(1,(\eta-1)4^{\eta-1}\right)} \min_{k\in\mathbb{N}}\|x_k^\delta-x^\dagger\|\\
= &C_{q,\eta,\varepsilon} \min_{k\in\mathbb{N}}\|x_k^\delta-x^\dagger\|
\end{align*}
with
\begin{equation}\label{bound}
C_{q,\eta,\varepsilon}:=(1+\varepsilon')\left(\sqrt{C_{q,\eta,\varepsilon'}^{1+q}} + c_{q,\eta,\varepsilon'}^\frac{-\eta}{2} \sqrt{\frac{\eta}{\eta-1}}\right)\sqrt{\max\left(1,(\eta-1)4^{\eta-1}\right)}.
\end{equation}
Finally, by Proposition \ref{prop5} the probability can be bounded as follows
\begin{align*}
\mathbb{P}\left(\Omega_\delta\right)&\ge 1 - \mathbb{P}\left(\sup_{m\ge c_{q,\eta,\varepsilon'}\kappa_\delta}\frac{1}{m}\left|\sum_{j=1}^m X_j^2-1\right|>\varepsilon\right)\\
&\qquad\qquad-\mathbb{P}\left(\sup_{m\ge k_\delta}\frac{1}{m}\left|\sum_{j=1}^m(y^\delta-y^\dagger,u_j)^2-\delta^2\right|> \varepsilon \delta^2\right) \\
&\ge 1-4 e^{-\frac{c_{q,\eta,\varepsilon'}\kappa_\delta}{2}\left(\varepsilon-\log(1+\varepsilon)\right)}= 1-4 e^{-\frac{c_{q,\eta,\varepsilon'}}{2}(\varepsilon-\log(1+\varepsilon)) \delta^{\frac{2(1-\eta)}{(q+\eta-1)(q+\eta)}}}\\
&=1-4 e^{-p_{q,\eta,\varepsilon} \delta^\frac{2(1-\eta)}{(q+\eta-1)(q+\eta)}},
\end{align*}
with
\begin{equation}\label{prob}
p_{q,\eta,\varepsilon}:=\frac{c_{q,\eta,\varepsilon'}\left(\varepsilon-\log(1+\varepsilon)\right)}{2}.
\end{equation}
\section{Numerical experiments}\label{sec4}
In this final section we will test the introduced method numerically. For that we took four test problems from the popular MATLAB toolbox by Hansen \cite{hansen1994regularization}. These are discretisations from one dimensional Fredholm integral equations, namely Phillips' test problem, an example from gravity surveying, the backwards heat equation and the determination of the second anti derivative. These problems cover varying degrees of ill-posedness and smoothness of the unknown solution. The discretisation dimension is set to $D=2^{11}=2048$. We corrupt the data by Gaussian white noise, with three different noise levels $\delta=1$ (here the noise dominates the data), $\delta=10^{-2}$ (noise and data are of equal size) and $\delta=10^{-4}$ (here the noise is dominated by the data). We compare the heuristic discrepancy principle to the modified discrepancy principle (with spectral cut-off) from \cite{jahn2021optimal,jahn2022probabilistic}.
The philosophy of the discrepancy principle is that the reconstruction is determined such that it explains the measured data up to the noise and hence requires knowledge of $\delta$. To obtain this, one would classically determine a truncation level $k$ such that $\|Kx_k^\delta-y^\delta\|\approx \|y^\dagger-y^\delta\|$ (since $Kx_k^\delta$ is our approximation for $y^\dagger$). Because of the white noise dilemma the aforementioned norms are infinity and thus it can be applied only after discretisation, similar to the heuristic discrepancy principle. Precisely, it is determined as follows. For a fixed fudge parameter $\tau>1$ and discretisation level $m\in\mathbb{N}$ we set
$$k_{\rm dp}^\delta(m):=\min\left\{0\le k\le m~:~ \sum_{j=k+1}^m(y^\delta,u_j)^2 \le \tau m \delta^2\right\},$$
and the final choice is trough $k^\delta_{\rm dp}:=\max_{m\in\mathbb{N}}k_{\rm dp}^\delta(m)$.
The modified discrepancy principle recently has been applied by the author to the iterative Landweber method (instead of spectral cut-off) in \cite{jahn2022discretisation} and we employ it in this numerical part too. The clear advantage of that method is that it does not require knowledge of the singular value decomposition. Consequently, the singular value decomposition cannot be used for discretisation either. Instead, we construct artificially lower dimensional discretisations from our initially given matrix $K\in \mathbb{R}^{D\times D}$, in that we average blocks of rows of $K$. Precisely, we define
$$P_{m}:=\frac{1}{\sqrt{D/m}} I_{m \times m} \otimes E_{1\times D/m}\in\mathbb{R}^{m\times D},$$
with $I_{m\times m}$ the identity on $\mathbb{R}^m$ and $E_{m \times n}\in \mathbb{R}^{m\times n}$ the matrix where all entries are one. Hereby, the Kronecker-product $\otimes$ of two matrices $A=(a_{ij})\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{s\times t}$ is given as $A\otimes B=(a_{ij}B)\in R^{ms\times nt}$. For example, for $D=8$ and $m=2$ one obtains
$$P_2=\frac{1}{\sqrt{4}} \begin{pmatrix} 1 & 1 & 1 & 1 & &&&\\ &&&&1 &1 & 1 & 1\end{pmatrix}.$$
Since $m$ has to be a divisor of $D$ we set $m=m_l=2^l$ for $l=1,2,...,11$. Starting with $x_0=0$ the Landweber iterates are given by
$$x_{k+1,m}^\delta:=x_{k,m}^\delta - \omega (P_mK)^*((P_mKx_{k,m}^\delta-P_my^\delta)$$
with $k=0,1,...$. We set the maximum number of Landweber iterations to $K_{\rm max}=5*10^6$. Again we formulate a discretised discrepancy principle (for Landweber iteration)
\begin{equation}\label{num:eq1}
k^{\delta}_{\rm dp}(m):=\min\left\{k\le K_{\rm max} ~:~\|P_mKx_{k,m}^\delta - P_my^\delta\|\le \tau \sqrt{m}\delta\right\}.
\end{equation}
The fudge parameter for the discrepancy principle is set to $\tau=1.5$. Note that in contrast to the modified discrepancy principle with spectral cut-off we now not only have to determine a final iteration index $k$, but also an accompanying discretisation dimension $m$. In light of the discrepancy principle for spectral cut-off one is tempted to maximise over $m\in\mathbb{N}$ again. Here however the situation is slightly more complicated. We use Algorithm 1 to determine the stopping index $k$ and the discretisation dimension $m$.
\begin{algorithm}\label{algorithm1}
\caption{Modified discrepancy principle for Landweber iteration}
\begin{algorithmic}[1]
\STATE Given Landweber iterates $x_{{k^\delta_{\mathrm{dp}}}(m_l),m_l}^\delta$ and $x_{2{k^\delta_{\mathrm{dp}}}(m_l),m_l}^\delta$ with ${k^\delta_{\mathrm{dp}}}(m_l)$ determined by the discrepancy principle , $m_l=2^1,2^2,...,2^{11}$ and $k=1,... K_{\rm max}$;
\STATE \textit{Initialisation}
\STATE Set $i=0$;
\STATE \textit{Calculate maximum and ratio}
\REPEAT
\STATE$i=i+1$
\UNTIL{${k^\delta_{\mathrm{dp}}}(m_i)=\max_{i\le l\le 11}{k^\delta_{\mathrm{dp}}}(m_l)$ and
$\frac{\left\|P_{m_i}Kx^\delta_{2{k^\delta_{\mathrm{dp}}}(m_i),m_i}-P_{m_i}y^\delta\right\|}{\left\|P_{m_i}Kx^\delta_{{k^\delta_{\mathrm{dp}}}(m_i),m_i}-P_{m_i}y^\delta\right\|}\ge \frac{1}{2}$}
\STATE $x^\delta_{{k^\delta_{\mathrm{dp}}},{m^\delta_{\mathrm{dp}}}}:=x^\delta_{{k^\delta_{\mathrm{dp}}}(m_i),m_i}$;
\end{algorithmic}
\end{algorithm}
Algorithm 1 again follows the paradigm from \cite{jahn2021optimal, jahn2022probabilistic} of maximising the stopping index $k$, as can be seen from the first condition in the repeat loop. The second condition hereby guarantees that the accompanying discretisation dimension is sufficiently large, see \cite{jahn2022discretisation} for details. We modulate the heuristic discrepancy principle for Landweber iteration in a similar fashion and set
\begin{equation}\label{num:eq2}
k^\delta_{\rm HD}(m_l):=\arg \min_{k\le K_{\max}} \sqrt{k}\|P_{m_l} K x_{k,{m_l}}^\delta-P_{m_l}y^\delta\|.
\end{equation}
The final choice of $k$ and $m$ is then determined in analogy with Algorithm \ref{algorithm1}.1 through Algorithm \ref{algorithm2}.2.
\begin{algorithm}\label{algorithm2}
\caption{Heuristic discrepancy principle for Landweber iteration}
\begin{algorithmic}[1]
\STATE Given Landweber iterates $x_{k^\delta_{\rm HD}(m_l),m_l}^\delta$ and $x_{2k^\delta_{\rm HD}(m_l),m_l}^\delta$ with $k^\delta_{\rm HD}(m_l)$ determined by the heuristic discrepancy principle, $m_l=2^1,2^2,...,2^{11}$ and $k=1,... K_{\rm max}$;
\STATE \textit{Initialisation}
\STATE Set $i=0$;
\STATE \textit{Calculate maximum and ratio}
\REPEAT
\STATE$i=i+1$
\UNTIL{$k^\delta_{\rm HD}(m_i)=\max_{i\le l\le 11}k^\delta_{\rm HD}(m_l)$ and
$\frac{\left\|P_{m_i}Kx^\delta_{2k^\delta_{\rm HD}(m_i),m_i}-P_{m_i}y^\delta\right\|}{\left\|P_{m_i}Kx^\delta_{k^\delta_{\rm HD}(m_i),m_i}-P_{m_i}y^\delta\right\|}\ge \frac{1}{2}$}
\STATE $x^\delta_{k^\delta_{\rm HD},m^\delta_{\rm HD}}:=x^\delta_{k^\delta_{\rm HD}(m_i),m_i}$;
\end{algorithmic}
\end{algorithm}
We present averaged (over 100 runs) relative errors, i.e., $e_*=\rm{mean}(\|x^\delta_{k^\delta_{*}}-x^\dagger\|)/\|x^\dagger\|$ for spectral cut-off and $e_*=\rm{mean}(\|x^\delta_{k^\delta_*,m^\delta_\cdot}-x^\dagger\|)/\|x^\dagger\|$ for Landweber iteration, with $*\in\{\rm{HD},\rm{DP}, \rm{opt}\}$. Hereby, the 'opt' subscript is used for the optimal choice given by
\begin{equation}
k^\delta_{\rm opt}:=\arg\min_{k\le D}\|x_k^\delta-x^\dagger\|
\end{equation}
for spectral cut-off and
\begin{equation*}
(k^\delta_{\rm opt},m^\delta_{\rm opt}) = \arg\min_{\substack{k\le K_{\max}\\l\le 11}}\|x_{k,m_l}^\delta-x^\dagger\|
\end{equation*}
for Landweber iteration. This choice is also known under the term oracle (see \cite{wasserman2006all}), since one could only determine its value if an oracle told one the exact $x^\dagger$ before. We display the results in Table \ref{tab:phillips}-\ref{tab:heat}.
We clearly see convergence as $\delta$ decreases. In fact, we could not see real outliers in the 100 runs, which shows the method is rather stable. Clearly, since the counter example for $L^2$ convergence from Lemma \ref{ex1} was based on rare events with exponentially small probability on which one could not control the error, its impact is not visible for this small number of runs. Moreover, the heuristic discrepancy principle yields errors close to the optimal one in most cases, for both spectral cut-off (treated rigorously in this manuscript) and Landweber iteration (borrowed from \cite{jahn2022discretisation}), showing that the optimality might hold in various practical settings and is not in particular restricted to the one from Theorem \ref{th2}. The only real exemption is for \texttt{phillips} and noise level $\delta=10^{-4}$ where the error of the heuristic discrepancy principle was $2.5*10^{-2}$ and thus more then three times larger than the optimal one (for spectral cut-off) with $7.6*10^{-3}$.
All in all, the small numerical study indicates that the modified heuristic discrepancy principle might be an interesting method for applications, since it seems to give good results in different scenarios under minimal requirements.
\begin{table}[hbt!]
\centering
\caption{Comparison between DP and HDP for \texttt{phillips}.\label{tab:phillips}}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{c|ccc|ccc|}
\toprule
\multicolumn{1}{c}{}&\multicolumn{3}{c}{Spectral Cut-off} & \multicolumn{3}{c}{Landweber} \\
\cmidrule(l){2-4} \cmidrule(l){5-7}
$\delta$ & $e_{\rm HDP}$ & $e_{\rm DP}$ & $e_{\rm opt}$ & $e_{\rm HDP}$ & $e_{\rm DP}$ & $e_{\rm opt}$\\
1e0 & 8.8e-1 & 9.3e-1 & 6.8e-1 & 8.7e-1 & 9.4e-1 & 6.3e-1 \\
1e-2 & 1.1e-1 & 7.2e-2 & 4.2e-2 & 8.3e-2 & 9.7e-2 & 3.9e-2 \\
1e-4 & 2.5e-2 & 1.2e-2 & 7.6e-3 & 2.3e-2 & 1.6e-2 & 1.3e-2 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[hbt!]
\centering
\caption{Comparison between DP and HDP for \texttt{deriv2}.\label{tab:deriv2}}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{c|ccc|ccc|}
\toprule
\multicolumn{1}{c}{}&\multicolumn{3}{c}{Spectral Cut-off} & \multicolumn{3}{c}{Landweber} \\
\cmidrule(l){2-4} \cmidrule(l){5-7}
$\delta$ & $e_{\rm HDP}$ & $e_{\rm DP}$ & $e_{\rm opt}$ & $e_{\rm HDP}$ & $e_{\rm DP}$ & $e_{\rm opt}$\\
1e0 & 8.2e-1 & 1.9e0 & 7.5e-1 & 1.4e0 & 1.3e0 & 8.6e-1 \\
1e-2 & 4.5e-1 & 3.8e-1 & 3.5e-1 & 6.0e-1 & 5.1e-1 & 4.4e-1 \\
1e-4 & 2.1e-1 & 1.9e-1 & 1.6e-1 & 2.6e-1 & 2.4e-1 & 1.9e-1 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[hbt!]
\centering
\caption{Comparison between DP and HDP for \texttt{gravity}.\label{tab:gravity}}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{c|ccc|ccc|}
\toprule
\multicolumn{1}{c}{}&\multicolumn{3}{c}{Spectral Cut-off} & \multicolumn{3}{c}{Landweber} \\
\cmidrule(l){2-4} \cmidrule(l){5-7}
$\delta$ & $e_{\rm HDP}$ & $e_{\rm DP}$ & $e_{\rm opt}$ & $e_{\rm HDP}$ & $e_{\rm DP}$ & $e_{\rm opt}$\\
1e0 & 1.9e-1 & 2.6e-1 & 1.7e-1 & 2.0e-1 & 2.4e-1 & 1.4e-1 \\
1e-2 & 3.3e-2 & 3.9e-2 & 2.6e-2 & 3.8e-2 & 4.3e-2 & 2.3e-2 \\
1e-4 & 7.1e-3 & 7.7e-3 & 5.0e-3 & 7.9e-3 & 9.4e-3 & 5.2e-3 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[hbt!]
\centering
\caption{Comparison between DP and HDP for \texttt{heat}.\label{tab:heat}}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{c|ccc|ccc|}
\toprule
\multicolumn{1}{c}{}&\multicolumn{3}{c}{Spectral Cut-off} & \multicolumn{3}{c}{Landweber} \\
\cmidrule(l){2-4} \cmidrule(l){5-7}
$\delta$ & $e_{\rm HDP}$ & $e_{\rm DP}$ & $e_{\rm opt}$ & $e_{\rm HDP}$ & $e_{\rm DP}$ & $e_{\rm opt}$\\
1e0 & 8.2e-1 & 8.4e-1 & 7.5e-1 & 8.6e-1 & 8.4e-1 & 6.9e-1 \\
1e-2 & 2.6e-1 & 2.5e-1 & 1.5e-1 & 1.5e-1 & 3.7e-1 & 1.3e-1 \\
1e-4 & 2.0e-2 & 2.9e-2 & 1.9e-2 & 1.8e-2 & 3.5e-2 & 1.7e-2 \\
\bottomrule
\end{tabular}
\end{table}
\bibliographystyle{siamplain}
|
1,116,691,497,808 | arxiv | \section{Introduction}
Quantum computer provide us more advantages to solve some problems efficiently than the classical Turing machine. Many efficient quantum algorithms towards some problems which are difficult to solve in the classical computer were discovered in the past decades (for example see the review \cite{childs}).
The polynomial equation solving problem is a typical hard but important problem with a lot of applications.
Different from numbers, polynomial is a little complicate to deal with in quantum computer now.
By Shor's work \cite{shor}, congruence equation can be solved efficiently in quantum computer.
The HHL algorithm \cite{harrow} seems to be another breakthrough related to this problem, although it can only give a state about the solution of linear system.
A work about diagonal equation with three terms in finite field was studied in \cite{dam}.
And recently, two works about polynomial interpolation in finite field were given in \cite{chen}, \cite{childs2}.
In this paper, we consider three types of polynomial equation solving problem, which seems cannot solved efficiently by the already existed quantum algorithms, especially Shor's algorithms.
The first equation is linear divisibility equation. It is a problem aims at finding a nontrivial factor of a positive integer $b$ with the form $ax+1$. This problem is a candidate of NPI problem \cite{adleman}.
It is a special type of binary-quadratic Diophantine equations, and was considered in \cite{adleman} to study P and NP problem.
It is proved to be $\gamma$-complete, but not known to be NP-complete.
A result in \cite{garey} shows that if
the linear divisibility equation can be solved in polynomial time, then NP $=$ co-NP.
Besides this, this problem has its own importance to study. Since it is a factoring problem, however, Shor's algorithm can only provide some partial helps.
The second equation studied in this paper is quadratic congruence equation with restriction in the solution. The general quadratic congruence equation $x^2\equiv a\mod b$ can be solved efficiently be Shor's algorithms. However, it is a NP-complete problem \cite{manders} if we want to find a solution satisfies $0\leq x\leq c$. Some people believes that quantum computer cannot solve NP-complete problem efficiently (for example see \cite{aaronson}). This might be a problem to study the potential of quantum computer.
The third equation considered in the paper is about exponential congruence equation in finite field, i.e., diagonal equation studied in \cite{dam}.
The importance of this equation was discussed in paper \cite{dam}. However, here we care about this equation is because of its relations to polynomial equation in finite filed.
As for these three types of polynomial equations, Grover's algorithm and Shor's algorithm can only provide partial helps. We cannot rely on these two algorithms to solve these equations efficiently in quantum computer. As for the first two equations, we can achieve a cubic speedup in quantum computer based on the two algorithms.
As for the third equation, we present a relatively simple quantum algorithm to it compared with \cite{dam}. But the worst complexity is not improved here. And some generalizations of this equation and the relation to general polynomial equation in finite field of this equation were also discussed in this paper.
\section{Linear divisibility problem}
\setcounter{equation}{0}
The linear divisibility problem \cite{garey} states that for any two positive integers $a,b$, does there exists a positive integer $x$, such that $ax+1$ divides $b$?
This is equivalent to decide whether or not the following Diophantine equation
\begin{equation} \label{eq1}
(ax+1)y=b
\end{equation}
contains positive solutions for $x,y$?
In the following, we try to find quantum algorithm towards this problem. We first consider in the case when $a$ is prime.
By Shor's quantum factoring algorithm \cite{shor}, we can completely factor $b=p_1^{d_1}p_2^{d_2}\cdots p_n^{d_n}$ in time $O(\textmd{poly}(\log b))$. Then (\ref{eq1}) is equivalent to:
Does there exist $0\leq t_i\leq d_i~(i=1,2,\ldots,n)$, not all equal to zero, such that
\begin{equation} \label{eq2}
p_1^{t_1}p_2^{t_2}\cdots p_n^{t_n}\equiv1\mod a.
\end{equation}
Since $a$ is prime, so $\mathbb{Z}_a^*$ has a generator $g$, which can be computed in time $O(\textmd{poly}(\log a))$ \cite{dubrois}. By Shor's quantum discrete logarithm algorithm, we can computer $\lambda_i~(i=1,2,\ldots,n)$ in time $O(\textmd{poly}(\log a))$, such that
\[p_i\equiv g^{\lambda_i}\mod a.\]
Thus (\ref{eq1}) is equivalent to the following linear equation with restrictions:
\begin{equation} \label{eq3}
\left\{
\begin{array}{ll} \vspace{.2cm}
\lambda_1t_1+\lambda_2t_2+\cdots+\lambda_nt_n\equiv0\mod (a-1), & \hbox{} \\ \vspace{.2cm}
0\leq t_i\leq d_i,~~i=1,2,\ldots,n, & \hbox{} \\
t_1+t_2+\cdots+t_n>0. & \hbox{}
\end{array}
\right.
\end{equation}
The total costs of the reduction from (\ref{eq1}) to (\ref{eq3}) is $O(\textmd{poly}(\log b))$.
In order to test whether (\ref{eq3}) contains a solution or not, a naive method is searching all possible cases of $t_i$. It contains $D=(d_1+1)(d_2+1)\cdots(d_n+1)$ cases. A classical result in number theory \cite{hua} states that for any $\epsilon$, we have $D=O(b^\epsilon)$, the omitted constant $A_\epsilon=(\frac{2}{\epsilon \ln(2)})^{2^{1/\epsilon}}$ in $O(b^\epsilon)$ depends on $\epsilon$. And when $\epsilon$ is small, $A_\epsilon$ can be very large. For example if $\epsilon=1/5$, then $A_\epsilon\approx 1.2\times 10^{37}$. Although $A_\epsilon$ is a constant, not good to be vary large in algorithm. Another classical result in number theory \cite{hua} states that for almost $b$, we have $D=O(\log b)$. So the average complexity of linear divisibility problem in quantum computer is $O(\textmd{poly}(\log b))$.
Direct searching by Grover's algorithm \cite{grover} only provides a quadratic speedup. However, based on the linear properties of (\ref{eq3}), we can actually achieve a cubic speedup, i.e., $O(b^{\epsilon/3}\textmd{poly}(\log b))$ in quantum computer, by the method introduced in \cite{brassard}. The basic idea is dividing $\{1,2,\ldots,n\}$ into two parts $\{1,2,\ldots,j\}\cup\{j+1,j+2,\ldots,n\}$, such that $(d_1+1)(d_2+1)\cdots(d_j+1)=O(D^{1/3})$. Then denote
\[S_1=\{\lambda_1t_1+\lambda_2t_2+\cdots+\lambda_jt_j\mod (a-1):0\leq t_i\leq d_i,1\leq i\leq j\},\]
and $S_2$ is the set by sorting $S_1$. The total complexity to get the set $S_2$ is $O(D^{1/3}\log D)$.
Then define a map $f$ as
\[\begin{array}{rll} \vspace{.2cm}
f:\mathbb{Z}_{1+d_{j+1}}\times\cdots\times \mathbb{Z}_{1+d_n} &\rightarrow& \mathbb{Z}_2 \\
f(t_{j+1},\ldots,t_n) &=& \left\{
\begin{array}{ll} \vspace{.2cm}
0, & \hbox{if $\exists~v\in S_2$, such that $\lambda_{j+1}t_{j+1}+\cdots+\lambda_jt_j\equiv -v\mod (a-1)$;} \\
1, & \hbox{otherwise}
\end{array}
\right.
\end{array}\]
The value of the map $f$ can be computed by binary searching in the set $S_2$, which costs $O(\log D)$. And the way to find a solution of $f$ can be achieved by Grover's searching with complexity $O(D^{1/3})$. Therefore, the total complexity to solve (\ref{eq3}) is $O(b^{\epsilon/3}\textmd{poly}(\log b))$.
In the above searching procedure, equation (\ref{eq3}) actually does not provide too much help to solve (\ref{eq1}). Since the above searching procedure can be done directly in (\ref{eq2}) and this does not depends on whether $a$ is prime or not. And most importantly, when $a$ is not prime, (\ref{eq3}) will be changed into a linear system, which will be difficult to solve. At least, the above searching algorithm does not works anymore. Anyway, we have
\begin{proposition}
The linear divisibility equation (\ref{eq1}) can be solved in quantum computer in time $O(b^{\epsilon/3}\emph{poly}(\log b))$ for some $\epsilon$ such that $A_\epsilon=(\frac{2}{\epsilon \ln(2)})^{2^{1/\epsilon}}$ is not too large.
\end{proposition}
\begin{remark}
The linear divisibility problem provides us a new problem about factoring. Given $a,b,N$, does $N$ contains a factor in the form $ax+b$. The direct generalization of Shor's algorithm seems impossible to solve this problem efficiently. Also, as we can see in (\ref{eq3}), if $a$ is prime, then the equation $(ax+b)y=N$ is equivalent to
\begin{equation} \label{eq30}
\left\{
\begin{array}{ll} \vspace{.2cm}
\lambda_1t_1+\lambda_2t_2+\cdots+\lambda_nt_n\equiv s\mod (a-1), & \hbox{} \\
0\leq t_i\leq d_i,~~1\leq i\leq n. & \hbox{}
\end{array}
\right.
\end{equation}
for some $\lambda_1,\ldots,\lambda_n,s$. If $d_i=1$ for all $i$, then this is the subset sum problem, which brings new evidence to the difficulty of linear divisibility problem.
\end{remark}
\section{Quadratic congruence}
\setcounter{equation}{0}
With the finding of efficient factoring and discrete logarithm quantum algorithm, quadratic congruence equation can be solved efficiently in quantum computer. But quadratic congruence equation with restrictions on the solution cannot solved easily by Shor's work. In \cite{garey}, it introduces the following problem: For any three positive integers $a,b,c$, does there exists $0<x<c$, such that $x^2\equiv a\mod b$? This problem is proved to be NP-complete \cite{manders}. A basic idea to solve this problem in quantum computer is as follows:
We first factor $b=p_1^{d_1}p_2^{d_2}\cdots p_n^{d_n}$ by Shor's algorithm. Then all the solutions of $x^2\equiv a\mod b$ can be computed from
$x^2\equiv a\mod p_i^{d_i}$ for $1\leq i\leq n$. Let $\pm z_i$ be two solutions of $x^2\equiv a\mod p_i^{d_i}$, then by Chinese Reminder Theorem, the solution of $x^2\equiv a\mod b$ can be written as
\begin{equation}\label{eq4}
x=\sum_{i=1}^n\mu_iz_i \alpha_i B_i \mod b,
\end{equation}
where $\mu_i=\pm 1,B_i=b/p_i^{d_i}$ and $\alpha_i$ satisfies $\alpha_i B_i\equiv1 \mod p_i^{d_i}$. Therefore, the problem reduces to find suitable $\mu_i\in\{1,-1\}$ such that
\begin{equation}
0<\sum_{i=1}^n\mu_iz_i \alpha_i B_i \mod b<c.
\end{equation}
The method analyzed above to linear divisibly problem still works here and achieves a cubic speedup. Since $n<\log b$, so
\begin{proposition}
There is a quantum algorithm that solves the problem whether or not there exists $0<x<c$, such that $x^2\equiv a\mod b$ in time $O(b^{1/3}\emph{poly}(\log b))$.
\end{proposition}
\begin{remark}
This special type of quadratic congruence induces a more general problem as follows: Let $a_1,a_2,\ldots,a_n,b,c$ be some positive integers, does there exists $x_1,x_2,\ldots,x_n\in\{1,-1\}$, such that
\[0<a_1x_1+a_2x_2+\cdots+a_nx_n\mod b<c.\]
\end{remark}
\section{Exponential congruence equation}
\setcounter{equation}{0}
Let $\mathbb{F}_q$ be a finite field, $g$ be the generator of $\mathbb{F}_q^*$. Given $a,b,c,d\in\mathbb{F}_q^*$ and $a,c$ are factors of $q-1$, then does there exists $x,y\in \mathbb{F}_q$ satisfies the following equation in $\mathbb{F}_q$:
\begin{equation} \label{eq5}
g^{ax+b}+g^{cy+d}=1.
\end{equation}
A direct method is combining Grover's searching and Shor's discrete logarithm algorithm, just like the algorithm given in \cite{dam}. The method provided in \cite{dam} contains some hard estimations about the number of solutions of the equation (\ref{eq5}). In the following, we give a much simpler quantum algorithm to this problem compared with \cite{dam}. The worst complexity given in \cite{dam} does not improved here, but we get some different results from \cite{dam}. And some generalizations and observations about polynomial equation in finite field are discovered.
Assume that $a\leq c$. A result \cite{mullen} about the number $N$ of solutions of equation (\ref{eq5}) states that
\begin{equation}
|N-q|\leq (a-1)(c-1)q^{1/2}<c^2q^{1/2}.
\end{equation}
Hence, $N\geq q-c^2\sqrt{q}$.
Case 1, if $c\leq\displaystyle\frac{1}{2}q^{1/4}$, then $\displaystyle N\geq q-\frac{1}{4}q=\frac{3}{4}q$. Note that if $(x_0,y_0)$ is a solution of (\ref{eq5}), then there will exist $x_1,\ldots,x_{a-1}$, such that $(x_i,y_0)~(1\leq i\leq a-1)$ are solutions of (\ref{eq5}).
So in the searching space $\mathbb{F}_q$ of $y$, we have a probability at least $\displaystyle\frac{3}{4}q/aq=3/4a$ to get the right $y$, such that by putting $y$ into (\ref{eq5}), we will get a $x$ such that $(x,y)$ is a solution of (\ref{eq5}). So direct searching will cost $\widetilde{O}(\sqrt{a})=\widetilde{O}(q^{1/8})$.
Case 2, if $q-c^2\sqrt{q}\leq 0$, i.e., $c\geq q^{1/4}$.
Since $a,c$ are factors of $q-1$, so we can set $q-1=as=ct$. Then we know that we only need to search $x$ and $y$ in space $\{0,1,\ldots,s-1\}$ and $\{0,1,\ldots,t-1\}$ respectively. Moreover, given any $y_0\in \mathbb{Z}_t$, if there is a $x_0\in\mathbb{Z}_s$ such that $(x_0,y_0)$ is a solution of (\ref{eq5}), then $x_0$ can be computed efficiently by Shor's algorithm. So we only need to search $y$ in space $\mathbb{Z}_t$ and then check whether or not the (\ref{eq5}) has a solution for $x$ by Shor's algorithm.
At this time, the complexity to find one solution of (\ref{eq5}) or decide whether (\ref{eq5}) has a solution is $\widetilde{O}(\sqrt{q/c})=\widetilde{O}(q^{3/8})$.
This result also holds for $c\geq \displaystyle\frac{1}{2}q^{1/4}+1$.
\begin{proposition}
Assume $a,b,c,d\in\mathbb{F}_q^*$ and $a\leq c$ are factors of $q-1$, then there is a quantum algorithm can decide whether or not the equation $g^{ax+b}+g^{cy+d}=1$ contains a solution in time $\widetilde{O}(\sqrt{q/c})$ if $c\geq \displaystyle\frac{1}{2}q^{1/4}+1$, and in time $\widetilde{O}(\sqrt{a})$ if $c<\displaystyle\frac{1}{2}q^{1/4}+1$. Moreover, if it has solutions, then with in the same complexity, the quantum algorithm will return one solution.
\end{proposition}
Besides the efficiency of this problem, we can do more generalizations about polynomial equation in finite field based on the above problem.
Note that (\ref{eq5}) is equivalent to diagonal equation $\lambda_1u^a+\lambda_2v^c=1$, here $\lambda_1=g^b,\lambda_2=g^d$ and $u=g^x,v=g^y$.
The general diagonal equation with $n$ terms in finite field $\mathbb{F}_q$ has the form
\begin{equation} \label{eq7}
a_1x_1^{k_1}+a_2x_2^{k_2}+\cdots+a_nx_n^{k_n}=b,
\end{equation}
where $a_1,\ldots,a_n,b\in \mathbb{F}_q^*$ and $k_1,\ldots,k_n\in\mathbb{Z}_{q-1}$. This equation can also written in the form (\ref{eq5}). If there is a $1\leq i\leq n$ such that $\gcd(k_i,q-1)=1$, then (\ref{eq7}) is trivial to solve by Shor's discrete logarithm algorithm. So we can assume $k_1,\ldots,k_n$ are factors of $q-1$ in (\ref{eq7}).
In the general case, the result in \cite{mullen} states that the number $N$ of solution of equation (\ref{eq7}) satisfies
\[|N-q^{n-1}|\leq(k_1-1)(k_2-1)\cdots(k_n-1)q^{(n-1)/2}<k_1k_2\cdots k_n q^{(n-1)/2}.\]
We can also assume $k_1\leq k_2\leq \cdots\leq k_n$, then $N\geq q^{n-1}-k_1k_2\cdots k_nq^{(n-1)/2}$. The following analysis is similar to the equation (\ref{eq5}).
Case 1, if $k_1k_2\cdots k_n<\displaystyle\frac{1}{2}q^{(n-1)/2}$, then $N>\displaystyle\frac{1}{2}q^{n-1}$. Since if $(x_{10},x_{20},\ldots,x_{n0})$ is a solution of equation (\ref{eq7}), then there exist $x_{11},\ldots,x_{1(k_1-1)}$, such that $(x_{1i},x_{20},\ldots,x_{n0})$ is a solution of equation (\ref{eq7}) for any $1\leq i\leq k_1-1$. The searching space of $(x_2,\ldots,x_n)$ is $\mathbb{F}_q^{n-1}$. So in $\mathbb{F}_q^{n-1}$, we have a probability larger than $\displaystyle\frac{1}{2}q^{n-1}/k_1q^{n-1}=1/2k_1$ to get the right $x_2,\ldots,x_n$, such that by putting them into (\ref{eq7}), we will find a $x_1$ such that $(x_1,x_2,\ldots,x_n)$ is a solution of equation (\ref{eq7}). Since $k_1^n\leq k_1k_2\cdots k_n<\displaystyle\frac{1}{2}q^{(n-1)/2}$, so the complexity is $\widetilde{O}(\sqrt{k_1})=\widetilde{O}(q^{(n-1)/4n})$.
Case 2, if $k_1k_2\cdots k_n\geq q^{(n-1)/2}$, then $N\geq 0$, which means equation (\ref{eq7}) may not contains a solution, so the above analysis does not works anymore. For any $i$, set $k_il_i=q-1$. Different from the above case, we only need to search each $x_i$ in the space $\mathbb{Z}_{l_i}$. And we only need to search $(x_2,\ldots,x_n)\in\mathbb{Z}_{l_2}\times \cdots\times \mathbb{Z}_{l_n}$, then check whether or not the equation (\ref{eq7}) has a solution for $x_1$. Therefore, at this time, the size of the searching space is
\[\frac{k_1q^{n-1}}{k_1k_2\ldots k_n}\leq \frac{k_1q^{n-1}}{q^{(n-1)/2}}=k_1q^{(n-1)/2}.\]
As we know, the quantum searching and discrete logarithm algorithm will provide a quadratic speedup to this case,
so the complexity is $\widetilde{O}(\sqrt{q^{n-1}/k_2\ldots k_n})=\widetilde{O}(\sqrt{k_1}q^{(n-1)/4})$. This also holds for $k_1k_2\cdots k_n\geq\displaystyle\frac{1}{2}q^{(n-1)/2}$.
\begin{proposition}
Assume $a_1,\ldots,a_n,b\in\mathbb{F}_q^*$ and $k_1\leq k_2\leq \cdots\leq k_n$ are factors of $q-1$. Then there is a quantum algorithm can decides whether or not the equation $a_1x_1^{k_1}+a_2x_2^{k_2}+\cdots+a_nx_n^{k_n}=b$ contains a solution in time $\widetilde{O}(\sqrt{q^{n-1}/k_2\ldots k_n})$ if $k_1k_2\cdots k_n\geq\displaystyle\frac{1}{2}q^{(n-1)/2}$,
and in time $\widetilde{O}(\sqrt{k_1})$ if $k_1k_2\cdots k_n<\displaystyle\frac{1}{2}q^{(n-1)/2}$. Moreover, if it has solution, then with in the same complexity, the quantum algorithm will return one solution.
\end{proposition}
\begin{remark}
Equation (\ref{eq5}) can be written in the form $g^{cy}=g^{-d}-g^{ax+b-d}$. From the point of discrete logarithm, this problem is equivalent to find $x$ such that the discrete logarithm $l$ of $g^{-d}-g^{ax+b-d}$ in finite field $\mathbb{F}_q$ must divisible by $c$. So we can consider a general discrete logarithm problem, i.e., try to solve the equation
\begin{equation} \label{eq6}
g^{cy}=f(x_1,\ldots,x_n)
\end{equation}
in finite filed $\mathbb{F}_q$, here $c\mid (q-1)$ and $f$ is any polynomial belongs to $\mathbb{F}_q[x_1,\ldots,x_n]$. This problem aims at finding suitable $x_{10},\ldots,x_{n0}\in\mathbb{F}_q$, such that the discrete logarithm of $f(x_{10},\ldots,x_{n0})$ in finite field $\mathbb{F}_q^*$ is divisible by $c$.
Direct generalization of Shor's discrete logarithm seems impossible to this problem.
This is because the following problem cannot be solved efficiently in quantum computer due to the optimality of Grover's algorithm:
Let $F(t_1,\ldots,t_n)$ be any function over a set, such as $\mathbb{Z}_M^n$, and given a number $d\in\mathbb{Z}_M$, then does there exists $s_1,\ldots,s_n\in\mathbb{Z}_M$,
such that $d$ is a factor of $F(s_1,\ldots,s_n)$?
\end{remark}
In finite field $\mathbb{F}_q$, a single polynomial equation in $n$ variables can always be changed into the form
\begin{equation} \label{eq8}
\lambda_1g^{a_{11}x_1+\cdots+a_{1n}x_n}+\lambda_2g^{a_{21}x_1+\cdots+a_{2n}x_n}+\cdots+\lambda_kg^{a_{k1}x_1+\cdots+a_{kn}x_n}=c.
\end{equation}
We can introduce new variables to equation (\ref{eq8}), for any $1\leq j\leq k$, define
\begin{equation}\begin{array}{rll} \vspace{.2cm}
d_j &:=& \gcd(a_{j1},\ldots,a_{jn}), \\
d_jy_j &:=& a_{j1}x_1+\cdots+a_{jn}x_n.
\end{array}\end{equation}
Then equation (\ref{eq8}) is changed into
\begin{equation} \label{eq9}
\lambda_1g^{d_1y_1}+\lambda_2g^{d_2y_2}+\cdots+\lambda_kg^{d_ky_k}=c.
\end{equation}
Note that, if there is a $1\leq i\leq k$, such that $d_i=1$, then (\ref{eq9}) always has a solution in $\mathbb{F}_q$.
Let $y_1,y_2,\ldots,y_k$ be a solution of (\ref{eq9}), then in order to recover one solution of (\ref{eq8}), we need to solve the following linear Diophantine system
\begin{equation} \label{eq10}
\left(
\begin{array}{cccc}\vspace{.2cm}
a_{11} &~~ a_{12} &~~ \ldots &~~ a_{jn} \\\vspace{.2cm}
a_{21} &~~ a_{22} &~~ \ldots &~~ a_{2n} \\\vspace{.2cm}
\vdots &~~ \vdots &~~ \ddots &~~ \vdots \\
a_{k1} &~~ a_{k2} &~~ \ldots &~~ a_{kn} \\
\end{array}
\right)\left(
\begin{array}{c}\vspace{.2cm}
x_1 \\\vspace{.2cm}
x_2 \\\vspace{.2cm}
\vdots \\
x_n \\
\end{array}
\right)=\left(
\begin{array}{c}\vspace{.2cm}
y_1 \\\vspace{.2cm}
y_2 \\\vspace{.2cm}
\vdots \\
y_k \\
\end{array}
\right) \mod (q-1).
\end{equation}
Denote the coefficient matrix of above linear system as $A$. Then by a result in \cite{dickson}
there exist $L\in SL(k,\mathbb{Z}), R\in SL(n,\mathbb{Z})$, $D=\textmd{diag}\{s_1,s_2,\ldots,s_r\}$ and $s_i\mid s_{i+1}$ for $1\leq i\leq r-1$, such that
\begin{equation}\label{smith}
LAR=\left(
\begin{array}{ll}\vspace{.2cm}
D ~~& 0_{r\times (n-r)} \\
0_{(k-r)\times r} ~~& 0_{(k-r)\times (n-r)} \\
\end{array}
\right)=:J.
\end{equation}
Denote ${\bf x}=(x_1,\ldots,x_n)^T,{\bf y}=(y_1,\ldots,y_k)^T$ and $\tilde{{\bf x}}=R^{-1}{\bf x}=(\tilde{x}_1,\ldots,\tilde{x}_n)^T, \tilde{{\bf y}}=L{\bf y}=(\tilde{y}_1,\ldots,\tilde{y}_k)^T$, then (\ref{eq10}) is equivalent to
\begin{equation}\label{eq11}
J\tilde{{\bf x}}=\tilde{{\bf y}}\mod (q-1).
\end{equation}
Equation (\ref{eq11}) has a solution if and only if
\begin{equation}
\gcd(s_i,q-1)\mid \tilde{y}_i,~\textmd{for}~i=1,2,\ldots,r;~\textmd{and}~\tilde{y}_j=0~\textmd{for}~j=r+1,\ldots,k.
\end{equation}
In (\ref{smith}), $J$ is called the \emph{Smith normal from} of $A$. As we can see $s_1,\ldots,s_r$ are totaly determined by $A$. They form the \emph{invariant factors} of $A$. If $\gcd(s_r,q-1)=1$ and $r=k$, then any solution of (\ref{eq9}) will give a solution of (\ref{eq8}). Combining the above analysis, we have
\begin{proposition}
Let $\mathbb{F}_q$ be a finite field, $A=(a_{ij})$ be a matrix of size $k\times n$ and its rank equals $k$. Suppose $s_1,s_2,\ldots,s_k$ are the invariant factors of $A$ and $\gcd(s_k,q-1)=1$.
Let $d_i$ be the greatest common divisor of elements in the $i$-th row of $A$ for $1\leq i\leq k$.
(1). Any solution ${\bf y}$ of (\ref{eq9}) will recover a solution ${\bf x}$ of equation (\ref{eq8}). And the costs from ${\bf y}$ to ${\bf x}$ is $O(\emph{poly}(k,n))$.
(2). If there is a $1\leq i\leq k$, such that $d_i=1$, then there is a quantum algorithm find the solution of polynomial equation (\ref{eq8}) in time $O(\emph{poly}(k,n,\log q))$.
\end{proposition}
So the study of equation (\ref{eq7}) has its own importance. It might be a starting point to study the general polynomial equation in finite field.
\section{Conclusion}
In this article, we have considered three different types of equations. Each equation has its own character, and provides more interesting problems which deserve to study in the future.
|
1,116,691,497,809 | arxiv | \section{\@startsection{section}{1}{0pt}%
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\begin{document}
\newtheorem{thm}{Theorem}
\title{Relational Abstract Domains for the\\
Detection of Floating-Point Run-Time Errors\thanks{This work was
partially supported by the ASTR\'EE RNTL project.}}
\author{Antoine Min\'e}
\institute{DI-\'Ecole Normale Sup\'erieure de Paris, France,\\
\email{[email protected]}}
\maketitle
\begin{abstract}
We present a new idea to adapt relational abstract domains to the analysis of
IEEE 754-compliant floating-point numbers in order to statically detect,
through Abstract Interpretation-based static analyses, potential floating-point
run-time exceptions such as overflows or invalid operations.
In order to take the non-linearity of rounding into account, expressions are
modeled as linear forms with interval coefficients.
We show how to extend already existing numerical abstract domains, such as the
octagon abstract domain, to efficiently abstract transfer functions based on
interval linear forms.
We discuss specific fixpoint stabilization techniques and give some
experimental results.
\end{abstract}
\section{Introduction}
It is a well-established fact, since the failure of the Ariane 5 launcher in
1996, that run-time errors in critical embedded software can cause great
financial---and human---losses.
Nowadays, embedded software is becoming more and more complex.
One particular trend is to abandon fixed-point arithmetics in favor
of floating-point computations.
Unfortunately, floating-point models are quite complex and features such as
rounding and special numbers (infinities, $\mi{NaN}$, etc.) are not always
understood by programmers.
This has already led to catastrophic behaviors, such as the Patriot
missile story told in \cite{patriot}.
Much work is concerned about the {\it precision} of the computations, that is
to say, characterizing the amount and cause of drift between a
computation on perfect reals and the corresponding floating-point
implementation.
Ours is {\it not}.
We seek to prove that an exceptional behavior (such as division by zero or
overflow) will not occur in any execution of the analyzed program.
While this is a simpler problem, our goal is to scale up to programs of
hundreds of thousands of lines with full data coverage and very few
(even none) false alarms.
Our framework is that of Abstract Interpretation, a generic framework for
designing sound static analyses that already features many instances
\cite{ai,ai2}.
We adapt existing relational numerical abstract domains (generally designed
for the analysis of integer arithmetics) to cope with floating-point
arithmetics.
The need for such domains appeared during the successful design of a
commissioned special-purpose prototype analyzer for a critical
embedded avionics system.
Interval analysis, used in a first prototype \cite{magic1}, proved too
coarse because error-freeness of the analyzed code depends on tests
that are inherently poorly abstracted in non-relational abstract domains.
We also had to design special-purpose widenings and narrowings to
compensate for the pervasive rounding errors, not only in the analyzed program,
but also introduced by our efficient abstractions.
These techniques were implemented in our second prototype whose overall
design was presented in \cite{magic2}.
The present paper focuses on improvements and novel unpublished ideas; it is
also more generic.
\section{Related Work}
\subsubsection{Abstract Domains.}
A key component in Abstract-Interpretation-based analyses is the abstract
domain which is a computer-representable class of program invariants together
with some operators to manipulate them: transfer functions for guards and
assignments, a control-flow join operator, and fixpoint acceleration
operators (such as widenings $\mathbin{\triangledown}$ and narrowings $\mathbin{\vartriangle}$) aiming at
the correct and efficient analysis of loops.
One of the simplest yet useful abstract domain is the widespread interval
domain \cite{ai}.
Relational domains, which are more precise, include Cousot and Halbwachs's
polyhedron domain \cite{poly} (corresponding to invariants of the form
$\sum c_i v_i\leq c$), Min\'e's octagon domain \cite{octagons}
($\pm v_i \pm v_j\leq c$), and Simon's two variables per inequality domain
\cite{tvpli} ($\alpha v_i + \beta v_j \leq c$).
Even though the underlying algorithms for these relational domains
allow them to abstract sets of reals as well as sets of integers, their
efficient implementation---in a maybe approximate but sound way---using
floating-point numbers remains a challenge.
Moreover, these relational domains do not support abstracting floating-point
expressions, but only expressions on perfect integers, rationals, or
reals.
\subsubsection{Floating-Point Analyses.}
Much work on floating-point is dedicated to the analysis of the precision of
the computations and the origins of the rounding errors.
The CESTAC method \cite{cestac} is widely used, but also much debated as it
is based on a probabilistic model of error distribution and thus cannot give
sound answers.
An interval-based Abstract Interpretation for error terms is proposed in
\cite{onera}.
Some authors \cite{goubault,martel} go one step further by allowing error
terms to be related in relational, even non-linear, domains.
Unfortunately, this extra precision does not help when analyzing programs
whose correctness also depends upon relations between variables and not only
error terms (such as programs with inequality tests, as in
Fig.~\ref{limiter}).
\subsubsection{Our Work.}
We first present our IEEE 754-based computation model (Sect.~3) and
recall the classical interval analysis adapted to floating-point
numbers (Sect.~4).
We present, in Sect.~5, an abstraction of floating-point expressions in
terms of interval linear forms over the real field and use it to refine the
interval domain.
Sect.~6 shows how some relational abstract domains can be efficiently
adapted to work on these linear forms.
Sect.~7 presents adapted widening and narrowing techniques.
Finally, some experimental results are shown in Sect.~8.
\section{IEEE 754-Based Floating-Point Model}
We present in this section the concrete floating-point arithmetics model that
we wish to analyze and which is based on the widespread IEEE 754-1985
\cite{ieee754} norm.
\subsection{IEEE 754 Floating-Point Numbers}
The binary representation of a IEEE 754 number is composed of three
fields:
\begin{mylist}
\item a 1-bit sign $s$;
\item an exponent $e-\mb{bias}$, represented by a biased $\mb{e}$-bit
unsigned integer $e$;
\item a fraction $f=.b_1\ldots b_\mb{p}$, represented by a $\mb{p}$-bit
unsigned integer.
\end{mylist}
The values $\mb{e}$, $\mb{bias}$, and $\mb{p}$ are format-specific.
We will denote by $\mb{F}$ the set of all available formats and
by $\mb{f}=\mb{32}$ the 32-bit {\it single} format
($\mb{e}=8$, $\mb{bias}=127$, and $\mb{p}=23$).
Floating-point numbers belong to one of the following categories:
\begin{mylist}
\item {\it normalized} numbers $(-1)^s\times 2^{e-\mb{bias}}\times
1.f$, when $1\leq e\leq 2^\mb{e}-2$;
\item {\it denormalized} numbers $(-1)^s\times 2^{1-\mb{bias}}\times
0.f$, when $e=0$ and $f\neq 0$;
\item $+0$ or $-0$ (depending on $s$), when $e=0$ and $f=0$;
\item $+\infty$ or $-\infty$ (depending on $s$), when $e=2^\mb{e}-1$ and
$f=0$;
\item error codes (so-called {\it NaN}), when $e=2^\mb{e}-1$ and $f\neq 0$.
\end{mylist}
For each format $\mb{f}\in\mb{F}$ we define in particular:
\begin{mylist}
\item $\mi{mf}_\mb{f}=2^{1-\mb{bias}-\mb{p}}$ the smallest non-zero positive number;
\item $\mi{Mf}_\mb{f}=(2-2^{-\mb{p}})2^{2^\mb{e}-\mb{bias}-2}$, the largest non-infinity
number.
\end{mylist}
The special values $+\infty$ and $-\infty$
may be generated as a result of operations
undefined on $\mathbb{R}$ (such as $1/{+0}$), or when a result's absolute
value overflows $\mi{Mf}_\mb{f}$.
Other undefined operations (such as ${+0}/{+0}$) result in a {\it NaN}
(that stands for {\it Not A Number}).
The sign of $0$ serves only to distinguish between
$1/{+0}=+\infty$ and $1/{-0}=-\infty$; $+0$ and $-0$ are indistinguishable
in all other contexts (even comparison).
Due to the limited number of digits, the result of a floating-point
operation needs to be rounded.
IEEE 754 provides four rounding modes: towards $0$, towards $+\infty$,
towards $-\infty$, and to nearest.
Depending on this mode, either the floating-point number directly smaller or
directly larger than the exact real result is chosen
(possibly $+\infty$ or $-\infty$).
Rounding can build infinities from non-infinities operands
(this is called {\it overflow}), and it may return zero when the absolute
value of the result is too small (this is called {\it underflow}).
Because of this rounding phase, most algebraic properties of $\mathbb{R}$, such
as associativity and distributivity, are lost.
However, the opposite of a number is always exactly represented
(unlike what happens in two-complement integer arithmetics), and
comparison operators are also exact.
See \cite{goldberg} for a description of the classical properties and
pitfalls of the floating-point arithmetics.
\subsection{Custom Floating-Point Computation Model}
\begin{figure}[t]
\begin{mycenter}\fbox{
$\begin{array}{lll}
R_{\mb{f},+\infty}(x)&=&\left\{\begin{array}{ll}
\mathrm{\Omega} & \quad\text{if $x> \mi{Mf}_\mb{f}$}\\
\min\{y\in\mathbb{F}_\mb{f}\;|\;y\geq x\} & \quad\text{otherwise}\\
\end{array}\right.
\\\\
R_{\mb{f},-\infty}(x)&=&\left\{\begin{array}{ll}
\mathrm{\Omega} & \quad\text{if $x< -\mi{Mf}_\mb{f}$}\\
\max\{y\in\mathbb{F}_\mb{f}\;|\;y\leq x\} & \quad\text{otherwise}\\
\end{array}\right.
\\\\
R_{\mb{f},0}(x)&=&\left\{\begin{array}{ll}
\min\{y\in\mathbb{F}_\mb{f}\;|\;y\geq x\} & \quad\text{if $x\leq 0$}\\
\max\{y\in\mathbb{F}_\mb{f}\;|\;y\leq x\} & \quad\text{if $x\geq 0$}\\
\end{array}\right.
\\\\
R_{\mb{f},n}(x)&=&\left\{\begin{array}{ll}
\mathrm{\Omega} &
\quad\text{if $|x|\geq (2-2^{-\mb{p}-1})2^{2^{\mb{e}}-\mb{bias}-2}$}\\
\mi{Mf}_\mb{f} & \quad\text{else if $x\geq \mi{Mf}_\mb{f}$}\\
-\mi{Mf}_\mb{f} & \quad\text{else if $x\leq -\mi{Mf}_\mb{f}$}\\
R_{\mb{f},-\infty}(x) & \quad\text{else if
$|R_{\mb{f},-\infty}(x)-x|<|R_{\mb{f},+\infty}(x)-x|$}\\
R_{\mb{f},+\infty}(x) & \quad\text{else if
$|R_{\mb{f},+\infty}(x)-x|<|R_{\mb{f},-\infty}(x)-x|$}\\
R_{\mb{f},-\infty}(x) & \quad\text{else if
$R_{\mb{f},-\infty}(x)$'s least significant bit is $0$}\\
R_{\mb{f},+\infty}(x) & \quad\text{else if
$R_{\mb{f},+\infty}(x)$'s least significant bit is $0$}\\
\end{array}\right.
\end{array}$
}\end{mycenter}
\caption{Rounding functions, extracted from \cite{ieee754}.}
\label{ieeeround}
\end{figure}
\begin{figure}[t]
\begin{mycenter}\fbox{
$\begin{array}{llll}
\lb{\mi{const}_{\mb{f},\mb{r}}(c)}\rho
&=& R_{\mb{f},\mb{r}}(c)\\
\lb{v}\rho &=& \rho(v)\\
\lb{\mi{cast}_{\mb{f},\mb{r}}(e)}\rho
&=& R_{\mb{f},\mb{r}}(\lb{e}\rho)
& \quad\text{if $\lb{e}\rho\neq\mathrm{\Omega}$}\\
\lb{e_1\odot_{\mb{f},\mb{r}}e_2}\rho
&=& R_{\mb{f},\mb{r}}((\lb{e_1}\rho) \cdot
(\lb{e_2}\rho))
& \quad\text{if $\lb{e_1}\rho,
\lb{e_2}\rho\neq\mathrm{\Omega},\;
\cdot\in\{+,-,\times\}$}\\
\lb{e_1\oslash_{\mb{f},\mb{r}}e_2}\rho
&=& R_{\mb{f},\mb{r}}((\lb{e_1}\rho) \slash
(\lb{e_2}\rho))
& \quad\text{if $\lb{e_1}\rho\neq \mathrm{\Omega},\;
\lb{e_2}\rho\notin\{0,\mathrm{\Omega}\}$}\\
\lb{\ominus e}\rho
&=& -(\lb{e}\rho)
& \quad\text{if $\lb{e}\rho\neq\mathrm{\Omega}$}\\
\lb{\mi{expr}_\mb{f}}\rho
&=& \mathrm{\Omega}
& \quad\text{in all other cases}\\
\end{array}$
}\end{mycenter}
\caption{Expression concrete semantics, extracted from \cite{ieee754}.}
\label{ieeesem}
\end{figure}
We focus our analysis on the large class of programs that treat floating-point
arithmetics as a practical approximation to the mathematical reals
$\mathbb{R}$: roundings and underflows are tolerated, but not overflows, divisions by
zero or invalid operations, which are considered run-time errors and halt
the program.
Our goal is to detect such behaviors.
In this context, $+\infty$, $-\infty$, and {\it NaN}\/s can never be
created and, as a consequence, the difference between $+0$ and $-0$
becomes irrelevant.
For every format $\mb{f}\in\mb{F}$, the set of floating-point numbers will be
assimilated to a finite subset of $\mathbb{R}$ denoted by $\mathbb{F}_\mb{f}$.
The grammar of floating-point expressions of format $\mb{f}$ includes
constants, variables $v\in\mc{V}_{\mb{f}}$ of format $\mb{f}$, casts (conversion from
another format), binary and unary arithmetic operators (circled in order
to distinguish them from the corresponding operators on reals):
\begin{mycenter}$\begin{array}{lcll}
\mi{expr}_\mb{f} & :== & \mi{const}_{\mb{f},\mb{r}}(c) & \quad c\in\mathbb{R}\\
& | & v & \quad v\in\mc{V}_{\mb{f}}\\
& | & \mi{cast}_{\mb{f},\mb{r}}(\mi{expr}_{\mb{f}'}) \\
& | & \mi{expr}_\mb{f} \odot_{\mb{f},\mb{r}} \mi{expr}_\mb{f} & \quad
\odot\in\{\oplus,\ominus,\otimes,\oslash\}\\
& | & \ominus \mi{expr}_\mb{f}
\end{array}$\end{mycenter}
Some constructs are tagged with a floating-point format $\mb{f}\in\mb{F}$
and a rounding
mode $\mb{r}\in\{n,0,+\infty,-\infty\}$ ($n$ representing rounding to
nearest).
A notable exception is the unary minus $\ominus$ which does not incur
rounding and never results in a run-time error as all the $\mathbb{F}_\mb{f}$s are
perfectly symmetric.
An environment $\rho\in\prod_{\mb{f}\in\mb{F}}(\mc{V}_\mb{f}\rightarrow\mathbb{F}_\mb{f})$ is
a function that associates to each variable a floating-point value of
the corresponding format.
Fig.~\ref{ieeesem} gives the concrete semantics
$\lb{\mi{expr}_\mb{f}}\rho\in\mathbb{F}_\mb{f}\cup\{\mathrm{\Omega}\}$ of the expression $\mi{expr}_\mb{f}$ in
the environment $\rho$: it can be a number or the run-time error $\mathrm{\Omega}$.
This semantics uses solely the regular operators $+$,$-$,$\times$,$\slash$
on real numbers and the rounding function
$R_{\mb{f},\mb{r}}:\mathbb{R}\rightarrow\mathbb{F}_\mb{f}\cup\{\mathrm{\Omega}\}$ defined in Fig.~\ref{ieeeround}.
It corresponds exactly to the IEEE 754 norm \cite{ieee754} where the overflow,
division by zero, and invalid operation exception traps abort the system
with a run-time error.
\section{Floating-Point Interval Analysis}
\subsubsection{Floating-Point Interval Arithmetics.}
The idea of interval arithmetics is to over-approximate a set of numbers by
an interval represented by its lower and upper bounds.
For each format $\mb{f}$, we will denote by $\mathbb{I}_\mb{f}$ the set of real intervals
with bounds in $\mathbb{F}_\mb{f}$.
As $\mathbb{F}_\mb{f}$ is totally ordered and bounded, any subset of $\mathbb{F}_\mb{f}$ can be
abstracted by an element of $\mathbb{I}_\mb{f}$.
Moreover, as {\it all rounding functions $R_{\mb{f},\mb{r}}$ are monotonic}, we
can compute the bounds of any expression using pointing-point operations only,
ensuring efficient implementation within an analyzer.
A first idea is to use the very same format and rounding mode in the
abstract as in the concrete, which would give, for instance, the
following addition ($\mathrm{\Omega}$ denoting a run-time error):
\begin{mycenter}$
{}[a^-;a^+]\oplus^\sharp_{\mb{f},\mb{r}}[b^-;b^+]=\left\{\begin{array}{l}
\mathrm{\Omega} \quad \text{if $a^-\oplus_{\mb{f},\mb{r}}b^-=\mathrm{\Omega}$ or $a^+\oplus_{\mb{f},\mb{r}}b^+=\mathrm{\Omega}$}\\
{}[a^-\oplus_{\mb{f},\mb{r}}b^-;a^+\oplus_{\mb{f},\mb{r}}b^+] \quad \text{otherwise}
\end{array}\right.
$\end{mycenter}
A drawback of this semantics is that it requires the analyzer to determine,
for each instruction, which rounding mode $\mb{r}$ and which format $\mb{f}$ are used.
This may be difficult as the rounding mode can be changed at run-time by a
system call, and compilers are authorized to perform parts of computations
using a more precise IEEE format that what is required by the
programmer (on Intel x86, all floating-point registers are 80-bit wide and
rounding to the user-specified format occurs only when the results are stored
into memory).
Unfortunately, using in the abstract a floating-point format different
from the one used in the concrete computation is not sound.
The following semantics, inspired from the one presented in \cite{goubault},
solves these problems by providing an approximation that is independent
from the concrete rounding mode (assuming always the worst: towards $-\infty$
for the lower bound and towards $+\infty$ for the upper bound):
\begin{mycenter}\begin{supertabular}{ll}
$\bullet$ &
$\mi{const}^\sharp_\mb{f}(c) = [\mi{const}_{\mb{f},-\infty}(c);\mi{const}_{\mb{f},+\infty}(c)]$\\
$\bullet$ &
$\mi{cast}^\sharp_\mb{f}(c) = [\mi{cast}_{\mb{f},-\infty}(c);\mi{cast}_{\mb{f},+\infty}(c)]$\\
$\bullet$ &
$[a^-;a^+]\oplus^\sharp_\mb{f}[b^-;b^+]=
{}[a^-\oplus_{\mb{f},-\infty}b^-;a^+\oplus_{\mb{f},+\infty}b^+]$\\
$\bullet$ &
$[a^-;a^+]\ominus^\sharp_\mb{f}[b^-;b^+]=
{}[a^-\ominus_{\mb{f},-\infty}b^+;a^+\ominus_{\mb{f},+\infty}b^-]$\\
$\bullet$ &
$[a^-;a^+]\otimes^\sharp_\mb{f}[b^-;b^+]=$\\
& $\quad [\min((a^+\otimes_{\mb{f},-\infty}b^+),(a^-\otimes_{\mb{f},-\infty}b^+),
(a^+\otimes_{\mb{f},-\infty}b^-),(a^-\otimes_{\mb{f},-\infty}b^-));$\\
& $\quad \;\max((a^+\otimes_{\mb{f},+\infty}b^+),(a^-\otimes_{\mb{f},+\infty}b^+),
(a^+\otimes_{\mb{f},+\infty}b^-),(a^-\otimes_{\mb{f},+\infty}b^-))]$\\
$\bullet$ & $[a^-;a^+]\oslash^\sharp_\mb{f}[b^-;b^+]=$\\
& $\quad \; \circ \; \mathrm{\Omega} \quad \text{if $b^-\leq 0 \leq b^+$}$\\
& $\quad \; \circ \;
{}[\min((a^+\oslash_{\mb{f},-\infty}b^+),(a^-\oslash_{\mb{f},-\infty}b^+),
(a^+\oslash_{\mb{f},-\infty}b^-),(a^-\oslash_{\mb{f},-\infty}b^-));$\\
& $\quad \quad
\;\max((a^+\oslash_{\mb{f},+\infty}b^+),(a^-\oslash_{\mb{f},+\infty}b^+),
(a^+\oslash_{\mb{f},+\infty}b^-),(a^-\oslash_{\mb{f},+\infty}b^-))]$\\
& $\quad \quad \text{otherwise}$\\
$\bullet$ & $\ominus^\sharp [a^-;a^+]=[\ominus a^+;\ominus a^-]$\\
$\bullet$ & return $\mathrm{\Omega}$ if one interval bound evaluates to $\mathrm{\Omega}$\\
\end{supertabular}\end{mycenter}
This semantics frees the analyzer from the job of statically determining
the rounding mode of the expressions and allows the analyzer to use,
in the abstract, less precise formats that those used in the concrete
(however, using a more precise format in the abstract remains unsound).
\subsubsection{Floating-Point Interval Analysis.}
Interval analysis is a non-relational Abstract Interpretation-based analysis
where, at each program point and for each variable, the set of its possible
values during all executions reaching this point is over-approximated by
an interval.
An abstract environment $\rho^\sharp$ is a function
mapping each variable $v\in\mc{V}_\mb{f}$ to an element of $\mathbb{I}_\mb{f}$.
The abstract value $\lb{\mi{expr}_\mb{f}}^\sharp\rho^\sharp\in\mathbb{I}_\mb{f}\cup\{\mathrm{\Omega}\}$ of an
expression $\mi{expr}_\mb{f}$ in an abstract environment $\rho^\sharp$ can be
derived by induction using the interval operators defined in the preceding
paragraph.
\begin{figure}[t]
\begin{mycenter}\fbox{
\begin{tabular}{l}
{\bf for} (n=0;n<N;n++) \{\\
\quad // fetch $X$ in $[-128;128]$ and $D$ in $[0;16]$\\
\quad $S=Y$; \quad $R=X \ominus_{\mb{32},n} S$; \quad $Y=X$;\\
\quad {\bf if} ($R\leq \ominus D$) $Y=S\ominus_{\mb{32},n}D$;\\
\quad {\bf if} ($R\geq D$) $Y=S\oplus_{\mb{32},n}D$;\\
\}\\
\end{tabular}
}\end{mycenter}
\caption{Simple rate limiter function with input $X$, output $Y$, and
maximal rate variation $D$. }
\label{limiter}
\end{figure}
An assignment $v\leftarrow \mi{expr}_\mb{f}$ performed in an
environment $\rho^\sharp$ returns
$\rho^\sharp$ where $v$'s value has been replaced by
$\lb{\mi{expr}_\mb{f}}^\sharp\rho^\sharp$ if it does not evaluate to
$\mathrm{\Omega}$, and otherwise by $\mathbb{F}_\mb{f}$ (the {\it top} value) and reports
an error.
Most tests can only be abstracted by ignoring them (which is sound).
Even though for simple tests such as, for instance, $X\leq Y\oplus_{\mb{f},\mb{r}}c$,
the
interval domain is able to refine the bounds for $X$ and $Y$, it cannot
remember the relationship between these variables.
Consider the more complete example of Fig.~4.
It is a rate limiter that given random input flows $X$ and $D$, bounded
respectively by $[-128;128]$ and $[0;16]$, computes an output flow $Y$
that tries to follow $X$ while having a change rate limited by $D$.
Due to the imprecise abstraction of tests, the interval domain will bound
$Y$ by $[-128-16n;128+16n]$ after $n$ loop iterations while in fact it is
bounded by $[-128;128]$ independently from $n$.
If $N$ is too big, the interval analysis will conclude that the limiter
may overflow while it is in fact always perfectly safe.
\section{Linearization of Floating-Point Expressions}
Unlike the interval domain, relational abstract domains rely on
algebraic properties of operators, such as associativity and distributivity,
that are not true in the floating-point world.
Our solution is to approximate floating-point expressions by
{\it linear expressions} in the {\it real} field with
{\it interval coefficients} and free variables in
$\mc{V}=\cup_{\mb{f}\in\mb{F}}\mc{V}_\mb{f}$.
Let $i+\sum_{v\in\mc{V}}i_v v$ be such a linear form; it can be viewed as
a function from $\mathbb{R}^\mc{V}$ to the set of real intervals.
For the sake of efficiency, interval coefficient bounds will be
represented by floating-point numbers in a format $\mb{fa}$ that is efficient
on the analyzer's platform: $i,i_v\in\mathbb{I}_\mb{fa}$.
Because, for all $\mb{f}$ and $\cdot\in\{+,-,\times,\slash\}$,
$\odot^\sharp_\mb{f}$ is a valid over-approximation of the
corresponding real interval arithmetics operation, we can define the following
sound operators $\boxplus^\sharp$, $\boxminus^\sharp$, $\boxtimes^\sharp$, $\boxslash^\sharp$
on linear forms:
\begin{mycenter}\begin{supertabular}{llll}
$\bullet$ & $(i+\sum_{v\in\mc{V}}i_vv)\boxplus^\sharp(i'+\sum_{v\in\mc{V}}i'_vv)$
&=& $(i\oplus^\sharp_\mb{fa} i')+\sum_{v\in\mc{V}}(i_v\oplus^\sharp_\mb{fa} i'_v)v$\\
$\bullet$ & $(i+\sum_{v\in\mc{V}}i_vv)\boxminus^\sharp(i'+\sum_{v\in\mc{V}}i'_vv)$
&=& $(i\ominus^\sharp_\mb{fa} i')+\sum_{v\in\mc{V}}(i_v\ominus^\sharp_\mb{fa} i'_v)v$\\
$\bullet$ & $i\boxtimes^\sharp(i'+\sum_{v\in\mc{V}}i'_vv)$
&=& $(i\otimes^\sharp_\mb{fa} i')+\sum_{v\in\mc{V}}(i\otimes^\sharp_\mb{fa} i'_v)v$\\
$\bullet$ & $(i+\sum_{v\in\mc{V}}i_vv)\boxslash^\sharp i'$
&=& $(i\oslash^\sharp_\mb{fa} i')+\sum_{v\in\mc{V}}(i_v\oslash^\sharp_\mb{fa} i')v$\\
\end{supertabular}\end{mycenter}
Given an expression $\mi{expr}_\mb{f}$ and an interval abstract environment
$\rho^\sharp$ as in Sect.~4, we construct the interval linear form
$\lp{\mi{expr}_\mb{f}}\rho^\sharp$ on $\mc{V}$ as follows:
\begin{mycenter}$\begin{array}{llll}
\bullet & \lp{\mi{const}_{\mb{f},\mb{r}}(c)}\rho^\sharp & = &
[\mi{const}_{\mb{f},-\infty}(c);\mi{const}_{\mb{f},+\infty}(c)]\\
\bullet & \lp{v_\mb{f}}\rho^\sharp & = & [1;1]v_\mb{f}\\
\bullet & \lp{\mi{cast}_{\mb{f},\mb{r}}(e)}\rho^\sharp & = &
\lp{e}\rho^\sharp \;\boxplus^\sharp\; \varepsilon_\f(\lp{e}\rho^\sharp ) \;\boxplus^\sharp\;
\mi{mf}_\mb{f}[-1;1]\\
\bullet & \lp{e_1\oplus_{\mb{f},\mb{r}}e_2}\rho^\sharp & = & \\
\multicolumn{4}{l}{\quad
\lp{e_1}\rho^\sharp \;\boxplus^\sharp\; \lp{e_2}\rho^\sharp \;\boxplus^\sharp\;
\varepsilon_\f(\lp{e_1}\rho^\sharp) \;\boxplus^\sharp\; \varepsilon_\f(\lp{e_2}\rho^\sharp)
\;\boxplus^\sharp\; \mi{mf}_\mb{f}[-1;1]}\\
\bullet & \lp{e_1\ominus_{\mb{f},\mb{r}}e_2}\rho^\sharp & = & \\
\multicolumn{4}{l}{\quad
\lp{e_1}\rho^\sharp \;\boxminus^\sharp\; \lp{e_2}\rho^\sharp \;\boxplus^\sharp\;
\varepsilon_\f(\lp{e_1}\rho^\sharp) \;\boxplus^\sharp\; \varepsilon_\f(\lp{e_2}\rho^\sharp)
\;\boxplus^\sharp\; \mi{mf}_\mb{f}[-1;1]}\\
\bullet & \lp{[a;b]\otimes_{\mb{f},\mb{r}}e_2}\rho^\sharp & = & \\
\multicolumn{4}{l}{\quad
([a;b]\boxtimes^\sharp\lp{e_2}\rho^\sharp) \;\boxplus^\sharp\;
([a;b]\boxtimes^\sharp\varepsilon_\f(\lp{e_2}\rho^\sharp))
\;\boxplus^\sharp\; \mi{mf}_\mb{f}[-1;1]}\\
\bullet & \lp{e_1\otimes_{\mb{f},\mb{r}}[a;b]}\rho^\sharp & = &
\lp{[a;b]\otimes_{\mb{f},\mb{r}}e_1}\rho^\sharp \\
\bullet & \lp{e_1\otimes_{\mb{f},\mb{r}}e_2}\rho^\sharp & = &
\lp{\iota(\lp{e_1}\rho^\sharp)\rho^\sharp\;\otimes_{\mb{f},\mb{r}}\;e_2}\rho^\sharp \\
\bullet & \lp{e_1\oslash_{\mb{f},\mb{r}}[a;b]}\rho^\sharp & = & \\
\multicolumn{4}{l}{\quad
(\lp{e_1}\rho^\sharp\boxslash^\sharp[a;b]) \;\boxplus^\sharp\;
(\varepsilon_\f(\lp{e_1}\rho^\sharp)\boxslash^\sharp [a;b])
\;\boxplus^\sharp\; \mi{mf}_\mb{f}[-1;1]} \\
\bullet & \lp{e_1\oslash_{\mb{f},\mb{r}}e_2}\rho^\sharp & = &
\lp{e_1\;\oslash_{\mb{f},\mb{r}}\;\iota(\lp{e_2}\rho^\sharp)\rho^\sharp} \\
\end{array}$\end{mycenter}
where the ``error'' $\varepsilon_\f(l)$ of the linear form $l$ is the following
linear form:
\begin{mycenter}$\begin{array}{lll}
\displaystyle
\varepsilon_\f\left([a;b]+\sum_{v\in\mc{V}}[a_v;b_v]v\right) &=&
(\max(|a|,|b|)\otimes^\sharp_\mb{fa}[-2^{-\mb{p}};2^{-\mb{p}}])\;+\\
&&\displaystyle
\sum_{v\in\mc{V}}(\max(|a_v|,|b_v|)\otimes^\sharp_\mb{fa}[-2^{-\mb{p}};2^{-\mb{p}}])v\\
\multicolumn{3}{l}{\text{{\it($\mb{p}$ is the fraction size in bits for the format
$\mb{f}$, see Sect.~3.1)}}}
\end{array}$\end{mycenter}
and the ``intervalization'' $\iota(l)\rho^\sharp$ function over-approximates
the range of the linear form $l$ in the abstract environment $\rho^\sharp$ as the
following interval of $\mathbb{I}_\mb{fa}$:
\begin{mycenter}$\begin{array}{l}
\displaystyle
\iota\left(i+\sum_{v\in\mc{V}}i_vv\right)\rho^\sharp =
i\oplus^\sharp_\mb{fa}\left(\sideset{}{^\sharp_\mb{fa}}\bigoplus_{v\in\mc{V}}i_v\otimes^\sharp_\mb{fa}
\rho^\sharp(v)\right)\\
\text{{\it (any summation order for $\oplus^\sharp_\mb{fa}$ is sound)}}
\end{array}$\end{mycenter}
Note that this semantics is very different from the one proposed by Goubault
in \cite{goubault} and subsequently used in \cite{martel}.
In \cite{goubault}, each operation introduces a new variable representing an
error term, and there is no need for interval coefficients.
\subsubsection{About $\iota$.}
Dividing a linear form by another linear form which is not reduced to an
interval does not yield a linear form.
In this case, the $\iota$ operator is used to over-approximate the divisor
by a single interval before performing the division.
The same holds when multiplying two linear forms not reduced to an
interval, but we can choose
to apply $\iota$ to either argument.
For the sake of simplicity, we chose here to ``intervalize'' the left
argument.
Moreover, any non-linear operator (such as, e.g., square root or sine) could
be dealt with by performing the corresponding operator on intervals after
``intervalizing'' its argument(s).
\subsubsection{About $\varepsilon_\f$.}
To account for rounding errors, an upper bound of
$|R_{\mb{f},\mb{r}}(x\cdot y)-(x\cdot y)|$ (where $\cdot\in\{+,-,\times,\slash\}$) is
included in $\lp{\mi{expr}_\mb{f}}\rho^\sharp$.
It is the sum an error relative to the arguments $x$ and $y$, expressed
using $\varepsilon_\f$, and an absolute error $\mi{mf}_\mb{f}$ due to a possible underflow.
Unlike what happened with interval arithmetics, correct error computation
does not require the abstract operators to use floating-point formats
that are no more precise than the concrete ones: the choice of $\mb{fa}$ is
completely free.
\subsubsection{About $\mathrm{\Omega}$.}
It is quite possible that, during the computation of
$\lp{\mi{expr}_\mb{f}}\rho^\sharp$, a floating-point run-time error $\mathrm{\Omega}$ occurs.
In that case, we will say that the linearization ``failed''.
It does not mean that the program has a run-time error, but only that
we cannot compute a linearized expression and must revert to the classical
interval arithmetics.
\subsubsection{Main Result.}
When we evaluate in $\mathbb{R}$ the linear form $\lp{\mi{expr}_\mb{f}}\rho^\sharp$ in
a concrete environment $\rho$ included in $\rho^\sharp$ we get a real
interval that over-approximates the concrete value of the expression:
\begin{theorem}
~\par\noindent\begin{tabular}{l}
\noindent
If $\lb{\mi{expr}_\mb{f}}^\sharp\rho^\sharp\neq\mathrm{\Omega}$ and the linearization does not fail and
$\forall v\in\mc{V},\;\rho(v)\in\rho^\sharp(v)$,\\
\noindent
then $\lb{\mi{expr}_\mb{f}}\rho\in\lp{\mi{expr}_\mb{f}}\rho^\sharp\:(\rho)$.
\end{tabular}
\end{theorem}
\subsubsection{Linear Form Propagation.}
As the linearization manipulates expressions symbolically, it is able to
perform simplifications when the same variable appears several times.
For instance $Z\leftarrow X\ominus_{\mb{32},n}(0.25\otimes_{\mb{32},n}X)$
will be interpreted as
$Z\leftarrow[0.749\cdots;0.750\cdots]X+2.35\cdots 10^{-38}[-1;1]$.
Unfortunately, no simplification can be done if the expression is broken into
several statements, such as in
$Y\leftarrow 0.25\otimes_{\mb{32},n}X; Z\leftarrow X\ominus_{\mb{32},n}Y$.
Our solution is to remember, in an extra environment $\rho^\sharp_l$,
the linear form assigned to each variable and use this information while
linearizing: we set $\lp{v_\mb{f}}(\rho^\sharp,\rho^\sharp_l)=\rho^\sharp_l(v)$ instead
of $[-1;1]v_\mb{f}$.
Care must be taken, when a variable $v$ is modified, to discard all occurrences
of $v$ in $\rho^\sharp_l$.
Effects of tests on $\rho^\sharp_l$ are ignored.
Our partial order on linear forms is flat, so, at control-flow joins,
only variables that are associated with the same linear form in both
environments are kept; moreover, we do not need any widening.
This technique is reminiscent of Kildall's constant propagation \cite{kildall}.
\subsubsection{Applications.}
A first application of linearization is to improve the precision of the
interval analysis.
We simply replace $\lb{\mi{expr}_\mb{f}}^\sharp\rho^\sharp$ by
$\lb{\mi{expr}_\mb{f}}^\sharp\rho^\sharp\;\cap\;\iota(\lp{\mi{expr}_\mb{f}}\rho^\sharp)\rho^\sharp$
whenever the hypotheses of Thm.~1 hold.
While improving the assignment transfer function (trough expression
simplification), this is not sufficient to treat tests precisely.
For this, we need relational domains.
Fortunately, Thm.~1 also means that if we have a relational domain that
manipulates sets of points with real coordinates $\mathbb{R}^\mc{V}$ and that is
able to perform assignments and tests of linear expressions with interval
coefficients, we can use it to perform relational analyses on floating-point
variables.
Consider, for instance, the following algorithm to handle an assignment
$v\leftarrow \mi{expr}_\mb{f}$ in such a relational domain (the procedure
would be equivalent for tests):
\begin{mylist}
\item If $\lb{\mi{expr}_\mb{f}}^\sharp\rho^\sharp=\mathrm{\Omega}$, then we report a run-time error and
apply the transfer function for $v\leftarrow\mathbb{F}_\mb{f}$.
\item Else, if the linearization of $\mi{expr}_\mb{f}$ fails, then we do not report
an error but apply the transfer function for
$v\leftarrow\lb{\mi{expr}_\mb{f}}^\sharp\rho^\sharp$.
\item Otherwise, we do not report an error but apply the transfer function for
$v\leftarrow\lp{\mi{expr}_\mb{f}}\rho^\sharp$.
\end{mylist}
Remark how we use the interval arithmetics to perform the actual
detection of run-time errors and as a fallback when the linearization cannot be
used.
\section{Adapting Relational Abstract Domains}
We first present in details the adaptation of the octagon abstract domain
\cite{octagons}
to use floating-point arithmetics and interval linear forms, which
was implemented in our second prototype analyzer \cite{magic2}.
We then present in less details some ideas to adapt other domains.
\subsection{The Octagon Abstract Domain}
The octagon abstract domain \cite{octagons} can manipulate sets of constraints
of the form $\pm x\pm y\leq c$, $x,y\in\mc{V}$, $c\in\mathbb{E}$ where $\mathbb{E}$ can
be $\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{R}$.
An abstract element $\mb{o}$ is represented by a half-square constraint matrix
of size $|\mc{V}|$.
Each element at line $i$, column $j$ with $i\leq j$ contains four constraints:
$v_i+v_j\leq c$, $v_i-v_j\leq c$, $-v_i+v_j\leq c$, and $-v_i-v_j\leq c$, with
$c\in\overline{\E}=\mathbb{E}\cup\{+\infty\}$.
Remark that diagonal elements represent interval constraints as
$2v_i\leq c$ and $-2v_i\leq c$.
In the following, we will use notations such as $\max_\mb{o}(v_i+v_j)$ to
access the upper bound, in $\overline{\E}$, of constraints embedded
in the octagon $\mb{o}$.
Because constraints in the matrix can be combined to obtain implied
constraints that may not be in the matrix (e.g., from $x-y\leq c$ and
$y+z\leq d$, we can deduce $x+z\leq c+d$), two matrices can represent the same
set of points.
We introduced in \cite{octagons} a Floyd-Warshall-based closure operator that
provides a normal form by combining and propagating, in $\mc{O}(|\mc{V}|^3)$
time, all constraints.
The optimality of the abstract operators requires to work on closed matrices.
\subsubsection{Floating-Point Octagons.}
In order to represent and manipulate efficiently constraints on real numbers,
we choose to use floating-point matrices: $\mathbb{F}_\mb{fa}$ replaces $\mathbb{E}$
(where $\mb{fa}$ is, as before, an efficient floating-point format chosen by the
analyzer implementation).
As the algorithms presented in \cite{octagons} make solely use of the $+$ and
$\leq$ operators on $\overline{\E}$, it is sufficient to replace
$+$ by $\oplus_{\mb{fa},+\infty}$ and map $\mathrm{\Omega}$ to $+\infty$ in these algorithms to
provide a sound approximation of all the transfer functions and operators
on reals using only $\overline{\FF}_\mb{fa}=\mathbb{F}_\mb{fa}\cup\{+\infty\}$.
As all the nice properties of $\mathbb{E}$ are no longer true in $\mathbb{F}_\mb{fa}$, the closure
is no longer a normal form.
Even though working on closed matrices will no longer guaranty the optimality
of the transfer functions, it still greatly improves their precision.
\subsubsection{Assignments.}
Given an assignment of the form
$v_k\leftarrow l$, where $l$ is a interval linear form, on an octagon $\mb{o}$, the
resulting set is no always an octagon.
We can choose between several levels of approximation.
Optimality could be achieved at great cost by performing the assignment
in a polyhedron domain and then computing the smallest enclosing octagon.
We propose, as a less precise but much faster alternative
($\mc{O}(|\mc{V}|$ time cost), to replace
all constraints concerning the variable $v_k$ by the following ones:
\begin{mycenter}$\begin{array}{llll}
v_k+v_i & \leq & u(\mb{o},l\boxplus^\sharp v_i) & \forall i\neq k\\
v_k-v_i & \leq & u(\mb{o},l\boxminus^\sharp v_i) & \forall i\neq k\\
-v_k+v_i & \leq & u(\mb{o},v_i\boxminus^\sharp l) & \forall i\neq k\\
-v_k-v_i & \leq & u(\mb{o},\boxminus^\sharp(l\boxplus^\sharp v_i)) \quad
& \forall i\neq k\\
v_k & \leq & u(\mb{o},l) \\
-v_k & \leq & u(\mb{o},\boxminus^\sharp l) \\
\end{array}$\end{mycenter}
where the upper bound of a linear form $l$ on an octagon $\mb{o}$ is approximated
by $u(\mb{o},l)\in\overline{\FF}_\mb{fa}$ as follows:
\begin{mycenter}$\begin{array}{l}\displaystyle
u \left(\mb{o},\;[a^-;a^+]+\sum_{v\in\mc{V}} [a^-_v;a^+_v] v\right) =
a^+ \;\oplus_{\mb{fa},+\infty}
\vspace*{-8pt}
\\\displaystyle
\qquad\quad
\left(
\sideset{}{_{\mb{fa},+\infty}}\bigoplus_{v\in\mc{V}}
\max (\begin{array}[t]{l}
\max_\mb{o} (v) \otimes_{\mb{fa},+\infty} a^+_v,\;
\ominus(\max_\mb{o} (-v) \otimes_{\mb{fa},-\infty} a^+_v),\\
\max_\mb{o} (v) \otimes_{\mb{fa},+\infty} a^-_v,\;
\ominus(\max_\mb{o} (-v) \otimes_{\mb{fa},-\infty} a^-_v)
)\end{array}
\right)
\vspace*{6pt}\\
\text{{\it (any summation order for $\oplus_{\mb{fa},+\infty}$ is sound)}}
\end{array}
$\end{mycenter}
and $\oplus_{\mb{fa},+\infty}$ and
$\otimes_{\mb{fa},+\infty}$ are extended to $\overline{\FF}_\mb{fa}$ as follows:
\begin{mycenter}$\begin{array}{l}
+\infty\oplus_{\mb{fa},+\infty}x = x\oplus_{\mb{fa},+\infty}+\infty=+\infty\\
+\infty\otimes_{\mb{fa},+\infty}x = x\otimes_{\mb{fa},+\infty}+\infty=
\left\{\begin{array}{ll}
0&\quad\text{if $x=0$}\\
+\infty&\quad\text{otherwise}
\end{array}\right.
\end{array}$\end{mycenter}
\begin{example}
\label{assign}
Consider the assignment $X=Y\oplus_{\mb{32},n}Z$ with $Y,Z\in[0;1]$.
It is linearized as $X=[1-2^{-23};1+2^{-23}](Y+Z)+\mi{mf}_\mb{32}[-1;1]$,
so our abstract transfer function will infer relational constraints
such as $X-Y\leq 1+2^{-22}+\mi{mf}_\mb{32}$.
\end{example}
\subsubsection{Tests.}
Given a test of the form $l_1\leq l_2$, where $l_1$ and $l_2$ are
linear forms, for all variable $v_i\neq v_j$, appearing in
$l_1$ or $l_2$, the constraints in the octagon $\mb{o}$ can be tightened by
adding the following extra constraints:
\begin{mycenter}\begin{supertabular}{lll}
$v_j-v_i$ &$\leq$&
$u(\mb{o},\;l_2\boxminus^\sharp l_1\boxminus^\sharp v_i\boxplus^\sharp v_j)$\\
$v_j+v_i$ &$\leq$&
$u(\mb{o},\;l_2\boxminus^\sharp l_1\boxplus^\sharp v_i\boxplus^\sharp v_j)$\\
$-v_j-v_i$ &$\leq$&
$u(\mb{o},\;l_2\boxminus^\sharp l_1\boxminus^\sharp v_i\boxminus^\sharp v_j)$\\
$-v_j+v_i$ &$\leq$&
$u(\mb{o},\;l_2\boxminus^\sharp l_1\boxplus^\sharp v_i\boxminus^\sharp v_j)$\\
$v_i$ &$\leq$&
$u(\mb{o},\;l_2\boxminus^\sharp l_1\boxplus^\sharp v_i)$\\
$-v_i$ &$\leq$&
$u(\mb{o},\;l_2\boxminus^\sharp l_1\boxminus^\sharp v_i)$\\
\end{supertabular}\end{mycenter}
\begin{example}
\label{test}
Consider the test $Y\oplus_{\mb{32},n}Z\leq 1$ with $Y,Z\in[0;1]$.
It is linearized as $[1-2^{-23};1+2^{-23}](Y+Z)+\mi{mf}_\mb{32}[-1;1]\leq [1;1]$.
Our abstract transfer function will be able to infer the constraint:
$Y+Z\leq 1+2^{-22}+\mi{mf}_\mb{32}$.
\end{example}
\begin{example}
The optimal analysis of the rate limiter function of Fig.~\ref{limiter} would
require representing interval linear invariants on {\it three} variables.
Nevertheless, the octagon domain with our approximated transfer functions
can prove that the output $Y$ is bounded by $[-136;136]$
independently from $n$ (the optimal bound being $[-128;128]$),
which is sufficient to prove that $Y$ does not overflow.
\end{example}
\subsubsection{Reduction with Intervals.}
The interval environment $\rho^\sharp$
is important as we use it to perform run-time error
checking and to compute the linear form associated to an expression.
So, we suppose that transfer functions are performed in parallel in the
interval domain and in the octagon domain, and then, information from
the octagon result $\mb{o}$ is used to refine the interval result
$\rho^\sharp$ as follows:
for each variable $v\in\mc{V}$, the upper bound of
$\rho^\sharp(v)$ is replaced by
$\min (\max \rho^\sharp(v),\max_\mb{o}(v))$ and the same is done for its lower bound.
\subsection{Polyhedron Domain}
The polyhedron domain is much more precise than the octagon domain as it
allows manipulating sets of invariants of the form $\sum_v c_v v\leq c$,
but it is also much more costly.
Implementations, such as the New Polka or the
Parma Polyhedra libraries \cite{parma}, are
targeted at representing sets of points with integer or rational coordinates.
They internally use rational coefficients and, as the coefficients usually
become fairly large, arbitrary precision integer libraries.
\subsubsection{Representing Reals.}
These implementations could be used as-is to abstract sets of points with
real coordinates, but the rational coefficients may get out of control, as
well as the time cost.
Unlike what happened for octagons, it is not so easy to adapt the algorithms
to floating-point coefficients while retaining soundness as they are much
much more complex.
We are not aware, at the time of writing, of any such floating-point
implementation.
\subsubsection{Assignments and Tests.}
Assignments of the form $v\leftarrow l$ and tests of the form $l\leq 0$
where $l=[a^-;a^+] + \sum_{v\in\mc{V}}[a^-_v;a^+_v]v$
seem difficult to abstract in general.
However, the case where all coefficients in $l$ are scalar except maybe the
constant one is much easier.
To cope with the general case, an idea (yet untested)
is to use the following transformation
that abstracts $l$ into an over-approximated linear form $l'$ where
$\forall v,\;a^-_v=a^+_v$ by transforming all relative errors into
absolute ones:
\begin{mycenter}$\begin{array}{lll}
l' &=&\displaystyle
\left(
[a^-;a^+]\oplus^\sharp_\mb{fa}
\sideset{}{^\sharp_\mb{fa}}\bigoplus_{v\in\mc{V}}
(a^+_v \ominus_{\mb{fa},+\infty} a^-_v)\otimes^\sharp_\mb{fa}
[0.5;0.5]\otimes^\sharp_\mb{fa}\rho^\sharp(v)\right)\;+
\\
&&\displaystyle
\sum_{v\in\mc{V}}
((a^-_v \oplus_{\mb{fa},+\infty} a^+_v)\otimes^\sharp_\mb{fa}[0.5;0.5]) v
\vspace*{4pt}\\
\multicolumn{3}{l}{
\text{{\it (any summation order for $\oplus^\sharp_\mb{fa}$ is sound)}}}
\end{array}$\end{mycenter}
\subsection{Two Variables per Linear Inequalities Domain}
Simon's domain \cite{tvpli} can manipulate constraints of the form
$\alpha v_i+\beta v_j\leq c$, $\alpha,\beta,c\in\mathbb{Q}$.
An abstract invariant is represented using a planar convex polyhedron for
each pair of variables.
As for octagons, most computations are done point-wise on variable pairs
and a closure provides the normal form by propagating and combining
constraints.
Because the underlying algorithms are simpler than for generic polyhedra,
adapting this domain to handle floating-point computations efficiently
may prove easier while greatly improving the precision over octagons.
This still remains an open issue.
\subsection{Ellipsoid and Digital Filter Domains}
During the design of our prototype analyzer \cite{magic2}, we encountered
code for computing recursive sequences such as
$X_i=((\alpha \otimes_{\mb{32},n} X_{i-1})\oplus_{\mb{32},n}
(\beta \otimes_{\mb{32},n}X_{i-2}))\oplus_{\mb{32},n}\gamma$
(1), or $X_i=(\alpha \otimes_{\mb{32},n}X_{i-1})\oplus_{\mb{32},n}
(Y_i\ominus_{\mb{32},n}Y_{i-1})$ (2).
In order to find precise bounds for the variable $X$, one has to consider
invariants out of the scope of classical relational abstract domains.
Case (1) can be solved by using the ellipsoid abstract domain of
\cite{magic2} that can represent non-linear real invariants of the form
$aX_i^2+bX_{i-1}^2+cX_iX_{i-1}\leq d$, while case (2) is precisely
analyzed using Feret's filter domains \cite{filters} by inferring
temporal invariants of the form $|X_i|\leq a\max_{j\leq i} |Y_j|+b$.
It is not our purpose here to present these new abstract domains
but we stress the fact that such domains, as the ones discussed in the
preceding paragraphs, are naturally designed to work with perfect reals, but
used to analyze imperfect floating-point computations.
A solution is, as before, to design these domains to analyze interval linear
assignments and tests on reals, and feed them with the result of the
linearization of floating-point expressions defined in Sect.~5.
This solution has been successfully applied (see \cite{filters} and
Sect.~8).
\section{Convergence Acceleration}
\label{widen}
In the Abstract Interpretation framework, loop invariants are described
as fixpoints and are over-approximated by iterating, in the abstract,
the body transfer function $F^\sharp$ until a post-fixpoint is reached.
\subsubsection{Widening.}
The widening $\mathbin{\triangledown}$ is a convergence acceleration operator introduced
in \cite{ai} in order to reduce the number of abstract iterations:
$\lim_i (F^\sharp)^i$ is replaced by $\lim_i E^\sharp_i$ where
$E^\sharp_{i+1}=E^\sharp_i\mathbin{\triangledown} F^\sharp(E^\sharp_i)$.
A straightforward widening on intervals and octagons is to simply discard
unstable constraints.
However, this strategy is too aggressive and fails to discover sequences
that are stable after a certain bound, such as, e.g.,
$X=(\alpha\otimes_{\mb{32},n}X)\oplus_{\mb{32},n}\beta$.
To give these computations a chance to stabilize, we use a staged widening
that tries a user-supplied set of bounds in increasing order.
As we do not know in advance which bounds will be stable, we use,
as set $\mathbb{T}$ of thresholds, a simple exponential ramp:
$\mathbb{T}=\{\pm 2^i\}\cap\mathbb{F}_\mb{fa}$.
Given two octagons $\mb{o}$ and $\mb{o}'$, the widening with thresholds
$\mb{o}\mathbin{\triangledown}\mb{o}'$ is obtained by setting, for each binary unit
expression $\pm v_i\pm v_j$:
\begin{mycenter}$
\max_{\mb{o}\mathbin{\triangledown}\mb{o}'} (C) =
\left\{\begin{array}{ll}
\max_{\mb{o}}(C) & \text{if $\max_{\mb{o}'}(C)\leq\max_{\mb{o}}(C)$}\\
\min \{t\in\mathbb{T}\cup\{+\infty\}\;|\;t\geq \max_{\mb{o}'}(C)\} &
\text{otherwise}\\
\end{array}\right.
$\end{mycenter}
\subsubsection{Decreasing Iterations.}
We now suppose that we have iterated the widening with thresholds up to an
abstract post-fixpoint $X^\sharp$: $F^\sharp(X^\sharp)\sqsubseteq X^\sharp$.
The bound of a stable variable is generally over-approximated by the
threshold immediately above.
One solution to improve such a bound is to perform some decreasing iterations
$X^\sharp_{i+1}=X^\sharp_i\sqcap F(X^\sharp_i)$ from
$X^\sharp_0=X^\sharp$.
We can stop whenever we wish, the result will always be, by construction,
an abstraction of the concrete fixpoint;
however, it may no longer be a post-fixpoint for $F^\sharp$.
It is desirable for invariants to be abstract post-fixpoint so that the
analyzer can check them independently from the way they were generated instead
of relying solely on the maybe buggy fixpoint engine.
\subsubsection{Iteration Perturbation.}
Careful examination of the iterates on our benchmarks
showed that the reason we do not get an abstract post-fixpoint
is that the {\it abstract} computations are done
in floating-point which incurs a somewhat non-deterministic extra rounding.
There exists, between $F^\sharp$'s definitive pre-fixpoints and
$F^\sharp$'s definitive post-fixpoints, a chaotic region.
To ensure that the $X^\sharp_i$ stay above this region, we replace
the intersection $\sqcap$ used in the decreasing iterations
by the following narrowing $\mathbin{\vartriangle}$:
$\mb{o}\mathbin{\vartriangle}\mb{o}'=\epsilon(\mb{o}\sqcap\mb{o}')$
where $\epsilon(\mb{o})$ returns an octagon where the bound of each
unstable constraint is enlarged by $\epsilon \times d$, where $d$ is the
maximum of all non $+\infty$ constraint bounds in $\mb{o}$.
Moreover, replacing $\mb{o}\mathbin{\triangledown}\mb{o}'$ by $\epsilon(\mb{o}\mathbin{\triangledown}\mb{o}')$
allows the analyzer to skip above $F^\sharp$'s chaotic regions
and effectively reduces the required number of increasing iterations, and so,
the analysis time.
Theoretically, a good $\epsilon$ can be estimated by the relative amount of
rounding errors performed in the abstract computation of one loop iteration,
and so, is a function of the complexity of the analyzed loop body,
the floating-point format $\mb{fa}$ used in the analyzer and the
implementation of the abstract domains.
We chose to fix $\epsilon$ experimentally by enlarging a small value
until the analyzer reported it found an abstract post-fixpoint for our program.
Then, as we improved our abstract domains and modified the analyzed program,
we seldom had to adjust this $\epsilon$ value.
\section{Experimental Results}
We now show how the presented abstract domains perform in practice.
Our only real-life example is the critical embedded avionics software of
\cite{magic2}.
It is a $132,000$ lines reactive C program
(75 KLoc after preprocessing) containing approximately $10,000$ global
variables, $5,000$ of which are floating-point variables, single precision.
The program consists mostly of one very large loop executed
$3.6\cdot 10^6$ times.
Because relating several thousands variables in a relational domain
is too costly, we use the ``packing'' technique described in \cite{magic2}
to statically determine sets of variables that should be related together
and we end up with approximately $2,400$ octagons of size
$2$ to $42$ instead of one octagon of size $10,000$.
\begin{figure}[t]
\begin{mycenter}
\begin{tabular}{c|c|c|c|c|c|c|c|}
\cline{2-8}
&\multicolumn{3}{|c|}{{\bf domains}}&
&{\bf nb. of}&&{\bf nb. of}\\
\cline{2-4}
&\;{\bf linearize}\;&\;{\bf octagons}\;&\;{\bf filters}\;&
\;\;{\bf time}\;\;&\;{\bf iterations}\;&\;{\bf memory}\;&\;{\bf alarms}\;\\
\cline{2-8}
(1)&
$\times$&$\times$&$\times$&
1623 s & 150 & 115 MB & 922 \\
\cline{2-8}
(2)&
$\surd$&$\times$&$\times$&
4001 s & 176 & 119 MB & 825 \\
\cline{2-8}
(3)&
$\surd$&$\surd$ &$\times$&
3227 s & 69 & 175 MB & 639 \\
\cline{2-8}
(4)&
$\surd$&$\times$&$\surd$&
8939 s & 211 & 207 MB & 363 \\
\cline{2-8}
(5)&
$\surd$&$\surd$&$\surd$&
4541 s & 72 & 263 MB & 6 \\
\cline{2-8}
\end{tabular}
\end{mycenter}
\caption{Experimental results.}
\label{results1}
\end{figure}
Fig.~\ref{results1} shows how the choice of the abstract domains influence
the precision and the cost of the analysis presented in \cite{magic2} on
our 2.8 GHz Intel Xeon.
Together with the computation time, we also give the number of abstract
executions of the big loop needed to find an invariant; thanks to our widenings
and narrowings, it is much much less than the concrete number of iterations.
All cases use the interval domain with the symbolic simplification
automatically provided by the linearization, except (1) that uses plain
interval analysis.
Other lines show the influence of the octagon (Sect.~6.1) and the specialized
digital filter domains (\cite{filters} and Sect.~6.4):
when both are activated, we only get six potential run-time errors
for a reasonable time and memory cost.
This is a sufficiently small number of alarms to allow manual inspection, and
we discovered they could be eliminated without altering the functionality
of the application by changing only three lines of code.
Remark that as we add more complex domains, the time cost per iteration
grows but the number of iterations needed to find an invariant decreases so
that a better precision may reduce the overall time cost.
\section{Conclusion}
We presented, in this paper, an adaptation of the octagon abstract domain in
order to analyze programs containing IEEE 754-compliant floating-point
operations.
Our methodology is somewhat generic and we proposed some ideas to adapt
other relational numerical abstract domains as well.
The adapted octagon domain was implemented in our prototype static analyzer
for run-time error checking of critical C code \cite{magic2} and tested
on a real-life embedded avionic application.
Practical results show that the proposed method scales up well and does
greatly improve the precision of the analysis when compared to the classical
interval abstract domain while maintaining a reasonable cost.
To our knowledge, this is the first time relational numerical domains are used
to represent relations between floating-point variables.
\subsubsection{Acknowledgments.}
We would like to thank all the members of the ``magic'' team:
Bruno Blanchet, Patrick Cousot, Radhia Cousot, J\'er\^ome
Feret, Laurent Mauborgne, David Monniaux, Xavier Rival, as well as
the anonymous referees.
\bibliographystyle{plain}
|
1,116,691,497,810 | arxiv | \part{title
\newboolean{Conference}
\setboolean{Conference}{false}
\newcommand{\NotConf}[1]{\ifthenelse{\boolean{Conference}}{}{#1}}
\newcommand{\IfConf}[2]{\ifthenelse{\boolean{Conference}}{#1}{#2}}
\begin{document}
\ititle{Forward Invariance of Sets for Hybrid Dynamical Systems}
\iauthor{
Jun Chai\\
{\normalsize [email protected]}\\
Ricardo Sanfelice\\
{\normalsize [email protected]}}
\idate{\today{}}
\iyear{2017}
\irefnr{012}
\makeititle
\tableofcontents
\newpage
\title{\Large \bf Forward Invariance of Sets for \\ Hybrid Dynamical Systems (Part I)}
\author{Jun Chai and Ricardo G. Sanfelice}
\maketitle
\begin{abstract}
In this paper, tools to study forward invariance properties with robustness to disturbances, referred to as {\em robust forward invariance}, are proposed for hybrid dynamical systems modeled as hybrid inclusions.
Hybrid inclusions are given in terms of differential and difference inclusions with state and disturbance constraints, for whose definition only four objects are required.
The proposed robust forward invariance notions allow for the diverse type of solutions to such systems (with and without disturbances), including solutions that have persistent flows and jumps, that are Zeno, and that stop to exist after finite amount of (hybrid) time.
Sufficient conditions for sets to enjoy such properties are presented.
These conditions are given in terms of the objects defining the hybrid inclusions and the set to be rendered robust forward invariant.
In addition, as special cases, these conditions are exploited to state results on nominal forward invariance for hybrid systems without disturbances.
Furthermore, results that provide conditions to render the sublevel sets of Lyapunov-like functions forward invariant are established.
Analysis of a controlled inverter system is presented as an application of our results.
Academic examples are given throughout the paper to illustrate the main ideas.
\end{abstract}
\section{Introduction}
\label{sec:intro}
\input{Sections/intro.tex}
\section{Preliminaries}
\label{sec:Hw}
\input{Sections/Hw.tex}
\section{Robust and Nominal Forward Invariance}
\label{sec:HwNotions}
\input{Sections/HwNotions.tex}
\section{Sufficient Conditions for Robust and Nominal Forward Invariance}
\label{sec:SuffConds}
\input{Sections/SuffConds.tex}
\input{Sections/HwSuffConds.tex}
\section{Nominal Forward Invariance of Sublevel Sets of Lyapunov-like Functions}
\label{sec:HwLya}
\input{Sections/HwLya.tex}
\section{Forward Invariance Analysis for a Controlled Single-Phase DC/AC Inverter System}
\label{sec:inverter}
\input{Sections/inverter.tex}
\section{Conclusion}
Forward invariance properties of sets that are uniform over the disturbances for hybrid inclusions with disturbance inputs are studied in this paper.
When a set $K$ enjoys such a property, solution pairs evolve within the set they started from, even under the effect of disturbances.
We formally define robust forward invariance of sets for hybrid systems $\HS\w$ modeled using differential and difference inclusions with state and disturbance constraints.
Among the four notions, two of them are considered stronger than the other two in the sense that all maximal solution pairs that start from the set of interest stay in it.
Sufficient conditions for each notion are presented in terms of the data of the hybrid system and require checking conditions involving its discrete and continuous dynamics within $K$.
In particular, when starting from an intersection involving $K$ and the jump set, the jump map ought to map the state back to $K$ to allow solutions evolving within $K$ during jumps; while when starting from an intersection involving $K$ and the flow set, the flow map needs to have vectors pointing inward of $K$ to allow solutions evolving within it during flows.
To guarantee the robust invariance notions, the flow map is required to enjoy a locally Lipschitz property on $C\w$ to avoid solutions from leaving $K$.
Such a property is also ensured by Assumption~\ref{assum:wbound}--a mild assumption on the inputs allowed by $C\w$ at points that are at the boundary of $K$ and in $C\w$.
To achieve the notions that require completeness of maximal solution pairs, a general result for $\HS\w$ that characterizes all possible ending behaviors of maximal solution pairs is presented.
The existence of nontrivial solution pairs from every point in $K$ is ensured by the assumption of zero disturbances being admissible during flows.
In addition, condition \textit{$\star$)} is introduced to exclude the case of solution pairs escaping to infinity in finite time, where two solution-independent conditions are proposed to verify \textit{$\star$)} for $\HS\w$.
The results on robust invariance are specialized to the nominal case, i.e., $\HS$.
Finally, one result to render sublevel sets of Lyapunov-like functions nominal forward invariant for $\HS$ is provided.
In particular, properties of the tangent cone to the sublevel of the Lyapunov-like function is exploited to derive mild conditions for invariance of the sublevel set.
These properties hold for free when the flow set is convex (locally) or the Lyapunov-like functions strictly decrease at the boundary of the set $K$.
In a second part of this work, which is being prepared, results on existence of invariance inducing state-feedback laws for $\HS\w$ using robust control Lyapunov functions for forward invariance will be provided.
In particular, controller synthesis that constructs state-feedback laws using a pointwise minimum norm selection scheme are under development.
Future research directions include studying optimality properties of feedback laws via inverse optimality analysis and the development of barrier certificates for hybrid systems $\HS$ and $\HS\w$.
\bibliographystyle{unsrt}
\subsection{Background and Motivation}
Forward invariance of sets for a dynamical system are key in numerous applications, including
air traffic management \cite{tomlin1998conflict},
obstacle avoidance in vehicular networks \cite{Qi.ea.15.CDC},
threat assessment in semi-autonomous cars \cite{Falcone.ea.11.TITS},
network control systems \cite{Pin.Parisini.09.NMPC}, and
building control \cite{Meyer.ea.16.Automatica}.
Techniques to verify such properties are vital in the design of many autonomous systems.
Such tools are even more valuable under the presence of disturbances.
Formally, a set $K$ is said to be forward invariant for a dynamical system if every solution to the system from $K$ stays in $K$.
This property is also referred to as flow-invariance \cite{fernandes1987remarks} or positively invariance \cite{blanchini1999set}.
When solutions are nonunique and invariance only holds for some solutions from each point in $K$, then $K$ is said to be weakly forward invariant -- the so-called {\em viability} in \cite{aubin.viability}.
In the presence of disturbances, one is typically interested in invariance properties that hold for all possible allowed disturbances, which has been referred to as {\em robust forward invariance}; see, e.g., \cite{Meyer.ea.16.Automatica}, \chai{\cite{li2016robust,sadraddini2016provably}}.
Tools to verify invariance of a set for continuous-time and discrete-time systems have been thoroughly investigated in the literature.
In the seminal article \cite{nagumo1942lage}, the so-called {\em Nagumo Theorem} is established to determine forward invariance (and weak forward invariance) of sets for continuous-time systems with unique solutions.
Given a locally compact set $K$ that is to be rendered forward invariant and a continuous-time system with a continuous vector field, the Nagumo Theorem requires that, at each point in the boundary of $K$, the vector field belongs to the tangent cone to $K$; see also \cite[Theorem 1.2.1]{aubin.viability}.
This result has been revisited and extended in several directions.
In \cite{yorke1967invariance}, conditions for weak invariance as well as invariance for closed sets are provided -- a result guaranteeing finite-time weak invariance is also presented.
In particular, one result shows that a closed set $K$ is forward invariant for a continuous-time system with unique solutions if and only if the vector field and its negative version are subtangential to $K$ at each point in it.
A similar result is known as the Bony-Brezis theorem, which, instead of involving a condition on the tangent vectors, requires the vector field to have a nonpositive inner product with any (exterior) normal vector to the set $K$ \cite{Bony.69.AlF, Brezis.70.CPAM}.
Taking advantage of convexity and linearity of the objects considered, \cite{bitsoris1988positive} provides necessary and sufficient conditions for forward invariance of convex polyhedral sets for linear time-invariant discrete-time systems.
Essentially, conditions in \cite{bitsoris1988positive} require that the new value of the state after every iteration belongs to the set that is to be rendered forward invariant.
This condition can be interpreted as the discrete-time counterpart of the condition in the Nagumo Theorem.
For the case of time-varying continuous-time systems, \cite{fernandes1987remarks} provides conditions guaranteeing forward invariance properties of $K$ given by a sublevel set of a Lyapunov-like function; see also \cite{Zanolin.87, chellaboina1999generalized, gorban2013lyapunov}.
The analysis of forward invariance of a set for systems under the effect of perturbations has also been studied in the literature; see \cite{Tarbouriech.Burgat.94.TAC} for the case when $K$ is a cone, \cite{Castelan.Hennet.91.CDC, Milani.Dorea.96.Automatica} when $K$ is a polyhedral.
The survey article \cite{blanchini1999set} and the book \cite{aubin.viability} summarize these and other analysis results for forward invariance of sets in continuous-time and discrete-time systems.
Thought outside the scope of this paper, the notion of robust controlled forward invariance has also been studied in the literature, see, e.g., \cite{Meyer.ea.16.Automatica, li2016robust, sadraddini2016provably}.
The study of forward invariance in systems that combine continuous and discrete dynamics is not as mature as the continuous-time and discrete-time settings.
Certainly, when the continuous dynamics are discretized, the methods for purely discrete-time systems mentioned above are applicable or can be extended without significant effort for certain classes of hybrid models in discrete time; see, in particular, the results for a class of piecewise affine discrete-time systems in \cite{rakovic2004computation}.
Establishing forward invariance (both nominal and robust) is much more involved when the continuous dynamics are not discretized.
Forward invariance of sets for impulsive differential inclusions, which are a class of hybrid systems without disturbances, are established in \cite{aubin.hs.viability}.
In particular, \cite{aubin.hs.viability} proposes conditions to guarantee (weak -- or viability -- and strong) forward invariance of closed sets and a numerical algorithm to generate invariant kernels.
For hybrid systems modeled as hybrid automata, safety specification is often recast as a forward invariance problem.
In such context, several computational approaches have been proposed for hybrid automata with nonlinear continuous dynamics, disturbances, and control inputs.
In \cite{tomlin2003computational}, a differential game approach is proposed to compute reachable sets for the verification of safety in a class of hybrid automata.
In \cite{platzer2008computing}, an algorithm is proposed to approximate invariant sets of hybrid systems that have continuous dynamics with polynomial right-hand-side and that can be written as hybrid programs.
\vspace{-0.1in}
\subsection{Contributions}
\vspace{-0.02in}
Motivated by the lack of results for the study of robust and nominal forward invariance in hybrid systems, we propose tools for analyzing forward invariance properties of sets.
In particular, formal notions of invariance and solution-independent conditions that guarantee desired invariance properties of sets are established for hybrid dynamical systems modeled as
\begin{align}\label{eq:Hw}
\HS\w \begin{cases}
(x, w_c) \in C\w & \dot{x} \ \ \in F\w(x, w_c)\\
(x, w_d) \in D\w & x^+ \in G\w(x, w_d)
\end{cases}
\end{align}
which we refer to as hybrid inclusions \cite{65} and where $x$ is the state and $w = (w_c, w_d)$ is the disturbance; see Section~\ref{sec:Hw} for a precise definition.
In the upcoming second part of this work, tools for the design of invariance-inducing controllers for hybrid system with disturbances are proposed based on the results presented in this paper.
The main challenges in asserting such forward invariance properties of a set $K$, subset of the state space, include the following:
\begin{enumerate}[leftmargin = 5mm]
\item {\it Combined continuous and discrete dynamics}:
given a disturbance signal and an initial state value, a solution to \eqref{eq:Hw} may evolve continuously for some time, while at certain instances, jump. As a consequence, the set $K$ must have the property that solutions stay in it when either the continuous or the discrete dynamics are active.
\item {\it Potential nonuniqueness and noncompleteness of solutions}:
the fact that the dynamics of \eqref{eq:Hw} are set valued and the existence of states from where flows and jumps are both allowed (namely, the state components of $C\w$ and $D\w$ may have a nonempty overlap with points from where flows are possible) lead to nonunique solutions.
In particular, at points in $K$ where both flows and jumps are allowed, conditions for invariance during flows and at jumps need to be enforced simultaneously.
\item {\it Presence of disturbances for systems with state constraints}:
for it to be interesting, forward invariance of a set $K$ for a hybrid system with disturbances
is an invariance property that has to hold for all possible disturbances.
In technical terms, for every $x$ such that $(x, w_c)$ belongs to $C\w$, the vectors in the set $F\w(x, w_c)$ need to be in directions that flow outside of $K$ is impossible for all values of $w_c$.
Similarly, for each $x$ such that $(x, w_d)$ belongs to $D\w$, the set $G\w(x, w_d)$ should be contained in $K$ regardless of the values of $w_d$.
\item {\it Forward invariance analysis of intersection of sets:} when provided a Lyapunov-like function, $V$, for the given system, conditions to guarantee forward invariance properties will need to take advantage of the nonincreasing property of $V$.
In such a case, the state component of the sets $C\w$ and $D\w$ will be intersected by sublevel sets of the given Lyapunov-like function, which require less restrictive conditions than for general sets.
\end{enumerate}
In this paper, we provide results that help tackle these key issues systematically.
For starters, we present a result to guarantee existence of nontrivial solutions to the system modeled as in \eqref{eq:Hw}, which also provides insights for solution behaviors based on completeness.
Then, we introduce the notions and sufficient conditions for forward invariance in hybrid dynamical systems.
The proposed notions of robust forward invariance are uniform over all possible disturbances, and allow for solutions to be nonunique and to cease to exist in finite (hybrid) time (namely, not complete).
For each notion, we propose sufficient conditions that the data of the hybrid inclusions and the set $K$ should satisfy to render $K$ robustly forward invariant.
Results for hybrid systems without disturbances are derived as special cases of the robust ones.
Compared to \cite{aubin.hs.viability}, which studies the nominal systems exclusively, we focus on a more general family of hybrid systems, for example, we do not always insist on the flow map to be Marchaud and Lipschitz; see, e.g., \cite{aubin.viability, aubin.hs.viability, aubin2009set}.
As an application of the results for generic sets $K$, we present a novel approach to verify forward invariance of a sublevel set of a given Lyapunov-like function intersected with the sets where continuous or discrete dynamics are allowed.
Such a result lays the groundwork for the second part of this paper.
Because of the nonincreasing properties of the given Lyapunov-like function along solutions, the developed conditions are less restrictive and more constructive when compared to the ones for a generic set $K$.
Moreover, our results are also insightful for systems with purely continuous-time or discrete-time dynamics.
In fact, because of the generality of the hybrid inclusions framework, the results in this paper are applicable to broader classes of systems, such as those studied in \cite{fernandes1987remarks,blanchini1999set,aubin.viability,yorke1967invariance, bitsoris1988positive, hu2003composite}.
\vspace{-0.2in}
\subsection{Organization and Notation}
The remainder of the paper is organized as follows.\footnote{Preliminary version of the results in this paper appeared without proof in the conference articles \cite{149} and \cite{120}. This work considers a more general class of disturbance signals than \cite{149}. Some conditions from \cite{149} to guarantee invariance are further relaxed in this paper. In addition, compared to \cite{120}, this work includes results to verify forward invariance of sublevel sets of Lyapunov-like functions.
}
Robust and nominal forward invariance are formally defined in Section~\ref{sec:HwNotions}.
Sufficient conditions to guarantee nominal and robust notions are presented in Section~\ref{sec:HSuffConds} and Section~\ref{sec:HwSuffConds}, respectively.
In Section~\ref{sec:HwLya}, a Lyapunov-like function is used to ensure forward invariance of sublevel sets.
An application of our results for the analysis of a controlled inverter system is presented in Section~\ref{sec:inverter}.
Academic examples are provided to illustrate major results.
\noindent{\bf Notation:}
Given a set-valued mapping $M : \reals^m \rightrightarrows \reals^n$, we denote the range of $M$ as $\rge M = \{ y \in \reals^n: y \in M(x), x \in \reals^m \}$ and the domain of $M$ as $\dom M = \{ x \in \reals^m : M(x) \neq \emptyset \}$.
The closed unit ball around the origin in $\reals^n$ is denoted as $\ball$.
Given $r \in \reals$, the $r$-sublevel set of a function $V: \reals^n \rightarrow \reals$ is $L_\Ly(\level): = \{x \in \reals^n: V(x) \leq r \}$, and $V^{-1}(r) = \{x \in \reals^n: V(x) = r \}$ denotes the $r$-level set of $V$.
Given a vector $x, |x|$ denotes the 2-norm of $x$.
We use $|x|_K$ to denote the distance from point $x$ to a closed set $K$, i.e., $|x|_K = \inf\limits_{\xi \in K} |x - \xi|.$
The closure of the set $K$ is denoted as $\overline{K}$.
The set of boundary points of a set $K$ is denoted by $\partial K$ and the set of interior points of $K$ is denoted by $\text{\normalfont int} K$.
Given vectors $x$ and $y$, $(x, y)$ is equivalent to $[x^\top \ y^\top]^\top$.
\subsection{Sufficient Conditions for Nominal Forward Invariance Properties for $\HS$}
\label{sec:HSuffConds}
\ConfSp{-1mm}
We present the sufficient conditions for forward invariance of a given set $K$ for $\HS$ that involve the data $(C, F, D, G)$.
For the discrete dynamics, namely, the jumps, such conditions involve the understanding of where $G$ maps the state to.
Inspired by the well-known Nagumo Theorem \cite{nagumo1942lage}, for the continuous dynamics, namely, the flows, our conditions use the concept of tangent cone to the closed set $K$.
The tangent cone at a point $x \in \reals^n$ of a closed set $K \subset \reals^n$ is defined using the Dini derivative of the distance to the set, and is given by\footnote{In other words, $\omega$ belongs to $T_K(x)$ if and only if there exist sequences $\tau_i \searrow 0$ and $\omega_i \rightarrow \omega$ such that $x + \tau_i \omega_i \in K$ for all $i \in \nats$; see also \cite[Definition 1.1.3]{aubin.viability}. The latter property is further equivalent to the existence of sequences $x_i \in K$ and $\tau_i > 0$ with $x_i \rightarrow x, \tau_i \searrow 0$ such that $\omega = \lim_{i\to\infty} (x_i - x)/\tau_i$.}
\ConfSp{-4mm}
\begin{align}\label{eq:tangent}
T_K(x) = \left\{ \omega \in \reals^n : \liminf\limits_{\tau \searrow 0} \frac{|x+\tau \omega|_K}{\tau} = 0 \right\}.
\end{align}
In the literature (see, e.g., \cite[Definition 4.6]{jahn1994tangent} and \cite{aubin.hs.viability}), this tangent cone is also known as the sequential Bouligand tangent cone or contingent cone.
In contrast to the Clarke tangent cone introduced in \cite[Remark 4.7]{jahn1994tangent}, which is always a closed convex cone for every $x \in K$, the tangent cone (possibly nonconvex) we consider in this work includes all vectors that point inward to the set $K$ or that are tangent to the boundary of $K$.\footnote{Note that, for a convex set, the Bouligand tangent cone coincides with the Clarke tangent cone.}
Our sufficient conditions for forward invariance require part of the data of $\HS$ and the set $K$ to satisfy the following mild assumption.
\begin{assumption}\label{assum:data}
The sets $K, C$, and $D$ are such that $K \subset \overline{C}\cup D$ and that $K \cap C$ is closed.
The map $F:\reals^n \rightrightarrows \reals^n$ is outer semicontinuous, locally bounded relative to $K \cap C$, and $F(x)$ is nonempty and convex for every $x \in K \cap C$.
\end{assumption}
The following result is a consequence of the forthcoming Theorem~\ref{thm:rwFI} in Section~\ref{sec:HwSuffConds} and its proof will be delayed to that section.
Sufficient conditions for a given set $K$ to be weakly forward pre-invariant and weakly forward invariant are presented.
\begin{theorem}(nominal weak forward pre-invariance and weak forward invariance)\label{thm:wfi}
Given $\HS= (C, F, D, G)$ as in \eqref{eq:H} and a set $K$, suppose $K, C, D,$ and $F$ satisfy Assumption~\ref{assum:data}.
The set $K$ is weakly forward pre-invariant for $\HS$ if the following conditions hold:
\begin{enumerate}[label = \ref{thm:wfi}.\arabic*), leftmargin = 12mm]
\item\label{item:wfi1} For every $x \in K \cap D$, $G(x) \cap K \neq \emptyset$;
\item\label{item:wfi2} For every $x \in \chai{\widehat{C}} \setminus D$, $F(x) \cap T_{K \cap C}(x) \neq \emptyset$;
\end{enumerate}
where $\chai{\widehat{C}} : = \partial(K \cap C) \setminus L$ and $L : = \{x \in \partial C: F(x) \cap T_{\overline{C}}(x) = \emptyset \}$.
Moreover, $K$ is weakly forward invariant for $\HS$ if, in addition\chai{, $K \cap L \subset D$ and, with $K^\star = K \setminus D$,}
\begin{enumerate}[label = N$\star$), resume, leftmargin = 9mm]
\item\label{item:N*} for every $\phi \in \sol_{\HS}( \chai{K^\star})$ with $\rge \phi \subset K$,
case \ref{item:b.2n} in Proposition~\ref{prop:existence} does not hold.
\end{enumerate}
\end{theorem}
\ConfSp{-1mm}
The next result, which is a consequence of Theorem~\ref{thm:rFI} provides sufficient conditions for a set $K$ to be forward pre-invariant and forward invariant for $\HS$.
\begin{theorem}(nominal forward pre-invariance and forward invariance)\label{thm:fi}
Given $\HS= (C, F, D, G)$ as in \eqref{eq:H} and a set $K \subset \reals^n$. Suppose $K, C, D,$ and $F$ satisfy Assumption~\ref{assum:data} and that $F$ is locally Lipschitz on $C$.
Let $\widehat{C}$ and $L$ be given as in Theorem~\ref{thm:wfi}.
The set $K$ is forward pre-invariant for $\HS$ if the following conditions hold:
\begin{enumerate}[label = \ref{thm:fi}.\arabic*), leftmargin=12mm]
\item\label{item:fi1} $G(K \cap D) \subset K$;
\item\label{item:fi2} For every $x \in \widehat{C}$, $F(x) \subset T_{K \cap C}(x)$.
\end{enumerate}
Moreover, $K$ is forward invariant for $\HS$ if, in addition, $K \cap L \subset D$ and, with $K^\star = K\cap C$, item \ref{item:N*} in Theorem~\ref{thm:wfi} holds.
\end{theorem}
\chai{
\begin{remark}
Some of the conditions in Theorem~\ref{thm:wfi} and Theorem~\ref{thm:fi} are weaker than those required by results in \cite{120}.
The construction of the set $L$ in items \ref{item:wfi2} and \ref{item:fi2} is inspired by the {\em viability domain} in \cite[Definition 1.1.5]{aubin.viability}.
Note that when \ref{item:N*} holds, completeness of maximal solutions is guaranteed by ensuring that
$K \cap L \subset D$, which guarantees that solutions can continue to evolve from $L$ via a jump.
\end{remark}
}
The following example is used to illustrate Theorem~\ref{thm:wfi} and Theorem~\ref{thm:fi}.
\begin{example}[solutions with finite escape time]\label{ex:pre}
Consider the hybrid system $\HS = (C, F, D, G)$ in $\reals^2$ with system data given by
\IfConf{
$F(x) :=(1 + x_1^2, 0)$ for every $x \in C:= \{ x \in \reals^2: x_1 \in [0, \infty), x_2 \in [-1,1] \}$ and $G(x) := (x_1+ \ball, x_2)$ for every $x \in D:= \{ x \in \reals^2: x_1 \in [0, \infty), x_2 = 0 \}.$
}{
\begin{align*}
& F(x) := \begin{bmatrix} 1 + x_1^2\\ 0 \end{bmatrix}
& \forall x\in C := \{ x \in \reals^2: x_1 \in [0, \infty), x_2 \in [-1,1] \},\\
& G(x) := \begin{bmatrix} x_1+ \ball \\ x_2 \end{bmatrix}
& \forall x\in D := \{ x \in \reals^2: x_1 \in [0, \infty), x_2 = 0 \}.
\end{align*}
}
Let $K = C$ and note the following properties of maximal solutions to $\HS$:
\begin{itemize}[leftmargin = 4mm]
\item For some $x \in K$, there exists $\phi = (\phi_1, \phi_2) \in \sol_\HS(x)$ with $\rge \phi \subset K$, but is not complete due to $\lim\limits_{t \searrow t^*}\phi_1(t,0) = \infty$ with $t^* <\infty$; for instance, from $x = (0, 1)$, the solution given by $\phi(t,0) = (\tan(t), 1)$ for every $(t,0) \in \dom \phi$ has its $\phi_1$ component escape to infinite as $t$ approaches $t^* = \pi/2$;
\item From points in $D \subset K$, there exist maximal solutions that leave $K$ and are not complete: such solutions end after a jump because their $x_1$ component is mapped outside of $K$.
\end{itemize}
\begin{figure}[h]
\centering
\includegraphics[width=\figsize]{pre_FI.eps}
\small
\put(-0.03,.33){$x_1$}
\put(-1.01,.65){$x_2$}
\put(-.98,.53){1}
\put(-.98,.05){-1}
\put(-.98,.26){0}
\put(-.7,.17){$C$}
\put(-.36,.35){$D$}
\put(-1,.35){$x_v$}
\put(-.83,.32){$T_{K\cap C}(x_v)$}
\put(-.62,.58){$x_u$}
\put(-.5,.5){$T_{K\cap C}(x_u)$}
\put(-.47,.03){$x_l$}
\put(-.35,.1){$T_{K\cap C}(x_l)$}
\caption{Flow and jump sets of the system in Example~\ref{ex:pre}.}
\label{fig:propwppfi}
\end{figure}
Thus, we verify weak forward pre-invariance of $K$ by applying Theorem~\ref{thm:wfi}.
The sets $K, C, D$ and the map $F$ satisfy Assumption~\ref{assum:data} by construction and condition \ref{item:wfi1} holds for $\HS$ by definition of $G, D$ and $K$.
Since $L = \emptyset$, condition \ref{item:wfi2} holds because for every $x \in \widehat{C}$, $F(x)$ points horizontally, and
\IfConf{
$$T_{K \cap C} (x) = \begin{cases}
\reals \times \reals_{\leq 0} & \text{\normalfont if } x_1 \in (0, \infty), x_2 = 1\\
\reals \times \reals_{\geq 0} & \text{\normalfont if } x_1 \in (0, \infty), x_2 = -1\\
\reals_{\geq 0} \times \reals_{\leq 0} &\text{\normalfont if } x = (0,1)\\
\reals_{\geq 0} \times \reals &\text{\normalfont if } x_1 = 0, x_2 \in (-1,1)\\
\reals_{\geq 0} \times \reals_{\geq 0} & \text{\normalfont if } x =(0,-1).
\end{cases}$$
}{
$$T_{K \cap C} (x) = \begin{cases}
\reals \times \reals_{\leq 0} & \text{\normalfont if } x \in \{ x \in \reals^2: x_1 \in (0, \infty), x_2 = 1\}\\
\reals \times \reals_{\geq 0} & \text{\normalfont if } x \in \{ x \in \reals^2: x_1 \in (0, \infty), x_2 = -1\}\\
\reals_{\geq 0} \times \reals_{\leq 0} &\text{\normalfont if } x = (0,1)\\
\reals_{\geq 0} \times \reals &\text{\normalfont if } x \in \{ x \in \reals^2: x_1 = 0, x_2 \in (-1,1)\}\\
\reals_{\geq 0} \times \reals_{\geq 0} & \text{\normalfont if } x =(0,-1).
\end{cases}$$
}
\noindent Tangent cones of $K \cap C$ at points $x_u, x_v$ and $x_l$ of $K$ are shown in Figure~\ref{fig:propwppfi}.
Now, consider the same data but with $G$ replaced by $G'(x) = G(x) \cap (\reals_{\geq 0} \times \reals)$ for each $x \in D$.
The set $K = C$ is forward pre-invariant for this system.
This is because maximal solutions are not able to jump out of $K$ as $G'$ only maps $x_1$ components of solutions to $[0, +\infty)$.
More precisely, the conditions in Theorem~\ref{thm:fi} hold: we have $G'(D \cap K) \subset K$, and Assumption~\ref{assum:data} and condition \ref{item:fi2} hold as discussed above.
\hfill $\triangle$
\end{example}
\begin{remark}
The hybrid inclusions framework allows for an overlap between the flow set $C$ and the jump set $D$.
As a result, the proposed conditions are not necessary to induce forward invariance properties of sets for $\HS$.
When existence of nontrivial solutions and completeness are not required for every point in $K$, as in the ``pre'' notions, some of these conditions are necessary.
In fact, suppose $K,C,D$, and $F$ satisfy Assumption~\ref{assum:data}:
\begin{itemize}[leftmargin = 4mm]
\item If $K$ is weakly forward pre-invariant for $\HS$, then for every
$x \in (K \cap D)\setminus C$, $G(x) \cap K \neq \emptyset.$
\item If $K$ is forward pre-invariant or forward invariant for $\HS$, then condition \ref{item:fi1} in Theorem~\ref{thm:fi} holds.
\item If $K$ is weakly forward invariant or forward invariant for $\HS$, then for every $x \in K \setminus D$, $F(x) \cap T_{K \cap C}(x) \neq \emptyset$.\footnote{A similar claim is presented in \cite[Proposition 3.4.1]{aubin.viability} for continuous-time system.}
\end{itemize}
Moreover, unlike \cite[Theorem 3]{aubin.hs.viability}, when the flow map $F$ is Marchaud\footnote{A map $F$ is Marchaud on $K \cap C$ when Assumption~\ref{assum:data} holds and $F$ has linear growth on $K \cap C$; see \cite[Definition 2.2.4]{aubin.viability}.} and Lipschitz as defined in Definition~\ref{def:lip}, condition $F(x) \subset T_{K \cap C}(x)$ for every $x \in K \setminus D$ is not necessary as the following example shows.
Consider $\HS$ in \eqref{eq:H} with data
$F(x) =
\begin{cases}
1 & \text{\normalfont if } x > -1\\
[-1,1] & \text{\normalfont if } x = -1
\end{cases}$
for each $x \in C : = [-1, 1]$, $G(x) := \{-1, 0\}$ for each $x \in D: = \{1\}$.
By inspection, the set $K = C$ is forward invariant for $\HS$ and $F$ is Marchaud and Lipschitz.
However, at $x = -1 \in K\setminus D$, $F(-1) \supset -1$ but $-1 \notin T_{K \cap C}(-1)$.
\end{remark}
\ConfSp{-4mm}
\subsection{Sufficient Conditions for N$\star$)}\label{sec:nstar}
In Theorem~\ref{thm:wfi} and Theorem~\ref{thm:fi}, item \ref{item:N*} excludes case \ref{item:b.2} in Proposition~\ref{prop:existence}, where solutions escape to infinity in finite time during flows.
In fact, when every solution $\phi$ to $\dot{x} \in F(x)$ with $\phi(0,0) \in K^\star$ does not have a finite escape time, namely, there does not exist $t^* < \infty$ such that $\lim\limits_{t \searrow t^*} |\phi(t)| = \infty$, item \ref{item:N*} holds for $\HS$ and $K^\star$ as defined in Theorem~\ref{thm:wfi} and Theorem~\ref{thm:fi}, respectively.
Although, in principle, such a condition is solution dependent, it can be guaranteed without solving for solutions when $F$ is single valued and globally Lipschitz.
Moreover, we provide several other alternatives in the next result.
\begin{lemma}(sufficient conditions for completeness)\label{lem:Nstar}
Given $\HS = (C, F, D, G)$ and a set $K\subset \reals^n$, suppose $K, C, D,$ and $F$ satisfy Assumption~\ref{assum:data}, \chai{set $D$ is closed} and item
\ref{item:wfi2} in Theorem~\ref{thm:wfi} holds.\footnote{When \ref{item:Nstar1} holds, condition \ref{item:wfi2} in Theorem~\ref{thm:wfi} is not required.}
Condition \ref{item:N*} holds if
\begin{enumerate}[label = \ref{lem:Nstar}.\arabic*), leftmargin=2.8\parindent]
\item\label{item:Nstar1} $K^\star$ is compact; or
\item\label{item:Nstar3} $F$ has linear growth on $K^\star$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\phi \in \sol_{\HS}(K^\star)$ with $\rge \phi \subset K$ be as described by \ref{item:b.2n} in Proposition~\ref{prop:existence}; namely, $t \mapsto \phi(t,J)$ defined on $I^J$, where $(T,J) = \sup \dom \phi, T+ J < \infty$ and, for some $t^J, I^J = [t_J, T)$.
Since $t \mapsto \phi(t,J)$ is locally absolutely continuous on $I^J$, $\lim\limits_{t \rightarrow T} \phi(t,J)$ is finite or infinity.
If it is finite, then $t \mapsto \phi(t,j)$ can be extended to $\overline{I^J}$, which contradicts with \ref{item:b.2n}.
Then, it has to be that $\lim\limits_{t \rightarrow T} \phi(t,J)$ is infinity.
When \ref{item:Nstar1} holds, $\lim\limits_{t \rightarrow T} \phi(t,J)$ being infinity is a contradiction since $K^\star$ is compact.
If \ref{item:Nstar3} holds, \cite[Lemma 1]{aubin.hs.viability}\footnote{The sets $X, K, C$ in \cite[Lemma 1]{aubin.hs.viability} are $C$, $K \cap C$, \chai{$D$ (or $\emptyset$ when applied for Theorem~\ref{thm:fi})}, respectively, in our context.}
implies that $t \mapsto \phi(t,J)$ is either:
\begin{itemize}[leftmargin = 4mm]
\item defined over $[t_J, \infty)$ with values in $K\cap C$;\footnote{\chai{This is the only case that applies for Theorem~\ref{thm:fi}.}} or
\item defined over $[t_J, T]$ with $T \geq t_J$, $\phi(T, J) \in D$ and $\phi(t, J) \in K \cap C$ for every $t \in [t_J,T]$.
\end{itemize}
In either case, we have a contradiction.
\end{proof}
The next example illustrates Theorem~\ref{thm:wfi}, Theorem~\ref{thm:fi} and Lemma~\ref{lem:Nstar}.
\begin{example}(weakly forward invariant set)\label{ex:wFI}
Consider the hybrid system $\HS = (C, F, D, G)$ in $\reals^2$ given by
\begin{align*}
&F(x):= (x_2, -x_1) & \forall x \in C;\\
&G(x):= (-0.9x_1, x_2) &\forall x \in D,
\end{align*}
where $C := \{x \in \reals^2: |x| \leq 1, x_2 \geq 0\}$ and
$D:= \{x \in \reals^2: x_1 \geq -1, x_2 = 0\}.$
\ConfSp{-3mm}
\begin{figure}[H]
\centering
\includegraphics[width=\figsize]{wFI.eps}
\small
\put(-0.05,.12){$x_1$}
\put(-.62,.5){$x_2$}
\put(-.26,.05){1}
\put(-.93,.05){-1}
\put(-.6,.05){0}
\put(-.7,.3){$C$}
\put(-.45,.03){$D$}
\caption{Sets and directions of flows/jumps in Example~\ref{ex:wFI}.}
\label{fig:wFI}
\end{figure}
The set $K_1 = \partial C$ is weakly forward invariant for $\HS$ by Theorem~\ref{thm:wfi}.
More precisely, for every $x \in K_1 \cap D$, $G(x) \in K_1$; and for every $x \in \partial (K_1 \cap C) \setminus (D \cup L) =$ $\{x \in \reals^2: |x| = 1, x_2 > 0\}$, since $\inner{\nabla( x^2_1 + x^2_2 - 1)}{F(x)} = 0$, applying item \ref{item:inner1} in Lemma~\ref{lem:innerTcone}, we have $F(x) \in T_{K_1 \cap C}(x)$.
In addition, $K_1 \cap C = \partial C$ is compact, which implies condition \ref{item:N*} holds by Lemma~\ref{lem:star}.
Thus, for every $x \in K_1$, there exists one complete solution that stays in $K_1$.
For example, for every $x \in [-1,1] \times \{0\}$, there exists one complete solution that is discrete and stays in $K_1$ (from the origin there is also a complete continuous solution that remains at the origin), but also there exist maximal solutions that flow inside $\{x \in \reals^2: |x| < 1 \}$ and leave $K_1$.
\chai{Now consider $K_2 = C$. It is forward invariant for $\HS$ by applying Theorem~\ref{thm:fi}.
In fact, using the observations above, item \ref{item:fi2} can be verified via Lemma~\ref{lem:innerTcone} since $\inner{\nabla( x^2_1 + x^2_2 - 1)}{F(x)} = 0$ for every $x \in \partial (K_2 \cap C) \setminus L = \{x \in \reals^2: |x| = 1, x_2 > 0\} \cup ([-1,0]\times \{0\})$.}
\hfill $\triangle$
\end{example}
Note that one can replace condition \ref{item:fi2} in Theorem~\ref{thm:fi} by
\begin{enumerate}[label = \ref{thm:fi}.2$^{'}$), leftmargin= 12mm]
\item\label{item:fi2'} For every $x \in \partial (K \cap C),$
\end{enumerate}
\begin{align}
& F(x) \subset T_{K \cap C}(x) &\text{\normalfont if } &x \notin \partial C \cap D\label{eq:fi2''}\\
&F(x) \cap (T_C(x) \setminus T_{K \cap C}(x)) = \emptyset &\text{\normalfont if } &x \in \partial C \cap D. \label{eq:fi2''s}
\end{align}
Note that assumption \eqref{eq:fi2''s} is important as in some cases, having item \eqref{eq:fi2''} only leads to solutions that escape the set $K$ by flowing as shown in the following example.
Consider the hybrid system $\HS$ on $\reals^2$ with
\begin{align*}
&F(x) =(x_2, -\gamma) & \forall x &\in C := \{x \in \reals^2: x_1x_2 \geq 0 \}\\
&G(x) = x & \forall x &\in D := \{x \in \reals^2: x_1 \geq 0, x_2 = 0 \},
\end{align*}
where $\gamma > 0$.
The set $K = \{x \in \reals^2: x_1 \geq 0, x_2 \geq 0 \}$ is weakly forward invariant, and the sets $K, C, D$ and the map $F$ satisfies \eqref{eq:fi2''}.
However, at the origin, we have $F(0) = (0, -\gamma)$ and
\ConfSp{-1mm}
\begin{align*}
&T_C(0) = (\reals_{\geq 0} \times \reals_{\geq 0} ) \cup (\reals_{\leq 0} \times \reals_{\leq 0}),\\
&T_{K \cap C}(0) = \reals_{\geq 0} \times \reals_{\geq 0}.
\end{align*}
\ConfSp{-5mm}
\noindent Hence, at the origin, one solution can flow into $C \setminus K$ (the third quadrant) because $F(0) \in T_C(0) \setminus T_{K \cap C}(0)$.
The following example is an application of Theorem~\ref{thm:fi} and Lemma~\ref{lem:Nstar}.
\begin{example}(forward invariant set)\label{ex:oscillator0w}
Consider the hybrid system given by
\IfConf{
\begin{align}\label{eq:ex_zerod}
\HS \begin{cases}
x \in C &\dot{x} = F(x) := (- |x_1|x_2, 0)\\
x \in D & x^+ = G(x) := x,
\end{cases}
\end{align}
}{
\begin{align}\label{eq:ex_zerod}
\HS \begin{cases}
x \in C &\dot{x} = F(x) := \begin{bmatrix}- |x_1|x_2 \\ 0
\end{bmatrix} \\
x \in D & x^+ = G(x) := x,
\end{cases}
\end{align}
}
where the flow set is $C = \{x \in \reals^2: |x| \leq 1, x_1x_2 \geq 0 \}$ and the jump set is $D = \{x \in \reals^2: |x| \leq 1, x_1x_2 \leq 0 \}$.
We observe that during flow, solutions evolve continuously within the unit circle centered at the origin; while at jumps, solutions remain at the current location.
Applying Theorem~\ref{thm:fi}, we show that the set $K_1 = C_1 \cup D_1$ is forward invariant for $\HS,$ where $C_1 = \{x \in \reals^2: x_1 \geq 0, x_2 \geq 0, |x| \leq 1\}$ and $D_1 = \{x \in \reals^2: x_1 \leq 0, x_2 \geq 0, |x| \leq 1\}$.
By construction, $K_1, C, D$ and $F$ satisfy Assumption~\ref{assum:data}.
Condition \ref{item:fi1} holds since $G$ maps the state to its current value.
Condition \ref{item:fi2} holds since
\begin{itemize}[leftmargin = 4mm]
\item for every $x \in \{x \in \partial C_1: |x| =1 \}$, since $x_1 x_2 \geq 0$
\IfConf{$\inner{\nabla (x_1^2 + x_2^2)}{F(x)} = -2|x_1|x_1x_2 \leq 0;$
}{
\begin{align*}
\inner{\nabla (x_1^2 + x_2^2)}{F(x)} = -2|x_1|x_1x_2 \leq 0;
\end{align*}
}
\item for every $x \in \{x \in \partial C_1: |x| \neq 1 \}$, $F(x) = (0,0)$,
which leads to $F(x) \in T_{K_1 \cap C}(x)$.
\end{itemize}
Finally, applying Lemma~\ref{lem:Nstar}, \ref{item:N*} holds since $K_1 \cap C$ is compact.
\hfill $\triangle$
\end{example}
In Example~\ref{ex:oscillator0w}, the set $K_1$ is forward invariant for $\HS$ as shown therein.
When arbitrarily small disturbances are introduced, solution pairs may escape the set of interest.
In particular, we revisit Example~\ref{ex:oscillator0w} with disturbances in the next section, and show that $K_1$ is only weakly forward invariant, uniformly in the given disturbances.
The forthcoming results in Section~\ref{sec:HwSuffConds} are useful to verify such properties of sets for hybrid systems $\HS\w$ given as in \eqref{eq:Hw}.
\subsection{Sufficient Conditions for Nominal Forward Invariance Properties for $\HS$}
\label{sec:HSuffConds}
We present the sufficient conditions for forward invariance of a given set $K$ for $\HS$ that involve the data $(C, F, D, G)$.
For the discrete dynamics, namely, the jumps, such conditions involve the understanding of where $G$ maps the state to.
Inspired by the well-known Nagumo Theorem \cite{nagumo1942lage}, for the continuous dynamics, namely, the flows, our conditions use the concept of tangent cone to the closed set $K$.
The tangent cone at a point $x \in \reals^n$ of a closed set $K \subset \reals^n$ is defined using the Dini derivative of the distance to the set, and is given by\footnote{In other words, $\omega$ belongs to $T_K(x)$ if and only if there exist sequences $\tau_i \searrow 0$ and $\omega_i \rightarrow \omega$ such that $x + \tau_i \omega_i \in K$ for all $i \in \nats$; see also \cite[Definition 1.1.3]{aubin.viability}. The latter property is further equivalent to the existence of sequences $x_i \in K$ and $\tau_i > 0$ with $x_i \rightarrow x, \tau_i \searrow 0$ such that $\omega = \lim_{i\to\infty} (x_i - x)/\tau_i$.}
\ConfSp{-5mm}
\begin{align}\label{eq:tangent}
T_K(x) = \left\{ \omega \in \reals^n : \liminf\limits_{\tau \searrow 0} \frac{|x+\tau \omega|_K}{\tau} = 0 \right\}.
\end{align}
In the literature (see, e.g., \cite[Definition 4.6]{jahn1994tangent} and \cite{aubin.hs.viability}), this tangent cone is also known as the sequential Bouligand tangent cone or contingent cone.
In contrast to the Clarke tangent cone introduced in \cite[Remark 4.7]{jahn1994tangent}, which is always a closed convex cone for every $x \in K$, the tangent cone (possibly nonconvex) we consider in this work includes all vectors that point inward to the set $K$ or that are tangent to the boundary of $K$.\footnote{Note that, for a convex set, the Bouligand tangent cone coincides with the Clarke tangent cone.}
Our sufficient conditions for forward invariance require part of the data of $\HS$ and the set $K$ to satisfy the following mild assumption.
\begin{assumption}\label{assum:data}
The sets $K, C$, and $D$ are such that $K \subset \overline{C}\cup D$ and that $K \cap C$ is closed \chai{$C$ is closed}.
The map $F: \reals^n \rightrightarrows \reals^n$ is outer semicontinuous, locally bounded relative to $K \cap C$, and $F(x)$ is nonempty and convex for every $x \in K \cap C$.
\end{assumption}
\chai{
In addition, for ease of representation, we denote the set that collects all points in the flow set that \ref{item:VC} does not hold as
$L : = \{x \in C: F(x) \cap T_C(x)\}$.
}
The following result is a consequence of the forthcoming Theorem~\ref{thm:rwFI} in Section~\ref{sec:HwSuffConds} and its proof will be delayed to that section.
Sufficient conditions for a given set $K$ to be weakly forward pre-invariant and weakly forward invariant are presented.
\begin{theorem}(nominal weak forward pre-invariance and weak forward invariance)\label{thm:wfi}
Given $\HS= (C, F, D, G)$ as in \eqref{eq:H} and a set $K$, suppose $K, C, D,$ and $F$ satisfy Assumption~\ref{assum:data}.
The set $K$ is weakly forward pre-invariant for $\HS$ if the following conditions hold:
\begin{enumerate}[label = \ref{thm:wfi}.\arabic*), leftmargin=2.8\parindent]
\item\label{item:wfi1} For every $x \in K \cap D$, $G(x) \cap K \neq \emptyset$;
\item\label{item:wfi2} For every $\chai{x \in \widehat{C} : = K \setminus (L\cup D)}$, $F(x) \cap T_{K \cap C}(x) \neq \emptyset$.
\end{enumerate}
Moreover, $K$ is weakly forward invariant for $\HS$ if, in addition\chai{, the following holds:}
\begin{enumerate}[label = N$\star$\arabic*), leftmargin=2\parindent]
\item\label{item:NL} $K \cap L \subset D$.
\item\label{item:N*} For every \chai{$\phi \in \sol_{\HS}(\widehat{C})$} with $\rge \phi \subset K$, case \ref{item:b.2n} in Proposition~\ref{prop:existence} does not hold.
\end{enumerate}
\end{theorem}
\vspace{-1mm}
The next result, which is a consequence of Theorem~\ref{thm:rFI} in Section~\ref{sec:HwSuffConds} provides sufficient conditions for a set $K$ to be forward pre-invariant and forward invariant for $\HS$.
\begin{theorem}(nominal forward pre-invariance and forward invariance)\label{thm:fi}
Given $\HS= (C, F, D, G)$ as in \eqref{eq:H} and a set $K \subset \reals^n$. Suppose $K, C, D,$ and $F$ satisfy Assumption~\ref{assum:data} and that
$F$ is locally Lipschitz on $C$.
The set $K$ is forward pre-invariant for $\HS$ if the following conditions hold:
\begin{enumerate}[label = \ref{thm:fi}.\arabic*), leftmargin=2.8\parindent]
\item\label{item:fi1} $G(K \cap D) \subset K$;
\item\label{item:fi2} For every \chai{$x \in \widehat{C} : = (K \cap C)\setminus L$}, $F(x) \subset T_{K \cap C}(x)$.
\end{enumerate}
Moreover, $K$ is forward invariant for $\HS$ if, in addition, \chai{\ref{item:NL} and \ref{item:N*} in Theorem~\ref{thm:wfi} hold.}
\end{theorem}
The following example is used to illustrate Theorem~\ref{thm:wfi} and Theorem~\ref{thm:fi}.
\begin{example}[solutions with finite escape time]\label{ex:pre}
Consider the hybrid system $\HS = (C, F, D, G)$ in $\reals^2$ with system data given by
\IfConf{
$F(x) :=(1 + x_1^2, 0)$ for every $x \in C:= \{ x \in \reals^2: x_1 \in [0, \infty), x_2 \in [-1,1] \}$ and $G(x) := (x_1+ \ball, x_2)$ for every $x \in D:= \{ x \in \reals^2: x_1 \in [0, \infty), x_2 = 0 \}.$
}{
\begin{align*}
& F(x) := \begin{bmatrix} 1 + x_1^2\\ 0 \end{bmatrix}
& \forall x\in C := \{ x \in \reals^2: x_1 \in [0, \infty), x_2 \in [-1,1] \},\\
& G(x) := \begin{bmatrix} x_1+ \ball \\ x_2 \end{bmatrix}
& \forall x\in D := \{ x \in \reals^2: x_1 \in [0, \infty), x_2 = 0 \}.
\end{align*}
}
Let $K = C$ and note the following properties of maximal solutions to $\HS$:
\begin{itemize}[leftmargin = 4mm]
\item For some $x \in K$, there exists $\phi = (\phi_1, \phi_2) \in \sol_\HS(x)$ with $\rge \phi \subset K$, but is not complete due to $\lim\limits_{t \searrow t^*}\phi_1(t,0) = \infty$ with $t^* <\infty$; for instance, from $x = (0, 1)$, the solution given by $\phi(t,0) = (\tan(t), 1)$ for every $(t,0) \in \dom \phi$ has its $\phi_1$ component escape to infinite as $t$ approaches $t^* = \pi/2$;\footnote{Note condition \ref{item:N*} does not hold for $\HS$ in Example~\ref{ex:pre}, and the solutions that have a finite escape time are maximal but not complete as described in case \ref{item:b.2n} in Proposition~\ref{prop:existence}.}
\item From points in $D \subset K$, there exist maximal solutions that leave $K$ and are not complete: such maximal solutions end after a jump because their $x_1$ component is mapped outside of $K$.
\end{itemize}
Thus, we verify weak forward pre-invariance of $K$ by applying Theorem~\ref{thm:wfi}.
\vspace{-3mm}
\begin{figure}[h]
\centering
\includegraphics[width=\figsize]{pre_FI.eps}
\small
\put(-0.03,.33){$x_1$}
\put(-1.01,.65){$x_2$}
\put(-.98,.53){1}
\put(-.98,.05){-1}
\put(-.98,.26){0}
\put(-.7,.17){$C$}
\put(-.36,.35){$D$}
\put(-1,.35){$x_v$}
\put(-.83,.32){$T_{K\cap C}(x_v)$}
\put(-.62,.58){$x_u$}
\put(-.5,.5){$T_{K\cap C}(x_u)$}
\put(-.47,.03){$x_l$}
\put(-.35,.1){$T_{K\cap C}(x_l)$}
\caption{Flow and jump sets of the system in Example~\ref{ex:pre}.}
\label{fig:propwppfi}
\end{figure}
\vspace{-2mm}
\noindent The sets $K, C, D$ and the map $F$ satisfy Assumption~\ref{assum:data} by construction and condition \ref{item:wfi1} holds for $\HS$ by definition of $G, D$ and $K$.
Condition \ref{item:wfi2} holds because for every $x \in K \cap C$, $F(x)$ points horizontally, and
\IfConf{
$$T_{K \cap C} (x) = \begin{cases}
\reals \times \reals_{\leq 0} & \text{\normalfont if } x_1 \in (0, \infty), x_2 = 1\\
\reals \times \reals & \text{\normalfont if } x_1 \in (0, \infty), x_2 \in (-1,1)\\
\reals \times \reals_{\geq 0} & \text{\normalfont if } x_1 \in (0, \infty), x_2 = -1\\
\reals_{\geq 0} \times \reals_{\leq 0} &\text{\normalfont if } x = (0,1)\\
\reals_{\geq 0} \times \reals &\text{\normalfont if } x_1 = 0, x_2 \in (-1,1)\\
\reals_{\geq 0} \times \reals_{\geq 0} & \text{\normalfont if } x =(0,-1).
\end{cases}$$
}{
$$T_{K \cap C} (x) = \begin{cases}
\reals \times \reals_{\leq 0} & \text{\normalfont if } x \in \{ x \in \reals^2: x_1 \in (0, \infty), x_2 = 1\}\\
\reals \times \reals & \text{\normalfont if } x \in \{ x \in \reals^2 : x_1 \in (0, \infty), x_2 \in (-1,1)\}\\
\reals \times \reals_{\geq 0} & \text{\normalfont if } x \in \{ x \in \reals^2: x_1 \in (0, \infty), x_2 = -1\}\\
\reals_{\geq 0} \times \reals_{\leq 0} &\text{\normalfont if } x = (0,1)\\
\reals_{\geq 0} \times \reals &\text{\normalfont if } x \in \{ x \in \reals^2: x_1 = 0, x_2 \in (-1,1)\}\\
\reals_{\geq 0} \times \reals_{\geq 0} & \text{\normalfont if } x =(0,-1).
\end{cases}$$
}
Tangent cones of $K \cap C$ at the boundary points $x_u, x_v$ and $x_l$ of $K$ are shown in Figure~\ref{fig:propwppfi}.
Now, consider the same data but with $G$ replaced by $G'(x) = G(x) \cap (\reals_{\geq 0} \times \reals)$ for each $x \in D$.
The set $K = C$ is forward pre-invariant for this system.
This is because maximal solutions are not able to jump out of $K$ as $G'$ only maps $x_1$ components of solutions to $[0, +\infty)$.
More precisely, the conditions in Theorem~\ref{thm:fi} hold: we have $G'(D \cap K) \subset K$, and Assumption~\ref{assum:data} and condition \ref{item:fi2} hold as discussed above.
\hfill $\triangle$
\end{example}
\begin{remark}
The hybrid inclusions framework allows for an overlap between the flow set $C$ and the jump set $D$.
As a result, the proposed conditions are not necessary to induce forward invariance properties of sets for $\HS$.
When existence of nontrivial solutions and completeness are not required for every point in $K$, as in the ``pre'' notions, some of these conditions are necessary.
In fact, suppose $K,C,D$, and $F$ satisfy Assumption~\ref{assum:data}:
\begin{itemize}[leftmargin = 4mm]
\item If $K$ is weakly forward pre-invariant for $\HS$, then for every
$x \in (K \cap D)\setminus C$, we have $G(x) \cap K \neq \emptyset.$
\item If $K$ is forward pre-invariant or forward invariant for $\HS$, then condition \ref{item:fi1} in Theorem~\ref{thm:fi} holds.
\item If $K$ is weakly forward invariant or forward invariant for $\HS$, then condition \ref{item:wfi2} in Theorem~\ref{thm:wfi} holds.\footnote{A similar claim is presented in \cite[Proposition 3.4.1]{aubin.viability} for continuous-time system.}
\end{itemize}
Moreover, unlike \cite[Theorem 3]{aubin.hs.viability}, when the flow map $F$ is Marchaud\footnote{A map $F$ is Marchaud on $K \cap C$ when Assumption~\ref{assum:data} holds and $F$ has linear growth; see \cite[Definition 2.2.4]{aubin.viability}.} and Lipschitz as defined in Definition~\ref{def:lip}, condition \ref{item:fi2} in Theorem~\ref{thm:fi} is not necessary
as the following example shows.
Consider $\HS = (C, F, D, G)$ in \eqref{eq:H} with data
$F(x) =
\begin{cases}
1 & \text{\normalfont if } x > -1\\
[-1,1] & \text{\normalfont if } x = -1
\end{cases}$
for each $x \in C : = [-1, 1]$, $G(x) := \{-1, 0\}$ for each $x \in D: = \{1\}$.
By inspection, the set $K = C$ is forward invariant for $\HS$ and $F$ is Marchaud and Lipschitz.
However, condition \ref{item:fi2} in Theorem~\ref{thm:fi} does not hold for every $x \in K \cap C$ -- it does not even hold on
$(K \cap C)\setminus D$, which are the points where flows are possible.
\end{remark}
\vspace{-4mm}
\subsection{Sufficient Conditions for N$\star$)}\label{sec:nstar}
In Theorem~\ref{thm:wfi} and Theorem~\ref{thm:fi}, item \ref{item:N*} excludes case \ref{item:b.2} in Proposition~\ref{prop:wexistence}, where solutions escape to infinity in finite time during flows.
Hence, when every solution $\psi$ to $\dot{x} \in F(x)$ with $x \in K \cap C$ does not have a finite escape time, namely, there does not exist $t^* < \infty$ such that $\lim\limits_{t \searrow t^*} |\psi(t)| = \infty$, then item \ref{item:N*} holds for $K$ and $\HS$.
Although, in principle, item \ref{item:N*} is a solution-dependent property, it can be guaranteed without solving for solutions when $F$ is single valued and globally Lipschitz.
Moreover, we provide several other alternatives in the next result.
\begin{lemma}(sufficient conditions for completeness)\label{lem:star}
Given $\HS = (C, F, D, G)$ and a set $K\subset \reals^n$, suppose $K, C, D,$ and $F$ satisfy Assumption~\ref{assum:data} and item \ref{item:wfi2} in Theorem~\ref{thm:wfi}.
Condition \ref{item:N*} in Theorem~\ref{thm:wfi} holds if
\begin{enumerate}[label = \ref{lem:star}.\arabic*), leftmargin=2.8\parindent]
\item\label{item:star1} \chai{$\widehat{C}$} is compact; or
\item\label{item:star2} $F$ is bounded on \chai{$\widehat{C}$}; or
\item\label{item:star3} $F$ has linear growth.
\end{enumerate}
\end{lemma}
\chai{Item \ref{item:star2} and \ref{item:star3} redundant? The following proof needs to be updated:
\begin{proof}
Let $\phi \in \sol_{\HS}(K \cap C)$ with $\rge \phi \subset K$ be as described by \ref{item:b.2} in Proposition~\ref{prop:existence}; namely, $t \mapsto \phi(t,J)$ defined on $I^J$, where $(T,J) = \sup \dom \phi, T+ J < \infty$ and, for some $t^J, I^J = [t_J, T)$.
Since $t \mapsto \phi(t,J)$ is locally absolutely continuous on $I^J$, $\lim\limits_{t \rightarrow T} \phi(t,J)$ is finite or infinity.
If it is finite, then $t \mapsto \phi(t,j)$ can be extended to $\overline{I^J}$, which would contradict \ref{item:b.2} being satisfied.
Then, it has to be that $\lim\limits_{t \rightarrow T} \phi(t,J)$ is infinity.
When \ref{item:star1} holds, $\lim\limits_{t \rightarrow T} \phi(t,J)$ being infinity is a contradiction since $K \cap C$ is compact.
If \ref{item:star2} holds, there exists $M > 0$ such that $\sup \{\eta: |\eta| \in F(x), x \in K \cap C\} \leq M$.
Thus, every solution to $\dot{x} \in F(x), x \in K \cap C$ does not exhibit a finite escape time.
If \ref{item:star3} holds, \cite[Lemma 1]{aubin.hs.viability},\footnote{The sets $X, K, C$ in \cite[Lemma 1]{aubin.hs.viability} are $C, K \cap C, \overline{K \cap C \cap D}$, respectively, in our context.} see also \cite[Theorem 3.3.5]{aubin.viability}, implies that $t \mapsto \phi(t,J)$ is either:
\begin{itemize}[leftmargin = 4mm]
\item defined over $[t_J, \infty)$ with values in $K\cap C$; or
\item defined over $[t_J, T]$ with $\phi(T, J) \in \overline{K \cap C \cap D}$ and $\phi(T, J) \in K\cap C$ for every $t \in [t_J,T]$.
\end{itemize}
In either case, we have a contradiction.
\end{proof}
}
Note that condition \ref{item:star3} is typically assumed in the study of viability and invariance of differential inclusions; see, e.g., \cite{aubin.viability, aubin.hs.viability, aubin2009set}.
Condition \ref{item:star1} and \ref{item:star2} do not require $F$ to be Marchaud, but impose boundedness of $F$ and extra properties on $K \cap C$.
In fact, besides \ref{item:star3} providing an alternative condition for \ref{item:N*}, linear growth is not imposed on $F$ in this paper.
The next example illustrates Theorem~\ref{thm:wfi} and Lemma~\ref{lem:star}.
\begin{example}(weakly forward invariant set)\label{ex:wFI}
Consider the hybrid system $\HS = (C, F, D, G)$ in $\reals^2$ given by
\begin{align*}
&F(x):= (x_2, -x_1) & \forall x \in C;\\
&G(x):= (-0.9x_1, x_2) &\forall x \in D,
\end{align*}
where $C := \{x \in \reals^2: |x| \leq 1, x_2 \geq 0\}$ and
$D:= \{x \in \reals^2: x_1 \geq -1, x_2 = 0\}.$
\vspace{-7mm}
\begin{figure}[H]
\centering
\includegraphics[width=\figsize]{wFI.eps}
\small
\put(-0.05,.12){$x_1$}
\put(-.62,.5){$x_2$}
\put(-.26,.05){1}
\put(-.93,.05){-1}
\put(-.6,.05){0}
\put(-.7,.3){$C$}
\put(-.45,.03){$D$}
\caption{Sets and directions of flows/jumps in Example~\ref{ex:wFI}.}
\label{fig:wFI}
\end{figure}
\vspace{-3mm}
\noindent The set $K = \partial C$ is weakly forward invariant for $\HS$ by Theorem~\ref{thm:wfi}.
More precisely, for every $x \in K \cap D$, $G(x) \subset K$; and for every $x \in K \setminus D = \{x \in \reals^2: |x| = 1, x_2 > 0\}$, since $\inner{\nabla( x^2_1 + x^2_2)}{F(x)} = 0$, $F(x) \subset T_{K \cap C}(x)$.
In addition, $K \cap C = \partial C$ is compact, which implies condition \ref{item:N*} holds by Lemma~\ref{lem:star}.
Thus, for every $x \in K$, there exists one maximal solution that is complete and stays within $K$.
One such maximal solution starts from $(0,1)$ and flows along $|x| = 1$, becomes eventually discrete after reaching $(1,0)$, while staying within $K$.
In fact, for every point in $\{x\in \reals^2: x_1 \in (-1, 1), x_2 = 0 \}$, there exists one maximal solution that is complete, discrete and stays in $K$ (from the origin there is also a complete continuous solution that remains at the origin), but also there exist maximal solutions that flow inside $\{x \in \reals^2: |x| < 1 \}$ and leave $K$.
\hfill $\triangle$
\end{example}
Note that because the set $K \cap C$ is closed and that for every $x \in \text{\normalfont int} (K \cap C)$, $T_{K \cap C}(x) = \reals^n$, condition
\begin{enumerate}[label = \ref{thm:fi}.2$^{'}$), leftmargin= 12mm]
\item\label{item:fi2'} For every $x \in \partial (K \cap C), F(x) \subset T_{K \cap C}(x)$
\end{enumerate}
implies condition \ref{item:fi2} in Theorem~\ref{thm:fi}.
In addition, one can replace condition \ref{item:fi2} in Theorem~\ref{thm:fi} by
\begin{enumerate}[label = \ref{thm:fi}.2$^{''}$), leftmargin= 12mm]
\item\label{item:fi2''} For every $x \in \partial (K \cap C),$
\end{enumerate}
\vspace{-6mm}
\begin{align}
& F(x) \subset T_{K \cap C}(x) &\text{\normalfont if } &x \notin \partial C \cap D\label{eq:fi2''}\\
&F(x) \cap (T_C(x) \setminus T_{K \cap C}(x)) = \emptyset &\text{\normalfont if } &x \in \partial C \cap D. \label{eq:fi2''s}
\end{align}
This replacement allows us to verify forward invariance of $K = C$ for the hybrid system $\HS$ in Example~\ref{ex:wFI}:
even though $F(x) \cap T_{K \cap C}(x) = \emptyset$ at every $x \in \{x \in \reals^2: x_2 = 0, 0 < x_1 \leq 1 \}$, the solutions can be extended by jumping when reaching the positive $x_1-$axis.
Note that assumption \eqref{eq:fi2''s} is important as in some cases, having item \eqref{eq:fi2''} only leads to solutions that escape the set $K$ by flowing as shown in the following example.
Consider the hybrid system $\HS$ on $\reals^2$ with
\IfConf{
system data given by
$F(x) =(x_2, -\gamma)$, $C := \{x \in \reals^2: x_1x_2 \geq 0 \}$,
$G(x) = x$ and $D := \{x \in \reals^2: x_1 \geq 0, x_2 = 0 \},$
}{
\begin{align*}
&F(x) =(x_2, -\gamma) & \forall x &\in C := \{x \in \reals^2: x_1x_2 \geq 0 \}\\
&G(x) = x & \forall x &\in D := \{x \in \reals^2: x_1 \geq 0, x_2 = 0 \},
\end{align*}
}
where $\gamma > 0$.
The set $K = \{x \in \reals^2: x_1 \geq 0, x_2 \geq 0 \}$ is weakly forward invariant, and the sets $K, C, D$ and the map $F$ satisfy condition \ref{item:fi2''}.
However, at the origin, we have $F(0) = (0, -\gamma)$ and
\vspace{-1mm}
\begin{align*}
&T_C(0) = (\reals_{\geq 0} \times \reals_{\geq 0} ) \cup (\reals_{\leq 0} \times \reals_{\leq 0}),\\
&T_{K \cap C}(0) = \reals_{\geq 0} \times \reals_{\geq 0}.
\end{align*}
\vspace{-5mm}
\noindent Hence, at the origin, one solution can flow into $C \setminus K$ (the third quadrant) because $F(0) \in T_C(0) \setminus T_{K \cap C}(0)$.
The following example is an application of Theorem~\ref{thm:fi} and Lemma~\ref{lem:star}.
\begin{example}(forward invariant set)\label{ex:oscillator0w}
Consider the hybrid system given by
\IfConf{
\begin{align}\label{eq:ex_zerod}
\HS \begin{cases}
x \in C &\dot{x} = F(x) := (- |x_1|x_2, 0)\\
x \in D & x^+ = G(x) := x,
\end{cases}
\end{align}
}{
\begin{align}\label{eq:ex_zerod}
\HS \begin{cases}
x \in C &\dot{x} = F(x) := \begin{bmatrix}- |x_1|x_2 \\ 0
\end{bmatrix} \\
x \in D & x^+ = G(x) := x,
\end{cases}
\end{align}
}
where the flow set is $C = \{x \in \reals^2: |x| \leq 1, x_1x_2 \geq 0 \}$ and the jump set is $D = \{x \in \reals^2: |x| \leq 1, x_1x_2 \leq 0 \}$.
We observe that during flow, solutions evolve continuously within the unit circle centered at the origin; while at jumps, solutions remain at the current location.
Applying Theorem~\ref{thm:fi}, we show that the set $K_a = C_1 \cup D_1$ is forward invariant for $\HS,$ where $C_1 = \{x \in \reals^2: x_1 \geq 0, x_2 \geq 0, |x| \leq 1\}$ and $D_1 = \{x \in \reals^2: x_1 \leq 0, x_2 \geq 0, |x| \leq 1\}$.
By construction, $K_a, C, D$ and $F$ satisfy Assumption~\ref{assum:data}.
Condition \ref{item:fi1} holds since $G$ maps the state to its current value.
Condition \ref{item:fi2} holds since
\begin{itemize}[leftmargin = 4mm]
\item for every $x \in \text{\normalfont int} (C_1)$, $T_{K_a \cap C}(x) = \reals^2$;
\item for every $x \in \{x \in \partial C_1: |x| =1 \}$, inner product between the gradient of $x^2_1 + x^2_2$ and $F(x)$ is nonpositive, i.e.,
\begin{align*}
\inner{\nabla (x_1^2 + x_2^2)}{F(x)} &= -2|x_1|x_1x_2 \leq 0
\end{align*}
because $x_1 x_2 \geq 0$, which implies $F(x) \in T_{K_a \cap C}(x);$
\item for every $x \in \{x \in \partial C_1: |x| \neq 1 \}$, we have that
$$T_{K_a \cap C}(x) =
\begin{cases}
\reals_{\geq 0} \times \reals & \text{\normalfont if } x_1 = 0, x_2 \notin \{0,1\}\\
\reals \times \reals_{\geq 0} & \text{\normalfont if } x_1 \notin \{0,1\}, x_2 = 0\\
\reals_{\geq 0}^2 & \text{\normalfont if } x = 0,
\end{cases}$$
which implies $F(x) \in T_{K_a \cap C}(x)$ holds.
\end{itemize}
Finally, item \ref{item:star1} in Lemma~\ref{lem:star} holds for $K_a$ and $C$, thus, \ref{item:N*} holds.
\hfill $\triangle$
\end{example}
In Example~\ref{ex:oscillator0w}, the set $K_a$ is forward invariant for $\HS$ as shown above.
When arbitrarily small disturbances are introduced, solution pairs may escape the set of interest.
In particular, we revisit Example~\ref{ex:oscillator0w} with disturbances in the next section, and show that $K_a$ is only weakly forward invariant uniformly in the given disturbances.
The forthcoming results in Section~\ref{sec:HwSuffConds} are useful to verify such properties of sets for hybrid systems $\HS\w$ given as in \eqref{eq:Hw}.
\subsection{Sufficient Conditions for Robust Forward Invariance Properties for $\HS\w$}\label{sec:HwSuffConds}
As an extension to the nominal notions, the robust forward invariance notions for $\HS\w$ in Definition~\ref{def:rwFI} - \ref{def:rFI} capture four types of forward invariance properties, some of which are uniform over disturbances $w$ for $\HS\w$.
In this section, Theorem~\ref{thm:rwFI} and Theorem~\ref{thm:rFI} extend Theorem~\ref{thm:wfi} and Theorem~\ref{thm:fi} to hybrid systems $\HS\w$ given in \eqref{eq:Hw}.
These results will be exploited in forward invariance-based control design for hybrid systems (with and without disturbances) in Part II of this work.
Similar to the results in Section~\ref{sec:HSuffConds}, throughout this section, the following version of Assumption~\ref{assum:data} with disturbances is assumed.
\begin{assumption}\label{assum:wdata}
The sets $K, C\w$, and $D\w$ are such that $K\subset \overline{\Pwc{C\w}}\cup \Pwd{D\w}$ and that $K \cap \Pwc{C\w}$ is closed.
The map $F\w$ is outer semicontinuous, locally bounded on $(K \times {\cal W}_c) \cap C\w$, and $F\w(x,w_c)$ is nonempty and convex for every $(x,w_c) \in (K \times {\cal W}_c) \cap C\w$.
For every $x \in \Pwc{C\w}, 0 \in \Psi^w_c(x)$.
\end{assumption}
Assumption~\ref{assum:wdata} guarantees that all points in the set to render invariant, namely, $K$, are either in the projections to the state space of $C\w$ and $D\w$, which is necessary for solutions from $K$ to exist.
The closedness of the set $K \cap \Pwc{C\w}$ and the regularity properties of $F\w$ are required to obtain conditions in terms of the tangent cone; see, also, \cite[Proposition 6.10]{65}.
The assumption of $0 \in \Psi^w_c(x)$ for every $x \in \Pwc{C\w}$ usually holds for free since systems with disturbances, such as $\HS\w$, typically reduce to the nominal system, in our case $\HS$, when the disturbances vanish.
A similar property could be enforced for the disturbance $w_d$, but such an assumption is not needed in our results.
Next, we propose sufficient conditions to guarantee robust weak forward pre-invariance and robust weak forward invariance of a set for $\HS\w$.
\begin{theorem}(sufficient conditions for robust weak forward (pre-) invariance of a set)\label{thm:rwFI}
Given $\HS\w = (\Cw, \Fw, \Dw, \Gw)$ as in \eqref{eq:Hw} and a set $K\subset \reals^n$, suppose $C\w, F\w, D\w$ and $K$ satisfy Assumption~\ref{assum:wdata}.
The set $K$ is robustly weakly forward pre-invariant for $\HS\w$ if the following conditions hold:
\begin{enumerate}[label = \ref{thm:rwFI}.\arabic*), leftmargin=3.2\parindent]
\item\label{item:rwFI1} For every $x \in K\cap\Pwd{D\w}$, $\exists w_d \in \Psi^w_d(x)$ such that $G\w(x, w_d) \cap K \neq \emptyset$;
\item\label{item:rwFI2} For every $x \in \chai{\Pwc{\widehat{C}\w}} \setminus \Pwd{D\w}$, $F\w(x, 0) \cap T_{K \cap \Pwc{C\w}}(x)$ $\neq \emptyset$;
\end{enumerate}
where $\chai{\widehat{C}\w} : = ((\partial (K\cap\Pwc{C\w}) \times {\cal W}_c) \cap C\w) \setminus L\w$ and $L\w : = \{(x, w_c) \in C\w: x \in \partial \Pwc{C\w}, F\w(x,w_c) \cap T_{\overline{\Pwc{C\w}}}(x) = \emptyset \}$.
Moreover, $K$ is robustly weakly forward invariant for $\HS\w$ if, in addition, \chai{$K \cap \Pwc{L\w} \subset \Pwd{D\w}$ and, with $\widetilde{K}^\star = ((K \setminus \Pwd{D\w}) \times {\cal W}_c)\cap C\w$,}
\begin{enumerate}[label = $\star$), leftmargin=6mm]
\item\label{item:*} For every $(\phi, w)\in \sol_{\HS\w}(\chai{\Pwc{\widetilde{K}^\star}})$ with $\rge \phi \subset K$, case \ref{item:b.2} in Proposition~\ref{prop:wexistence} does not hold.
\end{enumerate}
\end{theorem}
\begin{proof}
Given $C\w, F\w, D\w$ and $K$ satisfying Assumption~\ref{assum:wdata}, zero disturbance is always admissible to $\HS\w$ during continuous evolution of solution pairs.
We define a restriction of $\HS\w$ by $K$ with zero disturbance during flows as follows:
$\widetilde{\HS}_w = (\widetilde{C}, \widetilde{F}, \widetilde{D}_w, G\w)$,
where $\widetilde{C} := K \cap \Pwc{C\w}$, $\widetilde{F}(x) = F\w(x, 0)$ for every $x \in \Pwc{C\w}$ and $\widetilde{D}_w := (K \times {\cal W}_d)\cap D\w$.
Since $K \subset \overline{\Pwc{C\w}} \cup \Pwd{D\w}$, by Definition~\ref{def:HwSol}, there exists a solution pair to $\widetilde{\HS}_w$ from every $\xi \in K$.
Let $K_1 = \Pwd{\widetilde{D}_w}$,
\chai{$K_2 = K \setminus (\Pwd{\widetilde{D}_w} \cup \Pwc{L\w})$ and $K_3 = K \setminus (K_1 \cup K_2)$.
By definition, every $\xi \in K_3$ is such that $\xi \in \Pwc{L\w}\setminus \Pwd{\widetilde{D}_w}$ and $\widetilde{F}(\xi)\cap T_{\overline{\Pwc{C\w}}}(\xi) =\emptyset$.
Then, item (a) in \cite[Lemma 5.26]{65} and Definition~\ref{def:solution} imply there is only trivial solution from $\xi$ to $\widetilde{\HS}_w$, in which case we have $\rge \phi \subset K$.
Otherwise, in the case where $\phi(0,0) \in K_1 \cup K_2$, we show there exists $(\phi, w) \in \sol_{\widetilde{\HS}_w}$ that is nontrivial and it has $\rge \phi \subset K$ when \ref{item:rwFI1} and \ref{item:rwFI2} hold true.
To this end, we construct a nontrivial solution pair from every $\xi \in K_1 \cup K_2$.}
Since $K_1$ and $K_2$ are disjoint sets, we have following two cases:
\begin{enumerate}[label = \roman*), leftmargin = 4mm]
\item\label{item:rwFIi} when $\xi \in K_1$: since $K_1 \subset \Pwd{D\w}$, a jump is possible from every $\xi \in K_1$, i.e., from every $(\xi, w_d) \in \widetilde{D}_w$.
Let $\phi_a(0,0) = \xi$.
By condition \ref{item:rwFI1}, there exists $\widetilde{w}_d \in \Psi^w_d(\xi), \phi_a(0,1) \in G\w(\xi, \widetilde{w}_d)$, such that $\phi_a(0,1) \in K$.
\item\label{item:rwFIii} when $\xi \in K_2$:
since $K \subset \overline{\Pwc{C\w}} \cup \Pwd{D\w}$, $\xi \in \overline{\Pwc{C\w}}\setminus \Pwd{D\w}$ and solution pairs can only evolve by flowing from $\xi$.
Conditions enforced by Assumption~\ref{assum:wdata} imply that $\widetilde{C}$ is closed, $\widetilde{F}$ is outer semicontinuous, locally bounded and convex valued on $\widetilde{C}$.
\chai{Since $T_{\widetilde{C}}(x) = \reals^n$ for every $x \in (\text{\normalfont int} \widetilde{C})\setminus (\Pwd{\widetilde{D}_w} \cup \Pwc{L\w})$, item \ref{item:rwFI2} implies that $\widetilde{F}(x) \cap T_{\widetilde{C}}(x) \neq \emptyset$ for every $x \in K_2$.
Then, by an application of \cite[Proposition 6.10]{65}, there exists a nontrivial solution $\phi_b$ to $\widetilde{\HS}_w$ from every $\xi \in K_2$.
By item \ref{item:S1} in Definition~\ref{def:solution}, such a nontrivial solution $\phi_b$ is absolutely continuous on $[0,\varepsilon]$, for some $ \varepsilon > 0$, with $\phi_b(0) = \xi$, $\dot{\phi}_b(t) \in \widetilde{F}(\phi_b(t))$ for almost all $t \in [0,\varepsilon]$ and $\phi_b(t) \in \widetilde{C}$ for all $t \in (0,\varepsilon]$.
By closedness of $\widetilde{C}$, we have $\phi_b(t,0) \in K$ for every $t \in [0,\varepsilon]$.}
\end{enumerate}
The above shows that from every point in $K_1$, solution pairs to $\widetilde{\HS}_w$ can be extended via jumps with the state component staying within $K$ using the construction in case \ref{item:rwFIi}.
While from points in $K_2$, solution pairs can be extended using the construction in case \ref{item:rwFIii} with the state component staying within $K$.
As a consequence, from every point in $K$, there exists at least one $(\widetilde{\phi}, \widetilde{w}) \in \sol_{\widetilde{\HS}_w}$ with $\rge\widetilde{\phi} \subset K$.
Next, we prove that each such $(\widetilde{\phi}, \widetilde{w})$ is also maximal to $\HS\w$.\footnote{During flows, we have $(\widetilde{\phi}, 0)$.}
If $(\widetilde{\phi}, \widetilde{w})$ is complete, then it is already maximal and a solution pair to $\HS\w$.
Consider the case that $(\widetilde{\phi}, \widetilde{w})$ is not complete.
Proceeding by contradiction, suppose $(\widetilde{\phi}, \widetilde{w})$ is not maximal for $\HS\w$, meaning that there exists $(\phi, w)$ such that $\phi(t,j) = \widetilde{\phi}(t,j)$ and $w(t,j) = \widetilde{w}(t,j)$ for every $(t,j)\in \dom \widetilde{\phi}$ and $\dom \phi \setminus \dom \widetilde{\phi} \neq \emptyset$.
Let $(T,J) = \sup \dom \widetilde{\phi}$.
If $(T,J) \in \dom \widetilde{\phi}$, then, $\widetilde{\phi}(T,J) \in K$ and we have the two following cases:
\chai{
\begin{itemize}[leftmargin=4mm]
\item $\widetilde{\phi}(T,J) \in K_1 \cup K_2$, \ref{item:rwFI1} and closeness of $\widetilde{C}$ imply that, using the arguments in \ref{item:rwFIi} and \ref{item:rwFIii} above, it is possible for $\phi$ to satisfy $\phi(t,j) \in K$ for some $(t,j) \in \dom \phi \setminus \dom \widetilde{\phi}$.
By definition of solution pairs, this contradicts with maximality of $(\widetilde{\phi}, \widetilde{w})$ for $\widetilde{\HS}_w$.
\item $\widetilde{\phi}(T,J) \in K_3$, by definition of $L\w$, $F\w(\widetilde{\phi}(T,J), w_c) \cap T_{\overline{\Pwc{C\w}}}(\widetilde{\phi}(T,J)) =\emptyset$ for every $w_c \in \Psi^w_c{(\widetilde{\phi}(T,J))}$.
Hence, $\sup \dom \phi = (T,J)$, which contradicts with the assumption $\dom \phi \setminus \dom \widetilde{\phi} \neq \emptyset$.
\end{itemize}
}
If $(T,J) \notin \dom \widetilde{\phi}$, according to Proposition~\ref{prop:wexistence}, only \ref{item:b.2} holds.\footnote{Case \ref{item:a} does not hold due to $(\widetilde{\phi}, \widetilde{w})$ not being complete, while \ref{item:b.1} and \ref{item:c} do not hold because $(T,J) \not\in \dom \widetilde{\phi}$.}
In such a case, there is no function $z : \overline{I^J} \rightarrow \reals^n$ satisfying the conditions in \ref{item:b.2} of Proposition~\ref{prop:wexistence}, which are needed to have a $(\phi, w)$ such that $\dom \phi\setminus \dom \widetilde{\phi} \neq \emptyset$.
Thus, $K$ is robustly weakly forward pre-invariant for $\HS\w$.
The last claim requires to show that among these maximal solution pairs to $\HS\w$ that stay in $K$ for all future time, there exist one complete solution pair from every point in $K$ when, in addition, $(K \cap \Pwc{L\w}) \subset \Pwd{D\w}$ and item \ref{item:*} hold.
To this end, \chai{first, note that the existence of a nontrivial solution pair to $\HS\w$ from every $x \in K$ follows from $(K \cap \Pwc{L\w}) \subset \Pwd{D\w}$, which implies $K_3 = \emptyset.$}
Then, we apply Proposition~\ref{prop:wexistence} to complete the proof.
Proceeding by contradiction, given any $\xi \in K$, suppose every $(\phi^*,w^*)\in \sol_{\HS\w}(\xi)$ is not complete, i.e., $(T,J) = \sup \dom \phi^*, T + J < \infty$, and case \ref{item:a} in Proposition~\ref{prop:wexistence} does not hold.
Such a solution pair $(\phi^*, w^*)$ is not as described in case \ref{item:b.1.1} in Proposition~\ref{prop:wexistence} due to the closeness of $K \cap \Pwc{C\w}$.
Case \ref{item:c.1} does not hold for $(\phi^*, w^*)$ either, since $\rge \phi^* \subset K$ and $K \subset \overline{\Pwc{C\w}} \cup \Pwd{D\w}$.
Thus, by Proposition~\ref{prop:wexistence}, $(\phi^*, w^*)$ can only end as described by case \ref{item:b.1.2}, \ref{item:b.2} or \ref{item:c.2}.
\begin{itemize}[leftmargin= 4mm]
\item The solution pair ends because the functions described in case \ref{item:b.1.2} or \ref{item:c.2} of Proposition~\ref{prop:wexistence}, i.e., $\widetilde{z}$ does not exist for $(\phi^*(T,J), w^*(T,J))$.
However, using the same argument in item \ref{item:rwFIii} above with $\widetilde{w}_c \equiv 0$, for every $(x, 0) \in K_1 \times 0$ there exists $\widetilde{z}$ such that \ref{item:b.1.2} holds, which leads to a contradiction.
\item \chai{If $(\phi^*, w^*)$ is as described by case \ref{item:b.2}, $\phi^*(0,0) \notin \Pwc{\widetilde{K}^\star}$ by assumption \ref{item:*}.
More precisely, $\phi^*(0,0) \in K_1$, hence, the solution pair can be extended following the same construction in \ref{item:rwFIi} above, which contradicts with the maximality of $(\phi^*, w^*)$.}
\end{itemize}
\vspace{-5mm}
\end{proof}
Condition \ref{item:rwFI1} in Theorem~\ref{thm:rwFI} guarantees that for every $x\in K \cap \Pwd{D\w}$ \chai{such that} there exists $w_d \in \Psi^w_d(x)$, the jump map contains an element that also belongs to $K$.
Under the stated assumptions, condition \ref{item:rwFI2} implies the satisfaction of \ref{item:VCw} with zero disturbance $w_c$, which suffices for the purpose of Theorem~\ref{thm:rwFI} as it is about weak forward invariance notions.
While involving the tangent cone of $K \cap \Pwc{C\w}$ in condition \ref{item:rwFI2} is natural, such solution property is more than needed for robust weak forward pre-invariance of $K$ as defined in Definition~\ref{def:rwFI}.
\chai{
Similarly to Lemma~\ref{lem:Nstar}, solution-independent conditions that imply \ref{item:*} are derived for the disturbance case.
\begin{lemma}(sufficient conditions for completeness)\label{lem:star}
Given $\HS\w = (\Cw, \Fw, \Dw, \Gw)$ and a set $K\subset \reals^n$, suppose $K, C\w, D\w,$ and $F\w$ satisfy Assumption~\ref{assum:wdata}, set $\Pwd{D\w}$ is closed and item \ref{item:rwFI2} in Theorem~\ref{thm:rwFI} holds.
Condition \ref{item:*} in Theorem~\ref{thm:rwFI} holds if
\begin{enumerate}[label = \ref{lem:star}.\arabic*), leftmargin=3\parindent]
\item\label{item:star1} $\widetilde{K}^\star$ is compact; or
\item\label{item:star3} $F\w$ has linear growth on $\widetilde{K}^\star$.
\end{enumerate}
\end{lemma}
}
The following example illustrates Theorem~\ref{thm:rwFI}.
\begin{example}[robustly weakly forward invariant set] \label{ex:oscillator}
Consider a variation of hybrid system $\HS$ in Example~\ref{ex:oscillator0w} with disturbances given by
\IfConf{
\begin{align}\label{eq:ex_d}
\HS\w \begin{cases}
(x, w_c) \in C\w &
\dot{x} = F\w(x, w_c) := (-|x_1| x_2, w_c |x_1|x_1)\\
(x, w_d) \in D\w & x^+ \in G\w(x, w_d) :=\\
&\qquad \{R(\theta)x : \theta \in [w_d, -w_d] \},
\end{cases}
\end{align}
\vspace{-3mm}
}{
\begin{align}\label{eq:ex_d}
\HS\w \begin{cases}
(x, w_c) \in C\w &
\dot{x} = F\w(x, w_c) := |x_1|\begin{bmatrix}- x_2 \\ w_c x_1
\end{bmatrix} \\
(x, w_d) \in D\w & x^+ \in G\w(x, w_d) := \{R(\theta)x : \theta \in [w_d, -w_d] \},
\end{cases}
\end{align}
}
\noindent where $R(\theta) : =
\begin{bmatrix}
\cos \theta & \sin \theta\\
-\sin \theta & \cos \theta
\end{bmatrix}$ is a rotation matrix,
$C\w:= \{(x, w_c) \in \reals^2 \times \reals: 0 \leq w_c \leq |x| \leq 1, x_1x_2 \geq 0\}$, and $D\w:= \{ (x, w_d) \in \reals^2 \times \reals: x_1x_2 \leq 0, |x| \leq 1, -\frac{\pi}{4} \leq w_d \leq 0\}$.
As shown in Figure~\ref{fig:oscillator}, the projections of $C\w$ and $D\w$ onto $\reals^2$ are given by $\Pwc{C\w} = C_1 \cup C_2$ on $\reals^2$ with $C_2 = \{x \in \reals^2: x_1 \leq 0, x_2 \leq 0, |x| \leq 1\}$ and by $\Pwd{D\w} = D_1 \cup D_2$ with $D_2 = \{x \in \reals^2: x_1 \geq 0, x_2 \leq 0, |x| \leq 1\}$, respectively.\footnote{We use the same definitions for $K_1$, $C_1$, and $D_1$ as in Example~\ref{ex:oscillator0w}.}
Based on provided dynamics, solutions travel counter-clockwise during flows, while they either rotate clockwise or counter-clockwise during jumps.
As a result, solutions can evolve in any of the four quadrants in $\reals^2$, either by flow or jump.
\IfConf{
\vspace{-4mm}
\begin{figure}[H]
\centering
\setlength{\unitlength}{.3\textwidth}
\includegraphics[width = \unitlength]{oscillator.eps}
\put(-0.1,0.5){$x_1$}
\put(-0.5,0.9){$x_2$}
\put(-0.47,0.7){$C_1$}
\put(-0.7,0.12){$C_2$}
\put(-0.7,0.7){$D_1$}
\put(-0.47,0.12){$D_2$}
\put(-0.52, 0.39){0}
\put(-0.13, 0.39){1}
\put(-1.02, 0.39){-1}
\put(-1, 0.55){$\xi_1$}
\put(-0.84, 0.49){$\xi_2$}
\caption{Projection onto the state space of flow and jump sets of the system in Example~\ref{ex:oscillator}. The blue solid arrows indicate possible hybrid arcs during flow, while the red dashed arrows indicate possible hybrid arcs during jumps.}
\label{fig:oscillator}
\end{figure}
\vspace{-3mm}
}{
\begin{figure}[h]
\centering
\includegraphics[width = \figsize]{oscillator.eps}
\put(-0.1,0.5){$x_1$}
\put(-0.5,0.9){$x_2$}
\put(-0.47,0.7){$C_1$}
\put(-0.7,0.12){$C_2$}
\put(-0.7,0.7){$D_1$}
\put(-0.47,0.12){$D_2$}
\put(-0.52, 0.39){0}
\put(-0.13, 0.39){1}
\put(-1.02, 0.39){-1}
\put(-1, 0.55){$\xi_1$}
\put(-0.84, 0.49){$\xi_2$}
\caption{Projection onto the state space of flow and jump sets of the system in Example~\ref{ex:oscillator}. The blue solid arrows indicate possible hybrid arcs during flow, while the red dashed arrows indicate possible hybrid arcs during jumps.}
\label{fig:oscillator}
\end{figure}
}
First, we apply Theorem~\ref{thm:rwFI} to conclude robust weak forward invariance of the set $K_1 = C_1 \cup D_1$ for $\HS\w$.
Assumption~\ref{assum:wdata} holds for $K_1, C\w, D\w$ and $F\w$ by construction.
Since the set $L\w$ is empty, condition \ref{item:rwFI1} holds since for every $(x, w_d) \in (K_1 \times {\cal W}_d)\cap D\w$, the selection $x^+ = R(0)x$ always results in $x^+ \in K_1$.
Condition \ref{item:rwFI2} holds since, applying item \ref{item:inner1} in Lemma~\ref{lem:innerTcone}, for every $x \in \partial C_1 \setminus \Pwd{D\w}$, since $x_1x_2 \leq 0$, we have
\IfConf{
$$\inner{\nabla (x_1^2 + x_2^2)}{F\w(x, 0)} = -2x_1x_2|x_1| \leq 0.$$
}{
\begin{align*}
\inner{\nabla (x_1^2 + x_2^2 - 1)}{F\w(x, 0)} &= 2x_1(-x_2|x_1|) + 2x_2(w_cx_1|x_1|)\\
& = -2x_1x_2|x_1| \leq 0.
\end{align*}
}
Then, the robust weak forward invariance of $K_1$ follows from \ref{item:star3} in Lemma~\ref{lem:star} and Theorem~\ref{thm:rwFI}.
Note that the property is weak due to the following observations:
\begin{itemize}[leftmargin = 4mm]
\item Because of the set-valuedness of the map $G\w$, there exists a solution pair from a point $\xi_1 \in D_1$ that jumps to a point in $C_2$ that is not in $K_1$, as depicted in Figure~\ref{fig:oscillator}.
On the other hand, from the same point $\xi_1$, there exists a solution pair that keeps jumping from and to $\xi_1$, and stays within $D_1 \subset K_1$;
\item Because of the overlap between $\Pwc{C\w}$ and $\Pwd{D\w}$, there exists a solution pair that starts from a point $\xi_2 \in D_1$ and flows to a point in $C_2$ that is not in $K_1$, as depicted in Figure~\ref{fig:oscillator}.
On the other hand, the solution pair that jumps from and to $\xi_2$ from $\xi_2$ stays within $D_1 \subset K_1$. \hfill $\triangle$
\end{itemize}
\end{example}
To derive a set of sufficient conditions guaranteeing the stronger robust forward invariance property of $K$, i.e., every solution pair to $\HS\w$ is such that its state component stays within the set $K$, when starting from $K$, we require the disturbances $w$ and the set $K$ to satisfy the following assumption.
\begin{assumption}\label{assum:wbound}
For every $\xi \in (\partial K) \cap \Pwc{C\w}$, there exists a neighborhood $U$ of $\xi$ such that $\Psi^w_c(x) \subset \Psi^w_c(\xi)$ for every $x \in U \cap \Pwc{C\w}$.
\end{assumption}
The next result provides conditions implying robust forward pre-invariance and robust forward invariance of a set for $\HS\w$.
\begin{theorem}(sufficient conditions for robust forward \ (pre-) invariance of a set)\label{thm:rFI}
Given $\HS\w = (\Cw, \Fw, \Dw, \Gw)$ as in \eqref{eq:Hw} and a set $K\subset \reals^n$, suppose $C\w, F\w, D\w$ and $K$ satisfy Assumption~\ref{assum:wdata}.
Furthermore, suppose the mapping $x \mapsto F\w(x, w_c)$ is locally Lipschitz uniformly in $w_c$ on $C\w$.
The set $K$ is robustly forward pre-invariant for $\HS\w$ if the following conditions hold:
\begin{enumerate}[label = \ref{thm:rFI}.\arabic*), leftmargin=3.2\parindent]
\item\label{item:rFI1} For every $(x, w_d) \in (K \times {\cal W}_d)\cap D\w, G\w(x, w_d) \subset K$;
\item\label{item:rFI2} For every $(x, w_c) \in \chai{\widehat{C}\w}$, $F\w(x,w_c) \subset T_{K\cap \Pwc{C\w}}(x)$,
\end{enumerate}
where $\widehat{C}\w$ and $L\w$ be given as in Theorem~\ref{thm:rwFI}.
Moreover, $K$ is robustly forward invariant for $\HS\w$ if, in addition, \chai{$K \cap \Pwc{L\w} \subset \Pwd{D\w}$ and, with $\widetilde{K}^\star = (K \cap \Pwc{C\w})\times {\cal W}_c)\cap C\w$,} condition \ref{item:*} in Theorem~\ref{thm:rwFI} holds.
\end{theorem}
\begin{proof}
Since condition \ref{item:rFI1} and \ref{item:rFI2} imply condition \ref{item:rwFI1} and \ref{item:rwFI2}, respectively, under given conditions, which include the fact that $C\w, F\w, D\w$ and $K$ satisfy Assumption~\ref{assum:wdata}, the set $K$ is robustly weakly forward pre-invariant for $\HS\w$ by Theorem~\ref{thm:rwFI}.
Now we show that every $(\phi, w) \in \sol_{\HS\w}(K)$ has $\rge \phi \subset K$.
Proceeding by contradiction, suppose there exists a solution pair $(\phi, w) \in \sol_{\HS\w}(K)$ such that $\rge \phi \setminus K \neq \emptyset$.
Then, there exists $(t^*,j^*) \in \dom \phi$ such that $\phi(t^*,j^*) \not \in K$, i.e., $\phi$ eventually leaves $K$ in finite hybrid time.\footnote{Note that when $\rge \phi \subset K$ and $\lim\limits_{t+j \to \sup_t \dom \phi + \sup_j \dom \phi} \phi(t,j) = \infty$ (that is, $\phi$ stays in $K$ but escapes to infinity, potentially in finite hybrid time) corresponds to a solution that satisfies the definition of forward invariance for $K$.}
Then, we have the two following cases:
\begin{enumerate}[label = \roman*), leftmargin = 4mm]
\item\label{item:rFIi} In the case that $\phi$ ``left $K$ by jumping," namely, $\phi(t,j) \in K$ for all $(t,j) \in \dom\phi$ with $t+j < t^* +j^*$, $(\phi(t^*,j^*-1), w_d) \in D\w$ with $\phi(t^*,j^*) \notin K$ for some $w_d \in \Psi^w_d(\phi(t^*,j^*-1))$.
This contradicts item \ref{item:rFI1}.
More precisely, since $\phi(t^*,j^*-1) \in K \cap \Pwd{D\w}$, item \ref{item:rFI1} implies that $\phi(t^*, j^*) \in G\w(\phi(t^*,j^*-1), w_d(t^*,j^*-1)) \subset K$ for every $w_d \in \Psi^w_d(\phi(t^*, j^*-1))$.
Thus, $\phi$ did not leave $K$ by jumping.
Then, it must be the case that $\phi$ left $K$ by flowing, which is treated in the next item.
\item\label{item:rFIii} In the case that $\phi$ ``left $K$ by flowing," namely, there exists a hybrid time instant $(\tau^*,j^*) \in \dom \phi$ such that $\phi(t,j^*) \in \overline{\Pwc{C\w}}\setminus K$ for all $t \in (\tau^*,t^*]$ and $t^* - \tau^*$ is arbitrarily small and positive.
Moreover, by closedness of $K\cap \Pwc{C\w}$, we suppose that $\phi(\tau^*,j^*) \in (\partial K) \cap \Pwc{C\w}$.\footnote{By definition of solution pair, it is the case that $\phi$ left $K\cap \Pwc{C\w}$ and entered $\overline{\Pwc{C\w}}\setminus K$ passing through $(\partial K) \cap \Pwc{C\w}$.}
Let $t \mapsto \chi(t) \in K \cap \Pwc{C\w}$ be such that for every $t \in [\tau^*, t^*]$
\begin{align}\label{eq:chi}
|z(t)|_{K\cap \Pwc{C\w}} = |z(t)-\chi(t)|,
\end{align}
where $z(t) = \phi(t,j^*)$ for all $t \in [\tau^*, t^*]$.
Such points exist because of the closedness of $K \cap \Pwc{C\w}$.
By definition of solution pairs to $\HS\w$, the function $t\mapsto |z(t)|_{K\cap \Pwc{C\w}}$ is absolutely continuous.
Thus, for almost every $t \in [\tau^*, t^*]$, $\frac{d}{dt}|z(t)|_{K\cap \Pwc{C\w}}$ exists and equals to the Dini derivative of $|z(t)|_{K\cap \Pwc{C\w}}$.
Let $t$ be such that both $\frac{d}{dt}|z(t)|_{K\cap \Pwc{C\w}}$ and $\dot{z}(t)$ exist.
We have
\begin{align*}
&\frac{d}{dt}|z(t)|_{K\cap \Pwc{C\w}} \\
& \ = \liminf\limits_{h \searrow 0} \frac{|z(t)+h\dot{z}(t)|_{K\cap \Pwc{C\w}} - |z(t)|_{K\cap \Pwc{C\w}} }{h},
\end{align*}
which, by definition of $\chi(t)$ and \eqref{eq:chi}, satisfies
\IfConf{
\begin{align*}
&\frac{|z(t) + h\dot{z}(t)|_{K\cap \Pwc{C\w}} - |z(t)|_{K\cap \Pwc{C\w}}}{h} \\
& \ \leq \frac{|z(t) \scriptsize{-}\chi(t)| \scriptsize{+}|\chi(t)\scriptsize{+}h\dot{z}(t)|_{K\cap \Pwc{C\w}} \scriptsize{-}|z(t)|_{K\cap \Pwc{C\w}}}{h} \\
& \ = \frac{|\chi(t)+h\dot{z}(t)|_{K\cap \Pwc{C\w}}}{h}\\
&\ \leq \frac{|\chi(t)+h\omega|_{K\cap \Pwc{C\w}}}{h} + |\dot{z}(t) - \omega|,
\end{align*}
}{
\begin{align*}
&\!\!\!\!\!\!\!\!\! \frac{|z(t) + h\dot{z}(t)|_{K\cap \Pwc{C\w}} - |z(t)|_{K\cap \Pwc{C\w}}}{h} \\
& \leq \frac{|z(t) - \chi(t)| + |\chi(t)+h\dot{z}(t)|_{K\cap \Pwc{C\w}} - |z(t)|_{K\cap \Pwc{C\w}}}{h} = \frac{|\chi(t)+h\dot{z}(t)|_{K\cap \Pwc{C\w}}}{h}\\
&\leq \frac{|\chi(t)+h\omega|_{K\cap \Pwc{C\w}}}{h} + |\dot{z}(t) - \omega|,
\end{align*}
}
for every $\omega \in T_{K\cap \Pwc{C\w}}(\chi(t)).$
Moreover, for every such $\omega$,
$$\liminf\limits_{h \searrow 0} \frac{|\chi(t)+h\omega|_{K\cap \Pwc{C\w}}}{h} = 0$$
by definition of the tangent cone in \eqref{eq:tangent}.
Hence, we have
\IfConf{
\begin{align*}
\frac{d}{dt}&|z(t)|_{K\cap \Pwc{C\w}} \\
& \leq \liminf\limits_{h \searrow 0} \frac{|\chi(t)+h\omega|_{K\cap \Pwc{C\w}}}{h} + |\dot{z}(t) - \omega| \\
& = |\dot{z}(t) - \omega|.
\end{align*}
}{
\begin{align*}
\frac{d}{dt}|z(t)|_{K\cap \Pwc{C\w}}
\leq \liminf\limits_{h \searrow 0} \frac{|\chi(t)+h\omega|_{K\cap \Pwc{C\w}}}{h} + |\dot{z}(t) - \omega|
= |\dot{z}(t) - \omega|.
\end{align*}
}
Thus, for almost every $t \in [\tau^*, t^*]$,
$$\frac{d}{dt}|z(t)|_{K\cap \Pwc{C\w}} \leq |\dot{z}(t)|_{T_{K\cap \Pwc{C\w}}(\chi(t))}.$$
Since $K \cap \Pwc{C\w}$ is closed, by definition, $\chi(t) \in K \cap \Pwc{C\w}$ for every $t \in [\tau^*, t^*]$.
Condition \ref{item:rFI2} implies that for almost all $t \in [\tau^*, t^*]$, and every $w \in \Psi^w_c(\chi(t))$, we have
\begin{align}\label{eq:dini}
\frac{d}{dt}|z(t)|_{K\cap \Pwc{C\w}}
&\leq |\dot{z}(t)|_{T_{K\cap \Pwc{C\w}}(\chi(t))}\\
&\leq |\dot{z}(t)|_{F\w(\chi(t), w)}.
\end{align}
Since $t^* - \tau^*$ is positive and can be arbitrarily small, it is always possible to construct a neighborhood of $\chi(t)$ for every $t \in [\tau^*, t^*]$, denoted $U$, with $z(t) \in U$, and it is such that $\Psi^w_c(z(t)) \subset \Psi^w_c(\chi(t))$ by Assumption~\ref{assum:wbound}.
Then, because of that and the fact that the mapping $x \mapsto F\w(x, w_c)$ is locally Lipschitz uniformly in $w_c$ on $C\w$, we can construct a neighborhood $U'$ of $z(t)$ such that $\chi(t) \in U'$ for every $t \in [\tau^*, t^*]$ and for which there exists a constant $\lambda > 0$ satisfying
$$F\w(z(t), w_c) \subset F\w(\chi(t), w_c) + \lambda|z(t)- \chi(t)|\ball$$
for every $t \in [\tau^*, t^*]$ and every $w_c \in \Psi^w_c(z(t))$.
Hence, for every $t \in [\tau^*, t^*]$, every $w_c \in \Psi^w_c(z(t))$, and every $\eta \in F\w(z(t), w_c)$,
$$|\eta|_{F\w(\chi(t), w_c)} \leq \lambda|z(t)- \chi(t)|.$$
Moreover, since $\dot{z}(t) \in F\w(z(t), w_c)$, for every $w_c \in \Psi^w_c(z(t))$, together with \eqref{eq:dini} and \eqref{eq:chi}, we have that
\begin{align*}
\frac{d}{dt}|z(t)|_{K\cap \Pwc{C\w}} &\leq |\dot{z}(t)|_{F\w(\chi(t), w_c)}\\
&\leq \lambda|z(t) - \chi(t)| = \lambda|z(t)|_{K\cap \Pwc{C\w}}.
\end{align*}
Then, by the Gronwall Lemma (see \cite[Lemma A.1]{khalil.nonlinear.sys}), for every $t \in [\tau^*, t^*]$,
$$|z(t)|_{K \cap \Pwc{C\w}} = 0.$$
Since $K \cap \Pwc{C\w}$ is closed, $\phi(t^*,j^*) \in K \cap \Pwc{C\w}$, which contradicts the definition of $t^*$.
Thus, there does not exist maximal solution pair $(\phi, w) \in \sol_{\HS\w}(K)$ that eventually leaves $K\cap \Pwc{C\w}$ by flowing.
\end{enumerate}
Thus, the set $K$ is robustly forward pre-invariant for $\HS\w$.
\chai{
Following the proof of Theorem~\ref{thm:rwFI}, when $K \cap \Pwc{L\w} \subset \Pwd{D\w}$, with \ref{item:rwFI1} and \ref{item:rwFI2} satisfied, there exists a nontrivial solution pair $(\phi, w)$ with $\phi(0,0) = \xi$ to $\HS\w$ from every $\xi \in K$.
Then,} robust forward invariance of $K$ follows from the addition of condition \ref{item:*}.
As shown above, every $(\phi, w) \in \sol_{\HS\w}(K)$ has $\rge \phi \subset K$, thus, it suffices to show that every maximal solution pair to $\HS\w$ is complete.
We proceed by contradiction. Suppose there exists a maximal solution pair $(\phi^*, w^*) \in \sol_{\HS\w}(K)$ that is not complete, and $(T,J) = \sup \dom \phi^*$.
Because every $(\phi, w) \in \sol_{\HS\w}(K)$ has $\rge \phi \subset K$, by an application of Proposition~\ref{prop:wexistence}, $(\phi^*, w^*)$ only satisfies one of the cases described in item \ref{item:b.1.2}, \ref{item:b.2}, and \ref{item:c.2}.
In particular, condition \ref{item:*} eliminates case \ref{item:b.2} by assumption.
Then, condition \ref{item:rFI1} and condition \ref{item:rFI2} imply that $(\phi^*, w^*)$ can be extended within $K$ by jumps and flows, respectively.
More precisely, when $\phi^*(T,J) \in \Pwc{C\w}$, conditions in Assumption~\ref{assum:wdata} and item \ref{item:rFI2} imply the function $\widetilde{z} : [0, \varepsilon] \rightarrow \reals^n$ as described in \ref{item:VCw} in Proposition~\ref{prop:wexistence} exists with $\widetilde{w}_c(t) = 0$ for every $t \in [0,\varepsilon]$, and such $(\widetilde{z}, \widetilde{w}_c)$ can be used to extend $(\phi^*, w^*)$ to hybrid instant $(T+ \varepsilon, J)$, which contradicts the maximality of $(\phi^*, w^*)$.\footnote{Note that the resulting disturbance will be Lebesgue measurable and locally essentially bounded on interval $I^J$.}
When $\phi^*(T,J) \in \Pwd{D\w}$, jumps are always possible by virtue of condition \ref{item:rFI1}.
Therefore, the set $K$ is robustly forward invariant for $\HS\w$.
\end{proof}
\begin{remark}
In comparison to Theorem~\ref{thm:rwFI}, Lipschitzness of the set-valued map $F\w$ (uniformly in $w$) is assumed.
Together with Assumption~\ref{assum:wbound}, they are crucial to ensure that every solution pair stays in the designated set during flows.
Note that Assumption~\ref{assum:wbound} guarantees such property uniformly in $w_c$ (see the proof of Theorem~\ref{thm:rFI} for details).
We refer readers to the example provided below Theorem 3.1 in \cite{blanchini1999set}, which shows solutions leave a set due to the absence of locally Lipschitzness of the right-hand side of a continuous-time system.
\end{remark}
The following example shows an application of Theorem~\ref{thm:rFI}.
\begin{example}[Example~\ref{ex:oscillator} revisited]\label{ex:reoscillator}
Consider the hybrid system in Example~\ref{ex:oscillator}.
We apply Theorem~\ref{thm:rFI} to show the set $K_2 = \Pwc{C\w} \cup \Pwd{D\w}$ is robustly forward invariant for $\HS\w.$
Similar to Example~\ref{ex:oscillator}, $L\w = \emptyset$, Assumption~\ref{assum:wdata} and condition \ref{item:*} hold for $K_2, F\w, C\w$ and $D\w$.
Moreover, Assumption~\ref{assum:wbound} holds since $w_c \leq |x|$ for every $x \in \Pwc{C\w}$ and the map $F\w$ is locally Lipschitz on $C\w$ by construction. Then, condition \ref{item:rFI1} holds since for every $(x, w_d) \in (K_2 \times {\cal W}_d)\cap D\w$, the map $G\w$ only ``rotates" the state variable $x$ without changing $|x|$ within the unit circle centered at the origin. Condition \ref{item:rFI2} holds since
\begin{itemize}[leftmargin = 4mm]
\item for every $(x, w_c) \in (\partial K_2 \times {\cal W}_c)\cap C\w$, because $0 \leq w_c\leq |x| \leq 1$ and $x_1x_2 \geq 0$, we have
\begin{align*}
&\inner{\nabla (x_1^2 + x_2^2)}{F\w(x, w_c)} \\
&\quad = 2x_1(-x_2|x_1|) + 2x_2(w_cx_1|x_1|)\\
&\quad = 2x_1x_2 (w_c - 1)|x_1| \leq 0,
\end{align*}
which, applying item \ref{item:inner1} in Lemma~\ref{lem:innerTcone}, implies $F\w(x, w_c) \in T_{K_2 \cap \Pwc{C\w}}(x);$
\item for every $(x, w_c) \in ((\partial (\Pwc{C\w})\setminus \partial K_2) \times {\cal W}_c) \cap C\w$, \IfConf{the tangent cone $T_{K_2 \cap \Pwc{C\w}}(x)$ is given by
$$ \begin{cases}
\reals_{\geq 0} \times \reals & \text{\normalfont if } x \in C_1, x_1 = 0, x_2 \notin \{0,1\}\\
\reals_{\leq 0} \times \reals & \text{\normalfont if } x \in C_2, x_1 = 0, x_2 \notin \{0,-1\}\\
\reals \times \reals_{\geq 0} & \text{\normalfont if } x \in C_1, x_1 \notin \{0,1\}, x_2 = 0\\
\reals \times \reals_{\leq 0} & \text{\normalfont if } x \in C_2, x_1 \notin \{0,-1\}, x_2 = 0\\
\reals_{\geq 0}^2 \cup \reals_{\leq 0}^2 & x = 0,
\end{cases}$$
}{
we have
$$T_{K_2 \cap \Pwc{C\w}}(x) =
\begin{cases}
\reals_{\geq 0} \times \reals & \text{\normalfont if } x \in C_1, x_1 = 0, x_2 \notin \{0,1\}\\
\reals_{\leq 0} \times \reals & \text{\normalfont if } x \in C_2, x_1 = 0, x_2 \notin \{0,-1\}\\
\reals \times \reals_{\geq 0} & \text{\normalfont if } x \in C_1, x_1 \notin \{0,1\}, x_2 = 0\\
\reals \times \reals_{\leq 0} & \text{\normalfont if } x \in C_2, x_1 \notin \{0,-1\}, x_2 = 0\\
\reals_{\geq 0}^2 \cup \reals_{\leq 0}^2 & x = 0,
\end{cases}$$
}
which, applying item \ref{item:inner1} in Lemma~\ref{lem:innerTcone}, implies $F\w(x, w_c) \in T_{K_2 \cap \Pwc{C\w}}(x)$ holds true by definition of $F\w$.\footnote{We recall from Example~\ref{ex:oscillator} that $C_1 = \{x \in \reals^2: x_1 \geq 0, x_2 \geq 0, |x| \leq 1\}$ and $C_2 = \{x \in \reals^2: x_1 \leq 0, x_2 \leq 0, |x| \leq 1\}$.}
\end{itemize}
Thus, the set $K_2$ is robustly forward invariant for $\HS\w.$
\hfill $\triangle$
\end{example}
\subsection{Auxiliary Definitions and Results for Hybrid Systems}
\begin{definition}(solutions to $\HS$, \cite[Definition 2.6]{65})\label{def:solution}
A hybrid arc $\phi$ is a solution to the hybrid system $(C, F, D, G)$ if $\phi(0,0) \in \overline{C} \cup D$, and
\begin{enumerate}[label = (S\arabic*), leftmargin=2.3\parindent]
\item\label{item:S1} for all $j \in \mathbb{N}$ such that $I^j$ has nonempty interior
\begin{align*}
&\phi(t,j) \in C &\mbox{for all } &\quad t \in \text{\normalfont int} I^j,\\
&\frac{d\phi}{dt}(t,j) \in F(\phi(t,j)) &\mbox{for almost all } &\quad t \in I^j,
\end{align*}
\item\label{item:S2} for all $(t,j) \in \dom\phi$ such that $(t, j+1) \in \dom\phi$,
$$ \phi(t,j) \in D \qquad \phi(t,j+1) \in G(\phi(t,j)).$$
\end{enumerate}
\end{definition}
\begin{proposition}(\cite[Proposition 2.2]{120})\label{prop:existence}
Consider the hybrid system $\HS = (C,F,D,G)$. Let $\xi \in \overline{C} \cup D$. If $ \xi \in D$ or
\begin{enumerate}[label = (VC), leftmargin= 8mm]
\item\label{item:VC} there exist $ \varepsilon > 0$ and an absolutely continuous function $z: [0,\varepsilon] \rightarrow \reals^n$ such that $ z(0) = \xi, \dot{z}(t) \in F(z(t))$ for almost all $t \in [0,\varepsilon]$ and $z(t)\in C$ for all $t \in (0,\varepsilon]$,
\end{enumerate}
then there exists a nontrivial solution $\phi$ to $\HS$ with $\phi(0,0) = \xi$. If \ref{item:VC} holds for every $ \xi \in \overline{C} \setminus D$, then there exists a nontrivial solution to $\HS$ from every point of $ \overline{C} \cup D,$ and every $\phi \in \mathcal{S}_{\HS}$ satisfies exactly one of the following:
\begin{enumerate}[label = \alph*), leftmargin= 4mm]
\item $\phi$ is complete;
\item $\phi$ is not complete and ``ends with flow'', with $(T,J) = \sup\dom\phi$, the interval $I^J$ has nonempty interior; and either
\begin{enumerate}[label = b.\arabic*)]
\item $I^J$ is closed, in which case $\phi(T,J) \in \overline{C}\setminus(C \cup D)$; or
\item\label{item:b.2n} $I^J$ is open to the right, in which case $(T,J) \notin \dom\phi$, and there does not exist an absolutely continuous function $z : \overline{I^J} \rightarrow \reals^n$ satisfying $ \dot{z}(t) \in F(z(t))$ for almost all $t \in I^J, z(t) \in C$ for all $t \in \mbox{\normalfont int }I^J$, and such that $z(t) = \phi(t,J)$ for all $t \in I^J $;
\end{enumerate}
\item\label{item:cn} $\phi$ is not complete and ``ends with jump'': for $(T,J) = \sup\dom\phi$, one has $\phi(T,J) \notin \overline{C} \cup D$.
\end{enumerate}
Furthermore, if $G(D) \subset \overline{C} \cup D,$ then \ref{item:cn} above does not occur.
\end{proposition}
\subsection{Auxiliary Definitions and Results for Set-valued Maps}
\begin{definition}(outer semicontinuity of set-valued maps)\label{def:osc}
A set-valued map $S: \reals^n \rightrightarrows \reals^m$ is outer semicontinuous at $x \in \reals^n$ if for each sequence $\{x_i\}^{\infty}_{i=1}$ converging to a point $x \in \reals^n$ and each sequence $y_i \in S(x_i)$ converging to a point $y$, it holds that $y \in S(x)$; see \cite[Definition 5.4]{rockafellar2009variational}. Given a set $K \subset \reals^n$, it is outer semicontinuous relative to $K$ if the set-valued mapping from $\reals^n$ to $\reals^m$ defined by $S(x)$ for $x\in K$ and $\emptyset$ for $x \notin K$ is outer semicontinuous at each $x\in K$.
\end{definition}
\begin{definition}(Lipschitz continuity of set-valued maps)\label{def:lip}
Given a set-valued map $F: \reals^n \times {\cal W}_c \rightrightarrows \reals^n$, the mapping $x \mapsto F(x, w)$ is locally Lipschitz uniformly in $w$ at $x$, if there exists a neighborhood $U$ of $x$ and a constant $\lambda \geq 0$ such that for every $\xi \in U$
\begin{align*}
F(x, w) \subset F(\xi, w) + &\lambda |x - \xi|\ball\\
&\forall w \in \{w \in {\cal W}_c : (U \times {\cal W}_c) \cap \dom F\}.
\end{align*}
Furthermore, $x \mapsto F(x, w)$ is locally Lipschitz uniformly in $w$ on set $K \subset \dom F$ when it is locally Lipschitz uniformly in $w$ at each $x \in \Pwc{K}$.
\end{definition}
\chai{
\begin{lemma}(\cite[Theorem 2.9.10]{clarke1990optimization})\label{lem:innerTcone}
Given a set $S : = \{x: h(x) \leq 0\}$,
suppose that, for every $x \in \{x: h(x) = 0\}$, $h$ is directionally Lipschitz at $x$ with $0 \notin \nabla h(x) \neq \emptyset$ and the collection of vectors $Y : = \{y : \inner{\nabla h(x)}{y} <\infty \}$ is nonempty.
Then, the set $S$ admits a hypertangent at $x$ and
\begin{enumerate}[leftmargin = 4mm, label = \arabic*)]
\item\label{item:inner1} $y \in T_S(x) \ \ \text{\normalfont if } \inner{\nabla h(x)}{y} \leq 0$;
\item\label{item:inner2} $\exists y \in \text{\normalfont int} T_S(x)\cap \text{\normalfont int} Y$
such that $\inner{\nabla h(x)}{y} < 0$.
\end{enumerate}
\end{lemma}
}
|
1,116,691,497,811 | arxiv | \section{Introduction}
The study of $p$-brane solitons in low-energy superstring theories has
been the subject of detailed investigation recently [1--12]. With the
growing realisation that the duality symmetries of string theories can
bring about a unification of theories that were previously thought to
have been distinct, and that the non-perturbative spectrum of string
theory is within grasp, it becomes more and more important to achieve
a thorough understanding of the solitonic states, since they will have
an important r\^ole to play in the emerging theory. Many solitonic
$p$-brane solutions have been found, and in this review we shall not
attempt to cover all of the ground. Instead, we shall focus on an
approach to the problem of classifying the solutions that we have been
developing, which seems to have an advantage of simplicity. We also
find a simple way of determining the supersymmetry of the solutions.
The class of low-energy effective actions that we shall discuss are
those obtained by dimensional reduction of the effective action of
the ten-dimensional type IIA string theory. Equivalently, and more
simply, these maximal supergravity theories can be viewed as coming
from eleven-dimensional supergravity. As we shall see, all the
resulting lower-dimensional supergravity theories can be discussed in
a unified way, and the supersymmetry of the $p$-brane solutions in any
dimension can easily be determined by simply using the properties
of the eleven-dimensional transformation rules. In particular, there
is no need to examine the supersymmetry transformation rules of the
lower-dimensional theories themselves, and there is no need to
decompose the eleven-dimensional gamma matrices into products of
lower-dimensional ones.
The bosonic sector of the Lagrangian for $D=11$ supergravity is
${\cal L}= \hat e \hat R -\ft1{48} \hat e\, \hat F_4^2 +\ft16
\hat F_4\wedge\hat F_4\wedge \hat A_3$, where $\hat A_3$ is the 3-form
potential for the 4-form fields strength $\hat F_4$. Upon Kaluza-Klein
dimensional reduction to $D$ dimensions, this yields the following
Lagrangian:
\begin{eqnarray}
{\cal L} &=& eR -\ft12 e\, (\partial\vec\phi)^2 -\ft1{48}e\, e^{\vec a\cdot
\vec\phi}\, F_4^2 -\ft{1}{12} e\sum_i
e^{\vec a_i\cdot \vec\phi}\, (F_3^{i})^2
-\ft14 e\, \sum_{i<j} e^{\vec a_{ij}\cdot \vec\phi}\, (F_2^{ij})^2
\nonumber\\
&& -\ft14e\, \sum_i e^{\vec b_i\cdot \vec\phi}\, ({\cal F}_2^i)^2
-\ft12 e\, \sum_{i<j<k} e^{\vec a_{ijk} \cdot\vec \phi}\,
(F_1^{ijk})^2 -\ft12e\, \sum_{i<j} e^{\vec b_{ij}\cdot \vec\phi}\,
({\cal F}_1^{ij})^2 + {\cal L}_{\sst{FFA}}\ ,\label{dgenlag}
\end{eqnarray}
where $F_4$, $F_3^i$, $F_2^{ij}$ and $F_1^{ijk}$ are the 4-form, 3-forms,
2-forms and 1-forms coming from the dimensional reduction of $\hat F_4$ in
$D=11$; ${\cal F}_2^i$ are the 2-forms coming from the dimensional reduction
of the vielbein, and ${\cal F}_1^{ij}$ are the 1-forms coming from the
dimensional reduction of these 2-forms. The quantity $\vec\phi$ denotes an
$(11-D)$-component vector of scalar fields, which we shall loosely refer to
as dilatonic scalars that arise from the dimensional reduction of the
elfbein. These scalars appear undifferentiated, via the exponential
prefactors for the antisymmetric-tensor kinetic terms, and they should be
distinguished from the remaining spin-0 quantities in $D$ dimensions, namely
the 0-form potentials $A_0^{ijk}$ and ${\cal A}_0^{ij}$. These fields
have constant shift symmetries, under which the action is invariant, and are
thus properly thought of as 0-form potentials rather than true scalars. In
particular, their 1-form field strengths can adopt topologically non-trivial
configurations, corresponding to magnetically-charged sources for $p$-brane
solitons.
There are two subtleties that need to be addressed with regard
to the $p$-brane solutions following from (\ref{dgenlag}). Firstly, the
dimensional reduction of the $\hat F_4\wedge\hat F_4\wedge \hat A_3$ term in
$D=11$ gives rise to a term in $D$ dimensions, which we denote by
${\cal L}_{\sst{FFA}}$. This gives contributions to the equations of motion
which, in general, will be non-zero. The solutions that we wish to discuss,
in common with most of those in the literature, are ones for which this term
can be ignored. Thus we must check that our solutions are such that the
extra terms in the equations of motion following from
${\cal L}_{\sst{FFA}}$
vanish. The second complication is that the various field strengths occurring in (\ref{dgenlag})
are not, in general, simply given by the exterior derivatives of potentials.
There are extra terms, which we refer to loosely as ``Chern-Simons terms,''
coming from the process of dimensional reduction. Again, the solutions of
interest will be ones where these extra terms vanish, and so again, this
means that certain constraints must be satisfied in order for this to be
true. Both of these issues are discussed in detail in \cite{lp1} and so we
shall not consider them further here.
The only aspect of the bosonic Lagrangians (\ref{dgenlag})
that remains unexplained is the constant vectors $\vec a_{i\cdots j}$ and
$\vec b_{i\cdots j}$ appearing in the exponential prefactors of the kinetic terms for the
antisymmetric tensors. As was shown in \cite{lp1}, these ``dilatonic
vectors'' can be expressed as follows:
\begin{eqnarray}
&&F_{\sst{MNPQ}}\qquad\qquad\qquad\qquad\qquad\qquad
{\rm vielbein}\nonumber\\
{\rm 4-form:}&&\vec a = -\vec g\ ,\nonumber\\
{\rm 3-forms:}&&\vec a_i = \vec f_i -\vec g \ ,\label{dilatonvec}\\
{\rm 2-forms:}&& \vec a_{ij} = \vec f_i + \vec f_j - \vec g\ ,
\qquad\qquad\qquad \,\,\, \,\vec b_i = -\vec f_i\nonumber\,\\
{\rm 1-forms:}&&\vec a_{ijk} = \vec f_i + \vec f_j + \vec f_k -\vec g
\ ,\qquad\qquad\vec b_{ij} = -\vec f_i + \vec f_j\ ,\nonumber
\end{eqnarray}
where the vectors $\vec g$ and $\vec f_i$ have $(11-D)$ components
in $D$ dimensions, and are given by
\begin{eqnarray}
\vec g &=&3 (s_1, s_2, \ldots, s_{11-\sst D})\ ,\nonumber\\
\vec f_i &=& \Big(\underbrace{0,0,\ldots, 0}_{i-1}, (10-i) s_i, s_{i+1},
s_{i+2}, \ldots, s_{11-\sst D}\Big)\ ,
\end{eqnarray}
where $s_i = \sqrt{2/((10-i)(9-i))}$. It is easy to see that they satisfy
\begin{equation}
\vec g \cdot \vec g = \ft{2(11-D)}{D-2}, \qquad
\vec g \cdot \vec f_i = \ft{6}{D-2}\ ,\qquad
\vec f_i \cdot \vec f_j = 2\delta_{ij} + \ft2{D-2}\ .\label{gfdot}
\end{equation}
Note that the definitions in (\ref{dilatonvec}) are given for $i<j<k$, and
that the vectors $\vec a_{ij}$ and $\vec a_{ijk}$ are antisymmetric in their
indices. The 1-forms ${\cal F}_{\sst{M}i}^{(j)}$ and hence the vectors
$b_{ij}$ are only defined for $i<j$, but it is sometimes convenient to
regard them as being antisymmetric too, by defining $\vec b_{ij}=-\vec b_{ji}$
for $i>j$. Eqn.\ (\ref{dilatonvec}), together
with (\ref{gfdot}), contains all the necessary information about the dilaton
vectors in $D$-dimensional maximal supergravity.
In the absence of a $p$-brane, the equations of motion following from
(\ref{dgenlag}) admit a $D$-dimensional Minkowski vacuum, with $SO(1,D-1)$
Lorentz symmetry. The $p$-brane solutions we shall consider break this to
$SO(1,d-1)\times SO(D-d)$, where $d=p+1$ is the dimension of the world
volume of the $p$-brane. The metric tensor, compatible with this residual
symmetry, is given by
\begin{equation}
ds^2 = e^{2A}\, dx^\mu dx^\nu \eta_{\mu\nu} + e^{2B}\, dy^m dy^m\ ,
\end{equation}
where $A$ and $B$ are functions only of $r=\sqrt{y^m y^m}$. The coordinates
$x^\mu$ lie in the $d$-dimensional world volume, and $y^m$ lie in the
$(D-d)$-dimensional transverse space.
The solutions that we shall consider involve a subset of the
dilatonic scalars $\vec\phi$, and a subset of the antisymmetric tensor field
strengths $F$ and ${\cal F}$. The rank $n$ of all the participating fields
strengths in a given solution are all the same, although in dimensions
$D\le 7$ certain field strengths of higher rank will be dualised to field
strengths of lower rank. Each participating field strength can have
non-vanishing components given by either
\begin{equation}
F_{m\mu_1\ldots\mu_{n-1}} = \epsilon_{\mu_1\ldots\mu_{n-1}} (e^C)'\,
{y^m\over r}\qquad
{\rm or}\qquad
F_{m_1\ldots m_n} = \lambda \epsilon_{m_1\ldots m_np}\,
\fft{y^p}{r^{n+1}}\ ,\label{fansatz}
\end{equation}
where a prime denotes a derivative with respect to $r$. The first case
gives rise to an elementary $(d-1)$-brane with $d=n-1$, and the second
gives rise to a solitonic $(d-1)$-brane with $d= D-n-1$. These two field
configurations have electric charge
$u=\ft{1}{4\omega_{\sst{D}-n}} \int_{\partial\Sigma} *F$ or
magnetic charge $v=\ft{1}{4\omega_{n}}\int_{\partial\Sigma} F$ respectively,
where $\partial\Sigma$ is the boundary $(D-d-1)$-sphere of the transverse
space. The first case corresponds to an elementary, or fundamental,
$p$-brane solution, and the second case corresponds to a solitonic $p$-brane
solution. In a case where the degree $n$ of the field strength is such that
$D=2n$, the possibility arises that field strengths might have both electric
and magnetic contributions, involving both of the ans\"atze in
(\ref{fansatz}) simultaneously. In such cases, the configuration describes
a dyonic $p$-brane. It is useful to distinguish two different kinds of
dyonic configuration. In a dyonic solution of the first type, each
participating field strength has only electric or magnetic non-vanishing
components, but some of the field strengths are electric while others are
magnetic. In a dyonic solution of the second type, each participating field
strength has both electric and magnetic non-vanishing components. We shall
encounter both kinds of dyonic solution later.
In order to determine the supersymmetry properties of the various
$p$-brane solutions, it suffices to study the transformation laws of $D=11$
supergravity. In particular, from the commutator of the conserved
supercharges $Q_\epsilon = \int_{\partial\Sigma} \bar\epsilon \Gamma^{\sst{ABC}}
\psi_{\sst C}\, d\Sigma_{\sst{AB}}$, we may read off the $32\times32$
Bogomol'nyi\ matrix ${\cal M}$, defined by $[Q_{\epsilon_1},Q_{\epsilon_2}] =
\epsilon_1^\dagger {\cal M} \epsilon_2$, whose zero eigenvalues correspond to
unbroken components of $D=11$ supersymmetry. The expression for ${\cal M}$
for maximal supergravity in an arbitrary dimension $D$ then follows by
dimensional reduction of the expression in $D=11$. A
straightforward calculation shows that it is given by \cite{lp1}
\begin{eqnarray}
{\cal M} &=& m\rlap 1\mkern4mu{\rm l} + u\, \Gamma_{012} + u_i\, \Gamma_{01i} +
\ft12 u_{ij}\, \Gamma_{0ij} +p_i \Gamma_{0i}\nonumber\\
&&+ v\,\Gamma_{\hat1\hat2\hat3\hat4\hat5} + v_i \,
\Gamma_{\hat1\hat2\hat3\hat4i}+\ft12 v_{ij}\, \Gamma_{\hat1\hat2\hat3ij}
+ \ft16 v_{ijk}\, \Gamma_{\hat1\hat2ijk} + q_i\, \Gamma_{\hat1\hat2\hat3 i}
+\ft12 q_{ij}\, \Gamma_{\hat1\hat2ij}\ .\label{genbog}
\end{eqnarray}
The indices $0, 1,\ldots$ run over the dimension of
the $p$-brane worldvolume, $\hat1,\hat2,\ldots$ run over the transverse
space of the $y^m$ coordinates, and $i,j,\ldots$ run over the dimensions
that were compactified in the Kaluza-Klein reduction from 11 to $D$
dimensions. The quantities $u$, $u_i$, $u_{ij}$ and $p_i$ are the
electric Page charges associated with the field strengths $F_4$, $F_3^i$,
$F_2^{ij}$ and ${\cal F}_2^i$ respectively, while $v$, $v_i$, $v_{ij}$,
$v_{ijk}$, $q_i$ and $q_{ij}$ are the magnetic Page charges associated with
$F_4$, $F_3^i$, $F_2^{ij}$, $F_1^{ijk}$, ${\cal F}_2^i$ and ${\cal F}_1^{ij}$
respectively. The quantity $m$ is the mass per unit $p$-volume for the
solution, given by $\ft12 (A'-B')e^{-B} r^{\tilde d +1}$ in the limit
$r\rightarrow \infty$, where $\tilde d\equiv D-d-2$.
Once the mass per unit $p$-volume and the Page charges have been
determined for a given $p$-brane solution, it becomes a straightforward
algebraic exercise to calculate the eigenvalues of the Bogomol'nyi\ matrix, given by
(\ref{genbog}). The fraction of $D=11$ supersymmetry that is preserved by
the solution is then equal to $k/32$, where $k$ is the number of zero
eigenvalues of the $32\times32$ Bogomol'nyi\ matrix.
\section{Single-scalar solutions}
In the single-scalar $p$-brane solutions, the Lagrangian (\ref{dgenlag})
is consistently truncated to the simple form
\begin{equation}
{\cal L} = e
R -\ft12 e\, (\partial\phi)^2 -\fft1{2n!} e\, e^{a\phi} F_n^2 \ ,\label{genlag}
\end{equation}
where the scalar field $\phi$ is some linear combination of the dilatonic
scalars $\vec\phi$ of the $D$-dimensional theory, and $F_n$ is a single
canonically-normalised $n$-index field strength, to which all of the $N$
original field strengths that participate in the solution are proportional.
The constant $a$ appearing in the exponential prefactor can conveniently be
parameterised as
\begin{equation}
a^2 = \Delta - \fft{2d\tilde d} {D-2}\ ,\label{avalue}
\end{equation}
where $\tilde d\equiv D-d -2$ and $d\tilde d = (n-1)(D-n-1)$. The
quantity $\Delta$, unlike $a$ itself, is preserved under Kaluza-Klein
dimensional reduction.
If we write $\vec\phi=\vec n\, \phi + \vec\phi_\perp$, where $\vec n$ is
a constant unit vector and $\vec n\cdot \vec\phi_\perp=0$, then it is easy to
see that the conditions that must be satisfied in order for the truncation to
(\ref{genlag}) to be consistent with the equations of motion for
$\vec\phi_\perp$ are that $\vec a_\a\cdot \vec n =a$ for $\a$, where $\vec
a_\a$ ($1\le\a\le N$) denotes the dilaton vectors associated with the $N$
participating field strengths, and in addition, that
\begin{equation}
\sum_{\alpha} \vec a_{\alpha} F_{\alpha}^2 = a \vec n \sum_\alpha
F^2_\alpha \ , {\rm \ and\ hence\ }
\sum_\alpha M_{\alpha\beta} F_\alpha^2 = a^2 \sum_\alpha F_\alpha^2\ ,
\label{fequation1}
\end{equation}
where $M_{\alpha\beta}\equiv \vec a_\alpha \cdot \vec a_\beta$. If the
matrix $M_{\alpha\beta}$ is invertible, we therefore have
\begin{equation}
F_\beta^2 = a^2 \sum_\alpha (M^{-1})_{\alpha\beta} \sum_\gamma F_\gamma^2
\ ,\qquad
a^2 = \Big( \sum_{\alpha, \beta} (M^{-1})_{\alpha\beta} \Big)^{-1}
\ ,\label{avaluesol}
\end{equation}
which gives rise to a $p$-brane solution with $\Delta = \Big( \sum_{\alpha,
\beta} (M^{-1})_{\alpha\beta} \Big)^{-1} + 2d\tilde d/(D-2)$. In this
case, we can easily see that $a \ne 0$, and hence the unit vector $\vec n$
can be read off from eqn (\ref{fequation1}). If $M_{\alpha\beta}$ is
non-invertible, then it is clear from (\ref{fequation1}) that one solution is
is when $F_\a^2$ is a zero eigenvector of $M_{\alpha\beta}$, with the
constant $a$ being zero. It turns out that this is the only solution in the
singular case that does not simply reduce to an already-considered
non-singular case with a smaller
number $N$ of participating field strengths \cite{lp1}. It is also clear
that if the number of participating field strengths exceeds the dimension
$(11-D)$ of the dilaton vectors, then the associated matrix $M_{\a\beta}$
will be singular, and in fact it turns out that in all such cases, there is
no new solution \cite{lp1}. Thus in any dimension $D$, it follows that the
number $N$ of participating field strengths must always satisfy $N\le
11-D$.
Having reduced the Lagrangian (\ref{dgenlag}) to (\ref{genlag}) by the
above procedure, it is now a simple matter to obtain solutions for the
equations of motion that follow from (\ref{genlag}). An important point is
that one can reduce the second-order equations of motion for $A, B$ and
$\phi$ to first-order equations, by making an ansatz where they are all
proportional, and in which the
exponential factor in these variables, coming from the $F^2$ source term on
the right-hand sides of their equations of motion, is required to be
proportional to $A'$ \cite{lpss}. In the case of solutions where some
supersymmetry remains unbroken, these requirements are dictated by the
supersymmetry transformation rules. It also provides a way to obtain
solutions of the same general form, even in cases where there is no
residual unbroken supersymmetry. The solution is
\begin{eqnarray}
ds^2 &=& \Big (1+\fft{k}{r^{\tilde d}}\Big)^{-\ft{4\tilde d}{\Delta(D-2)}}
\, dx^\mu dx^\nu \eta_{\mu\nu} + \Big(1+ \fft{k}{r^{\tilde d}}\Big)^{
\ft{4d}{\Delta(D-2)}}\, dy^m dy^m\ ,\nonumber\\
e^\phi &=& \Big (1+\fft{k}{r^{\tilde d}}\Big)^{
\ft{2a}{\epsilon \Delta}}\ , \label{gensol}
\end{eqnarray}
where $\epsilon = 1$ and $-1$ for the elementary and the
solitonic solutions respectively, and $k= - \sqrt{\Delta}
\lambda/(2 \tilde d)$. In the elementary case, the function $C$ satisfies
the equation
\begin{equation}
e^C = \fft{2}{\sqrt\Delta} \Big ( 1+ \fft{k}{r^{\tilde d}}\Big)^{-1}\ .
\end{equation}
The mass of the solution is given by $\lambda/(2\sqrt\Delta)$.
Note that the dual of the solution for the field strength in the elementary
case is identical to the field strength of the solitonic case, and {\it vice
versa}. For this reason, we shall only consider solutions for field
strengths with $n\le D/2$.
In order to enumerate all the single-scalar $p$-brane solutions of this
type, one simply has to consider, for each dimension $D$ and each degree $n$
for the field strengths, all possible choices of the associated $N\le 11-D$
dilaton vectors, and then calculate the values of $a$, and the corresponding
ratios of participating field strengths, using the above equations. This is
easily done for $n=4$ (where there is always only one field strength) and
$n=3$ (where the number of field strengths is small). For $n=2$ and $n=1$,
where the numbers of field strengths grow significantly with decreasing
dimension $D$, the enumeration is most conveniently carried out by computer.
Substituting the results for the field strengths into (\ref{genbog}), it is
then straightforward, for each solution, to determine the fraction of the
original supersymmetry that is preserved. Most of the solutions turn out to
break all of the supersymmetry, but in certain cases, some of the
supersymmetry survives. Details may be found in \cite{lp1}; here, we
summarise the results:
\bigskip
\noindent{\it 4-Form solutions}
\medskip
Since there is only one 4-form field strength in any dimension, there is a
a unique solution for each dimension $D$, in which the scalar field $\phi$
appearing in (\ref{genlag}) is taken to be the canonically-normalised scalar
proportional to the entire exponent of the prefactor for the 4-form's kinetic
term. The value of $a$ for this solution corresponds to $\Delta=4$ for all
$D$. The eigenvalues of the Bogomol'nyi\ matrix (\ref{genbog}) turn out to be
$\mu=m \{0_{16}, 2_{16}\}$, where the subscripts denote the degeneracies
of each eigenvalue. Thus these solutions preserve $\ft12$ of the $D=11$
supersymmetry.
\bigskip
\noindent{\it 3-Form solutions}
\medskip
From (\ref{dilatonvec}) and (\ref{avaluesol}), it is easy to see that
there exist solutions involving $N$ participating field strengths, with
values of $\Delta$ given by $\Delta=2+2/N$, where, as always, $N\le 11-D$.
Substituting the solutions for the ratios of the field strengths, and hence
the ratios of the Page charges, into (\ref{genbog}), one finds \cite{lp1}
that all the supersymmetry is broken except when $N=1$, and hence
$\Delta=4$, and the eigenvalues are the same as for the case of 4-form
solutions, namely $\mu=m \{0_{16}, 2_{16}\}$. In this supersymmetric case,
any one of the 3-form field strengths can be used in constructing the
solution. In $D=6$, there are further solutions, associated with the
possibility of truncating the theory to self-dual supergravity. We shall
discuss these further in the next section.
\bigskip
\noindent{\it 2-Form solutions}
\medskip
As one can see from (\ref{dilatonvec}), the set of possible choices for
dilaton vectors is considerably larger for 2-form field strengths, and
correspondingly there are many more solutions. If only a single 2-form field
strength participates in the solution, then as usual we find $\Delta=4$.
For $N=2$, we can have $\Delta=3$ or $\Delta=2$. For $N=3$, there are always
solutions with $\Delta=\ft83$ and $\ft{12}7$. In addition, if $D\le 5$,
there is a further solution with $\Delta=\ft43$. As $N$ increases to its
maximum allowed value of $N=7$ in $D=4$, more and more solutions appear.
Full details may be found in \cite{lp1}. Here, we shall just give details
for the supersymmetric solutions. These occur for $N=1,2,3$ and 4
participating field strengths, with
$\Delta=4/N=4,2,\ft43$ and 1 respectively. The eigenvalues of the Bogomol'nyi\
matrix are as follows:
\begin{eqnarray}
\Delta=4: &&\mu=2 m \{0_{16},1_{16}\} \ ,\qquad\qquad D\le10\ ,\nonumber\\
\Delta=2: &&\mu=m \{0_{8},1_{16}, 2_{8}\} \ ,\qquad\qquad D\le9\ ,
\nonumber\\
\Delta=\ft43: &&\mu=\ft23 m \{0_{4},1_{12},2_{12},3_4\}\ ,
\qquad\qquad D\le5\ ,\label{n2eig}\\
\Delta=1': &&\mu=m \{0_{4},1_{24}, 2_{4}\}\ ,\qquad\qquad D=4 \ ,\nonumber
\end{eqnarray}
where we also indicate the maximum dimension $D$ in which each solution can
occur. We see that the $p$-brane solutions preserve $\ft12, \ft14,
\ft18$, and
$\ft18$ of the $D=11$ supersymmetry respectively. The prime on the value of
$\Delta$ for the fourth case is for later convenience, to distinguish it from
another $\Delta=1$ configuration that will arise in the discussion of 1-form
solutions. It should also be noted that the equations (\ref{avaluesol})
only determine the relations between the {\it squares} of the Page charges,
and thus the signs can be arbitrarily chosen. For the configurations with
$N\le3$ participating field strengths that give rise to supersymmetric
solutions, the eigenvalues of the Bogomol'nyi\ matrix are insensitive to these sign
choices. For the $N=4$ case, on the other hand, the choice of relative
signs does matter, and it turns out that there are exactly two possible sets
of eigenvalues that can result. One of these is the supersymmetric solution
with $\Delta=1'$ listed in (\ref{n2eig}); for the inequivalent choice of
relative signs, the eigenvalues are $\mu=m \{ 1_{16},3_{16}\}$, and thus
there is no supersymmetry. Note that any one of the 2-form field strengths
can be used to obtain the $\Delta=4$ solution. For $\Delta=2$, there are
various possible pairs of field strengths that can be chosen, the number of
such choices increasing with decreasing $D$. An example, which can be used
in all dimensions $D\le9$, is to choose $F_2^{12}$ and ${\cal F}_2^{1}$,
with Page charges given by $u_{12}=p_1=\lambda/(4\sqrt2)$. The
non-supersymmetric $\Delta=3$ solution is obtained for other choices of two
participating field strengths, for example $p_1=p_2=\lambda/(4\sqrt2)$.
Examples with $N=3$ and $N=4$ field strengths, yielding the $\Delta=\ft43$
and $\Delta=1$ supersymmetric solutions, are
$u_{12}=u_{34}=u_{56}=\lambda/8$, and $u_{23}=u_{46}=u_{57}=p_1^*=\lambda/8$
respectively. In the latter case, $p_1^*$ denotes an electric Page charge
for the dual of the field strength ${\cal F}_2^{1}$. As mentioned above, if
the signs of the charges in this $\Delta=1$ case are changed, then eight of
the sign choices give the same solution whilst for the other eight we still
get a solution, with the same metric, but with no supersymmetry preserved.
\bigskip
\noindent{\it 1-Form solutions}
\medskip
The situation for 1-form solutions is more complicated again; further
details may be found in \cite{lp1}. As $N$ increases up to 7, there is a
considerable proliferation of solutions, mostly non-supersymmetric. Here, we
shall just describe the supersymmetric $p$-brane solitons. There are in
total eight inequivalent field configurations that can give rise to
supersymmetric solutions, namely one for each value of $N$ in the range $1\le
N\le 7$, together with a second inequivalent possibility for $N=4$. In all
cases, the value of $\Delta$ is given by $\Delta=4/N$. Four of the eight
solutions have Bogomol'nyi\ matrices with identical eigenvalues to those given in
(\ref{n2eig}) for 2-form solutions (although the Bogomol'nyi\ matrices themselves are
of course different). These again arise for $N=1, 2, 3$ and 4 participating
field strengths. Examples of the Page charges that can give rise to these
four case are $q_{12}$, $\{q_{12}, v_{123}\}$, $\{q_{12}, q_{45}, v_{123}
\}$ and $\{ q_{12}, q_{45}, v_{123}, v_{345}\}$ respectively, where in each
case all the listed charges are equal. The remaining four solutions, which
cannot occur above $D=4$ dimensions, have eigenvalues as follows:
\begin{eqnarray}
\Delta =1:&& \{q_{12}, q_{34}, q_{56}, v_{127}\}\ ,\qquad
\mu =\ft12 m\{0_2, 1_8, 2_{12}, 3_8, 4_2\}\ ,\nonumber\\
\Delta = \ft45: && \{q_{12}, q_{34}, q_{56}, v_{127}, v_{347}\}\ ,\qquad
\mu= \ft25 m\{ 0_2, 1_2, 2_{12}, 3_{12},
4_2, 5_{2}\}\ ,\nonumber\\
\Delta = \ft23: &&
\{q_{12}, q_{34}, q_{56}, v_{127}, v_{347}, v_{567}\}\ ,\qquad
\mu=\ft13m \{ 0_2, 2_6, 3_{16}, 4_6,
6_2\}\ ,\label{n1eig}\\
\Delta =\ft47:&&
\{q_{12}, q_{34}, q_{56}, v_{127}, v_{347},v_{567}, v_7^*\}\ ,\qquad
\mu=\ft27 m \{0_2, 3_{14}, 4_{14}, 7_2\}\ ,\nonumber
\end{eqnarray}
where in each case the listed set of Pages charges provides an example of a
set that gives rise to the solution. In each case, the Page charges are all
equal. All of these four cases correspond to solutions that preserve
$\ft1{16}$ of the $D=11$ supersymmetry. In the first of these four
solutions, the eigenvalues are insensitive to the relative sign choices for
the Page charges, but in the last three cases, we again find the phenomenon
that there are precisely two inequivalent sets of eigenvalues for each
$\Delta$, depending on the relative signs of the Page charges. We have
given the choices that include zero eigenvalues. The other choices, for
which there is no supersymmetry, give rise to the sets of eigenvalues
\begin{eqnarray}
\Delta = \ft45: &&
\mu= \ft25 m\{ 1_8, 2_8, 3_8, 4_8 \}\ ,\nonumber\\
\Delta = \ft23: &&
\mu=\ft13m \{ 1_4, 2_8, 3_8, 4_8,
5_4\}\ ,\label{n1eigns}\\
\Delta =\ft47:&&
\mu=\ft27 m \{1_2, 2_6, 3_8, 4_8,5_6,6_2\}\ ,\nonumber
\end{eqnarray}
To conclude this section, we give a table that summarises the dimensions
in which the various supersymmetric $p$-brane solutions first occur:
\bigskip\bigskip
\centerline{
\begin{tabular}{|c|c|c|c|c|}\hline
&\phantom{for} 4-form\phantom{for} &\phantom{for} 3-form\phantom{for}
&\phantom{for} 2-form\phantom{for} & 1-form\\ \hline\hline
$D=11$ & $\Delta = 4 $ & & & \\ \hline
$D=10$ & & $\Delta =4$ & $\Delta=4$ & \\ \hline
$D=9$ & & & $\Delta =2$ & $\Delta =4$ \\ \hline
$D=8$ & & & & $\Delta =2$ \\ \hline
$D=7$ & & & & \\ \hline
$D=6$ & & $\Delta =2$ & & $\Delta = \ft43,1'$ \\ \hline
$D=5$ & & & $\Delta=\ft43$ & \\ \hline
$D=4$ & & &$\Delta = 1'$ & $\Delta = 1, \ft45, \ft23, \ft47$\\
\hline
\end{tabular}}
\bigskip
\centerline{Table 1: Supersymmetric $p$-brane solutions}
\bigskip
\section{Dyonic solutions}
In dimensions $D=2n$, the field strength $F_n$ can in principle have
components given by both the elementary and solitonic ans\"atze
(\ref{fansatz}) simultaneously. In this case, the equations of motion
can be reduced to the two independent differential equations
\begin{equation}
\phi'' + n \fft{\phi'}{r} =\ft12 a (S_1^2 - S_2^2)\ ,\qquad
A'' + n \fft{A'}{r} = \ft14 (S_1^2 + S_2^2)\ ,\label{phiadiff}
\end{equation}
together with the relations $B=-A$, $(e^C)' = \lambda_1 e^{a\phi + 2(n-1)A}
r^{-n}$, where $S_1$ and $S_2$ are given by
\begin{equation}
S_1 = \lambda_1 e^{\ft12 a \phi + (n-1)A} r^{-n}\ ,\qquad
S_2 = \lambda_2 e^{-\ft12 a\phi + (n-1)A} r^{-n}\ .\label{s1s2}
\end{equation}
By making a suitable ansatz of the kind discussed in the previous section,
which reduces the equations to first-order equations, one finds \cite{lp1}
that the equations (\ref{phiadiff}) admit a simple solution either when
$a^2=n-1$, and hence $\Delta=2n-2$, given by
\begin{equation}
e^{-\ft12 a \phi -(n-1) A} = 1 + \ft{\lambda_1}{a\sqrt2} r^{-n+1}\ ,\qquad
e^{+\ft12 a \phi -(n-1) A} = 1 + \ft{\lambda_2}{a\sqrt2} r^{-n+1}\ ,
\label{dyonicsol1}
\end{equation}
or when $a=0$, and hence $\Delta=n-1$, given by
\begin{equation}
\phi = 0\ ,\qquad e^{- (n-1) A} = 1 +\ft12
\sqrt{\ft{\lambda_1^2+\lambda_2^2}{(n-1)}} r^{-n+1}\ .\label{dyonicsol2}
\end{equation}
The possible dimensions for dyonic solutions are $D=8$, 6 and 4. As
discussed in \cite{lp1}, solutions of the kind we are discussing, where the
contributions from the ${\cal L}_{\sst{FFA}}$ term are assumed to vanish,
cannot occur in $D=8$. Furthermore, the solutions (\ref{dyonicsol1}) and
(\ref{dyonicsol2}) would require that the 4-form have a dilaton prefactor
with $\Delta=6$ or $\Delta=3$, whereas in fact it has $\Delta=4$. Thus we
are left with the cases $D=6$ and $D=4$.
\bigskip
\noindent{\it Dyonic solutions in $D=6$}
\medskip
In $D=6$, the constraints implied by the requirement that ${\cal
L}_{\sst{FFA}}$ not contribute imply that there cannot be dyonic solutions of
the first type, where each individual field strength has only electric or
magnetic components. However, dyonic solutions of the second type, where
each field strength has electric and magnetic components, can occur, for
$N\le5$ participating field strengths, with $\Delta=2+2/N$. The simple
dyonic solutions (\ref{dyonicsol1}) and (\ref{dyonicsol2}) require $\Delta=4$
and $\Delta=2$ respectively, and thus we see that there exist dyonic
solutions given by (\ref{dyonicsol1}) when $N=1$.
The mass per unit length of this
dyonic string is $m=\ft14(\lambda_1 +\lambda_2)$, and the Page charges are
$u=\ft14\lambda_1$ and $v=\ft14 \lambda_2$. The eigenvalues of the
Bogomol'nyi matrix are given by
\begin{equation}
\mu = m\pm u \pm v = \{0_8, (\ft12 \lambda_1)_8,
(\ft12 \lambda_2)_8, (\ft12 (\lambda_1 + \lambda_2))_8 \}\ ,\label{extra1}
\end{equation}
where, as usual, the subscripts on the eigenvalues indicate their
degeneracies. Thus the solution preserves $\ft14$ of the supersymmetry
\cite{dfkr}. When either $\lambda_1=0$ or $\lambda_2=0$, the solution
reduces to the previously-discussed purely solitonic and purely
elementary solutions, which preserve $\ft12$ of the supersymmetry. When
$\lambda_1=\lambda_2$, in which case the field strength becomes self-dual
and the dilaton vanishes, the solution is equivalent to
the self-dual string in $D=6$ self-dual supergravity, which we shall discuss
below. When $\lambda_1 =-\lambda_2$, the field strength is anti-self-dual,
and we have a massless string which preserves $\ft12$ of the supersymmetry;
however, the eigenvalues, given by (\ref{extra1}), for such a solution are
not positive semi-definite. In this case, the dilaton field does not
vanish, and hence the solution is distinct from the anti-self-dual string in
$D=6$ anti-self-dual gravity. It is worth remarking that the eigenvalues
(\ref{extra1}) for these dyonic solutions of the second type are quite
different from those for all the solutions we have discussed previously.
In those cases, the eigenvalues are non-negative as long as the mass per
unit $p$-volume is positive. However, for the dyonic solutions of the
second type, we see that the Page charges can be chosen so that eigenvalues
(\ref{extra1}) of the Bogomol'nyi matrix take both signs, even when the
mass is positive.
In the above discussion, we saw that the field strength of the solution
could be chosen to be either self-dual or anti-self-dual. In fact, one can
alternatively truncate the theory so as to retain a single 3-form
field strength on which a self-dual (or anti-self-dual) condition is imposed
\cite{romans}. In this case, the dilatonic fields are all consistently
truncated from the theory, implying that $a=0$ and hence $\Delta =2$. The
metric is given by (\ref{dyonicsol2}) with
$\lambda_1=\lambda_2=\lambda$. The mass per unit length is given by $m=\ft12
\sqrt{(\lambda_1^2 +
\lambda_2^2)/\Delta} = \ft12 \lambda$; the Page charges of the solution
comprise an electric charge $u$ and a magnetic charge $v$, with $u=v=\ft14
\lambda$ ($v=-\ft14 \lambda$ for the anti-self-dual case). The
eigenvalues of the Bogomol'nyi matrix are given by $\mu = \fft{\lambda}{4}
(2\pm 1\pm1)= m \{0_8,1_{16}, 2_8\}$, and so a quarter of the $D=11$
supersymmetry is preserved in this (anti)-self-dual case. Note that the
mass per unit $p$-volume of the self-dual (anti-self-dual) solution in the
previous paragraph is given by $m=u + v$, whilst the mass of these
solution in $D=6$ self-dual (anti-self-dual) supergravity is given by
$m=\sqrt{u^2 + v^2}$.
\bigskip
\noindent{\it Dyonic solutions in $D=4$}
\medskip
In $D=4$, the solutions (\ref{dyonicsol1}) and (\ref{dyonicsol2}) occur
when $\Delta=2$ and $\Delta=1$, corresponding to $a=1/\sqrt3$ and $a=0$.
As we discussed for the elementary and solitonic solutions in the previous
section, in the case $\Delta=1$, arising when there are 4 participating
field strengths, the relative signs of the Page charges, which are
undetermined by the field equations, affect the eigenvalues of the Bogomol'nyi\
matrix, giving two inequivalent outcomes, one with supersymmetry and the
other without. We can therefore now have three inequivalent sets of
eigenvalues, depending on whether the relative signs in each of the electric
and the magnetic sectors are chosen both to reduce to the previous
supersymmetric choices, or else one supersymmetric and the other not, or
finally both non-supersymmetric. (The occurrence of non-supersymmetric
solutions for certain sign choices for the Page charges has also been seen
recently in \cite{ko}.) Accordingly, we list the three possibilities for
$\Delta=1$ below, in this order:
\begin{eqnarray}
\Delta=1: && m=\ft12 \lambda\ ,\qquad
\{u_{57}, u_{46}, u_{23}, p_1^*, v_{57}, v_{46}, v_{23}, q_1^*\}=
\ft18 \{\lambda_1, \lambda_1, \lambda_1, \lambda_1,
\lambda_2, \lambda_2, \lambda_2, -\lambda_2\}\ ,\nonumber\\
&&\mu=\ft12 \lambda\, \{0_4, 1_{24}, 2_4\}\ ,
\nonumber\\
&&\nonumber\\
\Delta=1: && m=\ft12 \lambda\ ,\qquad
\{u_{57}, u_{46}, u_{23}, p_1^*, v_{57}, v_{46}, v_{23}, q_1^*\}=
\ft12 \{\lambda_1, \lambda_1, \lambda_1, \lambda_1,
\lambda_2, \lambda_2, \lambda_2, \lambda_2\}\ ,\nonumber\\
&&\mu=\ft12 \{(\lambda-\ft12 \lambda_2)_4,
(\lambda +\ft12 \lambda_2)_4,
(\lambda -\sqrt{\lambda_1^2+\ft14\lambda_2^2})_{12},
(\lambda +\sqrt{\lambda_1^2+\ft14\lambda_2^2})_{12} \}\ ,
\nonumber\\
&&\nonumber\\
\Delta=1: && m=\ft12 \lambda\ ,\qquad
\{u_{57}, u_{46}, u_{23}, p_1^*, v_{57}, v_{46}, v_{23}, q_1^*\}=
\ft12 \{\lambda_1, \lambda_1, \lambda_1, -\lambda_1,
\lambda_2, \lambda_2, \lambda_2, \lambda_2\}\ ,\nonumber\\
&&\mu=\ft14 \lambda\, \{1_{16}, 3_{16} \}\ ,
\nonumber\\
&&\nonumber\\
\Delta = 2:&&m=\fft{\lambda_1 + \lambda_2}{2\sqrt2}\ ,\qquad
\{p_1, u_{12}, q_1, v_{12} \} = \ft{1}{4\sqrt{2}}
\{\lambda_1,\lambda_1, \lambda_2, \lambda_2\}\ ,\label{dyonicd4}\\
&&\mu = \sqrt2\{(\lambda_1 + \lambda_2 - \lambda)_8,\,
(\lambda_1 + \lambda_2 + \lambda)_8,\,
(\lambda_1 + \lambda_2)_{16}\}\ ,\nonumber
\end{eqnarray}
where $\lambda\equiv \sqrt{\lambda_1^2 + \lambda_2^2}$.
The first $\Delta=1$ solution always preserves $\ft18$ of the
supersymmetry, regardless of the values of $\lambda_1$ and $\lambda_2$.
The second $\Delta=1$ solution breaks all the supersymmetry for generic
values of the Page charges, but gives rise to a supersymmetric elementary
solution when $\lambda_2=0$. The last $\Delta=1$ solution breaks all the
supersymmetry for all values of the Page charges. For $\Delta=2$, we can
have zero eigenvalues only for the following three cases: $\lambda_1=0$,
$\lambda_2=0$ or $\lambda_1 = -\lambda_2$. The first two cases correspond
to the purely solitonic and purely elementary solutions which preserve
$\ft14$ of the supersymmetry. The third case gives rise to a massless black
hole (which has been discussed in \cite{cvetic}), which preserves $\ft12$ of
the supersymmetry. However, some of the eigenvalues are negative in this
case.
\section{Multi-scalar solutions}
The third class of solutions that we shall describe here are ones where
the individual Page charges of the field strengths participating in a
solution are independent parameters (unlike the solutions discussed in the
previous two sections, where they are fixed in a specific set
of ratios in any given solution). This is
achieved by allowing more than one independent combination of the $(11-D)$
dilatonic scalar fields to be excited in the solution. In fact, the number
of independent combinations is precisely equal to the number $N$ of
independent Page charges. These combinations can be expressed as
$\varphi_\a=\vec a_\a\cdot \vec\phi$, where as usual $\vec a_\a$ denotes the
set of $N$ dilaton vectors for the $N$ participating field strengths. One
can easily verify from the equations of motion following from
(\ref{dgenlag}) that the remaining orthogonal combinations of the $\vec\phi$
fields can be consistently set to zero. A natural choice for the solutions
turns out to be to take $d A +\tilde d B=0$ and $A=- {\epsilon \tilde d \over D-2}
\sum_{\a,\beta} (M^{-1})_{\a\beta}\, \varphi_\beta$. The remaining equations
can then be solved by making an ansatz that is analogous to the one
discussed in section 2, which reduces the second-order equations of motion
to first-order equations \cite{lp2}. For this ansatz to be consistent, it
turns out that the matrix $M_{\a\beta}$ of dot products of dilaton vectors
must take the special form $M_{\a\beta}=4\delta_{\a\beta} -2 d \tilde d/(D-2)$.
In fact this is precisely the same as the form that $M_{\a\beta}$ takes in
the various supersymmetric single-scalar solutions that we discussed in
section 2. Thus we conclude that multi-scalar solutions of the type we are
discussing here have the interpretation of being generalisations of the
{\it supersymmetric} single-scalar solutions, in which the Page charges of
the individual participating field strengths, whose magnitudes were
previously required to be equal, become independent free parameters. The
solutions are given by \cite{lp2}
\begin{eqnarray}
&&e^{\ft12 \epsilon \varphi_\a - d A} = 1 + \fft{\lambda_\a}{\tilde d} r^{-\tilde
d}\ ,\nonumber\\
&&ds^2=\prod_{\a=1}^{N} \Big(1+ \fft{\lambda_\a}{\tilde d}
r^{-\tilde d}\Big)^{-\ft{\tilde d}{(D-2)}}\, dx^\mu dx^\nu \eta_{\mu\nu} +
\prod_{\a=1}^N \Big(1+ \fft{\lambda_\a}{\tilde d}
r^{-\tilde d}\Big)^{\ft{d}{(D-2)}}\, dy^m dy^m
\ .\label{gensol1}
\end{eqnarray}
We may now calculate the mass per unit $p$-brane volume and the Page
charges for the solution, finding
\begin{equation}
m = \ft14\sum_{\a=1}^N \lambda_\a\ ,\qquad P_\a = \ft14 \lambda_\a
\ .\label{mc2}
\end{equation}
Note that in our derivation of the solutions, we assumed that the matrix
$M_{\a\beta}$ is non-singular, which in general is the case. However, it can
be singular in two relevant cases, namely $D=5$, $N=3$ and $D=4$, $N=4$ for
the 2-form field strengths. In these cases, the analysis requires
modification; however, it turns out that (\ref{gensol1}) continues to solve
the equations of motion.
We have seen that multi-scalar solutions arise as generalisations of
the previous {\it supersymmetric} single-scalar $p$-brane solutions, in
which the Page charges that were previously equal become independent.
Thus we may view the supersymmetric single-scalar solutions with $N\ge2$
participating field strengths as starting points for such
generalisations. In the light of the findings described in section 2,
this means that multi-scalar solutions arise for 2-form field strengths,
with $2\le N\le 4$, and for 1-form field strengths with $2\le N\le 7$.
The supersymmetry of these multi-scalar solutions can determined
using the same approach as for the single-scalar solutions, by calculating
the eigenvalues of the Bogomol'nyi\ matrix. A full analysis of all these
solutions and their supersymmetry is contained in
\cite{lp2}, and we shall only summarise the results here.
For 2-form
multi-scalar solutions, the eigenvalues of the Bogomol'nyi\ matrix turn out to be
given by
\begin{eqnarray}
N=2:&& \{p_1, u_{12}\} = \ft14 \{\lambda_1, \lambda_2 \}\ ,
\qquad {\rm for}\,\, D\le 9\ ,\nonumber\\
\mu&=&\ft12 \{ 0, \lambda_1, \lambda_2, \lambda_1 + \lambda_2 \}
\ ,\nonumber\\
N=3:&& \{u_{12}, u_{34}, u_{56} \} = \ft14 \{ \lambda_1, \lambda_2,
\lambda_3 \}\ ,\qquad {\rm for}\,\, D\le 5\ ,\nonumber\\
\mu &=& \ft12 \{ 0, \lambda_1, \lambda_2, \lambda_3, \lambda_1 + \lambda_2,
\lambda_1 + \lambda_3, \lambda_2 + \lambda_3, \lambda_1 + \lambda_2 +
\lambda_3 \}\ ,\label{eigenvalues2f}\\
N=4:&& \{u_{12}, u_{34}, u_{56}, p_7^*\} = \ft14 \{\lambda_1,\lambda_2,
\lambda_3, \lambda_4\}\ ,\qquad {\rm for}\,\, D= 4\ ,\nonumber\\
\mu&=&\ft12 \{ 0, \lambda_1+\lambda_4, \lambda_2 + \lambda_4, \lambda_3 +
\lambda_4, \lambda_1 + \lambda_2,
\lambda_1 + \lambda_3, \lambda_2 + \lambda_3, \lambda_1 + \lambda_2 +
\lambda_3 +\lambda_4\}\ .\nonumber
\end{eqnarray}
Here a $^*$ on a Page charge indicates that the associated field strength is
dualised. Thus $p_7^*$ is the electric charge of the dualised field strength
$*{\cal F}_{\sst{MN}}^{(7)}$, and so it is the magnetic charge in terms of
the original undualised field strength ${\cal F}_{\sst{MN}}^{(7)}$. The
degeneracies of each eigenvalue in each set are equal.
For 1-form field strengths, there exist multi-scalar generalisations
for all the supersymmetric solutions described in section 2 with $2\le
N\le 7$. Rather than present them all here, we shall just give the
results for the case $N=7$ here, from which all the lower-$N$ cases can in
fact be derived by successively setting Page charges to zero.
The full details can be found in \cite{lp2}. Thus we have
\begin{eqnarray}
&&\{q_{12}, q_{34}, q_{56}, v_{127}, v_{347}, v_{567}, v^*_7\}
=\ft14 \{\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_4, \lambda_5,
\lambda_6, \lambda_7 \}\ ,\nonumber\\
\mu &=&
\{0,
\lambda_{126}, \lambda_{135}, \lambda_{234},
\lambda_{147},
\lambda_{257}, \lambda_{367}, \lambda_{456}, \lambda_{1245},
\lambda_{1346}, \label{eigenvalues1f}\\
&&\ \lambda_{2356}, \lambda_{1567}, \lambda_{2467},
\lambda_{3457},
\lambda_{1237},
\lambda_{1234567} \}\ ,\nonumber
\end{eqnarray}
where we define $\lambda_{\a\beta\cdots\gamma}= \lambda_\a + \lambda_\beta +
\cdots +\lambda_\gamma$. Note that we
presented only one representative set of Page charges among many
possibilities, since they all give identical eigenvalues. The degeneracy of
each eigenvalue is the same, with the total
number of eigenvalues being 32.
For generic values of the Page charges, we see that the numbers of
zero eigenvalues of the Bogomol'nyi\ matrices for all the multi-scalar solutions
given above are the same as for their single-scalar limits in
section 2. At certain special values of the Page charges, however, some
additional zero eigenvalues can arise, corresponding to an enhancement of
the supersymmetry. For example when $N=2$ for 2-form solutions, we see
from (\ref{eigenvalues2f}) that by setting $\lambda_2=-\lambda_1$, the
generic 8 zero eigenvalues are enlarged to 16, implying that $\ft12$ the
supersymmetry is now preserved instead of just $\ft14$. Since the final
eigenvalue in the set, $\ft12(\lambda_1+\lambda_2)$, is equal to twice the
mass, it follows that the solution becomes massless in this special case.
In $D=4$, this corresponds to a massless black hole. The solution appears
to suffer from two pathologies, however. Firstly, since one of
$\lambda_1$ or $\lambda_2$ must be negative in this special case, we can
see from (\ref{gensol1}) that there is a naked singularity. More
seriously, perhaps, we see from (\ref{eigenvalues2f}) that some of the
eigenvalues of the Bogomol'nyi\ matrix must also now be negative. The
non-negativity of the Bogomol'nyi\ matrix can be proved for classical solutions
(subject to some conditions that are evidently violated in this massless
example), and is required at the quantum level for all acceptable states,
since it arises as the square of the Hermitean supercharge operators. Thus
classical solutions with negative eigenvalues cannot form part of the true
quantum spectrum.
There are other examples of supersymmetry enhancements that can occur
while maintaining the non-negativity of the Bogomol'nyi\ matrix. For example, the
4-scalar solution for 2-forms can lead to three inequivalent enhancements,
given by
\begin{eqnarray}
\lambda_1=-\lambda\ ,&&\lambda_2 =\lambda\ ,\qquad
\mu = \ft12\{ 0_8, (\lambda_3\pm\lambda)_4, (\lambda_4\pm \lambda)_4,
(\lambda_3 + \lambda_4)_8\}\ ,\nonumber\\
\lambda_1=-\lambda\ ,&& \lambda_2=\lambda_3=\lambda\ ,\qquad
\mu = \ft12\{0_{12}, (2\lambda)_4, (\lambda_4- \lambda)_4,
(\lambda_4 + \lambda)_{12}\}\ ,\label{enhance2f}\\
\lambda_1=-\lambda\ ,&& \lambda_2=\lambda_3=\lambda_4=\lambda\ ,\qquad
\mu =\lambda \{0_{16}, 1_{16}\}\ ,\nonumber
\end{eqnarray}
where as usual the subscripts denote the degeneracies of each eigenvalue.
Thus these three cases preserve $\ft14$, $\ft38$ and $\ft12$ of the
supersymmetry respectively, in contrast to $\ft18$ for generic values of
the charge parameters. Note that in these cases, although the Bogomol'nyi
matrices have no negative eigenvalues when the supersymmetry enhancements
occur, the metrics of the solutions still seem to have naked singularities
since one of the Page charges $\lambda_\a$ is negative. If we relax the
condition that the eigenvalues of the Bogomol'nyi matrix should be
non-negative, then further enhancements are possible, in which $\ft58$ or
$\ft34$ of the supersymmetry is preserved. (This does not
violate the classification of supermultiplets given in \cite{fs}, since
non-negativity of the commutator of supercharges was assumed there.)
Similar supersymmetry enhancements can occur for the 1-form solutions, as
can easily be derived from (\ref{eigenvalues1f}).
Finally, we remark that the non-supersymmetric $p$-brane solutions that
are obtained by reversing the signs of certain Page charges in the
$\Delta=1', \ft45, \ft23$ and $\ft47$ single-scalar solutions also admit
multi-scalar generalisations. These can be summarised by presenting the
eigenvalues of the Bogomol'nyi\ matrix for $N=4$ scalars for 2-form solutions, and
$N=7$ scalars for 1-form solutions:
\begin{eqnarray}
N=4:&& \mu=\ft12 \{\lambda_1,
\lambda_2, \lambda_3, \lambda_4, \lambda_{123},
\lambda_{124}, \lambda_{134},\lambda_{234}\}\ ,\nonumber\\
N=7:&&\mu = \ft12\{\lambda_{7}, \lambda_{14}, \lambda_{25}, \lambda_{36},
\lambda_{123}, \lambda_{156}, \lambda_{246}, \lambda_{345},
\lambda_{1267}, \lambda_{1357}, \lambda_{2347}, \lambda_{4567},\nonumber\\
&&\,\qquad\quad\lambda_{12457}, \lambda_{13467}, \lambda_{23567},
\lambda_{123456}\}\ .
\end{eqnarray}
Note that in these cases none of
the eigenvalues is proportional to the mass, for generic $\lambda_\a$. A
detailed discussion of the various cases that arise by choosing special
values for the Page charges is given in \cite{lp2}.
To conclude, we present a table summarising the various multi-scalar
$p$-brane solutions that we have been discussing.
\bigskip
\centerline{
\begin{tabular}{|c||c|l||c|c|}\hline
{\phantom{D}Dim.\phantom{D}}&\multicolumn{2}{c||}
{\phantom{DDDDD} 2-Forms \phantom{DDDDD}} &
\multicolumn{2}{c|}{\phantom{DDDDD}1-Forms\phantom{DDDDD}} \\ \hline\hline
$D=10$& $\phantom{D}N=1\phantom{D}$ &\phantom{DD}$p=0,6$ & & \\ \hline
$D=9$ & $N=2$ &\phantom{DD}$p=0,5$ & $N=1$ & $p=6$ \\ \hline
$D=8$ & &\phantom{DD}$p=0,4$ &$N=2$&$p=5$ \\ \hline
$D=7$ & &\phantom{DD}$p=0,3$ & &$p=4$ \\ \hline
$D=6$ & &\phantom{DD}$p=0,2$ &$N=3,4'$&$p=3$ \\ \hline
$D=5$ & $N=3$ &\phantom{DD}$p=0,1$ & &$p=2$ \\ \hline
$D=4$ & $N=4$ &\phantom{DD}$p=0$ &$N=4,5,6,7$&$p=1$ \\ \hline
\end{tabular}}
\bigskip
\centerline{Table 2: Multi-scalar $p$-brane solutions}
\bigskip
\noindent Here we list the highest dimensions where $p$-brane solutions
with the indicated numbers $N$ of field strengths first occur. They then
occur also at all lower dimensions.
|
1,116,691,497,812 | arxiv | \section{Introduction}
Striped patterns abound in nature, with lamellar features observable at the micron scale within the cell walls of fruits \cite{pollia}, the chitinous exoskeleton of beetles \cite{beetle}, and the fruit fly embryo \cite{fruitfly}, as well as at much larger scales, such as on the skin of zebras, tigers, and certain fish species \cite{fish,Turingfish}. In the latter examples, the stripes arise from activating and inhibiting dynamics, characteristic of a Turing instability \cite{fish,Turingfish}. Other patterns seen in living systems, such as those at the surfaces of \textit{chiral, liquid crystalline} materials, arise due to a characteristic length scale, \textit{e.g.}, the cholesteric pitch $P_0$ \cite{pollia, beetle, boulig-biochol, bouligarches}. Striped patterns can further be influenced by geometrical confinement. For instance, stripes decorating a sphere necessarily have defects where they must collide or terminate due to the system's topology, seen in the presence of poles on a globe. Motivated by the ubiquity of curved, chiral materials, we study striped patterns at the free surfaces of cholesteric liquid crystal (CLC) shells. Using varying surfactant concentrations in the ambient aqueous medium, we show that the CLC shell surface develops transient, undulated stripe instabilities as the pitch axis reorients to accommodate changes in the anchoring conditions. The instabilities are recapitulated in numerical simulations and are generic features: the transient undulations occur under either homeotropic to planar anchoring transitions or under the reverse change. Fig.~\ref{fig:FCDunwind} illustrates this process for a cholesteric shell with diluting surfactant, triggering a homeotropic to planar anchoring transition. The initial striped, focal conic structure (double spiral, top row) unwinds and develops the secondary, bent stripes (bottom row).
\begin{figure}[htp]
\centering
\includegraphics[width=0.4\textwidth]{1-FCDunwind.eps}
\caption{\label{fig:FCDunwind} A thick cholesteric shell with 1\% wt PVA, 7 mM SDS in the surrounding aqueous phases has homeotropic anchoring on the inner and outer interfaces. With 1 M NaCl in the inner phase and 0.1 M NaCl in the outer phase, the shell osmotically swells from top to bottom, diluting the surfactants in the inner phase and reducing the inner homeotropic anchoring strength. The planar stripes become thicker as a consequence and a secondary periodicity within primary stripes is visible, while maintaining the overall double spiral structure of a focal conic domain.}
\end{figure}
Recent work \cite{TranPRX} has shown that when cholesteric shells are subjected to modifications in the anchoring conditions, the stripe pattern is altered dramatically. When the anchoring transforms from homeotropic to planar, the stripes unwind at the defects. The topologically required defects in the system evolve from being defects in the twisting direction ($\lambda^{\pm}$ pitch defect lines) to becoming defects in the molecular orientation of the liquid crystalline molecules (nematic defects with a total charge of $+2$). Along with this conversion of the defect structure, the stripes themselves widen and develop an instability that generates a secondary periodicity, shown in the bottom row of Fig.~\ref{fig:FCDunwind}. Our objective here is to understand this instability using numerical simulations and to describe the stripe undulations through the lens of the Helfrich-Hurault instability. We show that the secondary stripe periodicity is caused by strain in the cholesteric layers brought about by changes in the anchoring condition.
The paper is organized as follows: In the next section (Section~\ref{sec:setup}), we discuss the experimental setup and numerical simulations. Then, in Section~\ref{sec:Results}, we discuss our experimental and simulation results for both planar to homeotropic and homeotropic to planar anchoring transitions. In Section~\ref{sec:Analysis}, we present an analysis of the undulating stripe patterns using the framework of the Helfrich-Hurault instability and predict a stripe periodicity with this model that matches those observed in experiments and simulations. We conclude with Section~\ref{sec:Conclusion} by giving perspectives for future work.
\section{Methods\label{sec:setup}}
\subsection{Cholesteric shell preparation}
We use 4-cyano-4'-pentylbiphenyl (5CB, Kingston Chemicals Limited) doped with 2.8\% wt (S)-4-cyano-4'-(2-methylbutyl)biphenyl (CB15, EMD Performance Materials) to obtain a CLC with a pitch of $\sim$ 5 $\mu$m. The pitch is determined using a Grandjean-Cano wedge cell \cite{GJwedge, Cwedge}. Briefly, two glass slides are coated with a unidirectionally rubbed polyvinyl alcohol (PVA, Sigma-Aldrich) film to give planar anchoring. The slides are then clamped and glued with rubbing directions parallel to one another and a Mylar spacer on one side of the cell to create the wedge angle. The distance between disclination lines in the cell is measured and used, along with the angle, to determine the pitch.
CLC shells are produced using a glass capillary microfluidic device to generate double emulsions, similar to methods described in past works \cite{ufluidics,TranPRX,tll-sds,lagerwall-review}. Briefly, three different fluid phases are used to generate shells of CLC: an inner aqueous phase, the middle CLC phase, and an outer aqueous phase. The tapered circular capillary of the inner water phase is fitted into a tapered square capillary for the middle CLC phase, which is subsequently fitted into a larger circular capillary for injection of the outer aqueous phase. The middle, square capillary is treated with octadecyltrimethoxysilane (OTS, Sigma-Aldrich) to improve CLC wetting of the capillary. Three syringe pumps (Harvard Apparatus) are used to control the flow rates of each phases, with 40 mL/hr, 500 $\mu$L/hr, and 400 $\mu$L/hr as the flow rates corresponding to the outer, middle and inner phases. Both aqueous phases have 1\% wt polyvinyl alcohol (PVA, 87-89\% hydrolyzed, average $M_{w} = 13,000-23,000$) to stabilize the emulsions. After the double emulsions are collected from the microfluidic device, the shells are left to settle in the vial and equilibrate to the planar anchoring conditions induced by PVA and water.
We use sodium dodecyl sulfate as the surfactant (SDS, Sigma-Aldrich) in varying concentrations to induce homeotropic anchoring \cite{TranPRX,tll-sds,lagerwall-interface}. For homeotropic anchoring of the inner shell surface, SDS is included in the inner aqueous phase during the microfluidic production of CLC double emulsions. To induce homeotropic anchoring on the outer surface of the CLC shell, the double emulsions are pipetted into vials containing aqueous solutions of 1\% wt PVA, 0.1 M NaCl, and SDS with concentrations ranging from 7-10 mM \cite{TranPRX}. The shells are either viewed immediately to observe the planar to homeotropic anchoring transition or can be left overnight to obtain relaxed shells with homeotropic anchoring as an initial condition. To induce the homeotropic to planar anchoring transition, these homeotropic shells are pipetted into another solution with only 1\% wt PVA and 0.1 M NaCl, without SDS. After introducing shells to the appropriate outer aqueous phase for the desired anchoring transition, the sample vial is gently mixed before pipetting into a viewing chamber (Grace Bio-Labs). An upright microscope in transmission mode fitted with crossed polarizers (Zeiss AxioImager M1m) and a high-resolution color camera (Zeiss AxioCam HRc) is used to capture polarized micrographs of shells.
Sodium chloride (NaCl, Fisher Scientific) is added to aqueous phases, ranging in concentration from 0.1 to 1 M, in order to increase the SDS interfacial density \cite{abbsurf}. In some samples, a higher concentration of NaCl in the inner aqueous phase compared to the outer phase can trigger osmotic swelling, in which water permeates through the CLC shell towards the inner aqueous droplet, increasing the inner droplet volume and thinning the shell over time \cite{osswell,alex-waltz,TranPRX}. However, we found the osmotic swelling of CLC shells to occur over tens of hours, while stripe transformations from the introduction or removal of SDS occur over a span of tens of minutes. Therefore, we do not expect osmotic swelling to have an effect on the initial pattern formation that is the focus of this work. Indeed, the time and the manner of pattern formation for shells with and without osmotic swelling is comparable.
\subsection{Landau-de Gennes modeling}
The numerical simulations model the CLC molecular orientation via the Landau-de Gennes free energy where a symmetric, traceless $3 \times 3$ matrix $Q_{\alpha \beta}(\mathbf{x})$ represents the nematic orientation order parameter and is defined at each site $\mathbf{x}$ of a cubic lattice, with indices $\alpha,\beta=x,y,z$ indicating the Cartesian directions \cite{ravnik,sussman}. The
free energy in the bulk regions of the liquid crystal reads
\begin{equation}
f_{\mathrm{bulk}}= \frac{A}{2}\, Q_{\alpha \beta}Q_{\beta \alpha} + \frac{B}{2} \,Q_{\alpha \beta} Q_{\beta \gamma} Q_{\gamma \alpha} + \frac{C}{4} ( Q_{\alpha \beta}Q_{\beta \alpha})^2,
\end{equation}
with summation over repeated indices implied. The constants $A$, $B$, and $C$ are set by the thermodynamic properties of the liquid crystal. Note that the scalar order parameter is given in terms of these parameters as $S_0 = (-B+ \sqrt{B^2-24AC})/6C$. We also have an elastic component in the bulk which incorporates the energy penalties
associated with splay, twist, and bend distortions:
\begin{align}
f_{\mathrm{grad.}} & = \frac{L_1}{2}\,( \epsilon_{\alpha \gamma \delta} \partial_{\gamma} Q_{\delta \beta}+2 q_0 Q_{\alpha \beta})^2 + \frac{L_2}{2} ( \partial_{\beta} Q_{\beta \alpha})^2 \nonumber \\
& {} + \frac{L_{24}}{2} (\partial_{\alpha} Q_{\beta \gamma} \partial_{\gamma} Q_{\alpha \beta} - \partial_{\alpha} Q_{\alpha \beta} \partial_{\gamma} Q_{\beta \gamma}), \label{eq:LdGFT}
\end{align}
where $\epsilon_{\alpha \beta \gamma}$ is the Levi-Cevita symbol and $\partial_{\alpha} \equiv \frac{\partial}{\partial x_{\alpha}}$ are the partial derivatives along the three spatial directions. When we square the two terms in the first line of Eq.~\eqref{eq:LdGFT}, we are left with a sum over the remaining free indices $\alpha$ and $\beta$. More details are given in, \textit{e.g.}, \cite{fukuda}. We include a splay-bend coefficient $L_{24}$ that is a total derivative term, but will be important in our system since we will be interested in interfacial phenomena.
At interfaces, we also have an anchoring energy that we model using a Rapini-Papoular surface potential \cite{rapinipapoular}. For homeotropic anchoring strength $W_0$ and planar anchoring strength $W_1$,
the total interface free energy reads:
\begin{align}
f_{\mathrm{in.}} & = \int \mathrm{d} A \Big\{ W_0 (Q_{\alpha \beta}-Q^{\parallel}_{\alpha \beta})^2 \nonumber \\ & {} +W_1 [(\bar{Q}_{\alpha \beta}-\bar{Q}^{\perp}_{\alpha \beta})^2 + (Q_{\alpha \beta}Q_{\alpha \beta}-3S_0^2/2)^2\Big\} ,
\end{align}
where $Q^{\parallel}_{\alpha \beta} = 3 S_0 (\hat{s}_{\alpha} \hat{s}_{\beta}- \delta_{\alpha \beta}/2)$ is the orientation tensor
constructed from the interface's surface normal vector $\hat{\mathbf{s}}$ and where $\bar{Q}_{\alpha \beta}=Q_{\alpha \beta}+S_0 \delta_{\alpha \beta}/2$. The tensor $\bar{Q}^{\perp}_{\alpha \beta}$ is the projection of $\bar{Q}_{\alpha \beta}$ onto the plane of the interface, so $\bar{Q}^{\perp}_{\alpha \beta}=(\delta_{\alpha \beta} - s_{\alpha} s_{\delta})\bar{Q}_{\delta \gamma}(\delta_{\gamma \beta}-s_{\gamma}s_{\beta})$.
This free energy is then minimized over a computational domain using a conjugate gradient method in the ALGLIB package (\url{http://www.alglib.net/}). Note that the minimization here does not reflect the actual dynamics of a liquid crystalline system. In reality, the free energy would be minimized in a manner consistent with liquid crystal hydrodynamics. Nevertheless, we can obtain a qualitative idea about the evolution of various patterns by monitoring the states of the system during the minimization procedure from some specified initial condition while tracking any long-lived transient states. Often, the minimization procedure will end within local free energy minima (metastable states) which are also of interest.
In the experiments presented here, we consider 5CB mixed with a chiral dopant as our cholesteric liquid crystal. Therefore, we make use of some standard values for this compound in our simulations \cite{ravnik}. We set $A=-1$, as this can be set simply by choice of free energy units. We then utilized a one-constant approximation with $B=-12.33$, $C=10.06$, and $L_1=L_2=2.32$ (Fig.~\ref{fig:unwinding}), consistent with the parameters of 5CB and a lattice spacing $\Delta x \approx 4.5~\mathrm{nm}$ close to the nematic correlation length (size of nematic defect core). These particular choices were not important for determining the qualitative features of the simulation, including the formation of the undulation instability that is the focus of this work. For the other shell simulations in Figs.~\ref{fig:arches} and \ref{fig:HomToPla}, we used the two-constant approximation with $B=-1.091$, $C=0.6016$, $L_1=0.003805$, and $L_2=2L_{24}=0.01141$. These latter two constants were chosen to correspond to $K_1=K_3 \approx 10~\mathrm{pN}$ and $K_2 = 5~\mathrm{pN}$ for the characteristic splay, bend, and twist elastic constants, respectively, for 5CB \cite{elastic5CB}. The choice of constants yields a large lattice spacing $\Delta x \approx 30~\mathrm{nm}$. Note that the bulk free energy constants $B$ and $C$ are not so important in this case because we simulate deep in the nematic phase (low temperature) with a large $\Delta x$ so that all the lattice sites retain the equilibrium value $S_0$ of the scalar order parameter. The minimization procedure essentially only minimizes the elastic component as the bulk remains at a uniaxial minimum. For the simulations with a deformable CLC shell interface in Fig.~\ref{fig:freeinterface}, we tuned the cholesteric to an isotropic-nematic phase transition by setting $A=5.284$, $B=-11.2612$, and $C=0.88889$, with the elastic constants $L_1 = L_2 = 2L_{24}= 0.928$. The various corresponding dimensions of the shell are given in the figure captions in SI units.
\section{Results \label{sec:Results}}
We now examine the undulation instability of CLC stripes on shells for both homeotropic and planar anchoring transitions, using experiments and simulations. At one end of an anchoring transition, the equilibrated \textit{planar} state is absent of stripes, with director defects adding up to a topologically-required index of $+2$ \cite{alex-defects, alex-waltz}. An example planar shell with four +1/2 defect points is shown in Fig.~\ref{fig:FCDwinding}-i. In the planar state, the pitch axis is oriented radially, with the cholesteric layers forming concentric shells. At the other end of an anchoring transition, the \textit{homeotropic} state of a CLC shell typically stabilizes focal conic domains, seen as double spirals at the shell surface, in which the cholesteric twists along the shell interface (top row of Fig.~\ref{fig:FCDunwind} and Fig.~\ref{fig:surface}-i). The pitch axis in the homeotropic state is oriented parallel to the surface. When the CLC shell is left to equilibrate for a sufficiently long time, the stripes eventually arrange into lines of latitude on the shell, with stripes terminating at two focal conic domains, as forced in by the spherical topology \cite{TranPRX}. Converting between these two states requires tilting the pitch axis orientation with respect to the surface, yielding undulating stripes that are featured in the bottom row of Fig.~\ref{fig:FCDunwind} and are the subjects of interest in this paper.
\subsection{Planar to homeotropic transitions}
\begin{figure}[htp]
\centering
\includegraphics[width=0.4\textwidth]{2-FCDwinding.eps}
\caption{\label{fig:FCDwinding} An initially planar-anchored CLC shell (a) has 10 mM SDS added to the outer aqueous phase, detailed in Methods (Sec.~\ref{sec:setup}). As the surfactant adsorbs to the shell interface and interacts with the CLC, it begins to orient the CLC molecules perpendicular to the interface, tilting the pitch axis away from the radial direction. The change in anchoring generates large stripes with a periodicity around twice the pitch, analogous to those seen in Fig.~\ref{fig:FCDunwind} in the bottom panels. Arrows point out director defects. Video can be viewed at \cite{nikonsmallworld}.}
\end{figure}
We begin first with the planar to homeotropic anchoring transition, featured in Fig.~\ref{fig:FCDwinding}. Homeotropic anchoring is induced on the outer shell surface experimentally through the addition of a surfactant (SDS, see Sec.~\ref{sec:setup}) to the outer aqueous phase. The onset of the stripe instability is shown in Fig.~\ref{fig:FCDwinding}. Note that the undulated stripes do not conform to a discernible pattern, but run in different directions along the shell surface. The nematic defects in the planar to homeotropic transition do not appear to play a large role in ordering the undulating stripes, although the stripes can be observed terminating at the director defects, highlighted in Fig.~\ref{fig:FCDwinding}-iv.
\begin{figure}[htp]
\centering
\includegraphics[width=0.43\textwidth]{3-PlaToHom.eps}
\caption{\label{fig:arches} An initially planar shell (a) with a $6.6~\mu\mathrm{m}$ diameter and a $2.1~\mu\mathrm{m}$ thickness is minimized in the presence of moderate homeotropic boundary conditions ($W_0=0.04$ corresponding to about $2 \times 10^{-4}~\mathrm{J}/\mathrm{m}^2$). The cholesteric has a $1.2~\mu\mathrm{m}$ pitch. After $t=5000$ minimization steps, the concentric cholesteric layers begin to undulate and bend toward the surface, generating stripes shown in (b). The inset is the outer surface. A side view of (b) is plotted in (c). The planar defect is marked in red by the $\mathbf{n} \cdot \hat{\mathbf{r}}$ color map. As the system evolves, this region becomes increasingly homeotropic at both the inner and outer surfaces. Eventually, the bent, undulating CLC layers form the focal conic domains characteristic of the homeotropically-anchored state. We can see the beginning of these characteristic stripes after $t=60000$ minimization steps in (d). }
\end{figure}
We turn to simulations to elucidate the bulk CLC layer arrangements that bring about the undulating stripe patterns. The stripe behavior is simulated by taking a planar CLC shell configuration as the initial condition [Fig.~\ref{fig:arches}(a), left] and minimizing the free energy under homeotropic anchoring conditions. Large stripes are spawned at early stages of the transition [Fig.~\ref{fig:arches}(b)], reminiscent of those seen in experiments. Cross sections of the shell reveal the source of the large stripes to be undulation of the concentric CLC layers, evident from comparing the cross sections in Fig.~\ref{fig:arches}(a) and (b). During the minimization, homeotropic anchoring increases most significantly at the director defect, the area of highest distortion [red region in Fig.~\ref{fig:arches}(c)]. Interestingly, far from the director defect, the CLC layers are not as undulated as they are near the defect, evident in the cross section of Fig.~\ref{fig:arches}(c). That undulations are most pronounced near the director defect suggests that pitch axis reorientation occurs first at these local regions of disorder before propagating to the rest of the system. Eventually, the typical stripe pattern of homeotropic CLC shells develops on the surface, forming focal conic domains and indicating that the pitch axis completes its reorientation and becomes parallel to the shell surfaces. The simulation results further show that the distortions due to the changing anchoring are confined to regions closest to the shell interfaces, with the most bent and distorted cholesteric layers near the surface [Fig.~\ref{fig:arches}(d)].
\subsection{Homeotropic to planar transitions}
\begin{figure*}[htp]
\centering
\includegraphics[width=0.93\textwidth]{4-Figure-FromPRX.eps}
\caption{\label{fig:PRXunwind} A homeotropic CLC shell left to equilibrate for one month has two focal conic domains at its poles, with additional defects in the thinnest region of the shell (i). During the homeotropic to planar anchoring transition, the focal conic domains unwind and widen to create greater planar anchoring at the shell surface. As the transition continues, stripes begin to locally undulate with a $2P_0$ periodicity (iii-iv), where $P_0 = 5$ $\mu$m. At the end of the transition the planar state is reached, with remnant undulations on the shell. Reproduced from Tran \textit{et. al.}, Supplemental Video 6 \cite{TranPRX}. }
\end{figure*}
A similar stripe instability also occurs for a homeotropic to planar anchoring transition. However, in contrast to the reverse transition, the onset of the stripe instability induced by imposing planar anchoring appears to be sensitive to the initial configuration of the system. We examine first the transition for an equilibrated, homeotropic CLC shell before turning to the transition for metastable, homeotropic states with the characteristic focal-conic-like domains decorating the CLC shell surface.
A CLC shell left to equilibrate in a solution with 7 mM SDS, 1\% wt PVA, and 0.1 M NaCl for a month is shown in Fig.~\ref{fig:PRXunwind}(i), first presented in \cite{TranPRX}. It has two focal conic domains at it poles, with the pole in the thinnest region of the shell highlighted in Fig.~\ref{fig:PRXunwind}(i). Transferring the shell to an aqueous solution without SDS, results in slow removal of surfactant from the interface and a reduction in the homeotropic anchoring strength. The double spirals of the polar focal conic domains unwind first, resulting in stripes of a larger periodicity at the poles, seen in Fig.~\ref{fig:PRXunwind}(ii). As the stripes unwind, they also widen to accommodate greater regions of planar anchoring at the interface. At some point during the anchoring transition, stripes near the thinnest region of the shell begin to undulate with a periodicity of twice the pitch, shown in Fig.~\ref{fig:PRXunwind}(iii) \& (iv). The undulations extend along the stripes as the transition continues. Near the end of the transition to planar anchoring, most of the stripes have been removed from the system, with a few undulations remaining. One of the excess defects near the focal conic domain in Fig.~\ref{fig:PRXunwind}(i) becomes the topologically required director defect in Fig.~\ref{fig:PRXunwind}(v).
\begin{figure}[htp!]
\centering
\includegraphics[width=0.43\textwidth]{5-HomToPla.eps}
\caption{\label{fig:HomToPla} A shell at equilibrium with homeotropic anchoring in (a) is relaxed under planar anchoring conditions ($6.6~\mu\mathrm{m}$ diameter, $2.1~\mu\mathrm{m}$ thickness, and $W_1=0.004$ corresponding to about $2 \times 10^{-5}~\mathrm{J}/\mathrm{m}^2$). Top views of the shell during the anchoring transition to planar are plotted in (b). The double spiral unwinds as the system approaches its new energetically-favorable, planar-anchored state. Cross sections of the shells over the course of the transition, from $t=2000$ (left) and $t=10000$ (right) minimization steps, are plotted in (c). At $t=2000$, the layers start to undulate in the shell interior (seen most prominently in the red regions). At the equator, the layer undulates such that a layer pinch-off gives rise to two pitch defects (red circles) that eventually become the planar defects at the end of the transition. At $t=10000$, the undulated layers merge and become concentric. Eventually, a planar state is achieved, similar to Fig.~\ref{fig:arches}(a) (left).}
\end{figure}
The absence of undulating stripes during the majority of the anchoring transition is captured through simulations, plotted in Fig.~\ref{fig:HomToPla}, where an equilibrated, homeotropically-anchored cholesteric shell relaxes to a planar-anchored state. The starting condition, shown in Fig.~\ref{fig:HomToPla}(a), has two characteristic double-spirals at both poles that correspond to the topologically-required, focal-conic-like domains where the pitch axis has a defect. During relaxation, these double spirals unwind as the pitch axis tilts toward the shell interior. We see a top view of the unwinding in Fig.~\ref{fig:HomToPla}(b). Meanwhile, the layers on the interior of the shell begin to undulate, shown on the left of Fig.~\ref{fig:HomToPla}(c). The layers near the equator undulate in opposite directions and pinch off, creating another pitch defect that eventually becomes the topologically-required nematic defect in the planar-anchored shell. Such a mechanism was conjectured in \cite{TranPRX}, but is observed here explicitly in the simulation. Eventually, the pitch axis points radially, and we obtain a state close to the equilibrated planar configuration, seen on the right of Fig.~\ref{fig:HomToPla}(c). We can discern from the insets in Fig.~\ref{fig:HomToPla}(c) that one of the nematic defects (red spot on the outer surface) sits near the shell equator. Faint stripes are still observable near the director defect, shown through the inset of Fig.~\ref{fig:HomToPla}(c), right. Generally, a secondary stripe instability does not materialize, reminiscent of the experiment featured in Fig.~\ref{fig:PRXunwind}, where stripes are capable of becoming wider and more planar without undulating in the majority of the system.
\begin{figure}[htp]
\centering
\includegraphics[width=0.44\textwidth]{6-Unwinding.eps}
\caption{\label{fig:unwinding} A CLC shell simulation ($0.84~\mu\mathrm{m}$ diameter and $0.18~\mu\mathrm{m}$ thickness using the one-constant approximation, see Sec.~\ref{sec:setup}), with a color map of the radial component $\mathbf{n} \cdot \hat{\mathbf{r}}$ of the director for a cholesteric phase with a $0.18~\mu\mathrm{m}$ pitch. The insets show the outer surfaces. Here, an initially $\hat{z}$-axis oriented pitch axis configuration (a) is set to relax with planar anchoring of magnitude $W_1=0.25$, which corresponds to a strength of $2 \times 10^{-4}~\mathrm{J}/\mathrm{m}^2$. The minimization procedure finds a local minimum in the free energy corresponding to the partially unwound stripes shown in (b), with a top view on the left and a side view on the right. The poles develop undulated stripes, while the equatorial stripes partially unwind. }
\end{figure}
However, it is possible to observe the undulated stripe instability in simulations by using a thinner, smaller shell and slightly altering the initial equilibrium, homeotropic configuration, as shown in Fig.~\ref{fig:unwinding}. Using a uniform CLC ground state with a vertical pitch axis as the initial shell configuration creates distortion of the polar focal conic domains, shown on the left of Fig.~\ref{fig:unwinding}(a). The distorted focal conic domains are regions where deformation of the CLC is easiest, observed also in our prior work \cite{TranPRX}. With tilting of the pitch axis beginning at these regions near the poles, the centers of the stretched double spirals are also where the undulations first appear. Under the anchoring transition, the stretched double spirals unwind and widen only in the double spiral region, leaving behind an undulated, planar stripe, shown in Fig.~\ref{fig:unwinding}(b). The simulation is then trapped in this configuration, indicating a metastable state. Cross sections on the right of Fig.~\ref{fig:unwinding}(b) reveal that the undulated stripes are a direct result of CLC layer undulations. The left panel of Fig.~\ref{fig:unwinding}(b) further reveals that the stripe undulation appears on both the inner and outer surfaces of the shell. These modulations of the director, perpendicular to the initial stripes, mimic what is seen at the former focal conic domain of the experiment featured in Fig.~\ref{fig:PRXunwind}i-ii \cite{TranPRX}. As the shell adapts to planar anchoring in Fig.~\ref{fig:PRXunwind}, the double spiral region also develops undulated stripes, matching the simulated behavior of the distorted focal conic domains in Fig.~\ref{fig:unwinding}.
\begin{figure}[htp]
\centering
\includegraphics[width=0.42\textwidth]{7-Shellsurface.eps}
\caption{\label{fig:surface} A thick cholesteric shell in an aqueous solution with 7 mM SDS, 1\% wt PVA, and 0.1 M NaCl has a focal conic domain texture initially. The pitch is 5 $\mu$m. The shell is transferred to another, similar aqueous solution, but without SDS, and the evolution of the texture is observed (i-iv). As the outer interface loses homeotropic anchoring strength with surfactant removal to the surrounding solution, the planar anchoring stripes widen (i-ii). When the stripes widen to $\sim 2 P_0 \approx 10$ $\mu$m, the stripes fill with perpendicular stripes of a second periodicity also $\sim 2 P_0$. See Video 1 in Supplemental Material. }
\end{figure}
Curiously, undulating stripes in the homeotropic to planar anchoring transition are more pervasive in experimental systems where the initial homeotropic state is meta-stable and have many focal conic domains, exemplified in Fig.~\ref{fig:FCDunwind}-i and Fig.\ref{fig:surface}-i. As the surfactant is washed off, the stripes widen to accommodate larger regions of planar anchoring. When the stripes reach a width of approximately twice the pitch $P_0$, the stripe interior becomes filled in with secondary, arch-like stripes of a periodicity $\sim 2P_0$ that run along primary stripes, covering the entire shell surface (Fig.~\ref{fig:FCDunwind}-iii-iv). Primary stripes being filled in with undulating, secondary stripes also matches the simulation results of Fig.~\ref{fig:unwinding}. However, it remains unclear why stripes in metastable configurations more readily undulate than those in equilibrium configurations. The initial, seeded planar anchoring at the surface of the simulation in Fig.~\ref{fig:unwinding} allows for the undulated stripes to emerge during the anchoring transition. This implies that larger, local regions of planar anchoring may be necessary in experiments for secondary, undulated stripes to appear. This could possibly occur in experiments from the segregation of surfactants into regions of favorable anchoring, reported in \cite{lt-sciadv}. Shell thickness heterogeneity in experiments may additionally play a role, as the local undulations of Fig.~\ref{fig:PRXunwind} occur near the thinnest region of the shell. Indeed, the simulated shell of Fig.~\ref{fig:unwinding}, exhibiting pronounced, undulated stripes, is smaller and thinner than that of Fig.~\ref{fig:HomToPla}.
Despite slight differences between homeotropic and planar anchoring transitions, the emergence of undulating stripes with changes in anchoring is a general phenomenon, demonstrated through the above experiments and simulations of CLC shells. Simulation results reveal that the stripe instability is a consequence of CLC layer undulation, resulting from the pitch axis tilting to accommodate the new anchoring conditions. In the following section, we formulate a simple model through the lens of the Helfrich-Hurault instability to describe how incompatible anchoring can trigger undulations in the cholesteric bulk.
\section{Analysis\label{sec:Analysis}}
A key feature of CLCs is that anchoring changes are confined to a small boundary layer $\ell$ near the interface, while retaining the overall layer structure of the cholesteric bulk, seen in the simulation results of Fig.~\ref{fig:arches}. This is true even if the interface is allowed to deform, as would happen at the cholesteric-isotropic interface, simulated in Fig.~\ref{fig:freeinterface}. Here, a shell with concentrically-arranged cholesteric layers deforms the cholesteric-isotropic interfaces of the shell, with the layers bending most significantly near the interfaces. In larger systems, the bulk layers generally remain in their former configuration. A similar boundary layer may be found in cholesterics confined to hybrid-anchored cells \cite{archtexture}.
The instabilities considered here result from cholesteric layer strain imposed by a change in anchoring conditions. The strain is relieved via a periodic modulation of the layers, which manifests in the stripe instability described in Sec.~\ref{sec:Results}. This phenomenon is reminiscent of the classic Helfrich-Hurault mechanism, in which cholesteric layers are strained via an applied magnetic field \cite{helfrich,hurault1}. We develop a similar analysis of the instability observed during anchoring transitions.
We begin by considering the Frank free energy density of the cholesteric:
\begin{eqnarray}
f_{\mathrm{n}} &=& \frac{K_1}{2}[\hat{\mathbf{n}}(\nabla \cdot\hat{\mathbf{n}} ) ]^2+\frac{K_2}{2}[ \hat{\mathbf{n}} \cdot(\nabla \times \hat{\mathbf{n}})+q_0]^2 \nonumber \\
&&\quad + \frac{K_3}{2}[(\hat{\mathbf{n}} \cdot \nabla) \hat{\mathbf{n}}]^2 \label{eq:frank}
\end{eqnarray}
where $K_1$, $K_2$, and $K_3$ are the splay, twist, and bend elastic constants, respectively. This energy does not properly take into account defects in the nematic director, but it can describe the cholesteric layer structure. Minimization of the $K_2$ term yields the usual cholesteric ground state, characterized by a spiraling nematic director $\hat{\mathbf{n}}= \sin(\mathbf{q}\cdot \mathbf{x}-q_0u) \hat{\mathbf{q}}^1_{\perp}+\cos(\mathbf{q}\cdot \mathbf{x}-q_0u)\hat{\mathbf{q}}^2_{\perp}$, with $\hat{\mathbf{q}}$ being the pitch direction and $\hat{\mathbf{q}}^{1,2}_{\perp}$ being two orthogonal basis vectors spanning the subspace perpendicular to the pitch. We also have $|\mathbf{q}|=q_0=2 \pi/P_0$, with $P_0$ being the pitch. The phases $q_0 u$ are the distortions of the cholesteric layers and $u=0$ for a perfect cholesteric helix.
The field $u(\mathbf{x})$ may be interpreted at large length scales as a displacement of the cholesteric layer spacings.
\begin{figure}[htp]
\centering
\includegraphics[width=0.48\textwidth]{8-Freeinterface.eps}
\caption{\label{fig:freeinterface} (a) Cross section through a thick cholesteric shell with a free interface from contact with an isotropic phase, illustrating the distortions of the cholesteric layers near the surface due to the incompatible homeotropic anchoring. In the bulk, the layers settle into concentric shells. Arrows indicate the focal conic ``hills.'' The color indicates the director $\hat{\mathbf{n}}$ orientation relative to the radial direction $\hat{\mathbf{r}}$. The pitch $P_0$ to shell thickness $t$ ratio here is $t/P_0 \approx 2$. (b) The rough interface and the director distribution just underneath the surface is plotted. Note that the bumps are correlated with the double spirals of the focal conic domains (arrows). }
\end{figure}
It can be shown that small deviations $u$ away from the cholesteric ground state have a free energy analogous to a smectic (layered) liquid crystal \cite{dgennes-prost,RadzihovskyLubensky}:
\begin{equation}
F_e=\int \mathrm{d}^3\mathbf{x}\left[\frac{\bar{B}}{2}
\left(\frac{\partial u}{\partial z}-\frac{(\nabla_{\perp} u)^2}{2}\right)^2+\frac{\bar{K}}{2} \left( \nabla_{\perp}^2 u\right)^2 \right],
\label{eq:layerE}
\end{equation}
where we integrate over the entire sample volume, $\bar{K} \approx 3K_3/8$ is an effective layer bending modulus, and $\bar{B}=K_2q_0^2$ is an effective layer compression modulus \cite{LubenskyHydro,dgennes-prost,RadzihovskyLubensky}. The gradient $\nabla_{\perp}$ represents the derivatives perpendicular to the layer orientation. The free energy in Eq.~\eqref{eq:layerE} is consistent with a rotational invariance of the overall layer structure \cite{kleman-odl}.
In our experiments, we always apply an anchoring that favors a pitch axis tilting, either away from the interface normal for planar to homeotropic transitions, or toward the interface normal for homeotropic to planar transitions. In both cases, such an anchoring will introduce the following free energy contribution at the interface for small tilts $u$:
\begin{equation}
F_{\mathrm{a}}=-\frac{|W|}{4}\int \mathrm{d}^2 \mathbf{x}\,(\nabla_{\perp}u)^2, \label{eq:anchoring}
\end{equation}
where we integrate over the interface surface and $W$ is the strength of the tilt-inducing anchoring \cite{odlanchoring}. Much like an applied magnetic or electric field, this free energy contribution will \textit{break} the rotational symmetry implicit in the layer free energy in Eq.~\eqref{eq:layerE}, driving an undulation instability.
Let us now consider how the instability would work for a flat interface (\textit{\textit{i.e.}} a small patch of the CLC shell) located at the $z=\ell$ plane. We will assume that $\ell$ corresponds to the penetration depth of the deformations induced by the anchoring, so that there are no deformations $(u=0)$ at $z=0$. The characteristic wave vector of the instability may be found by looking at the lowest harmonic of the $u$ field, which we assume corresponds to a modulation along the $x$-direction: $u(x,z)=u_0 \sin(\pi z/2\ell) \cos(q_x x)$, where $\ell$ is a characteristic length over which the deformation occurs within the CLC bulk. For a CLC free interface, we expect the deformations to be confined to a length $\ell$ on the order of the cholesteric pitch or half-pitch \cite{freeCholAnchor}. Substituting in this ansatz and integrating Eq.~\eqref{eq:layerE} and \eqref{eq:anchoring}, we find $f_s$, a free energy per unit area of interface:
\begin{eqnarray}
f_{\mathrm{s}} &=& u_0^2 \left[\frac{ \pi ^2 \bar{B}}{32\ell}+\frac{\ell q_x^4}{512} \left(9 \bar{B}u_0^2+64 \bar{K}\right)-\frac{Wq_x^2 }{8} \right] . \label{eq:HH}
\end{eqnarray}
We find that when $W>W_c=\pi \sqrt{\bar{K}\bar{B}}$, the free energy in Eq.~\eqref{eq:HH} is minimized for a non-zero $u_0$ (\textit{i.e.}, an undulated state). Moreover, the critical wave vector associated with the modulation is $q_x^*=(\bar{B}/\bar{K})^{1/4}(\pi/2\ell)^{1/2}$. Putting in typical elastic constants for 5CB, and assuming a 5~$\mu$m pitch $P_0$ (setting $\ell=P_0$ equal to the pitch), we expect $W_c \approx10^{-5}~\mathrm{J}/\mathrm{m}^2 $ and a modulation wavelength $\lambda_x^* \equiv 2\pi/q_x^*\approx 2P_0$. This is a relatively weak anchoring strength, so we do generically expect to see a modulation under the experimental conditions presented here.
A modulation wavelength of about twice the pitch is also consistent with our observations, detailed previously in Sec.~\ref{sec:Results}. However, we note that this spacing can be larger than $2P_0$, especially when the initial configuration of the layers does not correspond to a uniform cholesteric helix running either along or perpendicular to the interface. Indeed, aspects not captured by this simple argument are other, more complex arrangements of the cholesteric layers with initially homeotropic anchoring. For instance, the cholesteric layers can bend to form u-shapes near the interface, resulting in a distribution of $\lambda^{\pm}$ defects \cite{TranPRX,freeCholAnchor}, leading to a stripe periodicity equal to the pitch, instead of a half-pitch as one would obtain for a uniform, undistorted helix oriented parallel to the interface.
\section{Discussion \& Conclusion \label{sec:Conclusion}}
We have now shown how undulation instabilities develop at free CLC interfaces when the anchoring changes from homeotropic to planar or vice-versa. The instability is driven by a strain in the layers due to the reorientation of the pitch axis near the interface. One typically finds that the modulations have a periodicity equal to twice the cholesteric pitch. We have shown that this instability is analogous to the Helfrich-Hurault mechanism, with the anchoring change playing the role of an ``applied field''. A basic argument yields a reasonable estimate of the undulation instability periodicity and the critical anchoring strength at which we might expect to see the instability.
Confinement of the CLC within a spherical geometry necessitates the presence of defects that appear to influence the conformation of the stripe instabilities, with secondary stripes terminating at director defects for homeotropic transitions and extending along primary stripes for planar transitions. For the homeotropic anchoring transition, the initial director defects serve as favorable sites for cholesteric layer rearrangements and undulations. For the planar anchoring transition, the focal conic domains, pitch defects, also act as locations of initial pitch axis reorientation, evident from stripes becoming wider at double spirals first. We note that although the topologically-required defects serve as regions of easy deformation in the system, the presence of defects is not necessary for the onset of undulations. Indeed, the Helfrich-Hurault instability requires only a local geometric incompatibility, instead of a global frustration, for undulations to occur. The type of defect, whether director or pitch, merely reflects the cholesteric layer orientation with respect to the confining boundaries.
We hypothesize that the difference in the appearance and ordering of the stripe instability in homeotropic versus planar transitions lies in the shortest path for the pitch axis reorientation. For the homeotropic transition, the pitch axis is initially radial. To conform to the new homeotropic anchoring condition, the pitch axis must tilt to become tangent to the surface. However, every tilt direction from radial is equivalent. The disorder of the stripe instability for the homeotropic transition then lies in the degeneracy of the pitch-axis tilt direction. On the other hand, for the planar transition, the pitch axis is initially tangent to the spherical surface. Therefore, tilting the pitch axis \textit{along} the direction of its initial orientation is the shortest path to pointing radially. This constraint prescribes a set direction for undulations to take place, along the initial (primary) stripes on the surface. We additionally note that the secondary, undulated stripes in the planar transition often have an arching shape in experiments, reminiscent of Bouligand arches that emerge when a surface cuts a CLC at an angle to the pitch axis \cite{boulig-biochol, bouligarches, boulig-spherulites}. The exact connection between the stripe instability and Bouligand arches remains to be further explored.
It would be interesting to develop a more faithful simulation that takes into account the liquid crystal dynamics more properly. Our free energy minimization procedure assumed an over-simplified relaxation mechanism for the nematic director. In the real system, hydrodynamic effects may be important. Moreover, simulating a true free interface could also shed light on the role of the interface in triggering or stabilizing the stripe instability. We are currently limited to looking at interfaces between a cholesteric and isotropic phase, which has a fixed, weak homeotropic anchoring for the cholesteric. Furthermore, a description that takes into account heterogenous shell thicknesses and distributions of anchoring-inducing, surface-active agents may also be needed to fully capture the states observed in experiments.
Another unexplored aspect is the relationship of this instability to the stripe instability observed in \textit{nematic} liquid crystals under hybrid anchoring \cite{han-str-1,han-str-2,han-str-3,han-str-4,han-str-5}. Our system is similar when we transition from a planar to a homeotropic shell configuration because most of the bulk cholesteric layers maintain a nematic director orientation parallel to the shell surface (\textit{i.e.}, a planar orientation) while the outer layers have homeotropic anchoring. A hybrid-anchored cholesteric is even more complicated than the nematic case, due to the interplay between the anchoring and the cholesteric twist \cite{dozov}. In the case of the hybrid-anchored nematic, boundary terms in the elastic free energy (\textit{e.g.}, the saddle splay) play an important role in determining the onset of modulations \cite{hybridK24,hybridK13}. Although we include such terms in our free energy, we do not study the effect of this term systematically. We do expect such terms to contribute to our CLC shell, as we also have a boundary-driven instability.
We have established how frustration between the surface and bulk ordering of a chiral material can drive an undulating instability, generating stripe patterns with a periodicity larger than that of the material itself. The Helfrich-Hurault model was expanded beyond applied fields and mechanical strains to encompass also changes in the surface anchoring as a source of the instability. This work lays the foundation for further study of the Helfrich-Hurault model in systems where boundary conditions can be freely adjusted and curved.
\qquad
\begin{acknowledgments}
We thank R. D. Kamien, F. Livolant, and T. Lopez-Leon for useful discussions. Computational support was provided by the University of Tennessee and Oak Ridge National Laboratory's Joint Institute for Computational Sciences. M.O.L. gratefully acknowledges partial funding from the Neutron Sciences Directorate (Oak Ridge National Laboratory), sponsored by the U.S. Department of Energy, Office of Basic Energy Sciences. L.T. acknowledges support from the Simons Society of Fellows of the Simons Foundation.
\end{acknowledgments}
|
1,116,691,497,813 | arxiv | \section{Introduction \label{sec_introduction}}
In a naive sense, a ``regular'' mathematical object in one which is described by functions. The space where these structural functions live is the ``regularity'' of the defined mathematical object. Thus, $n$-manifolds $M$ are ``regular'' because they are described by its charts $\varphi_i:U_i\rightarrow \mathbb{R}^n$. Furthermore, its regularity is given by the space where the transition functions $\varphi_{ji} =\varphi_j\circ \varphi^{-1}_i$ live. Similarly, tensors $T$, affine connections $\nabla$ and partial differential operators $D$ in $M$ are also ``regular'', because they are described by their local coefficients.
In \cite{eu_1} the authors proposed a formalization to this general notion of ``regularity'' and begin its study in the context of $C^k$-manifolds. In particular, we considered the existence problem of subatlases satisfying prescribed regularity conditions. In the present work we will continue this study by considering existence and multiplicity problems for regular affine connections on regular $C^k$-manifolds.
Such problems are of wide interest in many areas. For instance, in gauge theory and when one needs certain
uniform bounds on the curvature, it is desirable to work with connections
$\nabla$ whose coefficients are not only $C^{k-2}$, but actually
$L^{p}$-integrable or even uniformly bounded \cite{gauge,bounded_geometry_1}.
In other situations, as in the study of holomorphic geometry, one
considers holomorphic or Hermitian connections over complex manifolds
whose coefficients are holomorphic \cite{holomorphic_1,atiyah_class}.
If $M$ has a $G$-structure, one can also consider $G$-principal
connections $\omega$ on the frame bundle $FM$, which induce an affine
connection $\nabla_{\omega}$ in $M$, whose coefficients $\omega_{b}^{c}(\partial_{a})=\Gamma_{ab}^{c}$
are such that for every $a$, $\Gamma_{a}\in C^{k-2}(U)\otimes\mathfrak{g}$,
where $(\Gamma_{a})_{cb}=\Gamma_{ab}^{c}$ \cite{kobayashi}. Interesting
examples of these are the symplectic connections, which are specially
important in formal deformation quantization of symplectic manifolds
\cite{symplectic_connection,deformation_quantization}. As a final
example, a torsion-free affine connection defines a $L_{3}$-space
structure in a smooth manifold $M$ (in the sense of \cite{L3_space})
iff its coefficients satisfy a certain system of partial differential
equations \cite{L3_space_2}. Thus, the theory of $L_{3}$-spaces is about certain regular connections.
In the formulation of ``regular'' object of \cite{eu_1}, the spaces describing the regularity are modeled by nuclear Fr\'echet spaces, since their category $\mathbf{NFre}$ is a closed symmetric monoidal relatively to the projective tensor project $\otimes$, and since most of the function spaces arising from geometry belongs to $\mathbf{NFre}$. Furthermore, since one such space for every open set $U\subset M$, this leads to consider presheaves $B:\operatorname{Open}(\mathbb{R}^n)^{op}\rightarrow \mathbf{NFre}$. On the other hand, notice that if a mathematical concept is described by $C^k$-functions $f$, then the regularity of this object is typically described not only by the space where the functions $f$ live, but also by the space where their derivatives $\partial^i f$, $1\leq i\leq k$, belong. Thus, we should consider sequences of presheaves $B_i:\operatorname{Open}(\mathbb{R}^n)^{op}\rightarrow \mathbf{NFre}$, with $i=0,...,k$. But, for reasons which will be more clear later (see Remark \ref{important_remark}), it is better to allow sequences $B_i$ indexed in more general sets $\Gamma$. These are called \textit{presheaves of $\Gamma$-spaces}.
\begin{example}
The prototypical example of $[0,k]$-presheaf, where $[0,k]={0,...,k}$, is the sequence $C^{k-i}$. If $\beta:\Gamma \rightarrow [0,k]$ is any function we get the $\Gamma$-presheaf $C^{k-\beta(i)}$. One can also consider the sequences $L^i(U)$, $W^{i,p}(U)$, and so on.
\end{example}
We would also like to sum and multiply in the regularity spaces in a distributive way. We will consider presheaves of \textit{distributive $\Gamma$-spaces}, which are presehaves of $\Gamma$-spaces endowed with a distributive structure. On the other hand, recall that we will be interested in regularity structures on $C^k$-objets. Thus, we need some way to make sense of the intersections $B_{\alpha(i)}(U)\cap C^{k-\beta(i)}(U)$, where the index $\alpha$ is necessary to ensure that both are presheaves of $\Gamma$-spaces for the same $\Gamma$. This will be done with the help of an ambient category $\mathbb{X}$ with pullbacks (typically the category of $\Gamma$-graded real vector spaces) and of an ambient object $X$, in which both $B_i$ and $C^{k-\beta(i)}$ can be included, being the intersection $B_i\cap_X C^{k-\beta(i)}$ given by the pullbacks of the inclusions. In other words, we will need an \textit{intersection structure} between $B_i$ and $C^{k-\beta(i)}$, denoted by $\mathbb{X}$.
With all this stuff, we define a ``regular'' function $f:U\rightarrow \mathbb{R}$, here called \textit{$(B,k,\alpha,\beta)$-function in $\mathbb{X}$}, as a $C^k$-function such that $\partial^i f\in B_{\alpha(i)}(U)\cap_{X(U)}C^{k-\beta(i)}(U)$ for each $i$. Similarly, a ``regular'' $C^k$-manifold, called \textit{$(B^k_{\alpha,\beta},\mathbb{X})$-manifold} is one whose transition functions $\varphi_{ji}$ are regular functions, i.e., are $(B,k,\alpha,\beta)$-functions in $\mathbb{X}$. Finally, a ``regular'' affine connection in a $(B^k_{\alpha,\beta},\mathbb{X})$-manifold $M$ is a $C^k$ affine connection $\nabla$ whose coefficients $(\Gamma_\varphi)^{c}_{ab}$ in each regular chart $\varphi$ of $M$ are $(B,k,\alpha',\beta')$-functions in some other ambient $\mathbb{Y}$. We also say that $\nabla$ is an affine $(B^k_{\alpha',\beta'},\mathbb{Y})$-connection in $M$.
In this new language, the existence and multiplicity problems for regular affine connections on regular manifolds have the following description:
\begin{itemize}
\item \textbf{Existence.} Let $M$ be a $(B^k_{\alpha,\beta},\mathbb{X})$-manifolds. Given $\alpha'$ and $\beta'$ and $\mathbb{Y}$, can we find conditions on $B$, $k$, $\alpha$ and $\beta$ such that $M$ admits affine $(B^k_{\alpha',\beta'},\mathbb{Y})$-connections?
\item \textbf{Multiplicity.} What can be said about the dimension of the space $\operatorname{Conn}^k_{\alpha',\beta'}(M;\mathbb{Y)}$ of $(B^k_{\alpha',\beta'},\mathbb{Y})$-connections in $M$?
\end{itemize}
We prove that arbitrarily regular connection exist in sufficiently regular manifolds. The will be based on four steps:
\begin{enumerate}
\item existence of arbitrarily regular affine connections in each $U\subset M$ (Proposition \ref{prop_local_existence});
\item existence of global affine connections in $M$ whose coefficients $(\Gamma_\varphi)^c_{ab}$ are regular, but whose derivatives are not (Theorem \ref{thm_existence_connections});
\item proving a Regularity Globalization Lemma, ensuring that the derivatives $\partial^i(\Gamma_\varphi)^c_{ab}$ can be made regular (Lemma \ref{key_lemma});
\item putting together the previous steps to get a global regular connection (Theorem \ref{corollary_existence_connections}).
\end{enumerate}
Concerning the multiplicity problem, in Proposition \ref{space_affine_connections_is_affine} we show that
the space of regular affine connections is an affine space of the space of regular $\operatorname{End}(TM)$-valued 1-forms, and in Theorem \ref{multiplicity_thm} we prove that if two regular connections are locally additively different, then they are locally different.
The paper is organized as follows. In Section \ref{sec_background}
we present the needed background for our main results. In parts, it contain concepts introduced in \cite{eu_1}, but now described in a more concrete way. Subsection \ref{sec_globalization_lemma} and Subsection \ref{sec_globalization_lemma} are fully new in content if compared to \cite{eu_1}. In particular, in Subsection \ref{sec_globalization_lemma} the Regularity Globalization Lemma is stated and proved. Section \ref{sec_a_b_connections} deals with the existence problem and the four steps described above. Section \ref{sec_multiplicity} is about the multiplicity problem. The paper ends with an informal discussion of why we should believe that there exists a topology in which $\operatorname{Conn}^k_{\alpha',\beta'}(M;\mathbb{Y)}$ is a nonempty open dense subset of the space of tuples of $(B,k,\alpha',\beta')$-functions in $\mathbb{Y}$.
\begin{itemize}
\item \textbf{Convention}: remarks concerning notations and assumptions will be presented using a bullet.
\end{itemize}
\section{Background \label{sec_background}}
In this section we will present all that is necessary to formally state and prove the existence and multiplicity results described in the introduction. We begin by recalling in detail the definitions and results from \cite{eu_1} that will be needed in the next sections, which are basically the notion of $B^k_{\alpha,\beta}$-manifold and the underlying concepts.
\begin{itemize}
\item \textbf{Notations}. In the following $\Gamma$ denotes an arbitrary set, regarded as a set of indexes. Given $k,l\geq0$ with $k\leq l$, $[k,l]$ will denote the integer closed interval from $k$ to $l$. $\mathbf{NFre}$ will denote the category of nuclear Fr\'echet spaces with continuous linear maps, and $\otimes$ is the projective tensor product, which makes $\mathbf{NFre}$ a symmetric monoidal category \cite{nuclear}.
\end{itemize}
\subsection{$\Gamma$-Spaces}
In this section we will recall what are distributive $\Gamma$-spaces. For a complete exposition, see Section 2 of \cite{eu_1}.
\begin{definition}
A \emph{$\Gamma$-space} is a family $B=(B_i)_{i\in\Gamma}$ of nuclear Fr\'echet spaces. A \textit{morphism} $f:B\Rightarrow B'$ between two $\Gamma$-spaces $B$ and $B'$ is a family $f_i:B_i\rightarrow B'_i$ of morphisms in $\mathbf{NFre}$, with $i\in\Gamma$. Composition and identities are defined componentwise, so that we have a category $\mathbf{NFre}_{\Gamma}$.
\end{definition}
\begin{example}\label{example_1_gamma_spaces}
The fundamental examples are the $[0,k]$-spaces given by $B_i=C^i(M;\mathbb{R}^m)$, where $M$ is a $C^k$-manifold, with the standard family of semi-norms
\begin{equation}\label{frechet_ck}
\Vert f\Vert_{r,l}=\sum_{j}\sup_{\vert\mu\vert=r}\sup_{x\in K_{l}}\vert\partial^{\mu}f_{j}(x)\vert
\end{equation}
with $0\leq r\leq i$ and $(K_{l})$ some nice sequence of compact
sets of $M$. More generally, given a set $\Gamma$ and a function $\beta:\Gamma\rightarrow [0,k]$, we can define the $\Gamma$-space $B_i=C^{k-\beta(i)}(M;\mathbb{R}^m)$.
\end{example}
\begin{example}
Similarly, we can consider in $C^{k-\beta(i)}(U;\mathbb{R})$ other Banach norms, as $L^p$-norms or Sobolev norms.
\end{example}
\begin{example}
More generally, any sequence of Banach spaces defines a $\Gamma$-space, where $\Gamma$ is the set in which the sequence is indexed.
\end{example}
\begin{example}\label{example_2_gamma_spaces}
Given a $\Gamma$-space $B$ and a function $\alpha:\Gamma' \rightarrow \Gamma$ we define a new $\Gamma'$-space $B_{\alpha}$ by reindexing $B$, i.e., $(B_\alpha)_i=B_{\alpha(i)}$.
\end{example}
\begin{definition}\label{definition_distributive_space}
A \emph{distributive structure} in a $\Gamma$-space $B$ consists of:
\begin{enumerate}
\item maps $\epsilon,\delta:\Gamma\times\Gamma \rightarrow\Gamma$;
\item continuous linear maps $*_{ij}:B_{i}\otimes B_{j}\rightarrow B_{\epsilon(i,j)}$
and $+_{ij}:B_{i}\otimes B_{j}\rightarrow B_{\delta(i,j)}$, for every $i,j\in \Gamma$
\end{enumerate}
such that the following compatibility equations are satisfied for every $x\in B_{i}$, $y\in B_{j}$
and $z\in B_{k}$, and every $i,j,k\in\Gamma$ (notice that the first two equations, which describe left and right distributivity, respectively, makes sense only because
of the last two).
\begin{eqnarray*}
x*_{i\delta(j,k)}(y+_{jk}z) & = & (x*_{ij}y)+(x*_{ik}z)\\
(x+_{ij}y)*_{\delta(i,j)k}z & = & (x*_{ik}z)+(y*_{jk}z)\\
\epsilon(i,\delta(j,k)) & = & \delta(\epsilon(i,j),\epsilon(i,k))\\
\epsilon(\delta(i,j),k) & = & \delta(\epsilon(i,k),\epsilon(j,k)).
\end{eqnarray*}
We also require that $+_{ii}$ coincides with the sum $+_i$ in $B_{i}$.
\end{definition}
\begin{definition}
A \emph{distributive $\Gamma$-space} is a $\Gamma$-space $B$ endowed with distributive structure $(\epsilon,\delta,+,*)$. A \textit{morphism} between distributive $\Gamma$-spaces $(B,\epsilon,\delta,+,*)$ and $(B',\epsilon',\delta',+',*')$ is a pair $(f,\mu)$, where $\mu:\Gamma \rightarrow \Gamma$ is a function and $f=(f_i)_{i\in \Gamma}$ is a family of continuous linear maps $f_i:B_i\rightarrow B'_{\mu(i)}$ such that the following equations are satisfied (notice again that the first two equations, which describe left and right distributivity, respectively, makes sense only because
of the last two):
\begin{eqnarray*}
f(x*_{ij}y)&=&f(x)*'_{\epsilon'(\mu(i),\mu(j))}f(y) \\
f(x+_{ij}y)&=&f(x)+'_{\delta'(\mu(i),\mu(j))}f(y) \\
\mu(\epsilon(i,j))&=& \epsilon'(\mu(i),\mu(j))\\
\mu(\delta(i,j))&=& \delta'(\mu(i),\mu(j)).
\end{eqnarray*}
\end{definition}
\begin{example}\label{example_distributive_gamma_spaces}
With pointwise sum and multiplication, the $\Gamma$-spaces $C^{k-\beta(i)}(M;\mathbb{R})$ are distributive for $\delta(i,j)=k-\max(\beta(i),\beta(j))=\epsilon(i,j)$. H\"older's inequality implies that the $\mathbb{Z}_{\geq 0}$-spaces $L^i(U)$ also are distributive, with $\delta(i,j)=\min(i,j)$ and $\epsilon(i,j)=i\star j= (i\cdot j)/(i+j)$, now viewed as a $\Gamma$-space for $\Gamma\subset \mathbb{Z}_{\geq 0}$ being the set of all $i,j\geq 0$ such that $i\star j$ is an integer. From Yong's inequality one can considers a different distributive structure in $L^i(\mathbb{R}^n)$ given by convolution product and such that $\epsilon(i,j)= r(i,j)=(i\cdot j)/(i+j-(i\cdot j))$, where now $\Gamma \subset \mathbb{Z}_{\geq 0}$ is such that $r(i,j)$ is an integer.
\end{example}
\begin{definition}
Let $B$ and $B'$ be a $\Gamma$-space and a $\Gamma'$-space, respectively. The \textit{external tensor product} between them is the $\Gamma\times\Gamma'$-space $B\otimes B'$ such that $(B\otimes B')_{i,j}=B_i\otimes B'_j$.
\end{definition}
\begin{remark}\label{example_B_epsilon}
A distributive $\Gamma$-space $(B,\epsilon,\delta,+,*)$ induces two $(\Gamma\times\Gamma)$-spaces $B_\epsilon$ and $B_\delta$, defined by $(B_{\epsilon})_{ij}=B_{\varepsilon(i,j)}$ and $(B_{\delta})_{ij}=B_{\delta(i,j)}$. The family of maps $*_{ij}$ and $+_{ij}$ defining a distributive $\Gamma$-structure are actually morphisms of $\Gamma\times\Gamma$-spaces $*:B\otimes B\Rightarrow B_{\epsilon}$ and $+:B\otimes B\Rightarrow B_{\delta}$.
\end{remark}
\subsection{Abstract Intersections}
We now recall abstract intersection structures between presheaves of distributive $\Gamma$-spaces. For a complete exposition, see Section 2 of \cite{eu_1}.
As a motivation, recall that the intersection $A\cap B$ of vector subspaces $A,B\subset X$ of a fixed vector space $X$ can be viewed as the pullback between the inclusions $A\hookrightarrow X$ and $B \hookrightarrow X$. However, if $A$ and $B$ are arbitrary vector spaces, the ``abstract intersection'' $A\subset B$ is a priori not well-defined. In order to define it we first consider an ambient space $X$ in which both can be embedded and one fixes embeddings $\alpha:A\hookrightarrow X$ and $\beta:B\hookrightarrow X$. The \textit{abstract intersection} between $A$ and $B$ inside $X$, relative to the fixed embeddings $\alpha$ and $\beta$, is then defined as the pullback $\operatorname{pb}(\alpha,\beta)$ between $\alpha$ and $\beta$.
Notice that if the category $\mathbf{C}$ in which the objects $A$ and $B$ live does not have pullbacks, then the above notion of abstract intersection may not exist for certain triples $(X,\alpha,\beta)$. In these cases we first need to regard both $A$ and $B$ as objects of another category $\mathbf{X}$ with pullbacks. Thus, we take a faithful functor $\gamma:\mathbf{A}\rightarrow \mathbf{X}$, a fixed ambient object $X\in \mathbf{X}$ and fixed monomorphisms $\alpha:\gamma_{\mathbf{A}}(A)\hookrightarrow X$ and $\alpha:\gamma(A)\hookrightarrow X$ $\alpha:\gamma(B)\hookrightarrow X$. The abstract intersection can then be defined as the pullback $\operatorname{pb}(\alpha,\beta)$ between $\alpha$ and $\beta$, but now it depends additionally on the faithful functor $\gamma$. Furthermore, a priori it exists only as an object of $\mathbf{X}$, which is why we call it ``abstract''.
\begin{definition}
A \emph{$\Gamma$-ambient} is a tuple $\mathcal{X}=(\mathbf{X},\gamma,\gamma_{\Sigma})$,
where $\mathbf{X}$ is a category with pullbacks and $\gamma$ and
$\gamma_{\Sigma}$ are faithful functors $\gamma:\mathbf{NFre}_{\Gamma}\rightarrow\mathbf{X}$
and $\gamma_{\Sigma}:\mathbf{NFre}_{\Gamma\times\Gamma}\rightarrow\mathbf{X}$. We say that a $\Gamma$-ambient $\mathcal{X}$ \textit{has null objects} if the ambient category $\mathbf{X}$ has a null object $0$.
\end{definition}
\begin{example}\label{standard_ambient}
As discussed in \cite{eu_1}, the typical example of $\Gamma$-ambient is such that $\mathbb{X}$ is the category $\mathbf{Vec}_{\mathbb{R},\Gamma\times \Gamma}$ of $\Gamma\times \Gamma$-graded real vector spaces, $\gamma_\Sigma: \mathbf{NFre}_{\Gamma\times\Gamma}\rightarrow \mathbf{Vec}_{\mathbb{R},\Gamma\times \Gamma}$ is the forgetful functor and $\gamma:\mathbf{NFre}_{\Gamma}\rightarrow \mathbf{Vec}_{\mathbb{R},\Gamma\times \Gamma}$ is the composition of $\Gamma_\Sigma$ with the inclusion $\mathbf{NFre}_{\Gamma}\hookrightarrow \mathbf{NFre}_{\Gamma\times \Gamma}$ given by $\imath(B)_{i,j}=B_i\oplus 0$.
\end{example}
\begin{example}\label{vectorial_ambient}
More generally, it is interesting to consider the class of \textit{vector $\Gamma$-ambients} \cite{eu_1}. These are such that $\mathbf{X}=\mathbf{Vec}_{\mathbb{R},\Gamma\times\Gamma}$, $\gamma_{\Sigma}$ creates null objects and $\gamma=\gamma_{\Sigma}\circ F$, where $F$ is another functor that creates null objects.
\end{example}
\begin{definition}
Let $B,B'$ be two $\Gamma$-spaces and let $\mathcal{X}=(\mathbf{X},\gamma,\gamma_{\Sigma})$ be a $\Gamma$-ambient. An \textit{intersection structure} between $B$ and $B'$ in $(\mathbf{X},\gamma,\gamma_{\Sigma})$ is given by an object $X\in\mathbf{X}$ and monomorphisms $\imath:\gamma(B)\hookrightarrow X$ and $\imath':\gamma(B')\hookrightarrow X$. We write $\mathbb{X}$ in order to denote $(X,\imath,\imath')$ and we say that $X$ is the \textit{ambient object} of $\mathbb{X}$.
\end{definition}
Notice that in the definition above the functor $\gamma_{\Sigma}$, which is part of the $\Gamma$-ambient, was not used yet. It will be used when taking distributive structures into account.
\begin{definition}
Let $(B,\epsilon,\delta,+,*)$ and $(B',\epsilon',\delta',+',*')$ be distributive $\Gamma$-spaces. An \textit{intersection structure} between them, denoted by $\mathbb{X}$, consists of\footnote{Recall the definitions of $B_\epsilon$ and $B_\delta$ in Remark \ref{example_B_epsilon}.} \begin{enumerate}
\item a $\Gamma$-ambient $\mathcal{X}=(\mathbf{X}, \gamma,\gamma_{\Sigma})$;
\item an intersection structure $\mathbb{X}_0=(X,\imath,\imath')$ between $B$ and $B'$;
\item an intersection structure $\mathbb{X}_{*}=(X_{*},\jmath_{*},\jmath_{*}')$ between $B_\epsilon$ and $B'_{\epsilon'}$;
\item an intersection structure $\mathbb{X}_{+}=(X_{+},\jmath_{+},\jmath_{+}')$ between $B_\delta$ and $B'_{\delta'}$.
\end{enumerate}
The \textit{abstract intersection} between $(B,\epsilon,\delta,+,*)$ and $(B',\epsilon',\delta',+',*')$ in $\mathcal{X}$, relatively to $\mathbb{X}$, is:
\begin{enumerate}
\item the abstract intersection $\operatorname{pb}(X,\imath,\imath';\mathcal{X},\mathbb{X}_0)$ between $B$ and $B'$;
\item the abstract intersection $\operatorname{pb}(X_*,\jmath_{*},\jmath'_{*};\mathcal{X},\mathbb{X}_{*})$ between $B_{\epsilon}$ and $B'_{\epsilon'}$;
\item the abstract intersection $\operatorname{pb}(X_+,\jmath_{+},\jmath'_{+};\mathcal{X},\mathbb{X}_+)$ between $B_{\delta}$ and $B'_{\delta'}$;
\item the additional pullbacks between $\gamma_\Sigma(*) \circ\jmath_{*}$ and $\gamma_\Sigma(*') \circ\jmath'_{*}$, and between $\gamma_\Sigma(+) \circ\jmath_{+}$ and $\gamma_\Sigma(+') \circ\jmath'_{+}$, as below\footnote{We presented only the first of these pullbacks, since the second one is fully analogous.}, describing the abstract intersections between the multiplications $*$ and $*'$, and the sums $+$ and $+'$, respectively\footnote{Recall from Remark \ref{example_B_epsilon} that $*$ and $+$ can be viewed as morphisms defined on external tensor products.}, in $\mathcal{X}$ and relatively to $\mathbb{X}$.
\end{enumerate}
\end{definition}
\begin{equation}\label{diagram_abstract_intersection}
\xymatrix{\operatorname{pb}(\gamma_\Sigma(*) \circ\jmath_{*},\gamma_\Sigma(*') \circ\jmath'_{*}) \ar[rr] \ar[dd] && \gamma _{\Sigma} (B' \otimes B') \ar[d]^{\gamma _{\Sigma}(*')} \\
& \operatorname{pb}(\jmath_*,\jmath'_{*}) \ar[d] \ar[r] & \gamma _{\Sigma}(B'_{\epsilon}) \ar@{^(->}[d]^{\jmath_{*} '} \\
\gamma _{\Sigma} (B \otimes B) \ar[r]_{\gamma _{\Sigma} (*)} & \gamma _{\Sigma}(B_{\epsilon}) \ar@{^(->}[r]_-{\jmath_{*}} & X_*}
\end{equation}
\begin{itemize}
\item The abstract intersections between $*$ and $*'$, and between $+$ and $+'$, as above, will be denoted simply by $\operatorname{pb}(*,*';\mathcal{X},\mathbb{X}_*)$ and $\operatorname{pb}(+,+';\mathcal{X},\mathbb{X}_+)$, respectively.
\end{itemize}
\begin{example}
The abstract intersection usually depends strongly on the ambient object $X$. E.g, in the $\Gamma$-ambient of Example \ref{standard_ambient}, if $X=\gamma(B)\oplus\gamma(B')$, then $\operatorname{pb}(X,\imath,\imath';\mathcal{X},\mathbb{X})\simeq 0$ independently of $\imath$ and $\imath'$. On the other hand, if $X=\operatorname{span}(\gamma(B)\cup\gamma(B'))$, then the abstract intersection is a nontrivial graded vector space if $B_i\cap B'_i$ is nontrivial for some $i\in\Gamma$. See Examples 12-13 of \cite{eu_1}.
\end{example}
\begin{remark}
In typical examples, if there is some $i\in \Gamma$ such that $B_i$ or $B'_i$ is a nontrivial vector space, then the abstract intersection between $B$ and $B'$ is nontrivial too. This depends on how we ``concretify'' the abstract intersections: see Lemma \ref{vectorial_intersection}.
\end{remark}
\subsection{Concretization}
Here we will recall the concretization procedures for abstract intersection structures. See Section 2 of \cite{eu_1}.
In the last section we described how we can intersect two objects $A,B\in \mathbf{C}$ in a category without pullbacks by means of considering spans in an ambient category $\mathbf{X}$ with pullbacks via a faithful functor $\gamma:\mathbf{C}\rightarrow \mathbf{X}$. The resulting abstract intersection, however, exists a priori only in $\mathbf{X}$, but in many situations we actually need to work with it in $\mathbf{C}$. Thus we need some ``concretization'' procedure. An obvious approach would be to require that $\gamma:\mathbf{C}\rightarrow \mathbf{X}$ reflects pullbacks, but this would assume that $\mathbf{C}$ has pullbacks, which is precisely what we are avoiding. One could also assume the presence of an opposite directed functor $F:\mathbf{X}\rightarrow \mathbf{C}$ which preserves pullbacks, but again this assumes that $\mathbf{C}$ has pullbacks. A final obvious attempt would be consider only flat functors $\gamma:\mathbf{C}\rightarrow \mathbf{X}$, but this is too restrictive for our purposes. The exact condition that we need is the following\footnote{One can actually work in a more general setting, as those discussed in \cite{eu_1}. However, the present conditions are sufficient for essentially everything, including our main results.}:
\begin{definition}
Let $\gamma:\mathbf{C}\rightarrow \mathbf{X}$ be a faithful functor and $X\in \mathbf{X}$. A \textit{concreteness structure} for $\gamma$ in $X$ is given by a set $W_{\gamma}(A;X)$ of morphisms $\gamma(A)\rightarrow X$ for each $A\in \mathbf{C}$. Let
$$W_{\gamma}(X)=\bigcup_{A\in \mathbf{C}} W_{\gamma}(A;X).$$
We say $\gamma$ satisfies the \textit{concreteness property} in the concreteness structure $W_{\gamma}(X)$ if for every pair $f:\gamma(A)\hookrightarrow X$ and $f':\gamma(B)\hookrightarrow X$ in $W_{\gamma}(X)$ there exist:
\begin{enumerate}
\item an object $A\cap_{X,\gamma}B\in\mathbf{C}$ and an isomorphism $u:\gamma(A\cap_{X,\gamma}B)\simeq \operatorname{pb}(f,f')$;
\item morphisms $\theta_1:A\cap_{X,\gamma}B \rightarrow A$ and $\theta_2:A\cap_{X,\gamma}B \rightarrow B$,
\end{enumerate}
such that the diagram below commutes.
\begin{equation}\label{diagram_concreteness}
\xymatrix{\gamma (A\cap_{X,\gamma}B) \ar@/_{0.3cm}/[ddr]_{\gamma(\theta_1)} \ar[rd]^{u}_{\simeq} \ar@/^{0.3cm}/[rrd]^{\gamma(\theta_2)} \\
& \ar[d]_-{\pi_1} \operatorname{pb}(f,f') \ar[r]^-{\pi_2} & \gamma(B) \ar[d]^{f'} \\
& \gamma(A) \ar[r]_-{f} & X}
\end{equation}
\end{definition}
\begin{remark}
The cospan in $\mathbf{C}$ defined by $(\theta_1,\theta_2)$ is called \textit{concrete cospan} of the span $(f,f')$. Thus, the concreteness property allow us to replace a span in $\mathbf{X}$ by a not necessarily universal cospan (i.e., not necessarily a limit) in $\mathbf{C}$.
\end{remark}
\begin{lem}\label{lemma_monomorphisms}
If $\gamma$ satisfies the concreteness property in $W_{\gamma}(X)$ and if $f$ and $f'$ are monomorphisms, then the corresponding $\theta_1$ and $\theta_2$ are too.
\end{lem}
\begin{proof}
By the comutativity of the diagram above, we have $\gamma(\theta_i)=\pi_i\circ u$, with $i=1,2$. Since $f$ and $f'$ are monomorphisms, it follows that $\pi_i$ are monomorphisms too. Thus, $\gamma(\theta_i)$ also are. Since $\gamma$ is faithful, it reflect monomorphisms.
\end{proof}
We can now put the previous discussion in our context.
\begin{definition}
Let $\mathcal{X}=(\mathbf{X},\gamma,\gamma_\Sigma)$ be a $\Gamma$-ambient and let $X\in \mathbf{X}$. Define
\begin{enumerate}
\item $W_{\gamma}(B;X)$ as the collection of all monomorphisms $\imath: \gamma(B)\hookrightarrow X$, where $B\in\mathbf{NFre}_{\Gamma}$, and $W_{\gamma}(X)$ as the union of all of them;
\item $W_{\gamma_{\Sigma}}(B;X)$ as the collection of all morphisms $f:\gamma_{\Sigma}(B)\rightarrow X$, for $B\in \mathbf{NFre}_{\Gamma\times\Gamma}$ which are of the form $f=\imath\circ g$, where $\imath:\gamma_{\Sigma}(B')\hookrightarrow X$ is a monomorphism and $g:\gamma_{\Sigma}(B)\rightarrow \gamma_{\Sigma}(B')$ is an arbitrary morphism. Let $W_{\gamma_{\Sigma}}(X)$ be the union of all of them.
\end{enumerate}
We say that $\mathcal{X}$ is \textit{$\gamma$-concrete} (resp. \textit{$\gamma_{\Sigma}$-concrete}) in $X$ if the functor $\gamma$ (resp. $\gamma_\Sigma$) satisfies the concreteness property in $W_{\gamma}(X)$ (resp. $W_{\gamma_{\Sigma}}(X)$). We say that $\mathcal{X}$ is \textit{concrete} if it is both $\gamma$-concrete and $\gamma_{\Sigma}$-concrete.
\end{definition}
\begin{remark}\label{remark_need_concreteness}
The class $W_{\gamma_{\Sigma}}(B;X)$ contains morphisms of the form $f=\imath\circ g$ (instead of only monomorphism) precisely because we will need to concrectify diagrams like (\ref{diagram_abstract_intersection}), where $g=\gamma_\Sigma(*)$. Notice that if $g$ was assumed to be a monomorphism, then by the arguments of Lemma \ref{lemma_monomorphisms}, the multiplication $*:B\otimes B\rightarrow B$ would be a monomorphism too, which is not true in full generality.
\end{remark}
\begin{definition}
Let $\mathbb{X}=(\mathcal{X},X,\imath,\imath')$ be an intersection structure between two fixed $\Gamma$-spaces $B$ and $B'$. We say that it is \textit{$\gamma$-concrete}\footnote{In \cite{eu_1} this corresponds to the notion of \textit{proper} intersection structure.} (resp. \textit{$\gamma_\Sigma$-concrete} or concrete) if $\Gamma$-ambient $\mathcal{X}$ is $\gamma$-concrete (resp. $\gamma_\Sigma$-concrete or concrete) in $X$.
\end{definition}
\begin{definition}
Consider, now, an intersection structure $\mathbb{X}=(\mathcal{X},\mathbb{X}_0,\mathbb{X}_*,\mathbb{X}_+)$ between two distributive $\Gamma$-spaces $(B,\epsilon,\delta,*,+)$ and $(B',\epsilon',\delta',*',+')$. We say that $\mathbb{X}$ is \textit{concrete} if $\mathbb{X}_0$ is $\gamma$-concrete and $\mathbb{X}_*$ and $\mathbb{X}_+$ are $\gamma_{\Sigma}$-concrete. The \textit{concrete intersection} between $B$ and $B'$ in $\mathbb{X}$ is given by the concrete cospans of the spans\footnote{Take a look at diagrams (\ref{diagram_abstract_intersection}) and (\ref{diagram_concreteness}), and at Remark \ref{remark_need_concreteness}.} $(\imath,\imath')$, $(\jmath_*,\jmath_*')$, $(\jmath_+,\jmath_+')$, $(\jmath_*\circ \gamma_{\Sigma}(*),\jmath_*'\circ \gamma_{\Sigma}(*'))$ and $(\jmath_+\circ \gamma_{\Sigma}(+),\jmath_+'\circ \gamma_{\Sigma}(+'))$. The vertices of these cospans will be respectively denoted by $B\cap_{X,\gamma}B'$, $B_\epsilon\cap_{X_*,\gamma_{\Sigma}}B_{\epsilon'}$, $B_\delta\cap_{X_+,\gamma_{\Sigma}}B_{\delta'}$, $*\cap_{X_*,\gamma_{\Sigma}}*'$ and $+\cap_{X_+,\gamma_{\Sigma}}+'$.
\end{definition}
\begin{remark}
Notice that if an abstract intersection between $\Gamma$-spaces $B$ and $B'$ is concrete, then it can itself be regarded as a $\Gamma$-space. However, if an abstract intersection between \textit{distributive} $\Gamma$-space is concrete, then it is not necessarily a \textit{distributive} $\Gamma$-space; it is only a $\Gamma\times \Gamma$-space with further stuff. This is enough for our purposes.
\end{remark}
We close with a useful lemma which shows that for concrete vector ambients, concrete intersections of nontrivial distributive $\Gamma$-spaces are nontrivial too.
\begin{lem}\label{vectorial_intersection}
Let $\mathbb{X}$ be a concrete intersection structure between distributive $\Gamma$-spaces $(B,\epsilon,\delta,*,+)$ and $(B',\epsilon',\delta',*',+')$, whose underlying $\Gamma$-ambient $\mathcal{X}$ is vectorial. If there exist $i,j\in \Gamma$ such that $B_i$ or $B'_j$ are nontrivial, then $\mathbb{X}$ is nontrivial too.
\end{lem}
\begin{proof}
See Proposition 1, page 10, of \cite{eu_1}.
\end{proof}
\subsection{Extension to Sheaves}
In the last sections we discussed that in order to formulate a notion of ``regularity'' on a $C^k$-manifold $M$ we have to consider nontrivial intersections between the existing $C^k$ regularity and the additional one. Notice, on the other hand, that by the very notion of regularity, they are \textit{local} properties. This means that a manifold $M$ is ``regular'' when the regularity is satisfied on the local pieces $U_s\subset M$ in such a way that coherence conditions hold at the intersections $U_{ss'}=U_s\cap U_{s'}$. Thus, we do not need a \textit{global} $\Gamma$-space $B(M)$, but actually one $B(\varphi_s(U_s))$ for each local chart $\varphi_s:U_s\rightarrow \mathbb{R}^n$. Therefore, we have to work with presheaves $B:\operatorname{Open}(\mathbb{R}^n)^{op}\rightarrow \mathbf{NFre}_{\Gamma}$ of $\Gamma$-spaces.
\begin{example}\label{example_Ck_presheaf}
The examples of $\Gamma$-spaces in Example \ref{example_1_gamma_spaces} and Example \ref{example_2_gamma_spaces} extends, naturally, to presheaves of $\Gamma$-spaces. E.g, given $m,k$ and $\beta:\Gamma \rightarrow [0,k]$, we have the presheaf $U\mapsto (C^{k-\beta(i)}(U))_i$. It will be denoted by $C^{k-\beta}_m$. or simply $C^{k-\beta}$ if $m=1$. If, in addition, $\Gamma = [0,k]$ and $\beta=id$ it will be denoted by $C^{k-}$.
\end{example}
\begin{example}
Let $B$ be a presheaf of $\Gamma$-spaces. Then every function $\alpha:\Gamma' \rightarrow \Gamma$ defines a new presheaf of $\Gamma'$-spaces $B_\alpha$ by $(B_\alpha)(U)=B(U)_\alpha$.
\end{example}
Notice that all the discussion above can be internalized in the presheaf category
of presheaves in $\mathbb{R}^{n}$ by means of just taking a parametrization of definitions and results in terms of opens sets of $\mathbb{R}^n$. Thus, we can talk about presehaves of distributive $\Gamma$-spaces, presheaves of intersection structures between presheaves of distributive $\Gamma$-spaces, and so on. We refer the reader to Section 3 of \cite{eu_1} for more details to this extension to the presheaf setting.
\begin{remark}
To maintain compatibility with notations of \cite{eu_1}, throughout this paper a presheaf of intersection structures between presheaves of distributive $\Gamma$-spaces $B$ and $B'$ will be called an \textit{intersection structure presheaf} (ISP). Furthermore, it will be denoted simply by $\mathbb{X}=(\mathcal{X},X)$, where $\mathcal{X}=(\mathbf{X},\gamma,\gamma_{\Sigma})$ is the $\Gamma$-ambient and $X:\operatorname{Op}(\mathbb{R}^n)^{op}\rightarrow \mathbf{NFre}_{\Gamma}$ is the presheaf of ambient $\Gamma$-spaces.
\end{remark}
\subsection{$B^k_{\alpha,\beta}$-Presheaves}
In order to prove the existence of a geometric object on a smooth manifold one typically first proves local existence and then uses a partition of unity to globalize the construction. To extend the use of partition functions to the present regular setting we will work with presheaves of $\Gamma$-spaces which are \textit{$B^k_{\alpha,\beta}$-presheaves}. See Section 3 of \cite{eu_1} for more details.
\begin{itemize}
\item In the following: $C^{k}_b(U)\subset C^{k}(M)$ denotes the space of $C^k$-bump functions with the Fr\'echet structure induced by the seminorms (\ref{frechet_ck}).
\end{itemize}
\begin{definition}
We say that a set of indexes $\Gamma '$ has \textit{degree $r$} for some $0\leq r \leq k$ if it contains the interval $[0,r]$.
\end{definition}
\begin{itemize}
\item In the following we will write $\Gamma_r$ to denote a generic set with degree $r$, while $\Gamma$ will remain an arbitrary set.
\end{itemize}
\begin{definition}
Let $B=(B_i)_{i\in \Gamma}$ be a presheaf of $\Gamma$-spaces and let $\alpha:\Gamma' \rightarrow \Gamma$ and $\beta:\Gamma' \rightarrow [0,k]$ be functions. We say that the tuple $(B,\alpha,\beta,k)$ has \textit{degree $r$} if the domain $\Gamma'$ of the functions $\alpha$ and $\beta$ has degree $r$, i.e., if they are defined in some $\Gamma_r$.
\end{definition}
\begin{definition}\label{definition_C^k-presheaves}
Let $(B,\alpha,\beta,k)$ be a tuple as above. Let $\mathbb{X}=(\mathcal{X},X)$ be a concrete ISP between $B_{\alpha}$ and $C^{k-\beta}$. We say that $(B,k,\alpha,\beta)$ is a $B^{k}_{\alpha,\beta}$\textit{-presheaf} in $\mathbb{X}$ if it has degree $k$ and for every open set $U\subset\mathbb{R}^n$ and every $i\Gamma_k$ there exists the dotted arrow $\star_{U,i}$ making commutative
the diagram below, where $\theta_{2,U}$ is part of the concrete cospan\footnote{which therefore is a monomophisms due to Lemma \ref{lemma_monomorphisms}.}.
\begin{equation}{\label{diagram_star}
\xymatrix{C^{k}(U)\otimes C^{k-\beta(i)}(U) \ar[r]^-{\cdot_{U,i}} & C^{k-\beta(i)}(U) \\
\ar@{^(->}[u]^{\imath_U \otimes \theta_{2,U}} C^k_b(U) \otimes B_{\alpha (i)}(U) \cap_{X(U)} C^{k-\beta(i)}(U) \ar@{-->}[r]_-{\star_{U,i} } & B_{\alpha(i)}(U) \cap_{X(U)} C^{k-\beta(i)}(U) \ar@{^(->}[u]_{\theta_{2,U}} }}
\end{equation}
\end{definition}
Intuitively, a $B^k_{\alpha,\beta}$-presheaf is a presheaf of $\Gamma$-spaces $B(U)$, which, when regarded as a presheaf $B_{\alpha}$ of $\Gamma_k$-spaces via $\alpha$, has an intersection space with $C^{k-\beta}(U)$ that is closed under multiplication of $C^k$-bump functions.
\begin{example}\label{example_b_k_a_b_1}
For every $k,\beta,\alpha$ the presheaf of $C^{k-\alpha}$ is a
$B_{\beta\beta}^{k}$-presheaf in the ISP which is objectwise
the standard vectorial intersection structure of Example \ref{vectorial_ambient}.
\begin{example}\label{example_b_k_a_b_2}
Similarly, the presheaves $L_\alpha(U)_{i}=L^{\alpha(i)}(U)$ with the distributive structure given by pointwise sum and multiplication,
is a nice $B_{\beta}^{k,\alpha}$-presheaf in
the standard ISP, for $A(U)=C_{b}^{\infty}(U)$. An analogous
conclusion is valid if we replace pointwise multiplication with convolution
product.
\end{example}
\end{example}
We close this section with two remarks concerning Definition \ref{definition_C^k-presheaves}.
\begin{enumerate}
\item \label{nice_presheaves} It has a generalization where one requires invariance under multiplication only by a subvector space $A(U)\subset C^k(U)$ with a nuclear Fr\'echet structure. In this case, for instance, in Example \ref{example_b_k_a_b_2} we would get a different $B^k_{\alpha,\beta}$-structure in $L_\alpha$ taking $A(U)=\mathcal{S}(U)$ as the Schwartz space. Following \cite{eu_1}, a \textit{nice} $B^k_{\alpha,\beta}$-presheaf should be one such that $A(U)$ nontrivially intersects $C^k_b(U)$. In the following, however, we will work only with $A(U)=C^k_b(U)$, so that every $B^k_{\alpha,\beta}$-preshef will be nice.
\item Following the conventions of \cite{eu_1} a $B^k_{\alpha,\beta}$-presheaf in $\mathbb{X}$ whose presheaves of intersection spaces $B_\alpha \cap_X C^{k-\beta}$ are nontrivial is called a \textit{$C^k_{\alpha,\beta}$-presheaf}. In this paper we will not work directly with them. However, from Lemma \ref{vectorial_intersection}, for vectorial ISP $\mathbb{X}$, if $B_{\alpha}$ is nontrivial (i.e., if for every $U\subset \mathbb{R}^n$ there is $i_U\in \Gamma$ such that $B_{\alpha(i_U)}(U)\neq 0$), then every $B^k_{\alpha,\beta}$-presheaf in $\mathbb{X}$ is a $C^k_{\alpha,\beta}$-presheaf. In particular, the $B^k_{\alpha,\beta}$-presheaves of Examples \ref{example_b_k_a_b_1}-\ref{condition_2} are $C^k_{\alpha,\beta}$-presheaves.
\end{enumerate}
\subsection{$B^k_{\alpha,\beta}$-Functions}
As discussed in the introduction, in order to formalize the notion of regularity on manifolds we have to demand conditions not only on the transition functions $\varphi^{a}_{s's}:\varphi_{s}(U_{ss'})\rightarrow \mathbb{R}$, with $a=1,...,n$, but also on their derivatives $\partial^\mu\varphi^{a}_{s's}$, with $\mu=1,...,k$, where $\varphi_{s's}=\varphi_{s'}\circ \varphi^{-1}_s$.
In our context this is described as follows. See Section 5 of \cite{eu_1} for a complete exposition.
\begin{definition}
Let $(B,k,\alpha,\beta)$ be a $B^k_{\alpha,\beta}$-presheaf in a concrete ISP $\mathbb{X}$. Let $U\subset\mathbb{R}^n$ be an open set and let $f:U\rightarrow \mathbb{R}$ be a real $C^k$-function. Let $S\subset [0,k]\subset \Gamma_k$ be a subset\footnote{Here is where we are using that a $B^k_{\alpha,\beta}$-presheaf has degree $k$.}. We say that $f$ is a $(B^k_{\alpha,\beta}\vert S)$\textit{-function} (or $(B,k,\alpha,\beta|S)$\textit{-function}) in $\mathbb{X}$ if for every $i\in S$ we have $\partial^{i}f \in B_{\alpha(i)}(U)\cap_{X(U)}C^{k-\beta(i)}(U)$. A vectorial function $f:U\rightarrow\mathbb{R}^m$, with $m\geq 1$ is a $(B^k_{\alpha,\beta}|S)$\textit{-function} in $\mathbb{X}$ if its coordinate functions $f^a:U\rightarrow \mathbb{R}$, with $a=1,...,m$, are.
\end{definition}
Thus, for instance, if $\Gamma = [0,k]=\Gamma_k$ and $\alpha(i)=i=\beta(i)$, then $f$ is a $(B,k,\alpha,\beta|S)$-function in $\mathbb{X}$ precisely if $\partial^\mu f$ belongs to $B_i(U) \cap_X(U) C^{k-i}(U)$. This means that the set $S$ and the functions $\alpha$ and $\beta$ determine how the derivatives $\partial^\mu f$ of $f$ lose regularity when $\mu$ increases.
\subsection{$B^k_{\alpha,\beta}$-Manifolds}
We are now ready to define what is a $B^k_{\alpha,\beta}$-manifold. See Section 5 of \cite{eu_1}.
\begin{definition}
A $C^{k}$-\emph{manifold} is a paracompact Hausdorff topological
space $M$ endowed with a maximal atlas $\mathcal{A}$ with charts
$\varphi_{s}:U_{s}\rightarrow\mathbb{R}^{n}$, whose transition functions
$\varphi_{s's}=\varphi_{s'}\circ\varphi_{s}^{-1}:\varphi_{s}(U_{ss'})\rightarrow\mathbb{R}^{n}$
are $C^{k}$, where $U_{ss'}=U_{s}\cap U_{s'}$.
\end{definition}
\begin{definition}
Let $(M,\mathcal{A})$ be a $C^k$-manifold. A subatlas $\mathcal{A}'\subset \mathcal{A}$ is \textit{closed under restrictions} if restrictions of charts in $\mathcal{A'}$ to smaller open sets remains in $\mathcal{A'}$. More precisely, if $(\varphi,U)\in \mathcal{A}'$ and $V\subset U$, then $(\varphi\vert_V,V)\in \mathcal{A}'$.
\end{definition}
\begin{definition}
Let $(B,k,\alpha,\beta)$ be a $B^k_{\alpha,\beta}$-presheaf in a concrete ISP $\mathbb{X}$. A $(B_{\alpha,\beta}^{k},\mathbb{X})$\emph{-structure
}in a $C^{k}$-manifold $(M,\mathcal{A})$ is a subatlas $\mathcal{B}_{\alpha,\beta}^{k}(\mathbb{X})$ closed under restrictions
whose transition functions $\varphi_{s's}$ are $(B,k,\alpha,\beta)$-functions in $\mathbb{X}$. Explicitly, this means that if $\varphi_s\in \mathcal{B}_{\alpha,\beta}^{k}(\mathbb{X})$, then for each other $\varphi_{s'}\in \mathcal{B}_{\alpha,\beta}^{k}(\mathbb{X})$ such that $U_{ss'}\neq \varnothing$, we have
$$
\partial^{i}\varphi^a_{ss'}\in B_{\alpha(i)}(\varphi_s(U_{ss'}))\cap_{X(\varphi_s(U_{ss'}))}C^{k-\beta(i)}(\varphi_s(U_{ss'}))
$$
for every $i\in [0,k]\subset \Gamma_k$.
\end{definition}
\begin{definition}
A \emph{$(B_{\alpha,\beta}^{k},\mathbb{X})$-manifold
}is a $C^{k}$ manifold endowed with a $(B_{\alpha,\beta}^{k},\mathbb{X})$-structure.
\end{definition}
In \cite{eu_1} the authors define \textit{morphisms} between two $(B^k_{\alpha,\beta},\mathbb{X})$-manifolds as $C^k$-maps whose local representation by charts in the corresponding $(B^k_{\alpha,\beta},\mathbb{X})$-structures are $(B,k,\alpha,\beta)$-functions in $\mathbb{X}$. Here we will need only the particular case between $C^k$-manifolds.
\begin{definition}
Let $(M,\mathcal{A})$ and $(M',\mathcal{A}')$ be $C^k$-manifolds, let $\mathbb{X}$ a concrete ISP, $(B,k,\alpha,\beta)$ a $B^k_{\alpha,\beta}$-presheaf in $\mathbb{X}$ and $S\subset [0,k]\subset \Gamma_k$ a subset. A \textit{$(B,k,\alpha,\beta|S)$-morphism}, \textit{$(B,k,\alpha,\beta|S)$-function} or \textit{$(B^k_{\alpha,\beta}|S)$-morphism} between $M$ and $M'$ is a $C^k$-function $f:M\rightarrow M'$ such that for every $\varphi\in\mathcal{A}$ and $\varphi ' \in \mathcal{A'}$ the corresponding $C^k$-function $\varphi' \circ f \circ \varphi^{-1}$ is actually a $(B^k_{\alpha,\beta}|S)$-function in $\mathbb{X}$. If $S=[0,k]$ we say simply that $f$ is a \textit{$B^k_{\alpha,\beta}$-morphism}.
\end{definition}
\begin{remark}
From the examples above it is clear that a $C^k$-manifold does not necessarily admits any $(B^k_{\alpha,\beta},\mathbb{X})$-structure. Thus, there are obstructions which typically can be characterized as topological or geometric obstructions on the underlying $C^k$-structure. A complete characterization of these obstructions was not found yet. On the other hand, in \cite{eu_1} the authors give sufficient conditions on $B$, $\alpha$, $\beta$, $k$ and on the underlying $C^k$-structure ensuring the existence of $B^k_{\alpha,\beta}$-structures, specially in the case where $(B,k,\alpha,\beta)$ is a $C^k_{\alpha,\beta}$-presheaf.
\end{remark}
\subsection{Affine $B^k_{\alpha,\beta}$-Connections}\label{sec_affine_connections}
We now introduce the objects of study in this paper: affine $\nabla$ connections on $C^k$-manifolds $M$ whose local coefficients $\Gamma^c_{ab}$ satisfies additional regularity conditions, possibly depending on some regularity on the underlying manifold $M$. In other words, we will consider ``regular'' affine connections on $(B^k_{\alpha,\beta},\mathbb{X})$-manifolds. The obvious idea would then to be take affine connections whose coefficients $(\Gamma_\varphi)^c_{ab}:\varphi(U)\rightarrow \mathbb{R}$ in each $(\varphi,U)\in \mathcal{B}^k_{\alpha,\beta}(\mathbb{X})$ are $B^k_{\alpha,\beta}$-functions. This, however, is too restrictive to the global existence of such objects due to two reasons.
\begin{enumerate}
\item \label{condition_1} \textit{The regularity rate decay of the derivatives $\partial^\mu \Gamma^c_{ab}$, when $\mu$ grows, is typically different of that $\partial^{\mu}\varphi^a_{ji}$ of the transition functions $\varphi^a_{ji}$}. This leads us instead to consider affine connections whose coefficients are $(B^k_{\alpha',\beta'}|S)$-functions, where $\alpha',\beta'$ are functions typically \textit{different} from $\alpha,\beta$ in shape and/or in domain/codomain. For instance, $\varphi^a_{ji}$ are $C^k$, while $\Gamma^c_{ab}$ is $C^{k-2}$. Thus, if $\beta(i)=i$, then $\beta'$ must be $\beta'(i)=i+2$. Equivalently, we can regard $\beta'$ as defined in $[2,k]$ instead of in $[0,k]$ and take $\beta'(i)=i$.
\item \label{condition_2} \textit{The ISP appearing in the regularity of $\Gamma^c_{ab}$ is usually different from that describing the regularity of $\varphi_{ij}$}. For instance, we could consider intersections in the same $\Gamma$-ambient $(\mathbf{X},\gamma,\gamma_{\Sigma})$, but in different presheaves of ambient $\Gamma$-spaces $X$ and $Y$.
\end{enumerate}
\begin{definition}
Let $(B,k,\alpha,\beta)$ be a $B^k_{\alpha,\beta}$-presheaf in a concrete ISP $\mathbb{X}$. Let $(M,\mathcal{B}^k_{\alpha,\beta}(\mathbb{X}))$ be a $B^k_{\alpha,\beta}$-manifold in $\mathbb{X}$. Let $S \subset [0,k]\subset \Gamma_k$ be a subset and consider functions $\alpha':\Gamma_k \rightarrow \Gamma$ and $\beta':\Gamma_k \rightarrow [2,k]$. Let $\mathbb{Y}$ be a concrete ISP between $B_{\alpha'}$ and $C^{k-\beta'}$. An \textit{affine $(B^k_{\alpha',\beta'}|S)$-connection} in $M$, relative to $\mathbb{Y}$, is an affine connections $\nabla$ in the underlying $C^k$-manifold such that for every $(\varphi,U) \in \mathcal{B}^{k}_{\alpha,\beta}(\mathbb{X})$ the corresponding local coefficients $(\Gamma_\varphi)^c_{ab}:\varphi(U)\rightarrow \mathbb{R}$ are $(B,k,\alpha',\beta'|S)$-functions in $\mathbb{Y}$.
\end{definition}
\subsection{Regularity Globalization Lemma}\label{sec_globalization_lemma}
Condition \ref{condition_1} and the nice hypotheses on $B^k_{\alpha,\beta}$ suffice to ensure existence of connections $\nabla$ which are \textit{locally regular} in $\mathbb{X}$, following the classical construction of affine connections via partitions of unity. Condition \ref{condition_2} will be used to prove global regularity. More precisely, we show that if $B_{\alpha}$ admits some universal way to connect two ISP from it to $C^{k-\beta}$, then every affine connection $\nabla$ \textit{locally regular} in a given $\mathbb{X}$ induces another affine connection $\overline{\nabla}$ which is \textit{globally regular} in an ISP $\mathbb{Y}$ ``compatible'' with $\mathbb{X}$. Here we introduce these notions of universality and compatibility between two ISP.
\begin{definition}$\;$
\begin{enumerate}
\item Two ISP $\mathbb{X}$ and $\mathbb{Y}$ between presheaves of $\Gamma$-spaces $B$ and $B'$ are \textit{compatible} if they have the same underlying $\Gamma$-ambient and \textit{strongly compatible} if they also have the same presheaf of ambient spaces $X$;
\item A \textit{strongly compatible sequence of} ISP (or scISP, for short) between sequences $B=(B_i)$ and $B=(B'_i)$ of presheaves of $\Gamma$-spaces is a sequence $\underline{\mathbb{X}}=(\mathbb{X}_i)$ such that $\mathbb{X}_i$ and $\mathbb{X}_j$ are strongly compatible for every $i,j$;
\item Two sequences $\underline{\mathbb{X}}=(\mathbb{X}_i)$ and $\underline{\mathbb{Y}}=(\mathbb{Y}_i)$ of ISP between $B=(B_i)$ and $B'=(B'_{i})$ are \textit{compatible} if $\mathbb{X}_i$ and $\mathbb{Y}_i$ are compatible for every $i$.
\end{enumerate}
\end{definition}
\begin{definition}
Let $B$ and $B'$ be two presheaves of $\Gamma$-spaces and of $\Gamma'$-spaces, respectively. Let $\Gamma_k$ be a set of degree $k$. A \textit{$\Gamma_k$-connective structure} between $B$ and $B'$ is given by:
\begin{enumerate}
\item sets of functions $\mathcal{O}\subset \operatorname{Mor}(\Gamma_k,\Gamma)$ and $\mathcal{O}\subset \operatorname{Mor}(\Gamma_k,\Gamma')$;
\item functions $D_{\mathcal{O}}$ and $D_{\mathcal{Q}}$ assigning to each pairs $\theta,\theta'\in\mathcal{O}$ and $\vartheta,\vartheta'\in\mathcal{Q}$ corresponding morphisms $D_\mathcal{O}(\theta,\theta'):B_{\theta}\Rightarrow B_{\theta'}$ and $D_\mathcal{Q}(\vartheta,\vartheta'): B'_{\vartheta}\Rightarrow B'_{\vartheta'}$ which satisfy the composite laws $$D_\mathcal{O}(\theta,\theta'')=D_\mathcal{O}(\theta',\theta'')\circ D_\mathcal{O}(\theta,\theta'),$$ meaning that the following triangles are commutative:
\begin{equation}\label{composition_law_connections}
\xymatrix{
B_\theta \ar@{=>}[rrd]_{D_{\mathcal{O}}(\theta,\theta'')} \ar@{=>}[rr]^{D_{\mathcal{O}}(\theta,\theta')} && B_{\theta'} \ar@{=>}[d]^{D_{\mathcal{O}}(\theta',\theta'')} &&& \ar@{=>}[d]_{D_{\mathcal{Q}}(\vartheta',\vartheta'')} B_{\vartheta'}&& B_{\vartheta} \ar@{=>}[ll]_{D_{\mathcal{Q}}(\vartheta,\vartheta') } \ar@{=>}[lld]^{D_{\mathcal{Q}}(\vartheta,\vartheta'') } \\
&& B_{\theta''} &&& B_{\vartheta ''} }
\end{equation}
\item for every $(\theta,\vartheta)\in\mathcal{O}\times \mathcal{Q}$ a concrete ISP $\mathbb{X}_{\theta,\vartheta}$ between $B_\theta$ and $B'_{\vartheta}$ whose corresponding sequence $\underline{\mathbb{X}}=(\mathbb{X}_{\theta,\vartheta})$ is strongly compatible. Let $X$ denote the underlying presheaf of ambient spaces;
\item for every other scISP $\underline{\mathbb{Y}}=(\mathbb{Y}_{\theta,\vartheta})$ between $B$ and $B'$, indexed in $\mathcal{O}\times \mathcal{Q}$ and compatible with $\underline{\mathbb{X}}=(\mathbb{X}_{\theta,\vartheta})$, a morphism $\mathcal{D}(X;Y):X\Rightarrow Y$ of presheaves, where $Y$ is the presheaf of ambient spaces of each $\mathbb{Y}_i$, such that the diagram below commutes for every $\theta,\theta',\vartheta,\vartheta'$.
\begin{equation}{\label{connection_on_B}
\xymatrix{ \ar@{==>}[rd]^{\xi^{\theta,\vartheta}_{\theta'\vartheta'}} \ar@{=>}[dd] (B_{\theta} \cap _{X,\gamma} B'_{\vartheta}) \ar@{=>}[rr] && B'_{\vartheta} \ar@{=>}@/^{0.75cm}/[ddd]^{\jmath'_{\vartheta}} \ar@{=>}[d]_{D_{\mathcal{Q
}}(\vartheta ,\vartheta ')} \\
& \ar@{=>}[r] (B_{\theta '} \cap_{Y} B'_{\vartheta '}) \ar@{=>}[d] & B'_{\vartheta '} \ar@{=>}[d]^{\imath_{\vartheta'} '} \\
B_{\theta} \ar@{=>}[r]^{D_{\mathcal{O}}(\theta ,\theta ')} \ar@{=>}@/_{0.5cm}/[rrd]_{\jmath_{\theta}} & \ar@{=>}[r]_{\imath_{\theta'}} B_{\theta '} & Y
\\
&& X\ar@{=>}[u]^{\mathcal{D}(X;Y)}}}
\end{equation}
\end{enumerate}
\end{definition}
\begin{remark}\label{remark_composition_law_connections}
By the universality of pullbacks, there exists the dotted arrow $\xi^{\theta,\vartheta}_{\theta',\vartheta'}$ in diagram (\ref{connection_on_B}). Furthermore, the composition laws (\ref{composition_law_connections}) and the uniqueness of pullbacks imply a composition law for those dotted arrows:
$$
\xi^{\theta,\vartheta}_{\theta '',\vartheta ''}=\xi^{\theta',\vartheta'}_{\theta '',\vartheta ''}\circ \xi^{\theta,\vartheta}_{\theta ',\vartheta '}.
$$
\end{remark}
\begin{definition}
We say that the sequence $\underline{X}=(\mathbb{X}_{\theta,\vartheta})$ is the \emph{base} scISP of the $\Gamma_k$-connective structure.
\end{definition}
\begin{itemize}
\item In the following we will say ``consider a $\Gamma_k$-connective structure between $B$ and $B'$ in the scISP $\underline{\mathbb{X}}$'', meaning that $\underline{\mathbb{X}}$ is the base scISP of the refereed connective structure.
\end{itemize}
In the case where $B$ and $B'$ are presheaves of distributive $\Gamma$-spaces and $\Gamma'$-spaces, we need to require that the connection between them preserves sum and multiplication.
\begin{definition}\label{distributive_connection}
Let $B$ and $B'$ be presheaves of distributive $\Gamma$-spaces and $\Gamma'$-spaces, respectively. A $\Gamma_k$-connective structure between $B$ and $B'$ in $\underline{\mathbb{X}}=(\mathbb{X}_{\theta,\vartheta})$ is \textit{distributive} if for every $\theta,\theta'\in\mathcal{O}$ and every $\vartheta,\vartheta'\in\mathcal{Q}$ there are objective monomorphisms $(\gamma_*)_{\theta,\theta'}:B_{\epsilon(\theta,\theta)}\Rightarrow B_{\epsilon(\theta',\theta')}$
and $(\gamma'_{*'})_{\vartheta,\vartheta'}:B'_{\epsilon'(\vartheta,\vartheta)}\Rightarrow B'_{\epsilon'(\vartheta',\vartheta')}$
making commutative the diagram below, and also objectwise monomorphisms
$(\gamma_{+})_{\theta,\theta'}:B_{\delta(\theta,\theta)}\Rightarrow B_{\delta(\theta',\theta')}$
and $(\gamma'_{+'})_{\vartheta,\vartheta'}:B'_{\delta'(\vartheta,\vartheta)}\Rightarrow B'_{\delta'(\vartheta',\vartheta')}$
making commutative the analogous diagram for the additive structures.
$$
\resizebox{\displaywidth}{!}{
\xymatrix{\ar@{=>}[d]_{(\gamma' _{*'})_{\vartheta,\vartheta'}} B'_{\epsilon '(\vartheta,\vartheta)} & \ar@{=>}[l]_-{*'} B'_{\vartheta} \otimes B'_{\vartheta} & \ar@{=>}[d]_{\xi_{\theta ',\vartheta '}^{\theta, \vartheta}\otimes \xi_{\theta ',\vartheta '}^{\theta ,\vartheta}} (B_{\theta} \cap_X B'_{\vartheta})\otimes (B_{\theta} \cap_X B'_{\vartheta}) \ar@{=>}[r] \ar@{=>}[l] & B_{\theta} \otimes B_{\theta} \ar@{=>}[r]^-{*} & B_{\epsilon (\theta,\theta)} \ar@{=>}[d]^{(\gamma _{*})_{\theta,\theta'}} \\
B'_{\epsilon '(\vartheta ',\vartheta ')} & \ar@{=>}[l]^-{*'} B'_{\vartheta '} \otimes B'_{\vartheta '} & (B_{\theta'} \cap_{Y} B'_{\vartheta '})\otimes (B_{\theta '} \cap_Y B'_{\vartheta '}) \ar@{=>}[r] \ar@{=>}[l] & B_{\theta '} \otimes B_{\theta '} \ar@{=>}[r]_-{*} & B_{\epsilon (\theta ',\theta ')}}}
$$
\end{definition}
Let us now return to our context of $B^k_{\alpha,\beta}$-presheaves. First, a notation:
\begin{itemize}
\item given a set $\Gamma_k$ of degree $k$, a function $\beta_0:\Gamma_k \rightarrow [2,k]$ and $j\in \Gamma_k$, let $[\beta_0;j]_k$ denote the subset of all $i\in [0,k]\subset \Gamma_k$ such that $0\leq \beta_0(j)-i \leq k$, i.e., the interval $[0,k-\beta_0(j)]$.
\item Thus, by restriction each presheaf $B$ of $\Gamma'$-spaces and the presheaf\footnote{Recall the notation in Example \ref{example_Ck_presheaf}.} $C^{k-}$ of $[0,k]$ can both be regarded as presheaves of $[\beta_0;j]_k$-spaces.
\end{itemize}
\begin{definition}
Let $\mathbb{X}$ be a concrete ISP and let $(B,k,\alpha,\beta)$ be a $B^k_{\alpha,\beta}$-presheaf in $\mathbb{X}$. Given functions $\alpha_0:\Gamma_k\rightarrow \Gamma$ and $\beta_0:\Gamma_k \rightarrow [2,k]$, for each $j\in\Gamma_k$ regard $B$ and $(C^{k-i})_i$ as presheaves of $[\beta_0;j]_k$-spaces. A $(\alpha_0,\beta_0;j)$\textit{-connection} in $B$ is a connection between $B$ and $C{^k-}$ in some scISP $\underline{\mathbb{X}}$, such that:
\begin{enumerate}
\item $\alpha,\alpha_0\in \mathcal{O}$, $\beta,\beta_0\in \mathcal{Q}$ and $\mathbb{X}_{\alpha,\beta}=\mathbb{X}$;
\item $\alpha_{0,j}\in\mathcal{O}$ and $\beta_{\#,j}\in\mathcal{Q}$, where $\alpha_{0,j}(i)=\alpha_0(j)$ is the constant function and $\beta_{,j}(i)=\beta_0(j)-i$ is the shifting function;
\item if $\vartheta \mathcal{Q}$ is any function bounded from above by $\beta_0$, i.e., if $\vartheta(i)\leq \beta_0(i)$, then the morphism $D_{\mathcal{Q}}(\beta_0,\vartheta):C^{h-\beta_0}\Rightarrow C^{k-\vartheta}$ is the canonical inclusion.
\end{enumerate}
\end{definition}
\begin{definition}
A $(\alpha_0,\beta_0,j)$-connection in $B$ is \textit{distributive} if it is a distributive connection between $B$ and $C^{k-}$ in the sense of Definition \ref{distributive_connection}.
\end{definition}
The globalization of the local regularity of an affine connection in a $(B^k_{\alpha,\beta},\mathbb{X})$-manifold will depends on the existence of a distributive $(\alpha_0,\beta_0;j)$-connection in $B$. Furthermore, the extended regular will have local coefficients which are $(B,\theta,\vartheta,\mathbb{Y}|S)$-functions for functions $(\theta,\vartheta)\in \mathcal{O}\times \mathcal{Q}$ which are ``ordinary'' and for a set $S$ which is of a very special shape, in the the following sense.
\begin{definition}
An \textit{additive structure} in a set $\Gamma_k$ of degree $k$ consists of
\begin{enumerate}
\item a set $\Gamma_{2k}$ of degree $2k$ containing $\Gamma_k$;
\item a map $\overline{+}:\Gamma_k\times \Gamma_k \rightarrow \Gamma_{2k}$ extending sum of nonnegative integers, i.e., such that if $z,l\in [0,k]$, then $z\overline{+}l=z+l$.
\end{enumerate}
An \textit{additive set of degree $k$} is a set of degree $k$ where an additive structure where fixed.
\end{definition}
\begin{itemize}
\item To make notations simpler, in the following we will write $z+l$ instead of $z\overline{+}l$ even if $z,l$ are arbitrary elements of $\Gamma_k$.
\item Let $(\Gamma_k,+)$ be an additive set of degree $k$ and $\beta_0:\Gamma_k \rightarrow [2,k]$ be a map. Given $z\in [\beta_0;j]_k$, let $\Gamma_k[z] \subset \Gamma_k$ be the set of every $l\in \Gamma_k$ such that $z+l\in [\beta_0;j]_k$.
\end{itemize}
\begin{lem}\label{lema_S_set}
In the same notations above, for every $z\in [\beta_0;j]_k$ we have
\begin{equation}\label{dependence_S_z}
\Gamma_k[z]\subset (\Gamma_k - \Gamma_k\cap[\beta_0(j)-z+1,\infty)).
\end{equation}
\end{lem}
\begin{proof}
Straightforward: just a simple inequalities study.
\end{proof}
\begin{example}\label{example_S_set}
In particular, if $\Gamma_k=[0,k]$, then $\Gamma_k[z]\subset [0,\beta_0(j)-z]$. More specifically, if $z=0$, then $\Gamma_k[z]=[\beta_0;j]_k$, while if $z=k-\beta_0(j)$, then $\Gamma_k[z]=0$.
\end{example}
\begin{remark}[Important Remark]\label{important_remark}
Sets like $\Gamma_k[z]$ will be the sets $S$ in which the regular affine connections that we will build will have coefficients as $(B,k,\theta,\vartheta|S)$-functions. Thus, for larger $\Gamma_k[z]$ we will have more control on the regularities of higher order derivatives of the coefficient functions. On the other hand, from Lemma \ref{lema_S_set} we see that when $z$ increases, the set $\Gamma_k[z]$ becomes smaller. Furthermore, in Example \ref{example_S_set} we saw that if $\Gamma_k=[0,k]$ is the obvious set of degree $k$, then $\Gamma_k[z]$ strongly depends on $z$. In particular, for $z=k-\beta_0(j)$ we have $\Gamma_k[z]=0$ which would produce a regular affine connection with no control on the regularity of derivatives! This is one of the main reasons for working in the setup of general $\Gamma_k$-spaces: \textit{to have more control on the regularity of derivatives of the new affine connections, specially independing on $z$}. Indeed, notice that if $\Gamma_k$ increases, then the difference (\ref{example_S_set}) becomes closer to $\Gamma_k$ and, therefore, is independent of $z$.
\end{remark}
\begin{example}\label{example_smooth_case}
If $k=\infty$, then $[\beta_0;j]_\infty=[0,\infty)$ for every $\beta_0$. Thus $[0,\infty)\subset \Gamma_{\infty}[z]$ for every $z\in [\beta_0;j]_\infty=[0,\infty)$, which follows from the fact that the additive structure of $\Gamma_\infty$ is compatible with sum of integers.
\end{example}
\begin{definition}
Let $\Gamma_k$ be a set of degree $k$, $\Gamma$ a set and let $\mathcal{O}\subset\operatorname{Mor}(\Gamma_k;\Gamma)$ and $\mathcal{Q}\subset\operatorname{Mor}(\Gamma_k;[0,k])$. Let $X\subset \Gamma_r$ a subset. We say that a sequence of pairs $(\theta_{l},\vartheta_{l})_l$, with $\theta_l\in\mathcal{O}$ and $\vartheta_l\in \mathcal{Q}$, is \emph{ordinary in $X$} if each $l\in X$ and the corresponding pair $(\theta_{*},\vartheta_{*})$,
given by $\theta_{*}(l)=\theta_{l}(z)$ and $\vartheta_{*}(l)=\vartheta_{l}(z)$ also
belongs to $\mathcal{O}\times\mathcal{Q}$. Finally, we say that $(\theta,\vartheta)\in\mathcal{O}\times\mathcal{Q}$
is \emph{ordinary in $X$} if $\theta=\theta'_{*}$ and $\vartheta=\vartheta'_{*}$
for some $\theta'_{*}$ and $\vartheta'_{*}$, i.e, if it is the induced
pair of a sequence of ordinary pairs in $X$.
\end{definition}
We can now state and prove the fundamental step in the globalization of the regularity.
\begin{lem}[Regularity Globalization Lemma]
\label{key_lemma} Let $M$ be a $(B_{\alpha,\beta}^{k},\mathbb{X})$-manifold and suppose that $B$ admits a $(\alpha_{0},\beta_{0};j)$-connection in some \emph{scISP} $\underline{\mathbb{X}}$. Then, for every $U\subset\mathbb{R}^{n}$, every
$z\in [\beta_0;j]_k$ and every \emph{scISP} $\underline{\mathbb{Y}}=(\mathbb{Y}_{\theta,\vartheta})$ between $B$ and $C^{k-}$, compatible with $\underline{\mathbb{X}}$, the elements of $$B_{\alpha_{0}(j)}(U)\cap_{X(U)}C^{k-\beta_{0}(j)-z}(U)$$
can be regarded as $(B,k,\theta,\vartheta|\Gamma_k[z])$-functions in $\mathbb{Y}_{\theta,\vartheta}$, for every $(\theta,\vartheta)\in\mathcal{O}\times\mathcal{Q}$ ordinary in $\Gamma_k[z]$.
\end{lem}
\begin{proof}
Since the $B$ and $C^{k-}$ admits a $\Gamma_k$-connective structure in some $\underline{\mathbb{X}}$,
by definition it follows that for every scISP $\underline{\mathbb{Y}}$ compatible
with $\underline{\mathbb{X}}$ and every $\theta,\theta'\in\mathcal{O}$
and $\vartheta,\vartheta'\in\mathcal{Q}$ we have morphisms $\xi_{U,i}:B_{\theta(i)}(U)\cap_{X(U)}C^{k-\vartheta(i)}\rightarrow B_{\theta'(i)}(U)\cap_{Y(U)}C^{k-\vartheta'(i)}(U)$
making commutative the diagram (\ref{connection_on_B}). Since the
$\Gamma_k$-connective structure is actually a $(\alpha_{0},\beta_{0};j)$-connection we
have $\alpha_{0,j}\in\mathcal{O}$ and $\beta_{\#,j}\in\mathcal{Q}$.
Thus, for $\theta=\alpha_{0,j}$ and $\vartheta=\beta_{\#,j}$ we
see that for every $\theta'\in\mathcal{O}$, $\vartheta'\in\mathcal{Q}$,
and $i\in[\beta_0;j]_k$ we have a morphism from $B_{\alpha_{0}(j)}(U)\cap_{X(U)}C^{k-\beta_{0}(j)-i}(U)$
to $B_{\theta'(i)}(U)\cap_{Y(U)}C^{k-\vartheta'(i)}(U)$. Notice that
if $f\in B_{\alpha_{0}(j)}(U)\cap_{X(U)}C^{k-\beta_{0}(j)-i}(U)$,
then $\partial^{l}f\in C^{k-\beta_{0}(j)-(i+l)}(U)$ for every $l$
that $i+l\in[\beta_0;j]_k$. Thus, by universality,
for every such $l$ we can regard $\partial^{l}f\in B_{\theta_{l}'(i)}(U)\cap_{Y(U)}C^{k-\vartheta_{l}'(i)}(U)$,
where $\theta'_{l}\in\mathcal{O}$ and $\vartheta'_{l}\in\mathcal{Q}$.
If $z\in[\beta_0;j]_k$, then by definition
we have $z+l\in [\beta_0;j]_k$ for every $l\in \Gamma_k[z]$, so that the functions
$\theta(l)=\theta_{l}'(z)$ and $\vartheta(l)=\vartheta_{l}'(z)$
are defined on $\Gamma_k[z]$. Suppose that this pair $(\theta,\vartheta)$
belongs to $\mathcal{O}\times\mathcal{Q}$, i.e, suppose that it is
ordinary in $\Gamma_k[z]$. Thus, $\partial^{l}f\in B_{\theta(l)}(U)\cap_{Y(U)}C^{k-\vartheta(l)}(U)$,
meaning that $f$ can be regarded as a $(B,k,\theta,\vartheta|\Gamma_k[z])$-function
in $\mathbb{Y}$.
\end{proof}
The Regularity Globalization Lemma will ensure that in each open set $U\subset \mathbb{R}^n$ the local regularity can be globalized. The collage of these conditions when $U$ varies will be made gain, as expected, via partitions of unity. Since we are introducing a new structure (connections between presheaves of $\Gamma$-spaces in an scISP), we need to require some compatibility between them, the partitions of unity and the regular charts of the underlying regular manifold.
\begin{definition}\label{nice_connections}
Let $B$ be a $B^k_{\alpha,\beta}$-presheaf in a concrete ISP $\mathbb{X}$. We say that a $(\alpha_{0},\beta_{0};j)$-connection in some scISP $\underline{\mathbb{X}}$ is:
\begin{enumerate}
\item \emph{Support preserving} if for every pair $(\theta,\vartheta)\in \mathcal{O}\times \mathcal{Q}$, every scISP $\underline{\mathbb{Y}}$ compatible with $\underline{\mathbb{X}}$ and every $U\subset \mathbb{R}^n$, if $f\in B_{\alpha_0(i)}(U)\cap_{X(U)}C^k_b(U)$, then $\operatorname{supp}( \xi^{\alpha_0,\beta_0}_{\theta,\vartheta}(f))\subset \operatorname{supp}(f)$\footnote{Recall that $\alpha_0\in \mathcal{O}$ and $\beta_0\in \mathcal{Q}$, so that the map $\xi^{\alpha_0,\beta_0}$ above exists.}.
\item \textit{Bump preserving} if for every pair $(\theta,\vartheta)\in \mathcal{O}\times \mathcal{Q}$, every scISP $\underline{\mathbb{Y}}$ compatible with $\underline{\mathbb{X}}$ and every $U\subset \mathbb{R}^n$, the map $\xi^{\alpha_0,\beta_0}_{\theta,\vartheta}$ factors as below\footnote{Compare with Definition \ref{definition_C^k-presheaves}. See Remark \ref{nice_presheaves}.}.$$
\xymatrix{\ar[d]_{\xi^{\alpha_0,\beta_0}_{\theta,\vartheta}} B_{\alpha_0(i)}(U)\cap_{X(U)}C^{k-\beta_0(i)}(U)& B_{\alpha_0(i)}(U)\cap_{X(U)}C^k_b(U) \ar@{_(->}[l] \ar@{-->}[d] \\
B_{\theta(i)}(U)\cap_{Y(U)}C^{k-\vartheta(i)}(U)& B_{\theta(i)}(U)\cap_{X(U)}C^k(U) \ar@{_(->}[l] }
$$
\item \textit{Unital} if given $f\in B_{\alpha_0(i)}(U)\cap_{X(U)}C^{k-\beta_0(i)}(U)$ and $p\in U$ such that $f(p)=1$, then $(\xi^{\alpha_0,\beta_0}_{\theta,\vartheta}(f))(p)=1$ for every $(\theta,\vartheta)\in \mathcal{O}\times \mathcal{Q}$ and every scISP $\underline{\mathbb{Y}}$ compatible with $\underline{\mathbb{X}}$.
\item \textit{Nice} if it is support preserving, bump preserving and unital.
\end{enumerate}
Next lemma shows that distributive nice $(\alpha_0,\beta_0;j)$-connections preserve partitions of unity.
\begin{lem}\label{lemma_preservation_partition_unity}
Let $B$ be $B^k_{\alpha,\beta}$ in $\mathbb{X}$ and suppose that it admits a distributive and nice $(\alpha_0,\beta_0;j)$-connection in some \emph{scISP} $\underline{\mathbb{X}}$. In this case, if $(U_s)_s$ is a covering of $\mathbb{R}^n$ and $(\psi_s)_s$ is a $C^k$-partition of unity in $\mathbb{R}^n$ subordinate to it, then for every $(\theta,\vartheta)\in \mathcal{O}\times \mathcal{Q}$ the corresponding sequence $(\xi^{\alpha_0,\beta_0}_{\theta,\vartheta}(\psi_s))_s$ is a partition of unity subordinate to the same covering.
\end{lem}
\begin{proof}
Support preserving ensures that each $\xi^{\alpha_0,\beta_0}_{\theta,\vartheta}(\psi_s)$ is a bump function with support in $U_s$, which is $C^k$ due to the bump preserving condition. Support preserving and locally finiteness of $(\psi_s)_s$ imply that $(\xi^{\alpha_0,\beta_0}_{\theta,\vartheta}(\psi_s))_s$ is locally finite too. Furthermore, since the $(\alpha_0,\beta_0;j)$-connection is distributive and unital, we have $$\sum_s \xi^{\alpha_0,\beta_0}_{\theta,\vartheta}(\psi_s)(p)=\xi^{\alpha_0,\beta_0}_{\theta,\vartheta}(\sum_s\psi_s(p))=\xi^{\alpha_0,\beta_0}(1)=1,$$
so that $(\xi^{\alpha_0,\beta_0}_{\theta,\vartheta}(\psi_s))_s$ is a partition of unity, as desired.
\end{proof}
\end{definition}
\begin{itemize}
\item Let $(M,\mathcal{A})$ be a $C^k$-manifold. For every $0\leq r \leq k$, let $\partial^r\mathcal{A}(U)\subset C^{k-r}(U)$ be the set of $r$th derivatives $\partial^r \varphi^a_{ji}$ of transitions functions $\varphi_{ji}=\varphi_j\circ \varphi_i^{-1}$ by charts $\varphi_i,\varphi_j\in \mathcal{A}$ such that $\varphi_i(U_{ij})=U$. Note that if $\mathcal{B}\subset \mathcal{A}$ is a subatlas, then $\partial^r \mathcal{B}(U)\subset\partial^r\mathcal{A}(U)$ for each $r$.
\end{itemize}
\begin{definition}
Let $(M,\mathcal{B}^k_{\alpha,\beta}(\mathbb{X}))$ be a $(B^k_{\alpha,\beta},\mathbb{X})$-manifold and take $0\leq r\leq k$. We say that a $(\alpha_{0},\beta_{0};j)$-connection in $B$ in some scISP $\underline{\mathbb{X}}$ has \emph{degree $r$} if for every par $(\theta,\vartheta)\in \mathcal{O}\times \mathcal{Q}$ such that $i \leq \vartheta(i)$, with $0\leq i\leq r$, every scISP $\underline{\mathbb{Y}}$ compatible with $\mathbb{X}$ and every $U\subset \mathbb{R}^n$, there exists the dotted arrow below (we omitted $U$ to simplify the notation). In other words, $\xi^{\alpha,\beta}_{\theta,\vartheta}(\partial^l \varphi_{ji}) = \partial^l \phi_{ji}$, for every transition function $\varphi_{ji}$ by charts $\varphi_i,\varphi_j\in \mathcal{B}^k_{\alpha,\beta}(\mathbb{X})$, where $\phi_{ji}$ are transition functions of charts $\phi_i,\phi_j\in \mathcal{A}$.
\begin{equation}
\xymatrix{ \ar@{-->}[d] (B_{\alpha(i)}\cap_{X}C^{k-\beta(i)})\cap_{C^{k-\beta(i)}} \partial^i\mathcal{B}^k_{\alpha,\beta}(\mathbb{X}) \ar@{^(->}[r] & B_{\alpha(i)}\cap_{X}C^{k-\beta(i)} \ar[d]^{\xi^{\alpha,\beta}_{\theta,\vartheta}} \\
(B_{\theta(i)}\cap_{X}C^{k-\vartheta(i)})\cap_{C^{k-\vartheta(i)}} \partial^i\mathcal{A} \ar@{^(->}[r] & B_{\theta(i)}\cap_{X}C^{k-\vartheta(i)}}
\end{equation}
\end{definition}
\section{Existence \label{sec_a_b_connections}}
We are now ready to prove the existence theorem of $(B^k_{\alpha_0,\beta_0}|S)$-connections in $B^k_{\alpha,\beta}$-manifolds in the same lines described in the introduction, now presented in a much more precise form:
\begin{enumerate}
\item \textit{Existence of regular locally defined connections.} In Proposition \ref{prop_local_existence} we will show that every coordinate open set $U\subset M$ of a $C^k$-manifold admits a $B^k_{\alpha_0,\beta_0}$-connection for every $\alpha_0,\beta_0$.
\item \textit{Existence of weakly locally regular globally defined connections.} In Theorem \ref{thm_existence_connections} we will prove, using the first step and a partition of unity argument, that every $(B^k_{\alpha,\beta},\mathbb{X})$-manifold admits, for every given functions $\alpha_0,\beta_0$, affine connections $\nabla$ whose coefficients in each $(\varphi,U)\in \mathcal{B}^k_{\alpha,\beta}(\mathbb{X})$ belongs to \ref{B_a'_C_b'}, where $\alpha'_0$ and $\beta'_0$ are numbers depending on $\alpha_0$ and $\beta_0$.
\item \textit{Existence of locally regular globally defined connections.} The connections obtained in the last step are \textit{almost} locally regular because their local coefficients are \textit{not} $(B,k,\alpha',\beta')$-functions for certain $\alpha',\beta'$, but only belong to (\ref{B_a'_C_b'}). In order to be $(B,k,\alpha',\beta')$-functions we also need regularity on the derivative of the coefficients. It is at this moment that Lemma \ref{key_lemma} plays its role and, therefore, the moment when we need to add their hypotheses and additional objects ($(\alpha_0,\beta_0;j)$-connections). This is done in Proposition \ref{prop_a',b'}.
\item \textit{Existence of globally regular globally defined connections.} Finally, in Theorem \ref{corollary_existence_connections} we glue the locally regular connections in each $(\varphi,U)\in \mathcal{B}^k_{\alpha,\beta}(\mathbb{X})$ getting the desired $B^k_{\alpha',\beta'}$-connections. It is in this step that we will need to assume that the $(\alpha_0,\beta_0;j)$-connections are compatible with partitions of unity, i.e., that they are nice in the sense of Definition \ref{nice_connections}.
\end{enumerate}
\begin{proposition}[Regular Local Existence]\label{prop_local_existence}
Let $(M,\mathcal{A})$ be a $C^k$-manifold, with $k\geq 2$, and $(\varphi,U)\in\mathcal{A}$ a chart, regarded as a $C^k$-manifold. Let $\Gamma$ be a set, $S \subset [0,k] \subset \Gamma_k$ a subset, and consider functions $\alpha_0:S \rightarrow \Gamma$ and $\beta_0:S \rightarrow [2,k]$. Let $B$ be a presheaf of distributive $\Gamma$-spaces. Then $U$ admits a $(B^k_{\alpha_0,\beta_0}|S)$-connection $\nabla_{\varphi}$ relatively to every concrete ISP between $B_{\alpha_0}$ and $C^{k-\beta_0}$.
\end{proposition}
\begin{proof}
Recall that any family of $C^{2}$-functions $f_{ab}^{c}:\mathbb{R}^{n}\rightarrow\mathbb{R}$
defines a connection in $\mathbb{R}^{n}$ \cite{chern}. Thus, any
family of $C^{2}$-functions $f_{ab}^{c}:V\rightarrow\mathbb{R}$
in an open set $V\subset\mathbb{R}^{n}$ defines a connection $\nabla_{0}$
in $V$. If $\varphi:U\rightarrow\mathbb{R}^{n}$ is a chart in $M$
with $\varphi(U)\subset V$, pulling back $\nabla_{0}$ we get a connection
$\nabla_{\varphi}$ in $U$ whose coefficients are $f_{ab}^{c}\circ\varphi$.
Consequently, if $f_{ab}^{c}$ are chosen $(B,k,\alpha_{0},\beta_{0}|S)$-functions
in $\mathbb{X}$, then $\nabla_{\varphi}$ is a $(B_{\alpha_0,\beta_0}^{k},\mathbb{X}|S)$-connection
in $U$.
\end{proof}
\begin{itemize}
\item If $(*,\epsilon)$ is the multiplicative structure of $B$, let $$\epsilon^{r}(l,m)=\epsilon(l,\epsilon(l,\epsilon(l,\epsilon(...\epsilon(l,m)...))).$$
\end{itemize}
\begin{theorem}[Locally Weakly Regular Existence]
\label{thm_existence_connections} Let $\mathbb{X}$ be a concrete
ISP and let $M$
be a $(B_{\alpha,\beta}^{k},\mathbb{X})$-manifold, with $k\geq2$. Let $\Gamma_k$ be a set of degree $k$.
Given functions $\alpha_{0}:\Gamma_k \rightarrow \Gamma$ and $\beta_{0}:\Gamma_k \rightarrow [2,k]$, there exists
an affine connection $\nabla$ in $M$ whose coefficients in each $(\varphi,U)\in\mathcal{B}_{\alpha,\beta}^{k}(\mathbb{X})$
belong to
\begin{equation}
B_{\alpha_{0}'}((\varphi(U'))\cap_{X(\varphi(U'))}C^{k-\beta_{0}'}(\varphi(U'))\label{B_a'_C_b'}
\end{equation}
for some nonempty open set $U'\subset U$,
\begin{equation}
\alpha_{0}'=\delta(\epsilon^{3}(\alpha(1),\alpha_{0}(0)),\epsilon(\alpha(2),\alpha(1))),\quad\beta_{0}'=\max(\beta(1),\beta(2),\beta_{0}(0))\label{a'_b'}
\end{equation}
and $(\epsilon,\delta)$ are part of the distributive structure of $B$.
\end{theorem}
\begin{proof}
Let $(\varphi_{s},U_{s})_{s}$ be an open
covering of $M$ by coordinate systems in $\mathcal{B}_{\alpha,\beta}^{k}(\mathbb{X})$
and let $\nabla_{s}$ be an affine connection in $U_{s}$ with coefficients
$(f_{s})_{ab}^{c}$. Let $(\psi_{s})$ be a partition of unity of class $C^k$, subordinate
to $(\varphi_{s},U_{s})$, and recall that $\nabla=\sum_{s}\psi_{s}\cdot\nabla_{s}$
is a globally defined connection in $M$. We assert that if $\varphi_{s}$
are in $\mathcal{B}_{\alpha,\beta}^{k}(\mathbb{X})$, then $\nabla$
is a well-defined connection whose cofficients belong to (\ref{B_a'_C_b'})
for certain $U'$. Notice that the coefficients $(\Gamma_{s})_{ab}^{c}$
of $\nabla$ in a fixed $\varphi_{s}$ are obtained in the following
way: let $N(s)$ be the finite set of every $s'$ such that $U_{ss'}\equiv U_{s}\cap U_{s'}\neq\varnothing$.
For each $s'\in N(s)$ we can do a change of coordinates and rewrite
$(f_{s'})_{ab}^{c}$ in the coordinates $\varphi_{s}$ as given by
the functions
\begin{equation}
(f_{s';s})_{ab}^{c}=\sum_{l,m,o}\frac{\partial\varphi_{s}^{c}}{\partial\varphi_{s'}^{l}}\frac{\partial\varphi_{s'}^{m}}{\partial\varphi_{s}^{a}}\frac{\partial\varphi_{s'}^{o}}{\partial\varphi_{s}^{b}}(f_{s'})_{mo}^{l}+\sum_{l}\frac{\partial^{2}\varphi_{s'}^{l}}{\partial\varphi_{s}^{a}\partial\varphi_{s}^{b}}\frac{\partial\varphi_{s}^{c}}{\partial\varphi_{s'}^{l}},\label{change_coordinates_connection-1}
\end{equation}
where by abuse of notation $\varphi_{s}^{c}=(\varphi_{s}\circ\varphi_{s'}^{-1})^{c}$.
We then have
\begin{equation}
(\Gamma_{s})_{ab}^{c}=\sum_{s'\in N(s)}\psi_{s'}(f_{s';s})_{ab}^{c}.\label{_partition_unity-1}
\end{equation}
From Proposition \ref{prop_local_existence} we can take each $\nabla_s$ $(B^k_{\alpha_0,\beta_0},\mathbb{X})$-connection, so that $(f_{s'})_{mn}^{l}$ can be choosen $(B,k,\alpha_{0},\beta_{0})$-functions
in $\mathbb{X}$. Furthermore, since $\varphi_{s}\in\mathcal{B}_{\alpha,\beta}^{k}(\mathbb{X})$
we have
\[
\frac{\partial\varphi_{s}^{c}}{\partial\varphi_{s'}^{l}}\in(B_{\alpha(1)}\cap_{X}C^{k-\beta(1)})(U_{ss'}^{s'})\quad\text{and}\quad\frac{\partial^{2}\varphi_{s'}^{l}}{\varphi_{s}^{a}\partial\varphi_{s}^{b}}\in(B_{\alpha(2)}\cap_{X}C^{k-\beta(2)})(U_{ss'}^{s'})
\]
where $U_{ss'}^{s'}=\varphi_{s'}(U_{ss'})$. This is well-defined
since $\beta(2)\leq k$. Due to the compatibility between the additive/multiplicative
structures of $B_{\alpha}$ and $C^{k-\beta},$ the first and the
second terms of the right-hand sice of (\ref{change_coordinates_connection-1})
belong to
\[
(B_{\epsilon^{3}(\alpha(1),\alpha_{0}(0))}\cap_{X}C^{k-\max(\beta(1),\beta_{0}(0))})(U_{ss'}^{s'})
\]
and
\[
(B_{\epsilon(\alpha(2),\alpha(1))}\cap_{X}C^{k-\max(\beta(2),\beta(1))})(U_{ss'}^{s'}),
\]
respectively, so that the left-hand side of (\ref{change_coordinates_connection-1})
belongs to
\[
(B_{\tau(\epsilon^{3}(\alpha(1),\alpha_{0}(0)),\epsilon(\alpha(2),\alpha(1)))}\cap_{X}C^{k-\max(\beta(1),\beta(2),\beta_{0}(0))})(U_{N(s)})=(B_{\alpha_{0}'}\cap_{X}C^{k-\beta_{0}'})(U_{N(s)}),
\]
where $U_{N(s)}=\cap_{s'\in N(s)}U_{ss'}$. Since $B$ is a $B_{\alpha,\beta}^{k}$-presheaf
in $\mathbb{X}$, diagram (\ref{diagram_star}) implies that each product $\psi_{s'}(f_{s';s})_{ab}^{c}$ in the sum (\ref{_partition_unity-1}) belongs to the same space, so that the entire sum also belongs to that space. Repeating the process for all coverings in $\mathcal{B}_{\alpha,\beta}^{k}(\mathbb{X})$
and noticing that $\alpha'$ and $\beta'$ do not depend on $s,s'$,
we get the desired result.
\end{proof}
\begin{proposition}[Locally Regular Existence]\label{prop_a',b'} In the same notations and hypotheses of Theorem \ref{thm_existence_connections},
if $B$ admits a $(\alpha',\beta';j)$-connection in some \emph{scISP} $\underline{\mathbb{X}}$, where $\alpha':\Gamma_k \rightarrow \Gamma$ and $\beta':\Gamma_k\rightarrow [2,k]$ are functions such that $\alpha'(j)=\alpha'_0$ and $\beta'(j)=\beta'_0$, then for
every \emph{scISP} $\underline{\mathbb{Y}}$ compatible $\underline{\mathbb{X}}$ and every $z\in [\beta';j]_k$, the elements of (\ref{B_a'_C_b'})
can be regarded as $(B,k,\theta,\vartheta|\Gamma_k[z])$-functions
in $\mathbb{Y}_{\theta,\vartheta}$, for every pair
$(\theta,\vartheta)\in \mathcal{O}\times \mathcal{Q}$ which is ordinary in $\Gamma_k[z]$.
\end{proposition}
\begin{proof}
Direct application of Lemma \ref{key_lemma}.
\end{proof}
\begin{theorem}[Global Regular Existence]
\label{corollary_existence_connections} Let $M$
be a $(B_{\alpha,\beta}^{k},\mathbb{X})$-manifold, with $k\geq2$,
and, as above, suppose that the structural presheaf $B$
admits a \emph{$(\alpha',\beta';j)$}-connection in some \emph{scISP} $\underline{\mathbb{X}}$, where $\alpha'(j)=\alpha'_0$ and $\beta'(j)=\beta'_0$. Suppose, in addition, that this $(\alpha',\beta';j)$-connection is distributive, nice and has
degree $r\geq2$. Then, $M$ admits a $(B_{\theta,\vartheta}^{k},\mathbb{Y}_{\theta,\vartheta}|\Gamma_k[z])$-connection for every \emph{scISP} $\underline{\mathbb{Y}}$ compatible with $\underline{\mathbb{X}}$, every $z\in [\beta';j]_k$ and every pair $(\theta,\vartheta)$ ordinary in $\Gamma_k[z]$ such that $i\leq \vartheta(i)$.
\end{theorem}
\begin{proof}
From Theorem \ref{thm_existence_connections} there exists an affine
connection $\nabla$ whose coefficients (\ref{_partition_unity-1})
belong to (\ref{B_a'_C_b'}). By the first part of the hypotheses we can apply Proposition \ref{prop_a',b'}, so that for any
pair $(\theta,\vartheta)$ ordinary in $\Gamma_k[z]$, the coefficients $(\Gamma_s)^c_{ab}$ in (\ref{_partition_unity-1}) can be regarded as $(B,k,\theta,\vartheta,\mathbb{Y}_{\theta,\vartheta}|\Gamma_k[z])$-functions. But, from the proof of Lemma \ref{key_lemma} we see that this way to regard elements of an intersection space as regular functions is made by taking their image under $\xi^{\alpha',\beta'}_{\theta,\vartheta}$. Thus, Proposition \ref{prop_a',b'} is telling us precisely that, for every $\varphi_s$, the corresponding $\xi^{\alpha',\beta'}_{\theta,\vartheta}((\Gamma_s)^c_{ab})$ are $(B,k,\theta,\vartheta,\mathbb{Y}_{\theta,\vartheta}|\Gamma_k[z])$-functions. We will show that, under the additional hypotheses, the collection of these when varying $s$ also defines an affine connection in $M$, which will be a $(B_{\theta,\vartheta}^{k},\mathbb{Y}_{\theta,\vartheta}|\Gamma_k[z])$-connection by construction. Notice that the expression (\ref{change_coordinates_connection-1}) which describes how $(\Gamma_s)^c_{ab}$ changes when $s$ varies involves multiplications, sums, bump functions and derivatives of the transition functions. But the additional hypotheses are precisely about the preservation of these data by $\xi$! More precisely, for $i \leq \vartheta(i)$ we have:
\begin{eqnarray*}
\xi^{\alpha',\beta'}_{\theta\vartheta}((f_{s';s})_{ab}^{c}) & = & \xi^{\alpha',\beta'}_{\theta\vartheta}\Bigl(\sum_{l,m,o}\frac{\partial\varphi_{s}^{c}}{\partial\varphi_{s'}^{l}}\frac{\partial\varphi_{s'}^{m}}{\partial\varphi_{s}^{a}}\frac{\partial\varphi_{s'}^{o}}{\partial\varphi_{s}^{b}}(f_{s'})_{mo}^{l}+\sum_{l}\frac{\partial^{2}\varphi_{s'}^{l}}{\partial\varphi_{s}^{a}\partial\varphi_{s}^{b}}\frac{\partial\varphi_{s}^{c}}{\partial\varphi_{s'}^{l}}\Bigr)\\
& = & \sum_{l,m,o}\xi_{\theta\vartheta}\Bigl(\frac{\partial\varphi_{s}^{c}}{\partial\varphi_{s'}^{l}}\Bigr)\xi_{\theta\vartheta}\Bigl(\frac{\partial\varphi_{s'}^{m}}{\partial\varphi_{s}^{a}}\Bigr)\xi_{\theta\vartheta}\Bigl(\frac{\partial\varphi_{s'}^{o}}{\partial\varphi_{s}^{b}}\Bigr)\xi_{\theta\vartheta}((f_{s'})_{mo}^{l})+ \\
& & + \sum_{l}\xi_{\theta\vartheta}\Bigl(\frac{\partial^{2}\varphi_{s'}^{l}}{\partial\varphi_{s}^{a}\partial\varphi_{s}^{b}}\Bigr)\xi_{\theta\vartheta}\Bigl(\frac{\partial\varphi_{s}^{c}}{\partial\varphi_{s'}^{l}}\Bigr)\\
& = & \sum_{l,m,o}\frac{\partial\phi_{s}^{c}}{\partial\phi_{s'}^{l}}\frac{\partial\phi_{s'}^{m}}{\partial\phi_{s}^{a}}\frac{\partial\phi_{s'}^{o}}{\partial\phi_{s}^{b}}\xi^{\alpha',\beta'}_{\theta\vartheta}((f_{s'})_{mo}^{l})+\sum_{l}\frac{\partial^{2}\phi_{s'}^{l}}{\partial\phi_{s}^{a}\partial\phi_{s}^{b}}\frac{\partial\phi_{s}^{c}}{\partial\phi_{s'}^{l}},
\end{eqnarray*}
where we omitted the superior indexes in the maps $\xi$ to simplify the notation and $\phi_s$ are charts in the maximal $C^k$-atlas $\mathcal{A}$ of $M$. Furthermore, in the first step we used (\ref{change_coordinates_connection-1}) and
in the second one we used compatibility between the distributive structures
of $B$ and $C^{k-}$ and also the distributive properties of $\xi$ (which comes from the fact that the $(\alpha',\beta';j)$-connection is supposed distributive) and the composition laws (\ref{composition_law_connections}). In the third step we used that the $(\alpha',\beta';j)$-connection has degree $r\geq2$ (here is where we need $i\leq \vartheta(i)$). By the same kind of arguments,
\begin{eqnarray}
\xi^{\alpha',\beta'}_{\theta,\vartheta}((\Gamma_{s})_{ab}^{c}) & = & \xi^{\alpha',\beta'}_{\theta\vartheta}\Bigl(\sum_{s'\in N(s)}\psi_{s'}\cdot(f_{s';s})_{ab}^{c}\Bigr)\\
& = & \sum_{s'\in N(s)}\xi_{\theta,\vartheta}^{\alpha',\beta'}(\psi_{s'}) \Bigl(\sum_{l,m,o}\frac{\partial\phi_{s}^{c}}{\partial\phi_{s'}^{l}}\frac{\partial\phi_{s'}^{m}}{\partial\phi_{s}^{a}}\frac{\partial\phi_{s'}^{o}}{\partial\phi_{s}^{b}}\xi^{\alpha',\beta'}_{\theta\vartheta}((f_{s'})_{mo}^{l})+ \\
& & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + \sum_{l}\frac{\partial^{2}\phi_{s'}^{l}}{\partial\phi_{s}^{a}\partial\phi_{s}^{b}}\frac{\partial\phi_{s}^{c}}{\partial\phi_{s'}^{l}}\Bigr) \\
& = & \sum_{s'\in N(s)}\xi_{\theta,\vartheta}^{\alpha',\beta'}(\psi_{s'})((\Gamma_{s'})_{\theta,\vartheta})^c_{ab} \label{globalization_regular_connection},
\end{eqnarray}
where $((\Gamma_{s'})_{\theta,\vartheta})^c_{ab}$ are the coefficients of the affine connection in $U_{s'}$ defined by the family of functions $\xi^{\alpha',\beta'}_{\theta\vartheta}((f_{s'})_{mo}^{l})$. Since the $(\alpha',\beta';j)$-connection in $B$ is nice and distributive it follows from Lemma \ref{lemma_preservation_partition_unity} that $(\xi^{\alpha',\beta'}_{\theta,\vartheta}(\psi_s))_s$ is a $C^k$-partition of unity for $M$. Thus, (\ref{globalization_regular_connection}) is a globally defined affine connection in $M$, which by construction is a $(B^k_{\theta,\vartheta},\mathbb{Y}_{\theta,\vartheta}|\Gamma_k[z])$-connection.
\end{proof}
\section{Multiplicity \label{sec_multiplicity}}
The last result was about the existence of regular connections. In the following we will consider the multiplicity problem. More precisely, given a $B^k_{\alpha,\beta}$-manifold $M$ in an ISP $\mathbb{X}$, $S\subset \Gamma_k$, functions $\theta:S \rightarrow \Gamma$ and $\vartheta:S\rightarrow [2,k]$, and a concrete ISP $\mathbb{Y}$ compatible with $\mathbb{X}$, we will study the set $\operatorname{Conn}^k_{\theta,\vartheta}(M;\mathbb{Y}|S)$ of $(B_{\theta,\vartheta}^{k},\mathbb{Y}_{\theta,\vartheta}|S)$-connections in $M$.
\begin{itemize}
\item Let $(\mathcal{B}_{\operatorname{end}})^k_{\alpha,\beta}(\mathbb{X})\subset \mathcal{B}^k_{\alpha,\beta}(\mathbb{X})$ be the collection of all charts $(\varphi,U)\in \mathcal{B}^k_{\alpha,\beta}(\mathbb{X})$ which trivializes the endomorphism bundle $\operatorname{End}(TM)$.
\end{itemize}
\begin{definition}
Let $S\subset \Gamma_k$, $\theta:S\rightarrow \Gamma$, $\vartheta:S\rightarrow [2,k]$ and $\mathbb{Y}$ as above. An $\operatorname{End}(TM)$-valued $(B^k_{\theta,\vartheta},\mathbb{Y})$-form of degree one is an $\operatorname{End}(TM)$-valued $1$-form $\omega$ of class $C^k$ whose coefficients $(\omega_\varphi)^c_{ab}$ in every $(\varphi,U)\in (\mathcal{B}_{\operatorname{end}})^k_{\alpha,\beta}(\mathbb{X})$ are $(B,k,\theta,\vartheta|S)$-functions in $\mathbb{Y}$. Let
\begin{equation}\label{regular_1_forms}
\Omega^1_{\theta,\vartheta}(M;\operatorname{End}(TM),\mathbb{Y}|S)
\end{equation} denote the set of all of them.
\end{definition}
We begin with an easy fact: in the same way as the set of affine connections in a $C^k$-manifold is an affine space of the vector space of $\operatorname{End}(TM)$-valued 1-forms, the set of regular affine connections on a regular manifold is an affine space of the vector space of regular $\operatorname{End}(TM)$-valued 1-forms. More precisely,
\begin{proposition}\label{space_affine_connections_is_affine}
Let $M$ be a $(B^k_{\alpha,\beta},\mathbb{X})$-manifold. For every $S$, $\theta$, $\vartheta$ and $\mathbb{Y}$ as above:
\begin{enumerate}
\item \emph{(\ref{regular_1_forms})} is a vector subspace of $\Omega^1(M;\operatorname{End}(TM))$;
\item $\operatorname{Conn}^k_{\theta,\vartheta}(M;\mathbb{Y}|S)$ is an affine space of \emph{(\ref{regular_1_forms})}.
\end{enumerate}
\end{proposition}
\begin{proof}
For the first one, recall that in
a distributive $\Gamma$-space $B=(B_{i})$ the additive structure
$+_{ij}:B_{i}\otimes B_{j}\rightarrow B_{\delta(i,j)}$ is such that
$\delta(i,i)=i$ and $+_{ii}$ is the sum of $B_{i}$. For the second one, if $\nabla$ and $\overline{\nabla}$ are two affine $C^k$-connections
in $M$, then $\omega=\nabla-\overline{\nabla}$ has coefficients in $(\varphi,U)\in (\mathcal{B}_{\operatorname{end}})^k_{\alpha,\beta}(\mathbb{X})$ given by $(\omega_\varphi)_{ab}^{c}=(\Gamma_\varphi)_{ab}^{c}-(\overline{\Gamma}_\varphi)_{ab}^{c}$. Thus, again by $+_{ii}=+_i$, if $(\Gamma_\varphi)_{ab}^{c}$ and $(\overline{\Gamma}_\varphi)_{ab}^{c}$ are $(B,k,\theta,\vartheta,\mathbb{Y}|S)$-functions, then $(\omega_\varphi)_{ab}^{c}$ are so.
\end{proof}
We will now prove that $(B_{\theta\vartheta}^{k},\mathbb{Y}|S)$-connections $\nabla$ and $\overline{\nabla}$ which are ``locally additively different'' are actually
``locally different''.
\begin{definition}
Let $M$ be a $(B^{k}_{\alpha,\beta},\mathbb{X})$-manifold. Given $S\subset \Gamma_k$, $\theta$, $\delta$ and $\mathbb{Y}$ as above, a \textit{3-parameter $(\theta,\vartheta,\mathbb{Y}|S)$-family} in $M$ is just a collection $(\Omega_\varphi)^{c}_{ab}$ of $(B,k,\theta,\vartheta|S)$-functions in $\mathbb{Y}$, with $a,b,c=1,...,n$, for each $(\varphi,U)\in \mathcal{B}^k_{\alpha,\beta}(\mathbb{X})$.
\end{definition}
\begin{itemize}
\item If $(\Omega_\varphi)^c_{ab}$ is a 3-parameter family, let
$
(\Omega_\varphi)^c= \sum_{a,b}(\overline{\Omega}_\varphi)^{c}_{ab}(p)
$.
\end{itemize}
\begin{definition}
Two 3-parameter $(\theta,\vartheta,\mathbb{Y}|S)$-families $(\Omega_\varphi)^{c}_{ab}$ and $(\overline{\Omega}_\varphi)^{c}_{ab}$ are:
\begin{enumerate}
\item \textit{locally additively different} if for every chart $(\varphi,U) \in \mathcal{B}^k_{\alpha,\beta}(\mathbb{X})$ and every $c=1,...,n$ we have $(\Omega_\varphi)^c\neq (\overline{\Omega}_\varphi)^c$;
\item \textit{locally different} if for every chart $(\varphi,U) \in \mathcal{B}^k_{\alpha,\beta}(\mathbb{X})$ and every $a,b,c=1,...,n$ there exists a nonempty open set $U_{a,b,c}\subset U$ such that $(\Omega_\varphi)^c_{ab}(p)\neq (\overline{\Omega}_\varphi)^c_{ab}(p)$ for every $p\in U_{a,b,c}$.
\end{enumerate}
\end{definition}
\begin{definition}
Two $(B^k_{\theta,\vartheta},\mathbb{Y}|S)$-connections $\nabla$ and $\overline{\nabla}$ in $M$ are \textit{locally additively different} (resp. \textit{locally different}) if the corresponding 3-parameter $(\theta,\vartheta,\mathbb{Y}|S)$- families of their coefficients are locally additively different (resp. locally different).
\end{definition}
\begin{theorem}\label{multiplicity_thm}
If two 3-parameter $(\theta,\vartheta,\mathbb{Y}|S)$- families $(\Omega_\varphi)^{c}_{ab}$ and $(\overline{\Omega}_\varphi)^{c}_{ab}$ in a smooth $(B^{\infty}_{\alpha,\beta},\mathbb{X})$-manifold are locally additively different, then they are locally different.
\end{theorem}
\begin{proof}
For every chart $(\varphi,U)$ and every $c=1,..,n$, let $U_c\subset U$ be the subset in which $(\Omega_{\varphi})^c(p)\neq (\overline{\Omega}_\varphi)^c(p)$. Since the 3-parameters are locally additively different, the subsets $U_c$ are nonempty. On the other hand, they are the complement of the closed sets $[(\Omega_{\varphi})^c-(\overline{\Omega}_\varphi)^c]^{-1}(0)$. Thus, they are nonempty of sets. For each $p\in U_c$ we then have
\[
0<\vert\Omega_{\varphi}^{c}(p) - (\overline{\Omega}_\varphi)^c(p) \vert\leq\sum_{a,b}\vert(\Omega_{\varphi}){}_{ab}^{c}(p)-(\overline{\Omega}_{\varphi}){}_{ab}^{c}(p)\vert,
\]
implying the existence of functions $a_c,b_c:U_c \rightarrow[n]$
such that
\[
(\Omega_{\varphi}){}_{a_c(p)b_c(p)}^{c}(p)\neq(\overline{\Omega}_{\varphi}){}_{a_c(p)b_c(p)}^{c}(p).
\]
Regard $[n]$ as a $0$-manifold and notice that if two smooth functions
$f,g:V\rightarrow[n]$ are transversal, then they coincide in an open
set $V'\subset V$. Since $[n]$ is a $0$-manifold, any smooth map
$f:V\rightarrow[n]$ is a submersion and therefore transversal to
each other. In particular, $a_c$ (resp. $b_c$)
is transversal to each constant map $a'\in[n]$ (resp. $b'$), so
that for each $a'$ (resp. $b'$) there exists $U_{a',c}\subset U_c$
(resp. $U_{b',c}\subset U_c$) in which
\[
(\Omega_{\varphi}){}_{a'b_c(p)}^{c}(p)\neq(\overline{\Omega}_{\varphi}){}_{a'b_c(p)}^{c}(p)\quad(\text{resp.}\quad(\Omega_{\varphi}){}_{a_c(p)b'}^{c}(p)\neq(\overline{\Omega}_{\varphi}){}_{a_c(p)b'}^{c}(p))
\]
Taking $U_{a',b',c}=U_{a',c}\cap U_{b',c}$ and noticing that this intersection
is nonempty, the proof is done.
\end{proof}
\subsection{Discussion}\label{sec_discussion}
We close with a broad discussion which we would like to see formalized and enlarged in some future work.
\begin{itemize}
\item Let $\mathcal{C}_k(\theta,\vartheta;\mathbb{Y})$ denote the set of 3-parameter $(\theta,\vartheta,\mathbb{Y})$-families in some $(B^k_{\alpha,\beta},\mathbb{X})$-manifold $M$. Let $\xi$ be a nice and distributive $(\alpha',\beta';j)$-connection in $M$ with degree $r\geq0$.
\end{itemize}
The construction of Section \ref{sec_a_b_connections} shows that every 3-parameter $(\alpha_0,\beta_0,\mathbb{X})$-family $f=(f_\varphi)^c_{ab}$ in $M$ induces, for each ordinary pair $(\theta,\vartheta)$, a corresponding $(B^k_{\theta,\vartheta},\mathbb{Y}_{\theta,\vartheta})$-connection $\nabla^{f}_{\xi}$ in $M$, whose 3-parameter family of coefficients $(\Gamma^f_{\xi, \varphi})^c_{ab}$ is given by (\ref{globalization_regular_connection}). Suppose, for a moment, that we found a class $\mathcal{S}\subset \mathcal{C}^k_3(\alpha_0,\beta_0;\mathbb{X})$ of 3-parameter $(\alpha_0,\beta_0,\mathbb{X})$-families such that for every $f\in \mathcal{S}$ the corresponding regular connection $\nabla^f_{\xi}$ has coefficients $(\Gamma^f_{\xi,\varphi})^c_{ab} \in \mathcal{C}_k(\theta,\vartheta;\mathbb{Y}) $ which are locally additively different to each other $\Omega_\varphi\in \mathcal{C}_k(\theta,\vartheta;\mathbb{Y})$. Thus, by Theorem \ref{multiplicity_thm} it follows that they are also locally different. In particular, \textit{every 3-parameter $(\theta,\vartheta,\mathbb{Y})$-family is locally additively different to some $(B^k_{\theta,\vartheta},\mathbb{Y})$-connection and this ``difference'' is measured by elements in $\mathcal{S}$.} This is the typical shape of denseness theorems. More precisely, one could expect the existence of a topology in $\mathcal{C}_k(\theta,\vartheta;\mathbb{Y})$ whose basic open sets are parameterized by elements of $\mathcal{S}$ and such that $\operatorname{Conn}^k_{\theta,\vartheta}(M;\mathbb{Y})\subset \mathcal{C}_k(\theta,\vartheta;\mathbb{Y})$ is a dense subset.
Actually, the existence of $\mathcal{S}$ as above also implies that \textit{if $f\in\mathcal{S}$, then $\nabla^f _{\xi}$ is locally additively different to each other $\nabla\in\operatorname{Conn}^k_{\theta,\vartheta}(M;\mathbb{Y})$}, which is the shape of an openness theorem. Thus, we could expect that, if the topology described above exists, then $\operatorname{Conn}^k_{\theta,\vartheta}(M;\mathbb{Y})\subset \mathcal{C}_k(\theta,\vartheta;\mathbb{Y})$ should be open and nonempty due Theorem \ref{corollary_existence_connections}.
Finally, looking again at the expressions (\ref{change_coordinates_connection-1}) and (\ref{globalization_regular_connection}) defining $\nabla ^f_{\xi}$, one concludes that the problem of finding $\mathcal{S}$ consists in studying the complement in $\mathcal{C}_k(\theta,\vartheta;\mathbb{Y})$ of the solution spaces $\mathcal{X}_{\xi}(\varphi,\Omega)$, for each $\varphi\in \mathcal{B}^{k}_{\alpha,\beta}(\mathbb{X})$ and each $\Omega\in \mathcal{C}_k(\theta,\vartheta;\mathbb{Y})$, of the system of nonhomogeneous equations
\begin{equation}\label{final_equation}
F^c_{\xi}(f;\varphi,\Omega)=(\Omega_\varphi)^c,\quad \text{where}\quad F^c_{\xi}(f;\varphi,\Omega)= \sum_{a,b} \xi ^{\alpha',\beta'}_{\theta,\vartheta}((\Gamma ^f_{\xi,\varphi})^c_{ab}),
\end{equation}
whose coefficients belong to the space of $(B,k,\theta,\vartheta|\Gamma_k[z])$-functions. Since equations (\ref{final_equation}) strongly depend on the $(\alpha',\beta';j)$-connection $\xi$, the basic strategy to find $\mathcal{S}$ (and then to get the openness and denseness results described above) would be to introduce additional conditions in $\xi$ which simplify (\ref{final_equation}). In other words, would be to look at (\ref{final_equation}) as parameterized systems of equations in the space of all $(\alpha',\beta';j)$-connections.
\section*{Acknowledgments}
The authors thank the reviewer's valuable considerations, which undoubtedly made the text much more readable. The first author was supported by CAPES (grant number 88887.187703/2018-00).
|
1,116,691,497,814 | arxiv | \section{Introduction}
Video games are uniquely positioned in the space of software. They are designed to be an immersive experience for the user with a focus on narrative, interactivity and rich sensory presentation. However, the level of immersion a player experiences is drastically diminished if bugs are discovered during play. Although the industry invests significant effort in testing games before launch, it is still common for at least some bugs to slip through. One of the major contributing factors is the relative size and complexity of modern video games. It is becoming impractical to manually test every interaction and explore everything that a game has to offer even with a dedicated team of testers.
With recent developments in automated game playing, software agents may offer a solution to the testing automation problem. Automated bug detection (ABD) is an emerging field of study that is receiving increasing attention in the video games research community \cite{Ariyurek2021, Gordillo2021, Gudmundsson2018, Albaghajati2020, Wilkins2020, Pfau2017, zheng2019}. The focus of ABD research is on designing software agents capable of (a) \textit{exploration}, playing the game in order to discover bugs, and (b) \textit{identification}, to determine whether a particular situation contains a bug.
\renewcommand\wr{.22}
\begin{figure}%
\centering
\vspace{-.3cm}
\subfloat[Texture Missing\centering]{
\includegraphics[width=\wr\textwidth]{figs/bugs/TextureMissing.png}}
\hspace{0cm} \vspace{-.3cm}
\subfloat[Texture Corruption\centering]{
\includegraphics[width=\wr\textwidth]{figs/bugs/TextureCorruption.png}}
\hspace{0cm} \vspace{-.3cm}
\subfloat[Z-Fighting\centering]{
\includegraphics[width=\wr\textwidth]{figs/bugs/ZFighting.png}}
\hspace{0cm}
\subfloat[Z-Clipping\centering]{
\includegraphics[width=\wr\textwidth]{figs/bugs/ZClipping.png}}
\hspace{0cm}
\subfloat[Geometry Corruption\centering]{
\includegraphics[width=\wr\textwidth]{figs/bugs/GeometryCorruption.png}}
\hspace{0cm} \vspace{-.3cm}
\subfloat[Screen Tear\centering]{
\includegraphics[width=\wr\textwidth]{figs/bugs/ScreenTear.png}}
\hspace{0cm} \vspace{-.3cm}
\subfloat[Camera Clipping\centering]{
\includegraphics[width=\wr\textwidth]{figs/bugs/CameraClipping.png}}
\hspace{0cm}
\subfloat[Black Screen\centering]{
\includegraphics[width=\wr\textwidth]{figs/bugs/BlackScreen.png}}
\hspace{0cm}
\subfloat[Boundary Hole\centering]{
\includegraphics[width=\wr\textwidth]{figs/bugs/BoundaryHole.png}}
\hspace{0cm}
\subfloat[Geometry Clipping\centering]{
\includegraphics[width=\wr\textwidth]{figs/bugs/GeometryClipping.png}}
\hspace{0cm}
\caption{Some of the bugs implemented in WOB. Each agent observation (left) contains a bug which is labelled with a mask (right). }
\label{fig:VisualBugs}
\end{figure}
As is common in an emerging field, the literature is somewhat fractured when it comes to testing and comparing approaches to these problems. Without direct ties to industry it is difficult to obtain a development version of a game, meaning that researchers need to construct their own. Ironically, as bugs are hard to come by, the experimental setup cost is high. This has discouraged progress on certain problems associated with ABD, particularly on methods for identifying some of the more challenging bugs that video games exhibit.
With the aim of supporting further work in all aspects of ABD we have developed World of Bugs\ (WOB), an open experimentation platform built with the Unity3D engine. WOB\ provides a means of training and testing ABD agents in both exploration and bug identification. The core contribution of this work is in a growing collection of common video game bugs \cite{Levy2009GameDE, Lewis2010, Nantes2013} implemented in the platform, their integration with the abstractions provided by Unity's ML-Agents\ package \cite{Juliani2018} and support for learning-based bug identification. The platform also provides a collection of pre-built baseline game environments, agents and datasets which researchers can use to conduct experiments.
\begin{figure}
\centering
\subfloat[\centering Maze-v0]{\includegraphics[width=.24\textwidth, height=0.24\textwidth]{figs/envs/Maze-v0.png}}%
\hfill
\subfloat[\centering
GettingStuck-v0]{\includegraphics[width=.24\textwidth,height=0.24\textwidth]{figs/envs/GettingStuck-v0.png}}%
\caption{Example environments built in to WOB (from above). The agent appears as a white sphere. In (a) the agent is able to see through some walls by clipping the camera, constituting an \textit{invalid information access} bug. In (b) the agent can get stuck at the bottom of an elevator shaft after taking a particular sequence of actions. Both environments also include various perceptual bugs.}
\label{fig:TestEnvironment}
\end{figure}
\section{World of Bugs\ (WOB)}
\label{sec:World of Bugs}
We have focused our attention on logical and perceptual bugs in 3D first person games. Logical bugs include crashes, getting stuck, performing invalid actions, out-of-bounds and invalid information access bugs, among others. Perceptual bugs manifest visually to the player (see Fig. \ref{fig:VisualBugs}) and include geometry and texture corruption, clipping issues, shadow-acne, and other rendering related issues. This setup provides a substantial challenge but is not completely out of reach of modern machine learning algorithms for both the exploration and identification problems.
Of the approaches to ABD many focus on developing explorative agents. Works such as \cite{Bergdahl2020, Gordillo2021, zheng2019} focus on learning agents capable of finding simple logical bugs (e.g. getting stuck, or out-of-bounds) by extensive exploration of the game with rules or guards for identification. However, the sorts of bugs that game testers are looking for are often not easily identified with such methods. High-level behavioural bugs such as those related to progression, narrative, or involve complex interactions or game AI are difficult to identify with rules, and it is not yet clear how they may be tackled in general.
For perceptual bugs and the more complex logical bugs, which have scarcely been explored in conjunction with exploration, vision and learning based methods are a promising direction \cite{Wilkins2020, Nantes2013}. Progress on these bugs is an essential next step in the development of more general ABD agents that do not rely on hand-crafted rules for identification. The primary purpose of WOB\ is to support work in this area, but it also has the potential to be extended to support research into the more challenging behavioural bugs.
Details on the implementation of bugs, the tools for developing agents and environments, and some examples are given in the next sections. The platform is available as a Unity package on the open source unity package registry OpenUPM, and on GitHub. Links and further details can be found \href{\anon{}{https://benedictwilkins.github.io/world-of-bugs/}}{here}\footnote{\anon{LINK REMOVED FOR DOUBLE BLIND REVIEW }{\href{https://benedictwilkins.github.io/world-of-bugs/}{https://benedictwilkins.github.io/world-of-bugs/}}}.
\subsection{Agents}
An agent in WOB\ is created by default with three sensors: a main camera, which renders the view of the scene as the player would see it, a bug mask camera, which renders a mask over the scene showing in which regions there is a bug present, and a sensor which records various environment/agent properties such as the agent's position and rotation. The bug mask acts as a label for the agents observation and is instrumental in enabling machine learning to be applied to the bug identification problem. In an attempt to standardise the comparison of explorative agents, WOB\ also implements a set of common actions including movement, jumping, view changes and simple event-based interaction, which can be configured during setup. Agent behaviours, goals and bug identification models may be specified in Python or in C\# with the support of ML-Agents\ and associated Python packages.
WOB\ provides a simple navigational agent that uses Unity's built in navigation system inspired by \cite{Prasetya2020}. The agent's behaviour is to walk and look around, picking random points in the scene to navigate to.
The agent can be used in simpler environments where extensive exploration is not necessary to ease experimentation with approaches to bug identification. Additionally WOB\ provides player controlled agents that may be used for debugging new environments or bug implementations, or for generating training data. The intention is that with time the community will provide a collection of ABD agents that may serve as comparisons for further work.
\subsection{Bugs}
Bugs are implemented as collections of scripts and shaders, they can be added to the scene and enabled/disabled from our Python API. Newly implemented bugs will be automatically registered with the API when added to the scene. Implementing a bug requires two considerations: how to manifest the bug in the main camera and where to mask in the mask camera. The first is straightforward, and often is just a matter of writing a script to modify some properties of the environment (e.g. deleting a texture for a missing texture bug). The second is more challenging, and for this reason we have provided some scripts and shaders to ease the process. Every bug has a tag, a unique identifier which determines its colour in the bug mask. Bug tags can be used to label bugs that are associated with particular objects in the environment, such as missing textures, corrupted geometry and bad animations. Once tagged, these objects will be rendered in the bug mask. Bugs such as screen tearing, flickering, and freezing, are slightly more difficult to label but typically involve a post processing step in the rendering pipeline to directly modify the final bug mask. For details on how this may be implemented refer to the supplementary documentation. The bug mask shader also contains code for rendering bugs that are not associated with a particular object and do not effect the whole scene, a good example is camera clipping (seeing through walls). It will automatically render the back-side geometry as part of the mask, that is, if the agent can see inside a geometry, the inside faces will be rendered in the mask. Additionally, it will render the sky box below a certain height, this may be used to label so-called \textit{map-hole} or \textit{boundary bug} (see Fig. \ref{fig:VisualBugs}).
The choice to label bugs as part of a mask over the players observation (the final rendered screen) has been made with generality and extensibility in mind. All bugs manifest at this level in some form\footnote{Audio related issues are not considered in this work, but a similar idea could be applied there.}. In cases where the bug includes a temporal aspect, for example with a miss-behaving NPC, the mask can be extended through time. A mask provides the greatest flexibility for labelling newly implemented bugs and gives more information than a single scalar label.
Labelling a newly implemented bug is an important design choice. The most appropriate mask is one that gives the most information about what has gone wrong. In many cases this corresponds to masking the afflicted game object, for example in the case of a misbehaving NPC the simplest choice may be to highlight its body. If a cut-scene fails to show at a given point things are less clear, perhaps the whole observation should be masked. For bugs such as the so-called \textit{invalid information access} bug \cite{Lewis2010} in which a player gains access to information they are not supposed to have, the source (or a proxy indicator) of the information can be masked. A common example that is already supported by WOB\ is \textit{camera clipping} (looking through a wall). This bug may enable the player to see the structure or exit of an upcoming maze (see Fig. \ref{fig:TestEnvironment}.a). In such a case, backside faces may be visible and act as a proxy indicator.
The label mask can be used in a number of ways, for example, to test different inductive biases for learning models, for bug segmentation or classification, or simply in model evaluation. The platform is meant only as a means to experiment with different approaches to ABD. Practical deployment of these models is currently beyond the scope of this work, but is an important consideration going forward.
\subsection{Environments}
At this time WOB\ has a number of built-in environments, details for each can be found in the platform documentation. They provide a stable testing ground for ABD approaches and have been designed either to exhibit particular bugs (see Fig. \ref{fig:TestEnvironment}), or simplify certain aspects of the ABD problem. For example, one environment consists of a single static room in which the agent is able to move and look around. This environment should generally be used to test approaches to identifying perceptual bugs as exploration is trivial. In addition to the built-in environments currently available, new game environments can be built for testing specific kinds of behaviours or to support new kinds of bugs. The tools are available to make this process relatively straight forward. The current set of implemented bugs have been designed to be as generic as possible and can be applied directly to any newly implemented 3D environments.
\subsection{Python API}
To better manage experiments and open the platform up to the Python machine learning ecosystem, WOB\ exposes a Python API which makes heavy use of ML-Agents\ and OpenAI gym \cite{Brockman2016}. In addition to the interface provided by these packages, the WOB\ API allows users to enable and disable bugs, change agent behaviours and control other aspects of the environment and agents at run-time.
\subsection{Datasets}
To our knowledge there are currently no publicly available datasets that contain examples of video game bugs that are suitable for training/testing learning-based approaches to bug identification\footnote{\cite{Wilkins2020} provides a dataset, however the bugs are artificial and unrealistic.}. As it is often easier to work with a static dataset when training learning models, WOB\ has explicit support for generating datasets given an environment and a game playing agent. We have generated a collection of datasets using the built-in agents and environments which we hope will serve as baselines for bug identification research. They consist of large (up to 500k) collections of labelled observations and actions. Links to datasets can be found in the platform documentation.
\section{Limitations}
The focus of WOB\ is currently on visual issues, we have made no consideration of audio related problems. While not explicitly supported, many of the design decisions that have gone into visual issues could be reused for audio.
Although ML-Agents\ support parallel agents running in the same instance of Unity. This is not currently supported by WOB\ as our Python API makes use of OpenAI gym which supports only single agent interactions. It is however possible to create multiple instances of any WOB\ environment running as separate processes to enable parallel training.
\section{Conclusions \& Future Work}
The bugs implemented thus far are a small portion of the many diverse bugs that video games exhibit. There are plans to grow the list of available bugs and move beyond simpler logical and perceptual bugs into more challenging areas such as progression and narrative. For this to happen however, richer game environments are required. The game environments currently provided are simple when compared with games that ABD might eventually be applied to in practice. We hope that over time the community will provide some more interesting examples that may exhibit a wider variety of bugs. ABD has great potential in addressing some key issues in testing modern video games, but to do so requires a deeper exploration of both game playing and bug identification. WOB\ provides researchers with the tools required to make progress on some of the key problems in ABD.
\printbibliography
\end{document}
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